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

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.     

CAPM over the long run: 1926–2001

A conditional one-factor model can account for the spread in the average returns of portfolios sorted by book-to-market ratios over the long run from 1926 to 2001. In contrast, earlier studies document strong evidence of a book-to-market effect using OLS regressions over post-1963 data. However, the betas of portfolios sorted by book-to-market ratios vary over time and in the presence of time-varying factor loadings, OLS inference produces inconsistent estimates of conditional alphas and betas. We show that under a conditional CAPM with time-varying betas, predictable market risk premia, and stochastic systematic volatility, there is little evidence that the conditional alpha for a book-to-market trading strategy is different from zero.     

Testing conditional factor models: A nonparametric approach

Some recent studies of conditional factor models do not specify conditioning information but use data from small windows to estimate the time series of conditional alphas and betas. In this paper, we propose a nonparametric method using an optimal window to estimate time-varying coefficients. In addition, we offer two empirical tests of a conditional factor model. Using our new method, we examine the performance of the conditional CAPM and the conditional Fama-French three-factor model in explaining the return variations of portfolios sorted by size, book-to-market ratios, and past returns, for which recent literature has generated controversial results. We find that, although in general the conditional FF model outperforms the conditional CAPM, both models fail to explain well-known asset-pricing anomalies. Moreover, for both models, the failure is more pronounced for the equally-weighted portfolios than for the value-weighted ones.     

Does the Conditional CAPM Work? Evidence from the Istanbul Stock Exchange

Using a sample of common stocks traded on the Istanbul Stock Exchange from February 1997 to April 2008, we test whether the conditional capital asset pricing model (CAPM) accurately prices assets. In our empirical analysis, we closely follow the methodology introduced in Lewellen and Nagel (2006). Our results show that the conditional CAPM fares no better than the static counterpart in pricing assets. Although market betas do vary significantly over time, the intertemporal variation is not large enough to drive average conditional alphas to zero.     

Conditional risk and performance evaluation: Volatility timing, overconditioning, and new estimates of momentum alphas

Unconditional alphas are biased when conditional beta covaries with the market risk premium (market timing) or volatility (volatility timing). We demonstrate an additional bias (overconditioning) that can occur any time an empiricist estimates risk using information, such as a realized beta, that is not available to investors ex ante. Calibrating to U.S. equity returns, volatility timing and overconditioning can plausibly impact alphas more than market timing, which has been the focus of prior literature. To correct market- and volatility-timing biases without overconditioning, we show that incorporating realized betas into instrumental variables estimators is effective. Empirically, instrumentation reduces momentum alphas by 20-40%. Overconditioned alphas overstate performance by up to 2.5 times. We explain the sources of both the volatility-timing and overconditioning biases in momentum portfolios.     

An International Investigation of the Factors that Determine Conditional Gold Betas

We investigate the unconditional and conditional gold betas of four country‐based gold industry portfolios. First, we document the similarity of unconditional gold betas across countries. Second, we find that the factors affecting conditional gold betas are different in the Australian and South African gold sectors relative to their North American counterparts. Only the gold bullion return volatility shows a negative association with conditional gold betas in Australian and South African gold mining firms. Moreover, gold price does not appear to play a systematic role in determining Australian or South African conditional gold betas. We discuss possible explanations for these findings.     

How to Discount Cashflows with Time-Varying Expected Returns

While many studies document that the market risk premium is predictable and that betas are not constant, the dividend discount model ignores time‐varying risk premiums and betas. We develop a model to consistently value cashflows with changing risk‐free rates, predictable risk premiums, and conditional betas in the context of a conditional CAPM. Practical valuation is accomplished with an analytic term structure of discount rates, with different discount rates applied to expected cashflows at different horizons. Using constant discount rates can produce large misvaluations, which, in portfolio data, are mostly driven at short horizons by market risk premiums and at long horizons by time variation in risk‐free rates and factor loadings.     

