EDHEC-Risk Institute October 2016

Multi-Dimensional Risk and Performance Analysis for Equity Portfolios — October 2016

3. Applications of Fundamental Beta

view of the observed market returns. We do this by using the Fama and MacBeth (1973) procedure. We also apply it to the estimation of the alpha in the static CAPM, in which the beta is constant over time. In the static CAPM, the first step of Fama- MacBeth approach is to estimate the beta of each portfolio through a time-series regression: In the fundamental CAPM, this step is bypassed by estimating a time-varying beta along the lines of the procedure described in Section 2.3. In the static CAPM, the second step is to run a cross-sectional regression of stock returns on the estimated betas at each date, so as to obtain a time-series of estimates for the market premium: In the fundamental CAPM, this step is modified by regressing the cross- section of stock returns on the time- series average of the conditional betas. Taking the average of conditional betas is consistent with Equation (3.5), where it is the expected conditional beta that determines the unconditional expected return of a stock. The last step is to form estimators for the market premium and for the alpha of each portfolio. The estimators have the same form in both models: they are equal to the averages of the cross-sectional regression estimates: (3.7)

conditional model into an unconditional one. To do this, we take expectations in both sides of (3.4) to obtain: (3.5) In the static CAPM, this equation is replaced by: (3.6) is the unconditional beta. (3.5) and (3.6) are not equivalent: in the conditional model, what determines the unconditional expected return of a stock is its average conditional beta, rather than its unconditional beta. In fact, as shown in Appendix A1, the two quantities would be equal if both the conditional market premium and the conditional market variance were constant, but although we assume here that market returns have a constant first moment, we do not assume that they are homoskedastic (i.e. that they have a constant variance). where β i

Estimation and Measurement of Pricing Errors

Consider the following one-factor model associated with the fundamental CAPM. For each test portfolio, we have:

where β i,t-1 is the fundamental beta and the residuals are centred and uncorrelated from the market. Taking conditional expectations in both sides of this equation, we obtain:

Hence, Equation (3.4) holds if, and only if, α i = 0. Thus, we are interested in testing whether α i = 0. Second, we also want to estimate the unconditional market premium implied by the model in order to check that it is plausible in

and

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