Another Look at the Cross-section of Expected Stock Returns

Our examination of the cross-section of expected returns reveals economically and statistically significant compensation (about 6 to 9 percent per annum) for beta risk when betas are estimated from time-series regressions of annual portfolio returns on the annual return on the equally weighted market index. The relation between book-to-market equity and returns is weaker and less consistent than that in Fama and French (1992). We conjecture that past book-to-market results using COMPUS-TAT data are affected by a selection bias and provide indirect evidence.     

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.     

Systematic Risk and Empirical Research

We show here that risky asset returns generating processes stated in terms of factors which include both accounting and non-accounting based measures of risk (e.g. book to market ratios) imply, under fairly standard regularity conditions, that the Sharpe-Lintner-Black asset pricing model beta is a 'sufficient' statistic in the sense that it captures all important attributes of the returns generating process in a single number. We then derive the parametric relationship between betas based on inefficient index portfolios and betas based on the market or tangency portfolio. We demonstrate that the relationship between risky asset expected returns and betas computed on the basis of inefficient index portfolios is both consistent with the predictions of the Capital Asset Pricing Model and the multi-factor asset pricing models of Fama and French (1992, 1993, 1995 and 1996). The 'trick' is to realise that inefficient index portfolios are composed of the market portfolio and a collection of inefficient but self financing 'kernel' or 'arbitrage' portfolios. It then follows that there is a perfect linear cross sectional relationship between risky asset expected returns, betas based on inefficient index portfolios and the arbitrage portfolios. Hence, if we happen to stumble across variables that span the same subspace as the vectors representing the arbitrage portfolios, it is easy to create the illusion that risky asset expected returns depend on variables other than 'beta'.     

Beta,size, book-to-market equity and returns: A study based on UK data

This paper examines the role of beta, size and book-to-market equity as competing risk measurements in explaining the cross-sectional returns of UK securities for the period July 1980 through June 2000. The methodology of [Fama, E., French, K., 1992. The cross-section of expected stock returns. Journal of Finance 47, 427–467] and [Pettengill, G., Sundaram, S., Mathur, I., 1995. The conditional relation between beta and returns. Journal of Financial and Quantitative Analysis 30, 101–116] is adopted. Results show that, when adopting the methodology of [Pettengill, G., Sundaram, S., Mathur, I., 1995. The conditional relation between beta and returns. Journal of Financial and Quantitative Analysis 30, 101–116], where data is segmented between up and down markets, a significant relationship is found between beta and returns even in the presence of size and book-to-market equity. Size is not found to be a significant risk variable, whereas book-to-market equity is found to be priced by the market and is thus a significant determinant of security returns. This is the case irrespective of the methodology adopted.     

Size and Book-to-Market Factors in Earnings and Returns

We study whether the behavior of stock prices, in relation to size and book-to-market-equity (BE/ME), reflects the behavior of earnings. Consistent with rational pricing, high BE/ME signals persistent poor earnings and low BE/ME signals strong earnings. Moreover, stock prices forecast the reversion of earnings growth observed after firms are ranked on size and BE/ME. Finally, there are market, size, and BE/ME factors in earnings like those in returns. The market and size factors in earnings help explain those in returns, but we find no link between BE/ME factors in earnings and returns.     

An Empirical Re-Examination of the Cross-Section of Expected Returns: UK Evidence

This paper is a study of the Fama and French (1992) analysis in the UK context. Consistent with their findings, our results do not support a positive relationship between beta and average monthly returns. We find that book-to-market equity and market leverage are consistently significant in explaining UK average returns. Contrary to the Fama-French evidence, size has an insignificant effect on average returns. A puzzling negative beta-returns relationship is found in some monthly regressions,and results based on annual data reveal a reversal of betas for the smallest-size portfolios. Some possible explanations are offered for these findings.     

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.

