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The Capital Asset Pricing Model

Global Financial Management

Campbell R. HarveyFuqua School of Business

Duke Universitycharvey@mail.duke.edu

http://www.duke.edu/~charvey

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Overview

Utility and risk aversion» Choosing efficient portfolios

Investing with a risk-free asset» Borrowing and lending» The markt portfolio» The Capital Market Line (CML)

The Capital Asset Pricing Model (CAPM)» The Security Market Line (SML)» Beta» Project analysis

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Efficient Portfolios with Multiple Assets

E[r]

s0

Asset 1

Asset 2Portfolios ofAsset 1 and Asset 2

Portfoliosof otherassets

EfficientFrontier

Minimum-VariancePortfolio

Investorsprefer

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Utility in Risk-Return Space

Indifference curves

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20.00%

25.00%tau=0.5, Ubar=6%

tau=0.5, Ubar=8%

tau=0.5, Ubar=10%

tau=0.25, Ubar=6%

tau=0.25, Ubar=8%

tau=0.5, Ubar=10%

Risk

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turn

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Individual Asset Allocations5.

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Risk

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urn

xy

Point x is the optimal portfolio for the less risk averse investor (red line)

Point y is the optimal portfolio for the more risk averse investor (black line)

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Introducing a Riskfree Asset

Suppose we introduce the opportunity to invest in a riskfree asset.» How does this alter investors’ portfolio choices?

The riskfree asset has a zero variance, and zero covariance with every other asset (or portfolio).» var(rf) = 0.

» cov(rf, rj) = 0 for all j. What is the expected return and variance of a portfolio consisting

of a fraction (1- )a of the riskfree asset and a of the risky asset (or portfolio)?

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Risk and Return with a Riskfree asset

Expected Return

Variance and Standard Deviation

Hence, the risk-return tradeoff is:

E r E r rP j f 1

Var rP P j P j 2 2 2

E r r E r rP fP

jj j

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Expected Return

Standard Deviation

rf

sj

E(rj)

Asset j (a=1)

The line represents all portfolios depending on a

Risk and Return with a Riskfree asset

Riskfree asset (a=0)

0

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Investing with Borrowing and Lending

StandardDeviation

sM0

ExpectedReturn

M

r f

E rM[ ]

a=0

a =2

a=1

a=0 5.

Lending Borrowing

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Optimal Investing With Borrowing and Lending

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 280.00%

5.00%

10.00%

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20.00%

25.00%

tau=0.5, Ubar=8%

tau=0.25, Ubar=6%

Portfolio

Risk

Ret

urn

X

Y

Y = optimal risk-return tradeoff for risk-averse investor

X = optimal risk-return tradeoff for risk-tolerant investor

rf=4%

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The Capital Market Line

ExpectedReturn

M

r f

E rm[ ]

StandardDeviation

IBM

SystematicRisk

DiversifiableRisk

E rIBM[ ] A

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The Capital Market Line The CML gives the tradeoff between risk and return for portfolios

consisting of the riskfree asset and the tangency portfolio M.» Portfolio M is the market portfolio.

The equation of the CML is:

The expected rate of return on a risky asset can be thought of as composed of two terms.» The return on a riskfree security, like U.S. Treasury bills;

compensating investors for the time value of money.» A risk premium to compensate investors for bearing risk.

E(r) = rf + Risk x [Market Price of Risk]

E r rE r r

p f pM f

M

( )( )

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Everybody holds the Market

Everybody holds the tangency portfolio M» If all hold the same portfolio, it must be the market!

Nobody can do better than holding the market» If another asset existed which offers a better return for the

same risk, buy that!Can’t be an equilibrium

Write the weight of asset j in the market portfolio as wj. Then we have:

» Simply use expressions for multi-asset case

E r w E r r r

Var r w w Cov r r

M j j fj

j Nf

M i j j ij

j N

i

i N

1

11,

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All Risk-Return Tradeoffs are Equal

Hence, if you increase the weight of asset j in your portfolio (relative to the market), » Then expected returns increase by:

» Then the riskiness of the portfolio increases by:

» Hence, the return/risk gain is:

» This must be the same for all assets– Why?

E r rj f

wCov r r Cov r ri j ii

Nj M, ,

1

E r r

Cov r rj f

j M

,

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All Assets are Equal

Suppose that for two assets A and B:

» Asset A offers a better return/risk ratio than asset B– Buy A, sell B– What if everybody does this?

