Book-to-market and size effects: Risk compensations or market inefficiencies?

Book-to-market and size effects: Risk compensations or market inefficiencies? Abstract Are the size and book-to-market effects in US data related to risk factors besides the market risk? Are the portfolios, HML and SMB, proper representations of latent factors? We apply the orthogonal portfolio approach to model the risk premium when not all factors are known. The results show that the size effect is a compensation for a latent risk factor related to relative distress, but the book-to-market effect is mostly non-risk-based. Our results validate SMB as a proper factor-mimicking portfolio, while HML is successful in asset pricing tests since it shares the same non-risk-based components as the test portfolios. 1

There is a long research history, starting with Banz (1981), showing the existence of a size effect in stock returns in the sense that small firms have a higher average return than big firms and this phenomenon cannot be explained by their market beta. Analogous to this finding, the early studies of Basu (1983) and Rosenberg et al. (1985) show the importance of the so-called book-to-market effect, according to which firms with high ratios of the book equity to the market equity have higher average returns than firms with low ratios. Fama and French (1992, 1993) pull the two strands together and show the importance of the book-to-market ratio and firm size in explaining cross-sectional differences in expected stock returns. Fama and French (1993) construct two factor-mimicking portfolios based on size and book-to-market characteristics, SMB and HML respectively. The average returns of these portfolios are supposed to capture the premium for a latent state variable related to relative distress. These mimicking portfolios are now extensively employed in empirical research. There are a large number of studies that attempt to interpret and explain the observed size and book-to-market effects. According to Chan and Chen (1991), small firms (firms with low market value of equity) are usually the firms that lost market value because of poor performance. These firms are typically highly leveraged and are less likely to survive economic downturns. Accordingly, the cross-sectional differences in average return between small and large firms may be interpreted as a compensation for the risk connected to the relative distress of the small firms. Analogous to this idea, Fama and French (1995, 1996) and Davis et al. (2000) consider the variables book-to-market and size as proxies for relative distress. From this viewpoint, the asset pricing is rational and the observed higher returns for small firms or high book-to-market firms should be considered a compensation for additional undiversifiable sources of risk besides the market risk. This explanation has been rejected by several other studies. Lakonishok et al. (1994) and Haugen (1995) consider the observed 2

book-to-market effect as an irrational asset pricing resulting from systematic errors in investors expectations about future returns. Naive investors may be more willing to invest in low book-to-market firms, which are usually firms with strong fundamentals, because these firms performed well in the past. This behavior pushes up the prices and lowers the expected returns for stocks of strong firms. Similarly, investors lower attention toward the stocks of the historically weak firms undervalues these firms and results in higher ex post average returns for their stocks. Daniel and Titman (1997) follow the idea of investors behavioral irrationality and argue that the relatively higher average return of the high book-to-market firms is due to growth and distress characteristics rather than a compensation for firms risk loading. Finally, MacKinlay (1995) puts forward data-snooping bias and Kothari et al. (1995) use the survivorship bias as explanations for the observed CAPM anomaly in the US data. This debate continues (see for example Petkova and Zhang (2003) and Ang and Chen (2004)). Our purpose is to investigate whether the observed size and book-to-market effects in US data are related to some risk factors beside the market risk or if these effects are due to non-riskbased components. We also examine the validity of HML and SMB as factor-mimicking portfolios in an asset pricing context, i.e. whether the mean returns of these portfolios are able to capture the risk premium associated with latent state variables. These issues are tackled by employing a latent factor model that can identify the pricing impact of unknown factors. This approach enables us to estimate the maximum amount of the expected returns that can be derived from a linear asset pricing model for a given set of test portfolios. MacKinlay and Pastor (2000) construct a new framework for modeling asset prices by combining information in the covariance matrix with average returns to define an exact factor model that excludes all non-risk-based elements from the expected returns. The model includes some known and observed factors plus an unspecified factor that represents all potential risk factors that are missing from the model. This unspecified factor is constructed 3

with the help of the optimal orthogonal portfolio, which by construction is orthogonal to the observed factor/factors and is also optimal since it eliminates mispricing when added to the initial factor model. 1 Ideally, the estimated expected return from using this approach is supposed to include the risk premium due to all the risk factors and exclude all the components of the mean return that are non-risk-based. We develop a method for testing mispricing when not all factors are observed, which is based on the orthogonal portfolio approach of MacKinlay and Pastor (2000). MacKinlay and Pastor (2000) define a priori the importance of the optimal orthogonal factor. Despite the computational advantage of this approach, it eliminates the ability of the data to identify the latent factor/factors. In contrast, our approach identifies the orthogonal factor within the model. Consequently, the expected excess return estimated by our factor model captures the entire risk premium for each test asset. The risk-based explanation of the size and book-tomarket effect does not exclude the possibility of the non-risk-based components. Our approach makes it possible to divide the observed average returns of the assets into three parts: the component explained by the observed factor/factors, the component due to the unobserved factor/factors, and finally the non-risk-based component. We then perform a formal test for mispricing using a bootstrap approach. We use a restricted regression model with one observed factor (OFR). The observed factor is represented by the value-weighted market index and the orthogonal portfolio represents all potential missing factors. Since we are interested in analyzing the size and book-to-market effects, we presume that the market index is a sufficient proxy for the market risk and that it does not contain any non-risk-based component. The models are estimated for six Fama and French portfolios over the period 1927-2002. These portfolios are double-sorted according to firm size and book-to-market ratio. The choice of these double-sorted portfolios as test assets is motivated by their ability to separate the overlapping effect of the two characteristics. More 4

