Investment Analysis and Portfolio Management

Lecture 7

Gareth Myles

The Capital Asset Pricing Model (CAPM)

The CAPM is a model of equilibrium in the market for securities.

Previous lectures have addressed the question of how investors should choose assets given the observed structure of returns.

Now the question is changed to: If investors follow these strategies, how will

returns be determined in equilibrium?

The Capital Asset Pricing Model (CAPM)

The simplest and most fundamental model of equilibrium in the security market Builds on the Markowitz model of portfolio choice Aggregates the choices of individual investors

Trading ensures an equilibrium where returns adjust so that the demand and supply of assets are equal

Many modifications/extensions can be made But basic insights always extend

Assumptions

The CAPM is built on a set of assumptions Individual investors

Investors evaluate portfolios by the mean and variance of returns over a one period horizon

Preferences satisfy non-satiation Investors are risk averse

Trading conditions Assets are infinitely divisible Borrowing and lending can be undertaken at the

risk-free rate of return There are no taxes or transactions costs

Assumptions

The risk-free rate is the same for all Information flows perfectly

The set of investorsAll investors have the same time horizon Investors have identical expectations

Assumptions

The first six assumptions are the Markowitz model

The seventh and eighth assumptions add a perfect capital market and perfect information

The final two assumptions make all investors identical except for their degree of risk aversion

Direct Implications

All investors face the same efficient set of portfolios

pr

pMVP

MVPr

fr

Direct Implications

All investors choose a location on the efficient frontier

The location depends on the degree of risk aversion

The chosen portfolio mixes the risk-free asset and portfolio M of risky assets

pr

pMVP

MVPr

fr

Less riskaverseMore risk

averse

M

Separation Theorem

The optimal combination of risky assets is determined without knowledge of preferences All choose portfolio M This is the Separation Theorem

M must be the market portfolio of risky assets All investors hold it to a greater or lesser extent No other portfolio of risky assets is held There is a question about the interpretation of this

portfolio

Equilibrium

The only assets that need to be marketed are: The risk-free asset A mutual fund representing the market portfolio No other assets are required

In equilibrium there can be no short sales of the risky assets All investors buy the same risky assets No-one can be short since all would be short If all are short the market is not in equilibrium

Equilibrium

Equilibrium occurs when the demand for assets matches the supply This also applies to the risk-free Borrowing must equal lending

This is achieved by the adjustment of asset prices

As prices change so do the returns on the assets

This process generates an equilibrium structure of returns

The Capital Market Line

All efficient portfolios must lie on this line

Slope =

Equation of the line

pr

p

fr

M

fM rr

pM

fMfp

rrrr

Mr

M

Interpretation

rf is the reward for "time" Patience is rewarded Investment delays consumption

is the reward for accepting "risk"

The market price of riskJudged to be equilibrium rewardObtained by matching demand to supply

M

fM rr

Security Market Line

Now consider the implications for individual assets

Graph covariance against returnThe risk on the market portfolio is The covariance of the risk-free asset is zero The covariance of the market with the

market is

M

2M

Security Market Line

Can mix M and the risk-free asset along the line If there was a portfolio

above the line all investors would buy it

No investor would hold one below

The equation of the line is

pr

fr

iM2M

Mr

iMM

fMfi

rrrr

2

M

Security Market Line

Define

The equation of the line becomes

This is the security market line (SML)

2M

iMiM

iMfMfi rrrr

Security Market Line

There is a linear trade-off between risk measured by and return

In equilibrium all assets and portfolios must have risk-return combinations that lie on this line

pr

fr

iM

iMir

Market Model and CAPM

Market model usesCAPM uses is derived from an assumption about

the determination of returns it is derived from a statistical model the index is chosen not specified by any

underlying analysis is derived from an equilibrium theory

iI

iM

iI

iM

Market Model and CAPM

In addition: I is usually assumed to be the market index,

but in principal could be any indexM is always the market portfolio

There is a difference between theseBut they are often used interchangeablyThe market index is taken as an

approximation of the market portfolio

Estimation of CAPM

Use the regression equation

Take the expected value

The security market line implies

It also shows

ifMiMiMfi rrrr

fMiMiMfi rrErrE

fiMiI r 1

0iM

CAPM and Pricing

CAPM also implies the equilibrium asset prices The security market line is

But

where pi(0) is the value of the asset at time 0 and pi(1) is the value at time 1

iMfMfi rrrr

0

01

i

iii p

ppr

CAPM and Pricing

So the security market line gives

This can be rearranged to find

The price today is related to the expected value at the end of the holding period

fMiMfi

ii rrrp

pp

0

01

fMiMf

ii rrr

pp

1

10

CAPM and Project Appraisal

Consider an investment projectIt requires an investment of p(0) todayIt provides a payment of p(1) in a yearShould the project be undertaken?The answer is yes if the present

discounted value (PDV) of the project is positive

CAPM and Project Appraisal

If both p(0) and p(1) are certain then the risk-free interest rate is used to discount

The PDV is

The decision is to accept project if

fr

ppPDV

1

10

fr

pp

1

10

CAPM and Project Appraisal

Now assume p(1) is uncertainCannot simply discount at risk-free rate if

investors are risk averseFor example using

will over-value the projectWith risk aversion the project is worth less

than its expected return

fr

ppPDV

1

10

))1(())1(( pEUpU

CAPM and Project Appraisal

One method to obtain the correct value is to adjust the rate of discount to reflect risk

But by how much?The CAPM pricing rule gives the answerThe correct PDV of the project is

][1

)1()0(

fMpf rrr

ppPDV