Tobin's q

Advanced Portfolio Theory

(Lecture Notes)

October 2004

Prof. Dr. Thorsten Hens

[email protected]

i

TABLE OF CONTENTS Table of Contents i List of Symbols v

CHAPTER 1: INTRODUCTION

1 Introduction 1 2 Portfolio Theory as A Subfield of Finance 1 3 The Structure of Modern Financial Markets 1

3.1 Principal-Agent Problems 2 3.2 Time Scale of Investment Decisions 2

4 The Anticipation Principle 2 5 Reflexivity and the Game Structure of Financial Markets 4 6 Rational Investment Process versus Simple Heuristics 6 7 Building Theories 7

CHAPTER 2: FOUNDATIONS FROM PORTFOLIO THEORY

1 Introduction 9 2 The Mean-Variance Analysis 9

2.1 Diversification 10 2.2 The Efficient Frontier 11 2.3 Optimal Portfolio of Risky Assets with a Risk-less Security 11 2.4 Two Fund Separation Theorem 12 2.5 The Structure of the Tangent Portfolio 13 2.6 The Asset Allocation Puzzle 14

3 Market Equilibrium 14 3.1 Capital Asset Pricing Model 15 3.2 Application: Market Neutral Strategies 16

4 Some Open Issues in the Mean-Variance Framework 17

CHAPTER 3: FOUNDATIONS FROM ASSET PRICING

1 Introduction 19 2 The General Equilibrium Model: Basics 19

2.1 Complete and Incomplete Markets 21 3 The Principle of No Arbitrage 24 4 Geometric Intuition for the FTAP 25 5 First Welfare Theorem 27 6 Market Selection Hypothesis with Rational Expectations 28 7 Stock Prices as Discounted Expected Payoffs 29 8 Consequences of No Arbitrage 29

8.1 The Law of One Price 29

ii

8.2 Linear Pricing 30 8.3 Derivatives 30

9 Equivalent Formulations of the No Arbitrage Principle 32 10 Limits to Arbitrage 34

10.1 The Case of 3Com and Palm 34 10.2 The Case of Closed End Funds 35 10.3 The LTCM case 35 10.4 No Arbitrage with Short-Sales Constraints 36

11 Identifying the Likelihood Ratio Process 36 11.1 The Likelihood Ratio Process with CAPM 37 11.2 The Likelihood Ratio Process with APT 38 11.3 The Representative Agent 38

12 The Rationality Benchmark 41 12.1 Empirical Evidence 43

13 Summary 50

CHAPTER 4: RATIONAL CHOICE: EXPECTED UTILITY AND BAYESIAN UPDATING

1 Rational Choice 52 1.1 Preferences 52 1.2 Utility Functions 52 1.3 Concept of Revealed Preferences 53 1.4 Concept of Experienced Preferences 53 1.5 The Concept of Preferences in Finance 53

2 Expected Utility Theory 54 2.1 The Representation Theorem 55 2.2 The Allais Paradox 55 2.3 The Probability Triangle 57 2.4 Ellsberg Paradox 58 2.5 Ambiguity 59

3 Stochastic Dominance 59 3.1 First Order Stochastic Dominance (FSD) 59 3.2 Second-Order Stochastic Dominance (SSD) 60 3.3 State-Dominance and Stochastic Dominance 60

4 A Second Look at Mean-Variance 61 4.1 Mean-Variance Principle as a Special Case of Expected Utility 61

5 Measures of Risk Aversion 63 6 Rational Probabilities 64 7 Rational Time Preferences 66

7.1 Time Preference Reversal 66

iii

7.2 Hyperbolic Discounting 66 7.3 Discounting and Risk Aversion 66

CHAPTER 5: CHOOSING A PORTFOLIO ON A RANDOM WALK: DIVERSIFICATION

1 Introduction 70 2 Returns 70 3 Portfolio Choice 71

3.1 Dynamic Portfolio Choice with CRRA: The “No Time Diversification” Theorem 73 3.2 Rebalancing, Fix Mix and Volatility Pumping 73

4 Asset Pricing 75 5 The Equity Premium Puzzle 77 6 Summary 78

CHAPTER 6: BEHAVIORAL PORTFOLIO THEORY

1 Introduction 80 2 Descriptive versus Prescriptive Theories 80 3 Search, Framing and Processing of Information 81

3.1 Selection of Information 81 3.2 Framing 82 3.3 Processing of Information 83

4 Prospect Theory 85 5 Probability Weighting and FSD 88

CHAPTER 7: CHOOSING A PORTFOLIO ON A MEAN REVERTING PROCESS: TIME DIVERSIFICATION

1 Introduction 90 2 The Mean Reverting Process 90 3 Optimal Portfolio Choice With Mean Reversion 91 4 The Case considered in Samuelson (1991) 92 5 Asset Pricing and Mean reversion 96 6 Conclusion 98

CHAPTER 8: BEHAVIORAL HEDGE FUNDS

1 What are Hedge Funds? 101 2 Hedge Funds Strategies 101

2.1 Long/Short strategies 101 2.2 Arbitrage 101 2.3 Event driven strategies 102 2.4 Directional strategies 102

3 Value at Risk 104 4 Investment Strategies Based on Behavioral Finance 105

4.1 Underreaction 105

iv

4.2 Momentum and Reversal 107 4.3 Strategies based on co-moving assets 107 4.4 Strategies exploiting probability weighting 108

5 Conclusions 109

CHAPTER 9: CHOOSING A PORTFOLIO ON A GARCH PROCESS: RISK MANAGEMENT

1 Introduction 112 2 Evidence 112 3 Coherent Risk Measures 113 4 Crash Measures 115 5 Asset Pricing Models 116 6 Summary 117

CHAPTER 10: EVOLUTIONARY FINANCE: SURVIVAL OF THE FITTEST ON WALL STREET

1 Introduction 119 2 The Ecology of the Market 119 3 Survival of the Fittest on Wall Street 120 4 The Evolutionary Portfolio Model 121 5 A Simulation Analysis 125 6 The Main Results 129 7 Extensions 132

v

LIST OF SYMBOLS

is

is

i i i io S

t Ts Si I

w

c

U c c c

===

1

0,1,..., time periods

1,..., states of the world

1,..., investors

(exogenous) wealth of investor i in state s

consumption of investor i in state s

( , ,..., ) utility of investor ii i

ks

K

KS S

ik

w wk K k

a

a aA

a a

θ

== =

⎡ ⎤= ⎢ ⎥⎣ ⎦

0

11 11

period zero wealth

1,..., assets where 0 is the consumption good

payoff of asset k in state s (resale value and dividends)

payoff matrix (S-rows and K columns)

unik

k k ks t t

kk ss k

s f

q

a q D

aR

q

R R

R

+ += +

=

=

k

1 1

0

ts of asset k bought by investor i

price of asset k

D dividend of asset k

(resale value and dividends)

(gross) return of asset k in state s

( : for the risk free asset)K

KS S

k ks s

k ii kk i

R RR R

r R

r

qi k

wθλ

π

⎡ ⎤= ⎢ ⎥⎣ ⎦=

=

11 11

k ks s

s

matrix of (gross) returns

ln( ) logarithm of gross return

= ln(R ) logarithm of gross return

wealth share of investor invested in asset

state price, i.e.

s

p

E

pE

π

πππ

=

∑*

*s

s S

zz=1

present value of one unit of wealth in state s

martingale probability of state s

expectation with respect to martingale measure

physical probability of occurence of state s

expect

sprob

ation with respect to physical measure

alternative notation for probabilty of state s

vi

*

1

i 1

1

( ) ( ) expected utility where is time preference

risk aversion parameter, for example ( ) ( )

likelihood ratio of state s

( ) ( ) expectation w

i

i

s

Si i i i i i

o s sus

i i it t

s ps

S

p s ss

E u c p u c

U c c

l

x E x p x

α

π

γ γ

α

µ

=

−

=

= +

=

=

= =

∑

∑

1

2 2

1

.r.t. physical measure

( , ) ( ( ))( ( )) Covariance

( ) ( ) ( ( )) Variance

( , )( , ) Correlation between x and y

( ) ( )

( ( ), ( )) mean-variance utility

stoc

S

s s ss

S

s ss

i

COV x y p x x y y

x VAR x p x x

COV x yx y

x y

V c cSD

µ µ

σ µ

ρσ σ

µ σ

=

=

= − −

= = −

=

∑

∑

hastic dominance; first (FSD) and second (SSD) order

( , ) normal distribution; (0,1) standardized normal distribution

``( )( ) - absolute risk aversion

`( )``( )

( ) - relative risk ave`( )

Nu x

ARA xu xu x x

RRA xu x

µ σ Φ

=

=

2 21 1

rsion

( ) 1/ ( ) risk tolerance

i.i.d. identically and independently distributed

sequence of random disturbances

RW random walk, . . : where ( ) 0, ( ) , ( , ) 0

AR autoregressi

t

t t t t t t t

RT x ARA x

e g x x E E E

ε

ε ε ε σ ε ε+ +

=

= + = = =

1

t0 1

ve process : ; mean-reversion : ¦ ¦ 1

state of the world in period t.

( , ,..., ) path up to period t.

t t t

t

t

x ax aε

ω

ω ω ω ω

+ = + <

∈Ω

=

Chapter 1:

Introduction

Introduction 1

1 INTRODUCTION

These lecture notes provide a rough guide through the slides of my course Introduction to Financial Economics, that I teach at the universities of Zurich, Bergen, Paris and Marseille. They should be read while looking also at

the slides. The point of these notes is to provide the general picture and main intuitions. 11 exercises (with

solutions) are set up to penetrate the material. Very brilliant students will be able to fill the details on their own.

On demand I did provide them on the black board while teaching. Also, at the end of each chapter, I give the

references to the literature in which more details can be found. The focus of my course is on recent

developments in portfolio theory and asset pricing. Besides foundations from traditional finance, the course

covers also behavioral and evolutionary finance. I do not deal with corporate finance.

2 PORTFOLIO THEORY AS A SUBFIELD OF FINANCE

The research field Finance contains at least the following various topics.

• Public Finance

• International Finance

• Corporate Finance

• Derivatives

• Risk Management

• Portfolio Theory

• Asset Pricing

Portfolio theory is a descriptive and normative theory. On the one hand it studies how people should

combine assets (normative view). The leading idea can be summarized as the maxim “Do not put all eggs in

one basket”, which is known as the principle of diversification. On the other hand portfolio theory describes

how investors do combine assets (descriptive view). This second aspect has important implications for asset

pricing, because if one knows how investors form their portfolios then one can determine how assets are

priced. This is because prices are determined by the demand and supply generated by the investors`

portfolios decisions.

3 THE STRUCTURE OF MODERN FINANCIAL MARKETS

Modern financial markets are populated by various investors with different wealth, objectives and heterogeneous

beliefs. There are private investors with pension, housing and insurance concerns, firms implementing investment

and risk management strategies, investment advisors providing financial services, investment funds managing

pension or private capital and the government financing the public deficit. The investment decisions are

ultimately implemented by brokers, traders and market makers.

Introduction 2

3.1 Principal-Agent Problems

Many financial markets are dominated by large investors. On the Swiss equity market, for example, more than

75% of the wealth is managed by institutional asset managers providing services to private investors, insurance

funds and pension funds. Since the asset managers` investment abilities and effort are not observable by their

clients, the contract between the principle and the agent must be based on measurable variables such as relative

performance. Though, such contracts may generate herding behavior particularly among institutional investors.

In the words of Lakonishok et al (1992, page 25): “Institutions are herding animals. We wat h the same indicators and listen to the same prognostications. Like lemmings we tend to move in the same direction at thesame time. And that naturally exacerbates price movements.” Additionally, asymmetric information may create

“window dressing” effects, i.e. agents change their behavior upon the beginning of the reporting period.

c

tc

r r r

3.2 Time Scale of Investment Decisions

Investors differ in their time horizon, information process and reaction time. Day traders for example make

several investment decisions per day requiring fast information processing. Their reaction time is limited to some

seconds. Other investors have longer investment horizon (e.g. one or more years). Their investment decisions do

not have to be made “just in time”. On the contrary, popular investment advice for investors with a longer

investment horizon is: “Buy stocks and take a good long sleep”. Investors following this advice are expected to

have a different perception to stocks as Benartzi and Thaler (1995) claim with the following example: ”Compare two investors, Nick who calculates the gains and losses in his portfolio every day, and Dick who only looks a his portfolio once per decade. Since, on a daily basis, sto ks go down in value almost as often as they go up, Nick's loss aversion will make stocks appear very unattractive to him. In contrast, loss aversion will not have much effect on Dick's perception of stocks since at ten year horizons stocks offer only a small risk of losing money”.

At the other end of the time scale are the day traders. In the words of a day trader, interviewed by Wall Street

Journal (1988), the situation is like this: “Ninety percent of what we do is based on perception. It doesn‘t matter if that perception is right o w ong o real. It only matters that other people in the market believe it. I may know it‘s crazy, I may think it‘s wrong. But I lose my shirt by ignoring it. This business turns on decisions made in seconds. If you wait a minute to reflect on things, you‘re lost. I can‘t afford to be five steps ahead of everybody else in the market. That‘s suicide“.

Thus, intraday price movements reflect how the average investor perceives incoming events. In the very long run

price movements are determined by trends in fundamental data like earnings, dividend growth and cash flows.

How the short run aspects get washed out in the long run, i.e. how aggregation of fluctuations over time can be

modelled is rather unclear.

4 THE ANTICIPATION PRINCIPLE

As the following chart documents, stock prices are driven by the arrival of new information. In a sense the stock

market is a huge prediction machine that always tries to adjust its level to the events anticipated for the future.

However, by the very nature of the information process, news are those events that have not been anticipated

Introduction 3

before. Thus the changes in stock prices must be random. This wisdom was already known in the sixties. Cootner

(1964) expresses it quite clearly by writing: “The only price changes that would occur a e tho e that result from new information. Since there is no reason to expect in ormation to be non-random in appearance, the period-to-period price changes of a stock should be random movements, statistically independent of one another.” Hence rationally anticipated prices must be random!”

r sf

t

1

To see how this view of the stock market is sometimes abused, consider the following example. On September 30th 2002 Alan Greenspan decreased interest rates from 1.75 to 1.25 basis points which should be good news for the stock market since the alterna ive investment in bonds is now less attractive. However, the DJIA dropped from 7701 to 7591. A standard explanation is that the decrease in interest rates was already anticipated by the

market. In fact, from September 1990 to September 2002 Alan Greenspan changed interest rates on 46 days. In

27 cases on that day the DJIA moved in the wrong direction! Note that referring to anticipated events in this way

is a tautology, nicknamed “Neptun” by Popper. A tautology arises if one gives a reason to explain a fact and one

argues that the reason is true because the fact is observed. To illustrate this concept Popper refers to a fisher

man in ancient Greece. When he was asked why the see is so rough today he answered: “Because Neptun is

angry today.” And when asked how he did know that Neptun is angry today he answered: “Don’t you see how

rough the see is today?” In finance the concept of expectations is often used as such a Neptun. If some event

1 Note that Cootner (1964) assumes information to be random. A recent study of Damianova, Hens and Sommer (2004) shows that indeed the news as perceived by business specialist are white noise (i.i.d. and Gaussian).

Introduction 4

drives the stock prices in the “wrong” direction then one argues that it was already anticipated in the prices!

That is to say the concept of expectations used this way cannot be falsified! A rigorous test of the anticipation

principle can never be done this way! Rather one has to analyze the statistical properties of the price process

relative to the news process.

5 REFLEXIVITY AND THE GAME STRUCTURE OF FINANCIAL MARKETS

The concept of reflexivity is described by Soros (1998) in the following way:

“Financial markets attempt to predict a future that is contingent on the decisions people make in the present. Instead of just passively reflecting reality, financial markets are actively creating the reality that they, in turn, reflect. There is a two way connection between present decisions and the future events, which I call reflexivity”

and “Reflexivity is absent from natural science, where the connection between scientists‘explanations and the phenomenon tha they are trying to explain runs only one way”. Moreover, Soros writes: “Each market participant is faced with the task of putting a p esent value on a future course of events, but that course is contingent on the present values tha all market participan s taken together attribute to it”.

t

rt t

r r

s

One appearance of reflexivity is already described by Keynes (1936, page 156) in his famous beauty contest:

“Professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six p ettiest faces f om a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor hasto pick, not those faces which he himself finds prettie t, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view”.

From the perspective of modern economics, Keynes beauty contest is a coordination game, i.e. a game where the

participants get high (low) payoff if they choose the same (a different) action.

Though, if the stock market were a coordination game why do we observe so many stock market fluctuations?

The answer is maybe in the behavior of noise traders. Sometimes investors may have to sell stocks for reasons

exogenous to the market. Rational coordination may brake down because this introduces uncertainty to the

coordination game.

To see the importance of noise trading consider a game played as a sequence of stage games with the following

structure (see Gerber, Hens and Vogt (2003)). There are 5 participants betting 1 point each on “up” or “down”.

A fair dice randomly distributes 6 points on “up” or “down”. If the participant’s prediction is correct, he receives

20 ECU otherwise nothing. The movement of the stock goes up (down) if the majority of points including those

from the noise traders is bet on up (down). The stage game is repeated in two rounds with 100 periods each.

The following chart displays a typical outcome:

Introduction 5

Spiel 1.2.: Kurs vs. Würfel (Summen)

-15

-10

-5

0

5

10

15

20

Perio d en

Kurs (kum.)Würfel (kum)

Even though the noise is unsystematic stock prices show a clear pattern. They gain momentum and have long

periods of up and of down movements and suddenly they revert their direction. Hence stock prices show

momentum and reversal – the overall volatility (measured in terms of variance) is clearly higher than that of the

exogenous noise process. Why this pattern emerges can be seen from the behavior of the participants. Most of

the time they are nicely coordinated but every now and then the coordination is broken by an extreme move of

the exogenous noise. When asked who is responsible for the sudden reverses in stock price movements the

participants answer “The exogenous noise!” But when asked: “So why don’t you stick to the previous

coordination?” They respond because I cannot know that the others did understand that our coordination was

broken by the noise!”

Spiel 1.2.: Einzelne TeilnehmerInnen/ kurs Perioden 1-50

0

0.2

0.4

0.6

0.8

1

1.2

Periode

KursTeilnehmerIn 1TeilnehmerIn 2TeilnehmerIn 3TeilnehmerIn 4TeilnehmerIn 5

Certainly the coordination game structure is an important aspect of stock markets. Yet this majority game aspect

has to be complemented with a minority game aspect. This seemingly contradiction is resolved when one makes

a distinction between the values investors hold in their depots (the depot values) and the values they did realize

for consumption. With respect to depot values stock markets are a coordination game. The more I bet on the

idea that prices go up the higher the value in my depot. However when it comes to realizing these values then

the stock market is a minority game, i.e. a game that cannot be won by a majority. In a stock market bubble for

example depot values increase to record levels but when investors try to realize these gains then markets crash.

The tension between the coordination game aspect with respect to depot values and the minority game aspect

with respect to realized values creates the fluctuations on financial markets.

A simple experiment that has both aspects is the so called guessing game that we describe in the next section in

Introduction 6

order to analyze whether strategies based on the idea of common knowledge of rationality or rather simple

heuristics are more successful on financial markets.

6 RATIONAL INVESTMENT PROCESS VERSUS SIMPLE HEURISTICS

The Guessing Game, also called “Pick a Number Game” (cf. Nagel (1995)) has the following rules: each

participant picks a number between 0 and 100. The winner is the participant who is closest to 2/3 of the average

number. Clearly the unique Nash equilibrium is that all participants choose the number 0. Only in this case no

participant has an incentive to change his choice. However it is a robust observation that the typical winning

number is in the range of 17-22. Hence the winners are those who have the best guess about the average

rationality of the participants and then try to outsmart the average by choosing a number below their guess. A

typical distribution of numbers chosen in the guessing game is:

0

2

4

6

8

10

12

14

16

0 13.6 27.2 40.8 54.4 Mor e

Bin

Freq

uenc

y

Even though these ideas seem rather simplistic they are valuable to understand hedge fund disasters like the one

of Long Term Capital Management (LTCM). LTCM (run by Meriwether, Merton and Scholes) gathered billions of

dollars from investors to be bet on the idea that markets will eventually converge to where rational theory

predicts them to be. To give an example, consider their pair trading strategy on the stocks of Royal Dutch

Petroleum (RDP) and Shell Transport and Trading (STT). The LTCM managers discovered that the share price of

RDP at the London stock exchange and that of STT at the New York stock exchange do not reflect the parity in

earnings and dividends stated in the splitting contract between these two units of the Royal Dutch/Shell holding.

According to this splitting contract, earnings and dividends are paid in relation 3 (RDP) to 2 (STT), i.e. the

dividends of Royal Dutch are 1.5 times higher than the dividends paid by Shell. Though, for a long time, the

market prices of these shares did not follow this parity as the figure bellow shows and since LTCM did not have

enough liquidity when the spread between the two shares widened they made huge losses on this trade.

Introduction 7

Actually the LTCM managers did not believe Keynes who wrote: “Markets can remain irrational longer than you

can remain solvent!”

7 BUILDING THEORIES

Albert Einstein is known to have said that “there is nothing more practical than a good theory.” But what is a

good theory? First of all, a good theory is based on observable assumptions so to avoid the Neptun fallacy.

Moreover, a good theory should have testable implications – otherwise it is a religion which cannot be falsified.

This falsification aspect cannot be stressed enough. Steve Ross, the founder of the econometric Arbitrage Pricing

Theory (APT), for example, claims that “every financial disaster begins with a theory!” By saying this he means

that those who start trading based on a theory are less likely to react to disturbing facts because they are

typically in love with their ideas. Finally, a good theory is a broad generalization of reality that captures the

essential features of it. Note that a theory does not become better if it becomes more complicated. Since there is

(and according to the rational point of view there has to be!) so much noise in our observations it is really hard

to build a good theory of financial markets. A good advice is to start with empirically robust deviations from

random behavior of asset prices and then to try to explain it with testable hypotheses. Of course this presumes

that you do understand the “Null Hypothesis”, i.e. what a rational market looks like. Therefore a big part of this

course will deal with traditional finance that explains the rational point of view. One way of reducing the noise is

to also use laboratory experiments because then one at least has control about the exogenous noise. However,

the step back into the reality of financial markets may be huge.

REFERENCES

Benartzi and Thaler (1995): Myopic Loss Aversion, Journal of Political Economy.

Cootner (1964): The Random Character of Stock Prices, MIT-press.

Damianova, Hens and Sommer (2004) News as Perceived by Business Specialists, mimeo University of Zurich.

Gerber Hens and Vogt (2003): Rational Investor Sentiment, NCCR-research paper.

Keynes (1936): General Theory, edition from 1964.

Lakonishok et al (1992): The Structure and Performance of the Money Management Industry, Brookings Papers

on Economic Activity, Washington, D.C.: Brookings Institutions: 331-339.

Nagel (1995): Unraveling the Guessing Game. An Experimental Study, American Economic Review, 85, 1313-

1326.

Soros (1998): The Crisis of Global Capitalism, Little, Brown and Company.

Wall Street Journal (1988): Making Book on the Buck, Sept. 23, 1988, p. 17.

Chapter 2:

Foundations from

Portfolio Theory

Foundations from Portfolio Theory 9

1 INTRODUCTION

The mean-variance analysis goes back to H. Markowitz (1952). In his work “Portfolio Theory Selection” he

recommends the use of expected return-variance of return rule,

“…both as a hypothesis to explain well-established investment behavior and as a maxim to guide one‘s own action”.

Later, Jagannathan and Wang (1996) recognize the mean-variance analysis and the Capital Asset Pricing Model

as

“…the major contributions of academic research in the post-war era.”

Campbell and Viciera (2002) write on page 7:

„Most MBA courses, for example, still teach mean-variance analysis as if it were a universally accepted framewo k for portfolio choice“ r

2 THE MEAN-VARIANCE ANALYSIS

There are assets. The gross return of the assets is Kk ,...,2,1=1−

+=

k kk t tt k

t

q DRq

, where ktq is the market price

(or payoff) of asset k in time t, and are the dividends of asset k in time t. is the expected

return

ktD ( )k

k E Rµ = t

t2 and is the variance of the gross returns. All assets can be represented in a two-

dimensional diagram with expected return

2 ( )kk Var Rσ =

µ as a reward measure and standard deviation ( )ktVar Rσ = as the

risk measure on the axes.

µ

σ

.

.

.

.

..

.

.

.Rf

kkµ

kσ

µ

σ

.

.

.

.

..

.

.

.Rf

kkµ

kσ

Every asset k can be characterized by the mean and standard deviation of its returns. The risk free-asset for

example has an expected return of fR with a zero standard deviation. An investor who puts all of his money

into risky asset k expects to achieve a return of with a standard deviation . kµ kσ

2 Expected returns are usually calculated using historical return values or Asset Pricing Models.


2.1 Diversification

Combining two risky assets k and j gives an expected portfolio return of jk µλλµµλ )1( −+= , where λ is the

portion of investor’s wealth invested in asset k. The portfolio variance is

. jkjkk Cov ,22222 )1(2)1( λλσλσλσ −+−+=

The advantage of combining risky assets depends on the covariance term. The smaller the covariance the smaller

is the portfolio risk, the higher is the diversification potential of mixing these risky asset. There is no

diversification potential of mixing risky assets with the riskless security, since the covariance of the returns is

equal to zero.

To see how portfolio risk changes with covariance it is convenient to standardize the covariance with the

standard deviation of assets returns. The result is the correlation coefficient between returns of assets k and j:

jk

jkjk

Covσσ

ρ ,, = . The portfolio variance reaches its minimum, when the risky assets are perfectly negatively

correlated: 1, −=jkρ . In this case, the portfolio may even achieve an expected return, which is higher than the

risk free rate without bearing additional risk. The portfolio consisting of risky assets does not contain risk

because whenever the return of asset k increases, the return on stock j decreases, so if one invests positive

amounts in both assets, the variability in portfolio returns cancel out (on average).

µ

σ

.

.

.

..

.

.

.Rf

k

j

, 1k jρ = −, 1k jρ =

µ

σ

.

.

.

..

.

.

.Rf

k

j

, 1k jρ = −, 1k jρ =

Investors can build portfolios from risky and risk-free assets but also portfolios from other portfolios etc. The set

of possible µ σ -combinations offered by portfolios of risky assets that yield minimum variance for a given rate

of return is called minimum-variance opportunity set or portfolio bound (see the figure bellow).


µ

σ

.

.

.

..

.

.

.rf

µ

σ

( )σ µ

.

.

.

..

.

.

.

MVP

rf

The investor’s problem when choosing a portfolio is to pick a portfolio with the highest expected returns for a

given level of risk. Alternatively, he can minimize the portfolio variance for different levels of expected return.

Formally, this is equivalent to the following optimization problem:

,( ) : min

. .

1

λσ µ λ

λ µ µ

λ

=

=

=

λ∑∑

∑

∑

k k jk j

k kk

kk

Cov

s t

and

j

(1.1)

2.2 The Efficient Frontier

The solution of problem (1.1) gives the mean-variance opportunity set or the portfolio bound. In order to identify

the efficient portfolios in this set, one has to focus on that part of the mean-variance efficient set that is not

dominated by lower risk and higher return. This is the upper part of the portfolio bound, since every portfolio on

it has a higher expected return but the same standard deviation as portfolios on the lower part (see the figure

bellow).

Thus, all the portfolios on the efficient frontier have the highest attainable rate of return given a particular level

of standard deviation. The efficient portfolios are candidates for the investor’s optimal portfolio.

µ

σ

.

.

.

. .

.

.

. r f r

µ

σ

.

.

.

. .

.

.

. f

Rf

2.3 Optimal Portfolio of Risky Assets with a Risk-less Security

The best combination of risky assets for riskaverse -µ σ - investors lies on the efficient frontier. Every point on it

is associated with the highest possible return a given certain risk level.


If an investor desires to combine a risky asset (or a portfolio of risky assets) with a risk-less security, he must

choose a point on the straight line3 connecting both assets.

The best portfolio combination is found when this line achieves its highest possible slope4. This line is known as

the Capital Market Line. Its tangent on the efficient frontier gives the best portfolio of risky assets, which is

called Tangent Portfolio.

2.4 Two Fund Separation Theorem

The optimal asset allocation consisting of risky assets and a riskless security depends on investor’s preferences,

which are given by the utility function 2 2,( , )

2λ λ λα

λµ σ µ σ= −i

iiU where is a risk aversion parameter of

investor i. The higher this parameter, the higher is the slope of the utility function

0α >i

5. The higher the risk aversion,

the higher is the required expected return for a unit risk (required risk premium).

Different investors have different risk-return preferences. Investors with higher (lower) level of risk aversion

choose portfolios with a low (high) level of expected return and variance, i.e. their portfolios move down (up) the

efficient frontier.

If there is a risk-free security, the Separation Theorem of James Tobin states that agents should diversify between

the risk-free asset (e.g. money) and a single optimal portfolio of risky assets. Since the Tangency Portfolio gives

the optimal mix of risky assets, a combination with the risk-free assets means that every investor has to make an

investment decision on the Capital Market Line. Different attitudes toward risk result in different combinations of

the risk-free asset and the tangent portfolio. More conservative investors for example will choose to put a higher

fraction of their wealth in the risk-free asset; respectively, more aggressive investors may decide to borrow

capital on the money market (go short in risk-free assets) and invest it in the tangent portfolio.

3 Since , the portfolio variance ( , ) 0k fCov R R = λσ is a linear function of the portfolio weights.

4 The slope of this line is known as the Sharpe Ratio. It is equal to fRλ

λ

µσ− .

5 The risk aversion concept is often discussed in the expected utility context, where the (absolute) risk aversion is measured by the curvature of a utility function. We will come back to this point once the expected utility framework has been introduced.


Thus, the asset allocation decision of investor i is described by the vector of weights:

60 0( , (1 ) ), 1,...i i i T iλ λ λ λ= − = I K, where i K i Tλ λ λ+∈ ∈ ∈1

0R , R, and R .

µ

σ

Rf

T

i

i´

Aggressive Investor

Best MixModerate Investor

Conservative Investor

µ

σ

Rf

T

i

i´

Aggressive Investor

Best MixModerate Investor

Conservative Investor

„The striking conclusion of his Markowitz analysis is that all investors who care only about mean andstandard deviation will hold the same portfolio of risky assets.“

7

2.5 The Structure of the Tangent Portfolio

According to the Two-Fund Separation an investor with utility 2,( , )

22

λ λ λα

λµ σ µ σ= −i

iiU has to decide how to

split his wealth between the optimal portfolio of risky assets with a certain variance-covariance structure

(Tangent Portfolio) and the risk-less asset. The structure of the Tangent Portfolio can be found either by

maximizing the Sharpe Ratio subject to a budget constraint8 or by solving the simplest -µ σ maximization

problem. Note that the composition of the Tangent portfolio does not depend on the form of utility function.

1

2 2,

K

kk=0

( , )max 2

s.t. 1

λ λ λλ

αλµ σ µ σ

λ

+∈= −

=∑

K

ii

iR

U (1.2)

In this problem, 0λ denotes the fraction of wealth invested in the risk-less asset. 0λ can be eliminated from the

optimization problem by substituting the budget constraint 01

1K

kk

λ λ=

= −∑ into the utility function. Using the

definition of λµ and 2λ

σ we get:

6 Note: there is no index i on the Tangent Portfolio Tλ since this portfolio is the same for every investor. 7 Campbell and Viceira (2002), p.3. 8 This approach is cumbersome since the portfolio weight λ appears both in the numerator and in the denominator.


( ) 'max 2λ

αµ λ λ− −i

f λR COV (1.3)

where now λ is the vector of risky assets in the Tangent Portfolio, µ is the vector of risky assets returns, and

Cov is their variance-covariance matrix. The first order condition of the problem is:

11 ( )λ µα

= − fiCOV R

If there are no constraints on 1λ , then the solution is:

11

1 (λ µα

−= fiCOV R )− (1.4)

With short-sales constraints, 0λ ≥ , for example, one can apply standard algorithms for linear equation systems

to solve the problem.

Say, the solution to the first order condition is optλ , then the Tangency Portfolio can be found by a

renormalization:opt

T kk opt

jj

λλλ

=∑

. Note that the risk aversion parameter α i cancels after the renormalization,

which is the Two Fund Separation property.

Using more sophisticated functions than (1.2) will not change the result obtained in (1.4).

2.6 The Asset Allocation Puzzle

Canner, Mankiw und Weil (1997) studied investors’ portfolios managed by different banks and found that the

ratio of bonds to stocks differs substantially among investors with different risk attitude. This is a puzzle in the

sense that professional bank advisors seem not to follow the Two-Fund Separation Theorem as an asset

allocation rule, i.e. the portfolio structure of risky assets is not constant and varies with differences in client’s risk

aversion. For example, the ratio of the portfolio weight of S&P500 to bonds changes from 0.25 to 1.5 in

conservative, moderate and aggressive portfolios.

3 MARKET EQUILIBRIUM

If individual portfolios satisfy the Two-Fund-Separation then by setting demand equal to supply the sum of the

individual portfolios must be proportional to the vector of market capitalizations9 Mkλ .

0(1 )i i Tk

i i

Mk kλ λ λ λ⎡ ⎤

= − =⎢ ⎥⎣ ⎦

∑ ∑ Hence, the normalized Tangency Portfolio will be identical to the Market

9 The market capitalization of a company for example is the market value of total issued shares.


Portfolio.10

CML

j

Mλ

Rf

σ

µ

Curve j is obtained by combination of some asset j with the market portfolio.

To determine a risk measure in a financial market with -µ σ -investors we need an asset pricing model.

3.1 Capital Asset Pricing Model

To understand the link between the individual optimization behavior and the market, compare the slopes of the

Capital Market Line and the j-curve. By the tangency property of Mλ they must be equal (see the figure above).

The slope of the Capital Market Line can be calculated dividing:

0

( (1 ) )f Mf M

d R Rd λ

µ λ λµ µ

λ =

+ −= − by 0

( (1 ) )f MM

d R Rd λ

σ λ λσ

λ =

+ −= − .

The slope of the j-curve is

0

( (1 ) )j Mj

d R Rd λ M

µ λ λµ µ

λ =

+ −= − divided by

2

0

( (1 ) ) ( , )j M j M

M

d R R Cov R Rd λ

Mσ λ λλ σ=

+ − −=

σ

From the slope’s equality at point Mλ follows:

12

( )( , )j M M M

j M M MCov x xµ µ σ µ µ

σ σ− −

=−

or equivalently: ( )j f jM M fµ µ β µ µ− = −

2

( , )where j M

jMM

Cov R Rβ

σ=

The result is the Capital Asset Pricing Model (CAPM).

10 Note that this equality is barely supported by empirical evidence, i.e. the Tangent Portfolio does not include all assets. The reason for this mismatch could for example be that not every investor optimizes over risk and return as suggested by Markowitz.


SML

Mλ

Rf

β

µ

1

Mµ ( )k f kM M fµ µ β µ µ− = −

The difference to the mean-variance analysis is the risk measure. In the CAPM the asset’s risk is captured by the

factor β instead by the standard deviation of asset’s returns. It measures the sensitivity of asset j returns to

changes in the returns of the market portfolio. This is the so called systematic risk.

3.2 Application: Market Neutral Strategies

The Capital Asset Pricing Model has many applications for investment managers and corporate finance. Even

professionals dealing with alternative investments consider it for building portfolios. One example is a form of

Market Neu ral Strategy followed by some hedge funds. This strategy aims a zero exposure to market risk. To

exclude the impact of market movements, it takes simultaneous long and short positions on risky assets. These

assets have the same beta (as measure for market risk) but different market prices. Under the assumption that

market prices will eventually return to their fundamental value defined by the CAPM, hedge fund managers take

long positions in underpriced assets and short positions in overpriced assets. In terms of expected returns, the

long (short) positions are in assets with higher (lower) expected returns than in the CAPM.

t

11

β

µ

r

SML

.

