8.1 THE CAPM MODEL

Equilibrium model that underlies all modern financial theory◦ Provide benchmark rate of return for evaluating

possible investments◦ Make an educated guess as to the expected return

Derived using principles of diversification with simplified assumptions

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

Perfect competition assumption, Individual investors are price takers

Single-period investment horizon Investments are limited to traded financial

assets, may borrow or lend any amount at a fixed risk-free rate

No taxes nor transaction costs

Information is costless and available to all investors

Investors are rational mean-variance optimizers

Homogeneous expectations, analyze securities in the same way and share the same economic view of the world

◦ same input list to feed into the Markowitz model◦ Same E(r), covariance matrix, same efficient

frontier and unique optimal risky portfolio

All investors will hold the same portfolio for risky assets – market portfolio (M)◦ Market portfolio contains all securities and the

proportion of each security is its market value as a percentage of total market value

Capital market line (CML) is the best attainable CAL

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

(explained in P9)

2M f ME r r A

Risk premium on an individual security will be proportional to the risk premium on M and beta coefficient of the security relative to M

2

,i Mi

M

Cov r r

2

,i Mi f M f

M

i M f

Cov r rE r r E r r

E r r

Value of the aggregate risky portfolio will equal the entire wealth of the economy

Use identical Markowitz analysis, same universe of securities, same time horizon, same input list, must arrive at the same composition of the optimal risky portfolio

When all investors have same risky portfolio, it must be M.

1y

The market portfolio is the optimal risky portfolio (tangency portfolio)◦ To decide how much in risky portfolio, how

much in risk-free asset?

Net borrowing and lending among all investors must be zero, y=1

2

M f

M

E r ry

A

1y

2M f ME r r A

To measure the GM’s contribution to the risk of the market portfolio (the manner: sum of the contributions of each stock equals the total variance)◦ variance of the portfolio: sum over all the

elements of the covariance matrix◦ The contribution of GM to the market

portfolio variance is the sum of the GM covariance row

1 1 2 2, ,

, ,

GM GM

GM

GM GM GM n n GM

wCov r r w Cov r rw

w Cov r r w Cov r r

demonstration

' var ,GM GM MGM S contribution to iance w Cov r r

1

1

1

, ,

,

n

M k kk

n

GM M GM k kk

n

k GM kk

r w r

Cove r r Cov r w r

w Cov r r

Contribution of GM to the risk premium of the market portfolio

The reward-to-risk ratio for GM

'

' var ,

,

GM GM f

GM GM M

GM f

GM M

w E r rGM S contribution to risk premium

GM S contribution to iance w Cov r r

E r r

Cov r r

GM GM fw E r r

M is the tangency portfolio, the market price of risk:

When equilibrium, all investors have same reward-to-risk ratio

2var

M f

M

E r rmarket risk premium

market iance

2,

GM f M f

GM M M

E r r E r r

Cov r r

Beta measures the contribution of GM to the market variance as a fraction of the total variance of the market portfolio

2

,GM MGM f M f

M

Cov r rE r r E r r

2

,GM M

M

Cov r r

GM f GM M fE r r E r r

CAPM holds for the overall portfolio

2

2 2

,1M M M

MM M

Cov r r

M f M M fE r r E r r

1

n

P k kk

w

Beta measures stock’s contribution to variance of market portfolio, required risk premium to be a function of beta.

Security Market Line: ◦ The expected return-beta relationship be

portrayed◦ Individual asset risk (or portfolio) premium as a

function of asset risk, using beta as risk measurement

Capital Market Line:◦ The expected return-std deviation relationship◦ Risk premiums of efficient portfolios (composed

of market and risk-free asset) as a function of portfolio standard deviation.

SML provide benchmark for evaluation ◦ Risk measured as Beta, SML provide the required

rate of return◦ Fairly priced assets plot exactly on SML◦ Given the assumptions, all securities must lie on

the SML in market equilibrium◦ Under-priced plot above, over-priced below

Alpha ◦ difference between the fair and actually expected

rate of return on a stock Useful in capital budgeting decisions

◦ Required rate of return that the project need to yield based on its beta (IRR)

SML provide a benchmark for the evaluation of investment performance◦ provide the required rate of return◦ Fairly priced assets plot exactly on the SML◦ Under-priced plot above, over-priced below

Alpha ◦ difference between the fair and actually expected

rate of return on a stock

CAPM: expected return, ex ante

◦ Not feasible to construct M, difficult in testing variance efficiency of the CAPM market portfolio

◦ CAPM implies relationships among expected returns, to calculate the reward-to-volatility ratio, but no way to observe the expectations directly

Index model: Actual (realized) return, ex post

i f i M fE r r E r r

Index model: Actual (realized) return, ex post

Index model beta coefficient is the same beta in CAPM.

2

2

( , ) ( , )

( , ) ( , )

( , )

i M i i M i M

i M M i M

i M

i Mi

M

Cov R R Cov R e R

Cov R R Cov e R

Cov R R

i i i M iR R e

In CAPM: mean excess return of stock i relative to mean excess return of market portfolio

If the M in index model represent the true market portfolio, take expectation of index model

i f i M fE r r E r r

i f i i M fE r r E r r i i i M iR R e

( )i f i i M f ir r r r e

CAPM predict Alpha=0. Alpha of a stock is its expected return in excess of (or below) the fair

expected return as predicted by the CAPM

Comparison of CAPM and Index model◦ CAPM predicts that alpha should be zero for all assets,

if the stock is fairly priced its alpha must be zero (expected return)

◦ Index model representation of the CAPM holds that the realized value of alpha should average out to zero for a sample of historical observed returns (if estimate the index model for several firms for a sample period)