importantly, the portfolios direct relation to Fama and French s factor-mimicking portfolios makes it easier to analyze the spillover of the non-risk-based components from the test assets to the constructed factor-mimicking portfolios. The analysis is performed in several steps. In the first step, we use OFR to test for the existence of non-risk-based components in the test assets. This result is compared with the mispricing given by the three-factor model of Fama and French in order to assess to what extent SMB and HML coincide with the latent factors. In the second step, we analyze the cross-sectional differences and the time variation in the risk premiums due to the optimal orthogonal portfolio. We also look at its importance relative to the market factor over time. A further step is to analyze the weights of the orthogonal portfolio, which may give hints of the relationship between the characteristics and the missing factors. However, since the disclosure of the latent factors depends on the choice of the test assets, this analysis should not be taken as a complete analysis of all missing risk factors. Finally, we look at the components of the mimicking portfolios for size (SMB) and book-tomarket (HML), which are the most frequently utilized factor portfolios besides the market portfolio in asset pricing tests. In the conventional tests SMB and HML are supposed to represent risk factors, and as pure factor-mimicking portfolios they should contain only riskbased components. However, factor portfolios are constructed from the test assets. The portfolios may therefore be contaminated by non-risk-based components that can exist in realized mean returns. If these portfolios are used as factors in an asset pricing model, then their explanatory power may be due to the existence of the same non-risk-based components among the test assets. This phenomenon is more likely to appear if the sorting and the factor portfolio are constructed according to the same characteristic. Our reasoning is not a critique of the common use of factor portfolios and sortings, but that the result of the asset pricing 5

tests may be driven by non-risk-based components. It is therefore interesting to analyze the non-risk-based and the risk-based components of these factor portfolios. Our results confirm the existence of some latent risk factors related to firm characteristics, which cannot be captured by the market index. These latent factors are related to relative distress and are more important during recession periods. We find a strong relationship between the latent factor and firm size, while a large part of observed book-to-market effect has a non-risk-based explanation, in particular during the latter part of the sample. The analysis of the factor-mimicking portfolios, consistent with the finding above, verifies that SMB is a good candidate as a mimicking portfolio in asset pricing. However, HML can be considered as an improper factor-mimicking portfolio. Its success as an explanatory variable in asset pricing tests might be due to the existence of the non-risk-based components in this portfolio. The outline of the paper is as follows: section I discusses the definition of the factor models and the applied econometric methods; section II analyses the empirical results and section III concludes the study. I Econometric Model Consider the following linear K-factor pricing model for N assets: R = α + βf + ε, (1) t E [ R ] = µ t ( R µ )( R µ ) = V E t t E[ ε ] = 0, E [ ε ε ] = Σ t [ F t ] = F, E ( F t µ F )( Ft µ F ) = Ω F E µ t Cov[F kt, ε it ] = 0 for k and i t t t 6

where R t is an (N 1) vector of excess returns on N assets, α is an (N 1) vector of model mispricing, i.e. the part of the excess return that is not captured by the factors, β is an (N K) matrix of the sensitivities of N assets to the K factors, and F is a an (K 1) vector of excess returns for the K factors. In an exact factor pricing model expected returns are exclusively explained by the included risk factors, i.e. there are no non-risk-based components in expected returns, and all the elements of the vector α are zero. In this case the portfolio with the maximum Sharpe ratio (the tangency portfolio in the mean-standard deviation space) can be constructed as a linear combination of the K known factors. The vector of the mean returns is defined by: µ ˆ = βµ ˆ ˆ F, (2) where βˆ is the estimated sensitivities resulting from N time series regressions of the model in equation (1). Since the factor structure is assumed to take into account the entire cross covariances of the returns, it follows that the variance-covariance matrix of the residuals will be diagonal. The covariance matrix of the returns is: Vˆ = βω ˆ ˆ βˆ + Dˆ F, (3) where Dˆ is a (N N) diagonal matrix of the estimated residual variances for N time series regressions of the model in equation (1). The factor moments µˆ F and Ωˆ F are both estimated from the sample. On the other hand, if the K factors are not sufficient to explain the expected returns then some elements of α will differ from zero. In this case an exact factor pricing model can be achieved by adding an optimal orthogonal portfolio, h, as a complement factor to the K known factors (see MacKinlay (1995)): R = +, (4) t β hrht + βft ut 7

E[u t u' t ] = Φ E[R ht ] = µ h E[(R ht - µ h ) 2 2 ] = σ, Cov[R ht, F kt ] = 0 for k and t h where R ht is the excess return on the optimal orthogonal portfolio, h, and β h is an (N 1) vector of the sensitivities of N assets to the factor h. The orthogonality of h to the K known factors provides the possibility of adding it to the factor model without changing the sensitivity matrix β. This construction allows us to relate the mispricing vector, α, to the residual covariance matrix, Σ. Combining equation (1) with equation (4) gives: α = β h µ h (5) 2 2 σ h Σ = β β + Φ = αα h hσ h + Φ. 2 µ h MacKinlay and Pastor (2000) use this link to provide a joint estimate of α and Σ. These estimates are then used to estimate the mean and the covariance matrix of the returns, µ and V, as: µ ˆ = αˆ + βµ ˆ ˆ F (6) Vˆ = αα θˆ + βω ˆˆ βˆ + Φ ˆ. ˆˆ h F 2 σ h where θ h is 2 µ h, the inverse square Sharpe ratio of the orthogonal portfolio, which is unknown and should be estimated as a parameter. Again µˆ F and Ωˆ F are estimated from the sample, but αˆ, βˆ and Φˆ are estimated by maximizing the following log likelihood function: T T 1 ln L αβ,, θ, Φ det ααθ Φ R α βf αα θ Φ R α βf (7) 1 ( ) ( + ) ( ) ( + ) ( ) h h t t h t t 2 2 t= 1 8

The matrix Φ is restricted to be diagonal, which is justified by the assumption that all the covariations among asset returns are explained by the K known factors plus the optimal orthogonal portfolio. We follow MacKinlay and Pastor (2000) and restrict Φ to be equal to σ 2 I, i.e. the idiosyncratic risks among assets are equal. It can be shown that the higher the θ h, the more important the contribution of the observed factor in explaining expected returns. MacKinlay and Pastor (2000) presume that this parameter is known. Thus, their approach determines a priori the contribution of the latent factors to expected returns. In contrast, we let the data identify the optimal orthogonal portfolio h. The weight vector of the optimal mean-variance tangency portfolio, i.e. the portfolio with maximum Sharpe ratio (MSR), is ( ι Vˆ 1 1 µ ˆ ) Vˆ 1 µ ˆ w msr =, (8) where ι is a (N 1) vector of ones, and µˆ and Vˆ are defined in equation (6). The weight vector of the assets in the optimal orthogonal portfolio, h, is given by: ( ι Vˆ 1 1 αˆ ) Vˆ 1 αˆ w h =, (9) where αˆ and Vˆ are estimated from the restricted factor model. Above in equation (4) we presume a linear asset pricing model where the optimal orthogonal portfolio represents all potential missing factors that are necessary for pricing the existing test assets. A linear asset pricing model implies that the stochastic discount factor is linear in the same factors, and this stochastic discount factor prices all the chosen test assets. It might be simpler to think of the latent factors in relation to the stochastic discount factor, where the contribution from these latent factors to the stochastic discount factor is captured by the optimal orthogonal portfolio. It is well known that the number of factors is not important, since factors can always be combined to form a new factor without affecting the ability of the 9

stochastic discount factor in pricing the assets (see Cochrane (2001 chapter 7)). In our analysis, we will reduce a multifactor model to a two-factor model: the market portfolio plus the orthogonal portfolio. The estimation of the parameters of the restricted model requires solution of a difficult and nonlinear likelihood function. We therefore use the simulated annealing approach (see Goffe et al. (1994)) to estimate the likelihood function in equation (7). The advantages of this approach compared to the conventional methods are that it is very robust and fails seldom even for very complicated problems (see Corana et al. (1987) and Goffe et al. (1994)). The estimated parameters for each model are used to estimate the expected returns and the return covariance matrix. For each test asset, the expected excess return estimated by the restricted factor model should capture the entire risk premium for that asset. Therefore, the difference between the sample mean excess return and the estimated expected excess return for each portfolio should be considered to be non-risk-based. A bootstrap procedure is used to test for significance of the non-risk-based components of the test portfolios. We resample with replacement the original data 200 times and re-estimate the model using the new samples. The empirical confidence intervals are used for inferences. II Analysis Our analysis is based on a one-factor restricted model (OFR), where an orthogonal portfolio represents all risk factors besides the market index. We presume that the market index is a pure representation of the market factor and does not contain any non-risk-based components. We first analyze whether there are non-risk-based components in the test assets. Then we study the importance of latent factors, which are represented by the optimal orthogonal portfolio, for explaining the risk premiums of the test assets. Third, we investigate the composition of the optimal orthogonal portfolio in order to examine its relationship with the 10