.

.

.

.

.

.

.

.

.

.

..

.long

short

Market neutral

β

µ

r

SML

.

.

.

.

.

.

.

.

.

.

.

..

.long

short

Market neutral

11 When prices revert and increase (decrease) in order to reach their fundamental value, the expected returns are decreasing (increasing).


4 SOME OPEN ISSUES IN THE MEAN-VARIANCE FRAMEWORK

The assumption of mean-variance preferences postulates that investors’ preferences depend only on the

mean and variance of payoffs. Though, experimental research provides a different description of

investors’ behavior under uncertainty. It suggests that, investor’s preferences are not always defined over

absolute payoffs. Individual decisions are often motivated by relative changes, i.e. gains and losses

compared to a certain reference point. Additionally, mean-variance preferences show some

inconsistencies with the axioms of rational choice.

The assumption of homogeneous beliefs requires that all investors have the same expectations about the

return distribution. In this case, all investors will face the same investment opportunity set and the

minimum-variance set will be identical for all investors. Though, if there is a disagreement among

investors’ beliefs, the composition of the tangency portfolio will not be uniquely determined. Then, it is

not obvious whether CAMP will hold. The relevance of the model is further questioned in the presence of

background risks that generate a mismatch between incentives and diversification behavior (e.g. CEOs

hedging firm’s stock options; employees holding shares of the company they currently work for).

The mean-variance framework suggests two portfolios: the tangent portfolio and the market portfolio.

These are quite different portfolios since typically the market portfolio is inefficient and the tangent

portfolio is underdiversified.

The mean variance analysis is not a dynamical concept. The optimal portfolio over a long period of time

is not just the sequence of optimal portfolios over short periods of time. Summing up one-period

decisions over a long period of time includes the risk of error accumulation.

The standard deviation is not always an appropriate risk measure, because of left sided fat tails in return

distributions.

REFERENCES:

Textbooks:

Huang and Litzenberger (1988): Foundations for Financial Economics, North Holland,

Chapters 1-4

Campbell and Viceira (2002): Strategic Asset Allocation, Oxford University Press, Chapters 1-2

Research Papers:

Later, Jagannathan and Wang (1996): The Conditional CAPM and the Cross-Sections of Expected Returns,

Journal of Finance (51), 3-53

Campbell and Viciera (2002): Strategic Asset Allocation, OUP

Canner, Mankiw and Weil (1997): An asset Allocation Puzzle, American Economic Review (87), pp.181-191

Markowitz, H. (1952): Portfolio Theory Selection, Journal of Finance (7), 77-91

Chapter 3:

Foundations from Asset Pricing

Foundations from Asset Pricing 19

1 INTRODUCTION

In this chapter, we will extend the two-period equilibrium model asking what the equilibrium prices in a multi-

period setting are if agents are not restricted to be mean-variance optimizers as in the Capital Asset Pricing

Model. An equilibrium model in a sequence economy consists of a description of time and uncertainty, a

description of the real side of the economy (goods, agents, preferences, and technology), trading arrangements,

and a description of agents’ behavior. An equilibrium is described by the conditions under which the decisions of

all agents in an economy are mutually consistent.

2 THE GENERAL EQUILIBRIUM MODEL: BASICS

To find the equilibrium in a system that exists over more than two periods, it is necessary to define first the

uncertainty associated with time and information. Similar to Lucas (1978), our model is defined over discrete

time that goes to infinity, i.e. . The information structure is given by a finite set of realized states 0,1,2,...t =

t tω ∈Ω in each t. The uncertainty with respect to information decreases with the time since at every t only one

state is realized. The path of state realizations over time is denoted by the vector 0 1( , ,..., )t tω ω ω ω= . The time-

uncertainty can be described graphically by an event tree consisting of an initial date ( ) and 0t = tΩ states of

nature at the next date.

t=0 t=1 t=2

The probability measure determining the occurrence of the states is denoted by P. Note that P is defined over the

set of paths tω . We call P the physical measure since it is exogenously given by nature. We use P to model

the exogenous dividends process. If these realizations are independent over time, P can be calculated as a

product of the probabilities associated with the realizations building the vector 0 1( , ,..., )t tω ω ω ω= . For

example, the probability to get two time “head” by throwing a fair coin is equal the probability to get “head”

once (equal to 0.5) multiplied with the probability to get “head” in the second run (equal to 0.5).

In our model the payoffs are determined by the dividend payments and capital gains in every period. Let

denote the investors. There are 1,...,i = I K1,...,k = long-lived assets in unit supply that enable wealth

transfers over different periods of time. 0k = is the consumption good. This good is perishable, i.e. it cannot be

used to transfer wealth over time. All assets pay off in terms of good k=0. This clear distinction between means

to transfer wealth over time and means to consume is taken from Lucas (1978).


In a competitive equilibrium with perfect foresight every investor decides about his portfolio strategy

according to his consumption preferences12 over time ( )1

,00,1,... arg max

i K

i i it

RU

θt

θ θ+

=∈

= under the budget constraint

K K,0 , ,

11 1

( )i k i k k k i kt t t t t t

k kq D qθ θ θ −

= =

+ = +∑ ∑ , where are the total dividend payments of asset k and the prices

are exogenously given, all in re

conditi

Note that

ktD

0,1,...ktq = vestors ag e on the prices given the realized state.13 The market clearing

on I

i,k 1, 1,...,k Kθ = =∑ equalizes demand and supply. ti=1

, ( )θ ω ∈i k tt R is the number of assets k that agent i has in period t given the path tω .

1( )θ ω ∈i t kt

+R is the portfolio of assets that agent i has in period t given the path tω , and , 0,1, 2,..i kt tθ = s

trategy along the set of paths. Note also that we did normalized the price of ion

good to be one and we used the Walras law to exclude the market clearing condition for the consumption good.

We start the economy with some initial endowment of assets i

. i

the portfolio s the consumpt

1θ− such that I

i . Assets start paying -1i=1

1θ =∑

dividends in , i.e. the budget constraint at the beginning is i k i k k k i k

k kq q D0t = ,0 , ,

11

0 0 0 0 01

( )K K

θ θ θ−= =

+ = +∑ ∑ . We can

as tthink of 0t = he starting point of our analysis, i.e. i 1θ− can be i s that we

inherit from a previous period (“the past”). Hence, in sense the economy can be thought of as restarted at

0t = .

nterpreted as the allocation of asset

a

Instead of using the number of assets k hold in the portfolio of investor i in time t, the investors’ demand can be

expressed in terms of asset allocation or percentage of the budget value. This is ,

,i k i

i k t tt k

wλθ = . Equalizing tq

demand with supply, i.e. ,i k iI wλ I

average of the traders’ ass or asset rule follows from the simple equilibrium condition

hat demand is equal supply. No other assumptions are necessary to derive this result!

Since our analysis is focused on the development of the strategies wealth over time,

11t t

ki tq=

=∑ , gives k i k it t t

iq wλ

=

=∑ . In other words, the price of asset k is the wealth

et allocation f k. This pricing

t

it is useful to rewrite the

,

1

model in terms of ,i ktλ (i.e. strategy as a percentage of wealth) instead of ,i k

tθ (i.e. strategy in terms of asset

units). Thus,

12 Note that investors’ preferences are defined over his consumption and not over the depot value. The utility function representing investors’ time preferences and risk attitude determines the consumption, which is smoothed over the realized states. 13 Investors may disagree on the probability distribution of the states but they agree on the prices conditionally on the states.


Definition:

A tive equilibrium with perfect foresight is a list of portfolio strategies and competi it=0,1,... , 1,...,i Iλ =

a sequence of prices kt=0,1,...q , 1,...,k K= such that for all 1,...,i I= and 1,... for all t 0,=

( )1

,0 , 1arg max and λ λ λ ++= = ∈∆

k ki i i i k i kt tD qU w w 0,1,... 1 1

1 1

s.t. ( )λ

λ+

= − −∈∆ = −

⎡ ⎤⎢ ⎥⎣ ⎦∑

Kt

Ki i t

t t t tkk t

w wq

and markets clear:

In other words, in a competitive equilibrium all investors choose an asset allocation that maximizes their

poun

complete, i.e. if there are sufficient many assets to hedge

t

Ii,kt

i=1, 1,..., =1,2,...i k

t tw q k K tλ = =∑

0,1,...itλ =

utility over time under the restriction of a budget constraint with a stochastic com d interest rate.14 The

compatibility of these decision problems today and in all later periods and events is assured by the assumption of

perfect foresight. This equilibrium is therefore also called equilibrium in plans and price expectations.

2.1 Complete and Incomplete Markets

The model can considerably be simplified if markets are

all risks. With complete markets – as we will show below – the sequence of budget constraints can be reduced

to a single budget constraint.

Definition:

A financial market (D, q) is said to be complete if any consumption stream 0 ( )ttθ ω can be attained

with some initial wealth 0w , i.e. it is possible to find some trading strategy tθ such

t k t k t k t k t k tt t t t t t

k kq D qθ ω ω θ ω ω ω θ ω −

−= =

⎡ ⎤+ = +⎣ ⎦∑ ∑

A financial market is said to be incomplete if there are some consumption streams that cannot be

The ne

that for all periods

1,2,...t =

K K0 1

11 1

( ) ( ) ( ) ( ) ( ) ( )

achieved whatever the initial wealth is. This may arise if there are more states than insurance possibilities.

cessary and sufficient condition for a financial market to be complete is:

1

1,...,1 1 t( ) ( ) ( ) for all , 1, 2,...k Kk t k t trank D q tω ω ω ω ω ω=− −⎡ ⎤

( )

t t

tt t

t t t t t

A

ω

ω ω−

∈Ω+ =⎣ ⎦

Hence, if

Ω =

ttω ω ωΩ

t=1,2,...

tK < max ( ) for some , then markets are incomplete.

14 The budget constraint is defined over the wealth in period t and t-1, where the product ,1

1 1

k kKi kt ttk

k t

D qq

λ −= −

+∑ is the

compound interest rate .


For example, in a symmetric tree model, where the set of possible states is equal to S, a necessary condition for

market completeness is that the number of assets is not smaller than the number of possible states, i.e.

Asset payoffs (dividend payments and

indicates the target consumption

K S≥ .

To illustrate the concept of market completeness let us consider the following example:

[2,1]

prices) are give

[1,1]

[1 ,-1+0]

[2+3,1+2]

4

3

3

0

A

B

+1 [1,0]

1

n in brackets over every possible state. The number below0θ over the time. There are two assets, i.e. k = 1,2. An investor has to decide

how many units of asset 1 and 2 to buy at 0t = and 1t = in order to achieve the target consumption path.

Starting at the end of the period, a nvestor has to solve the following problem for node A: n i

1 1

( ) ( )

( )

4

3( )

2 A A

A A

θ θ

θ θ

+ =

+ =

In other words, the sum of the payoffs of asset 1 and 2 multiplied with the number of assets hold in the two

possible states in node A must be equal to the target consumption in these states. The solution of the equation

1 21 11 2

system is: 1 21 1( ) 1, ( ) 2A Aθ θ= = , i.e. in 1t = given that node A is realized an investor has to hold 1 units of

asset 1 and 2 units of asset 2. This portfolio costs are: 3 2*1 *2 7+ =

Applying the same calculation pr for node B, we get: ocedure

( ) (1 0 ) 1B Bθ θ+ =

Thus, , i.e. in given that node B is realized an investor has to hold 1 unit of asset 1

and no asset 2. The portfolio costs are:

1 21 1

1 21 1( ) 1, ( ) 0B Bθ θ= = 1t =

1 ( 1)*1 *0 1−+ =

Applying the above procedure for

0 0

( + ) ( ) 7

( +

3 2

1)

2

0

3

01 )

1

1 ( 1

θ θ+ + = +

which gives

n be done for any other target consumption. Hence, in this example markets are

complete. An alternative way to define market completeness is by saying that every new asset is redundant to

0t = , we get:

1 2

1 20 01 2θ θ+ + = +−

0 013 /11, =15/11θ θ= .

The same argument ca

the already existing assets.


Definition:

An asset is redundant if its payoffs are a linear combination of the existing assets. In that case it is

ble to find prices for the assets such that the introduction of the asset does not change the set of possi

attainable consumption streams. Respectively, any target consumption stream can be also achieved with

other assets.

Hence, an asset is redundant if it has payoffs 1KD + which are a linear combination of the existing assets

1,2,...,k K= : 1

1

KK k kD Dα+ =∑ ng to the following linear rule:

k=. Choosing prices accordi

1

1

KK k k

kq qα+

=

=∑ in every event tω we have:

1 1 1,...,) ( )t t

k t k Kt tq Dωω ω ω− =

∈Ω ⎡+ ⎣1 1

1,..

1

.1

1

,1

( ( ) ( )

( ) ( )t t

k t K t K tt t t t t t

k Kk t k tt t t t

rank D q

rank D qω

ω ω ω ω ω

ω ω ω ω

− − −+ +

=− −

∈Ω

⎡ ⎤ ⎤+⎣ ⎦ ⎦

⎡ ⎤= +⎣ ⎦

i.e. the rank of the payoff matrix that includes the payoffs of the redundant assets does not change, since the

additional column in the payoff matrix is a linear combination of other columns.

om the previous chapter. If the

c

ey do not create

One possibility to represent the concept of market completeness is to map the consumption stream from the

state-preference15 model (e.g. Lucas model) into the mean-variance framework fr

introduction of additional assets enlarges the efficient frontier, then the original economy could not have had

omplete markets. Why? If assets are redundant in the state-preference model, their payoffs must be perfectly

correlated with the payoffs of other assets16. Thus, their inclusion in the portfolio of risky assets would not

change the efficient frontier shape. Consider for example hedge funds. With investments in various derivative

products they can reduce the variance of their payoffs without cutting the expected returns. This reduces the

correlation with other risky assets with linear payoffs and increases the diversification potential of hedge fund

investing. As a consequence, the efficient frontier including hedge funds shifts to the left promising investors to

receive the same expected returns with lower standard deviation. If hedge funds change the mean-variance

opportunity set (they are not perfectly positively correlated with other assets), they cannot be redundant in the

state-preference context. Thus, the economy without these assets must have been incomplete.

To summarize, redundant asset in the state-preference model are also redundant in the mean-variance

framework and a market is complete if any other asset is redundant. If assets are redundant, th

additional insurance possibilities, the efficient frontier in the mean-variance framework and the rank of the

15 The state preference model is based on the idea that agents build their preferences over scenarios, or with respect to the occurrence of different states. In contrast, in the mean-variance framework, agents consider only the mean-variance profile of assets returns over the whole investment period. The occurrence of particular scenario gains an importance only if it influences the mean and the variance of the final payoff.

16 ( , ) ( )( , ) 1( ) ( ) ( )

Cov A A Var AA AA A Var A

α αρ ασ σ α α

= = =


payoff matrix in the state-preference model do not change. If there are non-redundant assets, i.e. assets that

change the rank of the payoff matrix and the efficient frontier, their inclusion creates additional insurance

possibilities, thus the market cannot have been complete.

3 THE PRINCIPLE OF NO ARBITRAGE

An arbitrage opportunity is a risk-less profit. In equilibrium, there c

this would conflict with the assumption that agents fin

annot be any arbitrage opportunities because

d an optimal portfolio. Adding an arbitrage opportunity

can improve any portfolio.

Definition:

An arbitrage is a self-financing trading strategy, i.e. there is some strategy t=0,1,...θ with 1= 0θ− such

that for all

( )K K

k kt t t

k kD q

= 0,1,2,...t

0 k k kt t tq 1

1 1θ θ θ −

= =

= +∑

and the resulting consumption is positive: ,

θ

The non-existence of arbitrage opportunities is fundamental for asset pricing. In the absence of arbitrage prices

can be represented as conditional expected values. Since these prices must be consistent with the optimization

+∑

0 0tθ >

i.e. tθ ω ω≥ ≠0 0( ) 0 for all and all t = 0,1,2,.... and 0t t

calculus of agents with different utility functions and different risk-aversions, we use the so called risk-neutral

measures t=1,2,..π to build expected values. The existence of this measure is guaranteed by the absence of

arbitrage. This result is known as the Fundamental Theorem of Asset Pricing (FTAP).

FTAP:

There is no arbitrage opportunity if and only if there is a state price process t=1,2,.. 0π such that

111 t ( )

(1(

)) ( ( ) ( ))k t k t k t

t t tt

tq D qπ ωω1 t tt ω

ω ω ωπ −

−− = +∑ (1.5)

− ∈Ω

where 1t ttω ω ω−= .

In other words, we can price assets by discounting their payoffs with respect to the state prices, which depend

on agen y rectly through the dependence on asset prices. Note that state prices are not t’s preferences onl indi

equivalent to the so called physical (or objective) probability measure p. The state prices are a theoretical

construct that help to find the fair17 price of payoffs. In particular, t=1,2,..π is the price of an elementary security

paying 1 only in state tω of the path tω .

17 In the insurance context, „fair“ means that the insurance premium must be equal to the expected damages.


Proof: (Risk-neutral measure implies no arbitrage)

Consider any self-financing strategy . s tω, i.e uppose that for all t=0,1,2, and for all the budget constructions are satisfied:

)t

k k

0 11( ) ( ) ( ) ( ( ) ( )) (

K Kt k t k t k t k k t

t t t t t tq D qθ ω ω θ ω ω ω θ ω1 1

−−+ = +∑ ∑

= =

Multiplying both sides with tt ( )π ω and adding across tω gives:

0 1t

1 1t 1( ) (( ) ( ) ( ) ( ( ) )) ( )) (

t t

K Kt k t k t k t k t k t

t t t t t tk k

t tq D qω ω

θ ω ω θ ω ω ωπ ω π ω θ ω −−

= =

⎛ ⎞ ⎛+ = +⎜ ⎟ ⎜

⎝ ⎠ ⎝∑ ∑ ∑ ∑ ⎞

⎟⎠

Using (1.5) we get:

0 1t

11 1

1 1( ) ( ) ( ) (( ) )) (

t

K Kt k t k t k t k t

t t tk

tt t

kq q

ω

θ ωπ ω ω θ ω ω θ ω− −− −

= =

⎛ ⎞+ =⎜ ⎟

⎝ ⎠∑ ∑ ∑

Adding this along all paths t0, 1, t=( ..., )ω ω ω ω gives:

k00 1

1t (( ) )

t

Kt k

tt k

t qω

π θ θω ω −=

=∑∑ ∑

Now, if and this would require0 ( ) 0ttθ ω > t ( ) 0tπ ω 1 0θ− > , saying that a positive payoff incurs positive costs, ruling out

arbitrage opportunities.

QED

f we did reduce the sequence of budget constraints to a single budget constraint in

terms of present values of all future expenditures and incomes. This single budget constraint always

needs to hold. If markets are incomplete then in addition one needs to make sure that the process of

4 GE

To give the

ists of a finite number of states

Note that in this proo

gaps between an agent’s expenditure and his income can be achieved with the set of available assets.

OMETRIC INTUITION FOR THE FTAP

geometric intuition of the Fundamental Theorem of Asset Pricing, we consider a simple case: the

one-period model. It simplifies the future significantly: it cons 1,...,s S= at the

second date.18 Thus, the optimization problem of a consumer is:

, 1,..., .

kK

i ks s k

kc D s Sθ

=

= =∑

where is the consumption in t=0 (this is a short-hand notation to

0 1

0 0

max ( , ,..., )

,

i i i iS

i ik k

U c c c

c w qθ

θ= −∑

1

0ic 0

iθ in the multi-period model), is the 0iw

18 In the two-period model we use the simple notation s=1,2,…,S for the states 1 1ω ∈Ω .


( ) 001

K kk k

kD q θ

=

+∑ , where 0k

θwealth an agent i starts with: is the portfolio inherited from the past.

In an equilibrium, arbitrage opportunities are excluded, i.e there is no vector of strategies KRθ ∈ such that

. In other words, there is no such vector of strategies ´

0qD

θ⎡ ⎤

>⎢ ⎥⎣ ⎦

KRθ ∈ that does not require

capital, ' 0q θ− ≥ , but provides a non-negative payoff, 0D θ ≥ , and a strictly positive payoff in minimum one

of the states including s=0. By the Fundamental Theorem of Asset Pricing, the no arbitrage

s

condition is

equivalent with the existence of a state prices SRπ ++∈ , such that q D k Kπ= =∑ . We already

showed the proof that prices calculated as expected payoff values with respect to the risk-neutral probabilities

are arbitrag e condi

nea

1s=

e-free. Now, we are going to show graphically that the no arbitrag tion requires the existence

of positive state prices, i.e. that the price of asset k is a positive li r function of its payoffs.

Consider two assets paying dividends in 1,...,s S

, 1,...,S

kk s s

= . The payoffs in these states can be represented by the

s

state, we first determine the bound, where the asset payoff,

vector D R∈ . For simplicity we consider the se of two states. Thus, the dividend payoff of the both assets in

can be represented by the vectors and . To find the set of non-negative payoffs given a particular

2 ca

1,2s = 1D 2D

sD θ , is equal to 0. This is a line orthogonal to the

payoff vector19. Plotting these orthogonal lines for the vectors 1D and 2D , we determine the set of non-

negative payoffs in both states as cros a of two half planes as shown in the figure below.

To determine the set of arbitrage opportunities, we have to find a strate

D2

D1

q

or ' 0q θ ≤

the price vector

with a positive payoff in at least one of the states. To find th

q so that conditions ' 0q θ ≤ and sD θ are satisfied.

19 The scalar product of two vectors is the angle build by these vectors. The scais smaller (greater) than 90 degrees. The scalar product of orthogonal vectors is

sing arearbitrage

gy requiring investments, i.e. ' 0q θ− ≥

e set of arbitrage portfolios we then plot

This is possible if and only if q does not

lar product is positive (negative) if the angle equal to 0.


belong to the cone of and , i.e. if there exists 1D 2D 1 0π> > such that

5 FIRST WELFARE THEOREM

An allocation of consumption streams

1 2(1 )q D Dπ π= + − .

,00,1,... 1

( )I

t t i

i=θ ω

= is attainable if each component is in the consumption set

of the agent and it does not use umption than is available from the dividend process:

In a financial market the allocation of consumption streams

more cons

,0

1 1( ) ( ) , 0,1,2,...θ ω ω ω

= =

= =∑ ∑ for every I K

i t k t tt t

i kD t

( ),00,... 1

( )Ii t

t iθ ω= =

is Pareto-efficient if and only if it is

attainable and there does not exist an alternative attainab consumption streams le allocation of ( ),00,... 1

ˆ ( )Ii t

t iωθ =

=,

such that no consumer is worse off and some consumer is better of

( )f:

( )θ ,0U ( ) U ( ) for all i and > for some .ˆ ii

ie oe some agents consume much more than thers. From the

perspective of fairness, this might not be optimal.

ω θ ω= =≥i i ,00,... 0,...

t i tt t

Note that market effic ncy d s not rule out that o

Proof: (The First Welfare Theorem):

( )( ) ( )

Question: Wh

I

( ),00,...

,, 0t t

0

y did the agents not choose ?

Since markets are complete the alternative allocation must have been too expensive:

( ) ( ) ( )

ˆ ( )

ˆ ( )t t i tt

i tt

i tt

θ ω

θ ωπ ω π ω θ ω

=

>∑∑ ∑∑

1 1

,0 ,0

,0

( ) )

,0

( ) ( ) ( )

( ) ( ) ( )

Which

ˆ ( )

is a contradict !

ˆ )

io

(

n

K Kk t k tt t

k k

t t i t

t t i

i

t

t

i t

D Dω ω

π ω π ωθ θ ω

π ω π ω

ω

θω ωθ

= =

>

⇔ >

∑ ∑

∑∑∑ ∑∑∑

∑∑ ∑ ∑∑ ∑

QED

,00,...

1

i i ,00,.

,00, .. ...

Suppose is cation that is Pareto-better than the financial marˆ ket allocation:

i.e. U U ( ) for all i and >

(

)

ˆ ( ) for some i.

i

i tt

i tt

i tt

θ ω

θω ωθ

= =

== ≥

an attainable allo

Adding across consumers givest tt tω ω

t t

t t

(

:

t t

t t

ti t i t

t

t

tt i t i

ω ω

ω ω

The above mentioned result only uses that the agents’ utility functions are increasing in ,0itθ . If the functions are

moreover smooth, i.e. continuously differentiable, and if boundary solutions are ruled out, Pareto efficiency can

be defined using the consumers’ marginal rates of substitution.

Then, if an allocation is Pareto-efficient the marginal rates of substitution needs to coincide across consumers:

0 0

0 0, ,

( ) ( )

( ) ( )s s

z z

i i j ji js z si i j i

U c U czMRS MRS

U c U cθ θ

θ θ

∂ ∂= = =∂ ∂

The graphical representation of this efficiency concept in the Edgeworth Box is:


jsc

jzc

j

i

isc

izc

efficient allocation

6 MARKET SELECTION HYPOTHESIS WITH RATIONAL EXPECTATIONS

We can use the Pareto-efficiency property to formulate a Market Selection Hypothesis that determines which

investor survives best in the dynamics of the market in terms of (relative) wealth over time. If every invest

are stochastic, then investor i will dominate investor j if his beliefs on the occurring of the states are more

Let investors maximize their expected utilities: t

or has

some expected utility function with possible different time preferences and different risk attitude and if payoffs

accurate. Note that investor’s dominance is not defined over his strategy but on his ability to make good

estimates.

( )0

( ) ( ( ))i

ti i i it tu

tE p u

ω

δ ω θ ω∞

=

=∑ ∑ they may differ with respect

to the personal discount factor iδ , the risk preferences and personal beliefs (.)iu ( )ip ω for the occurrence of a

ωparticular event . Consider, the marginal rate of substitution of two expected utilit

states s and z. Pareto-efficiency requires:

y investors between two

, ,( ) ( )( ) ( )

i i i j j ji js s s ss z si i i j j j

z z z z

p u w p u wzMRS MRS

p u w p u w∂ ∂

= = =∂ ∂

If investors differ in their beliefs (expectations) for state s, e.g. i js sp p> , there must also be some states such

that i jz zp p< , then w i j

s sw> and to ensure the equality above.20 Thus, the better the agent’s beliefs,

the more those agents get in the more likely states. Note that the only one requirement for our fitness criteria to

hold is decreasing marginal utilities as in the expected utility framework.

The result that investors get more wealth in those states they assign a higher probability goes back to Sandroni

with perfect

foresight that selects for consumers with correct beliefs. It says that in dynamically complete markets where the

w i jz zw<

(2000) and Blume and Easley (2002). They formulate a theorem for competitive equilibrium

dividend process is i.i.d. two expected utility investors i and j with different beliefs such that iP P= (investor i

20 This result utility.

follows from the decreasing marginal utility. The more wealth an investor receives the lower is his marginal


lim 1itjt

t

ww→∞

= , where itj

t

ww

has correct beliefs) and jP P≠ , then P almost surely is th

g wealth

7 STOCK PRICES AS DISCOUNTED EXPECTED PAYOFFS

Suppose the first asset is short-lived and risk free (e.g. a one period loan-saving contract).

applying (1.5) gives:

e relative wealth of investors

i and j. In other words, excludi impossible events in the long run, investor i is ier than investor j if he

makes better predictions. If markets are complete the degree of risk aversion is not essential for capital

accumulation.

n

1Then 10

t11( )

(1( ) ( ( ) ( ))

)t

t tt qω

π ωω

1

1 1

1tt t t

ttq D

πω ω ω

∈Ω− ∑−

−−

= + . Since the security is risk-less,

its payoff 1( )ttD ω is equal to 1 in all states. Under the assumption that the asset lives only one period, there is

1no price fo this period, i.e. qr it after ( )tω is equal to 0. t

This is equivalent to:

1t

1

11 1 1

1 1( )( )

( )) 1(

t

tt

tt

tt

frq

ω

πω

ωω

ωπ −

−

−−

∈Ω−= ≡

+∑

Using this and the condition 1

1

1

t

t k t t

t ωt11 ( )( ) (

(( )

)( ))k k

t tt

t tq D qπ ωπ ω

ω ω ω− −= +∑ , we get: −

∈− Ω

1 *1( ) ( )( ( ( )) )tk t k t k tq D qω ωπ ω1

1 11 ( )t

ttf t

t t tr ω

ωω −

−−

∈Ω+− = +∑

where * ( )( ) 0t

t tt

π ωπ ω = > is indeed a (risk-neutral) probability meast

tω ∈Ω

( ).tπ ω∑ure based on the information of

period t-1. Hence, asset prices can be presented as discounted expected payoffs, conditional on the information

ation. This is a sequence of events (or a path) realizedavailable at the time of valu from the beginning until 1t − .

8 CON ENCES OF NO ARBITRAGE S QU

8.1 The Law of One Price

The Law of One Price says that if from period t onward two assets have identical dividend processes, then in

period t-1 they must have the same price.

To see this, suppose that and . Buying

E

1 2-1 -1 t tq q< 1 1

=t,t+1,... =t,t+1,...D =Dτ τ1t-1θ units of the cheaper asset and selling

the same amount of the expensive asset gives: in t-1 and in all other periods the portfolio is

K K

k k

2 1 1-1 -1 -1( - ) 0t t tq q θ >

hedged, i.e. 11( ) ( ) ( ( ) ( )) ( )k t k t k t k t k t

t t t t tq D qω θ ω ω ω θ ω1 1

−−= +∑ ∑

= =


11since ( ) 0θ ω = ≠ for k 1,2 ant ( ) ( )θ ω θ ω −−=d for k=1,2k t k t k t

t t .

8.2 Linear Pricing

The second consequence of the no arbitrag e concept of linear pricing saying that if in period t-1

one buys and holds a portfolio

e principle is th

t-1θ then in the same period the price of the portfolio must be a linear

ation of the prices of its comcombin ponents:

t-1 t-1 1 1 1k=1

ˆ ˆq = k kt t tq qθ θ θ

Kˆ

− − −= ∑

To see this, suppose that ˆq qθ θ> . Then, buying 1ˆktθ − units of asset k and selling the portfolio 1tθ −

ˆt-1 t-1 1t− gives:

t−

ˆ( ) ( ) ( ( ) ( )) ( ) ( ( ) ( )) ( )K

t t k t k t k tt t t t t t t tq q D q D qθ θθ ω θ ω ω ω θ ω ω ω θ ω

ˆ ˆ 0q qθ θ− > in t-1 and otherwise the position is hedged: t-1 t-1 1

ˆ ˆˆ ˆK

k t k t tθ θ ˆ ˆ 1 1t t 1 1

1 1k k

− −− −+ = + + +∑ ∑ because

( ) ( ) ( ) ( )ω ω θ ω θ ω

= =

ˆ ˆ ˆθ 11

1

−−= =∑ and t

t t t tD D .

ntal Theorem of Asset Pricing is essential for the valu ion of redundant assets such as derivatives.

In general, there are two possible ways to determine the value of a derivative. The first approach is based on

ining the value of a hedge portfolio. This is a portfolio t delivers the same payoff as the

derivative. The second approach uses the risk-neutral probabilities in order to determine the current value of the

As

Kt k t k t k

=k

8.3 Derivatives

The Fundame at

determ of assets tha

derivative’s payoff.

Consider an example of the one-period binomial model. In this simplified setting, we are looking for the current

price of a call option on a stock S. sume that 100S = and there are two possible prices in the next period:

200Su = if 2u = and 50Sd = if 0.5d = . The riskless interest rate is 10%. The value of an option with

K if u and strike price is given by max( ,0)Su K− max( ,0)Sd K− id d is realized.

To determine the value of the call, we replicate its payoff using the payoffs of the underling stock and the bond.

If arbitrage is excluded, the value of the call is equal to the value of the hedge portfolio, which is the sum of the

he idea is t at

same price as the call that we look fo

Calculating the call values for each of the states (“up” and “down”), we get that the call value given state “up”

is respective Su K

values of its constituents. T hat a portfolio th has the same cash flow as the option must have the

r.

max( ,0) (200 100) 100Su K− = − = max( ,0) max(50 100,0) 0− = − hedge

portfolio requires then to borrow 1/3 and to buy 2/3 assets in order to replicate the call’s payoff in each of the

states:

= . The


" " 2 / 3*200 1/up −: 3*110 100down

=− =

where n is the number of stocks and m is the number of bonds needed to replicate the call payoff.

" ": 2 / 3*200 1/ 3*110 0

In general, we need to solve:

max( ,0)max( ,0)

u

d

C Su K nSu mBRC Sd K nSd mBR

= − = += − = +

( )u dC CnSd

−= is also called delta of the option,

Su − ( )d uSdCm SuC

BR Su Sd−

=

match (“up” a two securities (stock and

associated with “up” and “down” movements, which are already

h no premium for risk. In this case, we e probability of an “up” (“down”) move

equal to the risk-neutral probability

−.

In the binomial model, we need two equations to nd “down”) with

bond). In the trinomial model (if there is a state “middle”), we will need a third security in order to replicate the

call payoff etc.

The second approach to value derivatives is based on the FTAP result that in the absence of arbitrage we do not

consider the “objective” probabilities

considered in the equilibrium prices, instead we can value all securities “as if” we are in a risk-neutral world

wit can consider th ment as being *π ( *1 π− ). Thus, the expected value of the stock with respect to these risk

neutral probabilities is: d . In a risk-less world, this must be the same as investing S today

and receiving Sr after one period.

Then, or R .

up and down movements of the stock price

and the risk-free rate:

* *0 (1 )S Su Sπ π= + −

* *(1 )u dπ π+ − =* *(1 )Su Sd SRπ π+ − =

The risk-neutral probabilities are then defined over the size of the

* *, 0 1R du d

π π−= ≤ ≤

−. Using the risk-neutral measure we can calculate the current

* *(1 )C C* *(1 )Su SdSR

π π+ −= and the call: u dC

R= . value of the stock: π π+ −

To value an opt the binomial lattice model: ion in multi-period setting, we use

Note that the risk-neutral probability is a stationary measure, i.e. it remains the same at every node. To see this,

C

( )2max 0,ddC d S K= −

( )max 0,udC udS K= −

Cd

Cu

( )2= −m ax 0 ,u uC u S K


suppose that at some no ce the stock price is Z. Then, its expected value after 1 period is:

** *( ) (1 )E Z Zu Zd

ππ π= + − .

de of the binomial latti

In a risk-less world this value must be equal to ZR:

Zu Zdπ π= + − = . Since Z cancel, we get that the risk-neutral probability is constant over the * (E Z ZRπ

* *) (1 )

time and depends only on the size and the frequency of “up” and “down” movements. Consider an example of

a call option over the periods t=0,1,2. The value of this option in t=1 depends on the realized state (“up” or

“down”), i.e.:

( )

( )

1 * 1u uuC CR

π π⎡ ⎤*

1

udC

C

= ⋅ − ⋅+

* 1 *d ud ddC CR

π π

⎣ ⎦

⎡ ⎤= ⋅ ⋅

The value of the call at t=0 is:

+ −⎣ ⎦

( ) ( )222

1 2 * 1 * 1 *uu ud ddC C CR

π π π π⎡ ⎤= ⋅ + ⋅ ⋅ − ⋅ + − ⋅⎣ ⎦C respectively

( ) [ ]( )

2 2

2 2 2

* 0, 2 * 1 * 0,1

1 * 0,

Max u S K Max udS KC

R Max d S K

π π π

π

⎡ ⎤⎡ ⎤⋅ − + ⋅ ⋅ − ⋅ −⎣ ⎦⎢ ⎥=⎢ ⎥⎡ ⎤+ − ⋅ −⎣ ⎦⎣ ⎦

We can continue this argument for more and more periods to obtain the hypergeometric distribution, which in

limit gives the normal distribution (see Varian, 1989).

