Testing conditional factor models

Using nonparametric techniques, we develop a methodology for estimating and testing conditional alphas and betas and long-run alphas and betas, which are the averages of conditional alphas and betas, respectively, across time. The estimators and tests can be implemented for a single asset or jointly across portfolios. The traditional Gibbons, Ross, and Shanken (1989) test arises as a special case of no time variation in the alphas and factor loadings and homoskedasticity. As applications of the methodology, we estimate conditional CAPM and multifactor models on book-to-market and momentum decile portfolios. We reject the null that long-run alphas are equal to zero even though there is substantial variation in the conditional factor loadings of these portfolios.     

Macroeconomic risk and the cross-section of stock returns

We develop a conditional version of the consumption capital asset pricing model (CCAPM) using the conditioning variable from the cointegrating relation among macroeconomic variables (dividend yield, term spread, default spread, and short-term interest rate). Our conditioning variable has a strong power to predict market excess returns in the presence of competing predictive variables. In addition, our conditional CCAPM performs approximately as well as Fama and French’s (1993) three-factor model in explaining the cross-section of the Fama and French 25 size and book-to-market sorted portfolios. Our specification shows that value stocks are riskier than growth stocks in bad times, supporting the risk-based story.     

A Cross-Sectional Empirical Test of a Dual-State Multi-Factor Pricing Model

During empirical testing of the Capital Asset Pricing Model an assumption is typically made that risk is intertemporally constant. However, prior research finds that risk changes over time. We empirically test a conditional dual-state cross-sectional model allowing risk to change through prior identification of different market and economic states. We examine relationships between returns and conditional market and economic-factor betas, size, book-to-market equity, and earnings-price ratios. We find that relationships shift across regimes, suggesting the importance of a conditional, as opposed to unconditional, model. Relationships also change in January.     

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.     

The Conditional Beta and the Cross-Section of Expected Returns

We examine the cross-sectional relation between conditional betas and expected stock returns for a sample period of July 1963 to December 2004. Our portfolio-level analyses and the firm-level cross-sectional regressions indicate a positive, significant relation between conditional betas and the cross-section of expected returns. The average return difference between high- and low-beta portfolios ranges between 0.89% and 1.01% per month, depending on the time-varying specification of conditional beta. After controlling for size, book-to-market, liquidity, and momentum, the positive relation between market beta and expected returns remains economically and statistically significant.     

Cross-sectional analysis of Swedish stock returns with time-varying beta: the Swedish stock market 1983–96

This paper analyses the ability of beta and other factors, like firm size and book-to-market, to explain cross‐sectional variation in average stock returns on the Swedish stock market for the period 1983–96. We use a bivariate GARCH(1,1) process to estimate time-varying betas for asset returns. The estimated variances of these betas, derived from a Taylor series approximation, are used for correcting errors in variables. An extreme bound analysis is utilized for testing the sensitivity of the estimated coefficients to changes in the set of included explanatory variables.
Our results show that the estimated conditional beta is a more accurate measure of the true market beta than the beta estimated by OLS. The coefficient for beta is not significantly different from zero, while the variables book-to-market and leverage have significant coefficients, and the latter coefficients are also robust to model specification. Excluding the down turn 1990–92 from the sample shows that the significance of the risk premium for leverage might be considered as an industry effect during this extreme period. Finally, we find a close dependence between the risk premium for beta and that for size and book-to-market. The omission of each of these variables may cause statistical bias in the estimated coefficient for beta.     

U.S. Banking Sector Risk in an Era of Regulatory Change: A Bivariate GARCH Approach

This paper assesses the impact of regulatory change on the risk and returns of the U.S. banking industry. The impact of five major regulatory changes on banking sector risk was assessed using daily data for eighteen major U.S. regional banks, money center banks and savings and loan type depository institutions. Risk in this case was proxied via the use of an M-GARCH model which generates time dependent conditional beta estimates. The evidence obtained suggests that the impact of deregulation and reregulation on banking sector risk is case specific. Further, the results obtained show that the market model incorporating dummy variables, which has proven so popular amongst existing studies, discards important information about the variability of beta which the time varying conditional betas capture.     

Are the Latent Variables in Time-Varying Expected Returns Compensation for Consumption Risk?

Multibeta asset pricing models are examined using proxies for economic state variables in a framework which exploits time-varying expected returns to estimate conditional betas. Examples include multiple consumption-beta models and models where asset returns proxy for the state variables. When the state variables are not specified, the tests indicate two or three time-varying expected risk premiums in the sample of quarterly asset returns. Conditional betas relative to consumption generate less striking evidence against the model than betas relative to asset returns, but both the consumption and the market variables fail to proxy for the state variables.     

The Risk and Predictability of International Equity Returns

We investigate predictability in national equity market returns,and its relation to global economic risks. We show how to consistentlyestimate the fraction of the predictable variation that is capturedby an asset pricing model for the expected returns. We use amodel in which conditional betas of the national equity marketsdepend on local information variables, while global risk premiadepend on global variables. We examine single- and multiple-betamodels, using monthly data for 1970 to 1989. The models capturemuch of the predictability for many countries. Most of thisis related to time variation in the global risk premia.     

