*We thank Bill Christie (the Editor) and two other anonymous referees for their extremely helpful comments and suggestions. We also benefited from discussions with Hadiye Aslan, Ozgur Demirtas, Armen Hovakimian, Robert Whitelaw, and seminar participants at Baruch College, Graduate School, and University Center of the University of New York, and the 2007 Financial Management Association meetings. We also thank Kenneth French for making a large amount of historical data publicly available in his online data library*.

# The Conditional Beta and the Cross-Section of Expected Returns

Article first published online: 28 APR 2009

DOI: 10.1111/j.1755-053X.2009.01030.x

© 2009 Financial Management Association International.

Additional Information

#### How to Cite

Bali, T. G., Cakici, N. and Tang, Y. (2009), The Conditional Beta and the Cross-Section of Expected Returns. Financial Management, 38: 103–137. doi: 10.1111/j.1755-053X.2009.01030.x

#### Publication History

- Issue published online: 28 APR 2009
- Article first published online: 28 APR 2009

- Abstract
- Article
- References
- Cited By

### Abstract

- Top of page
- Abstract
- I. Data and Variable Definitions
- II. Average Returns and FF-3 Alphas on Beta Portfolios
- III. Firm-Level Cross-Sectional Regressions
- IV. Robustness Check
- V. Results from Size/BM/Beta Portfolios
- VI. Cross-Sectional Implications of the Conditional CAPM
- VII. Conclusion
- References

*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.*

The Sharpe (1964), Lintner (1965), and Black (1972) capital asset pricing model (CAPM) implies the mean-variance efficiency of the market portfolio in the sense of Markowitz (1959). The primary implication of the CAPM is that there is a positive linear relation between expected returns on securities and their market betas, and that variables other than beta should not capture the cross-sectional variation in expected returns. However, over the last three decades, many studies have tested the empirical performance of the static (or unconditional) CAPM in explaining the cross-section of realized average returns. The findings of these earlier studies indicate that firm size, book-to-market ratio, earnings-to-price ratio, liquidity, and momentum have significant explanatory power for average stock returns, but that market beta has little or no power.

Early tests of the CAPM are based on the cross-sectional regressions of average stock returns on estimates of individual stock betas. Two obvious problems with these tests are errors-in-variables and residual correlations. First, beta estimates for individual stocks are imprecise and generate a measurement error problem when they are used to explain average returns. To improve the accuracy of estimated betas, Blume (1970), Friend and Blume (1970), and Black, Jensen, and Scholes (1972) use portfolios instead of individual stocks in their cross-sectional tests. Since estimates of betas for diversified portfolios are more precise than estimates for individual stocks, using portfolios in the cross-section regressions of average returns on betas diminishes the errors-in-variables problem.

Second, the regression residuals have common sources of variation. Positive correlation in the residuals yields downward bias in the usual ordinary least squares (OLS) estimates of the standard errors of the cross-sectional regression slopes. Fama and MacBeth (1973) introduce a method for addressing the inference problem caused by correlation of the residuals in cross-sectional regressions. Rather than estimating a single cross-section regression of average monthly returns on betas, they estimate month-by-month cross-section of regressions of monthly returns on betas. The time-series averages of the monthly slopes and intercepts and their standard errors are used to test whether the average market risk premium is positive and the average intercept is equal to the risk-free rate.

In cross-sectional tests, Douglas (1969), Black, Jensen, and Scholes (1972), Miller and Scholes (1972), Blume and Friend (1973), and Fama and MacBeth (1973) find that the average slope coefficient on beta is less than the average excess market return and the intercept is greater than the average risk-free interest rate. In their widely cited study, Fama and French (1992) examine the static version of the CAPM and find both at the firm and portfolio level that the cross-sectional relation between market beta and average return is flat.^{1} They interpret this flat relation as strong empirical evidence against the CAPM.

As indicated by Jagannathan and Wang (1996), although a flat relation between the unconditional expected return and the unconditional market beta may be evidence against the static CAPM, it is not necessarily evidence against the conditional CAPM. The CAPM was originally developed within the framework of a hypothetical single-period model economy. The real world, however, is dynamic and hence, expected returns and betas are likely to vary over time. Even when expected returns are linear in betas for every time period, based on the information available at the time, the relation between the unconditional expected return and the unconditional beta could be flat.^{2}

There is substantial empirical evidence that conditional betas and expected returns depend on the nature of the information available at any given point in time and vary over time.^{3}

In this paper, we investigate whether time-varying conditional betas can explain the cross-section of expected returns at the firm and portfolio level. There is substantial empirical evidence that conditional betas and expected returns depend on the nature of the information available at any given point in time and vary over time. Earlier studies use either a single or rolling long sample of monthly data in estimating beta. Instead, we use daily returns within a month to compute realized beta for each stock trading at the New York Stock Exchange (NYSE), American Stock Exchange (Amex), and Nasdaq for our sample period of July 1963 to December 2004. We propose three alternative specifications of expected future beta based on the past information on realized beta using autoregressive, moving average, and generalized autoregressive conditional heteroskedasticity (GARCH)-in-mean models to obtain time-varying conditional betas for each stock.

We estimate conditional betas by using the entire history of returns on a stock. Hence, the high- and low-conditional beta portfolios we form cannot be exactly replicated by an investor at any given point in time. Our focus is more in the nature of a hypothesis test that has asymptotic validity. Thus, our approach is somewhat different from standard practice, which identifies ex ante measures of risk based on information available at a given point in time that a particular portfolio will earn a higher return on average than another portfolio.

For each specification of conditional beta, we find that stocks with high (low) market betas have, on average, high (low) average returns. Our portfolio-level analyses and the firm-level cross-sectional regressions indicate that the positive relation between the conditional betas and the cross-section of average returns is economically and statistically significant. Average portfolio returns increase almost monotonically when moving from low- to high-beta portfolios. The *R*^{2} values from the regression of average portfolio returns on average portfolio betas are in the range of 82% to 98% for 10, 20, 50, and 100 beta portfolios. When we form the equal-weighted decile portfolios by sorting the NYSE/Amex/Nasdaq stocks based on conditional beta, we find that the average return difference between decile 10 (high beta) and decile 1 (low beta) portfolios ranges between 0.89% and 1.01% per month, depending on the time-varying specification of conditional beta. For 20, 50, and 100 beta portfolios, the average return difference ranges from 1.01% to 1.23% per month.

To check whether our findings are driven by small, illiquid, and low-price stocks, we exclude the Amex and Nasdaq stocks and form the beta portfolios by sorting only the NYSE stocks based on the conditional betas. The results indicate that excluding the Amex and Nasdaq sample has almost no effect on our original findings. We also control for the well-known cross-sectional effects, including size and book-to-market (Fama and French, 1993, 1995, 1996), liquidity (Amihud, 2002; Pástor and Stambaugh, 2003), and past return characteristics (Jegadeesh and Titman, 1993). After controlling for these effects, we estimate the cross-sectional beta premium as being in the range of 0.86% to 1.46% per month.

The paper is organized as follows. Section I contains the data and variable definitions. In Section II, we discuss the average raw returns and the average risk-adjusted returns on beta portfolios, and in Section III we present the firm-level cross-sectional regression results. Section IV provides a battery of robustness checks, including portfolio-level cross-sectional regressions, testing the long-term predictive power of conditional betas, and some additional tests after controlling for liquidity and momentum, after excluding the Amex and Nasdaq sample, and after controlling for microstructure effects. In Section V, we investigate whether our main findings are robust for size/BM/beta portfolios. In Section VI, we discuss the cross-sectional implications of the conditional CAPM approach. Section VII concludes the paper.

### I. Data and Variable Definitions

- Top of page
- Abstract
- I. Data and Variable Definitions
- II. Average Returns and FF-3 Alphas on Beta Portfolios
- III. Firm-Level Cross-Sectional Regressions
- IV. Robustness Check
- V. Results from Size/BM/Beta Portfolios
- VI. Cross-Sectional Implications of the Conditional CAPM
- VII. Conclusion
- References

Our first data set comprises all NYSE, Amex, and Nasdaq financial and nonfinancial firms. We obtain this information from the Center for Research in Security Prices (CRSP) for the period from July 1963 through December 2004. We use the daily stock returns to generate the conditional beta measures. Our second data set is Compustat, which we use primarily to obtain the book values for individual stocks.

For each month from July 1963 to December 2004, we compute the following variables for each firm in the sample.

#### A. Size

Following other studies, we measure firm size (ME) by the natural logarithm of the market value of equity (a stock's price times shares outstanding in millions of dollars) for each stock.

#### B. Book-to-Market

Following Fama and French (1992), we compute a firm's book-to-market ratio (BE/ME) by using its market equity at the end of June of year *t*− 1 and the book value of common equity plus balance-sheet-deferred taxes for the firm's latest fiscal year ending in calendar year *t*− 1. To avoid giving extreme observations heavy weight in our analysis, like Fama and French (1992), we set the smallest and largest 0.5% of the observations on book-to-market ratio equal to the next largest and smallest values of the ratio (the 0.005 and 0.995 fractiles).

#### C. Realized Beta

To estimate the monthly beta for an individual stock, we assume a single-factor return-generating process in the form of a market model:

- (1)

where *R*_{i,d,t} is the daily return on stock *i* on day *d* of month *t*, *R*_{m,d,t} is the daily market return on day *d* of month *t*, ɛ_{i,d,t} is the residual term,^{4}α_{i,t} is the intercept, and β_{i,t} is the realized beta of stock *i* in month *t*. We define the realized beta as

- (2)

where is the average daily return on stock *i* in month *t*, is the average daily market return in month *t*, and *n* is the number of daily return observations in month *t*. In our empirical analysis, we measure *R*_{m,d,t} by the CRSP daily value-weighted index, that is, the daily value-weighted average returns of all stocks trading at the NYSE, Amex, and Nasdaq.

Although earlier studies generally use monthly returns to estimate beta and test the CAPM and other factor models, we use daily returns because in principle, we believe we can estimate betas more precisely with higher-frequency data, just as Merton (1980) observed for variances. In practice, using daily returns creates microstructure issues caused by nonsynchronous trading. Nonsynchronous prices can have a big impact on short-horizon betas. Lo and MacKinlay (1990) show that small stocks react with a significant (i.e., a week or more) delay to common news, so a daily beta will miss much of the small-stock covariance with market returns. To mitigate the problem, we exclude the Amex and Nasdaq stocks and control for the size effect using two-dimensional size/beta portfolios based on the NYSE sample. Also, following Dimson (1979), we use both current and lagged market returns in the regressions. In Equation (3) we estimate the realized beta as the sum of the slopes ( and ):

- (3)

where the sum of the slopes, , adjusts for nonsynchronous trading (see Scholes and Williams, 1977; Dimson, 1979).

We estimate the time-varying conditional betas based on the following autoregressive of order one AR(1), moving average of order one MA(1), and GARCH(1,1)-in-mean models:

- (4)

- (5)

- (6)

We drop the *i* subscript in Equations (4) to (6) to save space. Here, *E*(β_{t} | Ω_{t−1}) denotes the current conditional mean of realized beta estimated with the information set at time *t*− 1, Ω_{t−1}. In AR(1) and MA(1) models, we assume that the conditional variance of realized beta, denoted by *E*(ɛ^{2}_{t} | Ω_{t−1}), is constant.

