Capital Asset Pricing Model

Chapter 7

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Learning Objectives Explain the theory behind the Capital

Asset Pricing Model Construct the security market line and use

it to estimate cost of capital Understand the practical applications of

the CAPM

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Asset Pricing Models Capital Asset Pricing Model (CAPM )

Expected returns are proportional to an investment’s

systematic risk

A stock’s expected risk premium varies in

proportion to it’s β

CAPM has issues, but is widely used in

both corporate and investment settings

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It is the equilibrium model that underlies all modern financial theory

Derived using Markowitz’s principles of diversification with simplified assumptions

Sharpe (1964), Lintner (1965), and Mossin (1966) are researchers credited with its development

Capital Asset Pricing Model (CAPM)

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Assumptions of the CAPM

Individuals All have the same (Homogeneous) expectations Only care about mean and variance (Mean-variance optimizers) Single-period investment horizon Everyone can borrow or lend at the risk-free rate

Markets are Perfect All assets are publicly held and publicly traded All information is costless and available to all investors No taxes or transaction costs

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All investors will hold the same portfolio of risky assets – market portfolio Optimal Risky Portfolio

Market portfolio will contains all securities

Each security will be held in proportion to its market value as a percentage of total market value

Assumption Results

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The Efficient Frontier and the CML

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Passivity is Efficient CAPM implies that passively investing in the

market portfolio is efficientMutual fund theorem: A single mutual fund

composed of the market will satisfy the desires of all investors

Active investors holding anything other than the market are holding an inefficient portfolio

Conundrum: if everyone is passive, because it is costless and efficient, what makes the market portfolio efficient?

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Security’s Risk

The only risk we are concerned with is systematic which we measure with β

A function of a security’s returns covariance with the market

βD = D,M / M2 or βD = (ρDMD)/M

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βD = (ρDMD)/M

D – Measures asset ‘D’ total risk ρDM – Measures the proportion of D’s total risk

that is systematic ρDMD– Measures the systematic risk of asset ‘D’ M – Measures the total market risk

Which is??? So βD:

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Notes on β

β – tells us how sensitive a stock is to market movements“Average Stock” has a β of 1Stocks with β > 1? Stocks with 0 < β < 1? Stocks with negative β?

What is the β of the market? What is the β of the risk free asset?

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CAPM and β CAPM states that expected returns are

proportional to an investment’s systematic riskA stock’s expected risk premium varies in

proportion to it’s β

E(Ri) = Rf + βi (RM - Rf)

Stock’s risk premium

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CAPM Example What is the expected return for a stock with a

beta of 1.7, if the expected market risk premium is 12% and the risk free rate is 3%?

What is the expected return for a stock with a beta of 1.7, if the stock’s expected risk premium is 12% and the risk free rate is 3%?

What is the expected return for a stock with a beta of 1.7, if the expected return on the market is 12% and the risk free rate is 3%?

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The market risk premium depends on the risk aversion of the average investorIt will be a function of the risk of the market and the average investor’s risk aversion:

E(RM)-rf = Ᾱσ2M

Ᾱ is the average degree of risk aversion across investors σ2

M is the variance of the market portfolio

What happens as the average risk aversion increases? Decreases? Why?What happens to the SML?

Market Risk Premium: RM - Rf

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How Risk Averse is the Market?

The variance of the return on the market portfolio is .04 and the expected return on the market portfolio is 20%. If the risk-free rate of return is 10%, the market’s degree of risk aversion, A, is _________

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Market Risk Premium Is the risk pricing benchmark

How much must investor be compensated to bear systematic risk

The risk premium is the price that investors earn on their investment in the market portfolioWhen investors are scared the risk premium rises to

induce people into the marketWhen investors are confident the risk premium falls as

investors are less concerned with risk

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Risk Premium Example

A stock is currently selling at $56. Its expected rate of return is 12%, the risk free rate is 4%, and the market risk premium is 6%. If the stock’s β triples? Assume a constant dividendWhat is the stock’s new risk premium?What is the new fair rate of return?What is the new price of the stock?