According to the Fundamental Theorem of Asset Pricing, if a price process is arbitrage-free, there exists no

strategy that generates risk-free returns on q.

9 EQUIVALENT FORMULATIONS OF THE NO ARBITRAGE PRINCIPLE

This is equivalent to the existence of a market expectation or a risk-neutral probability such that

, 0,1,2,...t tθ =

* 1 11 ( ) 0,1,2,...

1 t

k k kt t t

t

q E D q tr π + += + =

+.

Applying forward iteration to the expression above in order to get a result fo which is not dependent on

future realizations, we get the Dividend Discount Model (DDM)21:

r ktq

*1

1 , 0,1,2,...1t

tk kt

tq E D t

r

τ

π ττ

−∞

= +

⎛ ⎞⎛ ⎞= =⎜ ⎟⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠∑

Thus, if a market is rational, price movements will depend only on movements of the risk-free interest rate and

the dividend payments. Assuming that the dividend process follows a random walk, we can conclude that

perfectly anticipated prices must be random, i.e. ) kt * 1 * 1( (1 ) ) (

t t

k kt tE q r q E Dπ π+ +− + = −

21 To simplify expressions we have assumed a constant interest rate.


Expressing the no-arbitrage condition in terms of excess returns (returns exceeding the risk-less return) we get:

0kE R R− =

ord, the net pr o the risk-neutral probability must be equal to 0.

Positive net present values are possible only if one uses a probability measure different from

* 1t t ftπ +

In other w esent value of a strategy with respect t

( )

*π . Though, in this

In terms of expected gains, the no arbitrage principle requires that they are martingales, i.e.

case, the probability measure used does not include all possible risks.

* 1( ) 0t t tE g gπ + − = , where 1 11

( )k k kD q qt rttg τ τ τ

1T

kτ

τ

θ+ ++ −⎡ ⎤

0= +

=∑ , which is consistent with the idea that “nobody can beat the market”, since nobody can beat

a martingale. Though, every time somebody the market. To see this, consider the simple equilibrium

condition that demand is equal supply:

+=

= ⎢ ⎥⎣ ⎦∑ . Hence the cumulated expected gains are:

∞

*1

( )t

tE gπ τ

τ

beats

I K K

k ki=1 k=1 k=1

q qk

i

ik k

MVW

θ θ=∑∑ ∑

If the market value increases by x% and investor i’s wealth increases by less than x%, then somebody else must

have beaten the market. However, the number of investors who beat the market can be smaller than the number

such as

of investors who lose to the market, so that the average number of investors does not beat the market. From a

dynamic perspective, the question is if there are investors who beat the market persistently.

Sometimes it is not useful to work with abstract probability measures, *π . In this case, it is helpful to

*Pπ

= , which is also called pricing kernel, ideal

rn of a strategy with respect to the “ob

change the measure using the likelihood ratio process:

security, or stochastic discount factor. The expected retu jective”

probability P is then22 , where the covariance of strategy returns to the likelihood

ratio represents th

th respect to the risk-neutral probability

: ,( ) cov ( , )t t

k ktP t f t P tE R R R= −

e strategy risk.

To summarize, the absence of arbitrage is equivalent to the conclusion that gains are martingales, prices are

random and do not generate excess returns wi *π . If we define expected

returns in terms of the physical probabilities, the risk can be measured by the covariance of returns to the

likelihood ratio process. In this sense, the state prices contain information on the market risk that prevents

arbitrage. And hence the risk neutral measure should better be called the risk-adjusted measure.

To see that prices calculated as expected values under the physical probability are typically not fair, consider the

following example. Consider a security S with a current value of 5. Say the physical probability of an

22 Note that *

*f *R ( ) ( ) ( Rk k k ks

s s s s s s s pk )E R R p R p l R E lππ= = = = =∑ ∑ ∑ .

s s sspπ


“up”/”down” movements are equal to 90% respective 10%. The price in the next period can be either equal to

9, given that state “up” is realized, or 4, given that state “down” is realized. The expected value of the security

rice

as an attractive investment opportunity. Though, it is not risk-less. There is a 10% probability that the current

n

payoff under the physical probability is equal to 8.5. Compared to the current p of 5, this can be considered

price of 5 can decrease to 4. We can explicitly consider the risk of a “down” movement by calculating the risk-

eural probabilities *π . These are the probabilities that equalize the current price with its expected value. In the

example above, the r s considerably, i.e. *isk-neutral probability change π in the “up” state is now 20%,

the current price the risk-neutral pr

respectively 80% in the “down” state. If there is no risk, i.e. there is no state that delivers a lower payoff than

obability must be negative.

get

air prices

itable units. They decided to sell 5% of its Palm

stocks and retain 95%. At the IPO day, the Palm stock price opened at 38$, achieved its high at 165$ and

Are the risk-neutral probabilities relevant for personal investment decisions? The risk-neutral probabilities

determine the equilibrium value of future payoffs. Since these prices go into the bud constraint, they affect

the optimization problem of the investors. Though, investors are also utility maximizers. If investor’s expected

utility is defined over a probability measure that differs from the portfolio of his endowments, then the portfolio

that maximizes this utility will differ from the portfolio resulting by solving the budget constraint under f

and the investor will decide to trade. In other words, the existence of no arbitrage does not imply that investors

should follow a passive investment strategy (buy and hold an index for example).

10 LIMITS TO ARBITRAGE

There exist strictly positive state prices if and only if security prices exclude unlimited arbitrage. Though, in the

presence of limits of arbitrage like short-sales constraints, the arbitrage is limited and even the law of one price

may fail in equilibrium. Let us consider first some examples.

10.1 The Case of 3Com and Palm

On March 2nd 2000 3Com made an IPO of one of its most prof

closed at 95.06$. This price movement was puzzling because the price of the mother-company 3Com closed that

day on 81.81$. If we would calculate the value of Palm shares per 3Com share, which is 142.59$23, and subtract

it from the end price of 3Com, we get 81.81$ 142.58$ 60.77$− = − . If we additionally consider the available

cash per 3Com share, we would come to a “stub” value for 3Com shares of -70.77$! Clearly, this result is a

contradiction of the law of one price since the portfolio value (the value of Palm shares, the rest of 3Com shares

and the cash amount), which is negative, differs from the sum of its constituents, which are positive.

Though, the relative valuation of Palm shares did not open an arbitrage strategy, since it was not possible to

short Palm shares. Also it was not easy to buy sufficiently many 3Com stocks and then to break 3com apart to

sell the embedded Palm stocks. The mismatch persisted for a long time (see figure below).

23 0.95*95.06#outstanding 3Com shares


10.2 The Case of Closed End Funds

The case of closed end funds24 is more ingredients are not only known but also

tradable. Though, on average, the prices of fund shares are still not equal to the sum of the prices of its

components as the figure below shows.

puzzling since the portfolio

The reason for this mismatch is the f ed-end funds and trade their

components on market pric osed-end fund and selling the

corresponding portfolio until pically do not pay out the

dividends of their assets be

As in the 3Com-Palm case the violation of the law of one price does not constitute an arbitrage strategy because

n deepen until maturity.

xample of the risks associated with seemingly arbitrage strategies. The

LTCM managers discovered that the share price of Royal Dutch Petrolium at the London exchange and the share

price of Shell Transport and Trading at the New York exchange do not reflect the parity in earnings and dividends

stated in the splitting contract between these two units of the Royal Dutch/Shell holding. According to this

splitting contract, earnings and dividends are paid in relation 3 (Royal Dutch) to 2 (Shell), i.e. the dividends of

act that no investor can unbundle the clos

es. Additionally, buying a share of an undervalued cl

maturity does not work because closed end funds ty

fore maturity.

the discount/premium of the closed end funds ca

10.3 The LTCM case

The prominent LTCM case is an excellent e

24 A closed end fund is a mutual fund with a fixed asset composition.


Royal Dutch are 1.5 times higher than the dividends paid by Shell. Though, the market prices of these shares did

not follow this parity for long time as the figure bellow shows.

rtfolio forever: Doing this one can cash

in a gain today while all future obligations in terms of dividends are hedged. There is however the risk that the

the parity.

This example is most puzzling because a deviation of prices from the 3:2 parity invites investors to either buy or

sell a portfolio with shares in the proportion 3:2 and then to hold this po

company decides to change

10.4 No Arbitrage with Short-Sales Constraints

To illustrate how limits to arbitrage enlarge the set of arbitrage-free asset prices, consider the case of non-

negative payoffs and short-sales constraints i.e. k is ka 0 and 0λ≥ ≥ . The short-sales restriction may apply to one

or more securities. Then, the Fundamental Theorem of Asset Pricing reduces to:

Theorem: (FTAM with Short-Sales Constraints)

There is no long-only portfolio 0θ ≥ that q 0θ ≤ and A 0θ > is equivalent to

Proof

q>>0 .

:

Suppose q>>0 q 0θ > and 0θ ≥ . For strategy θ with A 0θ > must be true that . In other words,

every long-only portfolio must cost something.

Suppose , then for some kkq 0≤ (0,...,1,...,0)k

θ = is an arbitrage, i.e.

Hence, ALL positive prices are arbitrage-free because sales restrictions deter rational managers to exploit

eventual arbitrage opportunities. Consequently, the no-arbitrage condition does not tell us anything and we

need to look at specific assumptions to determine asset prices.

market portfolio; APT in the case that there are more than one risk factor

A 0 and q 0θ θ> ≤ .

11 IDENTIFYING THE LIKELIHOOD RATIO PROCESS

To get some structure for the Likelihood Ratio Process, we can use the CAPM where is supposed to be in the

asset span of the risk-free rate and the


determining expected returns; Macro Finance based on the preferences of a representative agent.

As we already have seen above, in the absence of arbitrage, there is always a risk-return decomposition of the

11.1 The Likelihood Ratio Process with CAPM

risk-free rate possible such that ( ) ( , )f k kp pR E R COV l R= + . Additionally, we proved that in the CAPM the

SML nee esult by showing that in the CAPM the likelihood ratio

process t por

Consider the eneral case of incomplete markets and decompose into

ds to hold. Now, we want to rephrase this r

has to be collinear to the marke tfolio.

g ⊥= +D D, ∈< >D D , and

∈<D

Sup⊥⊥>D .25 pose that γ= + +D a bM1 ere Dγ ∈< > , , Mγ ∉<, wh >1 26 and

( ) ( ) 0E E M( 1)E γ γ γ= = . =

We will t for th condition 0γ = show tha e SML to hold, the must be satisfied.

( , ) ( ) ( ) ( ) 0Cov M E M E E Mγ γ γ= − = . Hence, under the above assumption ( , ) 0Cov Mγ = Recall that

The price of the market portfolio is:

[ ] 21 1 σ( ) ( ) ( , ) ( ) ( )Df fq M E M Cov l M E M b MR R

⎡ ⎤= + = +⎣ ⎦ , which allows us to solve for b:

fR ( ) ( )q M E M−2 ( )

bMσ

=

For every z D∈< > we can also write:

[ ] 0( ) ( ) ( , ) ( ) ( , )Df fq z E z Cov l z E z bCov w zR R

⎡ ⎤= + = + +1 1 ( , )i Cov zγ⎣ ⎦ . Using the expression above for b,

we get: ( ) 2

1 ( , )( ) ( ) ( ) ( ) ( , )f

Cov M zq z E z q M E M Cov zR M

γ γ⎡ ⎤

= + − +( )σ⎢ ⎥

⎣ ⎦.

On the other hand, SML implies that:

2

1 ( , ) 1Cov M z ⎛( ) ( ) ( ) ( )q z E z q M E M ⎞− = −⎜ ⎟

This requires t

( )f fR M Rσ ⎝ ⎠

hat ( , ) 0 Cov z z Dγ = ∀ ∈⟨ ⟩ or 2 ( ) 0σ γ = , γ which is equivalent to saying that is risk-less, i.e.

for some Rγ λ λ= ∈1 . This is a contradiction to the assumption that γ is not in the asset span of the risk-

less asset and the endowment: , Mγ ∉< >1 .

25 This decomposition is not necessary in the case of complete markets, where Dl l= .

26 1 denotes the risk-free pay off.


11.2 The Likelihood Ratio Process with APT

In the CAPM, the beta measures the sensitivity of the security’s returns to the market return. The model relies on

restrictive assumptions about agents’ preferences or security returns. The Arbitrage Pricing Theory (APT) is based

on a pricing relation similar to that of the CAPM but with severa rn.

Let

l factors replacing the market retu

1,..., Jf f

linearly inde

be contingent claims called factors. It is assumed that all factors have zero expectation and are

pendent, i.e. ( , ) 0 for j lCov f f j l= ≠ . The likelihood ratio process can be identified as

=j 1

α=∑J

j j re jf D∈< > . f whe Following the same approach as in the CAPM, we get:

( )( , )

( )

j kp

j

f R

f( ) ( )jk f j f

p p

CovE R R E f R

Var− = −

∑. This gives a richer economic regression. However this

ounded by an economic model.

nt

A fundamental issue in finance is the question under which conditions can prices, which are market aggregates

be generated by aggregate endowments (consumption) and some aggregate utility function. If this were

identify the likelihood ra rocess wit

utility function. Moreove

ndivi utility functions. Finally, we ask if it is possible to use the

aggregate decision problem to determine asset prices “out of sample”, i.e. after some change of e.g. the

this in the

case of a one-period model, recall that the no arbitrage condition requires the existence of some risk-neutral

probability such that

regression is not f

11.3 The Representative Age

possible, one could tio p h the marginal rates of substitution of some aggregate

r, we are interested in the question if it is possible to find an aggregate utility function

that has the same properties as the i dual

dividend payoffs.

To answer the first question, we use the “Anything Goes” theorem saying that for every arbitrage free prices

there exists an economy with a representative consumer maximizing some expected utility function such that the

prices are the equilibrium prices of the economy populated only with this representative agent. To see

* 0π *q Dπ= . Choosing the utility function ( , ,..., )U c c c c cπ= +∑ for the

representative agent, at prices q he will consume the aggregate endowments.27 Thus, if there is an artificial

agent with beliefs over all possible states, he will also generate the prices .

The second question we are interested in is rela

individuals and the representative agent. Individuals may have different beliefs or differ in their risk aversion

*0 1 0

1

SR

S s ss=

* k

ted to the inherent differences in the utility functions of

0π q

27 To see why this argument holds consider the*

following maximization problem: c c c q w c D0 0 0

s k kθ s.t. and max k k

s s k k s sπ θ θ+ + = =∑ ∑ ∑ .

The first order condition gives *k ks s

sq Dπ=∑ , which is equivalent to the no arbitrage condition required to hold in

equilibrium.


meanwhile the representative agent is assumed to be risk neutral with beliefs equal to * 0π . Thus, the

question is under which conditions the utility function of the representative agent RU has the same properties

as the individuals iU . Whereas the existence of RU is conditioned on the existence of arbitrage-free prices, the

requirement that RU is indeed representative ty functions of individual a oned on the

eight here is a trade-off between utili

that can be achieved. The social planer has then to choose the weights appropriately under the restriction that

ny he s determine which

for the utili gents is conditi

Pareto-efficiency of the equilibrium allocation in the economy. In other words, if equilibrium allocations are

Pareto-efficient, then there exists some aggregate utility function of the same type as the individual utility

function able to generate the observed asset prices from a single agent decision problem. To see why this

argument must hold, note that the Pareto-efficiency condition is equivalent to maximizing some welfare function

being the w sum of individual utilities. Under scarce resources t ty levels

the allocation achieved is attainable with the given resources. In a nutshell, to find RU as an appropriate

aggregation of ma we can represent RU as a weighted sum, whereby t w

ed

iU , eight

Pareto-efficient allocation is obtained. If we elaborate on this, we have to remind that Pareto-efficiency requires

that all agents in the economy have equal MRS over consumption today and consumption in a period ahead, and

all agents agree on the state prices, i.e.* *

1 11 1

* *1 1

0 0

( ) ( )... :( ) ( )

I I

I I

U c U c

U c U cπ∇ ∇

= = =∂ ∂

. In particular to ob e observed

equilibrium allocat from the welfare maximization problem we define 1

R

,..., 1 1U ( ) sup ( ) ¦

I

I Ii i i i

c c i iW U c c Wγ

= =

⎧ ⎫= =

tain th

ion ⎨ ⎬⎩ ⎭

w

∑ ∑

here i*

0

1

( )i iU cγ =

∂. The first order condition of this problem gives

* *1 1 1( ) ... ( ) :I I IU c U cγ γ λ∇ = = ∇ = and

RU ( )W λ∇ = . Hence,

*

1 0*

0

( )( ) ( ) 1( )

R 1U and ∇

∇ = ∂ =∂

i iR

i i

U cW U WU c

.

In terms of security prices, we can rewrite the problem as: *

Max ( ) s.t. θ

θ⎛ ⎞−

− ≤ ⎜ ⎟⎝ ⎠

R R R qU c c W

D and the first

order condition gives:

* *

* 1*

R Rq D 1

0 0

( ) ( )

( ) ( )

R R i i

i i

U c U c D DU c U c

* π∇ ∇=

∂ ∂. Hence

s single representative agent decision problem.

To see why the representative agent is of the same type as the agents in the economy, suppose that all agents in

i i i i i

= = asset prices can also be generated from

the economy have common beliefs and time preferences. In this case, i0

1( ) ( ) ( )

S

s ss

c u c p u cβ=

= + ∑ for all U

1,..,i I= . To show that the agent with 1

i( ) sup ( ) ¦ where I

R i i i iU W U c c Wγ γ= = =⎨ ⎬∑ ∑ *,..., 1 1

0

1

( )

I I

c c i ii i U c= =

⎧ ⎫

⎩ ⎭ ∂ is


really representative, rewrite his expected u

01 1

s ss i= =

tility using as

,..., 1( ) sup ( ) ( ) ¦

I

I S IR i i i i i i

c c iU W u c p u c c Wγ β

=

⎧ ⎫⎛ ⎞= + =⎨ ⎬

( )i iU c

1⎜ ⎟

⎩ ⎭∑ ∑ ∑ . Using that

0 0i=1

( ) max ( ) s.t. c i

IR i i i

cu W u c Wγ= =∑ ∑ and

⎝ ⎠

1i=0

Ii

0 0

Iis

1 i=1

( ) max ( ) s.t. c is

IR i i i

s s sc i

u W u c Wγ=

= =∑ ∑ we get

R , which is of the same form as the utility of agent i under the condition

ate endowment in each state.

r a representative agent that we have given so far are tautological. We

he utility of the representative agent.

necessary weights for the individual utilities. On the other hand, using these weights we derive the equilibrium

prices, i.e. we recover what we already needed to start the argument. In this sense, there is no additional

ough the analysis. For example, Lettau

along these lines that between 1952 and 1998 80% of the excess stock returns can be explained with

as a proxy for log consumption-wealth ratio. Note that this

From any investor’s perspective, the relevant question is under which conditions one can make conditional

estimates ext question requires defining the conditions under which we can u

a representative agent making “out of sample predictions”.

economy. If they have identical utilities and identical endowments, their aggregation is merely a scaling

procedure. If agents’ utilities are not identical but quasi-linea S S

also possible to aggregate them in order to make estimates. In this case the weights in the welfare function

01

( ) ( ) ( )S

R R R Rs s

sU W u W p u Wβ

=

= + ∑that the agent consumes the aggreg

Note however that both arguments fo

need the equilibrium prices and allocation in order to define t This gives the

information gained thr and Ludvigson (2001) show with an argument

movements of aggregated households’ wealth

argument is an “in sample” statement which is equivalent to saying that security markets are 80% Pareto-

efficient.

for future equilibrium. Thus, the n se

Obviously, the first condition under which this is possible is related to the heterogeneity of the agents in the

r, i.e. ( )( , ,..., ) ,...,i i i i i i i iU c c c c u c c= + , it is 0 1 0 1

*

0

1

( )i iU c∂are independent of the equilibrium allocations. This would however imply that the agents` asset

allocations do not change with their income. The following conditions that we describe are all based on the

pter. The third assumption then additionally expected utility hypothesis that we will elaborate on in the next cha

to the expected utility hypothesis requires agents to have common beliefs on the occurrence of the states. If

moreover if there is no aggregate risk, i.e. 1 1

,K K

k ks z

k k

D D s z= =

= ∀∑ ∑ , then the risk-neutral measure is equal to the

physical measure. In this case, the efficient allocation lies on the diagonal of the Edgeworth box ( s zc c= ) and

*

, *s s

s zz z

pMRSp

ππ

= = . Finally, if agents have common beliefs, markets are complete and agents have (identical)

constant relative risk aversion (CRRA) and collinear endowments, we can also use the representative agent for

“out of sample” predictions. Last but not least this is also possible if agents have quadratic utility functions and


the same beliefs, provided markets are complete28. they share

An example for the utility of the representative agent if there is no aggregate risk and agents share common

beliefs is: 0 1 01

( , ,..., ) ( ) ( ) for any concave S

R R R R RS s s

sU W W W u W p u W uβ

=

= + ∑ . Moreover, if beliefs and time

preferences are common and agents have quasi-linear quadratic preferences, i.e.

20 1 0

1 2

iSi i i i i i i

S s s ss

γ=

⎛ ⎞⎜ ⎟⎝ ⎠

( , ,..., ) ( )U c c c c p c cβ= + −∑ , then we get:

20 1 0

1 1

12

RS I

S s s s is i

γ( , ,..., ) ( ) where R R R R R R R RU c c c c p c cβ γγ= =

⎛ ⎞⎜ ⎟⎝ ⎠

i.e. if

0 1 01

Si i i i i i i i

S s ss

U c c c c p c Wβ δ= + =∑ i( , ,..., ) ln( ) ln( ) and w ,

then 0 1 01 1

S s s s ss i

U c c c c p c p p= =

( , ,..., ) ln( ) ln( ) where .

that asset prices can be derived in this strong form from a single utility function, we can

more the case of the one-period model:

= + − =∑ ∑ .

Finally, if agents have logarithmic utilities with common time preferences and their endowments are collinear,

β δ= + =∑ ∑

Having established

identify the likelihood ratio process from the representative agents` marginal rates of substitution. Consider once

=

S IR R R R R R R R i i

0 1 01( , ,..., ) ( ) ( )

SR R R R

S s sRU c c c u c p u cγ

= + ∑1

0 01

, , 1,..., .

s

s s kk

c w c D w s Sq

λ

λ

=

=

= = =

=

∑

Suppressing the index R, the first order conditio

0

. . 1

KR k Rk

K

kk

s t

λ

=∑

S

s=1 0

( )( )

k ks sp u cu cγ sD q′

=′∑n is: . Hence, the likelihood ratio process is

then:

0 0

( ) ( ) , 1,...,( ) ( )

s ss

u c u wl su c u wγ γ′ ′

= = =′ ′

S , i.e. fluctuations in aggregate consumption determine asset prices.

HE RATIONALITY BENCHMARK

To finish the thoughts on the foundations of asset pricing we show how under specific assumptions the concept

12 T

28 For more results along this line see Hens and Pilgrim (2003).


of perfect foresight equilibrium can be used to derive a very simple and compelling asset pricing implication.

Theorem:

se

• relative dividends are i.i.d. (i.e.

Suppo

st s( ) and p ( ) for all t t tS pω ω ωΩ = = ),

• agents are expected utility maximizers, i.e. 1( ) ( ) ( )i i t i iU c E u c∞

= ∑ , 1

i tiPt

γ=

• all are logarithmic or there is no aggregate risk ( ,

• all agents have rational expectations, i.e.

then

(.)iu1

Kk

tk

D=∑ is deterministic )

, 1,...,iP P i I= = ,

,0(1 )

ki k it P j

j

DED

λ λ⎛ ⎞⎜ ⎟= − ⎜ ⎟⎜ ⎟⎝ ⎠∑

is the unique equilibrium with perfect foresight. In particular then, the

relative prices must be also constant over time and must equal to the expected relative dividends.

ve this theorem, consider first the one-period optimization problem defined over the consumption: To pro

i( ) . . iu c s ti i i i i i0 1 0

1

i i i

i i

1( , ,..., ) ( )

, 1,..., .

S

S s sis

k k

k

U c c c u c p

w

c D w s S

γ =

= +

= =

∑

∑

rder condition is the Euler equation:

0 0iK

c λ

λ

=

1

i 1

s s kk

K

q

λ

=

=∑0k=

\S

\s=1 0

( ) , 1,...,( )

i ik ks ssi i i

p u c D q k Ku cγ

= =∑The first o , which is not only necessary

but o ction is concave.

Now u

als sufficient since the objective fun

, s ppose that i, k 0(1 )i s

s j

D 1,...,k

s sj

p iλ λ= − =∑ ID∑

. Using this to rewrite the consumption constraints

i0 0i ic wλ= nd a

i i0(1 )i wλ−

= , we get 10(1 )j jc

wλ−∑j

0(1 )kDi i i i s

k k s js Dsi j∑ ⎥⎦i

q w w pλ λ⎡ ⎤⎢ ⎥= = −⎢⎣

∑ ∑ ∑ and 0

1 0(1 )kj

wλ= −∑(1 ) , 1,...,

i iKi ks s j j

wc D s Sλ−= =∑


If there is no aggregate risk, then \

1( ) (1 )i i I

i iu c wλ= −∑ 0

RHSLHS

\10( )i i i

iu cγ =

, where RHS is the price and LHS does not depend

on the realized state. Does any isλ exist that satisfies the equation above? For LHS is smaller than RHS

and for the LHS becomes larger than RHS. Thus, if the utility function is continuous, there exists some

0 0iλ →

0 1iλ →

0iλ solving the first order condition. For example, in the case of a logarithmic utility function

S0

s=1

ik kssi i

s

p c D qcγ

=∑ .

With 0(1 )kDi i i i s

k k s js Dsi i j

q w w pλ λ∑

⎡ ⎤⎢ ⎥= = −⎢ ⎥⎣ ⎦

∑ ∑ ∑ and i i

i 0

1 0

(1 ) , 1,...,(1 )

Kk

s s j jk

j

wc D sw

λλ=

−= =

−∑ ∑S we get

0 0(1 )i i i i i

LHS RHS

w wλ γ λ= − iiγ

, which is solved for 0 1 iλγ

=+

.

Now consider the cas functions are time and state separable and since the e of multiple periods: Since the utility

physical probability measure does not change over time, we can consider any node and its predecessor. If assets

are short-lived then there are no capital gains but only dividend payments. The first order condition is then: \S ( )( )tc -

( 1)\( 1)s=1 ( )

i ik ks s

tsi i i st

s

p u D qu cγ

−

−

−−

=∑ where is the predecessor of in t. Hence the previous argument works

analogically.

Finally, in the case of long-lived assets we need also to consider capital gains. Thus, th r condition is

s s

e first orde\S

( )( 1)\

( 1)s=1

( ) ( )( )

i it k k ks s ts si i i st

s−

12.1 Empirical Evidence

n we want to see how good the three asset pricing models that we developed so far are do

p u c D q qu cγ − −

−+ =∑ . Again, inserting the claimed solution for the asset prices shows that at

those prices the claimed portfolio solves the FOCs.

QED

In this sectio ing on

actual data. The models we want to test are the SML from the CAPM, a macro-finance regression based on the

idea of a representative agent, and the equilibrium implications of perfect foresight equilibrium that we derived

ned in the previous theorem. Before we are ready to do so, we first

describe the macro-finance regression we want to consider: Recall the no-arbitrage condition in the macro-

finance model:

under the specific assumptions mentio

1 *( )( ( ))t k t1 1

1

1( ) ) (( )

t

k t t kt t t tf t

t

q D qR ω

ω π ω∑ ω ωω

−− −

∈Ω−

= + .

Where from the FOC of the representative agent we get:


* 1 t ( ) ( ( ))( ) ( )))

t tt f t p u cR 1 1( (t t tu c

ω ωπ ω ω − ′=

γ ω− −′, which then gives:

1 t1( )k t

tq 1( ) ( ( )) ( ( ) ( ))

( ( ))t

t tk t k tt tt

p u c D qu cω

ω ωω ω ωγ ω −

∈Ω

′+

′∑ .

Thus, to test the macro-finance model, we run a regression on past price levels tq

−− =

k1− against

1( ( )) ( )( ( ))

tk kt tt

u c D qu c

ωγ ω −

′+

′.

If we want to test the model for cross-sectional data, we have to run a regression on 1 ( )f kt P tR E R− − ag

since the risk-return decomposition of the macro-model predicts that

ainst

( , )kP t tCov l R

1 ( ) ( , )f k kt P t P t tR E R Cov l R− = + must hold.

pply the procedure discussed above to test the power of the macro-finance model for the

American stock market from 1998 to 2001. We have price data, stock market capitalization, and dividend

payments on stocks included in DJIA index, as well as G d interest rates. First, we run a cross-section

regression on the SML calculating individual excess returns and using beta based on the market capitalization of

the firms. Second, we run a time-series macro-finance regression. If we assume that

The difference between time series and cross-sectional analysis is important if we study aggregates or the market portfolio. The SML cannot tell us anything about the excess return of the market.29 Thus, if we want to analyze

the excess return of the market we have to apply time series models. On the other hand, if we are interested in

the excess returns of individual assets, we can determine specific risk factors and apply APT.

Now, we can a

DP data an

1-

( )1-cu c

α

α= , then we get

that has to regressed against 1ktq −

11 ( )k ktt t

t

c D qc

α

γ−⎛ ⎞

+⎜ ⎟⎝ ⎠

, where ct is the aggregate consumption in the economy

hich in the Lucas model is equal to total dividends.

Before we start with the regressions, we first look at the dividends data. The first observation we make is that

w

the aggregate dividends vary over time and grow exponentially.

29 The SML ( ( )k f k M fR Rµ β µ− = − ) is tautological for the market portfolio since 1Mβ = .


Moreover, aggregate dividends move with the nominal GDP.

Also, in contrast to one of dividends between

sectors are not consta

However, the fluctuatio lative dividends. Next, we

test the relationship between rela n as suggested by the strategy

the assumptions in the rational benchmark theorem, the relative

nt.

ns in aggregate dividends seem more severe than those in re

tive dividends and relative market capitalizatio*λ determined in the rationality benchmark. The first impression looking at the data confirms the notion that

there is a relationship between relative market values and relative dividends as the examples bellow show.

Relative Dividenen nach Branchen

0%

10%

20%

30%

40%

50%

60%

80%

90%

100%

04.0

1.19

73

04.0

1.19

74

04.0

1.19

75

04.0

1.19

76

04.0

1.19

77

04.0

1.19

78

04.0

1.19

79

04.0

1.19

80

04.0

1.19

81

04.0

1.19

82

04.0

1.19

83

04.0

1.19

84

04.0

1.19

85

04.0

1.19

86

04.0

1.19

87

04.0

1.19

88

04.0

1.19

89

04.0

1.19

90

04.0

1.19

91

04.0

1.19

92

04.0

1.19

93

04.0

1.19

94

04.0

1.19

95

04.0

1.19

96

04.0

1.19

97

04.0

1.19

98

04.0

1.19

99

04.0

1.20

00

04.0

1.20

01

04.0

1.20

02

Ant

el d

er e

inze

lnen

Firm

en a

nnd

en

Information technologyFinancialsUtilitiesNon-cyclical servicesCyclical servicesNon-cyclical consumer goodsCyclical consumer goodsGeneral industriesBasic industriesResources

Aggregate Div an

0

0.5

1

1980 1985 1990 1995 2000 2005

d GDP on DJIA

3.5

1.5

2

2.5

3

dividends GDP

70%

den

Div

ide


Now, we want to see if the model based on relative dividends provides a better description of the data than the

macro-finance model. To perform the latter we have chosen the utility parameters. The MSE is minimized in

sample for 10.12α = and 1.1γ = . Note that this is quite a high risk aversion – a phenomenon known as the

lative market value of McDonalds against its relative dividends provides a

ws:

The same result is obtained for any other firm (see slides!). If we sum over all stocks in DJIA, we get total values

for market capitalization and dividends. The relationship is presented in the next figure.

equality premium puzzle30. Doing this for McDonalds, for example, gives:

To compare, the regression on the re

much better fit over the time as the figure bellow sho

30 The equity premium puzzle is the observation that the FOC for a CRRA representative agent can only be satisfied on market data if we assign an unreasonably high risk aversion. Or in other words, in order to induce that the representative agent holds bonds and stocks he must be extremely risk avers because otherwise stocks are a too attractive investment.

Rel Div Rel MV MC Donalds 81-01

0.02

0.025

0.035

0.04

V 81

-01

0

0.005

0.01

0.015

0.03

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Rel Div 81-01

Rel

M

Mc Donalds R^2=2%

50000

60000

0

10000

20000

30000

40000

0 50000 100000 150000 200000 250000 300000 350000 400000

Merck

0

0.02

0.04

0.06

0.08

0.1

0.12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Rel MV

Rel Div

Procter & Gamble

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Rel MV

Rel Div


Again market values and dividend in one period move nicely with each other. So why is the macro-finance

ooking at changes in the total dividends and

changes in the market capitalization the correlation is also much worse:

To test if the explanator cross sectional perspective, we

run the SML-regressions testing both mo rt is from the year 1981. Charts for

other years give a similar picture (see e figure bellow, the SML persistently

explains less of the variance

regression much worse? The simple answer is that this regression relates market values and dividends of two

consecutive time periods and not just of one time period. Indeed, l

y power of the traditional finance model improves from

dels for each year. The following cha

slides). As one can easily see in th

in the observed variables than the model based on *λ , which we will later develop

also from our evolutionary theory.

REL DIV vs REL MV 2001 R^2 = 0.58

0

0.2

0.25

0 0.05 0.1 0.15 0.2 0.25

Rel Div

0.05

0.1

0.15

Rel

MV

Delta Market Values and Dividends

-100000

0

100000

200000

300000

400000

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

-1000000

0

1000000

2000000

3000000

4000000

5000000

-300000

-200000

-3000000

-2000000

Delta MVDelta DIV

Total MV and Total DIV

0

500000

1000000

1500000

2000000

2500000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 210

5000000

10000000

15000000

20000000

25000000

30000000

35000000

Total Market ValueTotal Dividends

Delta Market Values and Dividends

-200000

-100000

0

100000

200000

300000

400000

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

-1000000

0

1000000

2000000

3000000

4000000

5000000

Delta MVDelta DIV

-300000 -3000000

-2000000


This finding is actually true for all years, as the following summary statistic shows:

Though, is the predictive power of *λ also higher? To see this, we want to base our predictions on the notion

that prices are governed by a mean reversion process that force them to come back to the cross section

regression, the SML for the traditional finance model and the regression line of relative MV against relative Div

s of Litterman (2003), quoted before:”…we view the financial markets as having a center of gravity that is defined by the equilibrium of demand and supply”. The question then is which

regression line is more suitable for such a center of gravity

The differences between models are in the definition of this benchmark (or fundamental value). In the

evolutionary model, it is calculated as the difference between current and predicted market values. In the CAPM,

the benchmark is based on the difference in current and predicted returns. Comparing the

evolutionary model. Or in the word

2R of both models,

we can conclude that predictions based on market values determined by relative dividends (as in the rational

benchmark and later on in the evolutionary model) have better fit (higher 2R ) than predictions based on beta as

a risk measure.

SML 1981 R^2=0.03

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.3 0.5 0.7 0.9 1.1 1.3 1.5

exce

ss re

turn

-1

beta

R^2 EVOL vs CAPM 1981-2001

1.2

0

0.2

0.4

0.6

0.8

1

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

EVOLCAPM


Indeed, over 20 years, the evolutionary model provides an outperformance of 50% given that the estimates of

ev

stment strategy achieves 13% outperformance compared to the DJIA index. All this is well

known among traditional f es outperform the SML the

“value puzzle”. Since facts s well say that the

really puzzling thing here i ied models.

