Coskeweness Risk and the Pricing of Small
Stocks
Student: Alexandru Robu
Student number: 10621628
Programme: BSc Economics and Business (
BSc ECB)
Track: Finance and Organization
Supervisor: Dr Liang Zou
Date: 02.02.2016
Statement of Originality
This document is written by Student Alexandru Robu who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text
and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Abstract
This paper investigates whether augmenting the Fama and French (2015) five-factor model with a conditional coskewness factor would alleviate the asset-pricing problems of small stocks. The analysis focuses on pricing US stocks during the period July 1968 – December 2013. During the 20 years following the first analysis of coskewness, a reversal of the coskewness premium is observed and thus different dynamics emerge. One of the proposed models of the paper proves significantly superior to the five-factor model as an asset pricing framework. However, in terms of pricing the troublesome small-stock portfolios the improvement is only minimal.
1. Introduction
Ever since William Sharpe introduced his capital asset pricing model in 1964, the concept of considering risk factors for explaining returns has received intense scrutiny. The CAPM suggests that the beta, the sensitivity of returns to market fluctuations, is the only relevant factor that determines variations in average stock returns, the rest being diversifiable. In other words, in the spirit of Markowitz (1952), the only two moments necessary to determine the pricing
relationship are the mean and variance.
As several subsequent studies have concluded, the cross-sectional variation of average returns cannot be solely explained through the market beta alone. Extensive research has been conducted on this topic. Fama and French (FF, 1992, 1993, 1995) underline the importance of size (market equity) and book-to-market (B/M) ratio in explaining a large part of the cross-section of average stock returns. They posit that provided rational pricing, systematic differences in returns must be due to differences in risk (FF, 1995). Moreover, their work in 1993 confirms that by constructing portfolios that mimic the risk of the above-mentioned risk factors, they are able to significantly improve the traditional CAPM introduced by Sharpe (1964) and Lintner (1965). An important criticism of Fama and French’s work is that the size and value risk factors are regarded as “anomalies” (Hung et al., 2004) which implies the sheer lack of theoretical motivation behind otherwise persistent effects.
In 2015, Fama and French extended their earlier work by adding two other factors to their three-factor model, as they claim there is substantial evidence that profitability and investment add value to the descriptive property of B/M for the cross-section of stock returns. The
motivation behind this particular improvement is the observation that B/M is a “noisy proxy for expected returns” (p. 2, 2015) as the firm’s equity value is also moved by forecasts of earnings and investment. After identifying suitable proxies for expected earnings and investment, Fama and French (2015) proceed to attach them to their three-factor model in the form of an
investment (CMA) and profitability (RMW) factor.
RMW represents the difference between the returns on diversified portfolios of stocks with robust and weak profitability whereas CMA is the difference in returns on diversified portfolios of the stocks of low and high investment firms to which FF refer as conservative and aggressive respectively. They find that in all of their tested portfolios the five-factor model outperforms its three-factor counterpart. However the GRS statistic also easily rejects all the models considered suggesting the models are only incomplete representations of expected returns. They conclude that portfolios of small stocks with negative exposures to RMW and CMA are the biggest asset pricing problem in four of the six sets of portfolios examined. They express this observation by stating that the troublesome portfolios behave like small stocks that invest a lot despite low profitability. Fama and French further note that a behavioral explanation for the anomalously low average returns of these stocks faces the challenge that unexplained average returns of big stocks with similar characteristics are positive. In ending they note, once again, that most serious problems of asset pricing models are related to small stocks.
Since a behavioral explanation is unsuitable, we must shift the focus of our analysis to other areas of asset pricing theory in search of a sound explanation of the anomalous returns of small stocks. Harvey and Siddique (HS, 2000) state that the motivation behind developing their conditional coskewness-based, Harvey-Siddique, model was that certain empirical shortcomings of asset pricing models have originated from specific groups of securities such as the microcaps (smallest Size quintile) that have also appeared problematic in the FF (2015). Since such assets are also some of the ones with the most skewed returns, it directs us towards investigating whether augmenting the five factor model so that skewness is priced in would help to better explain asset pricing problems of small stocks.
significant improvement is observed, than this reinforces the idea that the new factors added by the new FF five-factor model already embody the effect of skewness. However, if the model will provide any significant improvement, this should prompt further investigation into the
underlying causes. The dynamic of the new FF factors (RMW and CMA) and the coskewness factor should come under thorough scrutiny.