variables book-to-market and size. Finally, we analyze the relevance of the factor-mimicking portfolios SMB and HML as factors in asset pricing models by measuring the non-risk-based and the risk-based components of these portfolios. The test assets are six portfolios double-sorted according to firm size and book-to-market ratio. These portfolios are sufficient for capturing the cross-sectional differences in average returns, and to use a large number of assets would demand an extensive and time-consuming estimation procedure. Market return is the value-weighted return on all NYSE, AMEX and NASDAQ stocks (from CRSP). The risk-free rate is the one-month Treasury bill rate (from Ibbotson Associates). Data cover all the NYSE, AMEX and NASDAQ stocks over the period 1927-2002 and are from the data library on Kenneth French s home page. The factor-mimicking portfolios are defined as: ( SL + SM + SH ) ( BL + BM + BH ) SMB = (10) 3 3 ( SH + BH ) ( SL + BL) HML =, 2 2 where the notation L, M and H is for low, medium and high book-to-market ratio respectively, while S and B are used for small and big firms respectively. These differences are computed for all the components of the expected returns. A quick glance in Table I at the sample statistics for the market index and the six doublesorted portfolios shows that all the mean excess returns are positive and highly significant over the entire sample. There is a negative relation between size and average return and a positive relation between book-to-market and average return. The estimated average return of HML is highly significant, while for SMB the mean return is almost significant at the 5% level. We also divide the sample into two subperiods, 1927-1964 and 1965-2002. All the mean excess returns are significant in the first period, while two portfolios with low book-to- 11

market ratio, SL and BL, do not have significant mean excess returns in the second period. HML is significant in both periods, but SMB is not significant in any period. Table I can be inserted here. A. Asset pricing test of double-sorted portfolios We analyze the non-risk-based components of six double-sorted portfolios, which is equivalent to testing for an exact factor model. Our starting point is the OFR model. The nonrisk-based component (mispricing) for each test asset is computed as the difference between the asset s mean excess return and its estimated risk premium according to this model. A bootstrapped confidence interval will be used to test if this component is significantly different from zero. To assess whether there are factors beside the market portfolio, we compare the results given by OFR with mispricing based on CAPM. To analyze if the two mimicking portfolios, SMB and HML, coincide with the latent factors we also estimate the mispricing given by Fama and French s three-factor model. The difference between the estimated mispricings from OFR and the other two models may indicate that the latter models do not contain a sufficient number of factors. However, for the case when OFR detects a significant mispricing but the three-factor model does not, there is an additional explanation: the mimicking portfolios of the latter model may contain the same non-risk-based components as the test assets. Table II shows the test results and Figure 1 compares the estimated risk premiums from the different models with the sample mean excess returns. For the entire period, the OFR model shows that the non-risk-based component is only significant for SL, i.e. small growth stocks. The three-factor model gives the same result as the OFR model for this portfolio. The negative and significant intercept for SL is in accordance with results from finer partitions of the test assets, as in Fama and French (1993) with 25 portfolios and Davis et al. (2000) with 9 portfolios. CAPM, on the other hand, finds no significant mispricing in SL but shows 12

significant alpha for SM and SH. This is due to the fact that OFR and the three-factor model account for an additional positive risk premium for small firms, which drives the estimated alphas down for all three small firm portfolios. The non-significant intercept for SL in CAPM follows from a positive risk premium of the excluded factor that offsets the true negative mispricing. Table II can be inserted here. Figure 1 can be inserted here. For the three big firm portfolios both CAPM and OFR give no significant mispricing, while the three-factor model has a positive and highly significant alpha for BL and negative and significant alpha for BH. These results may be explained by the negative loading of the BL and positive loading of the BH on the HML portfolio, which due to the large positive mean of HML results in a lower estimated risk premium for BL and vice versa for BH. Consequently, these two portfolios seem to be under/overpriced according to the three-factor model, while our model suggests no mispricing. One possible motivation for this difference is the fact that the factor-mimicking portfolios are not pure factor portfolios and the very large mean return of the HML portfolio might contain some non-risk-based components. We will discuss this issue later in the paper. To see if our finding is stable over time, we divide the sample into two equal subperiods, 1927-1964 and 1965-2002. In the first period, there is no significant mispricing in any model. Since the market index is included in all three models we may conclude that CAPM may be a sufficient model and that there are no non-risk-based components in this period. In the second subperiod, the OFR model shows significant alphas for all the small portfolios (SL, SM, SH) and BH. The risk premiums for the second period given by OFR exhibit small differences within the group of small and big portfolios respectively, but the risk premiums of the small firm portfolios are larger than those of the big firms (see Figure 1). This confirms the small 13

firm effect. The risk premiums estimated by the three-factor model for the second period track the sample means. However, several portfolios still have significant alphas, which are in accordance with Davis et al. (2000), since despite the small intercept estimated by the threefactor model the t-statistics are large due to the small residual variances. We now compare the portfolio frontiers given by the different models. Given that OFR filters all non-risk-based components of the sample mean excess returns, this comparison illustrates to what extent the other models filter these components. It is also interesting to compare the market index with the optimal portfolio of OFR, which may give an indication of the efficiency of the market portfolio. Figure 2 plots the portfolio frontier constructed based on the estimated expected returns and the covariance matrix given by the different models. The frontier for the Sample shows that the existence of non-risk-based components results in a larger spread in mean excess return and consequently in steeper slopes of the asymptotes for this frontier. As expected, the three-factor model cannot filter out all non-risk-based components and as a result these asymptotes are also steeper than the corresponding asymptotes of OFR. In the first subperiod we see that there is no difference between OFR and the three-factor model and these frontiers do not deviate substantially from the Sample frontier, which supports the result of non-significant alphas presented in Table II. Consistent with this result, the market portfolio is on the frontier and has a Sharpe ratio of 0.134, and this ratio is very close to that of the tangency portfolio of the three-factor model (0.136) and OFR (0.137). In the second subperiod, however, the market index is not on the constructed frontiers. The difference between this portfolio and the tangency portfolio of the OFR may suggest the existence of some priced factor/factors that are important for covariation in returns but are not captured by the market index. We compare these two Sharpe ratios in greater detail in the next subsection. Figure 2 can be inserted here. 14

The results above suggest a time-varying pattern in the non-risk-based component of the mean returns. For a more detailed analysis, we estimate the OFR model using a twenty-year moving window. It is very interesting to see that all the estimates of the SL portfolio from around 1960 and onwards are increasingly negative and significant (see Figure 3). This confirms our previous results, which showed a highly significant non-risk-based component for the second period only. Figure 3 can be inserted here. All in all, the results show significant non-risk-based components, in particular for SL after 1960s. We might explain this result by investor overreaction to the price increase of the small firms: buying small firms when their market price is relatively high (low BM) drives the price of these stocks up and the average return down. On the other hand, we also find evidence for the existence of some risk factors besides the market, which supports the conjecture of Fama and French (1996) that the asset pricing model is consistent with a multifactor model that is not reduced to the CAPM. B. The importance of the orthogonal portfolio We have shown above that there may exist latent risk factors besides the market portfolio. We will now assess the overall importance of the orthogonal portfolio in relation to the market portfolio and compare its contribution to the risk premiums across different test assets. The overall importance of the orthogonal portfolio is measured from the following relationship between the tangency portfolio and the factor portfolios: s s + s 2 2 2 q = m h, where 2 q s is the squared Sharpe ratio of the tangency portfolio, s 2 m is the squared Sharpe ratio of the observed factor, and s 2 h is the squared Sharpe ratio of the orthogonal factor (see MacKinlay (1995)). Our metric is the ratio s 2 2 h s q. The result for the overall sample is 0.28 and for two 15

subsamples the results are 0.08 and 0.56 respectively, which reaffirms the importance during the second period of latent risk factors that are not represented by the market. We now look at the time variation in the importance of the orthogonal portfolio relative to the market portfolio (see Figure 4). There is an obvious time pattern for the importance of the orthogonal portfolio: relatively important during the first twenty years and from the early 1970s and twenty years onwards. Since the estimation periods with a high contribution from the orthogonal portfolio include the recessions of the 1930s and the 1970s, we may relate the orthogonal portfolio to distress factors that are active during such periods. Figure 4 can be inserted here. Based on OFR we can divide the total risk premium into two parts. One part is explained by the observed factor, which is represented by the market index. The second part is explained by the latent risk factor/factors represented by the optimal orthogonal portfolio, which is the estimated alpha from the one-factor restricted model, i.e. the sensitivity to the orthogonal factor times the risk premium of this factor. The risk premiums due to the orthogonal portfolio over the entire sample are shown in Figure 5. There is a cross-sectional difference in the estimates of these risk premiums: small firms have relatively larger risk premiums than big firms and within each size group firms with high BM have larger risk premiums. 2 The same pattern can be seen for estimates from twenty-year overlapping windows: it is more apparent in the very beginning of the sample and in particular for long sequences during the latter half of the sample (see Figure 6). Thus, small firms and firms with high book-to-market have a higher exposure to the latent risk factors represented by the orthogonal portfolio. In the literature these firms are considered to be overrepresented in the marginal firms or distressed firms (see Chan and Chen (1991) and Fama and French (1996)). This result combined with an increasing overall importance of the orthogonal portfolio during periods 16

with unfavorable economic conditions points to the following conjecture: the orthogonal portfolio can be seen as a proxy for risk associated with relative distress of the marginal firms. To sum up: there is an important and time-varying relative contribution of the orthogonal portfolio; this time pattern together with the cross-sectional pattern of the risk premium due to the orthogonal portfolio imply that this portfolio represents a state variable that is related to relative distress. Figure 5 can be inserted here. Figure 6 can be inserted here. C. Composition of the optimal orthogonal portfolio The optimal orthogonal portfolio represents a linear combination of different latent factors, but this combination may vary with the test asset under consideration. To get an idea of the factors represented by the optimal orthogonal portfolio based on the test assets in this study, we look at the portfolio weights of the orthogonal portfolio. In Figure 7 we can see that the small firm portfolios, independent of their book-to-market ratio, receive positive weights while the large portfolios have negative weights, which confirms that there are latent risk factor(s) related to firm size. Book-to-market has a clear pattern conditional on size: within each size group the high book-to-market portfolio has a larger weight than the low book-tomarket portfolio. 3 To get an even closer look at the possible variation in the orthogonal portfolio weights over time, we calculate these weights for overlapping twenty-year windows. The results in Table III show that the pattern is stable over the windows; for example the portfolio SH has the highest rank in 84% of the cases and the small firm portfolios almost always have larger weights than the big firm portfolios. Thus these results, in agreement with the previous subsection, show a very clear relation between firm size and latent risk factors, particularly for small firms with high book-to-market. 17

Figure 7 can be inserted here. Table III can be inserted here. D. Factor mimicking portfolios In this section we first analyze the non-risk-based component of the two factor-mimicking portfolios, SMB and HML. We then divide the risk premiums of these mimicking portfolios into fractions related to the market factor and the optimal orthogonal portfolio respectively. The results in Table II point to the existence of non-risk-based components in the six doublesorted portfolios. Furthermore, the comparison of the frontier for the three-factor model with the other frontiers in Figure 2 suggests that the non-risk-based components of these assets may have infected the related factor-mimicking portfolios, i.e. SMB and HML. Therefore, we examine the non-risk-based components of these mimicking portfolios. The existence of significant non-risk-based components in these portfolios may imply that the sample returns of these portfolios are impure representations of the background factors. We use the risk premiums and the non-risk-based components given by the OFR for the six double-sorted portfolios to construct the corresponding components of the two factormimicking portfolios according to equation (10). Figure 8 and Table I show that the estimated average return of HML is significant for the entire period as well as the two subperiods and is larger than that of SMB. The average return of SMB is only weakly significant for the entire sample. Based on these sample results, we can conclude that there is a strong value premium while the size effect appears to be tiny. This result has been taken at face value in most previous research (see for example Davis et al. (2000)). However, to assess the value and the size premiums, we need to exclude the non-risk-based component of these portfolios. Figure 8 shows that for the entire period and the first subperiod the non-risk-based component is not significant for SMB and HML. However, there is a significant non-risk-based 18

component in the second subperiod for both mimicking portfolios, and in particular for HML. The non-risk-based component of the HML is also larger than its systematic component in the second sub-period, while for SMB the risk-based component is always larger than the nonrisk-based component. The twenty-year moving windows give a significant and increasing non-risk-based component for HML during most subperiods in the second half of the sample (see Figure 9). For SMB, there are only a few periods with a significant non-risk-based component. Due to the importance of the non-risk-based component in HML, this portfolio should be considered as a very impure factor-mimicking portfolio. Hence, to assess the importance of a mimicking portfolio it is not sufficient to rely solely on its sample average. Figure 8 can be inserted here. Figure 9 can be inserted here. We now investigate the risk components of the HML and SMB in order to assess to what extent these portfolios are related to the latent risk factors besides the market factor. Figure 10 shows the results of the analysis of the relative contribution of the optimal orthogonal portfolio and the market index. For the entire period there is a positive contribution from the orthogonal portfolio as well as from the market factor, while the contribution of the orthogonal factor is slightly above that of the market factor for both mimicking portfolios. However, the result for HML is slightly misleading, since the results from the two subperiods show that the absolute contribution from the market portfolio is greater than the contribution from the orthogonal portfolio: the contribution is positive in the first period and negative in the second period. The most important result is that the contribution due to the orthogonal factor is larger for SMB relative to HML and there is a considerable difference in the second period. This result is confirmed by examining the twenty-year moving windows: Figure 11 shows that the orthogonal component of HML is sometimes large, but it is always below that of SMB after 1950s. 19

Figure 10 can be inserted here. We can conclude that the significant sample average of HML is largely due to a non-riskbased component and its risk based component, its proper risk premium, is mainly driven by the market factor, while the sample average of SMB is mostly related to the background factors not captured by the market index. III Conclusion We use the optimal orthogonal portfolio approach to analyze to what extent the observed size and book-to-market effects on US stock returns are due to latent risk factors besides the market portfolio. In addition, we examine whether HML and SMB are proper candidates as factor-mimicking portfolios in an asset pricing context. We employ a restricted one-factor model (OFR) with the market index as the only observed factor and an orthogonal portfolio represents all latent risk factors. As test assets, we use six double-sorted portfolios based on size and book-to-market ratio. To analyze if the observed size and book-to-market effects on US stock returns are due to market inefficiencies, we start by testing for non-risk-based components in the test assets by applying the OFR model. To assess whether there are factors besides the market portfolio and to what extent the two Fama and French mimicking portfolios coincide with the latent factors, we compare the results given by OFR with CAPM and the three-factor model. The results show a significant negative non-risk-based component particularly for SL after the 1960s. However, at the same time our results from this analysis reveal the existence of other latent factors besides the market portfolio. To establish the influence of the latent factors revealed by the previous analysis we look at the overall effect of the orthogonal portfolio and its contribution to the risk premiums of the test assets. Our findings show an important contribution of the orthogonal portfolio during the 20

recessions of the 1930s and the 1970s. We also find that the so-called distressed firms, i.e. small firms and firms with high book-to-market have a higher exposure to the latent risk factors represented by the orthogonal portfolio. Combining these time and cross-sectional patterns suggests that this portfolio represents a state variable that is related to relative distress. To further analyze the relationship between the optimal orthogonal portfolio and the two characteristics, we look at the portfolio weights. This portfolio is long in small firms and short in big firms, while, only conditional on size, high book-to-market portfolios have larger weights than the low book-to-market portfolios. Thus, there is a clear-cut relation between firm size and latent risk factors. We have shown above both the existence of extra factors besides the market and non-riskbased components. Have these results any bearing on the usefulness of SMB and HML as factor-mimicking portfolios in asset pricing? More specifically, are the average returns of these portfolios driven by non-risk-based components and/or risk factors? Furthermore, is their risk based component related to the orthogonal portfolio? Our analysis of these factormimicking portfolios shows that the observed size effect represented by the sample average of SMB may be mainly explained as risk compensation. This compensation is due to a large extent to the risk factor/factors represented by the optimal orthogonal portfolio. However, the large sample mean excess return of HML for the whole sample is mainly due to its market risk before the 1960s and thereafter it has a non-risk-based explanation. Accordingly, SMB may be considered a good candidate as a mimicking portfolio in asset pricing models. This mimicking portfolio would not be able to explain the realized mean excess return of the test assets when these means contain non-risk-based components. However, this failure should not be considered a drawback for SMB, since a pure mimicking portfolio should not explain the irrational component of the average returns. On the other hand, HML can be considered an 21

unsuitable factor-mimicking portfolio. Its success as an explanatory variable in asset pricing tests might be due to the existence of the non-risk-based components in this portfolio. All in all, the risk-based story is confirmed since there is a significant contribution from both the market factor and the orthogonal portfolio. The latter contribution shows that there are other risk-based sources that are not captured by the market index. However, the large observed book-to-market effect after 1965 has mostly a non-risk-based explanation, which may be considered the result of market inefficiency, market overreaction or sampling error. 22

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Footnotes 1 The optimal orthogonal portfolio has its starting point in Roll (1980) and has been used by MacKinlay (1995) to discuss the ability of the conventional F-test of asset pricing models. 2 The same result is found for the two sub periods, but it is slightly stronger for the second period. The results are available on request. 3 These findings are also evident for the two sub periods, but the weights are more aggressive for the second sub period. The results are available on request. 25

Figure 1. The risk-based-component of the mean excess return according to the different factor models The figure plots the sample mean excess returns and the estimated risk premiums according to three different factor models: CAPM, Fama and French three-factor model and one-factor restricted model, OFR. The model is estimated for six double-sorted portfolios based on data over the entire sample, 1927-2002, as well as two subperiods, 1927-1964 and 1965-2002. 1.4% Entire period 1.2% 1.0% 0.8% 0.6% 0.4% Sample CAPM Three-factor model OFR 0.2% 0.0% SL SM SH BL BM BH 1.6% 1.4% 1.2% 1.0% 0.8% 0.6% 0.4% 0.2% 0.0% First period SL SM SH BL BM BH Sample CAPM Three-factor model OFR 1.2% 1.0% 0.8% 0.6% 0.4% 0.2% 0.0% Second period SL SM SH BL BM BH Sample CAPM Three-factor model OFR 1

Figure 2. The portfolio frontiers estimated by the factor models The figure plots the portfolio frontier of the one-factor restricted model, OFR, and the Fama and French three-factor model. The model is estimated for six double-sorted portfolios based on data over the entire sample, 1927-2002, as well as two subperiods, 1927-1964 and 1965-2002. The figure also plots the value weighted market portfolio and the optimal tangency portfolios. 2.0% Entire period 1.5% 1.0% 0.5% 0.0% OFR Three-factor model Sample Market portfolio -0.5% -1.0% 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 First period 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% -0.5% -1.0% -1.5% 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 OFR Three-factor model Sample Market portfolio 2.0% Second period 1.5% 1.0% 0.5% 0.0% OFR Three-factor model Samole Market portfolio -0.5% -1.0% 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 2

Figure 3. The estimated mispricing based on the OFR model a using twenty-year moving window The figure plots the estimated mispricing (non-risk-based component) from the one-factor restricted model, OFR, and their 95% bootstrapped confidence intervals. The test assets are six double-sorted portfolios. The estimations are based on a twenty-year moving window, which are updated in January each year. For each window, the estimate of the mispricing is the deviation of the sample mean excess return from the risk premium given by the OFR model. SL BL 0.8% 0.4% 0.4% 0.0% 0.2% -0.4% 0.0% -0.8% -1.2% -0.2% -1.6% -0.4% 27-46 32-51 37-56 42-61 47-66 52-71 57-76 62-81 67-86 72-91 77-96 82-01 SM 0.8% 0.4% 0.0% -0.4% -0.8% 27-46 32-51 37-56 42-61 47-66 52-71 57-76 62-81 67-86 72-91 77-96 82-01 SH 1.2% 0.8% 0.4% 0.0% -0.4% -0.8% -1.2% 27-46 32-51 37-56 42-61 47-66 52-71 57-76 62-81 67-86 72-91 77-96 82-01 27-46 32-51 37-56 42-61 47-66 52-71 57-76 62-81 67-86 72-91 77-96 82-01 BM 0.8% 0.4% 0.0% -0.4% -0.8% 27-46 32-51 37-56 42-61 47-66 52-71 57-76 62-81 67-86 72-91 77-96 82-01 BH 1.2% 0.8% 0.4% 0.0% -0.4% -0.8% -1.2% 27-46 32-51 37-56 42-61 47-66 52-71 57-76 62-81 67-86 72-91 77-96 82-01 3

Figure 4. The relative importance of the orthogonal portfolio The figure plots the squared Sharpe ratio of the orthogonal portfolio divided by the squared Sharpe ratio of the tangency portfolio given by the one-factor restricted model, OFR. This ratio is a measure of the relative importance of the optimal orthogonal portfolio. The OFR model is estimated for the six double-sorted portfolios. All the estimates are based on a twenty-year moving window. The estimation is updated in January each year. Ratio 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 27-46 30-49 33-52 36-55 39-58 42-61 45-64 48-67 51-70 54-73 57-76 60-79 63-82 66-85 69-88 72-91 75-94 78-97 81-00 Date 4

Figure 5. The risk premium due to the optimal orthogonal portfolio The figure plots the component of the risk premiums due to the optimal orthogonal portfolio estimated with a one-factor restricted model, OFR. The risk premium due to the orthogonal portfolio is equal to the estimated alpha from the OFR model. The model is estimated for six double-sorted portfolios based on data over the entire sample, 1927-2002. The notation L, M and H is for low, medium and high book-to-market ratio respectively, while S and B are used for small and big firms respectively. 0,30% 0,25% 0,20% 0,15% 0,10% 0,05% 0,00% -0,05% -0,10% SL SM SH BL BM BH 5

Figure 6. Time variation in the risk premium due to the optimal orthogonal portfolio over time The figure plots the component of the risk premiums due to the optimal orthogonal portfolio estimated with a one-factor restricted model, OFR. The risk premium due to the orthogonal portfolio is equal to the estimated alpha from the OFR model. The model is estimated for six double-sorted portfolios based on twenty-year overlapping windows. The estimation is updated in January each year. The notation L, M and H is for low, medium and high book-tomarket ratio respectively, while S and B are used for small and big firms respectively. 0.8% 0.6% 0.4% 0.2% SL BL 0.0% -0.2% 27-46 32-51 37-56 42-61 47-66 52-71 57-76 62-81 67-86 72-91 77-96 82-01 0.8% 0.6% 0.4% 0.2% SM BM 0.0% -0.2% 27-46 32-51 37-56 42-61 47-66 52-71 57-76 62-81 67-86 72-91 77-96 82-01 0.8% 0.6% 0.4% 0.2% SH BH 0.0% -0.2% 27-46 32-51 37-56 42-61 47-66 52-71 57-76 62-81 67-86 72-91 77-96 82-01 6

Figure 7. The optimal orthogonal portfolio weights The figure illustrates the optimal orthogonal portfolio weights given by the one-factor restricted model, OFR, for six double-sorted portfolios. The result is presented for estimates based on the entire sample, 1927-2002. The notation L, M and H is for low, medium and high book-to-market ratio respectively, while S and B are used for small and big firms respectively. 400% 300% 200% 100% 0% -100% -200% -300% -400% Optimal orthogonal portfolio weights SL SM SH BL BM BH 7

Figure 8. The components of the mean return of the Fama and French s factor mimicking portfolios The figure plots the components of the sample mean excess returns of the Fama and French factor mimicking portfolios, SMB and HML, and their 95% bootstrap confidence interval estimated by the one-factor restricted model, OFR, over the entire sample, 1927-2002, as well as two subperiods, 1927-1964 and 1965-200. The figure divides the mean excess returns into two components: the risk-based component, i.e. the part that is related to the market factor and all the latent risk factors represented by the optimal orthogonal portfolio, and the remaining part is the non-risk-based component of the mean excess return. Entire period 0.8% SMB 0.8% HML 0.6% 0.6% 0.4% 0.4% 0.2% 0.2% 0.0% 0.0% -0.2% -0.2% -0.4% Sample Risk-based Non-risk-based -0.4% Sample Risk-based Non-risk-based First period 0,8% SMB 0,8% HML 0,6% 0,6% 0,4% 0,4% 0,2% 0,2% 0,0% 0,0% -0,2% -0,2% -0,4% Sample Risk-based Non-risk-based -0,4% Sample Risk-based Non-risk-based Second period 0.8% SMB 0.8% HML 0.6% 0.6% 0.4% 0.4% 0.2% 0.2% 0.0% 0.0% -0.2% -0.2% -0.4% Sample Risk-based Non-risk-based -0.4% Sample Risk-based Non-risk-based 8