Factor risk premiums and invested capital: calculations with stochastic discount factors

Abstract

Factor portfolios with value, size, momentum, profitability, and low volatility stocks have historically exhibited high returns after adjusting for market risk. As the weights of these portfolios increase in the stochastic discount factor, the excess returns of these factor strategies should decrease. We compare weights of these factor portfolios in the efficient set relative to which there are no factor risk premiums with market capitalization weights.

Appendix: Completion portfolios

We show the equivalence of the CAPM as one particular pricing kernel in the more general framework of Hansen and Jagannathan (1991).

We start with the Euler condition:

$$ E\left( {mR} \right) = 1, $$

(7)

where m is the candidate pricing kernel and R is a gross return of a test asset. Equation (7) says that when evaluating the expectations of the return payoffs of the portfolios using the pricing kernel, we obtain the price of the portfolios today—which is $1 as we work in gross returns.

Hansen and Jagannathan (1991) show how to construct the pricing kernel that satisfies Eq. (7). The pricing kernel’s volatility must be greater than a certain minimum threshold, and the pricing kernel with the minimum required volatility, m*, that solves Eq. (7) is given by

$$ m^{*} = w^{*} \cdot R, $$

(8)

where the optimal weights, w*, on the test portfolios are given by

$$ w^{*} = \frac{1}{{R_{f} }} + \left( {R - \bar{R}} \right)^{{\prime }} \varSigma_{R}^{ - 1} \left( {1 - \frac{1}{{R_{f} }}} \right). $$

(9)

In terms of computation, R is a T × N matrix of T observations of N asset gross returns; $\bar{R}$ is the average gross return of the portfolios; R _f is a given gross risk-free rate; and $\varSigma_{R}^{ - 1}$ is the empirical covariance matrix of R. In practice, the pricing kernel tends to place large weights on portfolios with high excess returns and low weights on low excess return portfolios, adjusted for risk in the covariance matrix, Σ _R . Hansen and Jagannathan (1991) derive Eq. (9) as an ordinary least squares (OLS) projection.

We compute the portfolio alphas using

$$ \alpha = \left( {\bar{R} - R_{f} } \right) - E^{m} \left( {R - R_{f} } \right), $$

(10)

where $E^{m} \left( {R - R_{f} } \right)$ is the excess return implied by the choice of the pricing kernel m. We evaluate

$$ E^{m} \left( {R - R_{f} } \right) = \frac{{{\text{cov}}\left( {R,m} \right)}}{E\left( m \right)}, $$

(11)

which is the same as Eq. (3).

One way to derive the constants a and b for the CAPM in Eq. (4) is as follows. The constant a pins down the risk-free rate and solves $E\left( {m_{\text{mkt}} } \right) = 1/R_{f}$. The constant b prices the market portfolio itself, $E\left( {m_{\text{mkt}} R_{\text{mkt}} } \right) = 1$.

When the optimal weights w* are normalized using $w^{*} /{\mathbf{sum}}(w^{*} )$, they represent the weights in the efficient portfolio (the tangency portfolio of the intersection of the Capital Allocation Line and the mean–variance frontier). In this case, the alphas in Eq. (5) are equal to zero.