The rest of the paper is structured as follows: the next section will provide the theoretical background necessary to understand the following developments, with a subsection specifically dedicated to understanding Fama and French (2015). Section 3 will naturally follow to present the observations and anomalies that were noted by the authors. Section 4 will describe the dataset and the methodology employed, followed by a thorough analysis of the results in the following section. Section 6 will focus on a comprehensive discussion of the results whereas the last section will add our concluding remarks suggestions for further research.
2. Theoretical framework
This section will provide the necessary information in order to properly understand the concepts that represent the building blocks of asset pricing theory, as well as describe reasoning behind asset pricing principles employed. It will focus mainly on the motivation and process of developing a risk-factor based model as well as the rationale behind the inclusion of higher co-moments.
2.1 Evolution of higher-moment asset pricing
The classical Capital Asset Pricing Model (CAPM) developed by Sharpe (1964) and Lintner (1965) has been subjected to heavy criticism regarding two main aspects: the soundness of the normality assumption in regard to the distribution of asset returns and the exclusion of other risk factors such as size and book-to-market which may explain certain risks unaccounted for by the market portfolio (Misirli & Alper, 2009).
At the base of the distributional-characteristics criticism stand the large amounts of empirical evidence which suggest that, in general, assets display significant skewness in their returns as well leptokurtosis (Bekaert et al., 1998 and Chung et al., 2006). Consequently, the standard mean-variance framework of the CAPM is deemed incomplete, and a framework which accounts for higher moments is required.
also be priced in, in other words that the mean-variane efficiency proposition is incomplete and that investors are also concerned about the skewness and kurtosis of their portfolios (Hung et al., 2004). The paper of Kraus and Litzenberger (1976) thus introduces an extended version of the CAPM which includes the third moment, represented by the systematic skewness factor. This was later augmented by Fang and Lai (1997) to include the co-kurtosis. The resulting model is usually referred to as CAPMC (CAPM with higher order co-moments), which builds on the underlying assumption that there exist two systematic risk factors beside beta which cannot be reduced through diversification and must demand a premium.
Heaney et al. (2011) note that it is relatively common that throughout literature, co-moments are used instead of unconditional co-moments to reflect the concept that in equilibrium investors must be only compensated for non-diversifiable variance, skewness or kurtosis (Ingersoll, 1975). Moreover, with respect to skewness, the direction of the scientific discourse seems to suggest that conditional coskewness exhibits superior performance in explaining the cross-section of returns in comparison to its unconditional counterpart [see Harvey and Siddique (1999, 2000)].
2.2 Harvey-Siddique (2000) – the standard in terms of analyzing skewness in asset pricing Building on earlier literature, particularly on the extended model of Kraus and
Litzenberger, and aware of the criticism that surrounded the capital asset pricing model, Harvey and Siddique (2000) embarked to develop a zero-cost portfolio based on the contribution to portfolio skewness of an asset’s individual skewness, or coskewness.
For the intuition behind their analysis they reference the work of Merton (1982), which explains why unconditional returns cannot be characterized by mean and variance alone. Consequently, ceteris paribus, investors should prefer portfolios that are skewed to the right, consistent with Arrow-Pratt notion of risk aversion. Hence more left-skewed assets are less desirable and should require a higher expected return.
Some of the empirical shortcomings of the CAPM have stemmed from failures in explaining the returns of specific securities or groups of securities such as the smallest
capitalizations or returns of specific strategies such as those based on momentum. These assets are also the ones with the most skewed returns. Skewness may be important in investment decisions as it induces asymmetries in ex post (realized) returns. Two factors may induce such
asymmetries: limited liability – option like asymmetries (see, e.g., Black, 1972; Christie, 1982) and agency problems as discussed in Brennan (1993). A manager that has a call option with respect to his investment strategy may prefer portfolios with high positive skewness.
As noted by Harvey and Siddique (2000), as well as Kostakis (2007), the Harvey Siddique asset pricing framework presents two advantageous features when compared to the Kraus Litzenberger version (KLPM). To begin with, the standardized coskewness measure is constructed using residuals, which makes it independent by construction of the market return. Moreover, is similar to the CAPM beta, in that it is unit free and analogous to a factor loading.
