Capital Asset Pricing Model and Single-Factor Models 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 if siP < sP2 or

CAPM With Risk-Free Borrowing 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 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 rfL Expected Return M L Z(M) E[Z(M)] 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 ri to the macro factor.

Single-Factor Models ri = E(ri) + biF + 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) + biF + ei. = ai + bi E(rM) + bi[rM – E(rM)] + ei = ai + bi rM + ei

The Single-Index Model Models the excess returns Ri = ri – rf and 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) = bi bj sM2

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). bi is a slightly biased estimate of CAPM bi . 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 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.