We use the GARCH-in-mean model, which was originally introduced by Engle, Lilien, and Robins (1987), to model the conditional mean of asset returns as a function of the conditional volatility. In the GARCH-in-mean model, we assume that the conditional variance of realized beta follows the GARCH(1,1) model of Bollerslev (1986).We compare the conditional betas (β^{AR}_{t|t−1}, β^{MA}_{t|t−1}, β^{GARCH}_{t|t−1}) with the lagged realized beta β^{realized}_{t−1} in terms of their power to predict the cross-section of one-month-ahead average stock returns.

Earlier studies find significant persistence in the conditional beta estimates for industry, size, or book-to-market portfolios (e.g., Braun, Nelson, and Sunier, 1995; Ang and Chen, 2007). However, these studies do not estimate conditional betas at the firm level. Ang, Chen, and Xing (2006) compute realized beta at the firm level using daily returns over the past 12 months and propose alternative measures of downside risk based on the unconditional realized betas. According to their descriptive statistics, the average AR(1) coefficient of realized betas is in the range of 0.077 to 0.675 depending on their specification of downside risk. We generate conditional beta estimates for each stock using AR(1) and MA(1) models given in Equations (4) and (5).

Jagannathan and Wang (1996) examine the relation between unconditional betas and the cross-section of unconditional expected returns by assuming that the conditional CAPM holds period by period. As described in Jagannathan and Wang (1996), when the conditional CAPM is assumed to hold for each period, cross-sectionally, the unconditional expected return on any asset is a linear function of its expected beta and its beta-premium sensitivity. In other words, the standard static (or unconditional) CAPM leads to a two-factor unconditional asset pricing model, where the first factor is the unconditional market beta that measures average market risk and the second factor is the unconditional premium beta that measures beta-instability risk. According to this model, stocks with higher expected betas should have higher unconditional expected returns. Similarly, stocks with betas that are correlated with the market risk premium and hence are less stable over the business cycle should also have higher unconditional expected returns. Jagannathan and Wang (1996) indicate that the beta-premium sensitivity of an asset measures the instability of the asset's beta over the business cycle.

We model the conditional mean of market beta as a function of its conditional variance as in the GARCH-in-mean specification. Equation (6) models the current conditional mean and conditional variance of realized betas as a function of the information set at time *t*− 1.

To provide an alternative justification for our use of the GARCH-in-mean model, we compute the correlations between the realized beta (β_{t}) and the conditional standard deviation of realized beta (σ_{t}), the realized beta (β_{t}) and the conditional variance of realized beta (σ_{t}), the conditional mean of realized beta (*E*(β_{t} | Ω_{t−1})) and the conditional standard deviation of realized beta (σ_{t}), and the conditional mean of realized beta (*E*(β_{t} | Ω_{t−1})) and the conditional variance of realized beta (σ_{t}).

Table I presents the percentiles of the correlation measures for all stocks trading at the NYSE, Amex, and Nasdaq. The correlation statistics indicate a strong relation between the monthly realized betas and their conditional volatility, and a strong relation between the conditional mean of monthly realized betas and their conditional volatility. We also find that the estimated slope coefficients () in β_{t}=*c*_{0}+*c*_{1}σ^{2}_{t|t−1}+ɛ_{t} are statistically significant. This result provides further justification of our use of the GARCH-in-mean model.

Correlation | β_{t},σ_{t} | β_{t}, σ^{2}_{t} | E(β_{t}| Ω_{t−1}), σ_{t} | E(β_{t}| Ω_{t−1}), σ^{2}_{t} |
---|---|---|---|---|

1% | −0.625 | −0.650 | −0.988 | −0.989 |

5% | −0.390 | −0.422 | −0.971 | −0.973 |

10% | −0.272 | −0.301 | −0.944 | −0.946 |

20% | −0.150 | −0.168 | −0.838 | −0.826 |

30% | −0.069 | −0.080 | −0.505 | −0.474 |

40% | 0.002 | −0.004 | 0.073 | 0.066 |

50% | 0.067 | 0.067 | 0.583 | 0.549 |

60% | 0.130 | 0.135 | 0.791 | 0.773 |

70% | 0.196 | 0.204 | 0.882 | 0.877 |

80% | 0.273 | 0.287 | 0.929 | 0.931 |

90% | 0.390 | 0.412 | 0.963 | 0.967 |

95% | 0.493 | 0.520 | 0.979 | 0.981 |

99% | 0.696 | 0.711 | 0.992 | 0.993 |

Table II presents percentiles of the time-series mean and standard deviation of realized and expected conditional betas. The statistics presented in Panel A of Table II are based on realized betas that we compute by using daily returns over the previous month without lagged market return. The statistics shown in Panel B of Table II are based on realized betas that we compute by using daily returns over the previous month with the lagged market return. In both panels, we only report the sample mean of the realized betas because theoretically, the mean of conditional betas *is* the same as the mean of realized beta. Theoretically, the means should be the same, but the discrepancies are caused by the filtration of conditional beta. The standard deviation of realized betas is greater than the standard deviation of expected betas. In both panels, the standard deviation of conditional betas obtained from the GARCH-in-mean model is somewhat greater than the standard deviation of conditional betas that we obtain from the AR(1) and MA(1) models.

Panel A. Realized Beta Is Estimated without the Lagged Market Return | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | 1% | 5% | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 95% | 99% |

β^{realized}_{t} | −0.1426 | 0.0347 | 0.1112 | 0.2325 | 0.3503 | 0.4610 | 0.5642 | 0.6862 | 0.8163 | 0.9777 | 1.2420 | 1.4954 | 2.0480 |

SD | 1% | 5% | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 95% | 99% |

β^{realized}_{t} | 0.3929 | 0.5348 | 0.6360 | 0.7841 | 0.9236 | 1.0641 | 1.2175 | 1.3879 | 1.5970 | 1.8609 | 2.3190 | 2.8373 | 4.2165 |

β^{AR}_{t|t−1} | 0.0019 | 0.0111 | 0.0228 | 0.0463 | 0.0701 | 0.0965 | 0.1255 | 0.1590 | 0.2009 | 0.2615 | 0.3687 | 0.4846 | 0.8120 |

β^{MA}_{t|t−1} | 0.0020 | 0.0109 | 0.0220 | 0.0442 | 0.0657 | 0.0891 | 0.1154 | 0.1439 | 0.1830 | 0.2369 | 0.3378 | 0.4527 | 0.8169 |

β^{GARCH}_{t|t−1} | 0.0014 | 0.0101 | 0.0164 | 0.0454 | 0.0747 | 0.1071 | 0.1431 | 0.1880 | 0.2435 | 0.3251 | 0.4693 | 0.6275 | 1.1644 |

Panel B. Realized Beta Is Estimated with the Lagged Market Return | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | 1% | 5% | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 95% | 99% |

β^{realized}_{t} | −0.1450 | 0.0692 | 0.1744 | 0.3217 | 0.4587 | 0.5885 | 0.7111 | 0.8392 | 0.9726 | 1.1352 | 1.3826 | 1.6126 | 2.2350 |

SD | 1% | 5% | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 95% | 99% |

β^{realized}_{t} | 0.5428 | 0.7126 | 0.8439 | 1.0454 | 1.2325 | 1.4229 | 1.6222 | 1.8551 | 2.1360 | 2.4868 | 3.0675 | 3.6365 | 5.4468 |

β^{AR}_{t|t−1} | 0.0165 | 0.0395 | 0.0596 | 0.0920 | 0.1225 | 0.1539 | 0.1885 | 0.2312 | 0.2845 | 0.3641 | 0.4991 | 0.6627 | 1.1396 |

β^{MA}_{t|t−1} | 0.0026 | 0.0117 | 0.0234 | 0.0472 | 0.0723 | 0.0976 | 0.1260 | 0.1609 | 0.2071 | 0.2737 | 0.3971 | 0.5455 | 1.0561 |

β^{GARCH}_{t|t−1} | 0.0059 | 0.0101 | 0.0221 | 0.0461 | 0.0802 | 0.1189 | 0.1635 | 0.2201 | 0.2877 | 0.3853 | 0.5551 | 0.7714 | 1.5448 |

We compare the conditional betas (β^{AR}_{t|t−1}, β^{MA}_{t|t−1}, β^{GARCH}_{t|t−1}) with the lagged realized beta β^{realized}_{t−1} in terms of their power to predict the one-month-ahead realized beta, β^{realized}_{t}. Table III shows the percentiles of *R*^{2} values from the regression of one-month-ahead realized betas on the lagged realized beta and conditional betas. The performance of conditional betas in predicting the one-month-ahead realized beta is much higher than the lagged realized beta. The 1 percentile of *R*^{2} is 0.01% for β^{realized}_{t−1} and the 99 percentile of *R*^{2} is 26.82%. The corresponding figures are 1.14% and 33.87% for β^{GARCH}_{t|t−1}.

R^{2} | 1% | 5% | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 95% | 99% |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Panel A. Realized Beta Is Estimated without the Lagged Market Return | |||||||||||||

β^{realized}_{t−1} | 0.01% | 0.11% | 0.22% | 0.69% | 1.18% | 1.94% | 3.22% | 4.66% | 6.89% | 9.94% | 13.26% | 17.67% | 26.82% |

β^{AR}_{t|t−1} | 0.74% | 2.04% | 2.87% | 4.58% | 6.16% | 7.88% | 10.31% | 13.12% | 15.77% | 19.24% | 23.90% | 27.76% | 33.82% |

β^{MA}_{t|t−1} | 0.63% | 1.94% | 2.67% | 4.48% | 5.95% | 7.65% | 9.98% | 12.54% | 15.39% | 18.72% | 23.26% | 26.70% | 32.32% |

β^{GARCH}_{t|t−1} | 1.14% | 2.46% | 3.33% | 4.93% | 6.75% | 8.90% | 11.01% | 13.80% | 16.70% | 20.67% | 24.88% | 27.88% | 33.87% |

Panel B. Realized Beta Is Estimated with the Lagged Market Return | |||||||||||||

β^{realized}_{t−1} | 0.01% | 0.03% | 0.08% | 0.22% | 0.46% | 0.81% | 1.21% | 1.73% | 2.65% | 4.12% | 6.63% | 9.54% | 16.76% |

β^{AR}_{t|t−1} | 1.29% | 2.28% | 3.12% | 4.22% | 5.29% | 6.10% | 7.13% | 8.38% | 10.20% | 12.66% | 16.20% | 19.60% | 24.77% |

β^{MA}_{t|t−1} | 1.29% | 2.35% | 3.14% | 4.15% | 5.19% | 6.04% | 7.13% | 8.40% | 10.19% | 12.50% | 15.92% | 19.16% | 24.00% |

β^{GARCH}_{t|t−1} | 1.17% | 2.12% | 3.09% | 4.23% | 5.04% | 6.02% | 7.10% | 8.31% | 10.50% | 12.58% | 16.99% | 19.73% | 24.62% |

These results provide some explanation for why the earlier studies that use lagged realized beta or unconditional beta could not identify a positive and significant relation between market beta and expected stock returns. We think that to generate more accurate measures of expected futures betas and to explain the cross-sectional variation in stock returns, one needs to use conditional betas.

### II. Average Returns and FF-3 Alphas on Beta Portfolios

- Top of page
- Abstract
- I. Data and Variable Definitions
- II. Average Returns and FF-3 Alphas on Beta Portfolios
- III. Firm-Level Cross-Sectional Regressions
- IV. Robustness Check
- V. Results from Size/BM/Beta Portfolios
- VI. Cross-Sectional Implications of the Conditional CAPM
- VII. Conclusion
- References

This section presents univariate and bivariate portfolio-level analysis after controlling for size and book-to-market.

