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 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.     

On Stable Factor Structures in the Pricing of Risk: Do Time-Varying Betas Help or Hurt?

There is now considerable evidence suggesting that estimated betas of unconditional capital asset pricing models (CAPMs) exhibit statistically significant time variation. Therefore, many have advocated the use of conditional CAPMs. If we succeed in capturing the dynamics of beta risk, we are sure to outperform constant beta models. However, if the beta risk is inherently misspecified, there is a real possibility that we commit serious pricing errors, potentially larger than with a constant traditional beta model. In this paper we show that this is indeed the case, namely that pricing errors with constant traditional beta models are smaller than with conditional CAPMs.     

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.     

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.     

An Unconditional Asset-Pricing Test and the Role of Firm Size as an Instrumental Variable for Risk

In an intertemporal economy where both risk (stock beta) and expected return are time varying, the authors derive a linear relation between the unconditional beta and the unconditional return under certain stationarity assumptions about the stochastic process of size-portfolio betas. The model suggests the use of long time periods to estimate the unconditional portfolio betas. The authors find that, after controlling for the betas thus estimated, a firm-size proxy, such as the logarithm of the firm size, does not have explanatory power for the averaged returns across the size-ranked portfolios.     

Local Risk,Local Factors,and Asset Prices

Firm location affects firm risk through local factor prices. We find more procyclical factor prices such as wages and real estate prices in areas with more cyclical economies, namely, high “local beta” areas. While procyclical wages provide a natural hedge against aggregate shocks and reduce firm risk, procyclical prices of real estate, which are part of firm assets, increase firm risk. We confirm that firms located in higher local beta areas have lower industry‐adjusted returns and conditional betas, and show that the effect is stronger among firms with low real estate holdings. A production‐based equilibrium model explains these empirical findings.     

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.     

Notes on dynamic factor pricing models

These notes discuss three aspects of dynamic factor pricing (i.e., APT) models. First, the diversifiable component of returns is unpredictable in a no-arbitrage world. Second, conditional factor loadings or betas have an unconditional factor structure when returns follow an unconditional factor structure, which provides a link between conditional and unconditional factor pricing models. Third, the estimation of dynamic factor pricing models is easily simplified in large cross sections when returns follow an unconditional factor structure. These results aid in the interpretation of existing applications and identify some of the issues in the formulation and estimation of dynamic factor pricing models.     

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.     

Asset Pricing with Countercyclical Household Consumption Risk

We show that shocks to household consumption growth are negatively skewed, persistent, countercyclical, and drive asset prices. We construct a parsimonious model where heterogeneous households have recursive preferences. A single state variable drives the conditional cross‐sectional moments of household consumption growth. The estimated model fits well the unconditional cross‐sectional moments of household consumption growth and the moments of the risk‐free rate, equity premium, price‐dividend ratio, and aggregate dividend and consumption growth. The model‐implied risk‐free rate and price‐dividend ratio are procyclical, while the market return has countercyclical mean and variance. Finally, household consumption risk explains the cross section of excess returns.     

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.     

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.     

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 Share of Systematic Variation in Bilateral Exchange Rates

Sorting countries by their dollar currency betas produces a novel cross section of average currency excess returns. A slope factor (long in high beta currencies and short in low beta currencies) accounts for this cross section of currency risk premia. This slope factor is orthogonal to the high‐minus‐low carry trade factor built from portfolios of countries sorted by their interest rates. The two high‐minus‐low risk factors account for 18% to 80% of the monthly exchange rate movements. The two risk factors suggest that stochastic discount factors in complete markets' models should feature at least two global shocks to describe exchange rates.     

Stochastic volatility and time-varying country risk in emerging markets

This study suggests an alternative method to estimate time-varying country risk. We first apply a new multivariate stochastic volatility (SV) model to a set of emerging stock markets. To estimate the SV model, we use a Bayesian Markov chain Monte Carlo simulation procedure. By applying the deviance information criterion, we show that the new model performs well relative to alternative multivariate SV models. We then compute the conditional betas for the different markets and compare the results with an often-used procedure based on multivariate GARCH models. We show that the new multivariate SV model more accurately captures the time-varying nature of country risk. The conditional betas show signs of large variations, indicating the importance of taking time-varying country risk into consideration when managing emerging market portfolios.     

Can unpredictable risk exposure be priced?

We study the link between beta predictability and the price of risk. An investor who desires exposure to a certain risk factor needs to predict what next period’s beta will be. We use a simple model to show that an ambiguity averse agent’s demand is lower when betas are hard to predict, leading to a reduction in risk premiums. We test the implications for downside betas and VIX betas. We find that they have economically and statistically small prices of risk once we account for the fact that an investor cannot observe ex-post realized betas when determining asset demand.     

Two-pass estimation of risk premiums with multicollinear and near-invariant betas

This paper investigates the reliability of the two-pass (TP) estimators of factor risk prices when betas (multifactor loadings) have high levels of cross-sectional correlation (multicollinearity) and/or when some of them have small cross-sectional variations (near-invariance). Our simulation results show the following. First, the TP estimators can have biases larger than 100% of true risk prices when data are generated by the betas with high levels of multicollinearity and invariance that can be observed from actual data. Second, the t-tests for hypotheses related to risk prices and pricing intercepts have only limited power. The levels of multicollinearity and invariance of betas can vary depending on the assets and sample periods used in estimation. Thus, we propose use of two pre-diagnostic statistics to measure these levels. Many previous studies have investigated the finite-sample properties of the TP estimators using the data generated with the estimated betas from actual data. Our results indicate that simulation outcomes can lead to quite different conclusions, depending on the levels of multicollinearity and invariance of the betas used to generate the data.     

Time-varying market,interest rate,and exchange rate risk premia in the US commercial bank stock returns

This paper examines the role of market, interest rate, and exchange rate risks in pricing a sample of the US Commercial Bank stocks by developing and estimating a multi-factor model under both unconditional and conditional frameworks. Three different econometric methodologies are used to conduct the estimations and testing. Estimations based on nonlinear seemingly unrelated regression (NLSUR) via GMM approach indicate that interest rate risk is the only priced factor in the unconditional three-factor model. However, based on ‘pricing kernel’ approach by Dumas and Solnik [(1995). J. Finance 50, 445–479], strong evidence of exchange rate risk is found in both large bank and regional bank stocks in the conditional three-factor model with time-varying risk prices. Finally, estimations based on the multivariate GARCH in mean (MGARCH-M) approach where both conditional first and second moments of bank portfolio returns and risk factors are estimated simultaneously show strong evidence of time-varying interest rate and exchange rate risk premia and weak evidence of time-varying world market risk premium for all three bank portfolios, namely those of Money Center bank, Large bank, and Regional bank.     

Sampling interval and estimated betas: Implications for the presence of transitory components in stock prices

We provide a theoretical framework to explain the empirical finding that the estimated betas are sensitive to the sampling interval even when using continuously compounded returns. We suppose that stock prices have both permanent and transitory components. The discrete time representation of the beta depends on the sampling interval and two components labeled “permanent and transitory betas”. We show that if no transitory component is present in stock prices then no sampling interval effect occurs. However, the presence of a transitory component implies that the beta is an increasing (decreasing) function of the sampling interval for more (less) risky assets. In our framework, assets are labeled risky if their “permanent beta” is greater than their “transitory beta” and vice versa for less risky assets. Simulations show that our theoretical results provide good approximations for the estimated betas in small samples. We provide empirical evidence about the presence of negative serial correlation and mean reversion in the returns of the portfolios considered. We discuss why our model is better able to provide an explanation for this sampling interval effect than other models in the literature.