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Chapter 7 - Part ICapital Asset Pricing Model (CAPM)

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Efficient Frontier with RiskfreeLending and BorrowingE(rp)

σp

rf

efficient frontier with risky assets only

new efficient frontier with riskfree lending and borrowing

more risk averse investor

less risk-averse investor

optimalrisky

portfolio

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Capital Asset Pricing Model (CAPM)An equilibrium model underlying modern finance theoryAnswers the following question:

What determines the expected return of each security?

Who can claim credit for CAPM?Markowitz (Nobel Prize)Sharpe (Nobel Prize), Lintner and Mossin

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Assumptions of CAPMIndividual investors are price-takers

Individual action does not affect stock pricesCommon single-period investment horizon

Investors maximize expected utilityHomogeneous expectations

Investors agree on the return distributionsInvestors may have different risk aversions

Market is frictionlessNo taxes or transaction costsWhat you see is what you are going to get

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Equilibrium Implications of CAPMAll investors will hold the same portfolio of risky assets – market portfolio

Risk aversion only affects relative weights between riskfree security and the optimal risky portfolio

Market portfolio contains all securities The proportion of each security is the percentage of that security’s market cap in total market capThe weights of risky securities in your risky portfolio are the same as their weights in the market portfolio

Risk premium for an individual security depends on its contribution to the risk of market portfolio

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Capital Market Line (CML)

E[rP]

E[rM]

rf

MCML

σMσP

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Capital Market LineSlope and Market Risk Premium in CML

risk of priceMarket :][

risks)for ion (compensat premiumrisk Market :][

money) of value time(the rate freeRisk :portfolioMarket :

M

fM

fM

f

rrE

rrE

rM

σ−

−

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Expected Return of An Individual Security Based on CAPM

The risk premium of an individual security depends on

The security’s contribution to the risk of the market portfolio, or The security’s systematic risk measure

The contribution to market (systematic) risk is measured by Beta

)][(][

],[2

fMifi

M

Mii

rrErrE

rrCov

−=−

=

βσ

β

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Facts about BetaWe can think Beta as a normalized covarianceBeta of the market portfolio is ____Beta of a riskfree asset is ________Can a security have a negative beta? ____Can a risky security have an expected return lower than riskfree rate?

If yes, then how?

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Security Market Line (SML)Graphical Presentation of CAPM

E(ri)

E(rM)rf

SML

βiβM = 1.0

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CML vs. SMLBoth specify a risk-return relationship

Measure of riskCML: risk is measured by σi

SML: risk is measured by βi

Applicability:CML is applicable only to the efficient portfolios

Combinations of riskfree asset and market portfolio

SML is applicable to individual security and portfolios (efficient or inefficient ones)

)][(][:

][][:

fMifi

M

fMifi

rrErrESML

rrErrECML

−=−

−=−

βσ

σ

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An Example of SMLExpected return for Market portfolio is 11%, risk free rate is 3%, security X and Y have betas of 1.25 and 0.6 respectively.

Q: What are the expected returns of X and Y?

A:

%13%825.1%3)][(][ =×+=−+= fMxfx rrErrE β

%8.7%86.0%3)][(][ =×+=−+= fMyfy rrErrE β

%8%3%11][ :premiumrisk Market =−=− fM rrE

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Graph for the SML Example

E(r)

rx=13%

SML

ββM=1.0

rM=11%

ry=7.8%

rf=3%

βx=1.25βy=0.6

Market risk premium: 8%

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Beta of a Portfolio

Beta of a portfolio is the weighted average of the betas of its component securities

Beta of the efficient portfolio on the CML is0 < β < 1 if portfolio is long in riskfree bond (lending)β > 1 if portfolio is short in bond (borrowing)

∑=

=N

iiiP W

1ββ

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An Example of Portfolio Beta

Q1: What is the beta of a portfolio with a. 70% in market portfolio and 30% in riskfreebond? b. -50% in riskfree bond and 150% in market portfolio?

Q2: Stock X has a beta of 1.2 while stock Y has a beta of -0.2. X and Y have expected returns of 18% and 4% respectively. What is the expected return of the market portfolio?

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Market Vs. Non-Market RiskThe total risk of security i, measured by its variance and denoted σi

2, consists of two parts

σi2 = βi

2σM2 + σεi2

First component on the RHS is related to movements of the market portfolio (βi

2σM2)

Second component is unique to the ith security (σεi2 )

R2 = βi2σM

2 / σi2 measures how much market

risk explains the total risk of the security

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An Example of Risk DecompositionIf the std dev of the market portfolio is 20%, then how much does the market risk account for the total risk of the following portfolios:

Portfolio A is on CML with a Beta of 1.5?

Portfolio B has a standard deviation of 35% and a beta of 1.5?