As further analysis show, the results are not very sensitive to the frequency of rebalancing, i.e. an investor who

rebalances his portfolio daily achieves 1.4% more return than an investor performing monthly rebalancing.

R^2 Evol prediction vs R^2 CAMP prediction

00.10.20.30.40.50.6

UN

ITED

PHIL

IP

MC

DO

NAL

D'

INT'

L

GEN

ERAL

M

EXXO

N

CO

CA

-

BOEI

NG

AMER

ICAN

R^2 EVOLR^2 CAPM

relative dividends are correct.

8

10

12

14

16

CAPMStar (rebal.)

6

An “out of sample” analysis confirms the better predicting power of the olutionary model. Over 4 years, the

evolutionary inve

0

2

4

Dez 80

Dez 81

Dez 82

Dez 83

Dez 84

Dez 85

Dez 86

Dez 87

Dez 88

Dez 89

Dez 90

Dez 91

Dez 92

Dez 93

Dez 94

Dez 95

Dez 96

Dez 97

Dez 98

Dez 99

Dez 00

Dez 01

inance people. Fama calls the fact that simple value strategi

like those given here, can however not be disregarded one may a

s that traditional finance still holds on to its badly specif

0.9

1

98 98

1.1

1.2

1.3

1.4

1.5

01.0

1.

01.0

3.

01.0

5.98

01.0

7.98

01.0

9.98

01.1

1.98

01.0

1.99

01.0

3.99

01.0

5.99

01.0

7.99

01.0

9.99

01.1

1.99

01.0

1.00

01.0

3.00

01.0

5.00

01.0

7.00

01.0

9.00

01.1

1.00

01.0

1.01

01.0

3.01

01.0

5.01

01.0

7.01

01.0

9.01

01.1

1.01

one share per firm (buyandhold) div monthly rebal


13 SUMMARY

The traditional model in finance is based on complete rationality, i.e. all agents maximize their intertemporal

utility having perfect foresight of prices. In general, the model does not have any specific structure other than the

principle of no arbitrage. Under specific assumptions, such as rational expectations, which are justified by market

selection, absence of aggregate risk or the assumption that agents have logarithmic utilities, the model has nice

structure, which fits well with long-run averages. Though, there are several fluctuations around the long-run

average. The rational model attributes them to exogenous shocks and does not aim to explain them.

ure of Incomplete

Markets Models. Kluwer Academic Publishers, Boston/Dordrecht/London, 2002

LeRoy & Werner (2002):Principles of Financial Economics, Cambridge University Press

Research papers:

Lucas (1978): Asset Prices in an Exchange Economy, Econometrica

Varian (1989):The Arbitrage Principle in Economics, Journal of Economic Perspectives.

Lettau Ludvigson (2001):Consumption, Aggregate Wealth and Expected Stock Returns, The Journal of Finance,

Vol. LVI No 3, pp. 815-849.

Sandroni (2000): (2000) Do Markets Favor Agents Able to Make Accurate Predictions? Econometrica (lead

REFERENCES:

Books:

Eichberger Harper (1998): Financial Economics, Oxford University Press

Thorsten Hens and Beate Pilgrim (2002): General Equilibrium Foundations of Finance: Struct

article) 68-6, 1303-1341.

Chapter 4:

Rational Choice:

Expected Utility and

Bayesian Updating

Rational Choice: Expected Utility and Bayesian Updating 52

1 RATIONAL CHOICE

In much of microeconomic theory, individuals are assumed to be rational. Rationality is a normative concept

stating how people should make decisions. Though, the reality is often different. Large body of experimental

work shows that the concept of rationality does not describe real decisions properly. To deepen the

need first to establish a normative basis for decision making which will

are with real individuals’ behavior. Also for giving advice we will need to

.

en formulating a normative decision theory it is important to not that there is a difference between a “good”

cision and a lucky decision. In the words of Howard (1988), “…a good decision is an action we take that is gically consistent with alternatives we perceive, the information we have, and the preferences we feel.” (p.

tcome. The connection between choices and feelings is

1.1 Preferences

rence relation, which is denoted by . It

understanding of peoples’ behavior, we

then serve as a benchmark to comp

know how one should take decisions

Wh

de

lo682). In contrast, a lucky decision has a good ou

established through the utility function.

The objectives of the decision maker can be summarized in a prefe

allows the comparison of pairs of alternative outcomes ,x y X∈ .

The first hypothesis of rationality is embodied in two basic as

completeness and transitivity. The assumption that a prefer

has a well defined preference between two possible alternat

sumptions about the preference relation:

ence relation is complete says that the individual

ives, i.e. for all , ,x y X∈ we have that either x≽y

r y≽x (or both). The strength of this assumption should not be underestimated. In many circumstances it is hard

The transitivity assumption says for all

o

to evaluate alternatives that are far from common experience.

, , ,x y z X∈ if x≽y and y≽z, then x≽z, i.e. the decision maker does not face choices in which his preferences

appear to cycle: for example, feeling that coffee is at least as good as cappuccino and cappuccino is at least as

good as tee then also preferring tee over coffee.

1.2 Utility Functions

A utility function assigns a numerical value to each element in ( )u x X , ranking the elements of X in

accordance to the individual preferences. If preferences are summarized by the preference relation ≽ then f r all o

,x y X∈ we have x≽y if and only if .

To insure that a utility function exists, we need that the preference relation on X is also continuous, i.e the

preferences cannot exhibit “jumps”. For example, a preference relation on a set of lotteries is continuous if for

all lotteries p, q, and r, if p ≽ q ≽ r then there exists

( ) ( )u x u y≥

α ∈ (0,1) such that q p rα α+ − ∼(1 ) . In words,

continuity means that small changes in probabilities do not change the nature of ordering between lotteries.


Not that the utility function representin

This is an ordinal property. The numer

e g preferences is not unique. It is only the ranking of utilities that matters.

ical values associated with the alternatives and the magnitudes of any

or example, if an individual chooses alternative x when facing a choice

I

differences are cardinal properties.

1.3 Concept of Revealed Preferences

According to the concept of revealed preference, the outside observer does not know the preferences of an

agent. He can only try to infer them from his choices. An agent is not rational if his choices cannot be

“rationalized” by some preferences. F

between x, y and z, he won’t behave “rational” by choosing y when he is faced with a decision between x, y,

and z`. In other words, if x is revealed to be at least as good as y, then y should not be revealed preferred to x.

n consumer demand theory, the idea behind the axiom of revealed preferences can be expressed as follows:

when facing prices and wealth ( , )p w the consumer chooses consumption bundle ( , )=x B p w even though

bundle ( ', ')=y B p w was also affordable, we can interpret the choice as “revealing” preference for x over y. If

tent demand, he would choose x over y whenever both are affordable. Thus, consistency

in demand requires that at least one of the consumption bundles is not

the consumer has consis

affordable given a new price-wealth

situation as budget constraint B(p’,w’). An example for inconsistent behavior, where both consumption bundles

are affordable is given in the figure bellow.

x

y

B(p,w)

B(p‘,w‘)

1.4 Concept of Experienced Preferences

The concept of experienced preferences is based on the notion that the outside observer knows the preferences

cl

to have developed methods to elicit preferences through questionnaires and statistical methods of comparing

reference on these issues see Frey and Stutzer (2002) “Happiness and Economics”.

e. But it eliciting preferenc

but abstract settings which are then transferred to the real problem at hand. For example, some private bankers

of the agent and is able to tell when the agent will regret his choice. Behavioral economists like Bruno Frey aim

answers conditional on agents socioeconomic characteristics (age, income, sex, employment,…). As an exiting

1.5 The Concept of Preferences in Finance

Finance is mainly based on the concept of revealed preferenc is applied by es in simple

let their clients rank lotteries in order to determine the asset allocation, which is best for the agent given the risk


aversion disclosed in their answers.

Since estimating agents’ preferences is necessarily associated with errors, it is advisable to impose minimal

assumptions on them. Under the condition that preferences can be defined over monetary equivalents, we try to

a simple axiom on agent’s preferences, Axiom 0, which

hus, a behavior concerning risk and intertemporal asset

ference rank

than choosing (A). Though, people typically choose (A) with the

motivation that a sure gain is better than a gamble even if the expected payoff is higher. While people prefer

sure positive payoff, they prefer to gamble when they face a choice associated with a sure loss, i.e. people who

previously choose (A) prefer (D). Now look at the combined payoff of these decisions: (A) and (D) is equivalent to

a lottery of loosing 7600 with probability 75% and winning 2400 with probability 25%; (B) and (C) is equivalent

to a lottery of loosing 7500 with probability 75% and winning 2500 with probability 25%. Clearly, (B) and (C) is

the better combination in any case since its payoff is higher than the combination (A) and (D) chosen by most of

the people.

2 E U T

Define X as the set of possible choices. For example X can be a set of lotteries L over a set of consequences

derive more complex rationality requirements from

assumes that more money is better than less money. T

allocation will not be called irrational if it does not contradict this axiom. However, as we will see below, in a

well developed financial market the axiom may be used to derive far reaching conclusions.

Consider the following example. The outcomes of a symmetric dice follow two schemes A and B. Scheme A pays

600$, 700$, 800$, 900$, 1000$ and 500$ and Schemes B pays 500$, 600$, 700$, 800$, 900$, 1000$ for

each of the outcomes 1,2,…,6. Although the probability of getting each payoff is equally in both schemes, some

people prefer scheme A to B because in 5 out of 6 cases the payoff from scheme A is higher than from scheme B.

This choice is a violation of Axiom 0 because the same money should get the same pre ing.

The second example of violation of the simple axiom that more money is better than less is a combination of two

choices between lotteries. The first one requires a choice between a sure gain of 2400 (A) and a gamble (B) with

25% chance of winning 10000 and 75% of winning nothing at all. The second choice requires a choice between

a sure loss of 7500 (C) and a gamble (D) with 75% of losing 10000 and 25% of loosing nothing. Clearly, the

expected payoff of choosing (B) is higher

XPECTED TILITY HEORY

1,... ,...,i nC c c c= . p in L describes then the probability of the occurrence of the consequence c C∈ . A lottery

can also be represented from a combination of states. In this case, the probability of a consequence ic is defined

as 1

S

i ss

p probc cs i=

= ∑=

wing

0.3, 0.4 and 0.3. Let 5, 2, and 2 be the payoffs in

, i.e. the sum of state probabilities associated with this consequence. Consider the follo

example: there are three possible states with probabilities

each of the states. In the lottery approach, there will be two consequences: 5 and 2 occurring with probability

0.3 (equal to the probability the state-preference approach) and 0.7 (equal to the sum of probabilities of the

states with 2 as payoff, i.e. 0.4+0.3).


We already assumed that the decision maker has preference relation on the set of lotteries, i.e. a complete and

transitive relation allowing comparison of any pair of lotteries. The existence of an utility function requires that

the preference relation is continuous.

We need one additional assumption on the decision maker’s preferences in order to represent his preferences by

a utility function with the expected utility form. This is the independence axiom. The preference relation ≽

satisfies it if for all lotteries p, q, and r and 0 1α≤ ≤ we have p ≽ q if and only if

(1 )p rα α+ − ≽ (1 )q rα α+ − . In other words, if we mix each of two lotteries with the same third one, then

ed by an

the preference ordering of the two resulting mixtures is independent of the particular third lottery used.

2.1 The Representation Theorem

Let ≽ be a preference order that is complete, transitive and continuous then ≽ can be represent

expected utility function, i.e. ( ) ( )u up q E p E q⇔ > if and only if ≽ satisfies the independence axiom.

One of the advantages of the expected utility representation is that it provides a valuable guide for action.

People often find it hard to think systematically about risky alternatives. But if the individual believes that his

choices should satisfy the axioms on which the theorem is based (in particular, the independence axiom), then

the theorem can be used as a guide in the decision process. It facilitates decisions because it separates beliefs

from risk attitudes.

2.2 The Allais Paradox

As a descriptive theory, however, the expected utility theorem is not without difficulties. The Allais paradox

constitutes the most famous challenge to the expected utility theorem. It is a thought experiment. There is an urn

containing balls numbered 0,1,...,99. The decision maker has to make two decisions. The first consists of a

choice between a lottery of gaining 50 independent of the number of the ball (lottery A) and a lottery paying 0 if

r of the ball is between 1 and 10, and 50 if the number of the ball

is above 11 (Lottery B).

The second decision consist of a choice between a lotte w 10 and

nothing otherwise (lottery A’) and a lottery paying 250 if the number of the ball is between 1 und 10 and

the number of the ball is 0, 250 if the numbe

ry paying 50 if the number of the ball is belo

nothing otherwise.


02500

05050

0 1-10 11-99

A 505050

B 502500

d B`≽A`.

certainty of receiving 50 over a

erred to

getting only 50 with the slightly better chance of 11/100.

h expected utility because they violate the independence axiom.

A

should not reverse in the s cond stage since the choice be en A’

and B’ once the third alternative (the outcome 11-99) is eliminated since it is the same in both lotteries.

ed.

dence axiom can also be represented using the lottery approach. Represent the

lottery outcomes and the associated probabilities as follows:

A`

B`

It is typical for individuals to express the preferences A≽B an The first choice means that one prefers the

lottery offering a 1/10 probability of getting five times more but bringing it with a

very small risk of getting nothing. The second choice means that a 1/10 probability of getting 250 is pref

Though, these choices are not consistent wit

ccording to the independence axiom, if one prefers lottery A over lottery B in the first stage, his preferences

e tween A and B is equivalent to the choice betwe

The violation of the independence axiom can result in a violation of Axiom 0. If individuals prefer A over B in the

first stage, one can sell A to them and buy B from them, respectively sell B’ and buy A’ in the second stage

where B’ is preferred over A’. If individuals` preferences are strict they are willing to pay a positive amount for

these deals. However in any case that my occur the total payoff of such a deal is 0, i.e. the counterparty cashing

in the margin is hedg

The contradiction to the indepen

10%0%90%

011%89%

$0 $50 $250

A

B 10%89%1%

0%100%0%

A`

A = 0.11*(0.00,1.00,0.00) + 0.89*(0,1,0)

B = 0.11*(0.09,0.00,0.90) + 0.89*(0,1,0)

A´= 0.11*(0.00,1.00,0.00) + 0.89*(1,0,0)

B´= 0.11*(0.09,0.00,0.90) + 0.89*(1,0,0)

A≽B and B`≽A` is then a contradiction to the independence axiom since subtracting the common lotteries the

B`

Note that the lottery payoffs can be written as:


lottery A (B) does not differ from lottery A’ (B’).

2.3 The Probability Triangle

A nice way of illustrating the expected utility hypothesis can be done in a triangle in which each corner

corresponds to one of three possible outcomes. The distance to the corner measures the probability of not

getting the outcome. On the diagonal from the upper left to the lower-right corner for example we can represent

all probability distributions with zero probability for the outcome assigned to the lower left corner.

1

0 1pa

pb

Pc = 1-

E p p u c=∑

pa -p

b = 0

n

In the probability triangle, the indifference curves of 1

u i ii=

are parallel straight lines

since

( ) : ( )

( ) ( ) ( )( )( ) ( ) ( ) ( )

is equivalent to − −

= = +− −

c c aU b a

c b c b

const u c u c u cE q const p pu c u c u c u c

.

Representing the Allais paradox in the probability triangle we get the following figure:

$0 $50

$250

A

B0.1

0.01A‘

0.11

B´

Indifference curves

rence curve of the decision maker to be steeper than the straight line

ver means that indifference curves cannot be parallel straight lines

ing A and B and A` ad B` are identical.

ch choices consider the state preference model. We can define three

states, i.e. s1=0, s2=1-10, s3=11-99. Suppose there are three Arrow securities so that agents can transfer

The choice A≽B (B`≽A`) requires the indiffe

connecting A and B (B` and A`). This howe

since the slopes of the straight lines connect

To look into the market consequences of su


s

z

ππ

money between those states back and forth at given prices . Violations of the independence axiom then

at agents prefer more money to less, since all lotteries can be attained by

An even more likely violation of the expected utility axioms than in the Allais paradox are the results observed in

an experiment conducted by Daniel Ellsberg in 1961.

Subjects are presented with an urn containing 90 balls. 30 are red and the rest is either yellow or black. The

proportion of yellow to black balls is unknown. Subjects must bet on the color that will be drawn. The payoff

matrix is as follows:

0$

result in violations of Axiom 0 stating th

some combination of the Arrow securities and the portfolio short one unit A and B` and long one unit A` and B

has a zero payoff and should not be valued positive by the agent.

2.4 Ellsberg Paradox

50$50$0

050$50$

50$00

005

red yellow black

B

A`

It is very common to individuals to prefer lottery A to B since A gives 1/3 chance of winning 50$, whereas the

decision maker does not know how likely the payoff is in the latter, the proportion of black balls can be very

small. The second choice decision maker face is between A’ and B’. Most of the subjects in the experiment

choose B’, because it gives 2/3 chance of winning 50$ whereas A’ can only be certain of 1/3 chance of winning

50$ and the proportion of yellow balls can be very small. The choice of A over B and B’ over A’ violates the

independence axiom because ignoring the outcome yellow the two lotteries A and B and the two lotteries A` and

B` are identical. The explanation for the preference reversal is that people always prefer definite information to

indefinite: the urn may have more yellow balls than black, in this case lottery A’ is more attractive than lottery B’,

but people tend to prefer “the devil they know”, i.e. lottery B’ where they get 50$ with 50% probability.

As the Allais paradox, the Ellsberg paradox opens an arbitrage opportunity as well. If A is preferred to B, one can

t unbundles the yellow and

than less.

A

B`

sell A to investors and buy B. If B’ is preferred to A’, one can sell B’ and buy A’. The strategy is hedged and if

preferences are strict so that the agent would be willing to pay some money for this strategy one can exploit his

preference. Though, from a market perspective, we cannot expect that the marke

black balls since their proportion in the urn is ambiguous. In this case, the market may be incomplete. Most likely

there will only be the two assets (0,1,1) and (1,0,0) available for trade. Hence, the typical choice in experiments

is not irrational, because it cannot be refuted by a market strategy violating the axiom that more money is better


2.5 Ambiguity

probabilities are equivalent to objective probabilities. Knight (1920) was the first who made a distinction

between risk (lotteries with objective probabilities) and uncertainty (lotteries with subjective probabilities). In

particular, when having to form subjective probabilities people will not mak

The Ellsberg paradox casts doubt on the basic premise of subjective expected utility theory that subjective

e point estimates but consider a

possible. Moreover, they do not assign likelihoods to the members of this set and

then compute compounded probabilities to get point estimates. They rather look at the worst-case scenario and

∈∆

offs and specific utility functions, we can ask under what general conditions it is

possible to assert that one state or payoff is preferred over another. We answer this question by comparing

probability distributions while assuming standard properties of the individuals’ preferences, i.e. increasing

marginal utility and risk-aversion. In general, two individuals with different preferences have different rankings

on prospects. However, in some cases it is possible to get an ordering that holds for all individuals regardless of

their preferences. This is possible when a choice between prospects can be made using their distribution

functions and one distribution stochastically dominates the other.

3.1 First Order Stochastic Dominance (FSD)

whole set of probabilities to be

maximize against that, so that their utility function can be expressed as: max min ( )pE u x . px

3 STOCHASTIC DOMINANCE

Instead of considering pay

Consider two random variables, x, z, with the distribution functions ( ) ( )A y P x y= ≤ and ( ) ( )B y P z y= ≤ . A

first-order stochastic dominates B if and only if for all y ( ) ( )A y B y< .

0

AB

FSD

Intuitively, no matter what level of y we look at, B always has a greater probability mass in the lower tail than

does A. For example, a lower-tail cumulative probability occurs at higher levels of y for A than for B.

Note that although A FSD B implies that the mean of y under A, ( )ydA y∫ , is greater than the mean of z under

B, a ranking of the means does not imply that the one FSD the other, rather, the entire distribution matters. The

relevance of FSD for portfolio choice is obtained from:

The distribution A FSD B if for every monotone (increasing) function we have .. :u R R→ ( ) ( )u uE A E B>


3.2 Second-Order Stochastic Dominance (SSD)

Only under quite stringent conditions will one distribution be FSD over another. A more-powerful result, involving

the concept of second-order stochastic dominance, is reached if we additionally make use of the risk-aversion

property of investors’ preferences.

A SSD B if and only if for all y ( ) ( )y y

xdA x zdB z−∞ −∞

<∫ ∫ . Geometrically, A SSD B if, up to every point y, the area

under A is smaller than the corresponding area under B.

0

1

A

B

+

-

SSD but not FSD

Concerning portfolio choice we get: A SSD B if and only if all monotone (increasing) and concave functions u

( ) ( )u uE A E B> .

+

ibution represents the expected value of y, i.e. the

t is not necessary to limit the definition of SSD to

as a higher me n A then A canno

u A u x

Note that the area lying to the left of the cumulative distr

expected utility under the particular distribution. Note also that i

the subset of distributions with the same mean. However, if B h an tha t

dominate B in terms of SSD. Note that ( ) )A E x( ) for (µ = = . Hence, if (A)< ( )Bµ µ then for

and the criteria for SSD is not met. If we choose ( ) ( ) ( )u uu x x E A E B= < 2( ) ( )u y y µ= − − , which is

concave, and A SSD B then ( ) ( ) 2 2

A BE y E yµ µ− − > − − or 2 2

A Bσ σ− > − . So if A and B have the same

mean and A SSD B, then A has a smaller variance.

3.3 State-Dominance and Stochastic Dominance

Risky alternatives can be formulated in terms of probability distribution over outcomes (see above) or in the case

dominance, i.e. implies. U c U d> . Note that state dominance implies FSD and the reverse is

C

that the random outcomes are generated by the occurrence of particular state, we can define a state-dependant

expected utility representation. The assumption made in axiom 0 that more money is better than less, is the state i i i i i ic d> ( ) ( )

not true.

onsider the following example. There are 3 states occurring with equal probability and two random variables A

and B with payoffs as listed below.


121B

210A

S=3S=2S=1

1/31/31/3 1

1/3

2/3

0 1 2

AB

e a higher payoff in e

In this part we focus on the question whether the mean-variance decision principle satisfies the axioms of the

expected utility theory.

4.1 Mean-Variance Principle as a Special Case of Expected Utility

The mean-variance principle can be considered as a special case of the expected utility function if we restrict the

agents’ preferences to be represented by a quadratic utility function:

B FSD A although B does not hav very state.

4 A SECOND LOOK AT MEAN-VARIANCE

2 2 2( ) ( ) ( ) ( ) ( )(1 ( )) ( ) ( , )2 2 2 2α α α αµ µ µ µ σ µ σ= − = − = − − =∑u s s sE x p x x x x x x x V

Also under the assumption that returns are distributed normally we get a mean-variance utility function. In this

case, agents’ expected utilities are defined only on the first two moments of the distribution since higher

s

moments are irrelevant for the normal distribution.

^( ) ( ) ( ; , ) ( ) ( ) ( , )u

xE x u x dN x u d N x Vµµ σ µ σσ−

= = =∫ ∫

ions. If agents have qua tion, they

ng wealth. We will see below that the so called absolute risk aversion is

increasing. This feature does n

The assumption that asset returns are normally dist

However, both of these results have considerable limitat dratic utility func

become more risk averse with increasi

ot seem very plausible.

ributed is also questionable since data show the existence of

so called “fat tails”, i.e. compared to the shape of a normal distribution, we observe too many extreme

w so that py=co

observations on the left and right side of the distribution.

Last but not least, under certain circumstances, the mean-variance approach is not consistent with Axiom 0. The

most prominent example is the mean-variance paradox. To see how this works, consider a lottery with a positive

payoff 0y > in one of the states and zero payoff otherwise. The probability of the positive payoff is p>0. Now,

assume that y → hile p → nst. Then, the variance of the final payoff will also tend to

infinity,

∞ 0

σ →∞ , and since the expected payoff of the lottery remains constant any mean- variance maximizer

will eventually prefer the zero outcome to the lottery:


µ

σ00

Finally, we analyze if mean-variance preferences are always consistent with the expected utility concept by

d over the payoffs 2, 4

an variance preferences nor the set of mean-variance

ngle with mean-variance preferences which are not

nce pr

cted uti

Clearly, this is a contradiction to Axiom 0 that agents prefer more money to less.

plotting the indifference curves of a mean-variance investor in a probability triangle define

and 6. As the figure shows, the indifference curves are not linear as required by the expected utility theory. Thus,

mean-variance preferences are not necessarily compatible with expected utility concept.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−2.7805−2.5366

−2.2927

−1.8049

−1.561

−1.3171

−1.0732−0.82927

−0.58537

−0.58537

−0.34146

−0.097561

0.14

634

0.14634

0.14634

0.39

024

0.39024

0.39024

0.63

415

0.63415

0.63415

0.87

805

0.87805

805

1.12

2

1.122

1.122

1.36

59

1.3659

1.3659

1.60

98

1.6098

1.6098

1.85

37

1.8537

2.09

76

2.0976

2.34

15

2.5854

2.82

93

3.0732

3.3171

3.561

4.04

88

4.29

27

−2.0488

−1.561−1.3171

−1.0732

−0.82927

−0.34146

−0.097561

1.122

1.3659

1.6098

1.8537

2.09762.3415

2.58

54

2.82

933.

07323.

3171

3.56

1

3.80

49

0.87

As discussed above, the expected utility framework match mean-variance preferences only in the case of

quadratic preferences or normally distributed returns. Though, there are examples showing that neither the

expected utility framework can cover the set of me

preferences can wrap all preferences within the expected utility framework. For example, the mean-variance

paradox is a case, where mean-variance preferences imply a contradiction to the expected utility concept by

violating axiom 0. Another example is the probability tria

linear, hence incompatible with the expected utility concept. On the other hand, if preferences are polynomial,

defined over higher moments or logarithmical, they are well-matched by the expected utility but not by mean-

variance principle.

Thus we view the mean-varia inciple as an alternative to the expected utility hypothesis. If one considers

the expe lity as the formalization of rational choice then the mean-variance principle must be seen as a

behavioral rule that is often used in practice.

Rational Choice: Expected Utility and Bayesian Updating

63

5 MEASURES OF RISK AVERSION

In this section, we briefly summarize the definition of risk aversion and give some examples.

A general definition that does not presume an expected utility formulation is based on the idea of comparing

preferences over a lottery and a certain payoff. A decision maker exhibits risk aversion if a degenerated lottery

that pays a certain payoff is at least as good as the lottery itself. In the context of expected utility theory, risk

aversion is equivalent to the concavity of . Strict concavity means that marginal utility of money is

decreasing. Hence, at any level of wealth, the utility gain from an extra unit of money is less than the utility loss

f having a unit money less.

r (convex).

The certainty equivalent

(.)u

o

A decision-maker is risk-neutral (risk-loving) if the utility function of money is linea

( )c F

certai

of a gamble is the amount of money for which the individual is indifferent

between the gamble and a n amount, i.e. ( )( ) ( ) ( ) ( ):C

u c F u x dF x U F= =∫ .

The risk premium ( )q F is the difference between the expected payoff of the lottery ( ) ( )F xdF xµ = ∫ and

the certainty equivalent, i.e. ( ) ( ) ( )q F F c Fµ= − .

Risk neutrality is equivalent to the linearity of (.)u , i.e. ''( ) 0u x = for all x. Then, we can measure the degree of

risk aversion by the curvature of (.)u . One possible measure of the curvature is ''( )u x , though, it is not

invariant to positive linear transformations of the utility function. To make it invariant, the simplest modification

is to use ''( ) / '( )u x u x . If we change the sign to have a positive number for an increasing and concave (.)u , we

get the Arrow-Pratt measure of absolute risk aversion: ''( )u x( )'( )

ARA xu x

= − . It determine the asset allocation in s

terms of units as income changes. The risk tolerance is then the inverse ARA or 1( )

( )RT x

ARA x= .

The relative risk aversion is obtained by simply multiplying the absolute risk aversion with the income:

''( )( )'( )

u xRRA x xu x

= − . It determines the asset allocation in terms of shares when income changes.

0

U F u F( ) [ ( )]= µ

u x( )

u x( )2

u x( )1

x2x1 c F F( ) ( )µx 0

u F[ ( )]µ

u x( )2

u x( )1

x2x1 c F( )

u x( )

x 0

U F( )

u x( )2

u x( )1

x2x1 µ ( )F

u x( )

x

Risk-neutral [q(F)=0] Risk-averse[q(F)>0] Risk-lover [q(F)<0]

µ( )F

U F( )

c F( )

u F[ ( )]µ

0

U F u F( ) [ ( )]= µ

u x( )

u x( )2

u x( )1

x2x1 c F F( ) ( )µx 0

u x( )

x 0

U F( )

u x( )

u F[ ( )]µ

u x( )2 u x( )2

u x( )1

x2x1 c F( )

u x( )1

x2x1 µ ( )F

Risk-neutral [q(F)=0] Risk-averse[q(F)>0] Risk-lover [q(F)<0]

µ( )F

U F( )

c F( )

u F[ ( )]µ


Consider some examples:

• ( ) xu x e α−= − is associated with constant ARA equal to α or an increasing RRA

• 1 2( )u x x xγ= − means increasing ARA

• ( )u x xα= is related to a decreasing ARA or a constant RRA

The demand (in units) for risky assets increases with income if and only if ARA(x) decreases with income. If an

investor has constant ARA(x), he should hold the same amount of risky assets by increasing income.

The wealth share of risky assets increases with income if and only if RRA(x) decreases with income. If an investor

A broad class of utility functions that exhibit some nice properties in terms of asset allocation (allocation on risky

and riskless assets) is the HARA (Hyperbolic Absolute Risk Aversion)-family. The name comes from the hyperbolic

representation of ARA, i.e.

has constant RRA(x), he should hold the same shares of risky assets as his income increases.

=AARAB

, where A and B are constants and the domain is the set for which ARA is

nonnegative. The HARA utility functi lass can be represented by on c

11

1

( )( ) 1 (1 )

ii xu x

β

β

α β

β

−+

=−

. For 0β = , we get

the exponential function ( ) exp( )xi iiu x

αα= − − , and for 1β = we get the logarithmic function

nt( ) ln( )i iu x xα= + . If all age s in the economy have the same β , they will have the same asset allocation in

terms of risky and riskless asset, i.e. the Two-Fund Separation will hold31.

Note that all in the HARA class (with exception of linear utility) are strict increasing and strict concave.

re also risk averse.

BILITIES

min preferenc

• The probability of a certain event A given event B is

(.)iu

Thus, agents with HARA utilities like consumption (or its monetary equivalent) and they a

6 RATIONAL PROBA

Under uncertainty, agents’ decisions are deter ed by es and assessed probabilities. Rational agents

assess probabilities following some basic rules:

• The probability of joint events cannot be larger than the probability of a single event, i.e.

( and ) ( )P A B P A≤

( and )A B( | )( )

PP A BP B

=

• The joint probability of independent events is ( and ) ( )* ( )P A B P A P B=

31 For a proof assuming Pareto-efficiency see Magill and Qunizii (1996) Chapter 3.


( ) ( | )( | )( ) ( | )i i

ii ii

p c p A cp c Ap c p A c

• Probabilities updating follows the Bayes rule: =∑

To illustrate that not even the first post easy to follow, consider the following story: “Linda is 31,

single, outspoken and very bright. She majored in philosophy. As a student she was deeply concerned with

issu surrounding t more likely that Linda is a bank clerk or a

onsider the

oor in or

previous pick? Mathematically, the chances to get the prize by choosing the other door are higher. The first pick

door is shown to be a l

There is a 1/3 chance to hit the prize door, and a 2/3 chance that one misses the prize. If one decides to stay

with his previous pick, the probability to get the prize is 1/3. However, one missed (and this with the probability

d one of the remaining rther

open the empty on

ning increases to 2/3 as the following

calculation shows.

. At

the beginning of the game, the probability that the prize is behind some door . Now, suppose that

e the probability that the quiz master opens door 2 wherever the price

(there are losed doors), if the price is behind door 3 and

candidate pic master will open door 2 for sure 1

ulate is always

es equality and discrimination.” Now ask yourself: Is i

bank clerk and active in the feminist movement? Many people answer the second alternative is more likely!

To illustrate that it is not always straightforward to satisfy the logic of conditional probabilities c

following statement that many people would think is reasonable: “Lazy students wear old jeans. This student

wears an old jeans. He must be lazy.” Of course this statement is wrong because it is ill conditioning

Finally, to illustrate the pitfalls in updating probabilities consider the “Monty Hall Dilemma” as an example. The

candidate of a quiz show can choose between 3 doors. There is a prize behind one of the door. After the

candidate makes a choice, the quiz master opens one of the other doors and the candidate has the option to

switch to the other closed d der to find the prize. Should a rational candidate switch or stick with his

has one chance in three of being correct but do the odds rise when one ooser?

of 2/3) then the prize is behin two doors. Fu more, of these two, the quiz master will

e, leaving the prize door closed. To summarize, if one decide to stay with the first pick the

chance of winning is 1/3 whereas if he decides to switch the chance of win

More formally, let tD denotes “prize is behind door t” and tO the event that “quiz master opens door t”

( ) 1/ 3tP D =

the candidate chooses door 1. We calculat

is. The conditional probability that the quiz master opens door 2 given the price is behind door 1 and the

candidate picked it already is 2 1( ) 1/ 2P O D| = only c

ked door one, then the quiz 2 3( )P O D| = , he will never open the

door with the prize behind it: 0 . Thus, the probability that the quiz master opens door 2 is:

.

Applying the Bayes rule in order to determine the conditional probability that the prize is behind the door 1 and

quiz a

2 2( )P O D| =

2 2 1 2 2 2 3( ) ( ¦ ) ( ¦ ) ( ¦ ) (1/ 2 0 1)1/ 3 1/ 2P O P O D P O D P O D= + + = + + =

m ster opens door 2 is:

2 1 11 2

2

( ¦ ) ( ) 1/ 6( ¦ ) 1/ 3( ) 1/ 2

P O D P DP D OP O

= = =


Thus, the cha the quiz master open the second door is the same as at

the beginning of the game. Thus, the candidate should use the information behind the quiz master’s action and

make up his mind choosing door 3.

,100). The

idea is that steadily increasing profits are perceived to be better than high, but stable profits. In terms of losses,

30,70,20)

ount exponentially, hyperbolic discounting leading to a preference reversal can be exploited

ark

t t t T t τ

nces that the first pick is winning given that

7 RATIONAL TIME PREFERENCES

7.1 Time Preference Reversal

The preference of consumption now is not independent on the consumption order. For example, a CEO

comparing the profits over the next 3 years may have the following preferences (30,70,100)≽(50,50

CEOs preferences reverse, (50,50,20) ≽ (30,70,20), low but stable profits are better than high but volatile

profits. The drop from 70 to 20 seems to be too harsh.

These preferences can be rational if looked at in isolation. The decision is however not rational if an

improvement through a financial market is possible. In this case, the CEO can lend 20 from the first to the

second period, which makes (50,50,100) dominating (30,70,100). Hence, preferences (50,50,20) ≽ (

contradicts Axiom 0.

7.2 Hyperbolic Discounting

Laibson (1997) summarized the idea of hyperbolic discounting as follows: „From today’s point of view one may find an action to be taken tomorrow profitable but when it comes to tomorrow this is no longer so.“ Since

capital markets disc

in perfect capital m ets.

Consider the utility function 1

( , ,..., )T

t

tU C C C C ß C τ

τ1 δ −+

= +

= + ∑ , where 1 1 00 andβ δ≥ ≥ .