Harvey and Siddique (2000) conclude their paper by reinforcing the idea that skewness can explain a significant portion of the variation of returns. Furthermore, they suggest that the additional factors used besides the market, specifically SMB and HML, may prove to capture the same economic risks that underlie conditional coskewness. This is a crucial observation for our analysis, as by introducing the new Fama and French factors (RMW and CMA), the probability that the effect of coskewness is captured by the other factors also increases. This could represent the cause of coskewness no longer entering the asset pricing equation.
2.3 Fama and French
This section provides an overview of the work of Fama and French in regard to the development of their asset pricing framework. It begins by describing the intuition and thought process that led to the development of each of the factors, followed by their observations, the problems encountered and how they relate to this paper. We wish to stress that since this paper was conceived to specifically address some of the anomalies that persisted in Fama and French’s work, a proper understanding of their rationale is vital.
2.3.1 The three-factor model
The Fama French three-factor model has been developed over a series of papers (1992, 1993, 1996), the final version essentially asserting that the size and value of a firm represent additional risk factors which together with the market risk premium can be employed in explaining the cross-sectional variation of stock returns.
The fact that lower capitalization firms generate higher average returns than their large capitalization counterparts was one of the first empirical irregularities documented in financial
literature (Banz, 1981). Roll (1981, 1983, as cited in Durand et al., 2011) claims that the premium warranted by small capitalized firms stems from difficulties in estimating risk and return. Campbell et al. (2008) conclude that the difference in default risk is what transpires through the size and value effects. On the behavioral front, Kumar and Lee (2003, as cited in Durand et al. (2011)) posit that the size effect may be caused by fluctuations in investor
sentiment. An interesting observation is made by Sohn (2009) who suggests that factors such as the SMB and HML must be priced in since they convey important information with regard to future volatility.
Furthermore Fama and French (1993) introduce the HML factor, which accounts for the value-growth premium. In their work they consider value stocks those which have a higher ratio of book value of equity to market capitalization, whereas those with a lower ratio are considered growth stocks. From an economic perspective, the HML premium is believed to stem from the increased likelihood of bankruptcy of small firms (Vassalou and Xing, 2004; Campbell et al., 2008) to which Fama and French (1992) refer as the distress premium.
Fama and French (1993) note that “variables that have no special standing in asset pricing theory show reliable power to explain the cross section of average returns” acknowledging the somewhat arbitrary capacity of their factors. This has long represented a major point of criticism of their work.
2.3.2 The five-factor model
In 2015 Fama and French expanded their model with two new factors which they claim to significantly improve the explanatory power of their model. In their paper, they argue there is substantial evidence that profitability and investment add value to the descriptive property of B/M for the cross-section of stock returns.
For the intuition behind these factors they use the dividend discount model, in an attempt to avoid further criticism regarding the arbitrary choice of their factors. From analyzing the model they observe that provided pricing rationality, the future dividends of a stock with lower price must have higher risk, if the expected dividends are equal [see Fama and French (2015) for a detailed derivation]. Their conclusion is that B/M is a “noisy proxy for expected returns” (p. 2, 2015) because the firm’s equity value is also moved by forecasts of earnings and investment.
As mentioned before, RMW represents the difference between the returns on diversified portfolios of stocks with robust and weak profitability whereas CMA is the difference in returns
on diversified portfolios of the stocks of low and high investment firms to which FF (2015) refer as conservative and aggressive respectively.
To evaluate the relative performance of their models, Fama and French (2015) rely primarily on the GRS statistic developed by Gibbons, Ross and Shanken (1989). Their model-evaluation process will be analyzed in further detail in the methodology section. By rejecting the models, the GRS informs that all their models are incomplete representations of expected
returns. FF (2015, p.10) state that the overriding purpose must be the identification of the best, be that imperfect, “story for average returns on portfolios formed in different ways” and thus we should be interested in relative reductions in the GRS statistic. Despite universal rejection by the GRS test, the five-factor model performs well. The major exception is the earlier-mentioned portfolio formed by small stocks with negative exposures to RMW and CMA.
3. Observations and Anomalies that still persist in the Five Factor model
In this section a closer look is taken at some of the notes and anomalies observed by Fama and French throughout their work, however restricting the discussion to those that have persisted until their 2015 paper. The reason behind this endeavor is to establish the shortcomings of the Fama and French five-factor model in order to be able to better quantify the change prompted by the inclusion of the coskewness factor.
