Identifying Risk-Based Factors

Identifying Risk-Based Factors Anchada Charoenrook + and Jennifer Conrad ∗ August 2008 Abstract Existing studies find that size, book-to-market, mome...
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Identifying Risk-Based Factors Anchada Charoenrook + and Jennifer Conrad ∗ August 2008

Abstract Existing studies find that size, book-to-market, momentum and liquidity explain the crosssection of average returns, but debate continues over whether these variables are risk factors. We propose a new test of whether a candidate variable is a priced risk factor. Specifically, we test whether there is a relationship between the conditional mean and conditional variance of the return on the candidate variable’s factor-mimicking portfolio, and test whether risk, in the mean-variance frame-work, explains all of the return of a factor-mimicking portfolio. The test results on size- and liquidity-based portfolios are consistent with these variables being risk-based factors. The results on the book-to-market based portfolio are consistent with book-to-market being a risk-based factor only in the 1963-2003 subsample. The results on the momentum factor are not consistent with a risk-based explanation in the mean-variance setting. JEL classification code: G12, G14 Keywords: Asset pricing, Multi-factor models, Size, Book-to-market, Momentum, Liquidity factor

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The Owen Graduate School of Management, Vanderbilt University, 401 21st. Avenue South, Nashville, TN 37203. Email: [email protected] ∗ Kenan-Flagler Business School, University of North Carolina-Chapel Hill, CB # 3490, Chapel Hill, NC 27599-3490. We thank Kenneth French, Bruce Grundy, David Hirshleifer, and seminar participants at the University of Washington and Vanderbilt University for helpful comments. Anchada also thanks the Financial Market Research Center for financial support.

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I.

Introduction There is substantial empirical evidence that multi-factor asset pricing models that include factors

based on firm characteristics such as size, book-to-market equity ratios, momentum, and liquidity predict the cross-section of average returns better than the classic single-period Capital Asset Pricing Model (CAPM). The best-known of the multi-factor models is the three-factor model of Fama and French (1993), which, along with the market portfolio, includes the returns on two factor-mimicking portfolios designed to capture sensitivity to size and book-to-market factors. 1 Based on the finding of Jegadeesh and Titman (1993) that lagged one-year returns predict subsequent returns, Carhart (1997) suggests adding the momentum factor to the Fama and French model. Chordia, Subrahmanyam, and Anshuman (2001), Amihud (2002), and Pastor and Stambaugh (2003) present empirical evidence suggesting there is also a priced liquidity risk factor. Since many of these factors are motivated by empirical evidence that a particular firm characteristic is strongly associated with ex post cross-sectional mean returns, there is considerable debate over what ex ante economic mechanisms drive these factors. Broadly speaking, the debate surrounding these factors focuses on two competing categories: risk-based and non-risk-based explanations. Proponents of risk-based explanations contend that these factors represent systematic risks not captured by the CAPM model or by the misspecification of the market proxy as in Roll (1977). A number of researchers (e.g., Vassalou and Xing (2004)) argue, more specifically, that the returns of factor-mimicking portfolios, such as SMB and HML, are related to time-variation in the investment opportunity set as proposed in the Intertemporal Capital Asset Pricing Model (ICAPM). Some proponents of non-risk-based explanations argue that the relation between factors such as size or book-to-market and stock returns may be evidence of market frictions, an artifact of the empirical methodology, or a consequence of data-snooping biases (see, e.g., MacKinlay (1995)). Other researchers suggest that investor irrationality combined with market frictions that limit arbitrage activities may offer an explanation for the importance of these factors in explaining the cross-section of returns. As is evident in our description of the existing literature in Section 2, this debate is ongoing. In this paper, we propose a new test of whether a particular candidate variable is a priced risk. Following Cochrane (2001), we argue that there must be a link between the conditional mean and 1

SMB and HML represent the factor-mimicking portfolios related to size and book-to-market, respectively.

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conditional variance of any factor-mimicking portfolio if the proposed factor is priced. Specifically, with a linear factor structure, we note that the risk premium at time t on the factor-mimicking portfolio, like that of any asset, is related to the conditional covariance of the portfolio’s return with the pricing kernel. However, for a factor-mimicking portfolio, the conditional covariance with the pricing kernel is linearly related to the factor’s conditional variance, and hence to the portfolio’s conditional variance. Consequently, there must be a linear relation between the conditional mean and conditional variance of a factor-mimicking portfolio return, if that factor is a component of the pricing kernel. This relation gives us two testable necessary conditions for a variable to be a priced risk factor. One, the sign of the relation between the conditional mean and variance of the factor is given by the sign of the mean return of the factor-mimicking portfolio, which is also the sign of the risk premium of this factor. Two, under the mean-variance framework, the variance risk of a factor-mimicking portfolio should completely account for its mean excess return. We test these hypotheses for four factor-mimicking portfolios which have been proposed or used as factors in the literature: SMB, HML, momentum (UMD), and liquidity (LIQ). We begin by testing whether the conditional variance of these four factor mimicking portfolios are time-varying. Our test results strongly that they are for all factors. In addition, the periods of high and low conditional variance of these factor candidates appear to be correlated. We also test for the relation between conditional variance and conditional mean for each of our candidate factors in the overall period, as well as subperiods. For size, or SMB, we find evidence of a strong link between the conditional variance and the conditional mean return of the portfolio in all periods we examine. The sign of the relation is positive, which is consistent with size being a priced risk factor. In addition, we cannot reject the hypothesis that, for SMB, variance risk accounts for all the mean excess return of the portfolio. We find statistically significant evidence that book-to-market, or HML, is a priced risk factor only in the second subperiod, covering 1963-2003. In this subperiod, we find a positive and significant relation between the conditional variance and conditional mean return of the HML portfolio, and the variance risk accounts for all the excess mean return of the portfolio. During the first subperiod, covering 1927-1962, although the unconditional mean return for this portfolio is positive and large, the relation between conditional variance and conditional mean return is

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insignificant (although positive). The residual return, after accounting for variance risk, is smaller than the unadjusted return but remains significantly positive. The results for UMD are strikingly different. Although there is strong evidence that the conditional variance of the momentum factor varies through time and strong evidence that there is a significant relation between conditional mean and conditional variance, the relation is of the wrong sign. That is, the relation between the conditional variance and conditional mean is negative— momentum profits are high when the conditional variance of UMD is low. This relation should be positive if momentum is a priced risk factor, since the unconditional mean return and the risk premium of the momentum factor-mimicking portfolio is positive. Consequently, our results reject momentum as a risk-based factor for the overall period, as well as both subperiods. We also examine a liquidity factor constructed using Amihud (2002)’s liquidity measure. We find that this liquidity factor is priced and that it passes the necessary conditions for a risk-based factor. The return of the liquidity-factor-mimicking portfolio is completely accounted for by its variance. Pastor and Stambaugh (2003) report that a liquidity factor accounts for more than half of the profits from the momentum trading strategy implying that the momentum factor may be risk-based and related to liquidity risk. However, we document a small, statistically significant negative correlation between the liquidity and momentum factor-mimicking portfolios, and, as mentioned above, momentum is rejected as a risk-based factor while liquidity is not. This result appears inconsistent with momentum being a risk-based factor related to liquidity. The relations between conditional means and variances we examine are necessary, but not sufficient, conditions for the candidate variables to be priced risk factors. In further tests, we investigate whether the magnitude of these relations are economically plausible. The parameter estimate of the relation between the conditional mean and conditional variance of the factormimicking portfolio return, combined with information on the volatility of the factor, yields a Sharpe ratio for the factor. We find that the Sharpe ratios for the HML and SMB portfolios are lower than the Sharpe ratio of a market portfolio proxy. The Sharpe ratio of the liquidity factormimicking portfolio is higher than the market portfolio’s Sharpe ratio, although its magnitude is similar. In addition, the magnitude of the relation between the conditional mean and conditional variance for the mimicking portfolios imply reasonable coefficients of relative risk aversion. In an overview of the paper, we discuss related papers in Section II, derive the relation between conditional mean and conditional variance of factor-mimicking portfolios in Section III, describe the 4

methods and the data that we use in Section IV and V, and discuss the results of our tests in Section VI. Section VII contains the results of various robustness checks, in Section VIII we discuss the economic implications of our results and Section IX contains a short summary. II.

Related Literature Several existing empirical studies investigate whether SMB and HML are risk-based factors. In

proposing their pricing model, Fama and French (1996) estimate time series regressions of excess returns of a number of characteristic-sorted portfolios on the multi-factor model which includes the market portfolio, SMB, and HML. They find that SMB and HML explain a significant portion of the returns of their test portfolios and reduce the intercept terms in these regressions to an economically insignificant value. They argue that given this evidence, SMB and HML should be included in assetpricing models as risk-based factors. More recently, studies in this area have adopted the ICAPM as the premise and focus on whether size and book-to-market equity ratios are proxies for variables that might represent timevariation in the investment opportunity set at the macroeconomic level. The evidence is mixed. For example, Liew and Vassalou (2000) report that SMB and HML both forecast future economic growth. Lettau and Ludvigson (2001) and Vassalou (2003) show that accounting for macroeconomic and business cycle news subsumes or diminishes the cross-sectional relation of average returns and SMB beta or HML beta. In contrast, Chen (2003) develops an ICAPM model in which he uses the technique of substituting out aggregate consumption by the aggregate budget constraint as in Campbell (1993). A result of this model is that if a factor is related to the time-varying investment opportunity set, the factor should forecast either future returns or future volatility of returns. Chen tests the book-tomarket equity ratio and finds that the forecasting ability of this factor is not sufficient to explain the historical high returns of the book-to-market factor-mimicking portfolio. Moskowitz (2003) examines whether SMB, HML, and UMD can predict the covariance structure of returns. He uses a multivariate GARCH model to allow for time-variation in the covariance structure of returns of basis assets which include characteristic-sorted portfolios, industry portfolios and the market portfolio. He finds that the size factor helps predict volatility and covariation with other assets, but he finds no evidence that book-to-market or momentum is linked to the covariance structure. 5

The empirical method employed in this paper differs from those used in previous studies. First, our method does not require the identification of specific fundamental risks to which a factormimicking portfolio is related. Rather, we require that the effect of time-series variation in the volatility of the fundamental risk is reflected in both the conditional mean and conditional variance of a factor-mimicking portfolio. 2 While it is certainly of interest to identify fundamental risks, testing the relation of a candidate factor to a specific fundamental risk (e.g., such as distress) is limited by the accuracy with which this risk is measured. Imposing a more restrictive structure based on a particular asset pricing model, such as in Chen (2003), allows for sharper tests, but could also lead to erroneous results if the model is mis-specified. Our method offers a less restrictive alternative. Second, our test measures how much of the return of the factor-mimicking portfolio is explained by variance risk and how much is left unexplained.

Examining the link between

conditional mean and conditional variance of the factor-mimicking portfolio is different from Moskowitz (2003), who examines whether factors can predict the 2nd moments and cross-correlation of other portfolios’ returns. Finally, our method tests a broad class of risk-based explanations and it does not require the assumption that the market portfolio includes all fundamental risks. A risk-based factor in our context may be either a proxy for time-variation in the investment opportunity set, or a proxy for systematic risks not captured by the market portfolio (Roll, 1977). Tests along the lines of the ICAPM often assume that the market portfolio is the ‘wealth’ portfolio and do not control for the Roll (1977) critique. III.

Factors means and variances We start with a multi-factor beta pricing model that is the focus of a great deal of research in the

past decade E t [ Ri ,t +1 ] = α + ∑ j =1 λ j ,t β i , j , J

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(1)

The translation of a change in conditional variance to conditional mean occurs if we assume that factor loadings (in the pricing kernel) and betas (in the returns model) are constant. Alternatively, we can assume that the conditional volatility of fundamental risk varies at higher frequencies than do changes in loadings or betas; we perform subperiod analyses and a statistical test of parameter estimates across subperiods as a check on this assumption. We also perform simulations to examine circumstances in which fundamental risks may not be reflected in conditional means and variances; these cases can include, for example, situations in which factor-mimicking portfolios are mis-specified, reflect multiple fundamental risks or contain excessive idiosyncratic volatility.

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where Ri ,t +1 is the gross return of asset i at time t+1. The betas are the multiple regression coefficients of returns on fundamental factors, and the λ’s are the (possibly time-varying) riskpremia associated with those J factors. The expectation in equation (1) is taken with respect to the information set of investors at time t. Hence, realized returns are given as: Ri ,t +1 = Et [ Ri ,t +1 ] + ∑ j =1 β i , j f j ,t +1 + ε i ,t +1 ,

(2)

J

where f j ,t +1 is the shock to factor j at time t+1, Et [ε i ,t +1 ] = 0 , Et [ f j ,t +1 ] = 0 , and Et [ε i ,t +1 f j ,t +1 ] = 0 . We assume that the fundamental factors (although not necessarily the factor candidates) are orthogonal. 3 Cochrane (2001) shows that a pricing model such as that represented in equation (1) implies a linear factor model for the stochastic discount factor, or pricing kernel, M. 4 J

M t +1 = a − ∑ b j f j ,t +1 .

(3)

j =1

Equilibrium implies Et [M t +1Ri ,t +1 ] = 1 , and assuming there is a risk-free rate, R f , we have

(

)

Et [ Ri ,t +1 ] − R f = − R f Covt M t +1 , Ri ,t +1 .

(4)

Thus, an asset’s risk premium is linearly related to the covariance of its return with the pricing kernel. To begin, consider a firm characteristic X that has a positive loading or beta on factor j, a beta of 0 with respect to other factors and no residual risk. Typically, the factor-mimicking portfolio based on X is constructed by sorting firms on X, then buying (shorting) portfolios of firms which have the highest (lowest) X values. The long and short portfolios allow us to create a factor-mimicking portfolio that is well-diversified; returns related to firm-specific characteristics other than X, and (ideally) factors other than j cancel out, or are diversified away. In these circumstances, given (1) and (2), the return on this portfolio can be written as: R X ,t +1 = Et [ R X ,t +1 ] + β X , j f j ,t +1 . Combining this with (3) and (4), we can simplify the covariance term in equation (4), and we have Even if the underlying economic factors are not orthogonal, one can use them to construct a set of orthogonal factors (using, e.g., factor analysis) which also span security returns. Thus, the factor candidates may be written as a combination of either the true (economic) fundamental factors, or as a combination of orthogonal factors which are constructed from the economic factors. See Cochrane (2001) pp. 173-179. 4 Cochrane (2001) pages 108-110. 3

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Et [ R X ,t +1 ] − R f = R f b j β X , j Vart ( f j ,t +1 ) ,

(5)

because the covariance of its return with the pricing kernel reduces to the variance of factor j. 5 In addition, since portfolio X has a positive loading only on factor j, the relation between the variance of the factor and the portfolio is given by Vart [ RX ,t +1 ] = [ β X , j ]2 Vart [ f j ,t +1 ] and thus equation (5) becomes:

(

)

Et [ R X ,t +1 ] − R f = R f γVart R X ,t +1 ,

(6)

where γ ≡ b j / β X , j . For zero investment factor-mimicking portfolios, the returns are returns in excess of the risk-free rate by construction, and we have

(

)

Et [rX ,t +1 ] = R f γVart rX ,t +1 .

(7)

where rX ,t +1 denotes excess return. Equation (7) says that the conditional expected excess return of a (zero investment) factor-mimicking portfolio is linearly related to the portfolio’s conditional variance of return. If a factor is priced, then predictable variation in its variance should be related to predictable variation in its return. 6 Equations (5) and (7) also show that the sign of the parameter that relates the first and second moments of the factor-mimicking portfolio in equation (7) is determined by the sign of the portfolio’s conditional mean return, which is also the same sign as this factor’s risk premium estimated from the cross-section of asset returns. The intuition behind the derivation above relies on the nature of a well-specified factormimicking portfolio. Like any other asset, with a linear factor structure on the pricing kernel, the factor-mimicking portfolio’s conditional mean return is related to its conditional covariance with the pricing kernel. However, for a factor-mimicking portfolio, that conditional covariance is linearly related to the conditional variance of the factor, and so the conditional variance of the factormimicking portfolio itself. In a time-series setting, if γ is constant, we should observe that a higher forecast conditional variance of return to the factor-mimicking portfolio flows through to a higher forecast mean return to the portfolio. Related to this, note that the parameter that relates the expected return to the conditional variance of the factor-mimicking portfolio, φ ≡ R f γ , equals the We consider the case of a factor-mimicking portfolio with exposure to multiple factors, and correlated factor candidates, later in the paper. 6 The relation between the conditional first and second moments of factors is similar to the relation that Campbell (1987) derives for ‘benchmark’ portfolios; see in particular pp. 396-397. 5

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Sharpe ratio of the factor-mimicking portfolio divided by its standard deviation of return. That is, another way to view the relation in equation (7) is that, holding the compensation for bearing a particular factor risk constant, when the expected volatility of the factor risk increases, the expected excess return simply scales up accordingly. zNote that if γ is not constant through time, then a time-series test of whether there is a linear relation between conditional variance and conditional mean will be mis-specified.

This mis-

specification could show up in significant residual returns to the factor-mimicking portfolio, or timeseries predictability in the residuals. We could also observe significant differences in estimates of γ over subperiods. In our empirical work, we examine the results of these specification tests. Of course, any particular proposed factor-mimicking portfolio may not meet the specific criteria that leads to the relation in equation (7) above. For example, a proposed ‘factor candidate’ may be related to multiple fundamental risks, or it may not completely diversify idiosyncratic shocks. In the next section, we consider the consequences of the first setting; in simulations in Section 7, we consider both situations. B.

Correlated factor candidates Some researchers propose the use of multifactor asset pricing models in which the portfolios

used as factor-mimicking portfolios are not orthogonal to each other. These portfolios include for example SMB, HML, and UMD. Consider a world in which there are three fundamental risk factors f1 ,f2, , and f3 and X which is related to two of the factors f1 and f2,. We can use the firm characteristic X to create a factor mimicking portfolio that is has a non-zero loading on two fundamental factors, and a loading of 0 on all other factors. Note that even though fundamental factors f1 and f2 may bee orthogonal, the factor candidate X would have a non-zero correlation with another factor candidate which also had non-zero loadings on either f1 or f2. With the linear pricing kernel from equation (3), the expected return on the new factor candidate X can be written as: Et [ RX ,t +1 ] − R f = R f λtVart ( RX ,t +1 ) ,

or

Et [rX ,t +1 ] = R f λtVart ( rX ,t +1 ) .

where λt is defined as:

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(8)

λt ≡

b1 β X 1Vart ( f 1,t +1 ) + b2 β X 2Vart ( f 2,t +1 )

( β X 1 ) 2 Vart ( f1,t +1 ) + ( β X 2 ) 2 Vart ( f 2,t +1 )

.

(9)

(9)

If λt = λ ∀t , then the factor-mimicking portfolio’s conditional variance and conditional mean are still linearly related through a constant ‘price of risk’, where this price of risk is a weighted average of the factors’ importance and prices. If this condition does not hold, the estimated

λ depends on the time-series variation of both factor variances, and so may itself vary through time as shown in equation (9). Thus, the test of equation (8) may be mis-specified, and we would be less likely to find a relation between conditional mean and variance. Importantly, however, note that even in this case, the numerator of equation (9) has the same sign as the conditional mean of the factor mimicking-portfolio, and consequently the sign of the expected excess return and the meanvariance relation will be the same. That is, our first testable restriction on whether a factor candidate can be considered a factor still holds. In simulation experiments in Section 7, we examine the power of tests of equation (7) under various assumptions regarding the relation of fundamental risks to factor candidates, and the resulting correlation between proposed factors. We also consider the effect of residual idiosyncratic volatility on our results, and discuss the implications of these results for the construction of factor candidates. IV.

Methods The relation derived in equation (7) above is one between the conditional mean and variance

of the return on factor-mimicking portfolios. To test this relation, we begin by testing whether there is significant time-series variation in the variance of return on factor-mimicking portfolios of interest. Simultaneously, we estimate the relation between conditional mean and variance using GARCH-in-mean models. Specifically, we estimate: rX ,t +1 = a + δht +1 + η t +1 , q

p

i =0

j =0

ht +1 = ω + ∑ α iη t2−i + ∑ γ j h t − j ,

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(10)

where rX ,t +1 is the return on the mimicking portfolio for factor X at time t+1, ht +1 = Vart (rX ,t +1 ) is the expected conditional variance for the factor portfolio at time t+1, and η t +1 / ht +1 is normally distributed. In robustness checks, we estimate the variance process using alternate models and find similar results; these results are discussed in Section VII.A. We test three hypotheses related to equation (10). First, if X is a risk factor, then the relation between conditional mean and variance of the portfolio return, estimated with the coefficient δ in equation (10), should have the same sign as the conditional expected risk premium on X. That is,

sign(δ ) = sign( Et [rX ,t +1 ]) . Second, equation (7) implies that the expected risk premium for the portfolio should be given entirely by the portfolio’s conditional variance. That is, the intercept term in the conditional mean equation of (10) should be zero. A non-zero and significant intercept may represent a component of the factor-mimicking portfolio return that is unrelated to risk. Alternatively, a significant intercept may represent unmodeled time-variation in the price of (variance) risk, which we assume constant in our empirical analysis, mis-specification in the model of conditional variance, or compensation for residual risk related to higher order moments. Finally, if X is a proxy for risk, then the Sharpe ratio of the factor-mimicking portfolio should be plausible. For example, the Sharpe ratio of the factor-mimicking portfolio should be less than that of the ex ante tangency portfolio. V.

Data

A. The market portfolio, SMB, HML, and UMD Market excess portfolio returns are calculated using the Center for Research in Security Prices (CRSP) market indices minus the one-month return of the Treasury bill that is closest to 30 days to maturity. The indices include firms from the AMEX, NYSE, and the NASDAQ. The portfolios SMB, HML, and UMD (Up-Minus-Down) mirror those used in Fama and French (1993).8 Firms are sorted independently along two size and three book-to-market dimensions, resulting in six portfolios. The size breakpoint, which determines the Small and Big portfolios, is the median NYSE market equity. The book-to-market breakpoints, which determine 8

These portfolios are obtained from Kenneth French’s website.

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the Growth, Neutral, and Value portfolios, are the 30th and 70th NYSE percentiles. The portfolios are rebalanced every quarter. The returns of the portfolios are value weighted. SMB is defined as SMB= 1/3(Small Value + Small Neutral + Small Growth) – 1/3(Big Value + Big Neutral + Big Growth). HML is defined as HML= 1/2 (Small Value + Big Value)-1/2(Small Growth+ Big Growth). To form a momentum factor-mimicking portfolio, portfolios are formed monthly based on two independent sorts: (1) size at month t-1, and (2) returns over month t-12 through t-2. As before, six portfolios are formed: two based on size and three based on past returns. The monthly size breakpoint is the median NYSE market equity. The monthly past returns breakpoints are the 30th and 70th NYSE percentiles. UMD is defined as UMD = 1/2 (Small High + Big High) –1/2(Small Low+ Big Low). Our sample includes monthly data for SMB, HML, and UMD from January 1927 through December 2003. B. The liquidity-factor-mimicking portfolio The illiquidity variable is constructed using a procedure similar to Amihud (2002). Illiquidity is defined as the average ratio of the daily absolute return to the (dollar) trading volume on that day, Ritd / VOLDitd , where Ritd and VOLDitd denote the return of stock i and the dollar daily volume on

day d of month t. Intuitively, a higher average price response per dollar of trade implies a more illiquid security. The dollar volume is the daily volume of day d multiplied by the average of the daily closing prices of day d-1 and day d. For each month t and security i, we compute illiquidity using data from month t-12 through t-1 as ILLIQit =

1 Dty

Dty

∑R d =1

itd

12

/ VOLDitd ,

where Dty is the number of days that stock i traded during months t-12 through t-1. For an observation to be valid, we require that there are at least 200 trading days during t-12 through t-1. We also eliminate the 1% extreme high and low illiquidity observations. The average illiquidity of the market, AILLIQt , is the value-weighted cross-sectional average of illiquidity measures of all valid observations of ILLIQit for month t. Since average illiquidity varies over the years, our final firmspecific illiquidity measure for month t is the ratio of a firm’s illiquidity over the average illiquidity of the market, ILLIQMAit =

ILLIQit . AILLIQt

To construct the liquidity (or rather the illiquidity) factor-mimicking portfolio, we sort firms into decile portfolios based on ILLIQMAit . Thus, each month firms are ranked based on the ratio of their illiquidity measure over the previous year compared to that of the market average over the previous year. We calculate the value-weighted average return of each liquidity portfolio formed in month t, where the weights are based on the capitalization value at the end of month t-1. The liquidity factormimicking portfolio monthly return is the return on the portfolio with the highest market-adjusted illiquidity minus the return on the portfolio with the lowest market-adjusted illiquidity. The liquidity factor-mimicking portfolio is rebalanced every month.9 The liquidity factor-mimicking portfolio is constructed for stocks traded on the NYSE (excluding ADRs) using data from the daily and monthly databases of CRSP from July 1962 through December 2003. We only select firms from the NYSE to avoid effects caused by the difference in market microstructure among the stock exchanges. VI.

Results In Table 1, we present summary statistics for our factor candidates; for comparison, we also

report results for the market portfolio. As reported in earlier studies, the returns to holding these

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In robustness checks, we test two other methods of constructing the liquidity factor-mimicking portfolio. In the

first, we use the same measure but rebalance once a year instead of every month. In the second, we rank on the basis of

ILLIQit without normalizing by the market. In each case, our results are unchanged.

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factor-mimicking portfolios are statistically and economically significant. Over our sample period, the market, UMD, and LIQ (liquidity factor) have similar monthly returns of 0.66%, 0.76%, and 0.82% per month, with fairly similar standard deviations of 5.5%, 4.8%, and 4.5% per month. The portfolios SMB and HML have lower mean monthly returns over our sample period, of 0.26% and 0.35% per month; they also have lower standard deviations of 3.3% and 3.7% per month. In Panel B of Table 1, we present the correlations between factor-mimicking portfolio returns. With the exception of the momentum factor, UMD, the correlations are positive and all are significant. As mentioned above, note that the relation between conditional means and variances of portfolios which mimic factor amalgams, such as SMB and HML, which we derived in Section III holds under certain conditions even when factor candidates are not orthogonal. If these conditions are not met, the relation between the conditional variance and conditional mean may vary over time and so equation (7) will be mis-specified. We conduct statistical tests on the specification of our model in Section VII. Figures I through IV show the time-series of the monthly returns of these factor-mimicking portfolios. The volatility of these returns appears to vary through time, with runs of high and low volatility periods. In addition, the periods of high volatility seem to be positively correlated across factors, with SMB, HML and UMD all experiencing higher volatility around the Great Depression, in the early to mid ‘70s, and again in the 1999-2001 period (shaded bars represent recession periods). The runs in volatility observed in Figures I through IV are consistent with time-varying volatility in each of the factor candidates. The similarity in patterns of volatility across these factor candidates suggests that there could be some commonality in the “fundamental factors” that underlie them. We return to this point in Section VI, where we conduct a set of simulation experiments to investigate the effect of a variety of different specifications of the relation between fundamental factors and factor candidates. For each factor candidate, we estimate a GARCH-in-mean model and test the significance and sign of the relation between the conditional variance and the conditional mean; we also test whether the intercept term in the mean equation is significantly different from zero. Where the data allow, we estimate all relations over the entire sample period of 1927 through 2003, as well as two subperiods: 1927 through 1962, and 1963 through 2003. Tables 2 through 5 report GARCH-in-mean estimates for SMB, HML, UMD and LIQ, respectively.

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In Table 2, we see (in the bottom part of Panel B) that both the ARCH and GARCH coefficients are positive and highly significant. Consistent with the patterns observed in Figure 1, these results suggest that the conditional variance of SMB is time-varying and persistent. After controlling for this time-variation in variance, we see (in the upper portion of Panel B) that there is a statistically significant, positive relation between the estimated conditional variance and conditional mean of the SMB portfolio; the estimate of δ in the overall sample period is 3.8, with a p-value of 0.01. This is consistent with a positive mean return earned by the SMB portfolio and a positive risk premium for SMB reported in existing cross-sectional studies (see, e.g., Lettau and Ludvigson (2001)). In addition, the estimate of the intercept in the mean equation is not statistically significant, with a p-value of 0.234. The small magnitude and statistical insignificance of the intercept contrasts sharply with the unconditional mean return for the SMB portfolio, of 26 basis points per month (Panel A of Table 2). The lack of significance in the intercept suggests that all of the conditional risk premium of the SMB portfolio can be explained by risk, or the factor’s conditional variance scaled by the factor’s covariance with the pricing kernel. Moreover, the estimates of δ in the two subperiods show the same relation; the estimate of δ is always positive, although p-values over the two subperiods are somewhat larger, at 0.14 and 0.06, respectively. The intercept term is small in magnitude and never significantly different from zero, despite the fact that the unconditional risk premium is consistently positive, and stable, at 23 and 28 basis points across the two subperiods. In Table 3, we present GARCH estimates for HML. As with SMB, we see that the variance of this factor portfolio varies significantly through time; in the lower panel of Table 3, both the ARCH and GARCH coefficients are large, positive and highly significant. In the overall period, however, we see only weak evidence that this conditional variance is related to the conditional mean; the estimate of δ is relatively large and positive at 1.90, but the p-value is 0.13. In addition, although the intercept is smaller than the unconditional mean return (24 vs. 35 basis points per month, respectively), the intercept in the mean equation is statistically significant, with a p-value of 0.04. These results suggest that the book-to-market factor does not fully meet the conditions specified in equation (7) for the overall period. When we examine the subperiods, it is apparent that the weak evidence of a significant relation between conditional mean and variance for HML is driven by the early 1927-1962 subperiod. Although we continue to see strong evidence of time-variation in variance in this subperiod, there is

15

little evidence that conditional variance is related to conditional mean returns—δ is positive, but the associated p-value is 0.79. Interestingly, we also see evidence of higher unconditional volatility in the first subperiod; this can be observed in the higher p-value (0.099) associated with the unconditional mean return of the HML portfolio in the 1927-1962 period. In the later subperiod of 1963 through 2003, we see a positive and statistically significant coefficient on conditional variance in the mean equation ( δ =5.66 with a p-value of 0.04), and an insignificant intercept. Thus, in this later subperiod, the evidence from these tests is consistent with book-to-market being a priced risk. In Table 4, we report the GARCH estimates of UMD. As with the previous two factors, we see strong evidence that the conditional variance of the factor portfolio varies through time in the lower panel of Table 4. However, the estimated relation between the conditional variance and the conditional mean is negative, at -1.81, with a p-value of 0.01. Moreover, the estimated relation is negative and significant (at better than the 10% level) for both subperiods. Consistent with the negative coefficients on conditional variance in Table 4, the estimated intercept is positive and significant, and larger than the unconditional mean return of the portfolio in all cases. Specifically, the point estimate on the intercept is 0.91% per month, with p-values less than 0.01, in the overall sample period as well as the two subperiods; in contrast, the unconditional risk premium for the overall sample period, and the two subperiods, is 76, 66, and 86 basis points per month, respectively. Since the excess return on the UMD portfolio is positive, the negative coefficient estimate on the portfolio’s conditional variance in the GARCH-in-mean equation is inconsistent with momentum being a priced risk factor. Finally, in Table 5, we present test results for LIQ. Since the data for the liquidity factor begin only in 1963, we do not examine separate subperiod results for this factor. As with other factor candidates, we see strong evidence for time-varying variance. The ARCH and GARCH coefficients are statistically significant, with a p-value less than 0.001. There is also a positive and statistically significant relation between conditional variance and conditional mean for the liquidity factor candidate. The coefficient estimates of the conditional variance in mean equation is 5.25, with a p-value of 0.013. In addition, the intercept is not statistically different from zero—that is, once we control for the predictable variation in volatility (or liquidity risk), we find no significant evidence for residual compensation for liquidity in the average returns of the factor-mimicking portfolio for liquidity.This is a sharp contrast to the highly 16

significant unconditional risk premium of 0.82% per month observed in Panel A. These results are consistent with liquidity being a priced risk factor. That is, we observe that the high returns associated with the liquidity portfolio are earned when the predicted volatility of the portfolio is high. As a specification test, for each of the factor candidates we test whether δ differs across subperiods.10 We cannot reject the hypothesis that δ is constant for any of the factors; that is, while the volatility and (with the exception of HML in the first subperiod) the mean returns of the factormimicking portfolios vary significantly through time, δ, which is related to γ in equation (7) or

λ equation (9), does not vary significantly across the subperiods of our sample.

In another

specification test, we analyze whether there is significant time-series predictability in the residuals of the mean equation for each factor; the results of a correlation test up to 6 lags for the residuals are presented in the last line of each Table. Only HML’s mean equation has significant predictability, with a p-value of 0.02. However, in robustness tests reported in the next section, we find that relaxing our constraint of a constant variance model across factor candidates reduces the predictability in the residuals, and leaves our inferences on HML unchanged. Overall, these tests suggest that the mean-variance model is relatively well-specified.11 VII.

Robustness tests

A.

Different volatility processes We conduct robustness tests on our results. In the first test, we examine whether

our results are robust to alternative models of conditional variances of the factor-mimicking portfolios. We consider a number of alternative models. Altogether, for each factor-mimicking portfolio, we estimate ARCH (1) through ARCH (4), GARCH, GARCH AR(1), EGARCH, and GJR-GARCH Glosten, Jagannathan, and Runkle (1993). The result estimates of all the models selected for each factor-mimicking portfolio are qualitatively the same as the result estimates presented in this paper. Overall, allowing for different variance processes does not change the

For the liquidity variable, we test this hypothesis using two equal-length subperiods of 7/1963-12/1982 and 1/1983 – 12/2003. 11 In some of the results the sum of the ARCH and GARCH estimates is close to 1. We estimate IGARCH processes for all the factor-mimicking portfolios and find that a non-stationary variance process in not a concern. These results are available on request from the authors.. 10

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results.13 The positive relation between the conditional mean and variance for the factor-mimicking portfolios for SMB and LIQ is consistent with these characteristics being risk-based factors. HML passes the hurdle in the second subperiod, while UMD does not meet the condition to be considered a risk-based factor using this test method.14 B.

Time-series variance process As a further robustness check on the relation between conditional mean and conditional

variances of returns, we estimate conditional variance using a simpler time-series forecast than the GARCH specification, while controlling for other factors that might be related to changes in the investment opportunity set through time. Following Guo (2006), we estimate a vector autoregression in the variance of each of the SMB, HML, UMD, and LIQ factors. Consequently, the variance of last period’s returns is used to construct a forecast of this period’s conditional variance. Specifically, we estimate: (13)

rX ,t +1 = a1 + b1 rX ,t + c1 E t Var (rX ,t +1 ) + d 1 cay t + υ X ,t +1

where EtVar (rX ,t +1 ) is the fitted value from the following regression: Var (rX ,t +1 ) = a 2 + b2 rX , t +c 2Var (rX ,t ) + d 2 cay t + η X ,t +1

(14)

where t marks time in quarters, and rX ,t +1 denotes quarterly returns of a factor-mimicking portfolio, and Var (rX ,t +1 ) is measured each quarter using daily returns. The variable cay is the consumptionwealth ratio, measured as in Lettau and Ludvigson (2001). Gou (2006) shows that cay controls for hedging demands by investors. For each factor candidate, we include cay as an exogenous variable only when it is significant. The results are presented in Table A2 in the Appendix. The results are similar to those obtained when using GARCH or EGARCH models to estimate conditional variance. That is, the estimated relation between quarterly returns and conditional variance, forecast from last period’s intra-quarter returns variance, is positive and significant at the 10% level or better for SMB, HML, and LIQ, although the intercept term in the SMB returns equation is positive and significant, at 52 In other unreported robustness checks, we include a January dummy in both the mean and variance equation; results are qualitatively similar. 14 It is worth pointing out that, despite the fact that the factor-mimicking portfolios are large, well-diversified portfolios with positively correlated returns, the best-fitting variance processes we estimate for them have striking dissimilarities. This suggests that the factor-mimicking portfolios are capturing, at least in part, different fundamental factors. 13

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basis points a month. The estimated coefficient for UMD is negative and significant at the 10% level. The variable cay is significant only for UMD. Overall, using the past period’s variance of returns to forecast this period’s conditional variance, as well as other control variables such as cay to forecast changes in the investment opportunity set, gives similar results to using GARCH-type models. Our conclusions do not change. We also examine the possibility that the relation between conditional mean and conditional variance may change through business cycles. We re-estimate GARCH in mean processes for factor-mimicking portfoliosincluding an NBER recession dummy to control for business cycles as rX ,t +1 = a + δht +1 + cDrecession + η t +1 , q

p

i =0

j =0

ln ht +1 = ω + ∑ α i g ( z t −i ) + ∑ γ j ln h t − j . We find that our results do not change. These results are available on request from the authors. C.

Correlated factor candidates As Table 1 indicates, the set of proposed factor candidates in use in the literature are

significantly correlated.

This would occur if these factor candidates contain an overlap in

fundamental factors, which are themselves orthogonal.15 We conduct two sets of tests to examine the effect of this correlation on our results. First, we orthogonalize the set of factor candidates using two different methods and re-estimate our tests on the new, orthogonal factors. Second, we conduct a set of simulation experiments to analyze how various combinations of fundamental factors, and hence different correlations, between factor candidates affect our tests. C.1

Orthogonalization To construct the first set of orthogonalized factors, we regress the returns of the factor-

mimicking portfolios for SMB, HML, LIQ and UMD on the market return, and then subtract the estimated market-related component from each factor to construct a new mimicking portfolio return that is orthogonal to the market proxy. We re-estimate the GARCH relations on the resulting “market excess” return. In results presented in Table A3 of the Appendix, we continue to find evidence that there is significant time-variation in the conditional variance of these returns. We also Such an overlap is consistent with the arguments of Daniel and Titman (2005), who argue that “factors” such as bookto-market serve as “catch-all” variables which capture a firm’s exposure to several unidentified fundamental factors. 15

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find significant positive relations between conditional variance and the conditional mean of the (orthogonal) factor-mimicking portfolio return for SMB and LIQ; the evidence for a significant positive relation between conditional variance and conditional mean in HML is stronger, but still not significant at conventional levels with a p-value of 0.12. The orthogonalized UMD factor continues to have a negative and significant relation between conditional mean and volatility. In a second set of tests, we perform pair-wise orthogonalizations of each factor-mimicking portfolio’s return on every other factor-mimicking portfolio.

In general, the results suggest that

there is a positive and significant relation between the conditional mean and conditional variance of returns for SMB and LIQ, and the evidence for a relation between mean and variance is stronger for the orthogonalized HML factor, with p-values ranging from 0.019 to 0.079. For example, parameter estimates for δ for the orthogonalized SMB portfolios are roughly similar to the estimates for the raw SMB return, although δ becomes insignificant (with a p-value of 0.44) when SMB is orthogonalized relative to the liquidity factor. A similar effect occurs for orthogonalized HML portfolios; the relation between conditional mean and conditional variance remains positive, but declines in significance somewhat when HML is orthogonalized with respect to the liquidity factor. In contrast, the results for various orthogonalizations of LIQ change very little, andthe estimates for δ are significant for all pair-wise orthogonalizations. For UMD, in Panel D, the estimates of δ are always negative, although the p-value increases to 0.14 when UMD is orthogonalized with respect to LIQ. In general, the relation between conditional mean and variance is positive and significant for most of the pair-wise orthogonalizations for SMB, HML and LIQ, while they are negative for UMD. However, the significance of the results weakens when portfolios are orthogonalized with respect to the liquidity factor. That is, of all the factors we consider, the liquidity factor-mimicking portfolios appears to have the most robust relation between conditional mean and variance of any, and appears to be most strongly related to the results of other factor-mimicking portfolios. C.2

Simulation experiments We use Monte Carlo simulations to examine the effect of different levels of correlation

between the factor candidates when testing for a relation between conditional variance and conditional mean returns in the factor-mimicking portfolios. In our simulation experiments, we begin by creating three ‘fundamental’ factors. These three factors are generated independently from

20

one another; each is generated using a GARCH-in-mean process where the error terms are independent and normally distributed. The parameters of the GARCH-in-mean process are calibrated to be generally similar to those which are estimated from the SMB and HML factors in the data. The length of the time-series generated in each simulation is 10,000 observations. In the second step of the simulation, we use these three fundamental factors to form two simulated factor candidates. We call these simulated factor candidates SFC1 and SFC2 (for “simulated factor candidate 1” and “simulated factor candidate 2”.) Each of these simulated factor candidates is formed from two fundamental factors; one of the fundamental factors is common, and so generates a correlation between the two factor candidates. That is, SFC1 = F11 * f1 + F12 * f 2 SFC 2 = F22 * f 2 + F23 * f 3 where F11 , F12 , F22 and F23 are the loadings with respect to the fundamental factors f1 , f 2 and f 3 . By varying the factor loadings, we can generate a wide range of correlations between the simulated factor candidates. We consider three general cases with varying specifications that generate a wide range of correlation between these two factor candidates including correlation values similar to those in Table 1. The details of our simulations and specific numerical results are presented in the Appendix. Overall, our simulation results suggest that if factor candidates are positively correlated through their common association with a fundamental factor, and have a significant mean return associated with them, we should easily be able to detect a significant relation between conditional mean and conditional variance. In contrast, if factor candidates are negatively correlated through a common, but differently signed, association with a fundamental factor, and the factors comprising a factor candidate are roughly equal in importance, our tests have only a weak ability to detect a relation between conditional mean and conditional variance—that is, we are more likely to fail to identify a candidate as a risk-based factor when, in fact, it is related to fundamental risks in the economy. The intuition in this case is that the shocks to the two fundamental risks which affect or comprise the factor candidate tend to offset one another, and consequently make the relation between conditional mean and conditional variance more difficult to detect. 21

If factors are positively correlated because they have a common negative relation to a fundamental factor, and positive association with other factors, we again have little ability to detect a relation between conditional mean and conditional variance because the fundamental factors offset each other. Importantly, this ‘diversification’ effect occurs in both the variance and the mean—that is, the mean return of the simulated factor candidates in these cases is quite small. Since factor candidates frequently become candidates because of their ability to generate significant returns, however, we view this case as less important in practice. However, the intuition that underlies the difficulty in generating significant mean returns is important and related to the intuition behind our tests, as well as the result in MacKinlay (1995): any factor risk (or, in MacKinlay’s model, a factor risk-based deviation from a proposed pricing model) must be accompanied by a common component in residual variance. Finally, across all of our simulation specifications, we find that when the estimated relation between conditional mean and conditional variance of return is significant, the sign of the excess return associated with a factor candidate is consistent with the sign of the relation between conditional mean and conditional variance. Since our empirical tests of the data are able to detect an appropriately signed relation between conditional mean and conditional variance for SMB, HML (in the second subperiod) and liquidity, concerns about low power do not appear to be warranted in these cases, especially if we consider the simulations that generate correlations of similar magnitudes as our factor candidates. For the momentum factor, which is the only factor candidate that is negatively correlated with the other candidates, our rejection is not based on the fact that the relation between conditional mean and conditional variance is insignificant, but because sign of the relation is reversed from what we would expect. In contrast, in none of our simulation runs are we able to generate a single estimate of the relation between conditional mean and conditional variance of the candidates that is both significant and different in sign from the unconditional mean return. D. What about the market? In our tests to this point, we have examined only characteristic based factor candidates. In our results, we find an appropriately-signed relation between conditional variance and conditional mean excess return for SMB, HML (in the second subperiod) and LIQ.

Interestingly, when

researchers have tested for such a relation in the market portfolio, the results are mixed (see, e.g., Lundblad (2006) for an excellent discussion of the existing empirical evidence.)

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Even in our own sample, the evidence for a significant mean-variance relation for the market depends on the method used to estimate the relation. If we estimate equation (7) using the valueweighted market portfolio, we find a positive mean-variance relation with a p-value that is only very weakly significant (δ=1.25, p-value of 0.14) and a significant intercept of 54 bp per month. If we use the method described in Guo (2002), who includes the cay variable in the mean equation, our results are consistent with his: we find that the forecast market variance, estimated from last period’s variance, is significantly positively related to this period’s mean market return, while cay is positively related to the mean return as well, and the intercept is not significant. When we use the MIDAS method of Ghysels, Santa-Clara and Valkanov (2004), we find a positive and significant relation between mean and variance for the market. If the market is a factor, what accounts for the relative weakness of the mean-variance relation in its case, when our test results are relatively robust across different specifications for the factor candidates? Although a full reconciliation of the different results in this literature is outside the scope of this paper (given the different sample periods, return horizons and methods that various researchers have used), there are some differences in the test portfolios themselves that may be important in generating the different results. Specifically, we conjecture that the stronger mean-variance relation that we find in the factor-mimicking portfolios SMB, HML, UMD, and LIQ is related to the fact that these are long-short portfolios, which are designed to maximize the correlation with a limited number of fundamental factors, while diversifying away other non-related risks. This may provide a sharper test of the factor’s mean-variance relation compared to that of the market, if the number of factors represented in the market portfolio is large, and the ‘diversifiable’ risk is non-trivial compared to that in a long-short factor-mimicking portfolio. In fact, when we orthogonalize the market portfolio relative to SMB, HML and LIQ and re-estimate the relation between conditional mean and conditional variance, the estimate of δ increases from 1.25 to 4.0 (although the p-value increases slightly to 0.23), and the intercept is statistically insignificant. Thus, the relation seems to be stronger, although it is still not significant at conventional levels. There is also some evidence of this conjecture in the literature in our paper. Lundblad (2006) reports that when the market’s risk-return relation is estimated over long periods (1836-2003 in his paper), this relation is positive across all of the specifications he considers. He argues that, given noise in the market data, longer sample periods are required to estimate the relation precisely. There is also one more piece of confirming evidence for this conjecture in our simulations. If we

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add noise to the returns process, in the form of an additional fundamental factor with zero mean return, we find that the relation between conditional mean and conditional variance becomes more difficult to detect. However, the importance of this ‘omitted factor’ in the variation of factor candidates must be relatively large (at least four times the variation of factors f1 - f3) before the relation between volatility and mean is not significant. If the market portfolio contains more idiosyncratic risk than a long-short factor-mimicking portfolio, such idiosyncratic volatility may make the relation between conditional mean and conditional volatility more difficult to measure; note that in our data (Table 1, Panel A), the (unconditional) standard deviation of the market return is higher than any of the other factor candidates we consider. VIII. Sharpe Ratios The above tests examine necessary, but not sufficient, conditions for a candidate factor to be a risk-based factor. For the factor-mimicking portfolios that pass the necessary conditions, we further examine whether the magnitude of the relation between the conditional mean and variance is economically plausible. One way to interpret the magnitude of the coefficient on the conditional variance of the factormimicking portfolio is that it is analogous to a coefficient of relative factor risk- aversion, or the elasticity of marginal value with respect to the factor. Blume and Friend (1975) suggest that relative risk aversion coefficients in the range of 2 are plausible. The coefficients we have estimated of 1.9 (SMB in the full period), 5.7 (HML in the second subperiod) and 6.8 (for LIQ) are generally larger than this, but do not seem implausible. Another basis for comparison for the magnitude of these coefficients is to compare the implied Sharpe ratios for the factors (calculated as φ Vart (rt m+1 )

for each factor candidate X, where

φ ≡ R f γ ) to one’s priors on the maximum Sharpe ratio for the tangency portfolio. For example,

MacKinlay (1995) argues that a reasonable value for the Sharpe measure for the tangency portfolio is approximately 0.175 per month. We calculate the time series of Sharpe ratios of each factor-mimicking portfolio from our GARCH-in-mean estimation and compare them to the Sharpe ratio of the value-weighted excess market portfolio calculated using both a GARCH-in-mean model and the MIDAS model of

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Ghysels, Santa-Clara and Valkanov (2004).16 Table 7 reports the mean and standard error of the Sharpe ratio estimates.17 We also test the difference between the Sharpe ratio of each factormimicking portfolio and the market’s Sharpe ratio. The Sharpe ratio estimates for SMB and HML for the entire sample period are not significantly higher than the Sharpe ratio of the market portfolio estimated with either GARCH or MIDAS models. They are also lower than MacKinlay’s suggested benchmark figure of 0.175. The Sharpe ratios for both SMB and HML increase in the second subperiod (to 0.174 and 0.177, respectively); these ratios are larger than the market’s ex post Sharpe ratio for the same subperiod, but not significantly different from the MacKinlay maximum which suggest that the relation between the conditional means and variances of SMB and HML are economically reasonable. On the other hand, the mean Sharpe ratio estimate of LIQ, at 0.228, is statistically significantly higher than the Sharpe ratio of the value-weighted market portfolio. In general, the price of liquidity risk is substantially larger in our sample than the price of risks related to both size and book-tomarket. In addition, the price of liquidity risk is significantly larger than the Sharpe ratio of 0.175 which MacKinlay (1995) argues is a plausible maximum, although it is smaller than the maximum Sharpe ratio of the frontier formed from the Fama-French size and book-to-market portfolios.18 As a consequence, while the liquidity factor meets the “necessary” condition in terms of the relation between its conditional mean and variance of return, further work is required to explain the higher magnitude of the risk-return trade-off in its case. IX.

Conclusion In this paper, we derive a relation between the conditional mean and conditional variance of

return of a factor-mimicking portfolio that must hold if a factor is associated with risk. We test whether this relation holds for some of the ‘usual suspects’ for factors in the literature. Specifically, we test whether size, book-to-market, momentum and liquidity factors have significant timevariation in conditional variance, and whether any predictable variation in the volatility is linked to

This market Sharpe ratio is calculated from the sample 1963 to 1999, which is the same sample used in Ghysels, SantaClara and Valkanov (2004). 17 Note that the strong evidence of time-varying volatility and its relation to expected return for SMB, HML and LIQ also implies that an unconditional estimate of a portfolio’s Sharpe ratio, which does not take this time-variation into account, is biased. 18 For example, Ahn, Conrad and Dittmar (2005) use the 1959-2003 sample period and the 25 size- and book-to-market sorted portfolios and form an efficient frontier with a Sharpe ratio of 0.39. 16

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the conditional mean return of the factor-mimicking portfolio for those variables. We also test whether the relation between conditional mean and conditional variance is of the appropriate sign, and whether variance risk can explain all of the excess return of a factor-mimicking portfolio. Our results suggest that the conditional mean returns of factor-mimicking portfolios based on size and liquidity are strongly and significantly related to predictable time-variation in the variance of these portfolios. This is consistent with size and liquidity being priced risk factors. The book-tomarket equity factor evidences this relation during the latter part of the sample period (1963-2003). The relation between conditional variance and conditional mean for the momentum factor is not consistent with momentum being a risk factor in the mean variance setting—although there is strong evidence that the volatility of the momentum factor varies through time, the relation between conditional variance and conditional mean is of the wrong sign. Our analysis of the properties of our tests of factor-mimicking portfolios has implications for the design of factor candidates, as well as the interpretation of our results. Long-short portfolios which diversify idiosyncratic shocks, and are formed from a single fundamental factor or a small set of positively correlated factors, generate more powerful tests and more precise inferences. Less positively, to the extent that several factor candidates are dominated by a common, important fundamental risk, we would expect to see both positive correlation among the candidates and similar inferences from our test of the relation between conditional mean and conditional variance. Thus, the results that SMB, HML and LIQ have an appropriately-signed relation between conditional mean and conditional variance may not be independent test results; collinearity between these factor candidates may make it difficult to interpret results on economic, and not simply statistical, grounds. However, the results for the UMD portfolio, in which the tests reject momentum as a risk factor, stand out even more sharply in contrast. To our knowledge, this paper is the first to show that predictable variation in the returns of factor-mimicking portfolios is linked to their own portfolios’ conditional mean returns. If these factors are risks, then we have presented evidence of a risk-return trade-off at a factor level. If these factors are not risks, then non-risk based explanations must include room for the relation between conditional first and second moments, which is presented here. Similarly, researchers seeking to show that momentum is a risk factor must explain why the risk-return trade-off for this factor appears to be negative.

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Appendix Simulation experiments

Monte Carlo simulations are used to generate three fundamental factors (f1, f2 and f3) which follow a GARCH-in-mean process. The parameters of the GARCH-in-mean process for these factors are calibrated to be generally similar to those which are estimated from the SMB and HML factors in the data.

The length of the time-series generated in each simulation is 10,000

observations. Two factor candidates SFC1 and SFC2 are formed from the fundamental factors as: SFC1 = F11 * f 1 + F12 * f 2 SFC 2 = F22 * f 2 + F23 * f 3 In Case I, both factor loadings on the common factor (i.e., F12 for SFC1 and F22 for SFC 2 ) are positive. In Case II, the loading on the common factor is positive on SFC1 and negative on SFC 2 , and in Case III, both factor loadings on the common factor are negative.

For

convenience, in each case, we calibrate the variances of SFC1 and SFC 2 to match that observed in SMB and HML. Once the factor candidates are generated, we estimate the relation between the (simulated) conditional mean and conditional variance by using the GARCH-in-mean relation in equation (10). The results are presented in Table A1. Panel A contains the results from Case I, Panel B contains the results from Case II, and Panel C contains the results from Case III. The first set of results in each Panel shows the average estimates of δ across all the simulations for that case. For example, in Panel A, the average δ is 3.33 for SFC1 ; this estimate is highly significant, with fewer than 1% of the simulations generated having an estimate of δ that is not significant at conventional levels. We are interested in how the relation between the conditional mean and conditional variance changes as the correlation between the simulated factor candidates varies across the simulations. Consequently, in the bottom part of each Panel, we report the average results of estimating equation (11) after ranking the draws from the simulation into portfolios (portfolios 0-10) on the basis of the observed correlation between the two simulated factors.

By reading down the results across

correlation groupings, we can examine the range of generated correlations between the simulated factor candidates across the simulations. Consequently, we can observe how the ability to detect a significant relation between conditional mean and conditional variance changes as the correlation between the simulated factor candidate changes.

27

Note that for Case I, when all factor loadings are positive, the vast majority (23,050 out of 23,605 simulated cases) of the correlations generated between the factor candidates are positive. In addition, the estimated relation between conditional variance and conditional mean is easily observed in each of the correlation-sorted portfolios. That is, for the entire range of factor loadings we consider, even when the average correlation between the two simulated factor candidates is quite high (above 0.9 in the last portfolio), the average estimated δ is positive and significant, and we never reject either SFC1 or SFC 2 as a risk factor. In addition, the estimated mean return on the portfolio is roughly constant across all of the combinations of factor loadings that we consider in this Panel (and is generally similar to the magnitude of the mean return that we observe in the data.) In Panel B, when factor loadings on the common factor are positive for one simulated factor candidate ( SFC1 ), and negative on the other ( SFC 2 ), the majority of the correlations observed in the simulated factor candidates are negative. Note that in the real data (see Table 1), negative correlation between factor candidates is observed only in the case of UMD. As in Panel A, the average return observed for SFC1 is roughly similar to that observed in the real data, regardless of the particular combination of factor loadings used. However, the average return for SFC2 declines as the importance of the common factor increases; we return to this point in our discussion of Case III in Panel C. In this experiment, for the SFC1 factor candidate, the average estimated δ in each of the ten correlation-ranked portfolios is positive and significant, as it is in the overall sample; on average, we do not reject SFC1 as a factor in any of the ten portfolios. In contrast, the results for the second factor candidate, SFC 2 , are different. In this case, we see a significant number of cases for which we reject SFC 2 as a risk factor. The proportion of rejections is the largest for moderate levels of negative correlation. To understand the difference between the results for SFC1 and SFC 2 , recall that the factor loading on the common factor f

2

is positive for the first simulated factor candidate, SFC1 . The

loading on the other factor, f1 , is always positive for SFC1 . Since both loadings are positive, SFC1 in Case II behaves quite similarly to SFC1 for Case I: the average estimated δ is positive and significant, and we never reject SFC1 as a risk factor. However, for SFC 2 , we have one negative factor loading (on f 2 ) and one positive factor loading (on f 3 ). The negative factor loading on the common factor f

2

generates a negative correlation with the other simulated factor candidate

SFC1 , but it also plays another role—it serves to offset, or ‘diversify’ the shock associated with the

28

second factor f 3 in generating SFC 2 . This ‘diversification’ weakens our ability to estimate the relation between conditional mean and variance for SFC 2 . The fact that this weakening effect happens most strongly for ‘moderate’ levels of correlation between SFC1 and SFC 2 is due to the relative importance of f

2

and f 3 . As we move from Portfolio 0 to Portfolio 10, the relative

importance of f 2 , the common factor, is increasing. In the mid-range, at portfolio 6, the variation in f

2

is roughly equal in size to the variation in f 3 , and the ‘diversification’ effect is at its maximum.

In Panel C, we report the Case III results when factor loadings on the common factor for both SFC1 and SFC 2 are negative. The average correlation between the two factor candidates is

positive, since they respond to the common factor in the same direction. For both SFC1 and SFC 2 , it is difficult to detect a relation between conditional mean and conditional variance; the

intuition is identical to that discussed for SFC 2 above. Since both simulated factor candidates have a positive factor loading on one fundamental factor, and a negative factor loading on another, the ‘diversifying’ effect across factor shocks makes it more difficult to detect the link between conditional mean and conditional variance. However, there is one other important feature of ‘offsetting’ fundamental factors which Panel C makes very clear. Note that the time-series mean is, on average, much smaller than that observed in Panel A, as well as most cases in Panel B.. In fact, it is extraordinarily difficult to generate a significant mean return in the time-series when the factor candidates are comprised of fundamental factors which are offsetting each other. The reason that this occurs is important.

It is related to the pattern in mean returns observed (only) for SFC2 in

Panel B for Case II, and it is related to the underlying intuition of the relation presented in equation (7): if a fundamental factor contributes to the conditional variance of a factor candidate, then it should likewise contribute to the conditional mean. If two fundamental factors have effects which offset each other, then it is not surprising that this offset occurs both in the conditional variance and in the conditional mean return.

Note that this intuition is also similar to that described in

MacKinlay (1995); he points out that, to prevent an asymptotic arbitrage opportunity, any factor risk-based deviation from a proposed pricing model must be accompanied by a common component in the residual variance. Our proposed test argues that this link must be present in a conditional setting as well.19 In another experiment, we add an additional source of noise to returns, by including a fourth fundamental factor in the return generating process which has zero mean return. This factor is consequently an ‘omitted variable’ in the test of whether a factor candidate is a priced risk factor. Our results show that, not surprisingly, the addition of such a factor makes the relation between conditional mean and volatility more difficult to detect. However, the importance of this

19

29

omitted factor in the variation of factor candidates must be relatively large (at least four times the variation of factors f1f3) before the relation between volatility and mean becomes significantly more difficult to detect. Moreover, as with our other simulations, the relation between the sign of δ and the sign of the risk premium for any factor candidate remains intact.

30

Table A1: Simulation experiments In Case I (Panel A), factor loadings F11, F12, F22 and F23 are positive; in Case II (Panel B), loadings on the common factor f2 are positive for SFC1 and negative for SFC2; in Case III (Panel C), loadings on the common factor are negative for both SFC1 and SFC2. We estimation equation (11) for each resulting factor candidate. δ is the average GARCH-in-mean coefficient estimate. The variable p is the proportion of simulations in that group for which the δ estimate is not statistically significant at the 5% level when the unconditional mean is significant. The average results across all simulation are presented in the upper part of each panel; average results on portfolios ranked by the magnitude of the correlation between factor candidates are presented in the bottom of each panel. The columns ‘mean’ and ‘delta’ reports the mean of the variable estimate in the corresponding bin. Panel A: Positive factor loadings average Bin

corr

SFC1

SFC2

N

mean

delta

p (5%)

mean

delta

p (5%)

0

-0.005

555

0.003

2.911

0.000

0.004

2.858

0.000

1

0.043

13615

0.004

3.137

0.000

0.004

2.839

0.000

2

0.143

4311

0.004

3.477

0.000

0.005

3.137

0.000

3

0.243

1821

0.004

3.661

0.000

0.005

3.220

0.000

4

0.348

1020

0.004

3.777

0.000

0.005

3.338

0.000

5

0.448

687

0.004

3.871

0.000

0.005

3.401

0.000

6

0.547

583

0.004

3.913

0.000

0.005

3.426

0.000

7

0.648

476

0.004

3.829

0.002

0.005

3.332

0.000

8

0.745

304

0.004

3.665

0.000

0.005

3.226

0.000

9

0.843

167

0.004

3.434

0.000

0.005

2.992

0.000

10

0.926

66

0.004

3.119

0.000

0.004

2.726

0.000

Total

0.142

23605

0.004

3.325

0.004

2.991

Panel B: Positive and negative loadings average Bin

corr

SFC1

SFC2

N

mean

delta

p (5%)

mean

delta

p (5%)

0

0.007

56

0.0037

2.779

0.000

0.0030

2.400

0.000

1

-0.049

3969

0.0037

3.044

0.000

0.0023

2.241

0.104

2

-0.146

2469

0.0040

3.378

0.000

0.0014

1.511

0.294

3

-0.246

1542

0.0041

3.566

0.000

0.0010

1.038

0.350

4

-0.349

1200

0.0042

3.764

0.000

0.0006

0.695

0.463

5

-0.449

983

0.0043

3.834

0.001

0.0002

0.205

0.594

6

-0.547

831

0.0043

3.859

0.000

-0.0003

-0.417

0.674

7

-0.649

680

0.0042

3.812

0.001

-0.0009

-1.196

0.560

8

-0.747

428

0.0041

3.599

0.000

-0.0013

-1.725

0.386

9

-0.846

253

0.0040

3.405

0.000

-0.0019

-2.275

0.095

10

-0.926

89

0.0037

3.105

0.000

-0.0023

-2.513

0.000

Total

-0.265

12500

0.004

3.427

0.000

0.001

1.016

0.316

31

Panel C: Negative loadings average

SFC1

SFC2

Bin

corr

N

mean

delta

p (5%)

mean

delta

p (5%)

0

-0.004

73

0.003

3.118

0.000

0.003

2.998

0.000

1

0.048

3371

0.002

2.610

0.063

0.003

2.369

0.071

2

0.144

1630

0.001

2.020

0.236

0.002

1.791

0.237

3

0.246

1002

0.001

1.224

0.346

0.001

0.989

0.374

4

0.346

642

0.000

0.592

0.422

0.000

0.542

0.400

5

0.448

511

0.000

0.116

0.530

0.000

-0.080

0.575

6

0.549

498

0.000

-0.560

0.612

0.000

-0.503

0.631

7

0.652

514

-0.001

-1.190

0.632

-0.001

-1.238

0.597

8

0.753

536

-0.001

-1.918

0.360

-0.001

-1.823

0.377

9

0.845

401

-0.002

-2.390

0.127

-0.002

-2.264

0.110

10

0.935

197

-0.002

-2.752

0.010

-0.002

-2.555

0.000

Total

0.280

9375

0.001

1.127

0.252

0.001

0.976

0.258

32

Table A2: Time-series variance process Panel A of this table reports the vector auto regression SMB, HML, UMD, and LIQ factors on the expected volatility estimated from a VAR(1) process as Var (rt +X1 ) = a2 + b2 rt X + c2Var (rt X ) + d 2 cayt , where cay is the consumption-wealth ratio. We include cay as an exogenous variable only when it is significant. The variance is the quarterly variance of daily returns. The variables are quarterly non-overlapping data. Then we extimate rt +X1 = a1 + b1 X t +c1 Et [Var (rt +X1 )] + d1cayt , where rt +X1 is the quarterly factor-mimicking portfolio (excess return). Sample period of quarterly returns are from 1963 Q4 through 2003 Q4 when cay in not included and is from 1963 Q4 through 2001 Q4 when cay is included. *** ,**, and * denote significance at 1%, 5%, and 10% level. P-values are reported in parenthesis. Panel A Market Dependent variables (Mkt-rf)t+1 Var(Mkt-rf)t+1

constant

(Mkt-rf)t+1

Var(Mkt-rf)t+1

cay t 2.1482***

0.0004

0.1227

3.1269***

(0.961)

(0.141)

(0.007)

(0)

0.0037***

-0.0009

0.2175

-0.1232***

(0)

(0.876)

(0.218)

(0.001)

constant

SMB t

VAR(SMB) t

0.0052***

0.0157

3.7262**

(0)

(0.84)

(0.033)

0.0011***

0.0036

0.5009***

(0)

(0.24)

(0)

constant

HML t

VAR(HML) t 4.6315*

SMB

SMB t+1 VAR(SMB) t+1

HML Dependent variables HML t+1 VAR(HML) t+1

0.0037

0.1047

(0.538)

(0.196)

(0.088)

0.0004***

-0.0057***

0.7542***

(0.002)

(0.001)

(0)

constant

UMD t

VAR(UMD) t

cay t

0.0373***

-0.1852**

-3.1536*

-1.1713***

(0)

(0.026)

(0.068)

(0.007)

0.0006**

0.0116***

0.5653***

-0.0442***

(0.045)

(0)

(0)

(0.009)

constant

LIQ t

VAR(LIQ) t

UMD Dependent variables UMD t+1 VAR(UMD) t+1

LIQ Dependent variables LIQ t+1 VAR(LIQ) t+1

0.014

0.1543*

1.3051

(0.174)

(0.052)

(0.372)

0.0037***

-0.0019

0.1973**

(0)

(0.656)

(0.013)

33

Panel B: We estimate rX ,t +1 = a + δ ht +1 + cDrecession ,t +1 + ηt +1 , q

p

i =0

j =0

ln ht +1 = ω + ∑ α i g ( z t −i ) + ∑ γ j ln h t − j , where Drecession ,t +1 is the NBER recession indicator.

SMB

HML

UMD

LIQ

-0.0021

-0.002

0.0096***

0.0057

(0.264)

(0.312)

(0)

(0.414)

Mean equation intercept recession dummy delta

0.001

0.005***

-0.001

-0.009

(0.663)

(0.008)

(0.758)

(0.112)

3.7721**

2.0456*

-1.8382**

2.9463**

(0.012)

(0.094)

(0.013)

(0.041)

Conditional variance equation intercept ARCH1 GARCH1

0**

0.0001***

0***

0.0001**

(0.026)

(0)

(0)

(0.034)

0.1089***

0.1865***

0.1639***

0.1667***

(0)

(0)

(0)

(0)

0.8893***

0.784***

0.8322***

0.8157***

(0)

(0)

(0)

(0)

34

Table A3: GARCH estimates of factor-mimicking portfolios orthogonal to other factors The four columns of this table report the GARCH in mean model estimates of the factor-mimicking portfolio in excess of the other four factor-mimicking portfolios. Each of the factor mimicking portfolios is first regressed individually on the other factor mimicking portfolios, one at a time, to create a pair-wise orthogonaled factor. Each orthogonal factor is then used in the GARCH estimates. The first table in each panel reports the intercept term from the regressions. The second table reports the model that best fits the orthogonalized portfolio. We estimated ARCH(1) through ARCH(4), GARCH(1,1), EGARCH(1,1), and GARCH(1,1)AR(1).

Panel A: Tests of SMB orthogonal to other factors Orthogonal to Intercept

Orthogonal to

Market

HML

LIQ

UMD

0.0031 (0.226)

0.0022**

0.0021

0.0036***

(0.04)

(0.141)

(0.001)

Market

HML

LIQ

UMD

GARCH (1,1)

GARCH (1,1)

GARCH (1,1)

GARCH (1,1)

-0.0014

-0.0016

-0.0025

-0.0007

(0.234)

(0.208)

(0.35)

(0.599)

4.5732***

3.6172**

5.0809

3.6435**

(0.007)

(0.022)

(0.11)

(0.017)

Mean equation intercept delta Conditional variance equation intercept ARCH1 GARCH1

N

0**

0**

0.0001**

0**

(0.042)

(0.022)

(0.028)

(0.023)

0.085***

0.107***

0.121***

0.103***

(0)

(0)

(0.001)

(0.001)

0.91***

0.891***

0.821***

0.896***

(0)

(0)

(0)

(0)

924

924

486

924

35

Panel B: Tests of HML orthogonal to other factors Orthogonal to Intercept

Orthogonal to

Market

SMB

LIQ

UMD

0.0026***

0.0043***

0.0037**

0.0064***

(0.034)

(0.002)

(0.013)

(0)

Market GARCH (1,1)

SMB

LIQ

UMD

GARCH (1,1) AR(1) GARCH (1,1) AR(1) GARCH (1,1) AR(1)

Mean equation intercept delta

0.0026**

-0.0011

-0.0017

0.0005

(0.021)

(0.631)

(0.379)

(0.839)

2.057*

7.4706**

6.744**

7.0601*

(0.075) AR(1)

(0.019)

(0.045)

(0.066)

-0.1376***

-0.1614***

-0.1485***

(0.003)

(0.001)

(0.002)

Conditional variance equation intercept ARCH1 GARCH1

N

0***

0***

0***

0***

(0)

(0.004)

(0.005)

(0.005)

0.1618***

0.1332***

0.1573***

0.1308***

(0)

(0)

(0)

(0)

0.8161***

0.8224***

0.775***

0.8232***

(0)

(0)

(0)

(0)

924

924

486

924

36

Panel C: Tests of LIQ orthogonal to other factors

Orthogonal to

Market

SMB

HML

UMD

Intercept

0.006**

0.0064**

0.0076**

0.0073**

(0.019)

(0.012)

(0.004)

(0.006)

Orthogonal to

Market

SMB

HML

UMD

GARCH (1,1)

GARCH (1,1)

GARCH (1,1)

GARCH (1,1)

Mean equation intercept delta

-0.004

0

-0.003

-0.002

(0.235)

(0.893)

(0.475)

(0.575)

3.28**

2.496*

3.298**

3.133**

(0.014)

(0.058)

(0.014)

(0.021)

Conditional variance equation intercept

0

0**

0**

0**

(0.108)

(0.039)

(0.043)

(0.032)

ARCH1

0.161***

0.17***

0.154***

0.159***

(0)

(0)

(0)

(0)

GARCH1

0.836***

0.817***

0.833***

0.826***

(0)

(0)

(0)

(0)

486

486

486

486

N

37

Panel D: Tests of UMD orthogonal to other factors Orthogonal to Intercept

Orthogonal to

Market

SMB

HML

LIQ

0.0096***

0.0084***

0.0099***

0.0086***

(0)

(0)

(0)

(0)

Market

SMB

HML

LIQ

EGARCH (1,1)

EGARCH (1,1)

EGARCH (1,1)

EGARCH (1,1)

0.0157***

0.0125***

0.0153***

0.0106***

(0)

(0)

(0)

(0)

-3.7941***

-2.9661***

-3.5857***

-2.4193*

(0)

(0)

(0.004)

(0.097)

-0.1979***

-0.3179***

-0.5378***

0.0106***

(0.004)

(0)

(0)

(0.002)

ARCH1

0.2178***

0.2692***

0.3298***

0.2965***

(0)

(0)

(0)

(0)

GARCH1

0.9683***

0.9501***

0.9178***

0.9361***

(0)

(0)

(0)

(0)

THETA

0.5405***

0.5192***

0.4336***

0.7207***

(0)

(0)

(0)

(0)

924

924

924

486

Mean equation intercept delta Conditional variance equation intercept

N

38

Gou tests Market Dependent variables

constant

(Mkt-rf)t+1

Var(Mkt-rf)t+1

cay t 2.1482***

0.0004

0.1227

3.1269***

(0.961)

(0.141)

(0.007)

(0)

0.0037***

-0.0009

0.2175

-0.1232***

(0)

(0.876)

(0.218)

(0.001)

constant

SMB t

VAR(SMB) t

0.0052***

0.0157

3.7262**

(0)

(0.84)

(0.033)

0.0011***

0.0036

0.5009***

(0)

(0.24)

(0)

constant

HML t

VAR(HML) t

HML t+1

0.0037

0.1047

4.6315*

(0.538)

(0.196)

(0.088)

VAR(HML) t+1

0.0004***

-0.0057***

0.7542***

(0.002)

(0.001)

(0)

constant

UMD t

VAR(UMD) t

cay t

UMD t+1

0.0373***

-0.1852**

-3.1536*

-1.1713***

(0)

(0.026)

(0.068)

(0.007)

VAR(UMD) t+1

0.0006**

0.0116***

0.5653***

-0.0442***

(0.045)

(0)

(0)

(0.009)

constant

LIQ t

VAR(LIQ) t

0.0126***

0.0126**

0.51

-0.14

(0.112)

(0.076)

0.9798***

-0.006**

0.2921

-1.00

(0.617)

(0.02)

(Mkt-rf)t+1 Var(Mkt-rf)t+1

SMB

SMB t+1 VAR(SMB) t+1

HML Dependent variables

UMD Dependent variables

LIQ Dependent variables LIQ t+1 VAR(LIQ) t+1

39

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Jegadeesh, N., and S. Titman, 1993, Returns to buying winners and selling losers: implications for stock market efficiency, Journal of Finance, 48, 65-91. Lundblad, C., 2007, The risk-return tradeoff in the long run:

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41

Figure I: Time series of the SMB portfolio returns

Figure II: Time series of the HML portfolio returns

42

Figure III: Time series of the UMD portfolio returns

Figure IV: Time series of the LIQ portfolio returns

43

Figure V: Conditional mean and conditional volatility of SMB from the GARCH model and realized returns (dotted line) Realized returns range from -0.08 to 0.1 (axis not shown)

Figure VI: Conditional mean and conditional volatility of HML from GARCH model and realized returns (dotted line) Realized returns range from -0.3 to 0.38 (axis not shown)

44

Figure VII: Conditional mean and conditional volatility of UMD from GARCH model and realized returns (dotted line) Realized returns range from -0.55 to 0.2 (axis not shown)

Figure VIII: Conditional mean and conditional volatility of LIQ from GARCH model and realized returns (dotted line) Realized returns range from -0.4 to 0.6 (axis not shown)

45

Table 1: Summary statistics This table presents summary statistics for the excess return of the CRSP value-weight market portfolio (Market), the small-minus-big (SMB), the high-minus-low (HML), and the liquidity (LIQ) factormimicking portfolios. Portfolios SMB and HML are constructed as in Fama French (1993). Portfolio UMD is constructed from the momentum effect. The LIQ is the liquidity factor-mimicking portfolio constructed using Aminhud (2001)’s liquidity measure. Each month we construct 10 decile portfolios based on the average liquidity measure over the previous 12 months. Liq is constructed as the return of the most illiquid portfolio minus the return of the most liquid portfolio. The SMB, HML, and UMD are monthly observations from 01/1927 through 12/2003. The LIQ factor starts from 07/1963 through 12/2003. Panel A: Summary statistics

Market SMB HML UMD LIQ

Mean

STD

Max

Min

Skewness

Kurtosis

N

0.007 0.003 0.004 0.008 0.007

0.055 0.033 0.037 0.048 0.057

0.374 0.376 0.360 0.184 0.363

-0.284 -0.116 -0.208 -0.509 -0.142

0.231 2.079 1.528 -3.009 1.469

7.437 20.080 15.197 27.954 6.155

924 924 924 924 486

Panel B: correlations and their p-values (in italics)

Market

SMB

HML

SMB

HML

UMD

LIQ

0.329 0.000 (0)

0.221 0.000 (0)

-0.341 0.000 (0)

0.236 0.000 (0)

0.103 0.002 (0.002)

-0.199 0.000 (0)

0.186 0.000 (0)

-0.503 0.000 (0)

-0.037 0.411 (0.411)

UMD

0.005 0.913 (0.913)

46

Table 2: The SMB portfolio returns This Table reports the time series tests of the SMB factor-mimicking-portfolio constructed as in Fama and French (1993) for the entire time series and for two subperiods. Panel A presents the average monthly returns. Panel B reports the estimates of the GARCH in mean model as in Equation (8). The following are the mean term and the variance term estimates. P-values are reported in parenthesis. The correlation reported is the correlation test of the residual term up to six lags. ***, **, and * denote the significance level at 1%, 5%, and 10%, respectively. Panel A: Average return 192701-200312

Mean N

192701-196212

196301-200312

0.0026

0.0023

0.0028

(0.019)

(0.175)

(0.047)

924

432

492

Panel B: GARCH in mean estimates of SMB 192701-200312

192701-196212

196301-200312

GARCH (1,1)

GARCH (1,1)

GARCH (1,1)

-0.0014

-0.0011

-0.0028

(0.234)

(0.435)

(0.312)

3.795**

3.018

5.754*

(0.011)

(0.136)

(0.063)

Mean equation intercept delta Conditional variance equation intercept

0.109**

0

0*

(0.025)

(0.238)

(0.062)

ARCH1

0.108***

0.109***

0.116***

(0)

(0)

(0.001)

GARCH1

0.89***

0.898***

0.835***

(0)

(0)

(0)

AIC Correlation test

-3929.73 0.060

47

Table 3: The HML portfolio returns This Table reports the time series tests of the HML factor-mimicking-portfolio constructed as in Fama and French (1993) for the entire time series and for two subperiods. Panel A presents the average monthly returns. Panel B reports the estimates of the GARCH in mean model as in Equation (8). The following are the mean term and the variance term estimates. P-values are reported in parenthesis. The correlation reported is the correlation test of the residual term up to six lags. ***, **, and * denote the significance level at 1%, 5%, and 10%, respectively. Panel A: Average return 192701-200312

Mean N

192701-196212

196301-200312

0.0035

0.0034

0.0037

(0.004)

(0.099)

(0.011)

924

432

492

Panel B: GARCH in mean estimates of HML 192701-200312

192701-196212

196301-200312

GARCH (1,1)

GARCH (1,1)

GARCH (1,1)

0.0024**

0.0043**

-0.0008

(0.043)

(0.024)

(0.682)

1.902

0.447

5.655**

(0.126)

(0.788)

(0.037)

Mean equation intercept delta Conditional variance equation intercept

0***

0***

0***

(0)

(0.004)

(0.004)

ARCH1

0.17***

0.191***

0.163***

(0)

(0)

(0)

GARCH1

0.802***

0.802***

0.768***

(0)

(0)

(0)

AIC Correlation test

-3846.12 0.0154

48

Table 4: The UMD portfolio returns This Table reports the time series tests of the UMD factor-mimicking-portfolio constructed as in Fama and French (1993) for the entire time series and for two subperiods. Panel A presents the average monthly returns. Panel B reports the estimates of the GARCH in mean model as in Equation (8). The following are the mean term and the variance term estimates. P-values are reported in parenthesis. The correlation reported is the correlation test of the residual term up to six lags. ***, **, and * denote the significance level at 1%, 5%, and 10%, respectively. Panel A: Average return 192701-200312

Mean N

192701-196212

196301-200312

0.0076

0.0066

0.0086

(0)

(0.013)

(0)

924

432

492

Panel B: GARCH in mean estimates of UMD 192701-200312

192701-196212

196301-200312

GARCH (1,1)

GARCH (1,1)

GARCH (1,1)

0.0091***

0.0095***

0.0091***

(0)

(0)

(0)

-1.811***

-1.503*

-2.681*

(0.01)

(0.086)

(0.07)

0***

0***

0***

(0)

(0)

(0.009)

0.163***

0.156***

0.207***

(0)

(0)

(0)

0.833***

0.848***

0.77***

(0)

(0)

(0)

Mean equation intercept delta Conditional variance equation intercept ARCH1 GARCH1

AIC Correlation test

-3468.7398 0.8236

49

Table 5: The LIQ portfolio returns This table presents time series tests of the liquidity-factor-mimicking portfolio. The liquidity-factor-mimicking portfolio is constructed using Amihud (2001)’s liquidity measure. Each month we construct 10 portfolios based average lagged 12 months liquidity. The liquidity-factor-mimicking portfolio is the return of the most illiquid portfolio minus the return of the most liquid portfolio. Panel A presents the average monthly returns. Panel B reports the estimates of an ARCH (2), GARCH (equation (8)), and EGARCH (equation (9)) in mean models. ***, **, and * denote the significance level at 1%, 5%, and 10%, respectively. P-values are reported in parenthesis. Panel A: Average return 196307-200312

Mean

0.0074 (0)

N

486

Panel B: GARCH in mean estimates of LIQ

196307-200312

196307-200312

196307-200312

GARCH (1,1)

ARCH(2)

EGARCH (1,1)

-0.0025

-0.01

-0.0045

(0.506)

(0.033)

(0.27)

3.1605**

5.2189***

2.8569**

(0.022)

(0.001)

(0.034)

0.0001**

0.002***

-0.1628

(0.032)

(0)

(0.151)

0.1572***

0.3892***

0.2597***

(0)

(0)

Mean equation intercept delta

Conditional variance equation intercept ARCH1

(0) ARCH2

0.0422 (0.239)

GARCH1

0.8269***

0.9696***

(0)

(0)

Theta

-0.1449* (0.064)

AIC correlation test

-1464.4533

-1450.5805

-1471.2929

0.0062

0.0011

0.0018

50

Table 7: Sharpe ratios of the factor-mimicking portfolios Panel A presents the mean and standard error of the time series of the Sharpe ratio of factor-mimicking portfolios and the value-

[ ]

weighted market portfolio. The factors’ Sharpe ratio is φ Vart (rt +X1 ) from the GARCH estimate of E t rt +X1 = a + φVart (rt +X1 ) . The market Sharpe ratio from GARCH including the intercept is

[ ] m t +1

Et r

, and the market Sharpe ratio from MIDAS is

Vart (rt m+1 )

φ Vart (rt m+1 ) from the MIDAS model estimates of E t [rt m+1 ] = a + φVart (rt m+1 ) in Ghysels, Santa-Clara and Valkanov (2004).

1927-2003

1927-1962

1963-2003

Mean

Mean

Mean

0.1128

0.1310

0.0954

0.1829

0.2438

0.1264

SMB Sharpe ratio

0.1179

0.0982

0.1737

from GARCH model

(0.075)

(0)

(0)

[0]

[0]

[0.005]

0.0628

0.0168

0.1632

(0)

(0)

(0)

[0]

[0]

[0.135]

Market Sharpe ratio from MIDAS model* Market Sharpe ratio from GARCH model Including the intercept

HML Sharpe ratio from GARCH model

LIQ Sharpe ratio

0.1776

from GARCH model

(0) [0]

51

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