For example, taking an action in t will lead in a consumption in t of -5 and consumption in t+1 of 6. From point

2

≥ ≥

of view of time t-1 the action is preferred if ( 5 6 ) 0β δ δ− + > or if 5

action is preferred if 5 6

6δ > . From point of view of time t, the

0β δ− + > or if 5

> . Hence, for 6

βδ 1 5 5δβ

> > we get a preference reversal.6 6

The example shows clearly that irrational decisions can be represented as the solution of a maximization

7.3 Discounting and Risk Aversion

Note that with the standard expected utili presentation,

problem. Though, this does not make them rational in the sense that they may violate Axiom 0.

0( ) ( ) ( )s ss

U c u c p u cβ= + ∑ , risk and time ty re


preferences are closely linked. In the case of CRRA, 1( ) ( )s su c c α−= , time preferences are defined over the

marginal utility of consumption over two point of time: 1 1 1

0 0 0

( )( )

u c cu c c

α

β∂= ⎜ ⎟∂ ⎝ ⎠

where ⎛ ⎞

β is the discount factor of

over the marginal utility of consumption in two states s and z: future consumption. Risk preferences are defined

( )( )

s s s

z z zu c cu c c

α⎛ ⎞∂

) found an expected utility representation separating both aspects: the = ⎜ ⎟∂ ⎝ ⎠. Epstein and Zin (1989

elasticity of intertemporal substitution, defined as ψ , and the relative risk aversion α :

(1 ) 1 (1 )1

1 1(1 )( ) (1 ) ( ) where t t t tU C C E U

θα

αθ θ

ψ(1 )

α αδ δ θ−

−−

+

⎡ ⎤ −= − + ≡⎢ ⎥ −⎣ ⎦

hows an interesting co ion between time and risk

For some class of utility functions, this representation s nnect

preferences. For example, if the utility function is logarithmic time and risk preferences coincide. For CRRA they

are reciprocal to each other.

Elasticity of i

1ntertemporal substitution Ψ

e on Log utility

lativ

eα

Coe

ffic

ient

of r

risk

aver

si

1CRRA utility

1αψ

=

REFERENCES:

Eisenführ und Weber (1999): Rationales Entscheiden, Springer.

Huang and Litzenberger (1988) Chapter 1.

Textbooks:

Frey and Stutzer (2002) “Happiness and Economics”, Oxford University Press.

Magill and Qunizii (1996): Theory of Incomplete Markets; MIT-Press

Research Papers:


Arrow (1971): “Axiomatic Theories of Choice, Cardinal bjective Probability: a Review”; Opening

Epstein and Zin (1989):“Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset

Returns“, Econometrica (57), pp.937-69

Laibson, D. (1997): “Golden Eggs and Hyperbolic D scounting”, Quarterly Journal of co

Utility and Su

lecture for the Workshop in Economic Theory organized by the International Economic Association in Bergen.

Howard, R.A.: “Decision Analysis: Practice and Promise“, Management Science (34), June 1988, p.679-695

Keeney (1982): “Decision Analysis”, Operations Research (30), pp. 803-838.

i E nomics, 112, 443-447.

Chapter 5:

Choosing a Portfolio on a Random

Walk: Diversification

Choosing a Portfolio on a Random Walk: Diversification 70

1 INTRODUCTION

One of the main claims of classical finance is that assets have random returns. Evoking the anticipation principle,

as mentioned above, Cootner (1964), for example, has claimed that “the day-to-day changes of asset prices

must statistically independent of each other”. By the Central Limit Theorem32 returns over longer periods would

section we want to take two points of view: First we ask what would be

nal investor facing a random walk. In particular we analyze whether – as

vestor should follow a buy and hold strategy when facing a random walk?

Second, we ask – as it is also claimed many times – whether a random walk can be the outcome of an economy

h rationally acting agents. Summarizing, we are asking whether neoclassical finance, i.e. evoking the utility

ximization principle, is indeed compatible with classical finance.

In this chapter we will challenge this commonly held view and show that expected utility maximizers with CRRA

y and

ested

in each asset. Moreover, we show that assuming the economy can be represented by one agent with a CRRA-

too small as compared to those found in

is neoclassical model is not able to explain

the equity premium, i.e. the fact that on average stocks have 6% higher returns than the risk free rate.

2 RETURNS

Let us first consider the assumption of normally distributed returns in more detail: First, it is obvious that returns

on stocks can never be normally distributed because stock returns are bounded below whereas a normal

distribution has unbounded support. The correct modeling of returns is to assume that they are log-normally

distributed, i.e. that the log of gross returns is normally distributed. Indeed, as returns over longer horizons are

the product over short horizon returns, the central limit theorem can be evoked to the sum of the logs of short

run returns.

Hence the correct modeling of returns should be to assume that the quantities

then be lognormal distributed. In this

the best portfolio strategy of a ratio

many people claim - every rational in

wit

ma

should follow a rebalancing portfolio strategy rather than simply buy and hold. In contrast to a passive bu

hold strategy, where the units of assets are fixed, a rebalancing strategy fixes the percentage of wealth inv

utility function would imply that fluctuations of asset prices are much

the DJIA 81-03 data. Finally, for reasonable degrees of risk aversion th

1logt

ktk

qq+

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

are normally

distributed. To get an intuition for the adequacy of this property, we for example take the logarithm of weekly

gross returns of 3Com and General Motors shares, plot a histogram and compare it with the histogram of a

normal distribution with the same mean and standard deviation.33

32 The Central Limit Theorem shows that the sum of i.i.d. variables is normally distributed. 33 To run the Chi-quadrate test we need a minimum amount of observations in each bin. This is why you see “clusters” on both sides of the histogram. Since the test is performed to the similarly transformed normal distribution there is no bias in the test results.

71Choosing a Portfolio on a Random Walk: Diversification

General Motors3M200

0

50

100

150

0

30

60

90

120

150

The hypothesis of normal distribution can be rejected for both shares although the deviations in the GM

histogram to the normal distribution appear small. The 3M Returns are too often close to its mean and also too

often far away from the mean. That is to say the histogram has fat tails and is too na

180

rrow in the middle range.

Running statistical tests for all 30 companies included in the DJIA, we see that 29 out of 30 do not have

lognormal distributed returns. Hence the assumption of a random walk cannot be supported. Yet we may ask

whether at least the conclusions drawn under this assumption make sense.

Properties of lognormal returns

If a random variable is log-normally distributed, i.e. if log( ) ( , )X N µ σ∼ then

21log( ) log( ) (log( ))

2 2EX E X Var X

σµ= + = + . In other words, the higher the variance, the higher are the

expected returns. In contrast to the normal distribution, we cannot increase the variance without adjusting the

mean because the support of the lognormal distribution is bounded below by zero. Increasing the variance is

equivalent to shifting more weight on the right side, which increases the expected value of returns!

X0

The disadvantage of log-returns is the property that the sum of lognormal returns is not itself lognormal. This is a

problem because the log-normal property of individual assets cannot be extended to the portfolio, i.e. if each

asset return is lognormal, the portfolio return as a weighted average of lognormals is not necessary lognormal as

well. This problem can be avoided by considering short time intervals. In the limit of continuous time the problem

is gone.

3

PORTFOLIO CHOICE

In this section, we will show that with log-normally distributed returns a CRRA-expected utility maximizer should


choose a fix-mix strategy, i.e. he should fix his portfolio weights and then rebalance the units hold in his

portfolio. As a side product we see that the mean-variance principle can be modified to display the agent’s

optimal intertemporal asset allocation.

To this end, consider the following expected utility maximization problem, which describes the “myopic”

optimization from one period to the next:

1

11

1

1

. .

α

λ

,0

1

λ

α

λ

+

−+

∈

+

=

−

=

=∑tt t

K

k tk

w R w

where , , 10

λ λ +=

=∑t

K

k t k tk

R R and 0, 1 ,t f t

Kt

tt

R

wEMax

s t

R R+ = .

First, we transform the problem by taking the logarithm to:

1

11 1max WtE Eα α− −

+ +⇔

1

1 1max logK

tt

Wtt t

R α αλλ + − −∈

⎡ ⎤ ⎢ ⎥⎣ ⎦

This is equivalent to:

1

11max log log(1 )

Kt

t tR

E W α

λα

+

−+

∈− −

If next period wealth is log normal we can apply the property that:

1 11log log ( log )2t t t t t tE W E W Var W+ += + 1+

to get:

1

2

1 1(1 )max (1 ) log (log )

2Kt

t t tR

aE W VAR Wλ

α+ + +

∈

−− +

Since next period wealth is proportional to this period wealth, from this expression we already see that the asset

allocation is independent of wealth. Using the notation for a portfolio’s mean and variance of log-returns we get:

22(1 )max (1 )

2t ttt r

aE rλλλ

α σ−− +

1 α− and eliminating it from the maximization problem, yields: Dividing by

1

2(1 )max2tK t

t Rt r

aE rλλ

λσ

+∈

−+


where 2

trλσ is the conditional variance of the log portfolio return.

Using the budget constraint ,K

o t k ,k t1λ λ we get:

== ∑

, 1

2, , 1

1 1

1max (K

t

K K K K

k t t k t fR k k

aE r rλ

λ +∈ = =

−∑ , , , , , 1 , 11 1

) cov( , )2 2k tt k t r k t j t k t j t

k j

r rλ σ λ λ+ + +

= =

+ −∑ ∑∑

The solution of the problem is: 1 21 ,

11 1cov ( ) .

21

optt t t t f ti E r r tλ σ

α−

+

⎞⎛ ⎞⎛⎟⎜ ⎟⎜= − ⎟⎜ ⎟⎜

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+ where is the vector of asset log

returns and

1tr +

2tσ is the vector of variances of the k assets.

Note that except for the term 212 tσ this is the same solution as in the two period mean-variance model from the

t allocation to those assets with higher volatility. The section 2.2. will

3.1 Dynamic Portfolio Choice with CRRA: The “No Time Diversification” Theorem

A fundamental observation in finance is that in an efficient market an expected utility maximizer with CRRA does

allocation over time. To see this we need to extend the previous analysis for the case that

the investor cares about his wealth n periods from now,

second chapter. This term tilts the asse

provide an intuition for this.

not change his asset

t nW + . If all wealth is invested, the budget constraint is:

t+n 1 2W ...t n t n t n t tR R R RW+ + − + −= or in logs: wt+n 1 2...t n t n t n t tw r r r r+ + − + −= + + + + . Note that we continue to use

Suppose that the log-normal returns are i.i.d. over time. This implies that risky returns are not serially

correlated, ) , and the variance of risky assets is pro ortional to the holding period,

)tw nVAR w= . With i.i.d. returns all means and variances are scaled up by the same factor n. In

other words, both the short-term and long-term investor face the same mean-variance choice scaled up or down

by the factor n. Thus, if investors have CRRA utility functions, the asset allocation does not depend on wealth

and since expectations and covariances are constant over time, both the short-term and long-term investors

We will see later that in the case of a log utility, portfolio choice will be myopic even if asset returns are not i.i.d.

In any case, a CRRA-expected utility maximizer will not choose a buy and hold strategy but fix his asset mix and

then rebalance.

3.2 Rebalancing, Fix Mix and Volatility Pumping

The “No Time Diversification Theorem” of Samuelson and Merton shows that on i.i.d. process with log-normal

gross returns.

t+n t+1( ) (t n tE w nE w+ = p

t+n t+1( ) (t nVAR +

choose the same portfolio, i.e. the long-term investor acts myopically. This result is known as the “no time

diversification” theorem which goes back to Samuelson (1969) and Merton (1969).


t allocation that is invariant over time. Hence

when the price of an asset goes up, the investor sells part of his holdings of that asset and when the price goes

invested in this asset

returns an expected utility maximizer with CRRA chooses an asse

,i kλdown he will purchase more of this asset in order to keep the share of wealth fixed

over time. This strategy is also called “fix mix” and the corresponding behavior is called “rebalancing”. It is

illustrated in the figure bellow.

sell buy

0

1

2

5

6

7

8

9

3

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

The next example, “volatility pumping”, illustrates that in a sense a rebalancing behavior is more successful the

more volatile the assets are.

There are two assets, cash with zero interest and a stock that in each period has a 50:50 chance of either

doubling its value or of reducing it by one-half. The stochastic process is i.i.d. as for example a repeated coin

grows at a rate of about 6%:

tossing. An investment left in the stock will have a value that fluctuates a lot but has no overall growth rate. Also

the investment in cash has no growth rate. However if we rebalance, say a (½, ½) asset allocation, our wealth

1 1 1 1 12

( ) ln( 1) ln( ) 0.0592 2 2 4

gµ = + + + ≈ , where ( )gµ is the expected growth

bit more precise: Let

rate of wealth.

Since we will use these ideas later quite frequently, we take the opportunity of this simple example to make the

reasoning a , t H Tω ∈ be the state of the world in period t and t0 1( , ,..., )tω ω ω ω= be

t. Then, the evolution of wealth is given by the recursion: 1 1( ) ( ) (1 ) ( )t t tw A w

the path up to period

ω ω λ λ ω+ +⎡ ⎤= + −⎣ ⎦ , where 1( )tA ω + is 2 in the case of H and ½ in the case of T. Since tω

is i.i.d. 1( ) ( )ttA Aω ω+ = . By the Law of Large Numbers we get for the expected evolution of log-returns:

[ ]10

1ln( ( )) ( ) ln 2 (1 ) ( ) ln (1 ) ln ( )2

tE w p H p T wω λ λ λ λ ω+ ⎡ ⎤= + − + + − +⎢ ⎥⎣ ⎦.

Hence, the expected growth rate is: [ ]1 1 1(g( ))= ln 2 (1 ) ln (1 )2 2 2

µ λ λ λ λ λ⎡ ⎤+ − + + −⎢ ⎥⎣ ⎦.

As claimed above, the expected growth rate of putting all money into one of the two assets is zero:

[ ] [ ] [ ]1 1 1 1 1(g(1))= ln 2 ln = (g(0))= ln 1 ln 1 0µ µ⎡ ⎤+ + =⎢ ⎥ 2 2 2 2 2⎣ ⎦


However, rebalancing the portfolio ( ) 1 1,1- ,2 2

λ λ ⎛ ⎞= ⎜ ⎟⎝ ⎠

gives:

1 1 1 1 1 1(g ) ln 1 ln 0.0592 2 2 2 2 4

µ ⎛ ⎞ ⎡ ⎤ ⎡ ⎤= + + + ≈⎜ ⎟ ⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎣ ⎦ ⎣ ⎦.

The variance of the expected growth rates is:

[ ]2

22 1 1 1(g( ))= ln 2 (1 ) (g( )) ln (1 ) (g( ))2 2 2

σ λ λ λ µ λ λ λ µ λ⎡ ⎤⎡ ⎤⎡ ⎤+ − − + + − −⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦⎣ ⎦.

The growth rates of different portfolio strategies can be represented in a mean-variance diagram.

, an investor may however go for a smal

th rate.

t if buted, then all CRRA-

to ask two questions: First, can this vi de con

turns with pricing

model, i.e. a model that generates these returns out of reasonable assumptions?

First of all, there are good reasons to prefer utility functions in which relative r aversion35 does not depend on

wealth:

omy suggests that relative risk aversion cannot de trongly on wealth. Per capita consumption and wealth have increased greatly over the past two centuries. Since financial risks are

The highest growth is achieved by the strategy investing half of the wealth in stocks and the rest in cash34.

Depending on his risk aversion ler growth rate in order to reduce the

variance of the grow

4 ASSET PRICING

In the previous section we have seen tha log-returns are i.i.d. and normally distri

expected-utility-maximizers choose a fix-mix strategy with the same fund of risky assets. In this section we want

ew be ma sistent, i.e. is it possible to generate i.i.d. log-normally

distributed re CRRA-expected-utility-maximizers? Second, does this give a reasonable asset

isk

„The long-run behavior of the econ pend s

34 Be sure that you can prove this fact! 35 Remind that the coefficient of relative risk aversion determines the fraction of wealth that an investor is ready to pay in order to avoid a gamble of a given size relative to his wealth.

EXP-g and VAR-g

0.02

0.06

0.08

EX

00 0.1 0.2 0.3 0.4 0.5 0.6

VAR-g

0.04P-g


ute scale of fi ased while the relative scale is t show idence of long-term trends in response to this

long-term growth; this implies that investors are willing to pay almost the same relative costs to avoid given relative risks as they did when they were much poorer, which is possible only if relative risk aversion is almost independent of wealth.“ (Campbell and Viceira, 2002), p. 24.

nancial variables under economic growth, we assume that agents

have utility functions with a constant relative risk aversion (CRRA).

To answer the question if the portfolio choice discussed above is compatible with an asset pricing model,

multiplicative, this means that the absol nancial risks has also increunchanged. Interest rates and risk premium do no any ev

Hence, to explain the rather limited range of fi

consider an equilibrium, where the demand of investor i in period t is: ,

i,ki k

it wλθ = . Normalizt ktq t ing the asset

supply to 1, we get the market clearing condition i,kt 1

iθ =∑ and as above the price of asset k is

, i.e. a weighted average of the strategies. Assuming two-fund separation36 for a common and k i,kt tq i

ti

wλ=∑

stationary strategy λ , the price of asset k is kt

1 1q k i kt ti rep

i iw wλ λ

α α= = i∑ ∑

prices are constant:

. Comparing the prices of asset

k and another asset j we see that relativek kt tj jt t

qq

λλ

= .

However, in reality, relative market values seem to deviate from this result as the figure bellow shows.

Extending the analysis to an aggregate level, we see that aggregate wealth is determined by aggregate

dividends i kt t

i kw D=∑ ∑ . Thus, prices will fluctuate with aggregate dividends, i.e. if aggregate dividends are

, empirically, the fluctuations of dividends are too small to explain the fluctuations of returns. In other

i.i.d. and Gaussian, the prices will be also i.i.d. and log-normal.

However

words, there is excess volatility.

36 Note that as shown above CRRA would imply two-fund separation.

Relative Market Values

1

0

0.1

0.2

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

0.3

0.4

0.5

0.6

0.7

0.8

0.9


5 THE EQUITY PREMIUM PUZZLE

ks is to show by how much

appreciated in t

Investing one single dollar in 1920 on DJIA would have given you 2586.52 dollars while it would only have given

s. Or to phrase this fact differentl

stock over bonds is about 6% p.a. in the US. In other countries this so called equity premium is similar: UK:

4

Of course, stocks have also been more volatile but the question remains whether these high excess returns can

ard asset

pricing model with a single representative consumer maximizing a CRRA-utility could generate such high risk

premium. To get some intuition why this model fails in this respect, or phrased less drastically why the equity

premium is a puzzle, consider the following simple question: Which X would make you indifferent to the

following lottery? Doubling your income with 50% probability and loosing X% of your income with 50%

probability. The typical answer: X = 23%

However an expected utility maximizing investor with CRRA who on realistic data chooses the risk free asset and

stocks would choose X = 4 % . Hence explaining the equity premium with this model would amount to choose

gue that for reasonable degrees of risk aversion,

the observed volatility in consumption is not high enough to generate the observed Sharpe ratios on stocks. This

A common argument to convince people to invest in stoc one`s wealth would have

he long run if it were invested in stocks as compared to bonds.

you 16.56 dollars if it were invested in bond y, the average excess return of

.6%, Japan 3.3%, Germany 6.6% France 6.3%.

Total DIV vs MV 81-03

2500000

3000000

3500000

4000000

4500000

40000000

50000000

60000000

1000000

1500000

2000000

20000000

0

500000

0

really be explained by the volatility of stock returns. Or, to be more precise, we ask whether the stand

unreasonable high degrees of risk aversion, i.e. such a representative consumer would not represent the average

risk aversion in the society.

Yet a different way of stating the equity premium puzzle is to ar

argument refers to the intertemporal utility maximization principle, as expressed in the first-order condition that

we derived above:

[ ] 1

0

, where i

sp m sf f i

SDF

uR l R l R

uβ⎛ ⎞′∂

= Ε = ⎜ ⎟′∂⎝ ⎠ and β is the time preference parameter.

elation coefficient is bounded above by 1 yields

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

10000000

30000000MVDIV

Using the definition of the covariance and the fact that the corr


the famous Hansen-Jagannathan volatility bound:

[ ]( )( ) ( ) ( )( )ln

1m fp

sm f

r rl c

r rσ ασ

σ

Ε −≤ ≈ ∆

+

Hence the excess return of the market divided by its standard deviation and the gross interest rate is bounded

above by the standard deviation of the likelihood ratio process. An approximation of the latter is the product of

the risk aversion coefficient and the volatility of changes in the growth rate of consumption, which is however

too small for the observed size of the left-hand side of the inequality.

6 S Y

In this chapter we have seen that a Random Walk has log-normal i.i.d. returns, which delivers nice results using

sing the properties of log-normal

with this standard model of neoclassical finance.

conomy 99: 225-262.

Samuelson (1969):“Lifetime Portfolio Selection by Dynamic Stochastic Programming“, Review of Economics and

Statistics (51), pp.239-246.

UMMAR

the Mean-Variance framework. Under these conditions however the best portfolio strategy is not necessarily buy-

and-hold but fix-mix –at least for the standard class of utility functions. Also, u

returns, we see that the more volatile the markets the higher are the expected returns!

Empirically, the assumption of log-normal i.i.d. returns is questionable. Relative market values do fluctuate and

the fluctuations in aggregate dividends are too small to explain the fluctuations in returns (excess volatility). Also

the observed equity premium cannot be reconciled

REFERENCES:

Textbooks:

Campbell and Viceira (2002): Strategic Asset Allocation, chapter 2.

Cootner, P. (1964). The Random Character of Stock Market Prices. The M.I.T. Press, Cambridge.

Luenberger (1998): Investment Science, chapter 15.

Research Papers:

Campbell and Viceira (2001): „Stock Market Mean Revision and the Optimal Equity Allocation of a Long-Lived

Investor“, European Finance Review.

Hansen LP, Jagannathan R. 1991. “Implications of Security Market Data for Models of Dynamic Economies”.

Journal of Political E

Behavioral Portfolio Theory

Chapter 6:


80

“No rational argument will have a rational effect on a man who does not want to adopt a rational attitude.”

(Karl Popper)

As concluded in previous chapters, neither the assumptions nor the conclusions of classical finance match the

lity. For example, the postulate from the efficient markets hypothesis that returns must follow a random walk

ot confirmed by the data. Additionally, even if there is some structure in the dynamics of economic variables

ically represented by the Euler equation, it does not represent the real economy very well. A natural way to

r

complete rationality

lity.

The field of behavioral finance provides a new set of explanations of observed empirical regularities documented

l finance. It also provides a new set of predictions. The goal of this chapter is to present

some of the central ideas of behavioral finance in order to develop a better understanding for the conclusions in

the following chapters.

2 DESCRIPTIVE VERSUS PRESCRIPTIVE THEORIES

Individuals’ behavior has been studied by many authors. Bubbles for example became representative for

investors’ behavior being inconsistent with prevalent normative theories. Regarding the South Sea Bubble in

1720, Sir Isaac Newton said: “I can calculate the movements of heavenly bodies but not the madness of people”. For Keynes (1936), people’s behavior is often driven by animal spirits: “Most… decisions to do something positive can only be taken as a result of animal spirits-of a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities.”

Already Veblen (1900) suggested that bubbles can be better understand if one understands the psychology of

decision making: “Market values being a psychological outcome, it follows that pecuniary capital, an aggregate of market values, may vary in magnitude with a freedom which gives the whole an air of caprice – such as psychological phenomena, particularly the psychological phenomena of crowds, frequently present, and such as becomes strikingly noticeable in times of panic or of speculative inflation”.

Kahneman and Tvesky are more specific by introducing the idea of heuristics: “In making predictions and judgments under uncertainty, people do not appear to follow the calculus of chance or the statistical theory of prediction. Instead, they rely on a limited number of heuristics which sometimes yield reasonable judgments and

1 INTRODUCTION

rea

is n

typ

start developing a theory that better matches the reality is to observe agents` behavior in the economy eithe

empirically (e.g. observing individual portfolios) or experimentally. The assumption of

underlying the efficient market hypothesis is wishful thinking that may be quite off from rea

as puzzles in the classica


81

sometimes lead to severe and systematic errors”.

Thou

psychology that might be relevant for the formation of investor sentiment, and no obvious way of deciding which psychological biases are the most important”.

dditionally, Raiffa (1994) raises the question of the usefulness of this research. One possibility is to use the

eas and build better theories either drawing direct inferences or by modifying normative theories to include

cognitive concerns. Though, Howard (1988) expresses concerns regarding this idea: “Some decision theorists weaken the norms until the normative behavior agrees

3 SEARCH, FRAMING AND PROCESSING OF INFORMATION

requires in the first step a selection of relevant issues. In complex, uncertain

environments, decision makers under time pressure often use rules of thumb or heuristics. However, applying

a

m A

necessarily le

a investment returns than experts by applying a simple heuristic: buy the shares of companies

whose name you recognize.

e ,

1

o ther

go left or right. Everyone has a private imperfect signal (e.g. "judgment" or "opinion"). For simplicity, let

o r ht

the signa eryone's signal

person to cho

gh, it is ambiguous which of the phenomena are robust. In the words of Shleifer (2000): “There is a lot of

A

id

have questioned the normative concepts. They desire towith the descriptive behavior of human beings. A moment of reflection shows that if we have a theory that is both normative and descriptive, we do not need a theory at all. If a process is natural like breathing, why would you even be tempted to have a normative theory?”. Another possibility is to use the research results and provide

better training for decision makers.

In the following we will provide a systematic discussion about the patterns of decision making as a basis for

deriving implications.

3.1 Selection of Information

Information processing

them may result in poor decision outcomes.

For example, using the recognition heuristic following the rule that if one of two objects is recognized and the

other is not, then one should infer that the recognized object has the higher value, may lead to false conclusions.

To illustrate the phenomena consider the results of the following example. In an experiment, people have been

sked which US city has more inhabitants: San Diego or San Antonio. Participants chose the city they have heart

ost about – San Diego – although it had less inhabitants. s noted above applying heuristics does not

ad to bad decision outcomes. Gigerenzer and Todd (1999) show for example that non-experts can

chieve higher

Many decisions under uncertainty in particular under asymmetric information, where everybody beliefs that

verybody has better information, induces imitative behavior. This behavior is known as herding. As (Banerjee

992) has pointed out herding may also result under complete rationality. Let us presume that you and a lot of

ther people have to find your way to a new destination, and you come to a crossway where you can only ei

everyone have a private signal "left" ("right") with probability 1/2 if the true best ch ice is to go left ( ig ). So,

l helps but it is not perfect. Ev is equally good. Assume further that you are the third

ose, and you first saw a man and then a woman go left. It is optimal for you to go "left" even if


82

y u know that the man must have had "left" signal, because he

"

w eft signal, say in this case she tosses a fair coin. She might have

walked the one or the other way. Having seen both the man and the woman walk "left," you know that the

"

i

i

ow that alternative descriptions of a decision problem often give rise to different

nces contrary to the principle of variance underling a rational choice. The on behind this normative

ould not be relevant for the

there are 60’000 SFr. at risk. One group of respondents

e n rescue 20’000 SFr. (alternative A) or rescue the total amount with

to loose nothing with probability 1/3 and loose everything with

ds.

our private signal/intuition says "right". Why? Yo

went left. The woman saw the man go "left." She would have figured out that the first individual's signal was

left". If her private signal was "left", she would have surely walked left, too. If her signal was "right", she

ould have been aware of one right and one l

man had a "left" signal and the woman had a better than even chance of having had a "left" signal. Even if

your private information is a "right" signal, you should choose "left" if you are acting rationally. The

consequence is that million rational individuals may walk "left" just because the first two individuals walked

left", even if the true best choice was "right" called. This situation is called informational cascades. Note that

n an informational cascade everybody acts rationally. Though, even if all participants as a collective have

nformation in favor of the correct action, each participant may take the wrong action.37

3.2 Framing

People’s behavior may depend on the way their decision problems are presented or framed. Tversky and

Kahneman (1986) sh

prefere in intuiti

concept is that variations of the form that do not affect the actual outcomes sh

choice. Consider the following example. Assume that

receiv s the following problem. They ca

probability 1/2 and loose the total amount with probability 2/3 (alternative B). A second group of respondents

receives the same problem but with a different formulation. Now, alternative A is the sure loss of 20’000 SFr.

and alternative B is the uncertain outcome

probability 2/3. The majority choice of the first group is risk-averse: the prospect of certainly saving 20’000 is

more attractive than a risky prospect with the same expected value. In contrast, the majority choice in the

second group is risk taking: the certain loss of 20’000 SFr. is less acceptable than the two-of-three chance to

loose everything. The preferences of both groups illustrate that choices involving gains are often based on risk-

aversion while choices involving losses are often risk-taking.

Benartzi and Thaler (1998) observe that employees planning their pension savings tend to split the funds equally

(1/n) over the alternatives offered by their pension fund. As a consequence, asset classes offering more choices

receive significantly more fun

37 For a survey on rational herding see Andrea Devenow, Ivo Welch (1996) Rational Herding in Financial Economics. European Economic Review 40:3-5, 603-615.


83

ve the winners in their

portfolio.

3.3 Processing of Information

Another example for the impact of framing on investment decisions is the practice of selling shares that have

sharply declined in value on the end of reporting period, i.e. window dressing.38 By selling the losers portfolio

managers want to dress their window, i.e. to show in their brochures that they ha

There are several biases in the processing of information.

• The availability bias can be best illustrated considering the following answers of the following questions:

What is a more likely death in the US – being killed by a falling airplane parts or by a shark? (Plous,

1993). Most people judge the probability of dying by an event, which receives more publicity and it is

easier to recall as higher against the statistical evidence.

• The gambler’s fallacy is a phenomenon where people inappropriate predict reversals in a random

sequence. For example, in a roulette after a run of order at most 7 people think it is time for the other

color to come again although the outcomes are i.i.d. Asking students to estimate the probability of runs

of different order (1 to 6) by throwing a fair coin 100 times, we observe that people underestimate the

frequency of long runs of the random walk process.

00.10.20.30.40.50.6

1 2 3 4 5 6 7 8

people random walk

• Violation of rational choice theory is commonly observed in simple repeated binary choice problems. In

such tasks people often tend to ‘match’ probabilities, i.e. they allocate their responses to the two

R. Vishny (1991) 38 Josef Lakonishok, A. Shleifer, R. Thaler, and


84

options in proportion to their relative payoff probabilities. Suppose that a monetary payoff of fixed size is

given with tossing a coin with probability 0.49 for “head” and with probability of 0.51 for “tail”.

Probability matching refers to behavior in which “head” is chosen on about 49% of trials and “tail” on

51%. On any coin toss, the expected payoff for choosing “tail” is higher than the expected payoff for

choosing “head”. Thus, the optimal choice is always to always select “tail”. But the rational behavior

would not match the randomness of the choice problem. From investors perspective, the decision

whether to invest in an asset whose price increase with “head” and decrease with “tail” is not to invest

if the coin is fair and there are transaction costs, respectively to follow a “buy and hold” strategy if the

at an object A belongs to a class B, they often use the representativeness heuristic. This means that

they evaluate the probability by the degree to which A reflects the essential characteristics of B. In many

is a helpful heuristic, but it can generate many severe biases. Consider

r known to beat the market in 2 of 3 years.

Given the sequences BLBBB, LBLBBB and LBBBBB (B: beat; L: loose), people typically choose the

• Also closed-end-funds represent a puzzle. They typically start with a discount of 10%, which is a natural

result of underwriting and start-up costs. The discount rate fluctuates widely and upon termination share

prices rise and discount shrinks. Several explanations have been suggested. One of them is based on the

notion that people may not be able to stop spending capital gains, much like being not able to stop

overeating. Since closed-end funds retain the profits and do not provide regular payoffs which could be

coin is worn as described above.

• When people try to determine the probability that a data set A was generated by a model/process B, or

th

circumstances, representativeness

the following example. Assume that there is a fund manage

sequence LBLBBB as it is representative for the probability 2/3. Though, the sequence BLBBB is more

likely than the sequence LBLBBB because the latter includes it. The probability for the sequence LBLBBB

is the conditional probability for the sequence BLBBB given the additional realization L in one of the

years. , the correct answer is LBBBBB39.

• Even though in one person decision problems additional constraints can only make one worse-off,

people often impose themselves rules to overcome self-control deficits. Examples include getting your

stomach removed to prevent yourself from eating too much. Perhaps an appropriate model for this

phenomenon should involve two agents the one who has the tendency to overeat and the other hinder

the overeater to do so.

39 If outcomes are independent then the cumulated probability for each sequence is where 11

( ,..., ) ( )n

n ii

P x x p x=

=∏ ix

ation in a sequence and ( )ip x is its probability. The probability of the first sequence is is a realiz1 4

11

2 41 2.., )nx ⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥ , the second sequence has probability 21

1 2( ,..., )nP x x3 3⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥( ,.

3 3P x

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ and the third

1 53

11 2( ,..., )3 3n ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

P x x ⎡ ⎤ ⎡ ⎤= .


85

•

• The Ellsberg paradox provided evidence that people dislike ambiguity or situations where they are not

4 PROSPECT

Large b that people systematically violate the assumptions underlying

exp

the theo

tries to

The the

wealth.4 ction has a kink at the origin, indicating a greater sensitivity to losses than to

gain a

and con

losses,

transfor

spent for consumption, one would nee to sell shares i.e. to start “dipping into capital” if one wants to

finance consumption. This however is risky since people may then dip into capital too much. Hence

closed end funds sell at discount, which makes their price lower than the sum of the prices of their

components.40

There is strong empirical evidence that investors do not diversify sufficiently, instead they hold primarily

shares of domestic companies. This phenomenon called home bias is associated with the notion that

people prefer familiar situations. They feel that they are in a better position than others to evaluate

particular decision problem.

sure what the probability distribution of a gamble is. If people feel more comfortable in situations of risk

than in situations of uncertainty then this could explain why they prefer investing in domestic companies.

Moreover, if investors are concerned that their model of stock returns is misspecified, they would charge

a substantially higher equity premium as compensation for the perceived ambiguity in the probability

distribution.

THEORY

ody of experimental work has shown

ected utility theory choosing among risky gambles. The Prospect Theory of Kahneman and Tversky is one of

ries trying to match the experimental evidence. Their theory has no aspiration as a normative theory: it

capture people’s attitude to risky gambles.

ory consists of four building blocks. First, utility is defined over gains and losses rather than over final 1 Second, the value fun

s, feature known as loss aversion. Third, the shape of the value function is concave in the domain of gains

vex in the domain of losses. In other words, people are risk averse over gains and risk-seeking over

i.e. they gamble to avoid losses. The final piece of prospect theory is the nonlinear probability

mation: small probabilities are over-weighted.

closed end fund40 As Martin Zweig, a famous manager showed, the discount can be eliminated by paying out dividends. 41 This idea was first proposed by Markowitz!


86

Probability W eightening W (P)

1

1.2

0

0.2

0.4

0

0.6

0.8

0 .2 0 .4 0 .6 0 .8 1 1.2W(P)

P

To illustrate the first result, consider a lottery paying 200 with probability 50% and -100 with probability 50%.

1 1Evaluating the lottery using the expected utility approach requires comparing ( 100) (200)

2 2U U− + with

(0)U . Typically people would not play this lottery because they are loss averse, i.e. losses are weighted with the

factor 2.25 more than gains. If lotteries are not evaluated according to gains and losses but according to final

wealth, then the lottery is evaluated by comparing 1 1( 100) ( 100)2 2

U w U w− + + with ( )U w , where 0w > is

the initial wealth. Every expected utility maximizer, with CRRA for example, ay

the lottery.

