EDHEC-Risk Institute October 2016

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

3. Applications of Fundamental Beta

As a result, (3.8) can be rewritten as:

is constant. This is more realistic, as it is well documented that some variables, including notably the dividend yield and the default spread, have power to predict stock returns, at least at long horizons (Fama and French, 1988, 1989; Hodrick, 1992; Menzly, Santos and Veronesi, 2004). As Jagannathan and Wang (1996) show, introducing a time-varying market premium implies that the unconditional expected return of a stock depends on its average conditional beta – as it already does in the conditional CAPM of Section 3.3.1 – but also on the covariance between the conditional beta and the conditional market premium. We thus want to test whether this covariance term improves the ability of the fundamental CAPM to explain the returns of portfolios sorted on size, book-to-market or short-term past return with respect to the case where the premium is constant. Derivation of Unconditional Model When the conditional market risk premium, λ t , varies over the business cycle, conditional expected stock returns are given by (see Equation (3.4)):

(3.9)

where the “beta-prem sensitivity” of an asset is defined as the sensitivity of its beta with respect to the predictive variable:

,

and the coefficient c prem is given by: .

Equation (3.9) says that the unconditional expected return on a stock is a linear function of its expected beta and its beta-prem. Ex-ante, stocks with higher average market exposures earn higher returns, which is the same intuition as in the static CAPM. But stocks that are more exposed to the market when the expected market return is higher, i.e. stocks that have a higher beta-prem, earn also higher expected returns: indeed, a large covariance between the conditional beta and the market premium makes a stock be like a “market follower”, and this increased market exposure requires compensation for bearing systematic risk. The coefficient c prem is the marginal premium earned for an additional unit of covariance between the beta and the market premium. To model the time-varying market premium, we follow Jagannathan and Wang (1996) in relating the expected market return to the yield spread between BAA and AAA bonds. This choice is motivated by the literature that has linked expected stock returns to the business cycle (Keim and Stambaugh, 1986; Fama and French, 1989), and the work of Stock and Watson (1989), which finds that the default spread is a good predictor of market conditions.

Taking expectations in both sides, Jagannathan and Wang (1996) obtain the following model for unconditional expected returns: (3.8)

where

Here

is the expected market risk

premium, and is the expected conditional beta. The model in (3.8) reduces to the one in Equation (3.5) if the market premium is uncorrelated from the beta.

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