Investor learning and mutual fund family

In this paper we revisit the cross-fund learning method suggested by Jones and Shanken (2005) and construct a linear hierarchical model to consider the learning across funds within the fund family during the performance evaluation. We provide a full Bayesian treatment on all the factors of the pricing model and allow both the fund family and the individual manager to have dependent prior information regarding funds' alphas. The simulation results suggest that returns from peer funds within the family significantly affect investors' updating on fund alphas since the posterior distribution on fund alphas experiences a faster shrinkage than those reported in the previous literature. The model can also be simulated with specific prior belief on different factors of the pricing model, i.e. fund alphas, betas and factor loadings of each pricing benchmark, to better address the learning issue.     

The conditional CAPM does not explain asset-pricing anomalies

Recent studies suggest that the conditional CAPM holds, period by period, and that time-variation in risk and expected returns can explain why the unconditional CAPM fails. In contrast, we argue that variation in betas and the equity premium would have to be implausibly large to explain important asset-pricing anomalies like momentum and the value premium. We also provide a simple new test of the conditional CAPM using direct estimates of conditional alphas and betas from short-window regressions, avoiding the need to specify conditioning information. The tests show that the conditional CAPM performs nearly as poorly as the unconditional CAPM, consistent with our analytical results.     

Bias in estimating the systematic risk of extreme performers: Implications for financial analysis, the leverage effect, and long-run reversals

We show how bias can arise systematically in the beta estimates of extreme performers when long-run return reversals are present and partly, or wholly, due to sign changes in unanticipated factor realizations. Our evidence is consistent with this bias being responsible for the large shifts in the beta estimates of extreme performers, more so than the leverage effect, which has been the predominant explanation in prior literature. Bias in these contemporaneous realized betas, estimated with the same returns that are to be risk adjusted, arises due to the general problem of “overconditioning,” where betas are estimated conditional on information that is not yet known. Several methods for conditioning betas on out-of-sample returns are evaluated and found to be lacking, although some offer improvement under certain circumstances. We also show evidence of this bias in the Fama-French Three-factor loadings of extreme performers. Our findings indicate not only that previous studies of long-run reversals understate contrarian profits but that bias is prevalent in the OLS beta estimates of extreme performers, and this has implications for estimating the cost of capital and measuring long-run performance. We offer recommendations for identifying when this bias is likely present, as well as general methods to correct for it.     

Changes in Expected Security Returns,Risk, and the Level of Interest Rates

Regressions of security returns on treasury bill rates provide insight about the behavior of risk in rational asset pricing models. The information in one-month bill rates implies time variation in the conditional covariances of portfolios of stocks and fixed-income securities with benchmark pricing variables, over extended samples and within five-year subperiods. There is evidence of changes in conditional “betas” associated with interest rates. Consumption and stock market data are examined as proxies for marginal utility, in a general framework for asset pricing with time-varying conditional covariances.     

Asset Pricing with Conditioning Information: A New Test

This paper presents a new test of conditional versions of the Sharpe-Lintner CAPM, the Jagannathan and Wang (1996) extension of the CAPM, and the Fama and French (1993) three-factor model. The test is based on a general nonparametric methodology that avoids functional form misspecification of betas, risk premia, and the stochastic discount factor. Our results provide a novel view of empirical performance of these models. In particular, we find that a nonparametric version of the Fama and French model performs well, even when challenged by momentum portfolios.     

Measuring performance in a dynamic world: Conditional mean–variance fundamentals

We develop conditional alpha performance measures that are consistent with conditional mean–variance analysis and the magnitude and sign of the implied true conditional time-varying alphas. The sequence of conditional alphas and betas is estimable from surprisingly simple unconditional regressions. Other common performance measures are derivable from the conditional investment opportunity set based on its conditional asset return moments. Our bootstrap analysis of Morningstar mutual fund returns data demonstrates that the differences between existing conditional alpha measures and our proposed alpha are substantive for typical parameterizations.     

THE EFFECT OF MARKET PROXY REBALANCING POLICIES ON DETECTING ABNORMAL PERFORMANCE

In this paper, effects on the measured abnormal performance of test portfolios are compared against market proxies having the same or different rebalancing policies. Results show that the common practice of comparing buy-and-hold test portfolios with equally weighted market proxies produces lower Jensen [ 7 ] alphas and lower alpha t-values. Comparing buy-and-hold test portfolios with value-weighted market proxies produces higher portfolio betas and alphas, but lower alpha t-values. Finally, comparing buy-and-hold test portfolios with buy-and-hold market proxies produces the most powerful tests of abnormal performance.     

