Borrowing in period 1 Intertemporal Trades. Intertemporal Trades Impatient preferencesPatient preferences.

1c

Borrowingin period 1

Intertemporal Trades

0I

21 +)+1( mmr

r 1

{ }210 ,= mmE

•

•

21 ccA ,

1C

2C

2m

2c

1m( )r

mm

+1+ 2

1

Intertemporal Trades

1C

2C

21 = CC

1C

2C

21 = CC

Impatient preferences Patient preferences

Optimal Holding Period for an Asset

$0

$50

$100

$150

$200

$250

$300

$350

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Time

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

22%

24%

tFV

tPV

Rate of return from holding asset

t*

FV(t) = 100 + 6t + 2t2 – 0.1t3

Asset Markets: Debt

Asset Markets: Debt

1/3/2007 1/3/2008 1/3/2009 1/3/20100

200

400

600

800

1000

1200

1400

1600

1800

2007Return: 3.67%Volatility: 16.02%

2008Return: - 38.52%Volatility: 41.10%

2009Return: 23.44%Volatility: 27.27%

Risky Assets: Equities

S&P 500

1/3/2007 1/3/2008 1/3/2009 1/3/20100.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

Malkiel Portfolio S&P 500

Risky Assets: Portfolios

2007Return: 3.67%Volatility: 16.02%

Return: 17.92%Volatility: 18.05%

2008Return: - 38.52%Volatility: 41.10%

Return: - 35.97%Volatility: 33.14%

2009Return: 23.44%Volatility: 27.27%

Return: 37.35%Volatility: 20.81%

S&P 500 and Malkiel Portfolio

Capital Asset Pricing Model

[ ]portfolio a ofreturn = Er

fr

xr

mr

m

fm rr

-

mx

Xm

fm

fx

rrrr

-

+=

[ ]security a ofreturn = Er

i

fr

mr

fm rr -

ifmfi rrrr -

1

Capital Market Line Security Market Line

( )( ) Beta,

m

mx

r

rr

var

,cov≡

Beta as a Measure of Relative Risk

2004 2005 2006 2007 2008 2009

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

SP500 VTI (Beta = 1.03) FMAGX (Beta = 1.25)FLSAX (Beta = 1.46) NCICX (Beta = 0.75)

Capital Asset Pricing Model

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50

5

10

15

20

25

Beta

Ret

urn

s

3-Year 5-Year 10-YearMutual Fund Name Symbol Beta Returns Beta Returns Beta ReturnsAmerican Century Heritage A ATHAX 1.44 20.50 1.17 19.26 0.96 8.42Fidelity Advisor Equity Growth T FAEGX 1.18 8.31 1.16 11.20 1.16 3.34Fidelity Magellan FMAGX 1.33 6.88 1.03 10.42 1.04 3.53Putnam International Growth & Income PNGAX 1.07 12.55 1.03 20.56 0.96 6.90Fidelity Diversified International FDIVX 1.08 14.57 1.02 22.18 0.96 10.85Templeton Growth A TEPLX 0.77 5.78 0.85 14.81 0.80 7.01Vanguard 500 Index VFINX 1.00 5.72 1.00 11.18 1.00 3.43Vanguard Total Stock Market Index VTSMX 1.04 6.19 1.04 12.27 1.01 3.89Vanguard PRIMECAP VPMCX 1.01 9.63 1.06 15.78 1.08 8.50Janis Growth & Income JAGIX 1.13 6.69 1.05 11.22 0.98 5.84Dreyfus Premier Balanced B PRBBX 0.98 4.05 0.90 6.59 0.87 1.43Dreyfus Founders Balanced A FRIDX 0.98 3.71 0.88 7.21

ri = 3 + 5bi

A theory that asset prices reflect all publicly available information about the value of an asset.

Strong Form: Asset prices reflect all information, public and private, and no one can earn excess returns

Semi-Strong Form: Asset prices adjust very rapidly to publicly available new information and in an unbiased fashion, such that no excess returns can be earned by trading on that information. Semi-strong-form efficiency implies that neither fundamental analysis nor technical analysis will be able to reliably produce excess returns.

Weak Form: Future asset prices cannot be predicted by analyzing price from the past. Excess returns can not be earned in the long run by using investment strategies based on historical share prices or other historical data.

Efficient Market Hypothesis

Expected Return of a Portfolio (2 investments):

Expected Variance of a Portfolio (2 investments):

E[rx]= x1E[r1] + x2E[r2] (x1 + x2 = 1)

s1,22 = x1

2s12 + x2

2s22 + 2x1x2s1,2

= x12s1

2 + x22s2

2 + 2x1x2r1,2s1s2

Diversification and Portfolio Theory

Diversification and Portfolio Theory

Portfolio Example

0.5 0.5

A B PortfolioState Prob. Return Return Return

1 0.2 -5.00% 15.00% 5.00%2 0.2 0.00% 10.00% 5.00%3 0.2 5.00% 5.00% 5.00%4 0.2 10.00% 0.00% 5.00%5 0.2 15.00% -5.00% 5.00%

Expected Return: 5.00% 5.00% 5.00%Variance: 0.63% 0.63% 0.00%Std. Deviation: 7.91% 7.91% 0.00%Covariance(A,B) -0.0050Correlation(A,B) -1.0000

E[rx]= x1E[r1] + x2E[r2]

s1,22 = x1

2s12 + x2

2s22 + 2x1x2s1,2

= x12s1

2 + x22s2

2 + 2x1x2r1,2s1s2

Weights:

Was the yen a negative beta asset in 2007 – 2008?