» Hence, in equilibrium, all return/risk ratios must be equal for all assets

E r r

Cov r r

E r r

Cov r rA f

A M

B f

B M

, ,

E r r

Cov r r

E r r

Cov r rA f

A M

B f

B M

, ,

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The Capital Asset Pricing Model If the risk-return tradeoff is the same for all assets, than it is the

one of the market:

This gives the relationship between risk and expected return for individual stocks and portfolios.» This is called the Security Market Line.

where

E r r

Cov r r

E r r

Cov r r

E r r

Var rA f

A M

B f

B M

M f

M

, ,

E r r

Cov r r

Var rE r r r E r rA f

A M

MM f f A M f

,

AA M

M

Cov r r

Var r

,

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Capital Asset Pricing ModelA Graphical Illustration

ExpectedReturn

ExpectedMarketReturn

Risk freerate

0 0.5 1.0 Beta

Expectedmarket riskpremium

Expectedreturn =

Risk freerate +

Betafactor x

Expected marketrisk premium

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The Intuitive Argument For the CAPM

Everybody holds the same portfolio, hence the market. Portfolio-risk cannot be diversified. Investors demand a premium on non-diversifiable risk only,

hence portfolio or market risk. Beta measures the market risk, hence it is the correct measure

for non-diversifiable risk.

Conclusion:

In a market where investors can diversify by holding many assets in their portfolio, they demand a risk premium proportional to beta.

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The SML and mispriced stocks

Suppose for a particular stock:

Remember the definition of expected returns:

Then P0 falls, so that E(rj) increases until disequilibrium vanishes and the equation holds!

E r

E P D P

Pjj j j

j

1 1 0

0

E r r

Cov r r

Var rE r rj f

j M

MM f

,

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Stock j is overvalued at X: » price drops, » expected return rises.

At Y, stock j would be undervalued!» expected return falls» price increases

Expected Return

rf

E(rM)

b=1bj

X

Y

The SML and mispriced stocks

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The CML and SML

E(r)

E(r)

rf

M ME(rM)

1.0M

rf

CML

SML

IBM

IBM

IBM

E(rIBM)

IBM,M/M

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The Capital Asset Pricing Model

The appropriate measure of risk for an individual stock is its beta. Beta measures the stock’s sensitivity to market risk factors.

» The higher the beta, the more sensitive the stock is to market movements.

The average stock has a beta of 1.0. Portfolio betas are weighted averages of the betas for the individual

stocks in the portfolio. The market price of risk is [E(rM)-rf].

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Using Regression Analysis to Measure Betas

Rate of Return on the Market

Rate of Return on Stock A

x x

x

x

x

x

x

x

xx

x

x

x

x

x

Jan 1995

Slope = Beta

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Calculating the beta of BA

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-40

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-10

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Beta

Return on the market index

Return on BA

Beta is the slope of a regression line which best fitsthe scatter of monthly returns on the share and onthe market index.

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Betas of Selected Common Stocks

Stock Beta Stock Beta

AT&T 0.96 Ford Motor 1.03

Boston Ed. 0.49 Home Depot 1.34

BM Squibb 0.92 McDonalds 1.06

Delta Airlines 1.31 Microsoft 1.20

Digital Equip. 1.23 Nynex 0.77

Dow Chem. 1.05 Polaroid 0.96

Exxon 0.46 Tandem 1.73

Merck 1.11 UAL 1.84

Betas based on 5 years of monthly returns through mid-1993.

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Beta and Standard Deviation

Risk of a

Share (Variance)

Market risk Specific risk

of the shareof the share

Risk of a

portfolio

Market risk of

the portfolio

Specific risk of

the portfolio

Beta ofshare

Risk ofmarket

Beta ofPortfolio

Risk ofmarket

This is the majorelement of a share's risk

This is negligiblefor a diversified portfolio

= +

x

= +

x

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Testing the CAPMBlack, Jensen and Scholes

Fitted Line

TheoreticalLine

Beta

AverageMonthlyReturn

•

••

•

•

•

•

•

•

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Estimating the Expected Rate of Return on Equity

The SML gives us a way to estimate the expected (or required) rate of return on equity.

We need estimates of three things:» Riskfree interest rate, rf.

» Market price of risk, [E(rM)-rf].

» Beta for the stock,bj.

E r r E r rj f j M f

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Estimating the Expected Rate of Return on Equity

The riskfree rate can be estimated by the current yield on one-year Treasury bills.» As of early 1997, one-year Treasury bills were yielding about 5.0%.

The market price of risk can be estimated by looking at the historical difference between the return on stocks and the return on Treasury bills.» This difference has averaged about 8.6% since 1926.

The betas are estimated by regression analysis.

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Estimating the Expected Rate of Return on Equity

Stock E(r) Stock E(r)AT&T 13.3% Ford Motor 13.9%Boston Ed. 9.2% Home Depot 16.5%BM Squibb 12.9% McDonalds 14.1%Delta Airlines 16.3% Microsoft 15.3%Digital Equip. 15.6% Nynex 11.6%Dow Chem. 14.0% Polaroid 13.3%Exxon 9.0% Tandem 19.9%Merck 14.5% UAL 20.8%

E(r) = 5.0% + (8.6%)b

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Example of Portfolio Betas and Expected Returns

What is the beta and expected rate of return of an equally-weighted portfolio consisting of Exxon and Polaroid?