First of all, FF (1995) note that high B/M (value) stocks tend to have low profitability and investment, whereas low B/M (growth) stocks tend to be profitable and invest more aggressively (especially large low B/Ms). To disentangle their effects FF (2015) use various joint sorts on Size, B/M, OP and Inv. To avoid poorly diversified portfolios a 3x3x3x3 sort is avoided, settling for a 2x4x4 sort (32 portfolios) using distinct NYSE breakpoints for small and big stocks in the sorts on B/M, OP and Inv.
Secondly, they observe that average return usually falls from small stocks to big stocks (see Table 1), otherwise referred to as the size effect. Here it can already be observed that the low B/M (extreme growth) stocks is the only exception, moreover, the glaring outlier seems to be the strikingly low average return of the microcap portfolio (whereas for the other four there is no obvious relation between size and average return).
The relation between average return and B/M (the value effect) is consistent as can be seen from Table 1. For every size quintile, average return increases with B/M, the effect being the strongest in the smallest stocks quintile as is suggested by previous research.
Table 1 Average monthly percent returns for 25 portfolios formed on Size and B/M; July 1968–December 2013.
Size-B/M Low 2 3 4 High
Small 0.05 0.68 0.73 0.90 1.01
2 0.37 0.65 0.82 0.87 0.93
3 0.43 0.72 0.73 0.79 1.01
4 0.57 0.55 0.67 0.76 0.80
Big 0.42 0.56 0.48 0.54 0.62
For the 25 value-weighted Size-B/M sorted portfolios they note that as in FF (1993, 2012) the portfolios of extreme growth stocks are an issue as portfolios of small extreme growth stocks produce negative three-factor intercepts and large extreme growth stocks produce positive intercepts. Fama and French (2015, p. 13) note that “by itself, the three-factor intercept for this portfolio (-0.49% per month (t=-5.18)) is sufficient to reject the three-factor model as a
description of expected returns on the 25 Size-B/M portfolios”. The five-factor reduces these problems, since the intercept for extreme growth microcaps rises to -0.29 and the intercepts of three other extreme growth portfolios move towards zero. Nevertheless, the pattern in the extreme growth intercepts survives in the five factor model. The RMW and CMA slopes say the portfolio is dominated by microcaps whose returns behave like those of unprofitable firms that grow rapidly. Furhtermore, for the 25 Size-Inv sorted portfolios there are quite a few interesting observations. The average return in the lowest investment quintile is much higher than that of the highest quintile. Moreover, the highest investment quintile also does not present the expected size effect and the microcap portfolio in this quintile has the lowest average excess return in the matrix (0.35% per month). FF show, through their five factor regression, that the stocks in this portfolio are similar to the microcaps in the lowest B/M quintile, specifically strong negative RMW and CMA slopes. They go one step further to point out that the low average returns on these portfolios are lethal for the five factor model.
An interesting observation is that the five factor model fails to improve the description of average returns from the four-factor model that excludes HML. This is mainly because the average HML return is captured by the exposures of HML to the other factors in the model. In the spirit of Huberman and Kandel (1987), adding HML does not improve the mean-variance-efficient tangency portfolio, thus HML becomes redundant. An orthogonalized version of the factor is proposed (HMLO) which will allow one to adequately analyze portfolio tilts towards the
factor premiums.
In ending, FF (2015) mention that augmenting the model with momentum and liquidity factors have yielded regression slopes close to zero and so produce trivial changes in model performance.
4. Data and Methodology
This section will provide an overview of the methodological approaches employed in this paper. As a detailed evaluation of estimation methods is beyond the scope of this paper, we restrict our discussion to the methods used by the authors of our papers of interest.
In order to obtain the coskewness factor we follow Harvey and Siddique (2000). For each individual stock we must first compute the direct measure of coskewness, defined as:
Where, , and is the market return minus its mean value. It must also be noted that the direct measure is calculated using exclusively excess returns, for which the risk free rate is taken as the 1 month Treasury-Bill return, as retracted from Ken French’s database. Consequently represents the contribution of a particular stock to the
coskewness of a broader portfolio. A negative value of this direct measure would thus imply that the asset decreases the skewness of the market portfolio and thus it should have a higher
expected return [see Harvey and Siddique (2000) for a stochastic discount factor (SDF) derivation].
To compute this factor, we use monthly U.S equity returns from the CRSP database. We extract all the stocks listed on the NYSE, AMEX, and NASDAQ between July 1963 and
December 2013 (the period is chosen identically with that analyzed by Fama and French (2015) in order to provide adequate comparison values). As the market portfolio we use the CRSP value-weighted return index. For the Fama and French five factors, as well as the risk free-rate, we use Ken French’s database to extract the monthly returns for the period July 1963 –
December 2013.
Following Harvey and Siddique (2000) we proceed to form value-weighted portfolios that capture the effect of coskewness. First of all, we exclude stocks that have less than 61 consecutive monthly returns. In terms of missing values, three or less months of consecutive
returns are accepted, anything more being strictly excluded. Using 60 months of returns, we compute the direct coskewness measure for each of the stocks contained in our dataset. We proceed to rank them based on their past coskewness values and then divide them using the same cut points used by HS (2000). Accordingly, three value weighted portfolios are formed: 30 percent with the most negative coskewness ( ), the middle 40 percent ( ), and the most positive 30 percent ( ). The 61st month value-weighted returns on these portfolios are then used to proxy for the systematic skewness. Aside from the individual returns on these portfolios, a hedge portfolio is also formed by computing the spread between the returns on the and
portfolios, we will also call this SKS. In consequence, the coskewness for any risky assets in our analysis will be represented by the betas with these factors.
The virtue of this particular method stems from its conditional nature as well as from the functional form of its factors. The conditionality argument is related to the intuition behind the concept itself. The overriding idea is that a risk-averse investor would have a positive preference regarding higher values of coskewness of its investments. Thus, as noted by Harvey and
Siddique (2000), nonincreasing absolute risk aversion (an essential property of the risk-averse individual according to Arrow (1964)) implies that in a portfolio, increases in total skewness are preferred. Consequently, an investor would be better off using historical data to determine the coskewness of its potential investments and act according to its preferences. By using an
estimation period of 60 months, it is assumed that the investor adjusts its portfolio monthly using the last 60 months of returns to determine the values of coskewness. The form of the HS
conditional coskewness factors is also advantageous for analysis purposes. In their paper Harvey and Siddique demonstrate that the hedge portfolio is analogous to the factor loading on the SMB portfolio of the Fama-French model. Therefore it is fair to consider an asset pricing model which is a combination of the Fama-French multifactor models combined with a non-linear component derived from the third moment.
Now that we established the composition of the new model and the reasoning that stands behind it, we must go further and establish the methods which we will employ to determine the relative performance of the model for both addressing the research question, that is examining the effect for the small stocks, as well as evaluating the overall performance of the model when compared to other important asset pricing models.
endeavor, particularly due to the lack of literature on this issue. However, throughout their 2015 paper, Fama and French often mention the pricing shortcomings of their model by purely analyzing the significance of the intercepts when regressing portfolios formed to reflect a specific trait on the three and five factors respectively. The t-values of the intercepts are then compared and thus conclusions are drawn about the performance of their model in pricing that particular asset class. We question the soundness of this approach, however due to lack of literature on this subject, as well as for comparison purposes, we are left with no choice and thus we employ the same method.
In order to test the overall performance of the model, several methods have been
considered. Throughout asset pricing literature the most common method employed is the Fama-MacBeth (1973) two-stage estimation method. In this case a rolling time-series regression of historical returns is performed in order to estimate the betas followed by a cross-sectional regression on these respective betas in order to obtain the risk premia associated with each of these factors. However, this approach has been subjected to heavy criticism as it ignores the dependence across portfolios as well as the impact of autocorrelation and homoskedasticity (see Shanken, 1992; Jagannathan and Wang, 1995). As a response, several other methods have been developed that address some of these issues, however none have been established as the
standard, which results in different scholars employing a variety of methods such as: GMM, FIML, MLE, Shanken’s modified two-pass, GLS etc. Nevertheless, since our comparison of model performance will be largely with that of the Fama French five-factor model, it is reasonable to consider the same measures for model performance as those employed in their 2015 paper. Consequently the main evaluation tool will be the GRS statistic, as developed by Gibbons, Ross and Shanken (1989) which uses an F-statistic to jointly test whether the intercepts, , are different than zero, where F≈(N, T-N-1). Moreover, several other absolute intercept tests proposed by Fama and French will be computed.
