Lecture 8

Capital Asset Pricing Modeland

Single-Factor Models

Outline

Beta as a measure of risk. Original CAPM. Efficient set mathematics. Zero-Beta CAPM. Testing the CAPM. Single-factor models. Estimating beta.

Beta

Consider adding security i to portfolio P to form portfolio C.

E[rC] = wiE[ri] + (1-wi)E[rP]

C2 = wi

2i2+2wi(1-wi)iP +(1-wP)

2P2

Under what conditions would C2 be

less than P2?

Beta

The value of wi that minimizes C2 is

22

2

2 PiPi

iPPiw

wi > 0 if and only if iP < P2 or

012 .P

iPiP

CAPM With Risk-FreeBorrowing and Lending

Security Market Line

E(ri) = rf + [E(rM) – rf]i

The linear relationship between expected return and beta follows directly from the efficiency of the market portfolio.

The only testable implication is that the market portfolio is efficient.

Efficient Set Mathematics

If portfolio weights are allowed to be negative, then the following relationships are mathematical tautologies.

1. Any portfolio constructed by combining efficient portfolios is itself on the efficient frontier.

Efficient Set Mathematics

2. Every portfolio on the efficient frontier (except the minimum variance portfolio) has a companion portfolio on the bottom half of the minimum variance frontier with which it is uncorrelated.

Efficient Set Mathematics

Expected Return

StandardDeviation

P

Z(P)E[rZ(P)]

Value of iP

Efficient Set Mathematics

3. The expected return on any asset can be expressed as an exact linear function of the expected return on any two minimum-variance frontier portfolios.

PQP

PQiPQPQi rErErErE 2

Efficient Set Mathematics

Consider portfolios P and Z(P), which have zero covariance.

iPPZPPZ

P

iPPZPPZi

rrErE

rrErErE

2

The Zero-Beta CAPM

What if

(1) the borrowing rate is greater than the lending rate,

(2) borrowing is restricted, or

(3) no risk-free asset exists?

CAPM With Different Borrowing and Lending Rates

Expected Return

rfL

E[Z(M)]

rfB

M

B

L

Z(M)

StandardDeviation

Security Market Line

The security market line is obtained using the third mathematical relationship.

iMZMMZ

M

iMMZMMZi

rrErE

rrErErE

2

CAPM With No Borrowing

Expected Return

rfL

E[Z(M)]

ML

Z(M)

StandardDeviation

CAPM With No Risk-Free Asset

Expected Return

E[Z(M)]

M

Z(M)

StandardDeviation

Testing The CAPM

The CAPM implies that

E(rit) = rf + i[E(rM) - Rf]

Excess security returns shouldincrease linearly with the security’s systematic risk andbe independent of its nonsystematic risk.

Testing The CAPM Early tests were based on running

cross section regressions.

rP - rf = a + bP + eP Results: a was greater than 0 and b

was less than the average excess return on the market.

This could be consistent with the zero-beta CAPM, but not the original CAPM.

Testing The CAPM

The regression coefficients can be biased because of estimation errors in estimating security betas.

Researchers use portfolios to reduce the bias associated with errors in estimating the betas.

Roll’s Critique

If the market proxy is ex post mean variance efficient, the equation will fit exactly no matter how the returns were actually generated.

If the proxy is not ex post mean variance efficient, any estimated relationship is possible even if the CAPM is true.

Factor Models Factor models attempt to capture the

economic forces affecting security returns.

They are statistical models that describe how security returns are generated.

Single-Factor Models

Assume that all relevant economic factors can be measured by one macroeconomic indicator.

Then stock returns depend upon (1) the common macro factor and (2) firm specific events that are uncorrelated with the macro factor.

Single-Factor Models The return on security i is

ri = E(ri) + iF + ei.

E(ri) is the expected return. F is the unanticipated component

of the factor. The coefficient i measures the

sensitivity of ri to the macro factor.

Single-Factor Models

ri = E(ri) + iF + ei.

ei is the impact of unanticipated firm specific events.

ei is uncorrelated with E(ri), the macro factor, and unanticipated firm specific events of other firms.

E(ei) = 0 and E(F) = 0.

Single-Factor Models

The market model and the single-index model are used to estimate betas and covariances.

Both models use a market index as a proxy for the macroeconomic factor.

The unanticipated component in these two models is F = rM - E(rM).

The Market Model

Models the returns for security i and the market index M, ri and rM , respectively.

ri = E(ri) + iF + ei.

= i + i ErM) + i[rM – E(rM)] + ei

= i + i rM + ei

The Single-Index Model

Models the excess returns Ri = ri – rf and RM = rM – rf .

Ri = E(Ri) + i F + ei.

= i + i ERM) + i [RM – E(RM)] + ei

= i + i RM + ei

CAPM Interpretation of i

The CAPM implies that E(Ri) = iE(RM).

In the index model i = E(Ri) – iE(RM) = 0.

In the market model

i = E(ri) – iE(rM)

= rf + i[E(rM) – rf] - iE(rM)

= (1 – i)rf

Estimating Covariances

ei is also assumed to be uncorrelated with ej.

Consequently, the covariance between the returns on security i and security j is

Cov(Ri, Rj) = i j M2

Estimating and Using the Single-Index Model

The model can be estimated using the ordinary least squares regression

Rit = ai + biRMt + eit

ai is an estimate of Jensen’s alpha.

bi is the estimate of the CAPM i .

eit is the residual in period t.

Estimates of Beta R square measures the proportion of

variation in Ri explained by RM. The precision of the estimate is

measured by the standard error of b. The standard error of b is smaller

(1) the larger n,

(2) the larger the var(RM), and

(3) the smaller the var(e).

The Distribution of b and the 95% Confidence Interval for Beta

Hypothesis Testing t-Stat is b divided by the standard

error of b. P-value is the probability that Test the hypothesis that = using

the t-statistic

706322990

017851

2

t~..

..t

nt~bErrorStd

bt

Estimating And Using The Market Model

The model can be estimated using the ordinary least squares regression

rit = ai + birMt + eit

ai equals Jensen’s alpha plus rf (1–i. bi is a slightly biased estimate of

CAPM i .

eit is the residual in period t.

Comparison Of The Two Models

Estimates of beta are very close. Use the index model to estimate

Jensen’s alpha. The intercept of the index model is an

estimate of . The intercept of the market model is an

estimate of + (1 – )rf

The Stability Of Beta A security’s beta can change if

there is a change in the firm’s operations or financial condition.

Estimate moving betas using the Excel function

=SLOPE(range of Y, range of X).

Adjusted Betas

Beta estimates have a tendency to regress toward one.

Many analysts adjust estimated betas to obtain better forecasts of future betas.

The standard adjustment pulls all beta estimates toward 1.0 using the formula

adjusted bi = 0.333 + 0.667bi .

Non-synchronous Trading

When using daily or weekly returns, run a regression with lagged and leading market returns.

Rit = ai + b1Rmt-1 + b2Rmt + b3Rmt+1

The estimate of beta is

Betai = b1 + b2 + b3.