#### A. Univariate Portfolio-Level Analysis

Table IV presents the equal-weighted average returns of decile portfolios that are formed by sorting the NYSE/Amex/Nasdaq stocks based on the lagged realized beta, and the conditional AR(1), MA(1), and GARCH-in-mean betas. We base the results in Panel A on the realized betas that we compute using daily returns over the previous month without lagged market return.

Panel A. Realized Beta Estimated without the Lagged Market Return | ||||||||
---|---|---|---|---|---|---|---|---|

Decile | β^{realized}_{t−1} | β^{AR}_{t|t−1} | β^{MA}_{t|t−1} | β^{GARCH}_{t|t−1} | ||||

Average Return | Average Beta | Average Return | Average Beta | Average Return | Average Beta | Average Return | Average Beta | |

1 Low β | 1.56 | −1.65 | 1.09 | 0.08 | 1.07 | 0.02 | 1.11 | 0.00 |

2 | 1.36 | −1.37 | 1.21 | 0.24 | 1.21 | 0.25 | 1.15 | 0.23 |

3 | 1.38 | −0.01 | 1.30 | 0.37 | 1.31 | 0.37 | 1.27 | 0.36 |

4 | 1.38 | 0.23 | 1.41 | 0.48 | 1.36 | 0.48 | 1.29 | 0.47 |

5 | 1.35 | 0.47 | 1.37 | 0.58 | 1.41 | 0.59 | 1.38 | 0.58 |

6 | 1.41 | 0.72 | 1.44 | 0.69 | 1.46 | 0.70 | 1.41 | 0.69 |

7 | 1.34 | 1.01 | 1.51 | 0.81 | 1.46 | 0.82 | 1.49 | 0.82 |

8 | 1.27 | 1.37 | 1.53 | 0.96 | 1.54 | 0.96 | 1.51 | 0.97 |

9 | 1.25 | 1.91 | 1.58 | 1.16 | 1.60 | 1.15 | 1.67 | 1.18 |

10 High β | 1.07 | 3.40 | 1.83 | 1.55 | 1.85 | 1.62 | 2.03 | 1.71 |

High to low | −0.49** | 0.74** | 0.78** | 0.92*** | ||||

(−2.53) | (2.33) | (2.47) | (2.65) | |||||

Alpha | −0.48*** | 0.50** | 0.53** | 0.60** | ||||

(−2.85) | (2.17) | (2.33) | (2.60) |

Panel B. Realized Beta Estimated with the Lagged Market Return | ||||||||
---|---|---|---|---|---|---|---|---|

Decile | β^{realized}_{t−1} | β^{AR}_{t|t−1} | β^{MA}_{t|t−1} | β^{GARCH}_{t|t−1} | ||||

Average Return | Average Beta | Average Return | Average Beta | Average Return | Average Beta | Average Return | Average Beta | |

^{}***Significant at the 0.01 level. ^{}**Significant at the 0.05 level. ^{}*Significant at the 0.10 level.
| ||||||||

1 Low β | 1.44 | −2.25 | 1.13 | 0.03 | 1.13 | 0.05 | 1.10 | 0.02 |

2 | 1.32 | −0.55 | 1.15 | 0.33 | 1.15 | 0.33 | 1.19 | 0.32 |

3 | 1.33 | −0.08 | 1.27 | 0.47 | 1.25 | 0.48 | 1.25 | 0.46 |

4 | 1.39 | 0.24 | 1.38 | 0.60 | 1.38 | 0.60 | 1.28 | 0.59 |

5 | 1.42 | 0.53 | 1.36 | 0.71 | 1.37 | 0.72 | 1.39 | 0.71 |

6 | 1.41 | 0.85 | 1.41 | 0.83 | 1.42 | 0.83 | 1.40 | 0.83 |

7 | 1.36 | 1.21 | 1.49 | 0.95 | 1.49 | 0.95 | 1.44 | 0.95 |

8 | 1.38 | 1.65 | 1.57 | 1.08 | 1.58 | 1.08 | 1.55 | 1.09 |

9 | 1.25 | 2.32 | 1.62 | 1.27 | 1.64 | 1.26 | 1.69 | 1.29 |

10 High β | 1.08 | 4.25 | 2.02 | 1.74 | 2.00 | 1.70 | 2.11 | 1.83 |

High to low | −0.33* | 0.89*** | 0.87** | 1.01*** | ||||

(−1.92) | (2.66) | (2.54) | (2.83) | |||||

Alpha | −0.35** | 0.63*** | 0.60** | 0.71*** | ||||

(−2.36) | (2.76) | (2.62) | (3.06) |

When we sort portfolios based on the lagged realized beta, β^{realized}_{t−1}, the average return difference between decile 10 (high beta) and decile 1 (low beta) is about −0.49% per month with the Newey-West (1987)*t*-statistic of −2.53. Although this result does not support the empirical validity of CAPM, it is not conclusive, because as discussed earlier, β^{realized}_{t−1} is not a precise estimator of β^{realized}_{t}. The static CAPM predicts a contemporaneous positive relation between expected stock returns and market betas. However, we cannot use the current realized beta in empirical tests because of the statistical problems indicated by Miller and Scholes (1972) and Fama and MacBeth (1973).

When we examine the cross-sectional predictive power of the conditional beta measures, we see that they are more accurate estimators of β^{realized}_{t}. When we sort decile portfolios based on β^{AR}_{t|t−1}, β^{MA}_{t|t−1}, and β^{GARCH}_{t|t−1}, the average return difference between decile 10 (high beta) and decile 1 (low beta) is in the range of 0.74% to 0.92% per month and highly significant.

In addition to the average raw returns, Panel A of Table IV also shows the magnitude and statistical significance of the intercepts (FF-3 alphas) from the regression of the equal-weighted portfolio returns on a constant, excess market return, small minus big (SMB), and high minus low (HML) factors. If the conditional CAPM is right and FF-3 alphas do not adequately capture time variations in betas, then conditional-beta-sorted portfolios will have alphas different from zero. Panel A shows that the 10-1 difference in the FF-3 alphas is negative for β^{realized}_{t−1}, but it is positive and highly significant for the AR(1), MA(1), and GARCH-in-mean betas.

Our results in Panel B of Table IV are based on the realized betas that we compute using daily returns over the previous month with the lagged market return. When we sort portfolios based on β^{realized}_{t−1}, the average return difference between high- and low-beta portfolios is negative but marginally significant. When decile portfolios are sorted based on β^{AR}_{t|t−1}, β^{MA}_{t|t−1}, and β^{GARCH}_{t|t−1}, the average return difference between high- and low-beta portfolios is positive, in the range of 0.89% to 1.01% per month, and highly significant. Panel B also shows that the 10-1 difference in the FF-3 alphas is negative for β^{realized}_{t−1} but positive and highly significant for the AR(1), MA(1), and GARCH-in-mean betas. Overall, the results in Table IV indicate that the strong positive relation between the conditional betas and expected returns is robust to the measurement of realized betas.^{5}

Due to space considerations, in the following sections we report results only from the realized beta measures estimated with the lagged market return.

#### B. Controlling for Size and Book-to-Market

We test whether there is a positive relation between conditional beta and expected returns after we control for size and book-to-market. We control for size by first forming decile portfolios ranked based on market capitalization. Then, within each size decile, we sort stocks into decile portfolios, which we rank based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. Panel A of Table V shows that in each size decile, the highest (lowest) beta decile has a higher (lower) average returns. The column labeled “Average Returns” averages across the 10 size deciles to produce decile portfolios with dispersion in market beta but containing all sizes of firms. This procedure creates a set of decile beta portfolios with near-identical levels of firm size, and thus these decile beta portfolios control for differences in size. After controlling for size, the average return difference between high- and low-beta portfolios is 1.41% per month with the Newey-West (1987)*t*-statistic of 3.48. Thus, market capitalization does not explain the high (low) returns to high (low) beta stocks.

Panel A. Equal-Weighted Returns on Beta Portfolios after Controlling for Size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Small Size | Size 2 | Size 3 | Size 4 | Size 5 | Size 6 | Size 7 | Size 8 | Size 9 | Big Size | Average Returns | t-statistic | |

1 Low β | 2.85 | 0.61 | 0.48 | 0.63 | 0.68 | 0.78 | 0.88 | 0.83 | 0.84 | 0.80 | 0.94*** | 5.07 |

2 | 2.12 | 0.98 | 1.03 | 0.83 | 0.88 | 0.86 | 1.11 | 1.00 | 1.05 | 0.88 | 1.07*** | 5.26 |

3 | 2.63 | 0.98 | 1.06 | 0.86 | 0.81 | 0.84 | 0.99 | 1.10 | 0.96 | 0.93 | 1.12*** | 4.76 |

4 | 2.40 | 1.31 | 1.13 | 1.05 | 0.96 | 0.98 | 1.12 | 1.07 | 1.03 | 0.99 | 1.20*** | 4.67 |

5 | 2.93 | 1.29 | 1.20 | 1.19 | 1.13 | 1.12 | 1.24 | 1.24 | 1.01 | 1.02 | 1.34*** | 4.72 |

6 | 3.18 | 1.51 | 1.17 | 1.18 | 1.15 | 1.18 | 1.16 | 1.13 | 1.06 | 0.97 | 1.37*** | 4.58 |

7 | 3.39 | 1.79 | 1.55 | 1.29 | 1.17 | 1.17 | 1.19 | 1.11 | 1.28 | 0.99 | 1.49*** | 4.67 |

8 | 3.49 | 1.83 | 1.62 | 1.65 | 1.53 | 1.40 | 1.27 | 1.34 | 1.11 | 1.01 | 1.63*** | 4.54 |

9 | 4.46 | 2.18 | 1.88 | 1.63 | 1.72 | 1.69 | 1.28 | 1.26 | 1.28 | 1.00 | 1.84*** | 4.56 |

10 High β | 5.36 | 3.04 | 2.31 | 2.23 | 2.05 | 2.13 | 2.03 | 1.75 | 1.51 | 1.08 | 2.35*** | 4.55 |

High β to low β | 1.41*** | 3.48 |

Panel B. Equal-Weighted Returns on Beta Portfolios after Controlling for BM | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Low BM | BM 2 | BM 3 | BM 4 | BM 5 | BM 6 | BM 7 | BM 8 | BM 9 | High BM | Average Returns | t-statistic | |

^{}***Significant at the 0.01 level.
| ||||||||||||

1 Low β | 0.35 | 0.40 | 0.77 | 0.86 | 0.82 | 1.15 | 1.21 | 1.42 | 1.35 | 1.91 | 1.02*** | 4.55 |

2 | 0.19 | 0.65 | 0.72 | 0.90 | 1.03 | 1.05 | 1.22 | 1.49 | 1.31 | 1.95 | 1.05*** | 4.62 |

3 | 0.51 | 0.79 | 0.93 | 1.04 | 1.02 | 1.16 | 1.19 | 1.32 | 1.60 | 1.83 | 1.14*** | 4.79 |

4 | 0.59 | 0.70 | 1.11 | 1.10 | 1.05 | 1.22 | 1.47 | 1.49 | 1.90 | 2.28 | 1.29*** | 5.14 |