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Portfolios Portfolio β is a weighted average of the

component stocks’ β Because the portfolio return is a weighted

average of the component stocks returnReturns are a function of β

For the Market:

P

( ) ( ) andP k kk

k kk

E r w E r

w

( ) ( )M f M M fE r r E r r

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Portfolio β Example You invested 40% of your money in asset A,

βA is 1.5 and the balance in asset B, βB is 0.5. What is the portfolio beta?

If the risk free rate is 3% and the market risk premium is 11%, what are Stock A & B’s expected return?

What is the portfolio’s expected return?

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Security Market Line

The graphical representation of the CAPM equation

Graphs individual asset’s returns as a function of their systematic risk

Alpha: is the deviation from fair return

The SML and a Positive-Alpha Stock

rf

What is the slope of the SML?

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α Example

Risk free is 5% & the market risk premium is 7% If a stock has a β of 1.6, what is its fair return? If the stocks expected return is 19%, what is α?

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Dis-Equilibrium SML

A

B

C

E(r

i)

β

SML

Applications of CAPM Use SML as benchmark for fair return on risky

asset SML provides “hurdle rate” for internal

projects

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Example 1The T-bill rate is 1% and the market risk premium is 6%.

What is the fair return for each company? Which stocks are overvalued and which are

undervalued?

Stock Forecast Return

Beta Fair Return

GS 9.5% 1.38 ?

C 18% 2.59 ?

JPM 8.0% 1.33 ?

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Example 2The T-bill rate is 1% and the market risk premium is __%.

Assume that all stocks are fairly priced, fill in the missing blanks

Stock Forecast Return

Beta Fair Return

GS 9.5% ? ?

C ? ? 18.0%

JPM 8.0% 1.33 ?

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Example 3

A stock is selling for $125, its β is 1.1, and is expected to pay a $5.50 dividend next year. At what price does the investor expect to sell the stock for after the dividend, if the risk free rate is 4% and the expected return on the market is 18%?

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Example 4

The market price of a security is $56. Its expected rate of return is 16%. The risk-free rate is 5%, and the market risk premium is 9%. What will the market price of the security be if its beta increase by 50%? Assume the stock is expected to pay a constant

dividend in perpetuity

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Example 5

Consider two stocks, A and B. Stock A has an expected return of 10% and a beta of 1.2. Stock B has an expected return of 14% and a beta of 1.8. The market expected return is 9% and the risk-free rate is 5%. Which stock is a better buy? Why?

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CAPM in practice

CAPM deals with expectations and the “market” We CANNOT observe expectation so we have

to use realized returns We CANNOT observe the market so we have to

use an index → Have to cast CAPM as an Index Model

Expect versus Realized Returns

Actual return = Expected return + the effect of surprises

ri = Actual return earn on the security E(ri )= Expected return on the security βi= Market Beta F = Surprise in the market return (+/-) ei = Firm specific events 31

E(ri)

ri

( )i i i ir E r F e

Expect v Actual Return Example

If the market is expected to return 15% over the next year, what is the expected return for a stock with a β of 0.9? The risk free is expected to be 3%.

If the actual market return was 21%:What is the market surprise?What was the actual return earned over the year?

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CAPM as an Index Model

rit - rft = αi + βi (rMt – rft) + eit

rit: Asset i’s HPR i: Asset t: Time

αi: SCL intercept

βi: SCL slope

rMt: Index return at time t

eit: Firm-specific return

Google, 01/06-12/10

Google vs. S&P 500, 01/06-12/10

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CAPM and the Real World

CAPM is based on fundamentally false assumption, and has some issues empirically

The principals underlying CAPM are VALIDInvestors should diversifyWe should only care about systematic riskA well-diversified risky portfolio is suitable for the

majority of investors