Value at Risk and Expected Stock Returns

Transcription

1 Value at isk and Expected Stock eturns August 2003 Turan G. Bali Associate Professor of Finance Department of Economics & Finance Baruch College, Zicklin School of Business City University of New York 17 Lexington Avenue, Box New York, New York Phone: (646) Fax : (646) Turan_Bali@baruch.cuny.edu Nusret Cakici Professor of Finance Department of Economics City College of New York City University of New York Convent Avenue at 138 th Street New York, New York Phone: (212) Fax : (212) ncakici@ccny.cuny.edu Key words: cross-section of expected returns, value at risk, beta, size, liquidity, total risk JEL classification: G10, G11, C13

2 Value at isk and Expected Stock eturns ABSTACT This paper provides empirical evidence that firm size, liquidity, and Value-at-isk (Va) explain the cross-sectional variation in expected returns, while market beta and total volatility have almost no power to capture the cross-section of expected returns at the firm level. The strong positive relation between average returns and Va turns out to be robust across different investment horizons and loss probability levels. In addition to the cross-sectional regressions at the firm level, this study tests the empirical performance of Va at the portfolio level using a time-series approach. The results based on the 25 size/book-to-market portfolios of Fama-French (1993) indicate that Va has additional explanatory power after controlling for the characteristics of market return, size, book-to-market ratio, and liquidity. Mini-Abstract This paper provides empirical evidence that firm size, liquidity, and Value-at-isk (Va) explain the cross-sectional variation in expected returns, while market beta and total volatility have almost no power to capture the cross-section of expected returns at the firm level. The results based on the 25 size/bookto-market portfolios of Fama-French (1993) indicate that Va has additional explanatory power after controlling for the characteristics of market return, size, book-to-market ratio, and liquidity.

3 DIGEST In addition to several firm characteristics, previous empirical studies use other factors such as total risk and diversifiable risk to explain the cross-section of expected returns. Value-at-isk has so far not been considered as an alternative risk factor that adds to the explanation of stock returns. An important contribution of this paper is to test whether the maximum likely loss measured by Value-at- isk (Va) can explain the cross-sectional and time-series differences in expected returns. This study provides evidence from the monthly and annual Fama-MacBeth (1973) regressions that size, liquidity and Va can capture the cross-sectional differences in expected returns on the NYSE, AMEX, and NASDAQ stocks for the period of January 1963 to December 2001, while market beta and total volatility have almost no power to explain average stock returns at the firm level. The paper also compares the relative performance of size, beta, and Va in explaining the cross-sectional variation in portfolio returns. The results show that all the risk factors considered in the paper can capture the crosssectional differences in portfolio returns, but Va has the best performance in terms of 2 values. The strong positive relation between stock (or portfolio) returns and Va turns out to be robust across different investment horizons and loss probability levels. In addition to the cross-sectional regressions in a Fama-French (1992) asset-pricing framework, a time-series regression approach is used to evaluate the empirical performance of Va at the portfolio level. The paper introduces an alternative factor HVAL (high Va minus low Va), which is meant to mimic the risk factor in returns related to Value-at-isk, and is defined as the difference between the simple average of the returns on the high Va and the low Va portfolios. The relative performance of total volatility and Va along with liquidity is investigated in terms of their ability to capture time-series variation in stock returns using the 25 portfolios of Fama-French (1993). Monthly returns on a portfolio of stocks are regressed on the returns to a market portfolio of stocks and the portfolios for size, book-tomarket, liquidity, and Va. The results based on the 25 portfolios of Fama-French indicate that Va can capture substantial time-series variation in stock returns, and provides an additional explanatory power after controlling for the characteristics of market return, size, book-to-market ratio, and liquidity. The results also imply that the relation between Va and expected stock returns is not a reversal in long-term returns, liquidity or volatility effect. The modern theory of portfolio choice determines the optimum asset mix by maximizing the expected risk premium per unit of risk in a mean-variance framework or the expected value of some utility function approximated by the expected return and variance of the portfolio. In both cases, market risk of the portfolio is defined in terms of the variance (or standard deviation) of expected portfolio returns. Modeling portfolio risk with the traditional volatility measures implies that investors are concerned only about the average variation (and co-variation) of individual stock returns, and they are not allowed to treat the negative and positive tails of the return distribution separately. The standard risk measures determine the volatility of unexpected outcomes under normal market conditions, which corresponds to the normal functioning of financial markets during ordinary periods. However, neither the variance nor the standard deviation can yield an accurate characterization of actual portfolio risk during highly volatile periods. Therefore, the set of mean-variance efficient portfolios may produce an inefficient strategy for maximizing expected return of the portfolio while minimizing its risk. Our results suggest a new approach to optimal portfolio selection in a Va framework. A mean-va approach can be introduced to allocate financial assets by maximizing the expected value of some utility function approximated by the expected return and Va of the portfolio as well as the investor s aversion to Va. 1

4 2 Value at isk and Expected Stock eturns I. Introduction The capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965), and Black (1972) implies the mean-variance efficiency of the market portfolio in the sense of Markowitz (1959). The primary implication of the Sharpe-Lintner-Black (SLB) model is that there exists a positive linear relation between expected returns on securities and their market betas, and variables other than beta should not capture the cross-sectional variation in expected returns. However, over the last two decades, many researchers have found that idiosyncratic factors such as firm size, book-to-market equity, and earningsprice ratio have significant explanatory power for average stock returns, while beta has little or no power at the firm level. In addition to several firm characteristics, previous empirical studies use other factors such as total risk and diversifiable risk to explain the cross-section of expected returns. 1 Value-at-isk has so far not been considered as an alternative risk factor that adds to the explanation of stock returns. An important contribution of this paper is to test whether the maximum likely loss measured by Value-at- isk (Va) can explain the cross-sectional differences in expected returns. The empirical findings indicate that there is a positive and statistically significant relation between Va and returns on the NYSE, AMEX, and NASDAQ stocks for the January 1963 December 2001 period. This result is robust across different investment horizons and loss probability levels. We provide the time-series averages of the Fama-MacBeth (1973) slope coefficients from monthly and annual cross-sectional regressions to check the sensitivity of our findings to alternative investment horizons. The Fama-MacBeth regression results indicate that Va and firm size have significant explanatory power for expected returns, while market beta has almost no power at the firm level. However, the results turn out to be slightly different at the portfolio level when we compare the relative performance of size, beta, and Va in explaining the cross-sectional variation in portfolio returns. All the risk factors considered in the paper can capture the cross-sectional differences in portfolio returns, but Va has the best overall performance in terms of 2 values. In addition to the cross-sectional regressions in a Fama-French (1992) asset-pricing framework at the firm level, this paper uses the time-series regression approach to evaluate the empirical performance of Va at the portfolio level. Monthly returns on a portfolio of stocks are regressed on the returns to a market portfolio of stocks and mimicking portfolios for size, book-to-market equity (BE/ME), and Value-at-isk. The results based on the 25 size/book-to-market portfolios of Fama-French (1993) indicate that Va can capture substantial time-series variation in stock returns.

5 3 The relationship between liquidity and expected stock returns has also received considerable attention [e.g. Amihud and Mendelson (1986, 1991), Amihud (2002), Pastor and Stambaugh (2002), Chordia, oll, and Subrahmanyam (2002), and the references therein]. One concern is that the Va measure may be proxying for liquidity risk. To investigate whether the maximum likely loss proxies for liquidity, we compute the correlation of Va with liquidity, and test the additional explanatory power of Va after controlling for the 3 Fama-French (1993) factors and liquidity. The results indicate that the relation between Va and expected stock returns is not a liquidity effect. DeBondt and Thaler (1985) find a reversal in long-term returns; stocks with low long-term (three- to five-year) past returns tend to have higher future returns. One concern is that if firms with the highest Va tend to be firms that also have the lowest cumulative returns over the period, then the results of this paper may be related to those of the reversal literature. To investigate whether the maximum likely loss measured by Va proxies for reversal in long-term returns, we measure the magnitude of a relationship between value-at-risk and the cumulative returns over past three years. The cross-sectional correlations and their time-series averages indicate a weak association between Va and reversal in long-term returns. The paper is organized as follows. Section II describes the data and variable definitions. Section III tests if Va can explain the cross-sectional variation in expected returns at the firm and portfolio level. Section IV determines whether Va can capture time-series variation in stock returns. Section V concludes the paper. II. Data and Variable Definitions The data include all New York Stock Exchange (NYSE), American Stock Exchange (AMEX), and NASDAQ nonfinancial firms from the Center for esearch in Security Prices (CSP) for the period from January 1958 through December In December of each year starting from December 1962, the following variables are computed for each firm in the sample: 3 SIZE: The most prominent empirical contradiction of the SLB model is the size effect of Banz (1981) who finds strong negative relation between size and average return. Following the existing literature, the firm size is measured by ln(me), the natural logarithm of the market value of equity (a stock s price times shares outstanding) for each stock as of the sample selection date (December of each year). BETA: Following Fama and French (1992), first all NYSE stocks on CSP are sorted by firm size (ME) to determine the NYSE decile breakpoints for ME. Second, NYSE, AMEX, and

6 4 NASDAQ stocks are allocated to 10 size portfolios based on the NYSE decile breakpoints. Then we subdivide each size decile into 10 portfolios on the basis of historical or pre-ranking βs for NYSE stocks. The pre-ranking βs are estimated with two to five years of monthly returns (as available) ending in December of year t. After assigning each stock in the sample to one of ten ME deciles and one of ten pre-ranking β deciles, we compute the equally-weighted monthly returns on the resulting 100 portfolios for the next 12 months, from January of year t to December of year t. This procedure yields 468 post-ranking monthly returns for each of 100 portfolios from January 1963 to December Finally, post-ranking βs are estimated using the full-sample (468 months) of post-ranking returns on each of the 100 portfolios, with the CSP value weighted index used as the proxy for the market. Following Fama and French (1992), the pre-ranking βs are estimated as the sum of the slopes from a regression of monthly returns on the current and prior month s returns on the CSP value-weighted index: j, t = β 0, j + β1, jm, t + β2, j m, t 1 + e j, t where j,t is the monthly return on stock j in period t, m,t is the monthly return on the CSP value-weighted index in period t, and β 1,j + β 2,j is the pre-ranking β for stock j. Va: There are three main decision variables that are required to estimate Va the confidence level, a target horizon, and an estimation model. In this paper, we use three confidence levels (99%, 95%, 90%) to check the robustness of Va as an explanatory variable for the expected stock returns. The time horizon is 1 month. Estimation model is based on the lower tail of the actual empirical distribution. We use 24 to 60 monthly returns (as available) to estimate the mean, mean, and the cut off return at the 99%, 95%, and 90% confidence levels ( 99%, 95%, 90% ) from the empirical distribution. The 1%, 5%, and 10% Vas are measured by the first-lowest, third-lowest, and sixth-lowest observation of 60 monthly returns in December of each year. 4 It should be noted that the original Va measures are multiplied by 1 before running our regressions. The original maximum likely loss values are negative since they are obtained from the left tail of the distribution, but the variables 1% Va, 5% Va, and 10% Va used in our regressions are defined as 1 the maximum likely loss. Therefore, the slope coefficients turn out to be positive, which gives the central result of the paper that there is a statistically significant positive relation between the Va and the cross-section of expected returns, i.e, the more a stock can potentially fall in value the higher should be the expected return.

7 5 III. Does Value-at-isk Explain the Cross-Section of Expected eturns? An earlier version of this paper presents a theoretical framework in which a portfolio s risk (measured by Va) is shown to be positively related to the portfolio s expected return. This section tests whether the 1% Va, 5% Va, and 10% Va portfolios can produce large and statistically significant cross-sectional variation in expected returns. In December of each year, all NYSE stocks on CSP are sorted by 1% Va, 5% Va, and 10% Va to determine the NYSE decile breakpoints for each Va measure. Then, NYSE, AMEX, and NASDAQ stocks are allocated to ten 1%, 5%, and 10% Va deciles based on the NYSE breakpoints. Decile 1 is the 10% of the total sample of stocks with the lowest Va. Decile 10 is the stocks with the highest Va. Table I shows the average returns of the Va portfolios from deciles 1 through 10, the average return differential between deciles 10 and 1, and the corresponding t-statistics. The results indicate that when common stock portfolios are formed on the 1%, 5%, and 10% Va, average stock returns are positively related to Va. In other words, firms with the lowest (highes maximum likely loss have the lowest (highes average returns. Going from the lowest 1% Va decile to the highest 1% Va decile, average returns on Va portfolios increase from 1.08% per month to 2.04% per month monotonically. The average return differential between the portfolios of the highest decile Va stocks and the lowest decile Va stocks is 0.96% per month (11.52% per annum), and statistically significant at the 5% level (t-stat = 2.34), using Newey-West (1987) standard errors with 3 lags. A strong positive relation is also detected in average returns on Va portfolios and the 5% and 10% Vas. Specifically, the average return differential between the portfolios of the highest decile Va stocks and the lowest decile Va stocks is 0.96% per month (11.52% per annum) for the 5% Va, and 0.97% per month (11.64% per annum) for 10% Va. These average return differentials are statistically significant at the 5% level: t-statistics are 2.18 and 2.20 for the 5% and 10% Va portfolios. These results imply that the more a stock can potentially fall in value the higher should be the expected return. In Figure 1, we graph the relationship between the 1% (5%) [10%] Va levels and the average returns from the decile portfolios. It is clear that the portfolios of higher Va companies tend to produce rates of return that are greater than the returns from portfolios of lower Va companies. To measure the degree of strong positive relation between average stock returns and value-at-risk, we regress the average returns from the decile portfolios on the average level of 1% (5%) [10%] Vas. The results indicate that the positive coefficients on Va are highly significant, and the 2 values are in the range of 88% to 95%. Table 1 presents time-series averages of the slopes from the month-by-month Fama-MacBeth (1973) (FM) regressions of the cross-section of realized returns on size, beta, 1% Va, 5% Va, and 10% Va. The average slopes provide standard FM tests for determining which explanatory variables on

8 6 average have non-zero expected premiums during the January 1963 to December 2001 period. Monthly cross-sectional regressions are run for the following econometric specifications: = ω + γ ln( ME) + ε jt, t t jt, jt, = ω + γ BETA + ε jt, t t jt, jt, = ω + γ Va( α ) + ε jt, t t jt, jt, where j = 1, 2,, N t (= the number of firms in month, j,t is the realized return on stock j in month t, BETA j,t is the full-sample post-ranking beta for the ME/pre-ranking β portfolios for firm j in month t, ln(me) j,t is the natural logarithm of market equity for firm j in month t, Va(α) j,t is 1 times the maximum likely loss (Va) for firm j in month t with the loss probability level α = 1%, 5%, and 10%, and ε j,t is the residual series from cross-sectional regressions. The null hypothesis is that the time series average of the monthly regression slopes, γ t, is zero, and statistical significance is established using a standard t-test. Panel A of Table 1 reports the time series averages of γ t, the t-statistics, and the time-series averages of the determination coefficients ( ) over the 468 months in the sample. The t-statistics 2 t shown in parentheses are the time-series average of γ t divided by its time-series standard error. As expected, the 2 values are very small at the firm level. Specifically, the 2 measures are in the range of 1.80% to 2.45% for Va, whereas the 2 equals 1.75% for size and 1.82% for BETA. The results indicate a significantly negative relation between monthly stock returns and firm size. The average slope from the monthly regressions of realized returns on size, ln(me), is about 0.20 with a t-statistic of In contrast to the persistent explanatory power of size, the FM regressions indicate that BETA does not capture the cross-sectional variation in expected returns for the January 1963 December 2001 period. The average slope, γ t, from the univariate regressions of monthly security returns on BETA is about 0.32 with a t-statistic of This result statistically confirms that the relation between average returns and beta is flat at the firm level, and is consistent with Fama and French (1992). Panel A provides empirical evidence that the maximum likely loss of the companies in the sample explains the crosssection of expected returns. The average monthly slopes from the univariate regressions of returns on 1% Va, 5% Va, and 10% Va are found to be statistically significant at the 10% level. Specifically, the average monthly slopes on 1% Va, 5% Va, and 10% Va are estimated to be about 0.02 (1.79), 0.03 (1.77), and 0.04 (1.83) with t-statistics in parentheses; i.e., the positive relation between j and Va(α) j turns out to be robust across different loss probability levels.

9 7 To check the robustness of estimated risk premia on size, BETA, and Va, we decide to present the time-series averages of the slope coefficients from the annual Fama-MacBeth regressions. In addition, Levy (1972) indicates that the correlation matrix will vary across investors for different investment holding periods. His results suggest that if an investor plans to hold the portfolio for one month, then the monthly update as in the standard FM procedure will be more appropriate for calculating average slopes. But, if she plans to hold it for a year, then the annual update that uses the annual rates of return on stocks will be more relevant. By presenting the results from monthly and annual regressions, we check the sensitivity of estimated slopes to alternative investment horizons. Panel B of Table 1 displays the time series averages of the annual slope coefficients with their t- statistics during the period (39 years). Here the dependent variable is the annual stock return. The results from the annual regressions turn out to be similar to our earlier findings from the month-bymonth FM regressions: Only size and Va can capture the cross-sectional differences in expected returns while BETA has almost no power to explain realized stock returns at the firm level. More specifically, the average slope from the annual regressions of realized returns on ln(me) is 0.29 with a t-statistic of In other words, the size effect [smaller (larger) stocks have higher (lower) average returns] is found to be robust across different investment horizons. The average slope, γ t, from the regressions of annual security returns on BETA is about 0.57 with a t-statistic of 1.56, confirming the flat relation between average returns and beta at the firm level. The average annual slopes from the regressions of annual returns on 1% Va, 5% Va, and 10% Va are about 0.04, 0.06, and 0.07, and they are all statistically significant at the 5% level. In other words, there is a stronger positive relation between annual expected returns and Vas compared to the relation between monthly returns and Vas. A notable point in Panel B is that similar to our earlier findings the average 2 values are very small at the firm level, but Va(α) j has a slightly better performance compared to size and BETA in explaining the cross-section of expected returns. Specifically, the 2 measures are in the range of 2.46% to 3.22% for Va, whereas the 2 equals 2.77% for size and 2.02% for BETA. It is important to note that the preceding analyses are conducted at the firm level, and the results turn out to be slightly different at the portfolio level. Panel C of Table I presents the relative performance of risk factors (size, BETA, Va) in explaining the cross-sectional variation in portfolio returns. Here the dependent variable is the annual return of the portfolio. The results from the portfolio regressions indicate that, in addition to size and Va, BETA also captures the cross-sectional differences in expected returns at the portfolio level. More specifically, the average slope from the annual regressions of portfolio returns on ln(me) is 0.13 with a t-statistic of The average slope on BETA is about 0.24 with a t-statistic of 23.21, consistent with the primary implication of the (SLB) model. The average

10 8 annual slopes from the regressions of portfolio returns on 1% Va, 5% Va, and 10% Va are about 0.03, 0.04, and 0.05, and they are all statistically significant at the 1% level. In other words, there is a stronger positive relation between average returns and Vas at the portfolio level. As expected, the average 2 values are considerably high at the portfolio level. A notable point in Panel C is that Va(α) j has a much better performance compared to size and BETA in explaining the cross-sectional variation in portfolio returns. Specifically, the 2 measures are in the range of 73.88% to 76.79% for Vas, whereas the 2 equals 56.72% for size and 50.06% for BETA. In the empirical asset pricing literature, the 25 portfolios of Fama and French (1993), in which firms are sorted according to size and book-to-market, are now the benchmark portfolios in testing and evaluating asset pricing models. Following Fama and French (1993), we form 25 portfolios according to size, BETA, Va. First, all the NYSE, AMEX, and NASDAQ stocks are sorted according to their market capitalization and then allocated to 25 size portfolios based on the NYSE decile breakpoints. Then, the relation between size and the cross-section of portfolio returns are determined using an OLS regression. Second, all the NYSE, AMEX, and NASDAQ stocks are sorted according to post-ranking betas and then allocated to 25 beta portfolios. The relation between post-ranking βs and the crosssection of portfolio returns are determined using a univariate regression. Third, all the NYSE, AMEX, and NASDAQ stocks are sorted according to 1% Vas and then allocated to 25 Va portfolios. The relation between the 1% Va and the cross-section of portfolio returns are determined based on the univariate OLS regression. The same procedure is repeated for the 5% and 10% Vas. We should note that at an earlier stage of the study we increased the number of portfolios to 50 in order to check the robustness of our findings. The qualitative results turn out to be very similar, i.e., the relative performance of the risk factors is not affected. Another notable point is that we test the empirical performance of size, BETA, and Va within size-portfolios, BETA-portfolios, and Va-portfolios. For example, within 25 size-portfolios, the average slope from the regressions of portfolio returns on ln(me) is 0.11 with a t-statistic of The average slope on BETA is about 1.10 with a t-statistic of The average slope on 1% Va is about 0.07 with a t-statistic of The 2 measures are about 70% for Vas, whereas the 2 equals 53% for size and 35% for BETA. IV. Does Value-at-isk Explain the Time-Series Variation in Expected eturns? In the empirical asset pricing literature, the 25 portfolios of Fama and French (1993), in which firms are sorted according to their market capitalization (size) and the ratio of book value of equity to its market value (book-to-market ratio), are now benchmark portfolios in tests and evaluations of asset pricing models. To study common risk factors in stock returns, Fama and French use six portfolios

11 9 formed from sorts of stocks on ME and BE/ME. As described in Fama-French (1993), in June of each year t from 1963 to 1991, all NYSE stocks on CSP are ranked on size. The median NYSE size is then used to split NYSE, AMEX, and NASDAQ stocks into two groups, small and big (S and B). They also break NYSE, AMEX, and NASDAQ stocks into three book-to-market equity groups based on the breakpoints for bottom 30% (Low), middle 40% (Medium), and top 30% (High) of the ranked values of BE/ME for NYSE stocks. In this way, they construct six portfolios (S/L, S/M, S/H, B/L, B/M, B/H) from the intersection of the two ME and three BE/ME groups. In a series of papers, Fama and French (1993, 1995, 1996) document the importance of SMB (small minus big: the difference between the return on a portfolio of small size stocks and the return on a portfolio of large size stocks) and HML (high minus low: the difference between the return on a portfolio of high BE/ME stocks and the return on a portfolio of low BE/ME stocks). In addition to SMB and HML, Fama and French use the excess market return, M-F, as proxy for the market factor in stock returns. M is the return on the value-weighted portfolio of the stocks in the six size-be/me portfolios, plus the negative-be stocks excluded from the portfolios. F is the one-month Treasury bill rate. Fama-French (1993) use excess returns on 25 portfolios, formed on size and book-to-market equity, as dependent variables in the time-series regressions. The 25 size-be/me portfolios are formed much like the six size-be/me portfolios discussed above [see Fama-French (1993) for more details]. To test the empirical performance of Va based on the 25 portfolios of Fama-French (1993), this paper introduces an alternative factor HVAL (high Va minus low Va), which is meant to mimic the risk factor in returns related to Value-at-isk, and is defined as the difference between the simple average of the returns on the high Va and the low Va portfolios. The construction of 1% Va portfolios is similar to Fama-French s size portfolios: in December of each year t from 1963 to 2001, all NYSE stocks are ranked on 1% Va. The median 1% Va figure is used to split NYSE, AMEX, and NASDAQ stocks into two groups, high Va and low Va. To find the direction and magnitude of a relationship between HVAL and the three Fama- French factors, we calculate the correlations between HVAL and M-F, SMB, and HML. Correlations HVAL M-F SMB HML HVAL 1 M-F SMB HML

12 10 This table shows that HVAL is positively correlated with M-F and SMB, whereas it is negatively correlated with HML. A notable point is that the positive relation between HVAL and SMB is much stronger than the negative relation between HVAL and HML. Table 1A in Appendix provides strong evidence that HVAL, when used alone, captures substantial time-series variation in stock returns. Based on the 2 values given in Table 1A, HVAL has much more explanatory power than SMB and HML for the 25 portfolios of Fama-French (1993). To compare the relative performance of HVAL, M-F, SMB, and HML, we calculate the correlations between the 25 portfolios returns and alternative factors. The following table shows, not surprisingly, that the excess return on the market portfolio of stocks, M-F, captures more common variation in stock returns than HVAL, SMB, and HML. 5 The average correlation between the 25 portfolios returns and M-F is about A notable point is that the average correlation between HVAL and monthly returns on the 25 portfolios is about 0.59, whereas the average correlation is about 0.47 for SMB and for HML. The table below indicates that HVAL as a single-factor is superior to SMB and HML in explaining the time-series variation in stock returns. Correlations HVAL M-F SMB HML S1B S1B S1B S1B S1B S2B S2B S2B S2B S2B S2B S3B S3B S3B S3B S4B S4B S4B S4B S4B S5B S5B S5B S5B S5B Average

13 11 At an earlier stage of the study, the role of alternative factors (M-F, SMB, HML, HVAL) in returns is analyzed in four steps. We examine (a) one-factor models: regressions that use only M-F, SMB, HML, or HVAL to explain excess stock returns, (b) two-factor models: regressions that use M- F along with SMB, HML, or HVAL as explanatory variables, (c) three-factor model: regressions that use M-F, SMB and HML, and (d) four-factor model: regressions that use M-F, SMB, HML, and HVAL as explanatory variables. As shown in the Appendix, the results indicate that Va can capture substantial time-series variation in stock returns in the one-factor and two-factor models. More important, Va has additional explanatory power after controlling for the characteristics of M-F, SMB, and HML in the four-factor model. 6 One concern is that Va may be proxying for DeBondt and Thaler s (1985) reversal in long-term returns. DeBondt and Thaler find that when portfolios are formed on long-term (three- to five-year) past returns, losers (low past returns) have high future returns and winners (high past returns) have low future returns. If firms with the highest Va tend to be firms that also have the lowest cumulative returns over the period, then the results of this paper may be related to those of the reversal literature. To investigate whether the maximum likely loss measured by Va proxies for reversal in long-term returns, we run a series of monthly cross-sectional regressions from January 1963 to December 2001: α= 1% eversal j,t = ωt + γ tva j, t + ε j, where j = 1, 2,, N t (= the number of firms in month, eversal j,t is the cumulative return over the t past three years on stock j in month t, Va is the 1% Va of firm j in month t, and ε j,t is the residual α= 1% j, t series from the cross-sectional regressions. For each month from January 1963 to December 2001, we compute the determination coefficients (or 2 s) to measure the proportion of the total variation in the cumulative returns that can be explained by the 1% Vas. 7 Figure 2 plots the 2 values for each month from January 1963 to December The time-series average of the 2 s is about 15.63%, implying a weak association between Va and reversal in long-term returns. 8 Another concern is that the Va measure may be proxying total volatility or liquidity risk. We should note that we do not assume normality or any other distribution (e.g., Student-t or GED) in our empirical analyses. In other words, value at risk of an individual asset is not measured based on the formula: Va = Ω ( α) σ, where i and σ i are the mean and standard deviation of returns on asset i i i i, and Ω( α ) is the critical value based on a density function. Instead, the 1%, 5%, and 10% Va estimates are obtained from the actual empirical distribution, and therefore they are not perfectly correlated with the standard deviation of stock returns. In addition, there is substantial empirical evidence that the

14 12 distribution of financial returns is typically skewed and leptokurtic. That is, the unconditional return distribution shows high peaks, fat tails and more outliers on the left (or righ tail. This implies that extreme events are much more likely to occur in practice than would be predicted by the thin-tailed normal (or log-normal) distribution. This also suggests that the normality assumption can produce Va numbers that are inappropriate measures of the true risk faced by financial firms. To account for skewness and kurtosis in the data, we use the historical empirical distribution that account for the nonnormality of returns and higher moments of the frequency distribution. If returns were normally or stock prices were log-normally distributed, then there would be a close relationship between Va and total volatility. However, total volatility and Va used in our empirical analyses are not the same because the returns (prices) are not normally (log-normally) distributed. Moreover, we do not use any parametric distribution such as the normal or log-normal to calculate Va. Therefore, the 1%, 5%, and 10% Va estimates do not depend only on the standard deviation of the normal or log-normal distribution. Instead, they are obtained from the empirical distribution, which implies that the Va measures depend on the mean, standard deviation, skewness, kurtosis, and higher moments of the distribution. We calculate the correlation between total volatility and Va across firms and time. The average correlation is found to be around 0.7, but this does not necessarily mean that Va is proxying for total volatility in forecasting expected stock returns. To asses whether it does, we run cross-sectional regressions of the monthly and annual returns on Va and total volatility. We find that total volatility by itself does not predict the realized stock returns. The regression coefficient is insignificant, and when we include both Va and total volatility, the coefficient on Va is almost unchanged from Table I and retains its significance, while the coefficient on total volatility is close to zero. Total volatility (TV) is measured by the historical standard deviation which is calculated for each stock in December of each year using 24 to 60 monthly returns (as available). Following the usual Fama- MacBeth (1973) procedure, the cross-section of realized returns is regressed on total volatility, and 468 monthly slope coefficients are obtained from j, t ω j, t + γ j, ttvj, t + ε j, t =. The time-series average of the monthly slopes, γ j,t, is found to be with a t-statistic of , i.e., total volatility (TV) has almost no power to explain the cross-section of expected returns at the firm level for the period. Similarly, the time-series average of the slope coefficients from the annual Fama-MacBeth regressions of the cross-section of realized annual returns on total volatility turns out to be with a t-statistic of The results indicate that Va has more power than total volatility in explaining the cross-section of expected returns at the firm level.

15 13 The relationship between liquidity and expected stock returns has received considerable attention. Liquidity generally implies the ability to trade large quantities quickly, at low cost, and without inducing a large change in the price level. In this paper, following Amihud (2002), we measure stock illiquidity as the ratio of absolute stock return to its dollar volume, ILLIQ i, t i, t / VOLDi, t =, where i,t is the return on stock i in month t, and VOLD i,t is the monthly volume in dollars. This ratio gives the absolute percentage price change per dollar of monthly trading volume. As discussed in Amihud, ILLIQ i,t follows the Kyle s (1985) concept of illiquidity, i.e., the response of price to the associated order flow or trading volume. To investigate whether the maximum likely loss proxies for illiquidity, we calculate the correlation of Va with illiquidity across firms and time. The average correlation is found to be about 0.1, and not significant. The results based on the cross-sectional regressions indicate that the relation between Va and expected stock returns is not a liquidity effect. Before we test the additional explanatory power of total volatility and Va after controlling for the 3 Fama-French (1993) factors and liquidity, we test whether expected stock return is an increasing function of stock illiquidity. Following the Fama-MacBeth (1973) procedure, the cross-section of realized returns is regressed on illiquidity measure defined above, and 468 monthly slope coefficients are obtained from j, t ω j, t + γ j, t ILLIQ j, t + ε j, t =. The time-series average of the monthly slopes, γ j,t, is found to be 0.83 with a t-statistic of Similarly, the time-series average of the slope coefficients from the annual Fama-MacBeth regressions of the cross-section of realized annual returns on ILLIQ turns out to be 0.08 with a t-statistic of These results are consistent with Amihud (2002), and indicate a strong positive relation between stock illiquidity and expected returns at the firm level for the period. To test the empirical performance of total volatility, illiquidity, and Va based on the 25 portfolios of Fama-French (1993), we introduce a factor for TV and ILLIQ very much like HVAL (high Va minus low Va). The factor for total volatility (HTVL) is defined as the difference between the simple average of the returns on the high TV and the low TV portfolios. The construction of TV portfolios is similar to Fama-French s size and our 1% Va portfolios: in December of each year t from 1963 to 2001, all NYSE stocks are ranked on total standard deviation (volatility). The median total volatility figure is used to split NYSE, AMEX, and NASDAQ stocks into two groups, high TV and low TV. The factor for illiquidity (HILLIQL) is defined as the difference between the simple average of the returns on the high illiquidity and the low illiquidity portfolios. The construction of ILLIQ portfolios is based on the monthly illiquidity measures for each stock: in each month t from 1963 to 2001, all NYSE stocks are ranked on illiquidity, and the median ILLIQ is used to split NYSE, AMEX, and NASDAQ stocks into two groups, high ILLIQ and low ILLIQ.

16 14 Table II presents the parameter estimates, t-statistics, adjusted 2 s, and standard error of estimates (S.E.E.) from the time-series regressions of excess stock returns on M-F, SMB, HML, HTVL, and HILLIQL. The results indicate that all of the coefficients on the 3 Fama-French factors (M- F, SMB, HML) are highly significant. As expected, the slopes on SMB and HML are related to size and BE/ME factors, respectively. In every book-to-market quintile, the slopes on SMB decrease monotonically from smaller- to bigger-size quintiles. In every size quintile of stocks, the HML slopes increase monotonically from strong negative values for the lowest-be/me quintile to strong positive values for the highest-be/me quintile. 17 of the 25 slope coefficients on HTVL, and 12 of the 25 slopes on HILLIQL are found to be statistically significant. 18 of the 25 2 values are 0.90 and above. The relative performance of additional factors (HTVL, HILLIQL) is tested on the basis of the likelihood ratio test by comparing the standard error of estimates (SEE) of the 3-factor and 5-factor 2 2 models. The likelihood ratio test (L) statistic is calculated as L = n ln( σ * / σ ), where σ * is the value of SEE under the null hypothesis (or for the 3-factor model), and σ is the value of SEE under the alternative (or for the 5-factor model). This statistic is distributed as Chi-square with degrees of freedom equal to the number of restrictions under the null hypothesis. Although not presented in the paper, the 3-factor asset pricing specification is rejected in favor of the 5-factor model in most cases, implying that HTVL and HILLIQL help explain the time-series variation in stock returns. Table III shows the time-series regression estimates of excess stock returns on M-F, SMB, HML, HVAL, and HILLIQL. The results indicate that all of the coefficients on M-F, SMB, HML are highly significant. Only 10 of the 25 slope coefficients on HILLIQL are found to be significant, whereas 20 of the 25 slopes on HVAL (high Va minus low Va) are statistically significant. Adjusted 2 values are in the range of 0.81 and 0.96, and 18 of the 25 2 s are above The results indicate considerable additional explanatory power of Va after controlling for the 3 Fama-French (1993) factors and liquidity. To test how well the average premiums for the five proxy risk factors explain the cross-section of average returns on stocks, following Fama-French (1993), we concentrate on the intercepts in the timeseries regressions. The dependent variables in the regressions are excess returns. The explanatory variables are excess returns on the market (M-F) and returns on zero-investment portfolios (SMB, HML, HVAL, HTVL, HILLIQL). Suppose the explanatory returns have minimal variance due to firmspecific factors, so they are good mimicking returns for the underlying state variables or common risk factors of concern to investors. Then, the multifactor asset-pricing models of Merton (1973) and oss (1976) imply a simple test of whether the premiums associated with any set of explanatory returns are

17 15 sufficient to describe the cross-section of average returns: the intercepts in the time-series regressions of excess returns on the mimicking portfolio returns should not be different from zero. 9 By examining the intercepts, we test whether the mimicking returns of SMB, HML, HVAL, HTVL, and HILLIQL absorb the size, book-to-market, Va, total volatility, and liquidity effects in average returns. The first panel in Table V shows that most of the intercepts in the 3-factor model are close to zero, implying that the regressions that use M-F, SMB, and HML to absorb common timeseries variation in returns do a good job in explaining the cross-section of average stock returns. A notable point in Table V is that 8 of the 25 intercepts in the 3-factor model differ from zero, whereas in the 5-factor models only 4 of the 25 intercepts are statistically different from zero. The intercepts in Table V indicate additional explanatory power of HVAL, HTVL, and HILLIQL in capturing the crosssectional variation in expected returns after controlling for the effects of market return, size and BE/ME. V. Conclusions This paper provides empirical evidence that firm size, liquidity, and Value-at-isk explain the cross-sectional variation in expected returns on the NYSE, AMEX, and NASDAQ stocks for the period January 1963 to December The empirical results do not support the central prediction of the SLB model since average stock returns are not positively related to market beta at the firm level. We provide the time-series averages of the slope coefficients from monthly and annual cross-sectional regressions to check the sensitivity of our findings to alternative investment horizons. The results from the annual regressions turn out to be similar to our findings from the month-by-month FM regressions: size, liquidity and Va can capture the cross-sectional differences in expected returns, while beta and total volatility have almost no power to explain average stock returns at the firm level. We compare the relative performance of size, beta, and Va in explaining the cross-sectional variation in portfolio returns. The results indicate that all the risk factors considered in the paper can capture the crosssectional differences in portfolio returns, but Va has the best performance in terms of 2 values. The strong positive relation between stock (or portfolio) returns and Va turns out to be robust across different investment horizons and loss probability levels. In addition to the cross-sectional regressions in a Fama-French (1992) asset-pricing framework, a time-series regression approach is used to evaluate the empirical performance of Va at the portfolio level. The relative performance of total volatility and Va along with liquidity is investigated in terms of their ability to capture time-series variation in stock returns using the 25 portfolios of Fama-French (1993). Monthly returns on a portfolio of stocks are regressed on the returns to a market portfolio of stocks and the portfolios for size, book-to-market, liquidity, and total volatility/value-at-risk. The results

18 16 based on the 25 portfolios of Fama-French indicate that Va can capture substantial time-series variation in stock returns, and provides an additional explanatory power after controlling for the characteristics of market return, size, book-to-market ratio, and liquidity. The results also imply that the relation between Va and expected stock returns is not a reversal in long-term returns, liquidity or volatility effect. We thank Linda Allen, Archishman Chakraborty, Jay Dahya, Gayle Delong, Charlotte Hansen, Armen Hovakimian, John Merrick, ui Yao, and especially Haim Levy and two anonymous referees for their extremely helpful comments and suggestions. An earlier version of this paper was presented at the Baruch College, City College, Graduate School and University Center of the City University of New York. The financial support from the PSC-CUNY esearch Foundation of the City University of New York is also gratefully acknowledged. All errors remain our responsibility.

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