Fama and French (2015) present two distinct interpretations of their zero-intercept hypothesis. The first posits that the mean-variance-efficient tangency portfolio combines the risk-free asset, the market portfolio, as well as the SMB, HML, RMW, and CMA. The second, more ambitious, interpretation is that the four variables lead to risk premiums that are not captured by the market factor. According to this view, SMB, HML, RMW, and CMA are not state variable mimicking portfolios, rather similarly to Fama (1996) the factors are just
diversified portfolios that provide different combinations of exposures to the unknown state variables.
The empirical tests proposed examine whether the proposed models help explain average returns on portfolios formed to produce large spreads in Size, B/M, profitability and investment. The prevalence of using sorted portfolios for testing asset pricing models has increased to unparalleled levels to a point that nowadays virtually every paper uses such portfolios to test or at least perform robustness checks. The original motivation for creating these test portfolios was provided by Blume (1970) as a way to reduce the errors-in-variables problem of estimated betas as regressors. The intuition behind this statement concerns the estimation errors that would arise from estimating alphas and betas, which provided a large number of assets, would tend to offset each other and thus allow for more accurate estimates.
5. Analysis
This section will present in detail the analytical process which was followed in order to provide a sensible answer to the research question as well as evaluate the relative performance of the proposed asset pricing models. Furthermore, a subpart will be dedicated to investigating how the new added factors influence some of the anomalies that persisted in the five-factor model of FF (2015).
5.1 Summary statistics and over-time factor dynamics
We begin our analysis with a snapshot of the data. As mentioned above, Harvey and Siddique (2000) found substantial evidence for the contribution of both the SKS factor as well as the negative skewness factor to the explanation of average returns. Their conclusions were also in line with what the data suggested as they preemptively observed there was a substantial premium required for stocks with negative coskewness. Therefore we must first investigate whether this effect has persisted for the next 20 years and to decide whether such expectations remain valid.
As can be observed from Table 2, a lot has changed in the 20 years since Harvey and Siddique’s analysis. Particularly striking is the complete reversal of the excess return on the SKS portfolio for the 6 years following the period analyzed by Harvey and Siddique (2000).
Table 2
Summary statistics for monthly factor percent returns for various periods.
The table shows average monthly returns (Mean), the standard deviations of monthly returns (Std.) and the t-statistics for the average returns (t-value).
SMB HML RMW CMA SKS July 1968 – Dec 1993 Mean 0.367 0.164 0.501 0.200 0.415 1.237 0.728 0.510 Std. 4.70 2.94 2.72 1.58 1.88 5.30 4.46 2.51 t-value 1.36 0.97 3.22 2.21 3.86 4.09 2.86 3.55 July 1968 – Dec 2013 Mean 0.477 0.196 0.383 0.278 0.379 1.310 0.954 0.356 Std. 4.62 3.09 2.98 2.21 2.03 4.86 4.56 2.57 t-value 2.41 1.49 3.00 2.95 4.37 6.30 4.88 3.23 Jan 1994 – Dec 2013 Mean 0.618 0.238 0.232 0.378 0.334 1.403 1.242 0.161 Std. 4.52 3.27 3.28 2.81 2.20 4.25 4.69 2.65 t-value 2.12 1.13 1.10 2.08 2.35 5.11 4.11 0.94 Jan 1994 – Dec 1999 Mean 1.388 -0.408 -0.395 0.163 -0.063 1.876 2.027 -0.150 Std. 4.05 3.08 2.88 1.73 2.14 4.04 4.01 2.35 t-value 2.91 -1.12 -1.16 0.80 -0.25 3.94 4.29 -0.54
For the period1 investigated by Harvey and Siddique, this factor has exhibited a positive mean excess return (0.510, t-value=3.55) whereas for the period that concerns our investigation (1968-2013), the mean excess return on SKS is roughly 30% lower, 0.356 (also t=3.23). Since this represents quite a significant change in the return on the SKS factor, we turn to Table 2 for a more detailed image of the period after 1993. For the period January 1994 – December 2013 we observe a very drastic change as the factor stands 70% lower than for the HS-examined period. Further investigation reveals that for the 6 years following December 1993, the monthly return on the SKS factor is negative on average, -0.150, whereas for the year 1999, the return on SKS is especially extreme standing at -1.20% monthly. The positive return on this hedged portfolio (SKS) represents a significant part of Harvey and Siddique’s argument for why coskewness
1
We note that due to our reluctance to compromise with regard to the length of the estimation period for the direct coskewness measure, the reproduced results are not the exact ones of HS (2000) as we exclude the 5 years for estimation purposes. Nevertheless all the relevant results of their paper remain valid for the period July 1968-December 1993 and thus this period provides us with sound values for comparison.
