Learning about beta: Time-varying factor loadings, expected returns, and the conditional CAPM

Abstract

We amend the conditional CAPM to allow for unobservable long-run changes in risk factor loadings. In this environment, investors rationally "learn" the long-run level of factor loadings from the observation of realized returns. As a consequence of this assumption, we model conditional betas using the Kalman filter. Because of its focus on low-frequency variation in betas, our approach circumvents recent criticisms of the conditional CAPM. When tested on portfolios sorted by size and book-to-market, our learning-augmented conditional CAPM passes the specification tests. * Federal Reserve Bank of New York and Swiss Finance Institute at the University of Lugano, respectively. Fama and French (1992) present compelling evidence that the unconditional captial asset pricing model (CAPM) does not account for returns on portfolios sorted by size and book-to-market (B/M). Since then, the literature on asset pricing has developed alternative theories, which depart from the original model along several dimensions. Promising avenues of research, which preserve the single-factor structure, have been conditional versions of the CAPM. The idea behind this approach is that, even though CAPM holds conditionally on time t information, it does not hold unconditionally. Accordingly, the poor empirical performance of CAPM might be due to its failure to account for time variation in conditional moments.Among the many studies of the conditional CAPM, recent implementations are put forward by Jagannathan and Wang (1996), Ferson and Harvey (1999), and Lettau and Ludvigson (2001). Earlier papers include Ferson, Kandel, and Stambaugh (1987), Harvey (1989). In all cases, the authors explicitly model the evolution of the conditional distribution of returns as a function of lagged state variables.They specify the covariance between the market return and portfolio returns as affine functions of these variables. This specification is estimated as a multifactor model, in which the additional factors are the interactions between the market return and the state variables.A recent paper by Lewellen and Nagel (2006) casts doubts on the empirical success of this approach. While acknowledging that betas vary considerably over time, these authors suggest that the covariation between sample estimates of betas and the market risk premium is not large enough to justify the deviations from the unconditional CAPM observed for value and momentum portfolios (Fama and French (1993); Jegadeesh and Titman (1993)).They argue that the good empirical performance of previous conditional studies is due to their cross-sectional design-which ignores key theoretical restrictions on the estimated slope coefficients-and suggest time-series regressions instead.In this paper, we complement the conditional CAPM literature by modeling a new type of time variation in conditional betas. There is substantial evidence that the risk of some asset classes has experienced long-run movements. For example, CAPM regressions on CRSP 1 data suggest that value stocks had higher be...

Factor-Loading Uncertainty and Expected Returns

“…Adrian and Franzoni (2009) find that persistence in betas range from -0.26 for their large-growth portfolio to 0.35 for their large-value portfolio and that the variance in shocks to betas range from 0.01 for their large-growth portfolio to 0.66 for their small-value portfolio. Moreover, while mapping these empirical estimates to our model parameters, one must keep in mind that these estimates are based on portfolio returns, which are likely to substantially underestimate the volatility of firm-level factor loadings.…”

“…They endogenously derive a conditional CAPM in which the market risk premium, market volatility, and systematic risk of stocks depend on the information available to investors. Adrian and Franzoni (2009) extend the conditional CAPM by introducing unobservable longrun changes in factor loadings. They show that when investors learn about these long-run changes over time, the model generates low-frequency variation in betas that can help explain 6 Pástor and Veronesi (2003) show that there is no effect of uncertainty about profitability on expected returns in the case of no dividends, and a small, local effect in the presence of dividends.…”

“…The specification in equation (21) is a special case of Adrian and Franzoni (2009) applied to our log-linear setting, and, as discussed in their paper, nests a number of conditional CAPM models as special cases (including Jagannathan and Wang 1996;Lettau and Ludvigson 2001;Jostova and Philipov 2005;Ang and Chen 2007).…”

Intertemporal Information Acquisition and Investment Dynamics

“…It is not the shadow price of a constraint to pay attention to information that an individual already 1 Other papers on robust control are e.g., Hansen, Sargent and Tallarini [3], Hansen and Sargent ( [4], [5]) and Hansen, Sargent and Wang [6]). Learning is also a central concept to other parts of the recent finance literature such as Pastor and Veronesi [7] (learning about profitability), Lewellen and Shanken [8] (learning and asset pricing tests), Adrian and Franzoni [9] (learning about time-varying factor loadings). 2 Wang [15] considers a dynamic setup with asymmetric information but keeps information exogenous.…”

Formal Identification of Sentiment Effects in Asset Markets

“…4 Indeed, Buss, Schlag and Vilkov (2009) show that a simple CAPM can be defended when sentiment effects are either absent or controlled for. 5 Recent studies by Kumar, Srescu, Boehme and Danielsen (2008) and Adrian and Franzoni (2009) demonstrate that once the estimation risk associated with beta and the risk premium are accounted for, the conditional CAPM has significant explanatory power in the cross-section of stock returns.…”