An Alternative View of Risk and Return: The Arbitrage Pricing Theory Chapter 12 Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.

An Alternative View of Risk and Return: The Arbitrage Pricing Theory

Chapter 12

Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

12-2

Key Concepts and Skills Discuss the relative importance of systematic

and unsystematic risk in determining a portfolio’s return

Compare and contrast the CAPM and Arbitrage Pricing Theory

12-3

Chapter Outline12.1 Introduction12.2 Systematic Risk and Betas12.3 Portfolios and Factor Models12.4 Betas and Expected Returns12.5 The Capital Asset Pricing Model and the Arbitrage

Pricing Theory12.6 Empirical Approaches to Asset Pricing

12-4

Arbitrage Pricing Theory

Arbitrage arises if an investor can construct a zero investment portfolio with a sure profit.

Since no investment is required, an investor can create large positions to secure large levels of profit.

In efficient markets, profitable arbitrage opportunities will quickly disappear.

12-5

Total Risk Total risk = systematic risk + unsystematic risk The standard deviation of returns is a measure

of total risk. For well-diversified portfolios, unsystematic

risk is very small. Consequently, the total risk for a diversified

portfolio is essentially equivalent to the systematic risk.

12-6

Risk: Systematic and Unsystematic

Systematic Risk: m

Nonsystematic Risk:

n

2

Total risk

We can break down the total risk of holding a stock into two components: systematic risk and unsystematic risk:

risk icunsystemat theis

risk systematic theis

where

becomes

ε

m

εmRR

URR

12-7

12.2 Systematic Risk and Betas The beta coefficient, , tells us the response of the stock’s return

to a systematic risk. In the CAPM, measures the responsiveness of a security’s

return to a specific risk factor, the return on the market portfolio.

)(

)(2

,

M

Mii R

RRCov

• We shall now consider other types of systematic risk.

12-8

Systematic Risk and Betas For example, suppose we have identified three

systematic risks: inflation, GNP growth, and the dollar-euro spot exchange rate, S($,€).

Our model is:

risk icunsystemat theis

beta rate exchangespot theis

beta GNP theis

betainflation theis

ε

β

β

β

εFβFβFβRR

εmRR

S

GNP

I

SSGNPGNPII

12-9

Systematic Risk and Betas: Example

Suppose we have made the following estimates:1. I = -2.30

2. GNP = 1.50

3. S = 0.50

Finally, the firm was able to attract a “superstar” CEO, and this unanticipated development contributes 1% to the return.

εFβFβFβRR SSGNPGNPII

%1ε

%150.050.130.2 SGNPI FFFRR

12-10

Systematic Risk and Betas: Example

We must decide what surprises took place in the systematic factors.

If it were the case that the inflation rate was expected to be 3%, but in fact was 8% during the time period, then:

FI = Surprise in the inflation rate = actual – expected

= 8% – 3% = 5%

%150.050.130.2 SGNPI FFFRR

%150.050.1%530.2 SGNP FFRR

12-11

Systematic Risk and Betas: Example

If it were the case that the rate of GNP growth was expected to be 4%, but in fact was 1%, then:

FGNP = Surprise in the rate of GNP growth

= actual – expected = 1% – 4% = – 3%

%150.050.1%530.2 SGNP FFRR

%150.0%)3(50.1%530.2 SFRR

12-12

Systematic Risk and Betas: Example

If it were the case that the dollar-euro spot exchange rate, S($,€), was expected to increase by 10%, but in fact remained stable during the time period, then:

FS = Surprise in the exchange rate

= actual – expected = 0% – 10% = – 10%

%150.0%)3(50.1%530.2 SFRR

%1%)10(50.0%)3(50.1%530.2 RR

12-13

Systematic Risk and Betas: Example

Finally, if it were the case that the expected return on the stock was 8%, then:

%1%)10(50.0%)3(50.1%530.2 RR

%12

%1%)10(50.0%)3(50.1%530.2%8

R

R

%8R

12-14

12.3 Portfolios and Factor Models Now let us consider what happens to portfolios of stocks when each

of the stocks follows a one-factor model. We will create portfolios from a list of N stocks and will capture the

systematic risk with a 1-factor model. The ith stock in the list has return:

iiii εFβRR

12-15

Relationship Between the Return on the Common Factor & Excess Return

Excess return

The return on the factor F

i

iiii εFβRR

If we assume that there is no unsystematic risk, then i = 0.

12-16

Relationship Between the Return on the Common Factor & Excess Return

Excess return

The return on the factor F

If we assume that there is no unsystematic risk, then i = 0.

FβRR iii

12-17

Relationship Between the Return on the Common Factor & Excess Return

Excess return

The return on the factor F

Different securities will have different betas.

0.1Bβ

50.0Cβ

5.1Aβ

12-18

Portfolios and Diversification We know that the portfolio return is the weighted

average of the returns on the individual assets in the portfolio:

NNiiP RXRXRXRXR 2211

)(

)()( 22221111

NNNN

P

εFβRX

εFβRXεFβRXR

NNNNNN

P

εXFβXRX

εXFβXRXεXFβXRXR

222222111111

iiii εFβRR

12-19

Portfolios and DiversificationThe return on any portfolio is determined by three sets of parameters:

In a large portfolio, the third row of this equation disappears as the unsystematic risk is diversified away.

NNP RXRXRXR 2211

1. The weighted average of expected returns.

FβXβXβX NN )( 2211

2. The weighted average of the betas times the factor.

NN εXεXεX 2211

3. The weighted average of the unsystematic risks.

12-20

Portfolios and DiversificationSo the return on a diversified portfolio is determined by two sets of parameters:

1. The weighted average of expected returns.

2. The weighted average of the betas times the factor F.

FβXβXβX

RXRXRXR

NN

NNP

)( 2211

2211

In a large portfolio, the only source of uncertainty is the portfolio’s sensitivity to the factor.

12-21

12.4 Betas and Expected Returns

The return on a diversified portfolio is the sum of the expected return plus the sensitivity of the portfolio to the factor.

FβXβXRXRXR NNNNP )( 1111

FβRR PPP

NNP RXRXR 11

that Recall

NNP βXβXβ 11

and

PR Pβ

12-22

Relationship Between & Expected Return

If shareholders are ignoring unsystematic risk, only the systematic risk of a stock can be related to its expected return.

FβRR PPP

12-23

Relationship Between & Expected Return

Exp

ecte

d re

turn

FR

A B

C

D

SML

)( FPF RRβRR

12-24

12.5 The Capital Asset Pricing Model and the Arbitrage Pricing Theory APT applies to well diversified portfolios and

not necessarily to individual stocks. With APT it is possible for some individual

stocks to be mispriced - not lie on the SML. APT is more general in that it gets to an

expected return and beta relationship without the assumption of the market portfolio.

APT can be extended to multifactor models.

12-25

12.6 Empirical Approaches to Asset Pricing Both the CAPM and APT are risk-based models. Empirical methods are based less on theory and

more on looking for some regularities in the historical record.

Be aware that correlation does not imply causality. Related to empirical methods is the practice of

classifying portfolios by style, e.g., Value portfolio Growth portfolio

12-26

Quick Quiz Differentiate systematic risk from unsystematic risk. Which type is essentially eliminated with well diversified portfolios? Define arbitrage. Explain how the CAPM be considered a special case of Arbitrage Pricing Theory?