One application of the loss aversion is the disposition effect, i.e. investors are reluctant to sell assets trading at a

loss relative to the price at which they were purchased. Odean (1998) finds that individual investors are more

likely to sell stocks which have gone up in value rather than stocks that have lost.

( ) au x x= where 0a > would pl


87

One further aspect of prospect theory that is similar to thinking in terms of gains and losses is narrow framing or

the tendency of agents facing new gambles to evaluate them separately, in isolation from other risks they

already have. In other words, agents derive utility directly from the outcome of the gamble instead of evaluating

its contribution to total wealth and miss the chance of diversification. Put differently, agents maximize utility

locally in an optimal manner, but by doing so they may come to a bad global outcome. For example, a lottery A

paying 60 by “head” and -59 by “tail” and a lottery B paying -59 by “head” and 60 by “tails” would be

rejected if evaluated separately. Though, playing both lotteries simultaneously is an arbitrage opportunity

delivers a sure profit of 1 (an arbitrage opportunity). Narrow framing my also arise from mental accounting, i.e.

from separating assets into categories like bonds, stocks and alternative investments. If one diversifies within

particul

losses.

The last building block of prospect theory is the proposition that individuals transform probabilities to decision

ox presented in previous chap plying the appropria

w u w u w u> + + or

or

Thus this choice while being a contradiction to the expected utility approach is consistent with prospect theory.

those categories but not across them then suboptimal allocations will result.

The third pillar of prospect theory implies that agents change their risk attitude when facing gains and losses. In

ar, they prefer to take additional risks and gamble in order to achieve sufficient gains that neutralize the

weights. This attitude can explain the Allais Parad ter. Ap te

transformation of the underlying probabilities we get: A≽B if and only if

(1) (50)w u (0.01) (0) (0.89) (50) (0.1) (250)

( (1) - (0.89)) (50) (0.01) (0) (0.1) (250)w w u w u w u> +

Applying the same procedure for the lotteries A’ and B’ we get B`≽ A`, if and only if

(0.9) (0) (0.1) (250) (0.89) (0) (0.11) (50)w u w u w u w u+ > +

( (0.9) - (0.89)) (0) ( .1) (250) (0.11) (50)w w u w o u w u+ >


88

5 PROBABILITY WEIGHTING AND FSD

Let there be three states, with payoffs (100, 100, 90.25). The true probabilities are

. Let the perceived probabilities be

1,2,3s =

(0.05,0.05,0.9)p = (0.1,0.1,0.85)w = . If the vNM utility function is

( )u x x=

since

, for example, then the prospect-theory-investor would prefer the lottery to a certain payoff of 100,

0.1 100 0.1 100 0.85 90.25 2 8.075 10 1 100+ + = + > =

Order Stochastic Dominance.42 To cure the problem of

Kahneman (1992) suggest defining the weighting function, say ( )T x

losses rather than on the probability density. The prospect utility

( ) ( ) ( )

. However, this preference violates First

FSD, the cumulative prospect theory of Tversky and

, on the cumulative probability density of

gains and function is obtained in general as

. Since s an increasing monoton formation the First Stochastic Dominance

property is preserved. In other words, if lottery A first order stochastically dominates lottery then T(A) first order

ic Review.

Howard, R.A. (1988): “Decision Analysis: Practice and Promise”, Management Science (34), p. 679-695

Kahneman and Tversky (1973): “On the Psychology of Prediction”, Psychological Review, 80

Kahneman and Tversky (1979): “Prospect Theory: an analysis of decision under risk”, Econometrica 47, 263-291

Tverky and Kahneman (1992): “Advances in Prospect Theory: Cumulative Representation of Uncertainty”,

Journal of Risk and Uncertainty, 5, pp. 297-323

PTv x v x dT x= ∫ ( )T x i e trans

B

stochastically dominates T(B).

REFERENCES:

Textbooks:

Campbell and Viceira (2002): “Strategic Asset Allocation”

Gigerenzer and Todd (1999): „Simple Heuristics That Make us Smart“, Oxford University Press

Plous, S. (1993): “The Psychology of Judgment and Decision Making”, McGraw-Hill

Simon, H.A. (1957): “Models of Man: Social and Rational”, Wiley

Shleifer (2000): “Inefficient Markets”

Veblen (1900): “Theory of the Leisure Class”

Research Papers:

DeGeorgi , Hens and Levy (2003): “Prospect Theory and the CAPM”, NCCR-working paper.

John Maynard Keynes (1936): “The General Theory of Employment, Interest and Money”, Harcourt

De Bondt (1998): “Anatomy of the individual investor”, European Econom

42 Since the preference (100,100,90.25) >(100,100,100) is a violation of the state dominance.

sing a Portfolio on a

Chapter 7:

Choo

Mean Reverting Process:

Time Diversification

Choosing a Portfolio on a Mean Reverting Process: Time Diversification

90

1 INTRODUCTION

As discussed in previous chapters, in efficient markets, mean-variance analysis is sufficient to solve investment

decision problems. However, there is ample evidence that markets are not as efficient as one would wish.

Concerning the issue of mean-reversion we for example observe that the longer the investment horizon the more

favorable is the ratio of return to variance43. The ratio of variance over an n-period horizon to n times the

variance of a one period horizon has been suggested as a test for mean-reversion. While this ratio should

alk, for mean-reverting processes it is significantly less than one. As one of

for many stocks in the DJIA. One may say that markets have a “memory”,

ntradicting the efficient market hypothesis as formulated

Fama (1970). In this chapter we discuss the question how to invest on mean-reverting processes. We find that

reasonable degrees of risk aversion a CRRA-expected utility maximizer will hold more risky assets when those

mean-reverting but he will also have to “time the market”, i.e. to adjust his asset allocation to the ups and

downs of the market. We conclude this chapter by asking once more whether the model can be closed, i.e.

to generate a mean-reverting

at are know under the term

“investor sentiment models”. Finally, we take a glance at investor sentiment indices as constructed by

ed as an autoregressive process of order one,

fluctuate around one in a random w

our exercises shows this is the case

i.e. prices in the future depend on past realizations, co

by

for

are

whether there are reasonable assumptions on the investors behavior that are able

asset price process. This leads us to behavioral finance asset pricing models th

practitioners.

2 THE MEAN REVERTING PROCESS

A mean reverting process is a stochastic process that can be describ

AR(1), i.e. as 1t t ty yµ ρ ε+ = + + where tε satisfies E( ) 0tε = , 2 2E( )tε σ= and. For = 1ρ , we get a unit

root process or a random walk. If < 1ρ the pr . If ocess is mean reverting > 1ρ the process is mean-averting.

As an example for an AR(1) process, consider a simple Markov process, where realizations at time t depend only

on the realizations in the previous period t-1 as summarized in the figure bellow.

1-cc-

1-bb+

-+tt-1

1-cc-

1-bb+

-+tt-1

The variables b and c denote the probabilities to reach state “+” in period t given that in the previous period the

state was “+” respectively “– “. If the process is mean reverting (averting), the probability to be in the same

state one period ahead is smaller (greater) than the probability for a “switch”. In contrast, these probabilities are

the same for a random walk, since the process has no memory. In a random walk with drift the probabilities

43 Recall that in a random walk both the return and the variance increase at the same rate with the investment horizon.


91

al for both cases “+” respectively “– “.

a strategic asset allocation plan over multiple periods and their asset allocation changes over time

as a response to market movements.

functions and a general mean-reverting process can be

excess stock return equals the unconditional mean. The optimal strategy of a long-term investor is the strategic

asset allocation. The line is slightly steeper than the tactical asset allocation line and it is shifted upward, so that

differ from ½ but are identic

3 OPTIMAL PORTFOLIO CHOICE WITH MEAN REVERSION

The results in the previous chapter show that betting on time diversification, i.e. holding more risky assets the

longer the investment horizon, does not make any sense if markets are efficient and assets returns are i.i.d. In

this chapter, we show that a rational investor with reasonable risk aversion will hold more stocks (diversify over

time) when returns are mean reverting and that she will have to time the market, i.e. she will have to adjust her

share of risky assets to the market conditions.

This point is made in the book of Campbell and Viceira (2002). Unfortunately these authors do use the term

“buy&hold” differently to the way we used it so far. For them a buy&hold strategy is a strategy in which the

asset allocation is hold fixed over time. We called this strategy a fix-mix or rebalancing strategy because it

cannot be followed by buying and then holding assets untouched. When percentages of wealth are fixed the

units of assets have to be rebalanced! In a second dimension Campbell and Viceira (2002) distinguish between

myopic, tactical and strategic decisions. Investors following a myopic buy-and-hold strategy plan over one period

of time and do not try to implement any timing strategies while building their asset allocations. The tactical asset allocation allows investors to react to market movements while planning ahead for one period only. Investors

implementing

The resulting portfolio rules for Epstein-Zin utility

represented in the following figure:

If an investor forms a tactical asset allocation then on average she will hold as much stocks as a myopic buy and

hold maximizer. But after a bull market she will reduce her share and after a bear market she will increase her

share of risky assets. I.e. the optimal share of risky assets is an increasing function of up-movements of the

market, the line has a positive slope. At the point where the line crosses the myopic buy-and-hold line, the


92

notion that long-term investors anticipate

that in the next period they will have the chance to change their decisions in order to exploit market

o ready to put more weight on

ods of time, high returns seem to be followed by periods of low

n

s

4 THE CASE CONSIDERED IN SAMUELSON (1991)

Markov process (markets have a memory over one period only), where the

total return can be high (“+”) or low (“–“). There are two assets: safe cash and a risky equity. In the case of a

random walk, regardless of the last period realization, the probability of observing high or low returns is the

same, equal to a where

the intercept is positive. The higher slope can be explained with the

opp rtunities, so that their aversion against the risky asset decreases and they are

the stock holdings in their portfolios.

This behavior of long-term investors has been discussed by Siegel (1994) as well. His advice that long-term

investors should hold more equities is only based on the reduced risk of stock returns at long horizons due to

mean reverting stock returns, ignoring the potential for timing: “Stocks have what economists call mean-reverting returns, meaning that over long perireturns and vice versa. On the other hand, over time, real returns on fixed income assets become relatively less certain. For horizons of 20 years or more, bonds are riskier than stocks.” Based on this fact Siegel recommends

holding more stocks the longer the investment horizon. The problem with this recommendation is that if stock

returns are mean reverting then they are predictable. Thus, investors can do even better by timing the market

and the buy-and-hold strategy is not optimal anymore.

Campbell and Viceira (2002) illustrate the proof of this claim by usi g an AR(1) process with log-normal returns

and Epstein-Zin utility. Samuelson (1991) gives the same intuition in a much simpler model based on the two-

tate Markov model introduced above and CRRA expected utility functions. For the sake of simplicity, we will

only present this case here.

Consider as before a simple 2-by-2

0 1a< < . If we assume that stocks rise on average in the long run, then 1 12

a a> > − .

If the process is mean reverting, then Prob 1/ 2 Probb c= + | + < < + | − = . If we want to recognize

additionally that stocks have an upward trend then the average of b and c must exceed ½. To simplify matters,

Samuelson has chosen the parameters without a trend in the following way:


93

1-cc-

1-bb+

-+tt-1

1-cc-

1-bb+

-+tt-1

1/21/2-

1/21/2+

-+tt-1

1/21/2-

1/21/2+

-+tt-1

1/32/3-

2/31/3+

-+tt-1

1/32/3-

2/31/3+

-+tt-1

2/31/3-

1/32/3+

-+tt-1

2/31/3-

1/32/3+

-+tt-1

cesses have the same unconditional expectationHence all pro 44. The return matrix that Samuelson has chosen is:

⎡ ⎤= ⎢ ⎥⎣ ⎦

. For the case of a logarithmic utility (or “Bernoulli” utility) we solve the problem

ln(1 2 ) (1 ) ln(1 )Max p pλ

1 31 0

R

λ λ+ + − − where λ is the percentage wealth invested in the risky asset.

* 3 1pThe solution of the problem is:

2λ = .

Myopic Investor

From the general solution giv

−

en just before, in the case of a random walk, where p=1/2 the optimal fraction of

risky assets is ¼. Conditioning upon knowing the “+” or “–“ that has occurred, the one-period Bernoulli

e next period, , if the previous

agent the optimal allocation does not change even if the problem is solved for T periods. To see this, let

investor who is permitted to time will choose to hold only cash in th * 0λ+ =

realization was good, or put half of its wealth in risky assets, 1/ 2λ− = , if the realization was bad. Thus, for a

myopic Bernoulli utility maximizer the long run average of the timing asset allocation coincides with the asset

allocation on a random walk that has the same long run probabilities as the Markov process. For a Bernoulli

,

*

tω ∈ + − be the real period t and t = ( ,..., )ization in 0 tω ω ω be the history up to that period. Then the

evolution of wealth can be written as ( )t tw 1 t+1 t t t1( ) = [1 ( )+R( ) ( )] t t

t wω λ ω ω λ ω+− ω+ where t( )tλ ω is the

decision taken at t and 1R( )tω + is the return at t+1. Evaluating the wealth at the end of the planning period we

see that the T-period panning problem decomposes into T separate one-period problems:

t )t

t t t

( )Max ( ¦ ) ln(1 2 ( ))+ ( ¦ ) ln(1 ( )

t

t tprob probλ ω

ω λ ω ω λ ω+ − − −

We get the same optimal solution as before. Thus, a strategic long-term Bernoulli utility maximizer chooses the

same asset allocation as the myopic Bernoulli utility maximizer as claimed above.

We can summarize the alternative portfolio rules for a Bernoulli utility maximizer as follows:

44 That is to say in the limit distribution of all three Markov matrices both states are equally likely. You may check this fact by iterating the matrices!


94

1( )U wIf we choose a more realistic utility function from the CRRA class as for example = −

problem becomes

w the optimization

1 1(1 2 ) (1 )(1 )Max p pλ

λ λ− −− + − − − and the solution is:

*

1 2 / 34

⎪ ⎪⎪

4 72 (1 ) 0 < 2/3

(3 2)

p

p pp

pλ

⎧ ⎫=

⎪⎪ ⎪+ −⎪ ⎪= <⎨ ⎬− ⎪⎪ ⎪

4⎪4 72 (1 )

2/3 < p <1 4(3 2)

p pp

− −

λ

ealth in

the same long run probabilities as the M

⎪ ⎪−⎪ ⎪⎩ ⎭

Thus, in the case of a random walk where p=1/2, the optimal allocation on risky assets is 0.1213...= If the

investor believes in mean reversion (p=1/3), he will put all his w the riskless asset, * 0λ+ = , after a good

realization and only part of it, * 1/ 4λ− = , after a bad realization. Thus, for a more realistic utility maximizer, the

long run average of the timing asset allocation is greater than the asset allocation on a random walk that has

arkov process, i.e.

*

* * *1 1 1 1 12 2 2 4+

The consequences for the long-term s g two

alternative three-period et the accumulated probabilities be as illustrated in the figure bellow.

because 0 0.1213...2λ λ λ− + > >

trategic asset allocation can be seen by considering the followin

cases. L

+ ⋅

+1/9

+ - +

-2/9

+ 2/9

4/9

-

Further, let 2 2( )- +λ λ be the optimal one-period choice in the second period after a -/+return. The first period

choice after a “+” return is determined as before:

Max -1/9 (1+2 ) (1+2 ) - 4/9 (1- ) (1+2 ) - 2/9 (1+2 ) (1- ) - 2/9 (1- ) (1- ) λ

λ λ λ λ λ λ λ λ+ − + +-1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1

-1 -1 9/25 16/25= 25/27 Max - (1+2 ) - (1- )

λλ λ


95

Hence, taking into account the second period optimization the odds for a good outcome have changed from 1 to

2 in the myopic case to 16 to 25 in the strategic case. Consequently, the investor invests more in the risky asset,

.

If we consider a strategic decision after a previous bear market then the probabilities change to:

1+ 0.0198193 0λ = >

- +

+

+-

-

2/9

1/9

-

2/9

4/9

we get:

+

Hence, taking into account the second period optimization the odds have changed to from 1 to 2 in the myopic

case to 18 to 26. Consequently, the investor invests more in the risky asset .

n

e summarized i

-1 2 -1 -1 2 -1 -1 2 -1 1 -1 -1 2 -1

-1 -1

4/9 2/9 1/9(1+2 ) (1+2 ) (1+2 ) (1- ) - (1- ) (1+2 ) (1- ) (1- ) (1- )

/26 /26

Max -2/9 - -

= 26/27 Max -18 (1+2 ) - 8 (1- )λ

λ

λ λ λ λ λ λ λ λ λ

λ λ

+ + − +

1+ 0.272078... 0.25λ = >

Thus, a strategic long term utility maximizer with a more realistic utility function chooses an asset allocatio that

has more risky assets than the myopic utility maximizer with the same utility function.

The alternative portfolio rules ar n the figure bellow.

To summarize, the investor with the more risk averse utility U(W) = -1/W has a smaller allocation of risky assets

than the Bernoulli case and the longer the time horizon the greater the proportion of risky assets. Similarly, a less

risk averse utility ( )U W W= has a greater allocation of risky assets than the Bernoulli case and the longer

zon the

the allocation of risky assets increases with the investment

horizon. The following figure summarizes the results of Samuelson (1991):

the time hori smaller the proportion of risky assets (see the slides). Finally, for the more risk averse utility

U(W) = -1/W we observe time diversification, i.e.


96

CRRA Utility One Period Two Periods

(No Timing) (Timing/TAA*) (No Timing) (Timi

Random Walk Mean Reversion Random Walk Mean Reversion

ng/SAA)

Bernoulli Utility

“fix-mix”

strategy ( ) ln( )u w w=

1 0.25RWλ = 1

1

0

0.5

λ

λ+

−

=

=2 20.5 0.5 0.25 RWλ λ λ λ

−+= + = =

2 10.25RW RWλ λ= =

“fix-mix” - strategy Average:

2

2

0

0.5

λ

λ+

−

=

=

2 20.5 0.5 0.25 RWλ λ λ λ−+= + = =

NO TIME DIVERSIFICATION

2 10.1213RW RWλ λ= =

“fix-mix” - strategy

2 1

2 1

0.0198193... 0

0.272078... 0.25

λ λ

λ λ+ +

− −

= >

= > =

=

Average:

1

1

0

0.25

λ

λ+

−

=

=

1 1 11 1

1( )u ww

= −

1 0.1213...RWλ =

“fix-mix” 2 2 11 1 0.145...

2 2 RWλ λ λ λ+ −

= +

= >0.1252 2 RWλ λ λ λ

+ −= + = > strategy

TIME DIVERSIFICATION

5 ASSET PRICING AND MEAN REVERSION

There are good reasons to believe that on aggregate dividends are not mean-reverting (see, for example

Campbell and Shiller (1988)). If however the dividend process is random (i.i.d. or a geometric Brownian motion),

then the rational portfolio choice of an expected utility maximizer with CRRA will not be able to generate a mean

reverting return process, because as we have shown above such an investor will choose a fix-mix strategy. To

build models where portfolio choice is consistent with mean reversion, we must look elsewhere or in the words

of Shiller (2000) “In sum, stock prices clearly have a life of their own; they are not simply responding to earnings or dividends. Nor does it appear that they are determined only by information ab t future earnings or dividends.

k the

intertemporal utility optimization (the

Euler equation). So far changes of believes and changes of risk attitudes have been proposed.

ouIn seeking explanations of stock price movements, we must look elsewhere.”

Investors Sentiment Hypothesis

If the input is a random wal n in a totally rational model the output should follow a random walk as well. To

derive mean reversion we therefore need to break with total rationality in the sense of expected utility

maximization with CRRA. In technical terms, in order to get mean reversion out of a representative agent model,

we need to introduce some reason for mean-reversion in the condition for


97

For example, Barberis, Shleifer and Vishny (1998) accept all traditional asset pricing assumptions including the

hypothesis that the representative investor applies the Bayesian rule for updat model of investor

sentiment, they assume that earnings follow a random walk but the investor believes the market switches

between two regimes: a ``momentum'' and a ``mean-reversion'' state. If the investor does Bayesian updating

every period then he switches between two moods called overreacti derreaction. Since there is only one

representative consumer, his marginal rate of substitution determines relative prices. Changes in the aggregated

dividends determine changes in relative prices so that the dividend yield should remains constant. There is no

a ertainty in th maximization problem besides of the investor’s expectations. Relative prices change

endogenous as n of investor’s beliefs to be in a momentum or me ver ime determined by a

Markov process.

Barberis, Huang, and Santos (2001) follow a different approach. They assume that not the beliefs determining

agent’s marginal rate of substitution but his risk preferences change over tim by Thaler and

Johnson (1990). The intuition is simple: after bad realizations of the dividend process the representative investor

faces losses that motivate her to take alizations she has a cushion of gains that she

will use to take more risks. This so called „house money“ effect should not be conf

prospect theory that investors are risk averse when facing gains when facing losses. The former

concerns the change of risk aversion over time while the latter concerns the evaluation of gains and losses of

lotteries at any point in time.

Investor Sentiment Index

One example of how to transfer the theories discussed above into practice are indices measuring the investors’

t for example is based on past price

i Random Walk with the same mean and standard

s driving market’s

mood: overconfidence and regret. The intuition is simple: the more profit investors make the likely it is that they

ing. In their

on and un

dditional unc e

ly a functio an re sion reg

e as initially predicted

less risk while after good re

used with the feature of

and risk seeking

sen iment. The sentiment index provided by Credit Suisse First Boston

real zations of the Swiss Market Index (SMI). Compared to a

deviation as the SMI, at the first glance, the Sentiment Index seems to provide a good fit as the next figure

shows. The problem with this index is the usual causality issue: Do the prices determine the index or the index

influences the prices.

The Sentiment Index of Merrill Lynch is based on the notion that there are two main factor


98

research suggests that most stocks and stock indices follow a mean-reversion

ty maximizer with CRRA would then time the market and hold a higher fraction of risky

Chan (2002): Time Series Applications to Finance, Wiley

Campbell and Viceira (2002): Strategic Asset Allocation

Shiller (2000): Irrational Exuberance

Siegel (1994): Stocks for the Long Run, McGraw-Hill

Research Papers:

Barberis, Shleifer and Vishny (1998): A Model of Investor Sentiment, Journal of Financial Economics,

(http://gsbwww.uchicago.edu/fac/nicholas.barberis/research/

become overconfident. On the contrary, the longer are the sequences associated with losses, the more likely it is

that investors would regret their investment choice and start selling their positions. The relative number of runs

associated with capital gains and losses determines which of these effects will dominate and determine future

market movement. One problem with this approach of measuring investors’ sentiment is that the length of the

runs is not determined endogenously by the model.

6 CONCLUSION

The main body of empirical

process. An expected utili

assets in his portfolio in order to diversify over time. However, his portfolio decision behavior cannot generate

mean reverting prices in an asset pricing model. In contrast, models based on Behavioral Finance can do this

trick.

REFERENCES:

Textbooks:

)

Barberis, Huang, and Santos (2001): Prospect Theory and Asset Pricing, Quarterly Journal of Economics,

(http://gsbwww.uchicago.edu/fac/nicholas.barberis/research/)

Campbell and Shiller (1988): The Dividend Price Ration and Expectations of Future Dividends and Discount


99

Thaler and Johnson (1990): “Gambling with the House Money and Trying to Break Even: The Effects of Prior

Outcomes on Risky Choice" Management Science, XXXVI, 643-660.

Factors, Review of Economic Studies 58, 495-514.

Samuelson (1991): Long-run Risk Tolerance When Equity Returns Are Mean Reverting: Pseudoparadoxes and

Vindication of “Businessman’s Risk”, in W.C. Brainard, W.D. Nordhaus, and H.W. Wattsleds, “Money,

Macroeconomics, and Economic Policy”, Essays in Honor of James Tobin, MIT Press, pp. 181-222

Chapter 8:

Behavioral Hedge Funds

Behavioral Hedge Funds 101

1 WHAT ARE HEDGE FUNDS?

Despite strong media attention, hedge funds still do not have a precise legal definition. Though, they have a

series of common characteristics that help distinguish them from other investment funds.

Hedge fund managers follow active investment strategies. They attempt to take advantage of a cheaper

alysis of investment opportunities, and regulatory unrestricted conditions of

on of flexible investment policies. Using the legal form of a limited

ent company, hedge funds avoid regulation and minimize their tax bills.

On the other hand, hedge funds have limited liquidity. The dates when investors can enter a hedge fund

are strictly specified. Additionally, there is a minimum time an investor is required to keep his money

invested in the hedge funds (lockup period). It is often required that investors give advance notice of

unds require both a management

atermark”, i.e. minimum rate of

return over the whole investment period above which the hedge funds is legitimated to change incentive

fees.

To align hedge fund manager’s interests with those of his investors, managers invest a significant

personal stake in their funds as partners.

Since manager’s skills and available investment opportunities are not scalable, size is not necessarily a

factor of success. Some hedge fund strategies have limited ability to absorb large sum of funds and

managers prefer to close them to new subscribers as soon as the assets under management achieve a

certain level.

2 HEDGE FUNDS STRATEGIES

2.1 Long/Short strategies

Long/short strategies involve the combined purchase of undervalued assets and the sale of overvalued securities.

The “fair” value is usually determined using the Arbitrage Pricing Theory (APT). The strategy is successful if the

purchased stock appreciates and/or the sold share looses value. This is the reason why long/short strategies

perform well in bear markets as well as bull markets. If both assets have different betas, the overall portfolio will

have some degree of market exposure. Though, a hedge fund may be neutral to a specific market factor, e.g.

sector index, series of interest rates etc.

2.2 Arbitrage

In the world of hedge funds, “arbitrage” has a different meaning than a risk-free trade that does not require any

cash flow and results in an immediate profit and no future losses as defined in the academic literature. In the

access to markets, better an

trade allowing the implementati

partnership offshore investm

their wish to cash in.

While traditional funds charge usually only a management fee, hedge f

fee and an incentive fee. Additionally, many funds include a “high w


rategy that profits from differences in prices of correlated financial

kets. The simultaneous purchase and sale of these instruments is

For

converti

on the o

the stoc nt

retu

Another

anomali

long/sho fixed income securities that are in some way interrelated but the relationship is

tempora

2.3 E

Event-d trategies focus on identifying securities that can benefit from the occurrence of extraordinary

events such as restruc

reaction

general appre ty or debt.

Distressed securitiesoperation

that are belo

most investors must sell securities of troubled companies because policy restrictions and regulatory constrains do

low credit ratings. The result is a pricing discount that reflects in

addition the uncertainty about the outcome of the event.

posure and boost returns.

Emerging markets hedge funds focus on equity or fixed income investments in so called emerging markets. The

y are associated with poor accounting, lack of proper legal systems, unsophisticated local

context of hedge funds, an arbitrage is a st

instruments usually traded on different mar

usually not risk-free. The strategy it attempts is to exploit discrepancies in the relative pricing of closely related

securities under the assumption that they will disappear over time.

example, managers of convertible arbitrage strategies attempt to buy undervalued instruments that are

ble into equity and then hedge the market risks by going short in the firm`s stock. The fair value is based

ptional characteristics in the convertible bond and the manager’s assumptions on the future volatility of

k. The risk is that the volatility may turn out as lower than expected. One of the main drivers of rece

rns in convertible arbitrage is generated by IPOs.

example for an arbitrage strategy is the fixed income arbitrage. This strategy seeks to exploit pricing

es within and across global fixed income markets and their derivatives. Usually, managers take offsetting

rt positions in similar

lly distorted by market events, investor preferences, exogenous shocks etc.

vent driven strategies

riven s

turings, takeovers, mergers, liquidations, bankruptcies etc. The strategy profits from the

of security prices, which is typically influenced more by the dynamics of the particular event than by the

ciation or depreciation of the firms equi

strategies for example invest in the debt or equity of companies experiencing financial or

al difficulties. The strategy exploits the fact that many investors are restricted from trading securities

w investment grade. In this sense, the strategy is a very good example of a regulatory arbitrage:

not allow them to hold securities with very

2.4 Directional strategies

Rather than hedging risks, direction strategies rely on the direction of market movement in order to achieve

profits.

Global macro funds for example do not hedge at all. They profit by exploiting extreme price/value changes by

taking large directional bets reflecting their forecasts on the future market movement. Leverage and derivative

products are used to hold large market ex

risks with this strateg

investors. The opportunities are due to undetected, undervalued securities.

Sector hedge funds focus on long and short investments in particular sector, e.g. life sciences (pharmaceuticals,


se they provide stable positive returns.

Another argument for investing in hedge funds is their diversification potential. Every Mean-Variance investor

the following figure.

ment risk.

biotechnology), real estate, and energy.

Dedicated short hedge funds seek to profit from the decline in stocks by taking short positions.

Hedge funds are interesting particularly in a downside market becau

would invest in hedge funds because of

100

150

200

250

300

0

50

31.0

1.19

94

31.0

5.19

94

30.0

9.19

94

31.0

1.19

95

31.0

5.19

95

30.0

9.19

95

31.0

1.19

96

31.0

5.19

96

30.0

9.19

96

31.0

1.19

97

31.0

5.19

97

30.0

9.19

97

31.0

1.19

98

31.0

5.19

98

30.0

9.19

98

31.0

1.19

99

31.0

5.19

99

30.0

9.19

99

31.0

1.20

00

31.0

5.20

00

30.0

9.20

00

31.0

1.20

01

31.0

5.20

01

30.0

9.20

01

31.0

1.20

02

31.0

5.20

02

30.0

9.20

02

Swiss M Ind.(r), USDarket

CSFB Tremont Hedge, USD

Higher moments of the return distribution of hedge funds strategies can be considered additionally as measures

for invest


3 VALUE AT RISK

Risk is primarily associated with losses. Measures based on the standard deviatio of returns are inappropriate

when returns are not normal for at least two reasons. First, standard deviation is a symmetric measure; it

roduced by practitioners, the measure of risk is an

amount

n

penalizes negative but also positive deviations from the mean. Moreover, deviations do not have a value

expression. With the Value at Risk (VaR) method initially int

qα such that the net worth of the position at some future date T is smaller than qα with probability α .

The number qα is the α -quantile of the return distribution. It is called the “α % VaR”. The figure bellow

illustrate the VaR concept graphically.

The biggest trouble with the VaR as a risk measure is that it neglects the impact of extreme negative events with

low probability. Additionally, the measure leads to erroneous conclusions when applied to a portfolio of risks.

We will come back to these issues in a later chapter.


4 INVESTMENT STRATEGIES BASED ON BEHAVIORAL FINANCE

Investment strategies based on Behavioral Finance research are primarily motivated by the evidence that

individuals do not update their expectations using the Bayesian rule. On the whole, one could expect that market

returns do not follow the predictions of traditional assets pricing models.

4.1 Underreaction

According to experimental evidence, individuals do not update their beliefs rationally using Bayesian rule.

Moreover, they systematically ignore part of the new evidence. The following example illustrates this. Consider

that there are 100 urns with 1000 balls in each of them. 45 of the urns have 700 black and 300 red balls and

r 0 black and 700 red balls. The question on the probability that a randomly selected

nswered correctly (45%). In the next question, individuals are provided with

the est of the urns have 30

urn has more black balls is usually a

additional information that from 12 balls drawn from the randomly selected urn 8 balls are black and 4 are red.

To answer the question the same question as before including the new information, individuals have to apply

Bayesian rule.

( ) (*/ )( /*)( ) (*/ ) ( ) (*/ )

p s p sp sp s p s p r p r+

where ( ) 45%p s= = and ( ) 55%p r =

After some calculations 12

(*/ )p s ⎛ ⎞= ⎜ ⎟ ( ) ( )8 40.7 0.3

8⎝ ⎠ and transformations

1( /*) ( ) (*/ )1( ) (*/ )

p s p r p rp s p s

=+

we get ( ) ( )

( ) ( )

8 4

8 4

120.3 0.7

855( ) (*/ )1245

0.7 0.38

p r p r

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠

411 0.3 0.0279 0.7

⎛ ⎞= =⎜ ⎟⎝ ⎠

and

1( /*) 96.04%1.027

p s ≈ ≈ .

Though, the typical answers are significantly lower (45% and 67%) indicating that on average people seam to

underreact to new information.

The implications for investment strategies are based on studies of market reaction upon earnings

announcements. Positive (negative) earnings surprises are associated with positive (negative) subsequent returns

as illustrated in the figure bellow. In the short run returns are predictable which is known as the post-earnings-

surprise announcement drift.

Behavioral Hedge Funds

106

The next figure shows the size of the cumulative returns after a quarter for growth and value stocks as a function

of the size of the surprise.

Though, from a cross-sectional perspective, the reaction to earnings surprises is much stronger for growth stocks

than for value stocks, i.e. the curve is still increasing but its slope is higher compared to the response function of

value stocks. Additionally, the impact of earnings surprises on subsequent returns is limited to about -20% by

negative surprises respective to about 13% by positive surprises and there is no return overshooting

(overreaction).

or example invest in stocks with positive standard unexpected earnings

Investment strategies based on the “post earnings announcement drift” are successful implemented in practice.

Fuller and Thaler Asset Management f

(SUE)45 and sell stocks with negative SUE.

45 The difference between actual earnings and the forecast is scaled by the historical standard deviation of the forecast errors. The forecast is a univariate first-order autoregressive model in seasonal differences.


Reversal

While most of the evidence indicates an underreaction to earnings announcements generating momentum in

stock prices, other evidence search for the sources of long-run mean reversion as first documented by DeBondt

and Thaler (1985). They showed that firms with prior extreme negative stock price performance (“losers”) seem

to outperfom those with prior extreme positive performance (“winners”) as if a part of the prior stock price

movement constitutes a deviation from fundamental value. There is still no consensus explanation for these

results. Though, what is clear is that even if overreaction exists, it appears to be to complicated to be

characterized as a simple function of recent earnings changes. In the words of Bernard (1985), a predictable

reversal of prior extreme price movements is consistent with a wide variety of market inefficiencies, including

random deviations of prices from fundamental values, and need not to be cause by any systematic

misinterpretation of earnings information.

Thus, the question if stock prices follow a momentum or a reversion is empirical. The largest p rt of the empirical

s the price-to-earnings (P/E) ratios. Since these rations are also used to characterize socks as

ing assets

To predict future asset price movements, one can use event studies indicating the amount of information

reflected in current prices, asset pricing models defining a fundamental value as the equilibrium in the long run

or just empirical evidence on a systematic price behavior without being concerned with any particular reasons.

Another possibility to derive profitable investment strategies is to go to the basics of asset pricing and bet that at

least in the long run the Law of One Price must hold. In the case of Royal Dutch and Shell Transportation for

example the market needed almost four years to recognize this.

-200%

0%

200%

400%

600%

800%

1000%

Mar-92

Dec-92

Sep-93

Jun-9

4

Mar-95

Dec-95

Sep-96

Jun-9

7

Mar-98

Dec-98

Sep-99

Jun-0

0

Mar-01

Dec-01

Sep-02

Jun-0

3

Mar-04

Small/Mid-Cap Growth (Net-of-Fees) Russell 2500 Growth

4.2 Momentum and

a

evidence indicates that momentum strategies (buying winners and selling the losers) are most successful over

medium formation and holding periods (3-12 months) and reversion strategies deliver highest returns over long

periods (3-5 years).

Reversal strategies can be defined not only over price movements but also over ratios including accounting

measures such a

“value” or “growth”, we can observe momentum and reversion in the returns of value stocks.