The blue line is FXY, an exchange-traded fund that tracks the yen. The red line is the S&P 500 index. Over the past year, the two time-series look like mirror images of each other. That is, holding yen seems to hedge U.S. stock-market risk.Source: http://gregmankiw.blogspot.com/ 29 May 2008

What does a negative beta asset look like?

Was the yen a negative beta asset in 2007 – 2008?

What does a negative beta asset look like?

2/12/2007 5/12/2007 8/12/2007 11/12/2007 2/12/2008 5/12/200860.00

70.00

80.00

90.00

100.00

110.00

120.00

130.00

140.00

FXY ^GSPC

Feb. 2007 to May 2008: FXY ^GSPC BLENDAverage Weekly Returns 0.19% 0.01% 0.10%Std. Dev. of Weekly Returns 1.60% 2.37% 0.92%

Annualized Returns 10.51% 0.33% 5.29%Annualized Volatility 11.56% 17.06% 6.62%

Was the yen a negative beta asset in 2007 – 2008?

What does a negative beta asset look like?

2/12/2

007

5/12/2

007

8/12/2

007

11/12/2

007

2/12/2

008

5/12/2

008

8/12/2

008

11/12/2

008

2/12/2

009

5/12/2

009

8/12/2

009

11/12/2

009

2/12/2

010

5/12/2

010

8/12/2

010

11/12/2

010

2/12/2

011

5/12/2

011

8/12/2

011

11/12/2

011

2/12/2

0120.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

160.00

FXY ^GSPC

Dealing With Risk: Diversification (Portfolio Theory)

Effect of Additional Investments / Assets on Diversification

Risk and Uncertainty: “Contingent Consumption Plans”

Purchase

Do notpurchase

Luckyday

Unluckyday

$100

$295

$95

Case 1:A person with an endowment of $100 is considering the purchase of a lottery ticket that costs $5. The winning ticket in the lottery gets $200. 40 tickets will be sold.

Case 2:A person with an endowment of $35,000 faces a 1% probability of losing $10,000. He is considering the purchase of full insurance against the loss for $100.

Purchase

Do notpurchase

Luckyday

Unluckyday

$35,000

$34,900

$34,900

Luckyday

Unluckyday $25,000

995$

900,34$

x

xE

0

900,34$

x

xE

Outcome A:

Outcome B:

31$

100$

x

xE

Pr(Lucky) = 0.025):

1

00 , bg CCE

•

•

BadC

GoodC

000,35$0 gC

KCC

C

gb

b

00

0 000,25$

( )KCC

C

gg

g

-=

900,34$=01

1

KKCC

C

bb

b

01

1 900,34$

11 , bg CCA

Risk and Uncertainty: “Contingent Consumption Plans”

Purchase

Do notpurchase

Luckyday

Unluckyday

$35,000

$34,900

$34,900

Luckyday

Unluckyday

$25,000

K = the “expected loss” ($10,000), and gK is the insurance premium.

1. Risk aversion is defined through peoples’ choices:

2. Non-linearity in the utility of wealth.

Given a choice between two options with equal expected values anddifferent standard deviations, a risk averse person will choose the optionwith the lower standard deviation:

Given a choice between two options with equal standard deviations anddifferent expected values, a risk-averse person will choose the optionwith the higher expected value:

212121 then , and ,XEXE If

212121 then ,XEXE and , If

Defining Risk Aversion

Risk Premium

$99,415

lB

Risk Premium

50099415992 ,,$ EUUU

Risk Aversion and the Marginal Utility of Money

$

Utility

$100,000$0

U1

lA

lC

U3

$50,000

000501 ,$UU 0001003 ,$UU

U($)

500992 ,EUU

$99,500

U2 lD

Modeling Different Risk Preferences

$

Utility

U($)

U($) U($)

Risk Aversi

on

Risk Seeking

Risk Neutra

l

Classification of Auctions

What is the nature of the good being auctioned?

What are the bidding rules?

Private values

Common value

English ascending bid

Dutch descending bid

Sealed bid

Vickrey second price

Evaluative Criteria for Auctions

Pareto Efficiency

Revenue or Profit Maximization

Does the auction design guarantee that the item will go to the bidder with the highest value?

Does the auction design guarantee the highest revenue (or profit) for the seller?

Types of Auctions and optimal bidding strategies

English (ascending bid)

Dutch (descending bid)

First-price, sealed bid

Second-price, sealed bid

n

Lvvb

bidders ofnumber

valuationpossiblelowest

bidder of valuationprivate

bid optimal

Where

*

n

L

v

b

vb

vb

Independent Private Values Auctions

n

Lvvb

Each bidder knows precisely how highly he/she values the item,and these values vary across all bidders.

Types of Auctions and optimal bidding strategies

Common (or Correlated) Values Auctions

The item being bid has an underlying objective value, but no bidder knows precisely what that value is.

Winning bids tend to come from those with the most optimistic estimates.

If estimate errors are randomly distributed around zero, then the winning bid will be greater than the true value of the item (the “winner’s curse”):

0

1

5 10 15

TrueValue

WinningBid

Distribution of bids: