The Capital AssetThe Capital AssetPricing ModelPricing Model

Chapter 9Chapter 9

Equilibrium model that underlies all modern financial theory

Derived using principles of diversification with simplified assumptions

Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development

Capital Asset Pricing Model (CAPM)Capital Asset Pricing Model (CAPM)

Individual investors are price takers Single-period investment horizon Investments are limited to traded financial

assets No taxes, and transaction costs

AssumptionsAssumptions

Information is costless and available to all investors

Investors are rational mean-variance optimizers

Homogeneous expectations

Assumptions (cont’d)Assumptions (cont’d)

All investors will hold the same portfolio for risky assets – market portfolio

Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value

Resulting Equilibrium ConditionsResulting Equilibrium Conditions

Risk premium on the market depends on the average risk aversion of all market participants

Risk premium on an individual security is a function of its covariance with the market

Resulting Equilibrium Conditions Resulting Equilibrium Conditions (cont’d)(cont’d)

Capital Market LineCapital Market LineE(r)

E(rM)

rf

MCML

m

M = Market portfoliorf = Risk free rate

E(rM) - rf = Market risk premium

E(rM) - rf = Market price of risk

= Slope of the CAPM

M

Slope and Market Risk PremiumSlope and Market Risk Premium

The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio

Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio

Expected Return and Risk on Expected Return and Risk on Individual SecuritiesIndividual Securities

Security Market LineSecurity Market LineE(r)

E(rM)

rf

SML

M

ßß = 1.0

= [COV(ri,rm)] / m2

Slope SML = E(rm) - rf

= market risk premium

SML = rf + [E(rm) - rf]

Betam = [Cov (ri,rm)] / m2

= m2 / m

2 = 1

SML RelationshipsSML Relationships

E(rm) - rf = .08 rf = .03

x = 1.25

E(rx) = .03 + 1.25(.08) = .13 or 13%

y = .6

e(ry) = .03 + .6(.08) = .078 or 7.8%

Sample Calculations for SMLSample Calculations for SML

Graph of Sample CalculationsGraph of Sample CalculationsE(r)

Rx=13%

SML

m

ß

ß1.0

Rm=11%

Ry=7.8%

3%

xß

1.25

yß.6

.08

Disequilibrium ExampleDisequilibrium ExampleE(r)

15%

SML

ß1.0

Rm=11%

rf=3%

1.25

Suppose a security with a of 1.25 is offering expected return of 15%

According to SML, it should be 13% Underpriced: offering too high of a rate of

return for its level of risk

Disequilibrium ExampleDisequilibrium Example

Black’s Zero Beta ModelBlack’s Zero Beta Model

Absence of a risk-free asset Combinations of portfolios on the efficient

frontier are efficient All frontier portfolios have companion

portfolios that are uncorrelated Returns on individual assets can be

expressed as linear combinations of efficient portfolios

Black’s Zero Beta Model FormulationBlack’s Zero Beta Model Formulation

),(

),(),()()()()(

2QPP

QPPiQPQi rrCov

rrCovrrCovrErErErE

Efficient Portfolios and Zero Efficient Portfolios and Zero CompanionsCompanions

Q

P

Z(Q)Z(P)

E[rz (Q)]

E[rz (P)]

E(r)

Zero Beta Market ModelZero Beta Market Model

2)()(

),()()()()(

M

MiMZMMZi

rrCovrErErErE

CAPM with E(rz (m)) replacing rf

CAPM & LiquidityCAPM & Liquidity

Liquidity Illiquidity Premium Research supports a premium for illiquidity

- Amihud and Mendelson

CAPM with a Liquidity PremiumCAPM with a Liquidity Premium

)()()( ifiifi cfrrErrE

f (ci) = liquidity premium for security i

f (ci) increases at a decreasing rate

Illiquidity and Average Illiquidity and Average ReturnsReturns

Average monthly return(%)

Bid-ask spread (%)