enters the pricing equation. They claim that investors have a preference for positive skewness and thus in order for them to hold relatively negatively skewed portfolios they would require a premium (which is reflected by a positive return on the SKS hedge portfolio). Due to the fact that the return on the SKS factor is on average negative for a substantial part of our period of concern we are skeptical whether the propositions of HS (2000) still hold for the three-factor model, let-alone for the five-factor one. Regarding the other two skewness factors, and , the same effect is observed. For the initial period, the HS expectations hold and the negative coskewness loaded portfolio requires a higher expected return to account for additional risk. However, for the following six-year period this effect is reversed. The three new factors introduced are very significant, with only the SKS factor losing significance during the reversal periods.
It is important to note that in the incipient stage of this paper, we strongly considered the idea that an effect such as skewness, which has received increased attention in the early 1990s, could have been somehow distorted by the market. This concept is a natural extrapolation from the work of several authors such as Kostakis (2007) which investigate how mutual funds increase their exposure to coskewness risk factors and present the returns as skill-related alpha. We must consider the possibility that these types of abnormal behavior may lead to drastic changes in the coskewness premium.
The reversal of the SKS premium is consistent with the findings of Smith (2007), who concludes all prices of risk are highly time-variable and that the strongest parameter instability is exhibited in models that include Fama and French factors. Moreover he suggests that several lagged variables may have lost predictive power since the 1990s. This observation is important when coupled with the assumptions made by Harvey and Siddique regarding the behavior of investors. According to the assumptions of HS (2000), investors must act rationally and exert their preference using a mean-variance-skeweness efficiency framework. Furthermore, since the skewness factors are conditional, an assumption is made that the estimated skewness values remain constant, specifically that the 60-month estimated values for coskewness will remain the same for the 61st month. The uncertainty of these assumptions combined with abnormal behavior of mutual funds could provide some insight into the reversal process of the SKS premium during the January 1994 – December 1999 period.
A comparison between the return on the SKS factor (Fig A2 of the Appendix) and the market skewness (Fig A1, as retrieved from Smith, 2007) also provide an interesting insight
which was also observed by Smith (2007). The comparison suggests that investors tend to act indifferently towards coskewness when the market is relatively negatively skewed. However when the market is positively skewed investors exhibit a very strong preference for positive skewness. Smith (2007) quantifies this effect, and concludes that investors are willing to sacrifice 7.87% annually for bearing coskewness risk in times of a positively skewed market. Overall, investors seem to be less averse to negative skewness when compared with their preference for positive skewness. This points us to expect that the factor which loads positive coskewness ( ), which for HS (2000) performed poorly, to have a greater effect since for the period after 1994 the market was more negatively skewed.
5.2 Model performance
Before we turn to addressing the main question of our research paper, we must first evaluate the performance of our proposed models relative to its predecessors. In addition to the already established Fama and French three and five-factor models (as well as some interesting variations), this paper will investigate the performance of three six-factor models, each of which will represent an augmentation of the five-factor model with one of the previously described, coskewness based factors: , and SKS.
In terms of dependent variables this paper will focus on weighted portfolios sorted by Size and B/M (we will use 2 x 3 x 3 sort of Fama and French, as there no significant differences observed by previous literature) which has showed one of the smallest improvements in
performance relative to the three-factor model. Moreover, a deciles sort on Size will be used to investigate, Fama and French’s (2015, p.19) claim that “most serious problems of asset pricing models are in small stocks”. Furthermore, for robustness we will also investigate the effects of coskewness on the 2 x 4 x 4 portfolio sort on Size, Operating Profitability and Investment which prompted the idea that the troublesome portfolios are comprised from small stocks with negative exposures to RMW and CMA, that is their returns behave similarly to the return of a firm that invests a lot despite low profitability.
Since this paper uses as the main avenue for comparison the FF (2015) paper, it is essential that in order to present comparable results, the means for comparison also include the ones employed by FF. Evaluating model performance has been a field in which the number of papers that propose different methods of evaluation is only matched by the number of papers dedicated specifically to disproving them. Throughout their work, FF have employed several of
these proposed methods as well as pioneer some methods themselves. Nevertheless, the overwhelming evidence has prompted them to restrict themselves to a few relatively
unsophisticated measures, namely the GRS statistic of Gibbons, Ross and Shanken (1989), a measure of average absolute intercept and a ratio based on this intercept. Consequently,
following FF (2015) we will proceed to judge the relative performance of the proposed models by providing a comparison between these models in terms of the above-mentioned statistics.