Portfolio Beta

Expected Rate of Return

How would you construct a portfolio with the same beta and expected return, but with the lowest possible standard deviation?

Use the figure on the following page to locate the equally-weighted portfolio of Exxon and Polaroid. Also locate the minimum variance portfolio with the same expected return.

p

p

( / )(. ) ( / )(. )

.

1 2 46 1 2 96

0 71

E rp( ) . ( . . ) . 5 0% 8 6%)(0 71 111%

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Graphical Illustration

E(r)

s b

E(r)

5.0%

M

sM

5.0%

CML SML

1.0

M

0.71

13.6%

11.1%

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Example

The S&P500 Index has a standard deviation of about 12% per year.

Gold mining stocks have a standard deviation of about 24% per year and a correlation with the S&P500 of about r = 0.15.

If the yield on U.S. Treasury bills is 6% and the market risk premium is [E(rM)-rf] = 7.0%, what is the expected rate of return on gold mining stocks?

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Example The beta for gold mining stocks is calculated as follows:

The expected rate of return on gold mining stocks is:

Question: What portfolio has the same expected return as gold mining stocks, but the lowest possible standard deviation?

Answer: A portfolio consisting of 70% invested in U.S. Treasury bills and 30% invested in the S&P500 Index.

gM

M

gM g M

M2 2

15 24

120 30

. (. )

..

E rg( ) . ( . . ) . 6 0% 7 0%)(0 30 71%

Beta

E r

Sd r

p

p

(. )( ) (. )( . ) .

( ) . ( . . ) .

( ) (. )( ) (. )( . .

7 0 3 10 0 30

6 0% 7 0%)(0 30 81%

7 0 3 12 0%) 3 6%

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Using the CAPM for Project Evaluation

Suppose Microsoft is considering an expansion of its current operations.

» The expansion will cost $100 million today

» expected to generate a net cash flow of $25 million per year for the next 20 years.

» What is the appropriate risk-adjusted discount rate for the expansion project?

» What is the NPV of Microsoft’s investment project?

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Microsoft’s Expansion Project

The risk-adjusted discount rate for the project, rp, can be estimated by using Microsoft’s beta and the CAPM.

Thus, the NPV of the project is:

r r E r rP f m f

rP 0 05 1 2 0 086. . * .

NPV milliontt

$25

.$100 $53.

115392

1

20

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Company Risk VersusProject Risk

The company-wide discount rate is the appropriate discount rate for evaluating investment projects that have the same risk as the firm as a whole.

For investment projects that have different risk from the firm’s existing assets, the company-wide discount rate is not the appropriate discount rate.

In these cases, we must rely on industry betas for estimates of project risk.

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Company Risk versusProject Risk

Suppose Microsoft is considering investing in the development of a new airline. » What is the risk of this investment?» What is the appropriate risk-adjusted discount rate for evaluating

the project?» Suppose the project offers a 17% rate of return. Is the investment

a good one for Microsoft?

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Industry Asset Betas

Industry Beta Industry BetaAirlines 1.80 Agriculture 1.00Electronics 1.60 Food 1.00Consumer Durables 1.45 Liquor 0.90Producer Goods 1.30 Banks 0.85Chemicals 1.25 International Oils 0.85Shipping 1.20 Tobacco 0.80Steel 1.05 Telephone Utilities 0.75Containers 1.05 Energy Utilities 0.60Nonferrous Metals 1.00 Gold 0.35Source: D. Mullins, “Does the Capital Asset Pricing ModelWork?,” Havard Business Review, vol. 60, pp. 105-114.

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Company Risk versusProject Risk

The project risk is closer to the risk of other airlines than it is to the risk of Microsoft’s software business.

The appropriate risk-adjusted discount rate for the project depends upon the risk of the project. If the average asset beta for airlines is 1.8, then the project’s cost of capital is:

r r E r rp f p m f

rp 0 05 18 0 086 20 5%. . . .

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Company Risk versusProject Risk

Required Return

b

SML

Company Beta

Company-wide Discount Rate

A

Project Beta

Project IRR

Project-specific Discount Rate

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Project Evaluation: Rules

The risk of an investment project is given by the project’s beta.» Can be different from company’s beta» Can often use industry as approximation

The Security Market Line provides an estimate of an appropriate discount rate for the project based upon the project’s beta.» Same company may use different discount rates for different

projects This discount rate is used when computing the project’s net present

value.

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Summary

Optimal investments depend on trading off risk and return» Investors with higher risk tolerance invest more in risky

assets» Only risk that can’t be diversified counts

If investors can borrow and lend, then everybody holds a combination of two portfolios» The market portfolio of all risky assets» The riskless asset

– Covariance with the market portfolio counts In equilibrium, all stocks must lie on the security market line

» Beta measures the amount of nondiversifiable risk» Expected returns reflect only market risk» Use these as required returns in project evaluation