class 8 the capital asset pricing model. efficient portfolios with multiple assets e[r] 0 asset 1...

Class 8

The Capital Asset Pricing Model

Efficient Portfolios with Multiple Assets

E[r]

s0

Asset 1

Asset 2Portfolios ofAsset 1 and Asset 2

Portfoliosof otherassets

EfficientFrontier

Minimum-VariancePortfolio

Utility in Risk-Return Space

Investor A

Investor B

ExpectedReturn

StandardDeviation

Higher Utility

Higher Utility

Individual Asset Allocations

Investor A

Investor B

ExpectedReturn

StandardDeviation

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

Introducing a Riskfree Asset

Expected Return

Variance and Standard Deviation

E r r E rp f f f j ( )1

Var rp p f j 2 2 21( )

SD rp p f j ( )1

Introducing a Riskfree Asset

Expected Return

Standard Deviation

rf

sj

E(rj)

Asset j

Introducing a Riskless Asset

ExpectedReturn

StandardDeviation

M

r f

sM

E rM[ ]

a=1

a=-1

a=0

a=0 5.

0

Individual Asset Allocations

ExpectedReturn

StandardDeviation

M

r fInvestor A

Investor B

HigherUtility

HigherUtility

The Capital Market Line

ExpectedReturn

M

r f

E rm[ ]

StandardDeviation

IBM

1 24 4 44 34 4 4 4 1 24 34SystematicRisk

DiversifiableRisk

E rIBM[ ] A

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.

The equation of the CML is:

Portfolio M is the market portfolio.

E r r E r rp f M fp

M

( ) [ ( ) ]

Historical Returns, 1926-1993

Portfolio

AverageAnnual Rate

of Return(Nominal)

St. Deviationof AnnualRates ofReturn

Ave. RiskPremium

over Treas.Bills

Treasury Bills 3.8% 3.3%

Treasury Bonds 5.2% 8.6% 1.4%

Corporate Bonds 5.8% 8.5% 2.0%

Common Stocks 12.4% 20.6% 8.6%

Relationship Between Risk and Return for Individual Assets

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. A risk premium to compensate investors for bearing

risk.

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

The Capital Asset Pricing Model

The tradeoff between risk and return for the market portfolio is:

This gives us the following relationship:

E r rM f

M

( )

2

E r rM f M( ) 2

The Capital Asset Pricing Model

This can be written explicitly as follows:

This implies that for each security j the following relationship must hold:

x E r r xj jj

N

f j jMj

N

[ ( ) ]

1 1

E r rj f jM( )

The Capital Asset Pricing Model Using the definition of l, we have the relationship

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

where

E r r E r rj f M f j( ) [ ( ) ]

jjM

M

2

The CAPM

ExpectedReturn

Beta

M

r f

1

E rm[ ]

Security Market Line

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].

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

Review of Intuition for the CAPM

Investors prefer high returns

In a diversified portfolio, the variance of the assetis irrelevant. Only the covariance with the returns

of the existing portfolio are relevant

Investors will dem and a higher return for holdingassets whose returns are highly correlated

with those of the m arket

The relevant m easure of r isk is the covariance ofthe return of an asset with the return of

the m arket

All investors assess individual assets accordingto their contribution to the risk of the

m arket portfolio

All investors spread their wealth over theentire m arket

Diversify as m uch as possibleto reduce risk

Investors are risk averse

From econom ics/uti lity theory

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

Betas of Selected Common Stocks

Stock Beta Stock BetaAT&T 0.96 Ford Motor 1.03Boston Ed. 0.49 Home Depot 1.34BM Squibb 0.92 McDonalds 1.06Delta Airlines 1.31 Microsoft 1.20Digital Equip. 1.23 Nynex 0.77Dow Chem. 1.05 Polaroid 0.96Exxon 0.46 Tandem 1.73Merck 1.11 UAL 1.84Betas based on 5 years of monthly returns throughmid-1993.

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 M f j( ) [ ( ) ]

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 1996, 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. Earlier we saw this difference has averaged about 8.6% since 1926.

The betas are estimated by regression analysis.

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

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

p

p

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

.

1 2 46 1 2 96

0 71

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

Example of Portfolio Betas and Expected Returns

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.

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%

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?

Example

The beta for gold mining stocks is calculated as follows:

The expected rate of return on gold mining stocks is:

Beta gM

M

gM g M

M

2 2

15 24

120 30

. (. )

..

E rg( ) . ( . . ) . 6 0% 7 0%)(0 30 81%

Example

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.

Beta

E r

Sd rp

p

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

( ) . ( . . ) .

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

7 0 3 10 0 30

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

7 0 3 12 0%) 3 6%

Using the CAPM for Project Evaluation

Suppose Microsoft is considering an expansion of its current operations. The expansion will cost $100 million today and is 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?

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:

NPV milliont

t

$25

( . )$100 $53.

115392

1

20

r r E r rp f p m f d irp 0 05 12 0 086 15 3%. . . .b g

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.

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?

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.

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 d irp 0 05 18 0 086 20 5%. . . .b g

Company Risk versusProject Risk

Required Return

b

SML

Company Beta

Company-wide Discount Rate

A

Project Beta

Project IRR

Project-specific Discount Rate

Summary

The risk of an investment project is given by the project’s beta.

The Security Market Line provides an estimate of an appropriate discount rate for the project based upon the project’s beta.

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