Time-varying betas and the cross-sectional return–risk relation: evidence from the UK

The seminal study by Fama and MacBeth in 1973 initiated a stream of papers testing for the cross-sectional relation between return and risk. The debate as to whether beta is a valid measure of risk was reanimated by Fama and French and subsequent studies. Rather than focusing on exogenous variables that have a larger explanatory power than an asset's beta in cross-sectional tests, the matrix of variances-covariances is assumed to follow a time varying ARCH process. Using monthly data from the UK market from February 1975 to December 1996, the cross-sectional return–risk relations obtained with an unconditional specification for assets’ betas are compared to those obtained when the estimated betas are based on an ARCH model. The approach taken by Pettengill, Sundaram and Mathur, which allows a negative cross sectional return–risk relation in periods in which the market portfolio yields a negative return relative to the risk free rate, was also investigated. These tests are also carried out on samples pertaining to a specific month and on samples from which a particular month is removed. Results suggest that the CAPM holds better in downward moving markets than in upward moving markets hence beta is a more appropriate measure of risk in bear markets.     

Positive return premia in Japan

This paper examines Jensen's [J. Finance, 1968, 23, 389–416] alphas and the time-varying return premia unexplained by standard risk factors in Japan and presents several new findings. First, in contrast to the US experience, positive alphas remain after Fama and French's three factors are applied to excess stock returns in Japan. Second, positive alphas remain in Japan, even if the Fama–French three factors combined with momentum and reversal factors are applied to excess stock returns. Third, the positive return premia unexplained by these five factors bear little relation to the dynamics of the Japanese macroeconomy. Fourth, the time series evolution of the positive return premia indicates autonomous dynamics with at least three regimes. Fifth, we can predict or time the acquisition of the positive return premia for small-size portfolios in Japan by observing the direction and effect of the return premia of large-size portfolios and high-book equity to market equity (BE/ME) portfolios. Finally, application of the self-exciting threshold autoregressive (SETAR) model shows that the size effects are stronger than the BE/ME effects in Japan, given that the return premia from small-size portfolios in the SETAR model are bounded by positive thresholds, while the return premia from high-BE/ME portfolios are bounded by negative thresholds.     

Equilibrium "Anomalies"

Many empirical “anomalies” are actually consistent with the single beta capital asset pricing model if the empiricist utilizes an equity‐only proxy for the true market portfolio. Equity betas estimated against this particular inefficient proxy will be understated, with the error increasing with the firm's leverage. Thus, firm‐specific variables that correlate with leverage (such as book‐to‐market and size) will appear to explain returns after controlling for proxy beta simply because they capture the missing beta risk. Loadings on portfolios formed on relative leverage and relative distress completely subsume the powers of the Fama and French (1993) returns to small minus big market capitalization (SMB) portfolios and returns to high minus low book‐to‐market (HML) portfolios factors in explaining cross‐sectional returns.     

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.     

Does beta react to market conditions? Estimates of ‘bull’ and ‘bear’ betas using a nonlinear market model with an endogenous threshold parameter

The authors use a logistic smooth transition market (LSTM) model to investigate whether ‘bull’ and ‘bear’ market betas for Australian industry portfolios returns differ. The LSTM model allows the data to determine a threshold parameter that differentiates between ‘bull’ and ‘bear’ states, and it also allows for smooth transition between these two states. Their results indicate that ‘bull’ and ‘bear’ betas are significantly different for most industries, and that up-market risk is not always lower than down-market risk. LSTM models indicate that the transition between ‘bull’ and ‘bear’ states is abrupt, supporting a dual-beta market modelling framework.     

Is beta still alive? Conclusive evidence from the Swiss stock market

Recent evidence from Fama and French (1992, 1996) and others shows that betas and returns are not related empirically. They interpret this as evidence against the validity of the capital asset pricing model and conclude that the beta is not a good measure of risk. This paper claims that usual tests do not leave much opportunity for beta to appear as a useful variable capable of explaining returns, because tests are often performed in periods where the average realized market excess return is not significantly different from zero. In order to assess the usefulness of beta, an alternative approach that dissociates results obtained in periods where the realized market excess is positive from those where it is negative is proposed. These new tests are then applied to a representative sample of the Swiss stock market over the period 1983–1991. The different results unambiguously support the fact that beta is a good measure of risk, because beta is strongly related to the cross-section of realized returns. These results also confirm that there are no arbitrage opportunities on this market.     

