EDHEC-Risk Institute October 2016

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

3. Applications of Fundamental Beta

Hence, we assume that λ t function of the spread:

is an affine

,

.

and

.

This implies that: 11

(3.10)

Estimates for the average alpha and the average market premium are shown in Table 9. Interestingly, the average alpha is smaller than for the three alternative pricing models considered in Section 3.3.1 (see Table 6): the average alpha is reduced by almost half (from 2.86% to 1.69%) with respect to the fundamental CAPM with a constant market premium. The estimated average market premium is greater than before (it grows from 7.14% to 8.59%) but it remains at a reasonable level. Overall, the introduction of a time- varying premium helps explain the cross- section of expected returns. Table 11 compares the distributions of alphas across the 30 portfolios for the four competing models. The distribution for the conditional CAPM with time- varying market premium shifts to the left and is closer to 0 than for the model with a constant premium. Furthermore, alpha becomes less significant in the fundamental CAPM with a time-varying premium.

where:

The coefficient is the beta of the conditional beta with respect to the default spread, and c sp is the sensitivity of the expected return of a stock with respect to its exposure to changes in the default spread. Model Estimation and Tests The conditional CAPM holds if, and only if, the α i in the following one-factor model is zero for each stock i : We again use Fama-MacBeth procedure. In the first step, we estimate the time-series of fundamental betas for each stock, and we compute the beta-prem sensitivity in Equation (3.10). In the second step, we regress at each date the cross-section of stock returns on the average fundamental beta and the beta-prem sensitivity, in line with Equation (3.9):

11 - Note that Jagannathan and Wang (1996) use a different form for the unconditional model corresponding to the conditional model of Equation (3.4). Starting from (3.10), they write the unconditional expected return as a function of the unconditional market beta and the beta with respect to the default spread. We work directly with (3.10) because we have an explicit model for the conditional beta (as a function of the stock’s attributes), which allows for a direct estimation of the sensitivity of the beta with respect to the predictive variable.

(3.11)

By averaging the estimates over time, we obtain estimates and for the average market premium and the coefficient c sp . t and t

,

73

An EDHEC-Risk Institute Publication