5 | 0.69 | 1.01 | 1.19 | 1.11 | 1.13 | 1.27 | 1.45 | 1.57 | 1.84 | 2.13 | 1.34*** | 5.04 |

6 | 0.45 | 0.92 | 0.90 | 1.20 | 1.20 | 1.36 | 1.57 | 1.64 | 1.71 | 1.85 | 1.28*** | 4.69 |

7 | 0.85 | 1.09 | 1.39 | 1.31 | 1.27 | 1.37 | 1.52 | 1.68 | 1.79 | 2.22 | 1.45*** | 4.87 |

8 | 1.00 | 1.19 | 1.34 | 1.40 | 1.17 | 1.67 | 1.54 | 1.70 | 1.91 | 2.26 | 1.52*** | 4.76 |

9 | 1.03 | 1.39 | 1.34 | 1.58 | 1.63 | 1.73 | 1.89 | 2.05 | 2.43 | 2.43 | 1.75*** | 4.93 |

10 High β | 1.64 | 1.62 | 1.68 | 1.90 | 2.04 | 2.07 | 2.44 | 2.31 | 2.94 | 3.16 | 2.18*** | 4.80 |

High β to low β | 1.16*** | 3.69 |

We also control for book-to-market (BM) by first forming decile portfolios ranked based on the ratio of book value of equity to market value of equity. Then, within each BM decile, we sort stocks into decile portfolios, which we rank based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. Panel B of Table V shows that in each BM decile, the highest (lowest) beta decile has a higher (lower) average returns. The last two columns report the average returns and Newey-West (1987)*t*-statistics of 10 beta portfolios after controlling for BM. The average return difference between high- and low-beta portfolios is 1.16% per month with the Newey-West *t*-statistic of 3.69. Thus, book-to-market ratio does not explain the high (low) returns to high- (low-) beta stocks.^{6}

Table VI shows the average return differences and FF-3 alphas on high-beta minus low-beta portfolios within each size and book-to-market decile. As shown in Panel A of Table VI, for all specifications of conditional beta, the average return differences and FF-3 alphas are positive and economically significant within each size decile. Except for the two biggest size portfolios (sizes 9 and 10), the average return differences and FF-3 alphas are also statistically significant at the 5% level or better. For example, for the smallest size decile, the average return difference between decile 10 (high beta) and decile 1 (low beta) is 2.49% per month for β^{AR}_{t|t−1}, 2.57% per month for β^{MA}_{t|t−1}, and 2.50% per month for β^{GARCH}_{t|t−1}. The corresponding FF-3 alphas are 2.02% per month for β^{AR}_{t|t−1}, 2.06% per month for β^{MA}_{t|t−1}, and 2.02% per month for β^{GARCH}_{t|t−1}. This strong positive relation between market beta and expected return is present for size 1 to size 9 portfolios, and the relation becomes somewhat weaker for the largest size portfolio (size 10).

Panel A. Average Return Differences and FF-3 Alphas within Size Deciles | ||||||
---|---|---|---|---|---|---|

Decile | β^{AR}_{t|t−1} | β^{MA}_{t|t−1} | β^{GARCH}_{t|t−1} | |||

High β to Low β | Alpha | High β to Low β | Alpha | High β to Low β | Alpha | |

Small size | 2.49*** | 2.02*** | 2.57*** | 2.06*** | 2.50*** | 2.02*** |

(5.79) | (5.54) | (6.09) | (6.05) | (5.89) | (5.35) | |

Size 2 | 2.17*** | 1.65*** | 2.22*** | 1.69*** | 2.43*** | 1.83*** |

(4.94) | (4.55) | (5.01) | (4.67) | (5.34) | (5.14) | |

Size 3 | 1.92*** | 1.40*** | 1.88*** | 1.33*** | 1.83*** | 1.27*** |

(4.00) | (3.55) | (3.88) | (3.36) | (3.82) | (3.40) | |

Size 4 | 1.67*** | 1.25*** | 1.68*** | 1.23*** | 1.60*** | 1.05*** |

(3.65) | (3.48) | (3.57) | (3.44) | (3.46) | (3.44) | |

Size 5 | 1.54*** | 1.14*** | 1.52*** | 1.13*** | 1.37*** | 0.94*** |

(3.46) | (3.56) | (3.40) | (3.57) | (3.06) | (3.05) | |

Size 6 | 1.49*** | 1.17*** | 1.51*** | 1.16*** | 1.35*** | 0.93*** |

(3.25) | (3.59) | (3.21) | (3.57) | (2.83) | (2.69) | |

Size 7 | 1.19*** | 1.01*** | 1.08** | 0.88*** | 1.15** | 0.88*** |

(2.68) | (3.26) | (2.39) | (2.87) | (2.56) | (2.91) | |

Size 8 | 0.72* | 0.64** | 0.82* | 0.73*** | 0.92** | 0.87*** |

(1.67) | (2.34) | (1.87) | (2.64) | (2.09) | (3.05) | |

Size 9 | 0.60 | 0.63** | 0.65 | 0.67** | 0.67 | 0.65** |

(1.33) | (2.09) | (1.39) | (2.13) | (1.45) | (2.13) | |

Big size | 0.24 | 0.42 | 0.24 | 0.43 | 0.28 | 0.50* |

(0.60) | (1.58) | (0.60) | (1.64) | (0.68) | (1.91) |

Panel B. Average Return Differences and FF-3 Alphas within Book-to-Market Deciles | ||||||
---|---|---|---|---|---|---|

Decile | β^{AR}_{t|t−1} | β^{MA}_{t|t−1} | β^{GARCH}_{t|t−1} | |||

High β to Low β | Alpha | High β to Low β | Alpha | High β to Low β | Alpha | |

^{}***Significant at the 0.01 level. ^{}**Significant at the 0.05 level. ^{}*Significant at the 0.10 level.
| ||||||

Low BM | 1.50*** | 1.60*** | 1.49*** | 1.54*** | 1.29*** | 1.36*** |

(3.75) | (4.59) | (3.74) | (4.50) | (3.27) | (4.21) | |

BM 2 | 1.27*** | 1.27*** | 1.31*** | 1.33*** | 1.22*** | 1.19*** |

(3.44) | (3.83) | (3.66) | (4.12) | (3.28) | (3.57) | |

BM 3 | 0.89** | 0.80*** | 0.82** | 0.70** | 0.91** | 0.73*** |

(2.58) | (2.71) | (2.37) | (2.37) | (2.61) | (2.67) | |

BM 4 | 0.98*** | 0.68*** | 0.93*** | 0.65** | 1.03*** | 0.67** |

(2.90) | (2.59) | (2.76) | (2.41) | (3.06) | (2.52) | |

BM 5 | 1.18*** | 0.71*** | 1.22*** | 0.75*** | 1.22*** | 0.79*** |

(3.48) | (2.71) | (3.48) | (2.73) | (3.22) | (2.65) | |

BM 6 | 0.90*** | 0.44* | 0.84*** | 0.39 | 0.92*** | 0.37 |

(2.90) | (1.73) | (2.66) | (1.56) | (2.83) | (1.53) | |

BM 7 | 0.90** | 0.47 | 0.90** | 0.46 | 1.24*** | 0.81** |

(2.46) | (1.53) | (2.39) | (1.45) | (3.08) | (2.38) | |

BM 8 | 0.82** | 0.41 | 0.74** | 0.30 | 0.89** | 0.37 |

(2.27) | (1.39) | (2.15) | (1.07) | (2.38) | (1.25) | |

BM 9 | 1.51*** | 1.01*** | 1.41*** | 0.94*** | 1.59*** | 1.14*** |

(4.09) | (3.25) | (3.86) | (2.98) | (4.10) | (3.52) | |

High BM | 0.60 | −0.15 | 0.49 | −0.24 | 1.25*** | 0.59* |

(1.62) | (−0.46) | (1.32) | (−0.73) | (3.21) | (1.75) |

Panel B of Table VI shows that for all specifications of conditional beta, the average return differences and FF-3 alphas are positive and economically significant within each BM decile. The average return differences and FF-3 alphas are also statistically significant at the 5% level or better. For example, for the lowest BM decile, the average return difference between decile 10 (high beta) and decile 1 (low beta) is 1.50% per month for β^{AR}_{t|t−1}, 1.49% per month for β^{MA}_{t|t−1}, and 1.29% per month for β^{GARCH}_{t|t−1}. The corresponding FF-3 alphas are 1.60% per month for β^{AR}_{t|t−1}, 1.54% per month for β^{MA}_{t|t−1}, and 1.36% per month for β^{GARCH}_{t|t−1}. Although there is no obvious pattern, the strong positive relation between market beta and expected return is more pronounced for BM 1 to BM 9 portfolios, and the relation becomes weaker for the highest BM portfolio.

### III. Firm-Level Cross-Sectional Regressions

- Top of page
- Abstract
- I. Data and Variable Definitions
- II. Average Returns and FF-3 Alphas on Beta Portfolios
- III. Firm-Level Cross-Sectional Regressions
- IV. Robustness Check
- V. Results from Size/BM/Beta Portfolios
- VI. Cross-Sectional Implications of the Conditional CAPM
- VII. Conclusion
- References

Here, we present the time-series averages of the slope coefficients from the cross-section of average stock returns on the lagged realized beta, conditional beta, size, and BM. The average slopes provide standard Fama-MacBeth (1973) tests for determining, on average, which explanatory variables have nonzero expected premiums. We run monthly cross-sectional regressions for the following econometric specifications:

- (7)

- (8)

- (9)

- (10)

In Equations (7) to (10)*R*_{i,t} is the realized return on stock *i* in month *t*, log*ME*_{i,t−1} is the natural logarithm of market equity for firm *i* in month *t*− 1, log(*BE*_{i,t−1}/*ME*_{i,t−1}) is the natural logarithm of the ratio of book value of equity to market value of equity for firm *i* in month *t*− 1, β^{realized}_{i,t−1} is the lagged realized beta of stock *i* in month *t*− 1, and β^{AR}_{i,t|t−1}, β^{MA}_{i,t|t−1}, and β^{GARCH}_{i,t|t−1} are the conditional expected beta of stock *i* in month *t* estimated with the information set at month *t*− 1.

Table VII reports the time-series averages of the slope coefficients γ* _{i,t}* (

*i*= 1, 2, 3) over the 498 months from July 1963 to December 2004. The Newey-West (1987) adjusted

*t*-statistics are given in parentheses. The results show a negative but insignificant relation between the lagged realized beta and the cross-section of average stock returns. The average slope, γ

_{1,t}, from the monthly regressions of realized returns on β

^{realized}

_{i,t−1}alone is about −0.07% with a

*t*-statistic of −1.19. The univariate regression results indicate a significant positive relation between average stock returns and conditional betas. The average slope, γ

_{1,t}, from the monthly regressions of realized returns on β

^{AR}

_{i,t|t−1}, β

^{MA}

_{i,t|t−1}, or β

^{GARCH}

_{i,t|t−1}is in the range of 0.44% to 0.50% and statistically significant at the 5% or better. These values imply a reasonable expected market risk premium of 5.33% to 5.87% per annum.