4.3 Strategies based on co-mov


4.4 Strategies exploiting probability weighting

Kahneman and Tversky (1979, 1992) documented that people give too much weight to small probabilities. To

illustrate this preference consider the following example. In an experiment, people are asked to decide between

two lotteries. They can either bet on a 2 to 5 shot and receive 40% profit or bet on a 20 to 1 shot and receive

200% profit in the case of a win. Although the expected return of the first lottery 5 2 91.47 7 7

⎞⎛ ⋅ + =⎜ ⎟⎝ ⎠

is higher,

the people prefer the latter 1 20 22221 21

⎞⎛ ⋅ + =⎜⎝ 21 ⎟⎠

.

at the racetrack and there are only small profits

on the deepest in-the-money puts. The losses are more and more as the puts gets more out-of-the-money just

Hodges, Tompkins, and W. T. Ziemba (2003) found that this result can be also observed by horse racetracks.

Applying the idea to index options on S&P500 and FTSE 100 futures they found that the favorite deep in-the-

money calls have positive expected value just like the favorites


like in the racetrack.

To exploit these regularities, the authors build an investment strategy that sells overpriced puts, hedge them

shorting futures and use the revenues of the put sale to buy calls. Its performance is presented in the figure

5 CONCLUSIONS

Investing is much more complicated as suggested by traditional finance. It is absolutely possible to achieve a

better performance than the market since there are systematical deviations from the so called “fundamental

value”. Some of these deviations are due to psychological biases extensively studied by experimental studies.

A good portfolio management has to stand up to the continuously changing process of existing anomalies.

Moreover, it must be based on a solid empirical analysis in order to recognize profitable market anomalies on

time.

bellow.


Price Reaction to Earnings Announcements, in Thaler: Advances in Behavioral Finance,

Sage.

Hodges and Ziemba (2003): The favorite-longshot bias in S&P500 futures options: the return to bets and the

cost of insurance, working paper, University of British Columbia

DeBondt and Thaler (1985): Does the Stock Market Overreact?, Journal of Finance 40 (3): 793-806

Froot and Deborah (1998): How are Stock Prices Affected by the Location of Trade, Journal of Financial

Economics (53).

Lhabitant (2002): Hedge Funds, Wiley.

Skinner and Sloan (1999): Earnings Surprises, Growth Expectations, and Stock Returns, Uni rsity of Michigan

REFERENCES:

Bernard (1985): Stock

ve

Working Paper

Chapter 9:

Choosing a Portfolio on a GARCH

Process: Risk Management

Choosing a Portfolio on a GARCH Process: Risk Management 112

1 INTRODUCTION

There is much ambiguity about risk nowadays. For micro-economists, risk usually has neutral meaning. In other

words, risk is associated with the uncertain (positive or negative) outcomes of decisions. For financial

economists, risk is related to something negative such as the possibility of losses. It is not surprising that the

least as unclear as the definition of risk. In this chapter, we will present

ir adequacy using axioms required for meaningful comparison between

nally, we will discuss the question if it is possible to generate processes

endogenously in order to explain phenomena such as the stochastic volatility.

EVIDENCE

Bachelier (1900) tried to convince his fellow students that stock prices are random by presenting them a set of

es.

more

irregular than those from a random walk. True returns show volatility that is stochastic, clustered and

e facts raise the issue

easure face various

difficulties interpreting the results of their analysis. To begin with we show that the volatility of real returns (e.g.

DJIA) is not constant, as required by a White noise process, but stochastic.

question of how to measure risk is at

various risk measures and discuss the

risky outcomes to be made. Additio

2

artificially generated truly random prices and prices from the DJIA. Hardly anyone could name the DJIA-pric

Looking however at the returns of these prices the true returns are easily identified because they are much

asymmetric. Moreover extreme returns associated with crashes are more frequent. Thes

how risk should be measured. Investors using the variance of asset returns as the risk m

S to c h a s t ic V o la t il i t y

D J IA - r e tu r n s R W - r e tu rn s

Second, volatility is asymmetric, i.e. on average, the volatility of asset returns increases in recessions and

decreases in boom phases. Intuitively, this result can be explained with the fact that investors become nervous in

down markets and place extreme trades. Hence, sharp decreases of asset prices are followed by sharp increases

in the volatility of returns. The effect is not country specific46.

46 Note however the exception that is given by Italy in this respect.


MSCI STOCKS INDEXSample Period

Mean up

Mean down

Volatility Up

Volatility Down Corr

Corr up

Corr Down

World 1970.1‐2003.1021.00% ‐15.00% 14.35 17.86 0.89 0.86 0.9USA 1970.1‐2003.1024.00% ‐17.00% 15.65 17.61 1 1 1GER 1970.1‐2003.1028.00% ‐23.00% 19.63 22.62 0.51 0.35 0.6UK 1970.1‐2003.1029.00% ‐21.00% 22.3 23.3 0.61 0.47 0.66CH 1970.1‐2003.1025.52% ‐19.90% 17.9 21.15 0.64 0.5 0.7CAN 1970.1‐2003.1025.73% ‐19.11% 17.41 19.66 0.71 0.65 0.72IT 1970.1‐2003.1035.89% ‐29.41% 25.45 23.84 0.35 0.2 0.41FR 1970.1‐2003.1032.94% ‐26.00% 21.71 23.45 0.54 0.4 0.6AUS 1970.1‐2003.1028.96% ‐23.53% 20.42 23.74 0.53 0.33 0.62JAP 1970.1‐2003.1027.45% ‐22.05% 19.42 20.4 0.36 0.24 0.37

Sample Period

Mean up

Mean down

Volatility Up

Volatility Down Corr

Corr up

Corr Down

MSCI STOCKS INDEX

World 1970.1‐2003.1021.00% ‐15.00% 14.35 17.86 0.89 0.86 0.9USA 1970.1‐2003.1024.00% ‐17.00% 15.65 17.61 1 1 1GER 1970.1‐2003.1028.00% ‐23.00% 19.63 22.62 0.51 0.35 0.6UK 1970.1‐2003.1029.00% ‐21.00% 22.3 23.3 0.61 0.47 0.66CH 1970.1‐2003.1025.52% ‐19.90% 17.9 21.15 0.64 0.5 0.7CAN 1970.1‐2003.1025.73% ‐19.11% 17.41 19.66 0.71 0.65 0.72IT 1970.1‐2003.1035.89% ‐29.41% 25.45 23.84 0.35 0.2 0.41FR 1970.1‐2003.1032.94% ‐26.00% 21.71 23.45 0.54 0.4 0.6AUS 1970.1‐2003.1028.96% ‐23.53% 20.42 23.74 0.53 0.33 0.62JAP 1970.1‐2003.1027.45% ‐22.05% 19.42 20.4 0.36 0.24 0.37

Even if one uses the implied volatility47, it also increases (decreases) with decreasing (increasing) returns as

shown in the figure bellow.

The third problem using the variance (volatility) as risk measure is that return distributions have “fat tails”, i.e.

there are too many extreme observations compared to the case of normally distributed returns. Examples of

extreme events are bubbles (New Economy bubble, Sought See bubble etc).

Time series with stochastic volatility can be parameterized using so called Generalized Autoregressive

Conditional Heteroscedastic (GARCH) models.

3 COHERENT RISK MEASURES

So far we introduced two risk measures: the variance and Value at Risk, VaR. Both measures have weaknesses.

The variance does not distinguish between positive and negative deviations from the mean. On the other hand,

f nt units or traders according to the portfolio

theory may lead to erroneous conclusions with respect to the risk exposure of the firm. This idea is illustrated in

the figure bellow.

using value at risk, we will accept positions as safe when in say, less than 1% of the cases, we get in trouble.

The value at risk has some dangers. When used we will accept positions that with high probability are good but

that with a very small probability lead to bankruptcy. Inside a financial institution, the use of value at risk is more

problematic. Additionally, adding up the value at risk of dif ere

olatility. 47 Note that the implied volatility is usually lower than the realized v

Choosing a Portfolio on a GARCH Process: Risk Management

114

A coherent risk measure satisfies a prescribed set of axioms required for meaningful comparison between risky

cash flows.

• The sub-additivity describes a diversification effect: )( ) ( ) (X Y X Yρ ρ ρ+ ≤ +

( ) ( )tX t Xρ ρ= • The homogeneity expresses a scaling property:

( ) ( ), if X Y X Yρ ρ≥ ≤ • The monotonicity connects state dominance with risk reduction:

• The risk-free condition captures the reduction of risk by investing in the risk-free payoff r :

( ) ( )X rn X nρ ρ+ = −

The only one measure that satisfies these axioms is the Conditional Value at Risk defined as

CVaR= E(x ( )) x VaR α≤ (Theorem by Artzner, Delbaen, Eber, and Heath (1999)). The CVaR is thus the

conditional expectation of the random variable x below the VaR-level. The main advantages of this measure

unt the size of losses and does not distort the risk exposure at

compared to VaR are that it takes into acco

portfolio level.

Computing the tangential portfolios with CVaR or with Variance as risk measure we see that on our data the

CVaR-portfolio is much better diversified than the portfolio based on variance.

DJIA 81-03 Mean-Variance Portfolio Rf=1.0023MALCOAALTRIA GP.AMERICAN EXPRESSAT & TBOEINGCATERPILLARCITIGROUPCOCA COLADU PONT E I DE NEMOURSEASTMAN KODAKEXXON MOBILGENERAL ELECTRICGENERAL MOTORSHEWLETT - PACKARDHOME DEPOTHONEYWELL INTL.INTELINTL.BUS.MACH.INTL.PAPERJOHNSON & JOHNSONJP MORGAN CHASE &.CO.MCDONALDSMERCK &.CO.MICROSOFTPROCTER & GAMBLESBC COMMUNICATIONSUNITED TECHNOLOGIESWAL MART STORESWALT DISNEY

DJIA 81-03 CVaR-Portfolio 95% Target 1.0023MALCOAALTRIA GP.AMERICAN EXPRESSAT & TBOEINGCATERPILLARCITIGROUPCOCA COLADU PONT E I DE NEMOURSEASTMAN KODAKEXXON MOBILGENERAL ELECTRICGENERAL MOTORSHEWLETT - PACKARDHOME DEPOTHONEYWELL INTL.INTELINTL.BUS.MACH.INTL.PAPERJOHNSON & JOHNSONJP MORGAN CHASE &.CO.MCDONALDSMERCK &.CO.MICROSOFTPROCTER & GAMBLESBC COMMUNICATIONSUNITED TECHNOLOGIESWAL MART STORESWALT DISNEY


Supposing normally distributed returns, these two portfolios would need to coincide because in that case the

CVaR is a linear function of mean and variance. Hence, maximizing for example CVaRλ λµ γ− and noting that

with normally distributed returns 2CVaRλ λ λαµ βσ= + we are actually maximizing a mean-variance utility.

However, the global minimum risk portfolio can be different to those based on CVaR directly.

Usin optimal portfolios differently on the risk-return

frontier, e.g. the global MVP is not efficient anymore.

be interested in the probability

for market crashes. The difference between the long term (30-years) Treasury bond yield and the earnings-to-

price ratio of the firms is a crash indicator used by the FED. The next figure shows if this indicator predicts S&P

500 crashes.

g CVaR (or Expected Shortfall) as a risk measure places

4 CRASH MEASURES

In addition to measuring the probability of realizing a certain loss investors may


5 ASSET PRICING MODELS

Asset pricing models based on the representative consumer paradigm are not able to generate stochastic

ts is

ls of the first generation (Arthur et al (1997), LeBaron et al

(1999), Brock, Hommes (1999), Lux (1998)) are able to generate GARCH effects. The following chart is taken

from Lux (2002) in which there are three types of agents, fundamentalists, trend followers and outsiders. The

fundamentalists buy (sell) if the current price of a risky asset is below (above) the fundamental value. The trend

followers buy (sell) if the prices have gone up (down). The outsiders get attracted to the market when prices

have gone up severely. They enter the market as trend followers. The relative proportion of these trader types is

determined by the success of the strategies. We will analyze these types of models in more detail the next

chapter. The following figure showing stochastic volatility is taken from Lux and Schornstein (2004):

volatility, bubbles and crashes48. Studying the interaction of agents (or strategies) on financial marke

therefore proposed. Indeed evolutionary finance mode

48 Unless on is willing to postulate that the representative consumer has beliefs or risk aversion that is changing stochastically over time.


6 SUMMARY

Empirical studies show that the volatility of stock returns is not constant but stochastic. Moreover, asset returns

have “fat tails”. For these among other reasons, the variance cannot be used as an appropriate risk measure.

The Value at Risk as an alternative measure has also disadvantages particular relevant in portfolio context. In

this case, the Conditional Value at Risk is the better risk measure because it considers the size of losses.

From investor’s point of view, crash measures provide additional information on the investment risk. Crashes are

important because they increase the volatility. Models with heterogeneous agents (strategies) are able to explain

stochastic volatility.

REFERENCES:

(1997) “Asset Pricing under Endogenous Expectations in an Artificial Arthur, Holland, LeBaron, Palmer, Taylor

Stock Market” in The Economy as an Evolving Complex System II, Arthur, Durlauf and Lane (eds), pp. 15-44,

Addison Wesley.

Artzner, Delbaen, Eber and Heath: Coherent Measures of Risk, Math. Finance 9 (1999), no. 3, 203-228

(http://www.math.ethz.ch/~delbaen/)

Arzner, Delbaen,Eber and Heath (1997):Thinking Coherently, Risk Vol. 10/ No 11, November 1997

Bachelier (1900): Théorie de la Spéculation", Annales de l'Ecole normale superiure (trans. Random Character of

Stock Market Prices).

Brock and Hommes (1998): Heterogeneous Beliefs and Routes to Chaos in a Simple Asset Pricing Model, Journal

of Economic Dynamics and Control ,Vol. 22, pp. 1235 – 274.

LeBaron, Arthur and Palmer (1999): Time Series Properties of an Artificial Stock Market, Journal of Economic

Dynamics and Control, Vol. 23, pp. 1487 – 516.

Lux (1998): The Socio-Economic Dynamics of Speculative Markets: Interacting Agents, Chaos, and the Fat Tails

of Return Distributions, Journal of Economic Behavior and Organization, Vol. 33, pp. 143 – 65.

Lux and Schornstein (2004): Genetic Learning as an Explanation of Stylized Facts of Foreign Exchange Markets,

forthcoming special issue on Evolutionary Finance of Journal of Mathematical Economics, Hens and Schenk-

Hoppe (eds).

Chapter 10:

Evolutionary Portfolio Theory:

Survival of the Fittest on Wall

Street

Evolutionary Portfolio Theory: Survival of the Fittest on Wall Street 119

“Whenever a theory appears to you as the only possible one, take this as a sign that you have neither understood the theory nor the problem which it was intended to solve.”

(Karl Popper)

e provide leading edge research that tries to give a synthesis of the

ditional and the behavioral finance point of view. We suggest a model of portfolio selection and asset pricing

t is based on the idea of heterogeneity of strategies. The strategies considered may result from completely

ional investors maximizing some inter-temporal expected utility, from simple heuristics, from behavioral

finance or from principal agent models describing incentive problems in institutions. Actually we do not consider

bservation is

epend on all

strategies that are in the market. Rationality therefore is to be seen as conditional on the market ecology. Our

t to be in a

based on

optimization and equilibrium – has borrowed a lot from classical mechanics, behavioral finance has borrowed

believe that it is time in finance to borrow from biology, in particular from evolutionary

es of natural selection and mutation as formulated by Charles Darwin, will be two fruitful

analogies that lead us surprisingly far in this chapter.

2 THE ECOLOGY OF THE MARKET

Before developing a theory of evolution of strategies it is most useful to carefully observe which strategies can be

found in a market. In biology Charles Darwin`s discovery of the principles of evolution would not have been

possible without a careful observation and classification of the plants and animals in the world. This major step

was provided to Darwin by Carl von Linee. We recall from the first chapter of this course that a financial market

is first of all characterized by an enormous heterogeneity of strategies that individual investors and institutions

apply. There is nothing like the single representative agent maximizing some artificial utility function. A

classification of strategies that will turn out to be useful is the following one:

• Staying Outside

o This is the stock market strategy that is still followed by the majority of people

o When severe upwards price movements happen then some people may give up this strategy

• News Trading Strategies

o This is the strategy the traditional theory would recommend:

1 INTRODUCTION

In this final chapter of the lecture w

tra

tha

rat

how strategies are generated but we ask how they perform once they are in the market. The first o

that there is nothing like “the best strategy” because the performance of any strategy will d

theory will be descriptive and normative as well, answering which set of strategies one would expec

market and how to find the best response to any such market. Whereas traditional finance –

from psychology. We

reasoning. The principl


o Try to guess the future returns of assets by gathering and processing all relevant news

• Valu

o Form your portfolio accor

o Price/Earnings, Cash-Flow/Price, Book / Market, Dividend-Yield etc

Technical Analysis

e suggest stratifying the financial markets not in terms of

individuals but in terms of strategies as listed above. For the market it is totally irrelevant who is investing

matters is how much money is invested according to such a

criterion. In biology strategies fight for resources like food. In finance strategies fight for market capital. In an

f wealth across strategies changes

over time. The following chart shows the evolution of wealth for the hedge fund strategies. This sector of the

market is re r lected in the way

that suits o ry: Wealth stratified by strategies: Figure 1 shows for every year how much wealth was

e Strategies

ding to value ratios like:

• Strategies based on

o Try to guess the momentum of prices by looking at moving averages of various length

o Try to guess the reversal of prices by looking at standard deviations from moving averages

• Experimentation

o Create new strategies by “backtesting”: Check what would have run nicely on realized data

o Be aware of “data snooping”!49

3 SURVIVAL OF THE FITTEST ON WALL STREET

The first point Charles Darwin made was to argue that for the evolution of the ecology we need not model the

interaction of the individuals but the interaction of the strategies played by individuals, i.e. the interaction of the

species.

From an evolutionary point of view the fate of a single individual animal counts nothing as compared to the

relative size of the population of its species. Hence, w

according to, say, P/E-ratios. The only thing that

evolutionary model there are two forces at play: The selection force reducing the set of strategies and the

mutation force enriching it. You see the selection force in financial markets when you realize that every loss

some strategy made by buying at high prices and selling at low prices must have generated an equally sized gain

for a set of counter-parties. The mutation force is clearly seen if you look back a bit in history and observe that

previously popular strategies like trying to corner a market are no longer so frequent while new strategies like

those followed by hedge funds have emerged. Ultimately, what the evolutionary finance model tries to explain is

how the ecology of the market evolves over time, i.e. how the distribution o

ma kable for our theory because this is the first sector in which data have been col

ur theo

49 On every given set of returns one can find strategies that would have done extremely well ex post. When backtesting a strategy one t the time. Moreover, when optimizing parameters o model on given data one should keep track of the many alternatives from which one has chosen the best set of parameters. See White (2000) for precise statements on this line.

has o make sure that one only uses the information that was known at f a


invested acc see that

ove w HFs has grown but some strategies like Long/Short have profited more than average

while others like Global Macro lost funds. Since the HF sector only accounts for 4-6% of wealth overall, this give

however on n found the evolutionary approach empirically.

This empiric l foundation for it. Note

that e ke strategies. We claim that for the vast majority of capital

the strategy according to which it is managed is in principle observable. This is because most capital is managed

by delegatio n o commit

to some str g e for

investors` money by advertising strategies they want to commit to. Moreover, the evolutionary approach takes a

“flow of f ve. It claims that understanding according to which principles wealth flows into

strategies is sed

this view (s -Warburg (2002)). UBS-Warburg claims, for example that the

daily level of the SOX-semiconductor index can be predicted by a cumulative flow index capturing the flow into

r daily data. Within one day or even within

HE VOLUTIONARY ORTFOLIO ODEL

e

ording to which HF-strategy: Global Macro, Long/Short-equity, Managed Futures ,… We

rall ealth invested in

ly a example of what one would need to do in order to

al work is currently in progress. In this lecture we will outline the theoretica

th evolutionary model is based on observables li

n a d in this process the principal (the investor) wants the agents (the wealth manager) t

ate y in order to simplify monitoring and verifiability problems. Indeed many banks compet

unds” perspecti

the key to understand where asset prices are going. Recently, some practitioners have also propo

ee for example the newsletter of UBS

this sector. We are more skeptical that the flows approach is useful fo

minutes prices can change drastically on the occurrence of strong news without any high volume of trade. For

the short run day to day changes we buy the news story given by the anticipation principle while in the longer

run – on monthly data for example, we got good predictions from the flows approach. In the next section we will

make these vague ideas precise.

T o t a l A s s e t s H i s t o r y

D e c e mb e r 1 9 9 3 - J u n e 2 0 0 2 ( U S D i n M i o . )

0

2 0 '0 0 0

4 0 '0 0 0

6 0 '0 0 0

8 0 '0 0 0

10 0 '0 0 0

12 0 '0 0 0

14 0 '0 0 0

19 9 3 19 9 4 19 9 5 19 9 6 19 9 7 19 9 8 19 9 9 2 0 0 0 2 0 0 1 2 0 0 2 Q2

Con ver t ible Ar bit r age

Dedicat ed Shor t Bias

M an aged Fut ur es

Emer gin g M ar ket s

Fixed I n come Ar bit r age

Even t Dr iven

Equit y M ar ket Neut r al

Global M acr o

Lon g/ Shor t Equit y

M ult i- St r at egy

4 T E P M

We base our evolutionary model on the Lucas (1978) asset pricing model. This has two advantages. First, th

Lucas model is a model that makes sense from an economic point of view. The time uncertainty structure is

simple enough to penetrate; budget equations and numeraire good properties are satisfied. Moreover, displaying


our new ideas in this traditional model will help to asses the differences of the two. Recall from the traditional

finance model that the Lucas (1978) model is defined over discrete time that goes to infinity, i.e. 0,1,2,...t = .

The information structure is given by a finite set of realized states t tω ∈Ω in each t. The uncertainty with

respect to information decreases with the time since at every t only one state is realized. The path of state

realizations over time is denoted by the vector 0 1( , ,..., )t tω ω ω ω= . The time-uncertainty can be described

graphically by an event tree consisting of an initial date ( 0t = ) and tΩ states of nature at the next date.

The probability measure determining the occurrence of the states is denoted by P. Note that P is defined over the

set of paths tω . We called P the physical measure since it is exogenously given by nature. We used P to

model the exogenous dividends p

t=2 t=1 t=0

rocess. In the model the payoffs are determined by the dividend payments and

denote the strategies. There arecapital gains in every period. Let 1,...,i I= 1,...,k K= long-lived assets in unit

supply that enable wealth transfers over different periods of time. 0k = is the consumption good. This good is

perishable, i.e. it cannot be used to transfer wealth over time. All assets pay off in terms of good k=0. This clear

distinction between means to transfer wealth over time and means to consume is taken from Lucas (1978). Hens

and Schenk-Hoppe (2004) show that this assumption is the evolutionary perspective: If the

consumption good were non-perishable and hence could al fer wealth over time then every

strategy that tries to save using the consumption good will be driv of the market by the strategy that does

not use the consumption good to transfer wealth over time but oth e uses the same investment strategy. As

in the traditional model, we use the consumption good as the n eraire of the system. Note that one of the

long-lived assets could be a bond paying a risk free dividend. Bonds may however be risky in terms of their resale

values, i.e. in terms of the prices .

The evolutionary portfolio model we propose studies the evolution of wealth over time as a random dynamical

stem (RDS). A dynamical system is an iterative mapping on a compact set that describes how a particle (in our

tive wealth among strategies) moves from one position to the next. In a random dynamical system

ks. An example of a random dynamical system is

essential when taking

so be used to trans

en out

erwis

um

11( )k t

tq ω ++

sy

model the rela

this movement is not deterministic but subject to random shoc

the famous Brownian motion describing the stochastic movement of a particle in a fluid. Note that in contrast to

traditional finance we assume that in order to “predict” the next position of the particle one is not allowed to

know the realizations of future data of the system. I.e. we do not allow for perfect foresight of prices! Yet it may


happen that under certain nice conditions the limit position of the RDS could also be described “as if” agents

were maximizing expected utility under perfect foresight. In particular some of our results give an e y

justification of the “rational benchmark-theorem” according to which i in proportion to expected relative

dividends is the unique perfect foresight equilibrium of some traditional economy with logarithmic expected

utility functions.

As in the traditional model, we start from the fundamenta uation ealth dynamics:

volutionar

nvesting

l eq of w

1 11 ,1 1

11

( ) ( )( ) ( ) ( )( )

k t k tKi t i k t i tt tt tk t

k t

D qw wq

ω ωtω λ ω ω

ω

+ ++ + +

+=

⎧ ⎫⎡ ⎤+⎪ ⎪= ⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭∑

From one period to the next the wealth of any strategy i is multiplied with the gross return it has generated by its

portfolio strategy , ( )i k ttλ ω

In every period asset price

executed in the previous period50. Returns come from two sources, dividends and

capital gains. s are determined by the equilibrium between demand and supply within

that period. Since was meant to denote the total dividends of asset k, we have normalized the

supply to one, brium prices are given by:

11( )k t

tD ω ++

and as before equili,

1

( ) ( ) ( )i kI

k t t i tt t

i t

q wω λ ω ω=

=∑ . Note that wealth,

an

of we relative

dividends and prices may all be subject to some growth rate like the rate at which nominal GDP is growing.

However, for alyzing what is the best way of splitting your wealth among the long-lived assets we can restrict

attention to relative wealth, relative dividends and relative prices getting rid of the absolute growth rates.

The fundamental equation alth evolution written in terms is given by: 1

1 0 1 ,1 1 11 1

1

ˆ( ) ( )( ) ( ) ( ) ( )ˆ ( )

k k tKi t t i k t it t t

t t t tk tk t

d qr rq

ω ω tω λ ω λ ω ωω

++ + + + +

+ +=

⎧ ⎫⎡ ⎤+⎪ ⎪= ⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭∑ ,

where ( )ˆ ( )

( )

k tk t tt i t

ti

Pqwωωω

=∑

,1

11 1

1 1

( )( )( )

k tk tt t j

t tj

DdD

ωωω

++

+ ++ +

=∑

,( )( )

( )

i ti t t

t i tt

i

wrwωωω

=∑

.

Here we did use the identity 0

( )( )

( )

k tt

i t kt t

i t

Dw

ωω

λ ω=∑

∑ which determines aggregate wealth by the ratio of

inflows (aggregate dividends) to outflows of wealth (the consumption rate). Details of the computation can be

found on one of the ppt-slides of this chapter. In deriving the fundamental equation of wealth evolution written

in relative terms we did however make one important assumption: All strategies have the same consumption

rate 0( )ttλ ω . The justification of this assumption is that we are searching for the best allocation of wealth

among the long-lived assets. It is clear that among two strategies with otherwise equal allocation of wealth

50 Note that up to now we did not make any assumption on how the portfolio strategy , ( )i k ttλ ω executed at tω is

determined!


among long-lived assets the one with a smaller consumption rate will eventually dominate. Written I relative

terms the asset pricing equation becomes really nice now. The relative asset prices is simply the convex

combination of the strategies in the market: ,

1

ˆ ( ) ( ) ( )i kI

k t t i tt t

i t

q rω λ ω ω=

=∑ . In terms of evolutionary game theory

this means that strategies are “playing t on any other strategy only via

e flow of wealth is not

already described by a dynamical system. We would like to have a mapping from the relative wealth in one

perio t

the field” i.e. one strategy has an impac

the average of the strategies. Careful reading of the wealth flow equation reveals that th

d 1( ) ( ( ),..., ( ))t t It t tr r rω ω ω=

d 1 1 1

to the relative wealth in the next

perio t+ . But 11 1 1( ) ( ( ),..., ( ))t t I

t t tr r rω ω ω+ ++ + += 1

1( )i ttr ω ++ also enters on the right hand side because capital

e

t

gy tλ ω+

gains do depend on the strategies played. Fortunately the dependence of capital gains of strategies wealth is

linear so that we can solve the wealth flow equation in th RDS-form. In the resulting equation, the inverse

matrix captures the capital gains. Note that the KxI matrix 1( )tω ++Λ is the matrix of portfolio strategies in

which portfolio strate 1t+ is one column.

Note that this equation is a first order stochastic difference equation describing a mapping from the simplex

1

1( )i

∆

into itself.

Before we start analyzing this RDS let us summarize the assumptions made so far. We used Lucas (1978)

distinction between means to transfer wealth over time and means to consume, we assumed that all strategies

have the same consumption rate and by writing the Lucas model as a RDS we assumed that strategies are not

allow e information that is not available at the time when executed. Every of these assumptions seems

well justified to us. Note that in contrast to many other economic models generating dynamics we did not make

ed to us

any simplifying assumptions like linear demand functions, usually justified by first order Taylor approximations.

One has to be very careful when making these seemingly innocuous assumptions. When iterating a dynamical

stem loo ere to the

the system based on the simplifying assumptions.

system terms of higher order may accumulate so that the real dynamics of the sy ks quite diff nt

dynamics of


5 A SIMULATION ANALYSIS

Since the model has been formulated as a dy ated period by period. For the sake of

simplicity, in this section we assume that all strategies are constant over time fix-mix strategies: ( )t

To get some first insight into the behavior of the RDS developed above it is useful to do a simulation analysis.

namical system it can be iterii tλ ω λ= .

We have seen in an earlier chapter that on an i.i.d. return process those strategies are the optimal portfolio rules

of expected utility maximizers with CRRA. Differences in the asset allocation could be attributed to differences in

beliefs and ri wing figure displays the exogenous dividends process that we use in our sk aversions. The follo

simulations:

These are the total dividends ( )k ttD ω , paid out from 1981 to 2001 by the stocks in the DJIA. First one observes

that there is a growth rate in dividends. On a logarithmic scale the dividends all assets` ( )k ttD ω are increasing

with linear rate over time. We take the dividends from 1981 to 2001 to gene

exogenous dividends of our model. To do so, we identify each year with one state

rate an i.i.d. process for the

1,2,...,21t tω ∈Ω = and

then randomly choose one of the states every period. “Randomly” meaning that in every period we select one of

the states from the same probability distribution with equally likely states. Thin of this process as being k

generated by repeatedly rolling a fair dice with 21 sides. Even though the exogenous random process is i.i.d. the

return process may however not be i.i.d. because in the returns the prices also enter and those are generated

endogenously. The next figure shows a typical run of a simulation with two strategies, a strategy generated from

mean-variance analysis (blue line) and the naïve diversification rule of fixing equal weights in the portfolio (red

line). Even though initially the wealth of the mean-variance rule accounts for 90% of the market wealth after a

few iterations the situation has reversed and the 1/n rule has 90% of the market wealth.

10

11

12

13

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

years

log

of d

ivid

end

14

15

16

s


.

This wealth dynamics is reflected in asset prices:

Asset prices initially reflect the mean-variance rule but rapidly converge to the 1/n rule. A careful reader may

and ret

lt in a competition with the

tations

wonder how we did compute the mean-variance rule. After all this is a return based rule urns are

endogenous. In our simulation we have even given the mean-variance rule perfect foresight and rational

expectations. I.e. we allowed it to know in advance which prices will ultimately resu

naïve diversification rule51. Note that the mean-variance principle is a mapping from expected returns to asset

allocations. Hence, one can also flip the mean-variance principle upside down and ask which return expec

one would need in order to generate a portfolio rule that is doing fine in competition with 1/n and the expected

relative dividends rule that is known to be the unique competitive equilibrium with perfect foresight.

51 We will later present analytical results showing that indeed 1/n will always win against mean-variance.


..

The figure shows that these expectations are far off from the observed returns when mean-variance competes

with 1/n. That is to say, adding some degree of sophistication to the mean-variance rule by adding learning rules

will also not help in competition with 1/n. Summarizing, the first simulation result shows that seemingly rational

portfolio rules like mean-variance can do quite poorly against seemingly irrational rules like 1/n – a result that

was first pointed at by DeLong, Shleifer, Summers and Waldman (1990).

The next simulations show that the fight for market capital, i.e. the endogenous changes in the wealth

distribution can ge

nerate endogenous uncertainty that exceeds the uncertainty given by the exogenous dividends

process. As an effect, asset prices shows “excess volatility” which was first pointed at by Shiller (1981). Consider

a situation with two strategies, the 1/n strategy and a mean-risk strategy in which the risk measure now is the

CVaR, explained above.

Wealth

0.4

0.5

0.6

0.7

0

0.1

0.2

0.3

1 45 89 133 177 221 265 309 353 397 441 485 529 573 617 661 705 749 793 837 881 925 969

Period

Mean-VarianceMean-CVaRExpected DividendsGrowth OptimalEqual WeightsProspect TheoryCAPM

MV return expectations to obtain...

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1lambda star portfolio1/n portfoliotrue returns at 1/n


In this setting the wealth evolution never settles. As the 1/n rule gains wealth it turns prices to its disadvantage

so that the CVaR can grow. But the same happens to the CVaR rule. As it grows it turns the prices to the

advantage of the 1/n rule. As an effect asset prices show volatility in excess of the volatility of the exogenous

dividends: The blue line above shows the prices of some asset the red line below shows the fluctuations in its

dividends.

situation in which all the market wealth is concentrated at the strategy λ*. The next figure shows a typical run of

prices and one based on λ* prices), the growth optimal rule maximizing the expected

logarithm of returns based on1/n prices, the equal weights or illusionary diversification portfolio 1/n, a portfolio

based on prospect theory. In the course we made available the simulation program with which the competition

of any set of simple strategies can be studied. We recommend running simulations on your own in order to get

some intuition for the market selection process. Our conjecture from these simulations is: Starting from any initial

distribution of wealth, on P-almost52 all paths the market selection process converges to λ*, if the dividend

process d is i.i.d..

We conclude our simulation analysis by including the expected relative dividends portfolio *, , 1,2,...,k k

PE d k Kλ = = in the market selection process. As a result the process always converges to the

01 16 31 46 61 76 91 106 121 136 151 166 181 196 211 226 241 256 271 286 301 316 331 346 361 376 391 406 421 436 451 466 481 496

0.05

0.1

0.15

0.2

0.25

0.3

the simulation. The final figure of this section shows the average run with a 1 standard-deviation band for each

strategy. Note that these bands do not widen but get tighter as time goes on, indicating that the process

converges. In these figures we did display the expected dividends rule λ* in competition with two mean-variance

rules (one based on 1/n

52 That is to say on allrobability measure P. For example if P- is i.i.d, every infinite sequence in which some state is not visited infinitely often has easure zero.

paths except for those that are highly unlikely, i.e. those that have measure zero according to the pm


In this section we explain the main analytical results that we were able to derive so far. These results are derived

Definition: ]

is called the single survivor of the market selection process if starting from any initial distribution of

wealth the process converges P-almost surely to

6 THE MAIN RESULTS

from three types of structural assumptions: Assumptions on the maturity of assets (short-lived or long-lived),

assumptions on the dividend process (i.i.d or stationary Markovian) and assumptions on the set of strategies

(simple or stationary adapted). Under these assumptions we derive two properties of the expected relative

dividends portfolio λ*: The single survivor property and the evolutionary stability property. Here is a precise

definition:

[Single-Survivor-Property

The strategy iIr∈∆ I

ir e∞ = ∈∆ , the i-th corner of the simplex.

So far we have been able to show the single survivor hypothesis of λ* only for the case of short-lived assets. In

this case there are no capital gains and the wealth evolution reduces to:

Mean +/- standard deviation

0

0.1

0.3

0.4

0.5

0.6

0.7

ealth

sha

re

0.2

0 20 40 60 80 100

Period

W

1 stern2 illu3 (m-s)(1)4 (m-s)(stern)5 gop (1)6 cpt7 (m-s)e (1)8 (m-s)e (stern)

SAMPLE RUN

0.5

0.6

0

0.1

0.2

0.3

0.4

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101

Period

Mean-Variance (1)Expected DividendsGrowth Optimal (1)Equal WeightsProspect TheoryMean-Variance (star)CAPM


1 ,1 11 0

1

( )( ) ( ) ( )

( ) ( )

kKi t i k t i tt t

t tk t i tk t t

i

dr r

rω

tω λ λλ ω ω

+ + ++

=

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥= ⎨ ⎬⎢ ⎥⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭

∑ ∑ω ω . The first result we get is:

Theorem: [Evstigneev, Hens, Schenk-Hoppé (2003)]

Suppose assets are short-lived and non-redundant, relative dividends d are i.i.d and consider λ simple (constant

in time). Then )

*

0 ((1 )k kpE d ωλ λ= − , k=1,..,K is the single survivor of the market selection process.