Is value riskier than growth?

We study the relative risk of value and growth stocks. We find that time-varying risk goes in the right direction in explaining the value premium. Value betas tend to covary positively, and growth betas tend to covary negatively with the expected market risk premium. Our inference differs from that of previous studies because we sort betas on the expected market risk premium, instead of on the realized market excess return. However, we also find that this beta-premium covariance is too small to explain the observed magnitude of the value premium within the conditional capital asset pricing model.     

Size,time-varying beta,and conditional heteroscedasticity in UK stock returns

The purpose of this study is to examine the relationship between firm size and time-varying betas of UK stocks. We extend the Schwert and Seguin (1990)(Journal of Finance 45, 1120–1155) methodology by explicitly modeling conditional heteroscedasticity in the market model residual returns. Our results show that the time-varying coefficient is not statistically significant for both small and large firm stock indexes. We also find that accounting for GARCH effects in the Schwert-Seguin market model yields beta estimates that are markedly differently from those when conditional heteroscedasticity is ignored. Event studies that ignore conditional heteroscedasticity may bias the abnormal returns of small and large firms, thereby leading to a different conclusion regarding the significance of an information event.     

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.     

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.     

The Conditional CAPM and the Cross-Section of Expected Returns

Most empirical studies of the static CAPM assume that betas remain constant over time and that the return on the value-weighted portfolio of all stocks is a proxy for the return on aggregate wealth. The general consensus is that the static CAPM is unable to explain satisfactorily the cross-section of average returns on stocks. We assume that the CAPM holds in a conditional sense, i.e., betas and the market risk premium vary over time. We include the return on human capital when measuring the return on aggregate wealth. Our specification performs well in explaining the cross-section of average returns.     

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.     

A semiparametric conditional capital asset pricing model

This paper proposes using a functional coefficient regression technique to estimate time-varying betas and alpha in the conditional capital asset pricing model (CAPM). Functional coefficient representation relaxes the strict assumptions regarding the structure of betas and alpha by combining the predictors into an index. Appropriate index variables are selected by applying the smoothly clipped absolute deviation penalty. In such a way, estimation and variable selection can be done simultaneously. Based on the empirical studies, the proposed model performs better than the alternatives in explaining asset returns and we find no strong evidence to reject the conditional CAPM.     

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.     

Are TIPS the “real” deal?: A conditional assessment of their role in a nominal portfolio

This paper documents predictable time-variation in the real return beta of US Treasury Inflation Protected Securities (TIPS) and in the Sharpe ratios of both indexed and conventional bonds. The conditional mean and volatility of both bonds and their conditional correlation first are estimated from predetermined variables. These estimates then are used to compute conditional real return betas and Sharpe ratios. The time-variation in real return betas and the correlation between TIPS and nominal bonds coincides with major developments in the fixed-income market. One implication of this predictability is that portfolio managers can assess more efficiently the risk of investing in TIPS versus conventional bonds. Conditional Sharpe ratios indicate that over the sample period, TIPS had superior volatility-adjusted returns relative to nominal bonds. This finding is striking in view of the absence of a major inflation scare during the sample period from February 1997 through August 2001, but is loosely consistent with the possibility that TIPS elevated rather than reduced Treasury borrowing costs. On the other hand, mean–variance spanning tests indicate that TIPS did not enhance the mean–variance efficiency of diversified portfolios.     

Estimating Systematic Risk Using Time Varying Distributions

This article proposes a dynamic vector GARCH model for the estimation of time-varying betas. The model allows the conditional variances and the conditional covariance between individual portfolio returns and market portfolio returns to respond asymmetrically to past innovations depending on their sign. Covariances tend to be higher during market declines. There is substantial time variation in betas but the evidence on beta asymmetry is mixed. Specifically, in 50% of the cases betas are higher during market declines and for the remaining 50% the opposite is true. A time series analysis of estimated time varying betas reveals that they follow stationary mean-reverting processes. The average degree of persistence is approximately four days. It is also found that the static market model overstates non-market or, unsystematic risk by more than 10%. On the basis of an array of diagnostics it is confirmed that the vector GARCH model provides a richer framework for the analysis of the dynamics of systematic risk.     

The cross-sectional relationship between stock returns and domestic and global factors in the Chinese A-share market

By using an extension of the Fama and MacBeth cross-sectional regression model, this analysis examines the relationship between stock returns and (i) a local beta, (ii) two global betas, and (iii) some firm-specific characteristics in the Chinese A-share market. The results of the analysis suggest that neither the conditional local beta nor the global betas has a significant relationship with stock returns in A-shares. Our findings indicate that firm factors, such as the book-to-market ratio and firm size, are important in explaining stock returns. However, the size effect is sensitive to the specification of the model. Finally, the results of sub-period tests indicate that the A-share market did not become increasingly integrated with either the world stock markets or the Hong Kong stock market over the period 1995–2002.