◦ JOF 1995, distribution of alpha is roughly bell shaped, mean is slightly negative but statistically indistinguishable from zero

◦ Not appear that mutual funds outperform the market index

Market model- applicable variation of index model◦ Return surprise of any security is proportional

to the return surprise of the market plus a firm-specific surprise

◦ If CAPM is valid, market model is identical to the index model

i i i M M ir E r r E r e

Role of CAPM in real-life investment◦ Notion that all alpha can be zero is feasible in principle,

but not expected to emerge in real markets (Grossman and Stiglitz, 1981)

such an equilibrium may be one that the real economy can approach, but not necessarily reach

actions of security analysts are forces that drive security prices to proper levels (alpha=zero)

◦ In the absence of security analysis , one should take security alphas as zero, obtain the best investment portfolio on the assumption that all alpha values are zero

Is the CAPM testable◦ Proxies must be used for the market portfolio, unobservable◦ CAPM fails the test, data reject the hypothesis that alpha are

uniformly zero Result of a failure of data, validity of the market proxy or statistical

method CAPM is still considered the best available description of security

pricing and is widely accepted

It is important to consider the econometric technique used for the model estimated

Statistical bias is easily introduced◦Miller and Scholes paper demonstrated how econometric problems could lead one to reject the CAPM even if it were perfectly valid

8.4 EXTENSIONS OF CAPM

EXTENSIONS OF CAPM Two kinds of extension to the simple version

of CAPM◦ Relax the assumptions ◦ Consider more risk factors other than the

uncertain value of the securities, such as unexpected changes in relative prices of consumer goods.

Extensions of the CAPM Assumption: all risky assets are traded Human capital and privately held corporations Consideration of labor income and non-traded

assets

2

, ,

,

Hi M i H

Mi M

HM M H

M

PCov R R Cov R R

PE R E R

PCov R R

P

Extensions of the CAPM Assumption: single-period

◦ Merton 1992, individuals optimize a lifetime consumption, continually adapt consumption. Sources of risk, changes of parameters describing investment opportunities, such as future risk-free rates , risk of market portfolio Prices of the consumption goods that can be purchased

with any amount of wealth Merton’s Multi-period Model and hedge portfolios

◦Incorporation of the effects of changes in the real rate of interest and inflation

1

K

i iM M iM Kk

E R E R E R

Extensions of the CAPM Continued

A consumption-based CAPM◦ Models by Rubinstein, Lucas, and Breeden

Investor must allocate current wealth between today’s consumption and investment for the future

C: consumption-tracking portfolio

i iC C fE R E r r

8.5 LIQUIDITY AND CAPM

Liquidity and the CAPM

Trading is of importance to investors (heterogeneous expectations), trading costs

Liquidity: ease and speed with which it can be sold at fair market value◦ Cost of engaging in a transaction, bid-ask spread◦ Price impact, adverse movement in price when attempt to

execute a larger trade◦ Immediacy, ability to sell the asset quickly without reverting to

fire-sale prices Illiquidity can be measured in part by the discount from fair

market value a seller must accept if the asset is to be sold quickly

Research supports a premium for illiquidity.◦ Amihud and Mendelson (1986)

Illiquidity premium◦ Begin with Ignorance of systematic risk, T-bill◦ N type of investors (n periods), investment horizon

LC

L LE r Ch

IC

II

CE r h

Liquid Liquid stock (L)stock (L)

Illiquid Illiquid stock (I)stock (I)

T-T-billbill

Liquidation Liquidation costcost

Expected Expected return net return net of of transaction transaction costcost

r

r

Stock L and stock I prices must fall, causing their return to rise

Suppose each gross return is higher by some fraction of liquidation cost

Liquid Liquid stock (L)stock (L)

Illiquid Illiquid stock (I)stock (I)

T-billT-bill

Gross Gross returnreturn

00

Net returnNet return rIr yC

LL

Cr xC h II

Cr yC h

Lr xC

When h increases, less impact. As horizon become large, per-period impact of the transaction cost approaches zero

Equilibrium illiquidity premium◦ for marginal investor with horizon ,

net return( I=L)

◦ Expected return on stock I

LIh

L IL I

LI LI

C Cr xC r yCh h

1 1I

LI L LI

Cy x

h C h

1I I L I L

LI

r r yC r xC C Ch

Expected return of stock L

Illiquidity premium of stock I versus stock L

L Lr r xC

1I L I L

LI

r r C Ch

for marginal investor with horizon , net return( T-bill=L)

Liquidity premium of stock L

rLh

LL

Cr xC rh

1L L

rL

r r Ch

Inventory management problem, bid-ask spread may be viewed as compensation for bearing the price risk involved in holding an inventory of securities

Component of the spread due to asymmetric information◦ Traders who post offers to buy or sell at limit prices need to be

worried about being picked off by better-informed traders Trading for tow reasons:

◦ non-informational motives, liquidity traders, noise trades, not motivated by private information, dealers earn profit from bid-ask spread

◦ Information traders, motivated by private information known only to the seller or buyer, impose a cost on both dealers and other investors

◦ Any traders posting a limit order is at risk from information traders, response is to widen spread to compensate for potential losses from trading with information traders

When consider common systematic risk factors, illiquidity premium is additive to the risk premium or the usual CAPM

is expected cost of illiquidity, beta is liquidity betas

1 2 3i i M M L L LE R kE C E R C

iE C

Three Elements of Liquidity

Sensitivity of security’s illiquidity to market illiquidity:

Sensitivity of stock’s return to market illiquidity:

Sensitivity of the security illiquidity to the market rate of return:

1

( , )

( )i M

LM M

Cov C C

Var R C

3

( , )

( )i M

LM M

Cov C R

Var R C

2

( , )

( )i M

LM M

Cov R C

Var R C