The assumption of simple strategies can be dispensed with if some regularity conditions on the process are

made. One needs to make sure that P-almost surely the relative dividends process is distinct from the simply

strategy of buying and holding the index (the CAPM-strategy, as we call it). In this case one can consider any

stationary and adapted portfolio rules. That is strategies that do not depend on time in an autonomous way and

that only depend on the paths tω generated by the time-uncertainty process – an assumption that is well know

from the literature on hedging derivatives. It does not allow to use information that is unknown at the time the

strategy is executed.

Theorem: [Amir, Evstigneev, Hens, Schenk-Hoppé (2004)]

Suppose assets are short-lived and non-redundant, relative dividends d are i.i.d and consider any λ stationary

and adapted to ω. Then )

*

0 ((1 )k kpE d ωλ λ= − , k=1,..,K is the single survivor of the market selection process,

provided it is sufficiently distinct from the CAPM-strategy.

The case of short-lived assets is however not really realistic. One may say that investing on assets is mainly done

a ulation) rather than because of dividends. Indeed the dividend yield on

k averages) is much smaller than return generated from capital gains (about 6-

: [Fix Point]

bec use of potential capital gains (spec

stoc s (about 3-4% on long term

7% on long-term averages). In the case of long-lived assets we cannot show the single survivor property of λ*.

However, we can show that λ* is the unique portfolio strategy that cannot be invaded by other strategies. We

call this property evolutionary stability. Before we do so we need however define a point of rest of the dynamical

system that can be perturbed by mutations.

Definition

A wealth distribution Ir ∈∆ is a fixed point of the random dynamical system ( , )F r ω if

( , ) for all F r rω ω= ∈Ω .

ince th describes a market selection process, eve phic population is a fix point. ThaS e RDS F ry monomor t is to say,

if all strategies are identical then the evolution of wealth does not change over time. A slightly deeper result is

two strategies differ then, since assets

that in the case of non-redundant assets the converse of this statement is also true: Every fix point of F is a

monomorphic population. The intuition of the proof is as follows. Suppose


are not redundant, there is a continuation path along which the wealth distribution changes, contradicting the

notion of a fix point.

Proposition: [Fix Points are Monomorphic Populations]

reover, if there are no-redundant assets then every fix point

Now we are in the position to define a stability notion of fix points that concerns the thought experiment of

t ith little initial wea fix points are

Every monomorphic population is a fix point of F. Moof F is a monomorphic population.

hrowing in w lth a new strategy into a market that is a fix point. Since

monomorphic we can thus restrict attention to markets in which only one strategy is the incumbent.

Definition: [Evolutionary Stability]

A trading strategy λi is called evolutionary stable if for all λj, starting from ,0 (1 , ) , for all 0< 1,i jr ε ε ε= − ∈∆ <

the market selection process co erges P-almost surely to ,(1,0) i jr∞nv = ∈∆ .

When an incumbent i jstrategy λ is evolutionary stable only with respect to mutant strategies λ that are local

s are close to each other, then we call the strategy λi

Theorem: [Evstigneev, Hens, Schenk-Hoppe (2003)]

S consider any λ stationary and adapted. Then λ*is

mutations of λi , i.e. in the strategy simplex the two strategie

locally evolutionary stable. The main result on evolutionary stability is:

uppose relative dividends d follow an i.i.d. process and

evolutionary stable. No other strategy is locally evolutionary stable.

The key to understand this result is given by the exponential growth rate of a strategy λi in a market determined

by strategy λj: ,

0 0 ( ) ,( , ) ln 1i k

i n kp n kg E d ω

λλ λ λ λλ

⎛= − +⎜ ⎟⎟⎜ ∑ . If some strategy has a higher exponential growth rate

than some other strategy then as the process unfolds, P-almost surely this strategy will have higher wealth. The

unique property of the strategy λ* is that it has, as shown in one of the ppt-slides, the highest growth rate

1

K

k =

⎞

⎠⎝

against itself. The following matrix, in which columns (rows) correspond to incumbents (mutants) summarizes the

t: If λ* is determining market prices then no other strategy can invade this market. If

some other strategy is the incumbent then there is always a potential invading strategy, which might not be λ*

itself (see the table below). The theorem can be generalized to relative dividends that are stationary Markov. The

evolutionary stability resul

expected relative dividends strategy is also the unique evolutionary stable strategy in this case: Let

1,..., , 1,2,...t S tΩ = = and let

11 1

1

S

ss

SS S

p pp p

p p

′

′

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥

be the Markov transition matrix then

⎣ ⎦


*0 0

1

(1 ) is the expected discounted dividends strategy n n

n

p dλ λ λ∞

=

= −∑1 11 1 1 1

* k k

d dλ λ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

1 1s s

K KS Sd dλ λ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

where and .

K K

S S

d dλ λ= =⎢ ⎥ ⎢ ⎥

Hence, a rational market in which prices are determined by expected relative dividends is more robust than any

ons reflect

ven an evolutionary justification of this robust finding.

irrational market. Occasionally, a big push of irrationality can drive prices away from the rational valuation. The

resulting markets are however more fragile and even a sequence of local mutations can bring the market back to

the rational valuation. This is why in the data presented in chapter 3 we see that market capitalizati

relative dividends. Hence we have gi

From our main result we can derive two corollaries that give a nice re-interpretation of mean-variance analysis

and the CAPM. First, it is easily seen that every under-diversified strategy like the mean-variance portfolio is,

ve a positive growth rate against the under-

notation this means that That is to say the CAPM strategy is a passive imitation

ill always mimic the average strategy o

dominates the market, the CAPM-strategy will automatically follow it. As an effect the relative market wealth of

XTENSIONS

to the time-uncertainty structure. Imagine for example a simple trend chasing strategy that buys (sells) assets

assume that switching basic strategies depends on their relative returns, which being endogenous may not be

stationary and adapted. Finally, assuming that str implies a market

microstructure that at first sight looks a bit unrealistic. This assumption amounts to modeling the market as a

cannot be evolutionary stable. Leaving out some asset is fatal for survival. Any other strategy that devotes some

arbitrary small fraction of wealth to the left out assets will surely ha

diversified strategy. The CAPM strategy on the other hand amounts to buy and hold the market portfolio. In our ,

strategy that w f the market and if there is one such strategy that

the CAPM strategy stays constant over time.

7 E

The main restriction of the results mentioned above is that strategies are assumed to be stationary and adapted

when prices have increased (decreased). Since the endogenous price process may not be stationary, this strategy

is not necessarily stationary and adapted. Similarly, imagine strategies are build by switching between basic

strategies which may be the stationary adapted strategies in the theorems from above. It is reasonable to

ategies are stationary and adapted

ˆ , 1,2,..., .k CAPM kt tq k Kλ = =

Evolutionary Portfolio Theory: Survival of the Fittest on Wall Street

133

batch auction in which every strategy submits a demand function in terms of its asset allocation shares

( )i ttλ ω and each auctioneer chooses prices so that his market clears:

,

1

( ) ( ) ( )i kI

k t t i tt t

i t

q wω λ ω ω=

=∑owed to depend on the prices of tha

.The implicit

assumption here is that the shares in the asset allocation are neither all t

market nor on the prices of any other market. Also the units bought ,

,, ( )t i k t

tq( ) ( )( )

i k t i ti k t t twλ ω ωθ ω = can not be

ary strategies in two

directions: Trend Chasing and Imitation.

Consider the following aggressive momentum or trend chasing strategy:

Aggressive Momentum Strategy:

Partition assets in: IN and OUT assets.

Asset Allocation Rule: = 1/n for all IN assets and = 0 for all OUT assets.

Partitioning Rule:

ω

based on limit orders, for example. However, one should keep in mind that in principle the strategies can be

revised in every period. I.e. allowing for changes in strategies if the resulting demand of assets is satisfactory will

help to overcome the limitations of this simple market microstructure, which again brings us back to non-

stationary strategies. In this section we will extend the model to allow for non-station

If the price of asset k has increased twice in the previous three periods then k joins IN.

If the price of asset k has decreased three times in the last three periods then k joins OUT.

The following figure presents the wealth dynamics of the aggressive momentum rule in interaction with λ*

The expected relative dividends rule is no longer evolutionary stable. For very many periods (from period 350 to

500, for example) it has conquered almost the entire market and yet in period 520 or so a little outburst of


activity by the momentum strategy happens. However, any such outburst is not sustainable and ultimately the

rges back to λ*. Note that in contract to our interpretatio t in the

A second way of modelling non-stationary strategies is to consider strategies as determining the flow of wealth

between a set of base strategies. To get some intuition for this point of view, consider the Hedge-Fund-strategies

attributed to two sources: Internally generated wealth from earning returns on the strategy and externally

generated wealth from switches of wealth between strategies. The next two figures show the decomposition of

relative wealth flows in these two components:

process conve n of the evolutionary stability resul

previous section here the shifts away from the rational market are generated endogenously.

as mentioned above as the base strategies. Now the evolution of wealth between those strategies can be

Assets History Through Internal Growth (relative)December 1993 - June 2002

0.000

0.100

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002Q2

0.600

0.500

Convertible ArbitrageDedicated Short BiasManaged FuturesEmerging MarketsFixed Income ArbitrageEvent DrivenEquity Market NeutralGlobal MacroLong/Short EquityMulti-Strategy

0.400

0.300

0.200


The key modelling assumption is then to link the external flow of wealth to the internally generated returns.

Recent empirical papers on Hedge Funds have found surprisingly robust external flow functions, say

for the various strategies. The following figure from Agarwal, Daniel and Naik (2003)

illustrates such functions:

Basically the flows follow the returns very procyclical: Higher returns trigger higher flows. However there are

some non-linearities. For some strategies the flow functions are convex for others they are concave, as

Getmansky (2003) has figured out:

[ ] [ ]: 0,1 0,1iH →

Assets Growth History Through In/Outflows (relative)

-0.4

-0.2

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002Q2

December 1993 - June 2002

0.8

0

0.2

0.4

0.6

Convertible ArbitrageDedicated Short BiasManaged FuturesEmerging MarketsFixed Income ArbitrageEvent DrivenEquity Market NeutralGlobal MacroLong/Short EquityMulti-Strategy

Evolutionary Portfolio Theory: Survival of the Fittest on Wall Street

136

retical paper on this view is Alos-Ferrer and Ania (2004). These authors consider

lived assets and use simple strategies as the base strategies between investors allocate thei

to the realized returns of the last period. Alos-Ferrer and Ania (2004) show that the wealth

ows a stationary Markov process the limit distribution of which puts most we

relative expected dividends

A first theo the case of short-

r wealth

proportionally

evolution foll alth on the simple

strategy of ˆ , 1,2,.., .k

k Pj

Pj

the relative expected dividends and the expected relative dividends portfolio are not large, as the

E D k KE D

λ = =∑

On the DJIA data the differences between

next figure

shows:

Summarizing these results we see that evolutionary portfolio theory confirms what traditional finance has to

not need to refer to the shock metaphor traditional finance

is using to excuse itself from not being able to model the off-equilibrium behaviour. The usefulness of the

volutionary approach will depend on how good one can assess the ecology of the market. We have started

fruitful collaborations on this with banks in Zürich.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

UN

ITEDTE

CH

NO

LOG

IES

PRO

CTE

R & G

AMBLE

CO

.

PHILIP

MO

RR

IS C

O

MIN

NE

SOTA

MIN

ING

/MFG

ME

RC

K & CO

., INC

.

MC

DO

NALD

'S C

OR

J.P. M

OR

GA

N

INTER

NA

TION

AL

PAPE

R

INT'L BU

SIN

ESS

MAC

HS

HEW

LETT -PA

CKA

RD

CO

.

GE

NER

ALM

OTO

RS

CO

RP

GE

NER

AL E

LECTR

ICC

O.

EXXO

N M

OB

IL CO

RP

.

EASTM

AN KO

DA

KC

O.

DU

PO

NT D

EN

EMO

UR

S

CO

CA

- CO

LAC

OM

PAN

Y

CATER

PILLA

R IN

C.

BOEIN

G C

OM

PAN

Y(TH

E)

AT&T CO

RP

.

AMER

ICAN

EXP

RES

SC

O.

ALCO

A INC

.

0.2

lambda^hat lambda^star

ld us:

Value strategies work! The twist of the evolutionary perspective is however that value only works in the long run.

Moreover since the evolutionary model is describing the off-equilibrium dynamics of the financial markets it is

also able to make medium run predictions and does

e


REFERENCES:

Agarwal, V., N. Daniel, and N. Naik (2003): “Flows and Performance in the Hedge Fund Industry”. Mimeo.

Alos-Ferrer and Ania (2004): “The Asset Market Game”, forthcoming in Special issue on Evolutionary Finance of

Journal of Mathematical Economics, Hens and Schenk-Hoppe (eds).

Amir, R., Evstigneev I., Hens, Th., Schenk-Hoppé, K-R., (2004). “Market Selection and Survival of Investment

Strategies”, NCCR-working paper no. 6, forthcoming in Special issue on Evolutionary Finance of Journal of Mathematical Economics.

Evstigneev, I. V., Hens, Th. and Schenk-Hoppé, K. R., Evolutionary Stable Investment in Stock Markets, NCCR

Hens, Th., and Schenk-Hoppé, K.R., Markets Do Not Select For a Liquidity Preference as Behavior Towards Risk,

change Economy," Econometrica, 46, 1429-1445.

Shiller, R.J. (1981): “Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?”,

American Economic Review (71), 421-436.

UBS-Warburg (2002): “Watching Flows”, Global equity research December 2002, Johansen and Ineichen.

White, H. (2000): “A Reality Check for Data Snooping”, Econometrica Vol. 68, No. 5, pp. 1097-1126.

De Long J., Shleifer A., Summers L. and R. Waldman (1990):”Noise Trader Risk in Financial Markets”, Journal of

Political Economy, Vol 98, pp. 703-738.

Getmansky, M. (2003): “The Life Cycle of Hedge Funds: Fund Flows Size and Performance”, Mimeo MIT.

working paper no. 84, June 2003.

NCCR working paper no. 21, December 2002.

Lucas, R. (1978): “Asset Prices in an Ex

Exercises

Exercises

EXERCISE 1: MEAN VARIANCE AND CAPM

There are two risky assets and one riskfree asset paying 2% return p.a. Investors can buy the riskfree

asset but not to sell it. The expected return of risky assets are

1,2k =

1 5%µ = and 2 7.5%µ = . The variance-covariance

matrix (in %) is:

1. Calculate the minimum-variance and the tangent portfolio

2. Assume a mean-variance investor facing the same variance-covariance matrix chooses a portfolio

COV−⎛ ⎞

= ⎜ ⎟−⎝ ⎠

2 1

1 4.

λ = (0.2,0.5,0.3) . What are the returns he implicitly expect?

3. Assume that the market portfolio is given by Mλ = (0.4,0.6) . Calculate the Beta factors of the risky

assets. Assume further that the excess return of the market portfolio is 3%. What is the CAPM expected

excess return of the risky assets?

EXERCISE 2: DATA ANALYSIS

1. Calculate the mean and standard deviation of the monthly net returns of asset classes and annualize

them

2. Make histograms

3. Calculate the covariance matrix

4. Calculate the efficient frontier with and without the Tremont index

5. Calculate the tangent portfolio with and without short sales constraints

6. Analyze the sensitivity of the tangent portfolio and its expected return

7. Show the SML based on the tangent portfolio without short sales

EXERCISE 3: THE GENERAL EQUILIBRIUM MODEL

Consider two-period financial economy without consumption in the first period. There are s=1,…,S states in the

second period, k=1,…,K assets and i=1,…,I consumers.

1. Define the financial equilibrium in this economy.

2. Assume that there are 3 states and 2 consumers such that

( ) ( )1 11 2 3 1 2, , ln ln , 0,1,2U x x x x x ω= + =

( ) ( )2 21 2 3 2 3 2,1,0x, , ln ln ,U x x x x ω= + =

*1 *2

2, 1/2, 3/2 , 0, 3/2, 1/2x x= = and ) is equilibrium.

Show that for the assets matrix

1 0

0 1

0 1

A⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

( ) ( ) ( ) (*1 *2

2, 1/2 , 2, 1/2θ θ= − = −

( )3 0,0,1A =3. Consider a third asset with the payoff . Is it possible to duplicate this asset from the assets

estions? in the previous qu

et? Is it unique

5.

EXERCISE 4: MARKET COMPLETENESS

nomy. There are 3 possible states and 2 assets with a payoff

A ⎜ ⎟= ⎜ ⎟⎜ ⎟

.

EXE

R and one risky share which price

increases by the factor and decreases by the factor in each period where . Use the

4. Calculate an arbitrage-free price for the third ass ?

Determine the equilibrium in the economy including the third asset.

Consider a two-period financial eco

1 0⎛ ⎞2 1

1 0⎝ ⎠

1. Is the market complete?

2. Assume that there is a third asset: a riskless bond. Is the market complete in this case?

RCISE 5: ASSET PRICING

1. Calculate the linear price rule using the risk-neutral probabilities

2. Consider a Binomial Model with a riskfree asset paying return

u d 0u R d> > >binomial tree over three periods 0,1,2t = to show how do assets values change over time.

3. Assume that there is a third asset: a hedge fund paying the highest payoff achieved over the time. Show

this payoff of the hedge fund in each of the states in every period.

4. Calculate the value of the hedge fund in 0t = using the Law of One Price

nt values in each period . Is

there a portfolio of assets that is able to duplicate the payoff of the hedge fund?

0u m d> > >5. Assume that the share can take three instead of two differe

EXERCISE 6: THE REPRESENTATIVE AGENT

Consider a one period economy (t=0,1) with two possible states in the second period (s=1,2). Assume that

consum There are t agents i=1,2 having the logarithmic expected utilities

u1(c1,c2)= 0.75 ln (c1) + 0.25 ln (c2) and u2(c1,c2)= 0.25 ln (c1) + 0.75 ln (c2), respectively. There are two

assets in unit supply: One riskless asset paying off 1 in both states and one risky asset paying off 2 in the first

state and 0.5 in the second state. The first (second) agent owns the first (second)

sources of income.

2. umer with logarithmic expected utility function whose demand could also

tate. Compute the new asset prices

using the representative consumer as determined in b).


This exercise is based on Based on US-DATA 1981-2001 (see homepage). Using DJIA Prices, DJIA-MV, DJIA-DIV,

GDP, Interest Rates:

s section regression of the SML: excess returns versus betas based on market

2. Do the time-series macro-finance regression:

ption only takes place in t=1. wo

asset. Asset are the only

1. Determine the competitive equilibrium.

Find a representative cons

generate the equilibrium prices found in a).

3. Suppose the payoff of the second asset increases to 3 in the first s

4. Compute the new equilibrium prices in the original economy with two agents

1. Do for every year the cros

capitalization.

1 t

1 against ( ),where and are the market capitalizak k k k k

t t t t tfq l D q q DR− +

1

tion and the total dividends of firm k,

and ss

t −

1-

t 11 1-

( ) c the likelihood ratio proce is l ,i.e. for u(c)= we

( )f tt

t

u cR

u c

α

αγ−−

∂=

∂1nst t

t1

1regress agai ( )k k k

t t tc

q D qα

−

⎛ ⎞+

cγ−⎜ ⎟

⎝ ⎠

EXERCIS

Consider the time-uncertai r coins. Suppose there are two assets

asso ff +1 es up H and asset 1 (2) pays

off

1.

E 8: EXPECTED UTILITY (1)

nty structure that is generated by tossing two fai

ciated with the two coins. Asset 1 (2) pays o if the first (second) coin com

-1 if the first (second) coin comes up T.

Draw the time uncertainty structure.

), (0,1), (0.5,0.5), (λ,1-λ), for any 0≤λ≤1, in a matrix with

ing four assets A,B,C, D with returns of 2 %, 4 %, and 6 % occuring with the probabilities

liste

A

B

C

D

1. Show that the ranking B>A>C>D is not compatible with expected utility.

investments A, B, C, D in a mean-standard-deviation-diagram

5. e order compatible with the expected utility hypothesis

chastic dominance concept

7. Find the men-variance optimal efficient portfolios and the optimal portfolio for investor with utility

8. How does the portfolio structure change with increasing income?

EXERCISE 10: EXPECTED UTILITY (3)

Consider an investor with utility

2. Display the payoffs of the strategies (1,0

states as rows and strategies as columns.

3. Show that every risk averse expected utility maximizer prefers the portfolio (0.5,0.5) to any other

portfolio.

EXERCISE 9: EXPECTED UTILITY (2)

Consider the follow

d below:

2 % 4 % 6 %

0.2 0 0.8

0 1 0

0.8 0 0.2

0.75 0.25 0

2. Represent the investments A, B, C, D in a probability triangle

3. Represent the

4. Show the indifference curves of an investor with the preferences B>A>C>D

Is this preferenc

6. Order the investments A, B, C, D using the sto

( , ) (10 ) for 10V µ σ µ σ σ= ⋅ − ≤

1. Draw the indifference curves in a mean-standard-deviation diagram

not change with the mean. Does the investor above have a

constant absolute risk aversion?

3. Show that this investor would prefer a sure payment of 10 units to a lottery paying either10 or 20 with

2. Consider a mean-variance investor with constant absolute risk aversion if for a given standard deviation

the slope of her indifference curve doe

4. rder stochastic dominance?

2. Test for Normality of the ln-returns.

3. Test aggregate dividends for normality of ln-returns.

4. For market totals, compare the variances of the ln-returns of dividends and market values.

E D ATI

You i

are rmine the gross return of the risky asset:

Suppose: R(1)=1.2 and R(2)=0.9 per period. Consider two expected utility maximizers, say A and B, with von

Neumann-Mo

the same probability

Is this preference consistent with the first-o

5. How would a risk-loving investor with a monotonic utility function decide between the sure payment and

the lottery?


1. Get the weekly returns for stocks listed on DJIA from the homepage.

5. Test for autocorrelation.

EXERCISE 12: TIM IVERSIFIC ON

r in tial wealth is $1000 which you can invest over two periods in a risk free asset and a risky asset. There

two states drawn with 50% probability in each period that dete

rgenstern utilities

1 1

1 1

2.25 ( ) if 1( ) and ( ) .

( ) if t t t t

t tt t t tt

u WW W W WW

− −

− −

= = ⎨ − ≥⎩ A B W W W W

u W− ⋅ − ≤⎧−

1. Compute the optimal share of risky assets 0 ≤ λ ≤1 for both utility functions when the investors are

2. Compute the optimal share of risky assets λ fixed at the outset for both utility functions when the

ut do not want to rebalance after the first period.

icy rule for shares of risky assets over time for both utility functions when the

eriod.

EXERC

Daniel g it cards and two purses. The probability to loose one of the purses is

25%. Assume that the probability to loose one of the purses is independent from the probability of loosing the

myopic, i.e. plan ahead for one period only. Do these shares depend on the realization of the states?

investor plan ahead for two periods b

3. Compute the optimal pol λ

investors plan ahead for two periods but do allow for rebalancing after the first p

ISE 13: BEHAVIORAL PORTFOLIO THEORY

oes on holiday. He has two cred

other one. The monetary loss of a credit card is -1 (the monetary loss of none of the cards is 0, respectively the

mon

Dan

the sam

1. Which strategy would Daniel choose if he maximizes his expected utility?

pect theory (assume that Daniel

3. Which strategy would Harry Markowitz choose as a mean-variance decision maker?

EXE

For the Data of the stock in the DJIA given on the web-page,

1. Compute the Variance Ratios:

etary loss of both cards is -2).

iel has two strategies: diversification (each card is in a different purse) and concentration (both cards are in

e purse).

2. Which strategy would Daniel choose if his decision is based on the pros

does not change the probability weights)?

RCISE 14: TESTING FOR MEAN REVERSION

( )( )( )

=n

t

t

VAR rVAR nnVAR r

2. Which stocks are mean-reverting, mean-averting or random walks?

3. Also compute the Variance Ratio for the DJIA-index.

EXERCISE 15: INTERTEMPORAL ASSET ALLOCATION

Consider an investment problem over two periods. The initial wealth is W=1. There are two possible states of

the world in each period being determined by a fair coin, tossed in each period (i.i.d.). There are two assets: a

riskless asset with gross return of 1 and a risky asset that has a gross return of 3 if the coin tosses up Heads and

of 0 h

period u

2.

EXERCIS

Take the

hout and with Hedge Funds included.

if t e coin tosses up Tails. The investor has an intertemporal expected utility without discounting and his per

tility is quadratic: u(w)=(w-1/3w2), for b>0 but sufficiently small to ensure monotonicity of u.

1. Determine the optimal asset allocation over time.

Is the optimal asset allocation independent of time?

E 16: HEDGE FUNDS

Hedge Fund and Index Data from the course web page and

1. Compute the Mean-Variance Efficient Frontier wit

2. Test the Indices and the HF-styles for Normality and Mean-Reversion.

3. Compute the 95% and the 99% VaR of the HF-styles.

4. Which risks may be hidden in Hedge Funds; i.e. which risks may not be covered by the Data on Hedge

Fund indices?

Supplement:

Statistical Properties of

Time-Series

Appendix Stochastic Processes - 147 - 147

true theories not only to account for but to predict phenomena.”

William Whewell

1 Introduction

ial economics as a discipline is based on uncertainty and its impact on the behavior of

are intimately related to this

the uncertainty factor from a

ground necessary to get an

intuition on the main empirical approaches used in testing financial models.

s

Why do financial economists formulate models using returns instead of prices? First, to the average investor,

estments in financial markets can be considered as a having constant returns-to-scale – the size of investment

es not have any impact on prices changes. Thus, in perfectly competitive financial markets, the return is a

ular, scale-free measure of investments. Second, returns have more attractive properties

n prices such as the stationary property of returns but not of prices.

• net return on the period to

“It is a test of

The very existence of financ

investors and ultimately on asset prices. Estimating and testing theoretical models

type of uncertainty on which those models are based. Previous chapters discussed

theoretical point of view. This supplement assists obtaining basic statistical back

2 Some Definition

inv

do

complete and, in partic

tha

1t − t : 1 1tt

t

PR

P−= −

• annualized return of returns over periods: k1/

1

1

( ) (1 ) 1k

k

t t jj

R k R−

−=

⎡ ⎤= + −⎢ ⎥⎣ ⎦∏ or

1

0

1( )

k

t tj

R k Rk

−

−=

j

⎡ ⎤≈ ⎢ ⎥

⎣ ⎦∑

• log returns: 11

log(1 ) log( )tt t t

t

Pr R p

P tp −−

≡ + = = − .

A random variable X is a real-valued function whose domain is the outcomes of an experiment. Intuitively, a

random variable assigns a number to an outcome of an experiment. Random variables come in two favors. A

discrete random variable is one that can take at most a countable number of possible values. A continuous random variable can take on any value over any interval or all of the real numbers, i.e. the set of possible values

is uncountable. Since the value of a random variable is determined by the outcome, we may assign probabilities

to the possible values of the random variable.

A probability function is a function that assigns to each value of a random variable the probability that the


va

fulue will be obtained. For a discrete random variable, we define the probability mass nction

In the case of a continuous random variable, we define the probability density function (pdf) with the

property that for any set B of real numbers: P X B f x dx∈ = ∫ .

The random variables are said to be independent, if for all a and b P X a Y b P X

: ( ) ( )p a P X a= = .

( )f x

( ) ( )B

, ( , ) ( ) ( )a P X b≤ ≤ = ≤

A stochastic process is a collection of random variables, i.e. for each

≤ .

t T∈ ( )X t is a random variable. If the

index t n is the state of the process at time t. The index set T can be countable

set and we have a tic process, or non-countable continuous set and we have a continuous-

3 The Normal Distribution

The random variable X is normally distributed with parameters mean

is interpreted as time, the

discrete-time stochas

( )X t

time stochastic process. In most cases, a stochastic variable has both a expected value term (drift term) and a

random term (volatility term). We can see the stochastic process forecasting for a random variable X, as a

forecasted value (E[X]) plus a forecasting error, where error follow some probability distribution. So:

X(t) = E[X(t)] + error(t).

µ and variance 2σ if the density of X is

2 2( ) / 2given by: 1

( )2

xf x e µ σ

πσ− −= . Although the distribution is completely defined by the first two moments,

the third (skewness) and the forth (kurtosis) moments are often used to test for normality.

The skewness or the normalized third moment of a random variable X is defined by:

3

3 0σ

=

( )( )

E XS X

µ−=

The kurtosis, or the normalized forth moment of X is defined by:

4( )E X µ−4( ) 3K X

σ= =

The nor l ewness equal to zero. Distributions skewed to

the left (right) have negative (positive) skewness.

ma distribution, as all other symmetric distributions, has sk


The normal distribution has kurtosis al to 3, in contrast, fat-tailed distributions with extra probability mass in

the tails have higher kurtosis.

4 The Lognormal Distribution

he concept of normally distributed returns suffers from at least two drawbacks. First, the normal distribution

support the entire real line, in c

equ

T

ontrast, the largest loss an investor can realize is the entire investment and no

more. The l normality. Second, if single-periods returns are normal

then returns over several periods cannot be normally distributed since they are the product of the single-periods

tive way to solve the last

ower bound of -1 is clearly violated by the

returns.

An alterna problem is to take the logarithmic returns and assume that they are normally

distributed it i ir N 2( , )µ σ∼ (simple gross returns are then lognormal distributed).

Under the lognormal model, the mean and variance of simple returns are given by:

2

2( )i

itE R eσ

µ += −1 and

2 22( ) ( 1)i iitVar R e eµ σ σ+= − , so that the fist problem is solved as well53.

5 The Martingale Model

A Martingale is a stochastic process satisfying the following condition:

E P P P P+ − = r equivalently

1 1( - | , ,...) 0t t t tE P P P P+ − = 1 1( | , ,...)t t t t o

If the price process is a Martingale, then the best estimate for the price in the next period is simply equal to this

period’s price conditioned on the history of the game. Alternatively, if the expected price changes at any stage

are zero conditioned on the history, the game is fair – it is not in favor to any player.

The Martingale property of asset prices was long considered to be a necessary condition for the efficiency of

capital markets, in which information contained in past prices is instantly and fully reflected in asset’s current

price. On an efficient market, the conditional expectation of future price changes, conditioned on the price

53 since (1 ) 0itR+ ≥ (1 ) exp( )it itR r+ =


history, must be zero. Furthermore, the more efficient the market, the more random is the sequence of price

changes. Thus, on efficient markets price changes must be completely random and unpredictable.

Though, the Martingale hypothesis places restrictions on expected returns but does not account for risk. Positive

expected prices changes do not contradict the fair game concept since they can be viewed as the reward

necessary to attract investors holding the asset and bear the associated risks. Once asset prices are properly

adjusted for risk, the martingale property holds. The transformation isolates the impact of time and state

preferences of investors on asset prices. The, the predictability of these preferences does not represent a violation

of the market efficiency.

The simplest version of a R

6. The Random Walk

andom Walk is given by:

21 and (0, )t t t tP P IIDµ ε ε σ−= + + ∼

As the Martingale, the Random Walk is also a fair game since the increments are independent.54 In fact, a

Random Walk is a Martingale with constant variance of the innovations.

Using the IID increments assumption (also called White Noise), we can calculate the conditional mean and

variance, conditional on some initial value at time 0: 0P

0 0( | )tE P P P mt= +

20( | )tVar P P tσ=

Thus, the Random Walk is a non-stationary55 since both its mean and variance are linear in time.

If we additionally assume that the increments are normally distributed, 2 (0, )t IID Nε σ∼ then the process is

equivalent t iener process or a Brownian Motion is derived from the

simple Rand by time series when time intervals become smaller and

A second version of a R ally distributed increments. This version

ple, processes where

o an arithmetic Brownian motion. W

om Walk, replacing the time sequence

approach zero.

andom Walk includes independent but not identic

is useful in particular by the explanation of time variation of volatility in financial assets returns.

The weakest form of a Random Walk is obtained by additionally relaxing independent assumption to include

processes with dependant but uncorrelated increments. For exam2 2( , ) 0 for all 0 but ( , ) 0 for some 0t t k t t kCov k Cov kε ε ε ε− −= ≠

≠ ≠ .

7. MARKET EFFICIENCY AND THE LAW OF ITERATED EXPECTATIONS

54 Independence requires that not only the increments but also any non-linear function of them are also uncorrelated, which is a much stronger requirement fort he fairness of the game than in the case of a Martingale.

ary if at least the mean and variance are constant over time. 55 Distribution is station


In an information efficient market, price changes cannot be forecasted if they are anticipated properly. If prices

adjust as rapidly as information become available then we would expect to see randomness in successive

rmation, th cast of the forecast one would make

if h ally:

transactions rather than great continuity, i.e. small movements in the same direction are very unlikely. On the

other hand, if prices are determined by discounted cash flows, one would expect that returns would become

deterministic. In fact, asset returns can be random even if security prices are determined by discounting future

cash flows. The key argument is the Law of Iterated Expectations.

To illustrate it, we define the information sets tI and tJ where t tI J⊂ . The Law of Iterated expectations says

that if one has limited info e best forecast he can make is the fore

e has superior information. Form [ ]( ) ( | ) | t t tI E E X J I= . In other words, if one has limited

information, he cannot predict the forecast erro would make if he would have superior information. Applied

implies that realized changes in asset prices as rational

expectations of some fundamental value

| E X

r he

to asset prices, the Law of Iterated Expectations *V (as discounted cash flow) are unforecastable given some

information set tI . Or if * *1 1 1( I ) resp. ( I )t t t t t t tP E V I E V P E V I E V+ + += = = = , the expectation in the price over

the next period is: * *1 1( ) ( I ) 0t t t t t tE P P E E V E V+ +− = − = . Thus, future prices changes are not forecastable as in

A Markov process is a stochastic process where all the values are drawn from a discrete set. In a first-order

Markov process only the most recent draw affects the distribution of the next one. Given the present, the future

is conditionally independent on the past.

Markov processes are usually defines by specifying the transition probabilities from one state to the state

ta bX

a Random Walk; future directions cannot be predicted on the base of past realizations.

8. MARKOV PROCESSES

next period. That is,

1 1 1Prob( is in A I , ,...) Prob( is in A I )t t t t tX X X X X+ − +=

For example, X 1t t ε+ = + + is a Markov process and for 0a = and 1b = it is also a Martingale.56

Mean reversion in contrast to a random walk describes the phenomenon that a variable appears to be pulled

The Mean Reversion Process is a log-normal diffusion process, but with the variance growing not proportionally

to t n value. The

stabilization of the variance is due the spring like effect of the mean-reversion. The pictures below illustrate this,

9. MEAN REVERSION

back to some long-run average level over time. Changes in the variable (growth rates, returns) must be

negatively serially correlated at some frequency for the correction to occur.

he time interval. The variance grows in the beginning and after sometime stabilizes on certai

56 1 1, 1, 1,( I , ,...) ( I , ,...) ( I , ,...) ( )t t t t t t t t t t t tE X X X E a bX X X a bE X X X E 2 for 0, 1, and (0, )t tX a b Nε ε+ − − −= + + = + + = ε σ= = ∼


for the "low" and “high” price cases:

Tobin's q

Documents

tboeingcaterpillarcitigroupcoca coladu pont

american expressat

event tree consisting

packardhome depothoneywell intl

oxford university press

expected utility maximizer

likelihood ratio process

tt ti tq