**Lecture 8** Capital Asset Pricing Model and 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] • sC2 = wi2si2+2wi(1-wi)siP +(1-wP)2sP2 • Under what conditions would sC2 be less than sP2?

**Beta** The value of wi that minimizes sC2 is wi> 0 if and only ifsiP< sP2 or

**CAPM With Risk-FreeBorrowing and Lending**

**Security Market Line** • E(ri) = rf + [E(rM) – rf]bi • 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. • Any portfolio constructed by combining efficient portfolios is itself on the efficient frontier.

**Efficient Set Mathematics** • 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** Value of biP Expected Return b>1 b=1 P 0<b<1 E[rZ(P)] Z(P) b=0 b<0 Standard Deviation

**Efficient Set Mathematics** • 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.

**Efficient Set Mathematics** • Consider portfolios P and Z(P), which have zero covariance.

**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** B Expected Return M L rfB Z(M) E[Z(M)] rfL Standard Deviation

**Security Market Line** • The security market line is obtained using the third mathematical relationship.

**CAPM With No Borrowing** Expected Return M L Z(M) E[Z(M)] rfL Standard Deviation

**CAPM With No Risk-Free Asset** Expected Return M Z(M) E[Z(M)] Standard Deviation

**Testing The CAPM** • The CAPM implies that E(rit) = rf + bi[E(rM) - Rf] • Excess security returns should increase linearly with the security’s systematic risk and be independent of its nonsystematic risk.

**Testing The CAPM** • Early tests were based on running cross section regressions.rP - rf = a + bbP + 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) + biF + ei. • E(ri) is the expected return. • F is the unanticipated component of the factor. • The coefficient bi measures the sensitivity of rito the macro factor.

**Single-Factor Models** • ri = E(ri) + biF + ei. • ei is the impact of unanticipated firm specific events. • eiis 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) + biF + ei. = ai + biE(rM) + bi[rM – E(rM)] + ei = ai + bi rM + ei

**The Single-Index Model** • Models the excess returns Ri = ri – rfand RM = rM – rf . • Ri = E(Ri) + bi F + ei. = ai + bi E(RM) + bi [RM – E(RM)] + ei = ai + bi RM + ei

**CAPM Interpretation of ai** • The CAPM implies that E(Ri) = biE(RM). • In the index model ai = E(Ri) – biE(RM) = 0. • In the market model ai = E(ri) – biE(rM) = rf + bi[E(rM) – rf] - biE(rM) = (1 – bi)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) = bibjsM2

**Estimating a and b 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 bi . • 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 b = 0. • Test the hypothesis that b = g using the t-statistic

**Estimating a And b 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–bi). • biis a slightly biased estimate of CAPMbi. • eitis 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 a. • The intercept of the market model is an estimate of a + (1 – b)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.