Reexamining the Robustness of the Market Value of Equity

We apply the methodology of Knez and Ready (KR) (1997) to data from the Japanese stock market and reexamine the robustness of the risk premium for the market value of equity (MVE). In particular, we compare two alternative explanations for the relation between stock returns and MVE: the one pointed out by Fama and French (FF) (1992) and the other proposed by Berk (1995). Consistent with results for the U.S. market, when we check FF's explanation for MVE, we find that the risk premium for MVE is not robust against extreme observations. Besides the evidence supporting KR's findings, we study the role of MVE proposed by Berk (1995), who points out that under controlled expected cash flows, MVE will be negatively correlated with expected returns. After showing that MVE negatively correlates with risk in the presence of expected cash flows, we test the robustness of the relation between returns and MVE. We find that the estimated risk premium for MVE is robust when realized cash flows (earnings plus depreciation) or book value of equity (BE) is used as a proxy for expected cash flows.     

Risk Interpretation of the CAPM's Beta: Evidence from a New Research Method

This study tests the validity of using the CAPM beta as a risk control in cross‐sectional accounting and finance research. We recognize that high‐risk stocks should experience either very good or very bad returns more frequently compared to low‐risk stocks, that is, high‐risk stocks should cluster in the tails of the cross‐sectional return distribution. Building on this intuition, we test the risk interpretation of the CAPM's beta by examining if high‐beta stocks are more likely than low‐beta stocks to experience either very high or very low returns. Our empirical results indicate that beta is a strong predictor of large positive and large negative returns, which confirms that beta is a valid empirical risk measure and that researchers should use beta as a risk control in empirical tests. Further, we show that because the relation between beta and returns is U‐shaped, that is, high betas predict both very high and very low returns, linear cross‐sectional regression models, for example, Fama–MacBeth regressions, will fail on average to reject the null hypothesis that beta does not capture risk. This result explains why previous studies find no significant cross‐sectional relation between beta and returns.     

Cross-sectional determinants of New Zealand share market returns

We present evidence of the cross-sectional relation between security returns, beta, firm size and book-to-market ratio over the period 1971 to 1993 on the New Zealand sharemarket. Our results suggest that the NZSE-40 market index is not a mean-variance efficient market proxy—the betas calculated with respect to it being of little use for explaining expected returns cross-sectionally. Also, there is a significant positive relation between book-to-market ratio and average return.     

The Conditional Relationship Between Beta and Returns: A Reassessment

Abstract:  Several recent empirical tests of the Capital Asset Pricing Model have been based on the conditional relationship between betas and market returns. This paper shows that this method needs reconsideration. An adjusted version of this test is presented. It is then demonstrated that the adjusted technique has similar, or lower, power to the more easily implemented CAPM test of Fama and MacBeth (1973) if returns are normally distributed.     

The risk–return relations in the Singapore stock market

This paper examines the risk–return relations in the Singapore stock market for the period April 1986 to December 1998. Though beta is significantly related to realized returns, the explanatory power is low. Adding other stock characteristics such as skewness and kurtosis provides limited incremental benefits. However, when a conditional framework based on up and down markets is introduced, the explanatory power increases more than 100 times and there is a significant positive (negative) relation between beta and returns when the market excess returns are positive (negative). The same relation applies when unsystematic risk, total risk and kurtosis are added separately to the beta–return relation during up and down markets with increased explanatory power. Our results indicate that other stock characteristics in addition to beta are also important in pricing risky assets and investors do not hold diversified portfolios. Our results are also checked and compared with another conditional model with time-varying betas conditional on a set of economic variables.     

Security returns, beta, size, and book-to-market equity: evidence from the Shanghai A-share market

The main purpose of this paper is to explore the cross-sectional relationship between security returns and beta, size and book-to-market equity in the Shanghai A-share market. This study takes place during the period January 1997–December 2006. The methodology of Fama and French (J Finance 51:55–84, 1992) and Pettengill et al. (J Financial Quant Anal 30:101–116, 1995) is adopted. The Results show no evidence of an unconditional relationship between beta and returns. However, a conditional relationship is found when the data is split into up and down markets. The relationship holds even in the presence of size and book-to-market equity. Both size and book-to-market equity is found to be priced by the market and thereby regarded as significant determinants of security returns.     

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.