Semiparametric estimation of a characteristic-based factor model of common stock returns

We introduce an alternative version of the Fama–French three-factor model of stock returns together with a new estimation methodology. We assume that the factor betas in the model are smooth nonlinear functions of observed security characteristics. We develop an estimation procedure that combines nonparametric kernel methods for constructing mimicking portfolios with parametric nonlinear regression to estimate factor returns and factor betas simultaneously. The methodology is applied to US common stocks and the empirical findings compared to those of Fama and French.     

The intertemporal relation between expected returns and risk

This paper explores the time-series relation between expected returns and risk for a large cross section of industry and size/book-to-market portfolios. I use a bivariate generalized autoregressive conditional heteroskedasticity (GARCH) model to estimate a portfolio's conditional covariance with the market and then test whether the conditional covariance predicts time–variation in the portfolio's expected return. Restricting the slope to be the same across assets, the risk-return coefficient is highly significant with a risk–aversion coefficient (slope) between one and five. The results are robust to different portfolio formations, alternative GARCH specifications, additional state variables, and small sample biases. When conditional covariances are replaced by conditional betas, the risk premium on beta is estimated to be in the range of 3% to 5% per annum and is statistically significant.     

Conditional beta pricing models: A nonparametric approach

We propose a two-stage procedure to estimate conditional beta pricing models that allows for flexibility in the dynamics of asset betas and market prices of risk (MPR). First, conditional betas are estimated nonparametrically for each asset and period using the time-series of previous data. Then, time-varying MPR are estimated from the cross-section of returns and betas. We prove the consistency and asymptotic normality of the estimators. We also perform Monte Carlo simulations for the conditional version of the three-factor model of Fama and French (1993) and show that nonparametrically estimated betas outperform rolling betas under different specifications of beta dynamics. Using return data on the 25 size and book-to-market sorted portfolios, we find that the nonparametric procedure produces a better fit of the three-factor model to the data, less biased estimates of MPR and lower pricing errors than the Fama–MacBeth procedure with betas estimated under several alternative parametric specifications.     

Testing Conditional Asset Pricing Models Using a Markov Chain Monte Carlo Approach

We use Markov Chain Monte Carlo (MCMC) methods for the parameter estimation and the testing of conditional asset pricing models. In contrast to traditional approaches, it is truly conditional because the assumption that time variation in betas is driven by a set of conditioning variables is not necessary. Moreover, the approach has exact finite sample properties and accounts for errors‐in‐variables. Using S&P 500 panel data, we analyse the empirical performance of the CAPM and the Fama and French (1993) three‐factor model. We find that time‐variation of betas in the CAPM and the time variation of the coefficients for the size factor (SMB) and the distress factor (HML) in the three‐factor model improve the empirical performance. Therefore, our findings are consistent with time variation of firm‐specific exposure to market risk, systematic credit risk and systematic size effects. However, a Bayesian model comparison trading off goodness of fit and model complexity indicates that the conditional CAPM performs best, followed by the conditional three‐factor model, the unconditional CAPM, and the unconditional three‐factor model.     

Conditional Asset Pricing and Stock Market Anomalies in Europe

This study provides European evidence on the ability of static and dynamic specifications of the Fama‐French (1993) three‐factor model to price 25 size‐B/M portfolios. In contrast to US evidence, we detect a small‐growth premium and find that the size effect is still present in Europe. Furthermore, we document strong time variation in factor risk loadings. Incorporating these risk fluctuations in conditional specifications of the three‐factor model clearly improves its ability to explain time variation in expected returns. However, the model still fails to completely capture cross‐sectional variation in returns as it is unable to explain the momentum effect.     

A NEW APPROACH TO CONTROLLING FOR THIN TRADING

Recent research has shown that thin trading can seriously bias beta estimates. Present techniques for controlling this bias in research designs involve the adjustment of OLS betas. This paper presents a new methodology that controls for this bias by forming portfolios where the level of thin trading is held constant, while the difference of another variable, pertinent to a specific research design, is maximized across the portfolios. Directly controlling the level of thin trading avoids reliance on beta adjustment techniques. Further, the linear programming model permits the control of the mean and higher moments of additional variables across portfolios.