β^{realized}_{t−1} | β^{AR}_{t|t−1} | β^{MA}_{t|t−1} | β^{GARCH}_{t|t−1} | logME_{t−1} | log(BE_{t−1}/ME_{t−1}) | R^{2} |
---|---|---|---|---|---|---|

^{}***Significant at the 0.01 level. ^{}**Significant at the 0.05 level.
| ||||||

−0.0678 | 1.04% | |||||

(−1.19) | ||||||

0.4443** | 2.06% | |||||

(2.09) | ||||||

0.4890** | 2.02% | |||||

(2.26) | ||||||

0.4565** | 2.13% | |||||

(2.32) | ||||||

−0.2373*** | 1.81% | |||||

(−4.75) | ||||||

0.4185*** | 1.00% | |||||

(5.98) | ||||||

−0.2029*** | 0.2396*** | 2.69% | ||||

(−3.83) | (3.21) | |||||

−0.0124 | −0.2229*** | 2.79% | ||||

(−0.21) | (−4.18) | |||||

0.7985*** | −0.2907*** | 4.34% | ||||

(3.07) | (−4.69) | |||||

0.8063*** | −0.2901*** | 4.31% | ||||

(3.09) | (−4.68) | |||||

0.8010*** | −0.2869*** | 4.27% | ||||

(3.05) | (−4.52) | |||||

−0.0357 | 0.4328*** | 1.85% | ||||

(−0.68) | (6.67) | |||||

0.6694*** | 0.4721*** | 2.83% | ||||

(3.34) | (7.39) | |||||

0.6796*** | 0.4718*** | 2.79% | ||||

(3.36) | (7.37) | |||||

0.6470*** | 0.4771*** | 2.89% | ||||

(3.41) | (7.44) | |||||

0.0045 | −0.1861*** | 0.2765*** | 3.50% | |||

(0.08) | (−3.31) | (4.03) | ||||

0.8967*** | −0.2587*** | 0.2745*** | 4.70% | |||

(3.65) | (−4.01) | (4.29) | ||||

0.9041*** | −0.2579*** | 0.2740*** | 4.70% | |||

(3.66) | (−4.00) | (4.28) | ||||

0.8929*** | −0.2526*** | 0.2872*** | 4.87% | |||

(3.60) | (−3.86) | (4.62) |

The univariate regression results also indicate a significant negative relation between average stock returns and firm size. The average slope, γ_{2,t}, from the monthly regressions of realized returns on log*ME*_{i,t−1} alone is about −0.24% with a *t*-statistic of −4.75. The parameter estimates show a significant positive relation between average stock returns and BM ratio. The average slope, γ_{3,t}, from the monthly regressions of realized returns on log(*BE*_{i,t−1}/*ME*_{i,t−1}) alone is about 0.42% with a *t*-statistic of 5.98. The findings of negative size and positive BM effect in Fama-MacBeth (1973) regressions are consistent with Fama and French (1992) and related studies.

We find that the strong positive relation between conditional beta and expected stock returns is robust across different econometric specifications. When we add size to the univariate regressions, the average slope coefficient on β^{AR}_{i,t|t−1}, β^{MA}_{i,t|t−1}, or β^{GARCH}_{i,t|t−1} is about 0.8% and statistically significant at the 1% level. When we add BM to the univariate regressions, the average slope coefficient on conditional betas is in the range of 0.65% to 0.68% and statistically significant at the 1% level. When we include both size and BM in the univariate regressions, the average slope coefficient on the conditional betas is about 0.9% and statistically significant at the 1% level.

The *R*^{2} values from the univariate regressions of realized returns on conditional beta are in the range of 2.02% to 2.13%. When we add size and BM to these univariate regressions, the *R*^{2} values increase to 4.7% to 4.87%. Although the *R*^{2} values from univariate and multivariate cross-sectional regressions are small, they are consistent with the earlier studies that report *R*^{2} for the firm-level cross-sectional regressions.

### IV. Robustness Check

- Top of page
- Abstract
- I. Data and Variable Definitions
- II. Average Returns and FF-3 Alphas on Beta Portfolios
- III. Firm-Level Cross-Sectional Regressions
- IV. Robustness Check
- V. Results from Size/BM/Beta Portfolios
- VI. Cross-Sectional Implications of the Conditional CAPM
- VII. Conclusion
- References

This section presents results from a battery of robustness checks.

#### A. Alternative Portfolio Partitions

We compute the equal-weighted average returns of 20, 50, and 100 portfolios that we form by sorting the NYSE/Amex/Nasdaq stocks based on the conditional AR(1), MA(1), and GARCH-in-mean betas. Although not presented in the paper, the average return difference between high- and low-beta portfolios is in the range of 0.83% to 1% per month for β^{AR}_{t|t−1}, 0.89% to 1.11% per month for β^{MA}_{t|t−1}, and 1.06% to 1.31% per month β^{GARCH}_{t|t−1}. All these average return differences are statistically significant at the 5% level or better. In addition to the average raw returns, we also find the magnitude and statistical significance of the FF-3 alphas. The 10-1 difference in the FF-3 alphas is positive and highly significant for the AR(1), MA(1), and GARCH-in-mean betas.

In addition to the firm-level Fama-MacBeth (1973) regressions, we examine the cross-sectional relation between conditional beta and expected returns at the portfolio level. We present the time-series averages of the slope coefficients from the cross-section of average portfolio returns on the conditional portfolio beta:

- (11)

- (12)

- (13)

In Equations (11) to (13), *R*_{p,t} is the realized return on portfolio *p* in month *t* calculated as the equal-weighted average returns of all stocks in portfolio *p*, and β^{AR}_{p,t|t−1}, β^{MA}_{p,t|t−1}, and β^{GARCH}_{p,t|t−1} are the conditional expected betas of portfolio *p* that we obtain from the equal-weighted average conditional beta of all stocks in portfolio *p* in month *t* estimated with the information set at month *t*− 1.

First, we form 10, 20, 50, and 100 portfolios by sorting the NYSE/Amex/Nasdaq stocks based on the conditional AR(1), MA(1), and GARCH-in-mean betas. Then, for each month from July 1963 to December 2004, we compute each portfolio's return as the equal-weighted average return of all stocks in the portfolio, and we calculate the portfolio's conditional beta as the equal-weighted average conditional beta of all stocks in the portfolio. We run the univariate regressions of average portfolio returns on the average conditional portfolio beta for each month from July 1963 to December 2004.

We calculate the time-series averages of the slope coefficients and the Newey-West (1987) adjusted *t*-statistics. The univariate regression results indicate a significant positive relation between average portfolio returns and average portfolio betas. Although not presented here, for 10 beta portfolios, the average slopes from the monthly regressions of average portfolio returns on β^{AR}_{p,t|t−1}, β^{MA}_{p,t|t−1}, and β^{GARCH}_{p,t|t−1} are about 0.44%, 0.47%, and 0.53%, respectively. These average slope coefficients have Newey-West *t*-statistics of 2.25, 2.37, and 2.61, respectively. A notable point is that for 20, 50, and 100 beta portfolios, the average slope coefficients are similar. In other words, the results are robust across different portfolio formations. These slope coefficients, which imply an expected market risk premium of 5.28% to 6.36% per annum, are also similar to our earlier findings from the firm-level cross-sectional regressions. The *R*^{2} values are much higher for the portfolio-level regressions: about 58% to 59% for 10 beta portfolios, 48% to 49% for 20 beta portfolios, 35% to 36% for 50 beta portfolios, and 25% to 26% for 100 beta portfolios.

In addition to the month-by-month Fama-MacBeth (1973) regressions, we take the time-series average of the monthly portfolio returns and the monthly portfolio betas and compute overall average portfolio return and overall average portfolio beta for 10, 20, 50, and 100 portfolios. We plot the average portfolio return against the average portfolio beta and find a strong positive relation between market beta and expected returns for all portfolio partitions. The *R*^{2} value is 97.94% for 10 beta portfolios, 95.97% for 20 beta portfolios, 88.47% for 50 beta portfolios, and 81.5% for 100 beta portfolios. We also find the slope coefficients for each portfolio partition. The results are similar to our earlier findings from the month-by-month firm-level and portfolio-level Fama-MacBeth regressions. The slope on average portfolio beta is almost identical for different portfolio partitions: 0.53% for 10, 50, and 100 beta portfolios, and 0.52% for 20 beta portfolios.

#### B. Long-Term Predictive Power of Conditional Betas

Table VIII presents the equal-weighted returns of decile portfolios that we form by sorting the NYSE/Amex/Nasdaq stocks based on the conditional GARCH-in-mean betas.^{7} The column labeled “β^{GARCH}_{t|t−1}” repeats our earlier result for one-month-ahead predictability: when we sort decile portfolios based on β^{GARCH}_{t|t−1}, the average return difference between decile 10 (high beta) and decile 1 (low beta) is 1.01% per month with the Newey-West (1987)*t*-statistic of 2.83. To test three-month-ahead predictability, we form decile portfolios by sorting stocks based on their conditional betas at time *t*− 2 obtained from the information set at time *t*− 3, β^{GARCH}_{t−2|t−3}. The average return difference between high- and low-beta portfolios is 0.93% per month with the *t*-statistic of 2.6. As shown in Table VIII, the conditional beta can predict up to 12 months ahead because the average return difference between high- and low-beta portfolios is 0.75% per month with a *t*-statistic of 2.09.

Decile | β^{GARCH}_{t|t−1} | β^{GARCH}_{t−1|t−2} | β^{GARCH}_{t−2|t−3} | β^{GARCH}_{t−3|t−4} | β^{GARCH}_{t−4|t−5} | β^{GARCH}_{t−5|t−6} | β^{GARCH}_{t−6|t−7} | β^{GARCH}_{t−7|t−8} | β^{GARCH}_{t−8|t−9} | β^{GARCH}_{t−9|t−10} | β^{GARCH}_{t−10|t−11} | β^{GARCH}_{t−11|t−12} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

^{}***Significant at the 0.01 level. ^{}**Significant at the 0.05 level. ^{}*Significant at the 0.10 level.
| ||||||||||||

1 Low β | 1.10 | 1.11 | 1.12 | 1.12 | 1.12 | 1.16 | 1.16 | 1.17 | 1.19 | 1.21 | 1.21 | 1.22 |

2 | 1.19 | 1.16 | 1.18 | 1.22 | 1.25 | 1.20 | 1.20 | 1.20 | 1.21 | 1.23 | 1.24 | 1.25 |

3 | 1.25 | 1.24 | 1.22 | 1.26 | 1.22 | 1.23 | 1.25 | 1.27 | 1.25 | 1.24 | 1.27 | 1.27 |

4 | 1.28 | 1.29 | 1.29 | 1.23 | 1.31 | 1.30 | 1.31 | 1.29 | 1.30 | 1.28 | 1.33 | 1.32 |

5 | 1.39 | 1.41 | 1.38 | 1.41 | 1.37 | 1.42 | 1.33 | 1.37 | 1.37 | 1.40 | 1.36 | 1.39 |

6 | 1.40 | 1.37 | 1.41 | 1.43 | 1.44 | 1.41 | 1.48 | 1.44 | 1.43 | 1.43 | 1.44 | 1.44 |

7 | 1.44 | 1.46 | 1.48 | 1.42 | 1.44 | 1.43 | 1.42 | 1.44 | 1.42 | 1.42 | 1.40 | 1.38 |

8 | 1.55 | 1.55 | 1.52 | 1.56 | 1.56 | 1.58 | 1.57 | 1.53 | 1.53 | 1.55 | 1.56 | 1.54 |

9 | 1.69 | 1.69 | 1.69 | 1.66 | 1.67 | 1.64 | 1.64 | 1.65 | 1.67 | 1.64 | 1.66 | 1.67 |

10 High β | 2.11 | 2.06 | 2.05 | 2.04 | 2.04 | 2.03 | 2.00 | 1.98 | 1.96 | 1.98 | 1.97 | 1.97 |

High to low | 1.01*** | 0.95*** | 0.93*** | 0.92** | 0.92** | 0.87** | 0.84** | 0.81** | 0.77** | 0.77** | 0.76** | 0.75** |

(2.83) | (2.69) | (2.60) | (2.55) | (2.54) | (2.42) | (2.33) | (2.23) | (2.13) | (2.16) | (2.12) | (2.09) | |

Alpha | 0.71*** | 0.64*** | 0.62*** | 0.60*** | 0.61*** | 0.56** | 0.53** | 0.49** | 0.46** | 0.44* | 0.43* | 0.43* |

(3.06) | (2.82) | (2.69) | (2.58) | (2.63) | (2.41) | (2.28) | (2.10) | (1.96) | (1.93) | (1.88) | (1.88) |

Table VIII also presents the magnitude and statistical significance of the FF-3 alphas from the regression of the equal-weighted portfolio returns on a constant, excess market return, SMB and HML factors. The 10-1 difference in the FF-3 alphas is positive and significant at the 5% level or better up to nine-month-ahead predictability. However, the economic and statistical significance of FF-3 alpha gradually reduce to 0.43% per month with the *t*-statistic of 1.88 for 12-month-ahead returns.

#### C. Controlling for Liquidity and Momentum

Following Amihud (2002), we measure stock illiquidity as the ratio of absolute stock return to its dollar volume:

- (14)

where *R _{i,t}* is the return on stock

*i*in month

*t*, and

*VOLD*is the respective monthly volume in dollars. This ratio gives the absolute percentage price change per dollar of monthly trading volume. As in Amihud (2002),

_{i,t}*ILLIQ*follows the Kyle's (1985) concept of illiquidity, that is, the response of price to the associated order flow or trading volume. The measure of stock illiquidity given in Equation (14) represents the price response associated with one dollar of trading volume. Thus, it serves as a rough measure of price impact.

_{i,t}We control for liquidity by first forming decile portfolios ranked based on Amihud's (2002) measure of illiquidity. Then, within each illiquidity decile, we sort stocks into decile portfolios, which we rank based on the GARCH-in-mean betas so that decile 1 (10) contains stocks with the lowest (highest) market beta. In each illiquidity decile, the highest (lowest) beta decile has a higher (lower) average returns. The column labeled “Illiquidity” in Table IX presents the average returns across the 10 illiquidity deciles to produce decile portfolios with dispersion in market beta. This procedure creates a set of decile beta portfolios with near-identical levels of illiquidity. Thus, these decile beta portfolios control for differences in illiquidity. After controlling for illiquidity, we find that the average return difference between high- and low-beta portfolios is 1.20% per month with the Newey-West (1987)*t*-statistic of 3.05. Thus, liquidity does not explain the high (low) returns to high- (low-) beta stocks.

Decile | Illiquidity | Volume | MOM6 | MOM12 |
---|---|---|---|---|

^{}***Significant at the 0.01 level.
| ||||

1 Low β | 0.99*** | 0.92*** | 0.86*** | 1.05*** |

(5.23) | (4.92) | (3.89) | (4.82) | |

2 | 1.13*** | 1.05*** | 0.94*** | 1.18*** |

(5.30) | (5.13) | (3.99) | (5.10) | |

3 | 1.18*** | 1.09*** | 1.02*** | 1.22*** |

(5.06) | (4.72) | (4.19) | (5.17) | |

4 | 1.19*** | 1.19*** | 1.16*** | 1.34*** |

(4.63) | (4.70) | (4.53) | (5.34) | |

5 | 1.29*** | 1.32*** | 1.30*** | 1.53*** |

(4.74) | (4.82) | (5.02) | (6.01) | |

6 | 1.38*** | 1.33*** | 1.36*** | 1.55*** |

(4.61) | (4.51) | (5.00) | (5.71) | |

7 | 1.37*** | 1.45*** | 1.39*** | 1.52*** |

(4.44) | (4.51) | (4.82) | (5.43) | |

8 | 1.57*** | 1.58*** | 1.45*** | 1.60*** |

(4.44) | (4.43) | (4.61) | (5.21) | |

9 | 1.75*** | 1.83*** | 1.57*** | 1.67*** |

(4.36) | (4.51) | (4.47) | (4.83) | |

10 High β | 2.20*** | 2.38*** | 1.85*** | 1.91*** |

(4.33) | (4.58) | (4.11) | (4.34) | |

High to low | 1.20*** | 1.46*** | 0.99*** | 0.86*** |

(3.05) | (3.68) | (2.89) | (2.74) |

When we measure liquidity of individual stocks using dollar trading volume, we obtain similar results. The column labeled “Volume” presents the average returns across the 10 volume deciles to produce decile portfolios with dispersion in Table IX market beta. After controlling for dollar trading volume, we find that the average return increase monotonically from 0.92% to 2.38% when moving from low- to high-beta portfolios. The average return difference between high- and low-beta portfolios is 1.46% per month with the Newey-West (1987)*t*-statistic of 3.68. Thus, trading volume does not explain the high (low) returns to high- (low-) beta stocks either.

We control momentum by first forming decile loser-winner portfolios ranked based on the past six-month average returns of individual stocks. Then, within each six-month momentum portfolio, we sort stocks into decile portfolios ranked based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. The column labeled “*MOM6*” in Table IX presents the average returns across the 10 momentum deciles to produce decile portfolios with dispersion in market beta. This procedure creates a set of decile beta portfolios with near-identical levels of past average six-month returns. Thus, these decile beta portfolios control for differences in momentum. After controlling for momentum, the average return difference between high- and low-beta portfolios is 0.99% per month with the Newey-West (1987)*t*-statistic of 2.89. Thus, momentum does not explain the high (low) returns to high- (low-) beta stocks. We obtain similar results when we form loser-winner portfolios based on the past 12-month average returns (*MOM12*). The average return difference between high- and low-beta portfolios is 0.86% per month with a *t*-statistic of 2.74.

After controlling for liquidity, momentum, size, and BM, we investigate whether the positive relation between conditional beta and the cross-section of expected returns holds in the firm-level Fama-MacBeth (1973) regressions.

Table X presents the time-series averages of the slope coefficients and the Newey-West (1987) adjusted *t*-statistics in parentheses. The regression results indicate a significant positive relation between average stock returns and the conditional GARCH-in-mean betas after controlling for illiquidity, trading volume, past average 6- and 12-month returns with and without size, and BM. The average slope coefficient on β^{GARCH}_{t|t−1} is about 0.51% with *ILLIQ*_{t−1} and 0.52% with *VOL*_{t−1}, and both coefficients are highly significant. The average slope coefficient on β^{GARCH}_{t|t−1} is about 0.41% and significant at the 5% level when we add *MOM6*_{t−1} or *MOM12*_{t−1} along with β^{GARCH}_{t|t−1} in the cross-sectional regressions.

β^{GARCH}_{t|t−1} | ILLIQ_{t−1} | VOL_{t−1} | MOM6_{t−1} | MOM12_{t−1} | logME_{t−1} | log(BE_{t−1}/ME_{t−1}) | R^{2} |
---|---|---|---|---|---|---|---|

^{}***Significant at the 0.01 level. ^{}**Significant at the 0.05 level.
| |||||||

0.5092*** | 6.4938*** | 2.87% | |||||

(2.45) | (3.41) | ||||||

0.5183*** | − | 2.64% | |||||

(2.46) | 6.3239*** | ||||||

(−2.57) | |||||||

0.4123*** | −3.6474** | 3.45% | |||||

(2.06) | (−2.33) | ||||||

0.4058*** | 1.3056 | 3.53% | |||||

(2.12) | (0.61) | ||||||

0.6453*** | 3.1506** | −4.6308*** | −0.1480** | 0.3141*** | 6.27% | ||

(3.00) | (2.20) | (−3.25) | (−2.51) | (4.46) | |||

0.6609*** | 3.2395** | −0.0913 | −0.1659*** | 0.3065*** | 6.33% | ||

(3.19) | (2.22) | (−0.05) | (−2.84) | (4.55) | |||

0.6235*** | −0.0044 | −4.7820*** | −0.1893*** | 0.3215*** | 6.14% | ||

(2.92) | (−0.01) | (−3.41) | (−2.98) | (4.58) | |||

0.6361*** | 0.1139 | −0.3081 | −0.2087*** | 0.3136*** | 6.20% | ||

(3.09) | (0.08) | (−0.16) | (−3.31) | (4.67) |

When we include alternative measures of liquidity and momentum along with β^{GARCH}_{t|t−1}, size, and BM, the average slope coefficient on β^{GARCH}_{t|t−1} becomes stable in the range of 0.62% to 0.66% for different specifications. As shown in for all these multivariate regressions with liquidity, momentum, size, and BM, the average slope coefficient on β^{GARCH}_{t|t−1} is statistically significant at the 1% level.

At an earlier stage of the study, we replicated our results presented in using the conditional beta estimates obtained from the AR(1) and MA(1) specifications. The results turn out to be similar to those from β^{GARCH}_{t|t−1}. We do not present our findings here from β^{AR}_{i,t|t−1} and β^{MA}_{i,t|t−1}. They are available on request.

#### D. Results from the NYSE Sample

To check the robustness of our findings, we exclude the Amex and Nasdaq stocks from our sample and form the beta portfolios by sorting only the NYSE stocks based on the conditional GARCH-in-mean betas. Table XI shows that for the univariate sort of NYSE stocks, the average return difference between high- and low-beta portfolios is about 0.86% with the Newey-West (1987)*t*-statistic of 2.79. The 10-1 difference in the FF-3 alphas is 0.37% with a *t*-statistic of 2.44.

Decile | Average Return (Univariate Sort) | Average Return (After Controlling for Size) | Average Return (After Controlling for BM) |
---|---|---|---|

^{}***Significant at the 0.01 level. ^{}**Significant at the 0.05 level.
| |||

1 Low β | 1.11 | 1.06 | 1.11 |

2 | 1.21 | 1.20 | 1.15 |

3 | 1.34 | 1.24 | 1.22 |

4 | 1.36 | 1.32 | 1.25 |

5 | 1.35 | 1.37 | 1.33 |

6 | 1.35 | 1.37 | 1.37 |

7 | 1.45 | 1.45 | 1.30 |

8 | 1.51 | 1.56 | 1.46 |

9 | 1.54 | 1.65 | 1.58 |

10 High β | 1.97 | 1.90 | 1.89 |

High to low | 0.86*** | 0.84*** | 0.78*** |

(2.79) | (2.81) | (2.94) | |

Alpha | 0.37** | 0.43*** | 0.34** |

(2.44) | (2.87) | (2.23) |

We further examine the cross-sectional relation by forming the beta portfolios within each size and BM decile. Table XI shows that the average return difference between high- and low-beta portfolios is 0.84% after we control for size and 0.78% after we control for BM. Both return differences are statistically significant at the 1% level. The 10-1 differences in the FF-3 alphas are also positive and highly significant. These results indicate that excluding the Amex and Nasdaq sample has almost no effect on our previous findings. These results remain the same for alternative specifications of conditional beta (β^{AR}_{i,t|t−1} and β^{MA}_{i,t|t−1}).

#### E. Controlling for Microstructure Effects and NYSE Breakpoint

Above, we excluded the Amex and Nasdaq stocks and presented the return/beta estimates from the portfolios of NYSE stocks formed based on the NYSE breakpoints. However, these results may be contaminated by microstructure effects because there is only a one-month gap between the conditional beta estimates and portfolio returns. Here, we follow Fama and French (1992) by skipping the month following portfolio formation to avoid microstructure effects and use the *NYSE breakpoints* to generate beta portfolios of NYSE/Amex/Nasdaq stocks with a relatively more balanced average market share. Since there are so many small-cap Nasdaq stocks, we determine portfolio breakdowns by using only NYSE stocks. Doing so enables us to avoid the beta portfolios that contain small stocks from being too small in terms of average market share.

Table XII presents the average returns on the beta portfolios of NYSE/Amex/Nasdaq stocks with NYSE breakpoints after we skip the month following portfolio formation. When we sort portfolios based on the lagged realized beta, the average return difference between high- and low-beta portfolios is economically and statistically nonsignificant. When we form portfolios based on the AR(1), MA(1), and GARCH-in-mean beta estimates, the average return difference between deciles 10 and 1 is about 0.7%, 0.72%, and 0.92% per month, respectively. Similar to our earlier findings, for all conditional beta estimates, the 10-1 differences in average returns and FF-3 alphas are positive and highly significant. Overall, the results in Table XII indicate that forming portfolios with CRSP or NYSE breakpoints and skipping the month following portfolio formation does not affect our main conclusions.

Decile | β^{realized}_{t−2} | β^{AR}_{t−1|t−2} | β^{MA}_{t−1|t−2} | β^{GARCH}_{t−1|t−2} | ||||
---|---|---|---|---|---|---|---|---|

Average Return | Average Beta | Average Return | Average Beta | Average Return | Average Beta | Average Return | Average Beta | |

^{}***Significant at the 0.01 level. ^{}**Significant at the 0.05 level.
| ||||||||

1 Low β | 1.31 | −1.56 | 1.17 | 0.21 | 1.15 | 0.22 | 1.14 | 0.19 |

2 | 1.31 | −0.17 | 1.30 | 0.52 | 1.32 | 0.53 | 1.24 | 0.50 |

3 | 1.36 | 0.18 | 1.40 | 0.65 | 1.38 | 0.66 | 1.33 | 0.64 |

4 | 1.36 | 0.45 | 1.39 | 0.76 | 1.41 | 0.77 | 1.43 | 0.75 |

5 | 1.38 | 0.70 | 1.44 | 0.86 | 1.43 | 0.86 | 1.36 | 0.85 |

6 | 1.40 | 0.96 | 1.45 | 0.95 | 1.48 | 0.95 | 1.43 | 0.94 |

7 | 1.38 | 1.24 | 1.58 | 1.04 | 1.53 | 1.04 | 1.49 | 1.03 |

8 | 1.35 | 1.59 | 1.57 | 1.14 | 1.59 | 1.14 | 1.57 | 1.14 |

9 | 1.31 | 2.09 | 1.65 | 1.29 | 1.67 | 1.28 | 1.68 | 1.30 |

10 High β | 1.23 | 3.82 | 1.87 | 1.73 | 1.87 | 1.69 | 2.06 | 1.80 |

High to low | −0.08 | 0.70** | 0.72** | 0.92*** | ||||

(−0.51) | (2.24) | (2.29) | (2.76) | |||||

Alpha | −0.09 | 0.48** | 0.49** | 0.63*** | ||||

(−0.82) | (2.32) | (2.38) | (3.07) |

### V. Results from Size/BM/Beta Portfolios

- Top of page
- Abstract
- I. Data and Variable Definitions
- II. Average Returns and FF-3 Alphas on Beta Portfolios
- III. Firm-Level Cross-Sectional Regressions
- IV. Robustness Check
- V. Results from Size/BM/Beta Portfolios
- VI. Cross-Sectional Implications of the Conditional CAPM
- VII. Conclusion
- References

When we construct beta portfolios, we control for size or BM ratio, but not both. Here, we test whether the significantly positive relation between conditional beta and expected returns remains intact after we control simultaneously for size and BM.

Table XIII presents the average returns and FF-3 alphas on the quintile portfolios of realized and conditional betas after we control for size and BM. At the beginning of each month *t* from July 1963 to December 2004, we first sort all NYSE/Amex/Nasdaq stocks into five size (market equity) portfolios. Then within each size portfolio, stocks are sorted into five BM (book-to-market equity ratio) portfolios. Finally, within each portfolio formed based on the intersections of five size and five BM portfolios, we sort stocks into five beta portfolios based on their realized and conditional betas in month *t*− 1.

Quintile | β^{realized}_{t−1} | β^{AR}_{t|t−1} | β^{MA}_{t|t−1} | β^{GARCH}_{t|t−1} |
---|---|---|---|---|

^{}***Significant at the 0.01 level.
| ||||

Low β | 1.3016 | 0.9565 | 0.9380 | 0.9351 |

(4.79) | (4.70) | (4.59) | (4.69) | |

2 | 1.3720*** | 1.1579*** | 1.1623*** | 1.1771*** |

(5.44) | (4.76) | (4.79) | (4.88) | |

3 | 1.3938*** | 1.3315*** | 1.3190*** | 1.2629*** |

(5.23) | (4.87) | (4.80) | (4.65) | |

4 | 1.4153*** | 1.5181*** | 1.5217*** | 1.5126*** |

(4.83) | (4.91) | (4.95) | (4.89) | |

High β | 1.3066*** | 1.9297*** | 1.9492*** | 1.9944*** |

(3.69) | (4.81) | (4.83) | (4.84) | |

Return diff. | 0.0050*** | 0.9731*** | 1.0112*** | 1.0593*** |

High β to low β | (0.04) | (3.75) | (3.84) | (3.83) |

Alpha diff. | −0.1138*** | 0.6903*** | 0.7272*** | 0.7552*** |

High β to low β | (−1.02) | (3.92) | (4.07) | (4.08) |

Table XIII shows that when we sort stocks in the 5 × 5 size/BM portfolios into five realized beta (β^{realized}_{t−1}) portfolios, the average return difference between high- and low-beta portfolios is about 0.5% per month with a *t*-statistic of 0.04. Similar to our earlier findings from the univariate and bivariate sorts, there is no significant relation between lagged realized beta and the cross-section of expected returns from trivariate sorts. When we sort the stocks in the 5 × 5 size/BM portfolios into five AR(1), MA(1), and GARCH-in-mean beta portfolios, the average return differences between high- and low-beta portfolios are about 0.97%, 1.01%, and 1.06% per month, respectively. These return differences are statistically significant at the 1% level. Moreover, for all conditional beta estimates, the 5-1 differences in FF-3 alphas are positive and highly significant. Overall, the results in Table XIII indicate that the significant positive relation between conditional beta and the cross-section of expected returns remains the same after we control simultaneously for both size and BM.

To provide further evidence for the significant positive link between conditional beta and expected returns on size/BM/beta portfolios, we run the Fama-MacBeth (1973) regressions using the 125 (5 × 5 × 5) portfolios of size, BM, and beta. First, we compute the monthly realized beta for each of the 125 portfolios, using daily returns within a month. Then, we generate the conditional beta estimates for each of the 125 portfolios using the AR(1), MA(1), and GARCH-in-mean specifications. We use the average firm size and average BM ratio of each portfolio as additional controls in Fama-MacBeth regressions. Table XIV shows that the average slope coefficients on conditional betas are positive and highly significant with and without controlling for the portfolios’ size and BM. Confirming the earlier findings from firm-level regressions, the average slopes on size and BM turn out to be significantly negative and positive, respectively. Overall, we can conclude that the Fama-MacBeth regressions at the firm level and at the portfolio level yield similar results on the relation between market beta and expected returns.

β^{realized}_{t−1} | β^{AR}_{t|t−1} | β^{MA}_{t|t−1} | β^{GARCH}_{t|t−1} | logME_{t−1} | log(BE_{t−1}/ME_{t−1}) | R^{2} |
---|---|---|---|---|---|---|

^{}***Significant at the 0.01 level. ^{}**Significant at the 0.05 level. ^{}*Significant at the 0.10 level.
| ||||||

−0.0759 | 7.98% | |||||

(−1.25) | ||||||

0.3507* | 13.88% | |||||

(1.82) | ||||||

0.3876** | 13.72% | |||||

(1.96) | ||||||

0.4359** | 14.41% | |||||

(2.13) | ||||||

−0.0183 | −0.1864*** | 23.05% | ||||

(−0.28) | (−3.43) | |||||

0.8091*** | −0.2673*** | 30.33% | ||||

(2.83) | (−3.97) | |||||

0.8580*** | −0.2698*** | 30.26% | ||||

(2.94) | (−3.99) | |||||

0.8093*** | −0.2583*** | 30.60% | ||||

(2.78) | (−3.88) | |||||

−0.0389 | 0.7236*** | 15.51% | ||||

(−0.69) | (6.64) | |||||

0.5709*** | 0.7860*** | 20.38% | ||||

(2.97) | (7.24) | |||||

0.6101*** | 0.7891*** | 20.28% | ||||

(3.10) | (7.25) | |||||

0.6356*** | 0.7856*** | 20.91% | ||||

(3.12) | (7.22) | |||||

−0.0074 | −0.1222** | 0.5424*** | 28.98% | |||

(−0.12) | (−2.17) | (4.91) | ||||

0.8472*** | −0.2081*** | 0.4918*** | 35.27% | |||

(3.04) | (−2.99) | (4.82) | ||||

0.8942*** | −0.2106*** | 0.4914*** | 35.23% | |||

(3.15) | (−3.01) | (4.81) | ||||

0.8451*** | −0.1973 *** | 0.5045*** | 35.36% | |||

(2.98) | (−2.88) | (5.11) |

To check whether the cross-sectional relation still holds after we control for the time-series relation between conditional betas and expected returns, we run the pooled panel regressions using both the cross-section and time series of 125 portfolio returns and betas. Table XV presents the parameter estimates and the *t*-statistics that are corrected for heteroskedasticity, first-order autocorrelation, and contemporaneous cross-correlations in the error terms. Similar to our earlier findings, the pooled panel regressions indicate a positive and highly significant relation between conditional beta and expected returns, but the relation between lagged realized beta and expected returns is not significant. These results hold after controlling for size and BM in cross-section and time-series setting.

β^{realized}_{t−1} | β^{AR}_{t|t−1} | β^{MA}_{t|t−1} | β^{GARCH}_{t|t−1} | logME_{t−1} | log(BE_{t−1}/ME_{t−1}) |
---|---|---|---|---|---|

^{}***Significant at the 0.01 level. ^{}**Significant at the 0.05 level.
| |||||

−0.0505** | |||||

(−2.44) | |||||

0.3067*** | |||||

(3.81) | |||||

0.3250*** | |||||

(4.03) | |||||

0.2906*** | |||||

(3.74) | |||||

−0.0196 | −0.2003*** | ||||

(−0.95) | (−8.33) | ||||

0.3061*** | −0.1960*** | ||||

(6.65) | (−8.01) | ||||

0.6251*** | −0.1887*** | ||||

(6.68) | (−7.60) | ||||

0.5147*** | −0.1781*** | ||||

(5.92) | (−7.13) | ||||

−0.0266 | 0.5279*** | ||||

(−1.30) | (11.77) | ||||

0.4372*** | 0.5546*** | ||||

(5.49) | (12.16) | ||||

0.4615*** | 0.5310*** | ||||

(5.79) | (11.46) | ||||

0.4103*** | 0.5378*** | ||||

(5.32) | (12.05) | ||||

−0.0115 | −0.1368*** | 0.4582*** | |||

(−0.56) | (−5.66) | (10.15) | |||

0.6194*** | −0.1283*** | 0.4704*** | |||

(6.83) | (−5.24) | (10.41) | |||

0.6435*** | −0.1227*** | 0.4436*** | |||

(6.89) | (−4.92) | (9.60) | |||

0.5360*** | −0.1098*** | 0.4578*** | |||

(6.18) | (−4.36) | (10.31) |

### VI. Cross-Sectional Implications of the Conditional CAPM

- Top of page
- Abstract
- I. Data and Variable Definitions
- II. Average Returns and FF-3 Alphas on Beta Portfolios
- III. Firm-Level Cross-Sectional Regressions
- IV. Robustness Check
- V. Results from Size/BM/Beta Portfolios
- VI. Cross-Sectional Implications of the Conditional CAPM
- VII. Conclusion
- References

The static (or unconditional) CAPM of Sharpe (1964), Lintner (1965), and Black (1972) indicates that there is a positive linear relation between expected returns on securities and their market betas:

- (15)

where *E*(*R*_{i,t}) is the unconditional expected excess return of asset *i*, *E*(*R*_{m,t}) is the unconditional expected excess return of the market portfolio, and β_{i}= Cov(*R*_{i,t}, *R*_{m,t})/Var(*R*_{m,t}) is the unconditional beta of asset *i*.

Fama and French (1992) and related studies find that the unconditional market beta cannot explain the cross-sectional variation in expected stock returns. The unconditional CAPM was derived by examining the behavior of investors in a hypothetical model in which they live for only one period, but in the real world, investors live for many periods. Hence, in an empirical examination of the CAPM that uses data from the real world, it is necessary to make certain assumptions. One of the most common assumptions in the static CAPM framework is that the betas of the assets remain constant over time. However, this assumption is not reasonable, because the relative risk of a firm's cash flow is likely to vary over the business cycle. As indicated by Harvey (1989), Shanken (1990), Jagannathan and Wang (1996), Ferson and Harvey (1991, 1999), and Lettau and Ludvigson (2001), betas and expected returns generally depend on the nature of the information available at any given point in time, and thus will vary over time.

The conditional version of the CAPM imposes the restriction that conditionally expected returns on assets are linearly related to the conditionally expected return on the market portfolio in excess of the risk-free rate. The coefficient in the linear relation is the asset's conditional beta or the ratio of the conditional covariance of the asset's return with the market to the conditional variance of the market:

- (16)

where *E*(*R*_{i,t+1} | Ω_{t}) is the conditional expected excess return of asset *i*, *E*(*R*_{m,t+1} | Ω_{t}) is the conditional expected excess return of the market portfolio, is the conditional market beta of asset *i*, and Ω_{t} denotes the information set at time *t*.

We can rewrite Equation (16) to simplify the follow-up expressions:

- (17)

where *A*_{m,t+1}=*E*(*R*_{m,t+1} | Ω_{t}) is the time *t*+ 1 conditional expected market risk premium.

Taking the unconditional expectation of both sides of Equation (17), we obtain the unconditional implication of the conditional CAPM:

- (18)

where Cov(*A*_{m,t+1}, β_{i,t+1}) denotes the unconditional covariance, and and are the unconditional means of the corresponding conditional estimates.

We note that the last term in Equation (18) depends only on the part of the conditional beta that is in the linear span of the market risk premium, which motivates Jagannathan and Wang (1996) to decompose the conditional beta of any asset *i* into two orthogonal components by regressing the conditional beta on the market risk premium. For each asset *i*, we run the following regression:

- (19)

where λ_{i}= Cov(*A*_{m,t+1}, β_{i,t+1})/Var(*A*_{m,t+1}) is the unconditional market beta-premium sensitivity that measures the sensitivity of conditional beta to the market risk premium.

Hence, cross-sectionally, the unconditional expected excess return on any asset *i* is a linear function of the unconditional average of its conditional market beta () and its unconditional market beta-premium sensitivity (λ_{i}). Equation (20) implies that stocks with higher expected betas have higher unconditional expected returns, as do stocks with betas that are prone to vary with the market risk premium and hence less stable over the business cycle. Hence, the one-factor conditional CAPM leads to a two-factor model for unconditional expected returns.

A complete test of the conditional CAPM specification requires that we estimate the expected beta () and beta-premium sensitivity (λ_{i}). Here, we use the average conditional beta estimates obtained from AR(1), MA(1), and GARCH-in-mean specifications as a proxy for . We estimate beta-premium sensitivity λ_{i} using the lagged market return as a proxy for the expected market risk premium, that is, λ_{i}= Cov(*R*_{m,t}, β_{i,t+1})/Var(*R*_{m,t}), where we use the lagged market return, *R*_{m,t}, as a proxy for the time *t*+ 1 conditional expected market risk premium, *A*_{m,t+1}=*E*(*R*_{m,t+1} | Ω_{t}) =*R*_{m,t}.

In Equations (21) to (23), , , and are the time-series average of β^{AR}_{i,t|t−1}, β^{MA}_{i,t|t−1}, and β^{GARCH}_{i,t|t−1}, respectively. Here, λ^{AR}_{i}, λ^{MA}_{i}, and λ^{GARCH}_{i} are obtained from the regression of β^{AR}_{i,t|t−1}, β^{MA}_{i,t|t−1}, and β^{GARCH}_{i,t|t−1} on *R*_{m,t}, respectively. The lagged return on the CRSP value-weighted index is our proxy for *R*_{m,t}.

We compute the time-series averages of the slope coefficients and their Newey-West (1987)*t*-statistics from the monthly cross-sectional Fama-MacBeth (1973) regressions of stock returns on their average conditional beta and beta-premium sensitivity. The average slopes on , , and are about 0.622, 0.6189, and 0.5705 with the Newey-West *t*-statistic of 2.89, 2.85, and 2.67, respectively. However, the average slope coefficients on beta-premium sensitivity are economically and statistically nonsignificant for all specifications of the conditional beta measures. The results indicate a significant positive relation between average conditional beta and the cross-section of expected returns within the conditional CAPM framework.

To provide further evidence on the correlation between conditional beta and market risk premium, we investigate the correlations between the conditional betas and the Chicago Fed National Activity Index (CFNAI), which is a weighted average of 85 existing monthly indicators of national economic activity constructed to have an average value of zero and a standard deviation of one. Since economic activity tends toward trend growth rate over time, a positive index reading corresponds to growth above trend, and a negative index reading corresponds to growth below trend. We expect to find a positive relation between expected stock returns and innovations in output (or growth above trend). Actual increases in real economic activity, if greater than expected (or greater than the trend), may increase agents’ expectations of future growth. Forecasts of higher economic growth should make stocks more attractive and thus cause an immediate jump in share prices. That is, the positive relation between expected returns and the CFNAI makes economic sense. Since there is a positive relation between conditional betas and expected returns, we also expect to find a positive link between the conditional beta and the CFNAI. Figure 1 shows that the sample correlations for almost all of the 125 portfolios are positive for the AR(1), MA(1), and GARCH-in-mean beta estimates. These results provide further evidence on the capability of conditional betas to predict the time-series and cross-sectional variation in stock returns.

### VII. Conclusion

- Top of page
- Abstract
- I. Data and Variable Definitions
- II. Average Returns and FF-3 Alphas on Beta Portfolios
- III. Firm-Level Cross-Sectional Regressions
- IV. Robustness Check
- V. Results from Size/BM/Beta Portfolios
- VI. Cross-Sectional Implications of the Conditional CAPM
- VII. Conclusion
- References

In this paper, we investigate the cross-sectional relation between conditional betas and expected stock returns for the sample period of July 1963 to December 2004. First, we use daily returns within a month to compute realized beta for each stock trading at the NYSE, Amex, and Nasdaq and then use autoregressive, moving average, and GARCH-in-mean models to obtain time-varying conditional betas for each stock.

For each specification of conditional beta, we find that the average portfolio returns increase almost monotonically when moving from low-beta to high-beta portfolios. The portfolio-level analyses and the firm-level cross-sectional regressions indicate that the positive relation between the conditional betas and the cross-section of average returns is economically and statistically significant. For the NYSE/Amex/Nasdaq sample, the average return difference between high- and low-beta portfolios is in the range of 0.89% to 1.01% per month, depending on the time-varying specification of conditional beta.

To check whether our findings are driven by small, illiquid, and low-price stocks, we exclude the Amex and Nasdaq stocks and form the beta portfolios by sorting only the NYSE stocks based on the conditional betas. The results indicate that excluding the Amex and Nasdaq sample has almost no effect on our original findings. We also control for the cross-sectional effects of size, BM, liquidity, and momentum. After controlling for these effects, we estimate the cross-sectional beta premium to be in the range of 0.86% to 1.46% per month. These results are robust across different measures of conditional beta. ▪

- 1
Jegadeesh (1992) obtains results similar to Fama and French (1992).

- 2
This is because an asset that is on the conditional mean-variance frontier need not be on the unconditional frontier, as Dybvig and Ross (1985) and Hansen and Richard (1987) point out. Also see Chan and Chen (1988) who indicate that even when betas vary over time, unconditional CAPM can hold.

- 3
An incomplete list includes Bollerslev, Engle, and Wooldridge (1988), Harvey (1989, 2001), Shanken (1990, 1992), Ferson and Harvey (1991, 1999), Fama and French (1997), Lettau and Ludvigson (2001), Campbell and Vuolteenaho (2004), Jostova and Philipov (2005), Petkova and Zhang (2005), Ang and Chen (2007), Lewellen and Nagel (2006), and Bali (2008).

- 4
French, Schwert, and Stambaugh (1987), Campbell, Lettau, Malkiel, and Xu (2001), Goyal and Santa-Clara (2003), and Bali, Cakici, Yan, and Zhang (2005) use within-month daily returns to estimate the monthly market variance or the monthly idiosyncratic or total volatility of each stock trading at the NYSE, Amex, and Nasdaq.

- 5
Brav, Lehavy, and Michaely (2005) use analysts’ expected rates of return instead of realized rates of return as a proxy for expected return and identify a positive, robust relation between expected return and market beta. Based on their experimental study, Bloomfield and Michaely (2004) find that market professionals expect firms with higher betas to be riskier investments and to generate higher returns. Harris, Marston, Mishra, and O'Brien (2003) estimate the relation between market beta and expected returns for S&P 500 stocks, and their findings indicate that for estimating the cost of equity, the choice between the domestic and global CAPM may not be a material issue for many large US firms.

- 6
- 7
We do not present the results from β

^{AR}_{i,t|t−1}and β^{MA}_{i,t|t−1}, which are similar to those in the table. They are available on request.

### References

- Top of page
- Abstract
- I. Data and Variable Definitions
- II. Average Returns and FF-3 Alphas on Beta Portfolios
- III. Firm-Level Cross-Sectional Regressions
- IV. Robustness Check
- V. Results from Size/BM/Beta Portfolios
- VI. Cross-Sectional Implications of the Conditional CAPM
- VII. Conclusion
- References

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