1 Today Risk and Return Portfolio Theory Capital Asset Pricing Model Reading Brealey, Myers, and Allen, Chapters 7 and 8.

1

Today

Risk and Return • Risk and Return • Portfolio Theory• Capital Asset Pricing Model

Reading• Brealey, Myers, and Allen, Chapters 7 and 8

2

Measuring Risk

• Variance - Average value of squared deviations from mean. A measure of volatility.

• Standard Deviation - Average value of squared deviations from mean. A measure of volatility.

• Variance measures ‘Total Risk’

3

Measuring Risk

• Unique Risk - Risk factors affecting only that firm. Also called “diversifiable risk.”

• Market Risk - Economy-wide sources of risk that affect the overall stock market. Also called “systematic risk.”

• Diversification - Strategy designed to reduce risk by spreading the portfolio across many investments.

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Measuring Risk

05 10 15

Number of Securities

Po

rtfo

lio

sta

nd

ard

dev

iati

on

Market risk

Uniquerisk

5

Portfolio Risk

22

22

211221

1221

211221

122121

21

σxσσρxx

σxx2Stock

σσρxx

σxxσx1Stock

2Stock 1Stock

The variance of a two stock portfolio is the sum of these four boxes

6

Portfolio Risk

Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The expected return on your portfolio is:

7

Portfolio Risk

2222

22

211221

2112212221

21

)3.27()40(.σx3.272.181

60.40.σσρxxCola-Coca

3.272.181

60.40.σσρxx)2.18()60(.σxMobil-Exxon

Cola-CocaMobil-Exxon

Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

8

Portfolio RiskExample

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

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Portfolio Return and Risk

)rx()r(x Return PortfolioExpected 2211

)σσρxx(2σxσxVariance Portfolio 21122122

22

21

21

10

Portfolio RiskThe shaded boxes contain variance terms; the remainder contain covariance terms.

1

2

3

4

5

6

N

1 2 3 4 5 6 N

STOCK

STOCKTo calculate portfolio variance add up the boxes

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Beta and Unique Risk

beta

Expected

return

Expectedmarketreturn

10%10%- +

-10%+10%

stock

Copyright 1996 by The McGraw-Hill Companies, Inc

-10%

1. Total risk = diversifiable risk + market risk2. Market risk is measured by beta, the sensitivity to market changes

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Beta and Unique Risk

Market Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the S&P Composite, is used to represent the market.

Beta - Sensitivity of a stock’s return to the return on the market portfolio.

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Beta and Unique Risk

2m

imi

Covariance with the market

Variance of the market

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Markowitz Portfolio Theory• Combining stocks into portfolios can reduce standard

deviation, below the level obtained from a simple weighted average calculation.

• Correlation coefficients make this possible.• The various weighted combinations of stocks that

create this standard deviations constitute the set of efficient portfoliosefficient portfolios.

15

Markowitz Portfolio Theory

Price changes vs. Normal distribution

Coca Cola - Daily % change 1987-2004

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

-9 -7 -5 -3 -1 0 2 4 6 7

Pro

port

ion

of D

ays

Daily % Change

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Markowitz Portfolio Theory

Standard Deviation VS. Expected Return

Investment A

0

2

4

6

8

10

12

14

16

18

20

-50 0 50

%

prob

abili

ty

% return

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Markowitz Portfolio Theory

Standard Deviation VS. Expected Return

Investment B

0

2

4

6

8

10

12

14

16

18

20

-50 0 50

%

prob

abili

ty

% return

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Markowitz Portfolio Theory

Exxon Mobil

Coca Cola

Standard Deviation

Expected Return (%)

40% in Coca Cola

Expected Returns and Standard Deviations vary given different weighted combinations of the stocks

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Efficient Frontier

Standard Deviation

Expected Return (%)

•Each half egg shell represents the possible weighted combinations for two stocks.

•The composite of all stock sets constitutes the efficient frontier

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Efficient Frontier

Standard Deviation

Expected Return (%)

•Lending or Borrowing at the risk free rate (rf) allows us to exist outside the

efficient frontier.

rf

Lending

BorrowingT

S

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Efficient Frontier

Example Correlation Coefficient = .4

Stocks % of Portfolio Avg Return

ABC Corp 28 60% 15%

Big Corp 42 40% 21%

Standard Deviation = weighted avg =

Standard Deviation = Portfolio =

Return = weighted avg = Portfolio =

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Efficient Frontier

Example Correlation Coefficient = .4

Stocks % of Portfolio Avg Return

ABC Corp 28 60% 15%

Big Corp 42 40% 21%

Standard Deviation = weighted avg =

Standard Deviation = Portfolio =

Return = weighted avg = Portfolio =

Let’s Add stock New Corp to the portfolio

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Efficient Frontier

Example Correlation Coefficient = .3

Stocks % of Portfolio Avg Return

Portfolio 28.1 50% 17.4%

New Corp 30 50% 19%

NEW Standard Deviation = weighted avg =

NEW Standard Deviation = Portfolio =

NEW Return = weighted avg = Portfolio =

NOTE: Higher return & Lower risk

How did we do that? DIVERSIFICATION

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Efficient Frontier

A

B

Return

Risk (measured as )

25

Efficient Frontier

A

B

Return

Risk

AB

26

Efficient Frontier

A

BN

Return

Risk

AB

27

Efficient Frontier

A

BN

Return

Risk

ABABN

28

Efficient Frontier

A

BN

Return

Risk

AB

Goal is to move up and left.

WHY?

ABN

29

Efficient Frontier

Return

Risk

Low Risk

High Return

High Risk

High Return

Low Risk

Low Return

High Risk

Low Return

30

Efficient Frontier

Return

Risk

Low Risk

High Return

High Risk

High Return

Low Risk

Low Return

High Risk

Low Return

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Efficient Frontier

Return

Risk

A

BNABABN

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Security Market LineReturn

Risk

.

rf

Risk Free

Return =

Efficient Portfolio

Market Return = rm

33

Security Market LineReturn

.

rf

Risk Free

Return =

Efficient Portfolio

Market Return = rm

BETA1.0

34

Security Market LineReturn

.

rf

Risk Free

Return =

BETA

Security Market Line (SML)

35

Security Market LineReturn

BETA

rf

1.0

SML

SML Equation = rf + β ( rm - rf )

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Capital Asset Pricing Model

R = rf + β ( rm - rf )

CAPM

37

Testing the CAPM

Avg Risk Premium 1931-2002

Portfolio Beta1.0

SML30

20

10

0

Investors

Market Portfolio

Beta vs. Average Risk Premium

38

Testing the CAPM

Avg Risk Premium 1931-65

Portfolio Beta1.0

SML

30

20

10

0

Investors

Market Portfolio

Beta vs. Average Risk Premium

39

Testing the CAPM

Avg Risk Premium 1966-2002

Portfolio Beta1.0

SML

30

20

10

0

Investors

Market Portfolio

Beta vs. Average Risk Premium

40

Return = a + bfactor1(rfactor1) + bf2(rf2) + …

Arbitrage Pricing Theory

Alternative to CAPMAlternative to CAPM

Expected Risk

Premium = r - rf

= Bfactor1(rfactor1 - rf) + Bf2(rf2 - rf) + …

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Arbitrage Pricing TheoryEstimated risk premiums for taking on risk factors

(1978-1990)

6.36Market

.83-Inflation

.49GNP Real

.59-rate Exchange

.61-rateInterest

5.10%spread Yield)(r

PremiumRisk EstimatedFactor

factor fr