As mentioned before the GRS statistic tests the hypothesis that the intercepts of certain portfolios are indistinguishable from zero, hypothesis which stems from the concept that a successful asset pricing model would completely capture expected returns. Since all the factor combinations (including the five-factor model) are rejected by the GRS statistic, FF (2015) provide some mediation by stating that asset pricing models are merely simplified propositions of expected returns and thus we must focus on their relative performance, in the hope of identifying their best, yet imperfect representation.
In addition to the average absolute intercept value, a ratio is also employed to judge the relative performance of the competing models by expressing an estimate of the proportion of the cross-section of expected returns left unexplained by the models. For this measure the absolute intercept value, a measure of dispersion of the intercepts, acts as the numerator, whereas the denominator measures the dispersion of the left-hand-side portfolios’ expected returns. Let be the time-series average excess return on portfolio i, and be the cross-sectional average of across all the portfolios in the particular sort. Then - , and the ratio equals , the average absolute intercept divided by the absolute average of .
There are several interesting observations that transpire from Table 3. First of all, we observe that for the period in question, HML is no longer completely redundant as all our metrics decrease with its inclusion. Secondly, we note that contrary to Harvey and Siddique’s conclusion
does not perform particularly well when it comes to reducing the intercept. This is only partly surprising as we have observed earlier that the premium that negative coskewness required in the past has decreased significantly. Nevertheless, remains still significant for certain types of portfolios on which we will elaborate in the following section. A similar effect is observed for the SKS factor however the performance is not quite as poor as that of which is to be expected (see section 5.1).
Table 3
Summary statistics of tests of asset pricing models for: July 1968- December 2013.
The table shows the factors that augment the and SMB in the regression model (for CAPM or other more simplistic models, see Table A1 of the Appendix).
a. 25 Size-B/M sorted portfolios GRS A| | b. 10 Size sorted portfolios GRS A| | HML 3.47 0.100 0.60 HML 2.28 0.041 0.84 HML RMW 3.07 0.095 0.57 HML RMW 1.50 0.031 0.63 RMW CMA 2.75 0.109 0.65 RMW CMA 1.20 0.031 0.65 HML RMW CMA 2.71 0.096 0.57 HML RMW CMA 1.17 0.033 0.67 HML RMW CMA 3.47 0.114 0.68 HML RMW CMA 1.90 0.045 0.93 HML RMW CMA SKS 2.99 0.099 0.59 HML RMW CMA SKS 1.37 0.034 0.71 HML RMW CMA 1.94 0.091 0.54 HML RMW CMA 0.86 0.031 0.64
The following results however are extremely interesting. The factor in particular shows very significant improvements in the GRS statistic. For the 25 Size-B/M sort (Table 1, a), the five-factor model reduces the GRS statistic by 22% when compared to its three-factor predecessor. However, the factor further reduces the GRS statistic by 28.4% singlehandedly (from 2.71 to 1.94), and for the first time the GRS statistic no longer rejects the model with absolute certainty. In the Appendix (Table A1) we present the GRS value for a Fama French three-factor model augmented only with the factor and the resulting improvement (33.4%) is very significant, in fact 1.5 times larger than that of RMW and CMA taken together. Similarly for the 10 portfolios sorted on market value alone (Panel b of Table 3), we can observe a
reduction of the GRS statistic of 26.5% (from 1.17 to 0.86) in which case the p-value (which we do not present to save space) is over 0.500 indicating that the model cannot be rejected at any reasonable significance level. The other metrics also indicate a reduction of unexplained cross-sectional variation however not as drastic.
To conclude this section, we concur that the results of the relative performance of the one of our three proposed six-factor models are very promising as they exhibit improved
performance by all the metrics considered by Fama and French (2015) when evaluating their own five-factor model. Nevertheless, further investigation into these effects is necessary and thus in the following sections we will attempt to break down the results in order to obtain a clearer picture of the situation.
5.3 25 Size-B/M portfolios
This section will provide a more in-depth look at how the new factors enter the asset pricing equation, specifically we will take a closer look at the coefficients for both the intercepts, in order to determine whether the troublesome portfolios persist, as well as the other relevant coefficients, in order to get a better understanding of the kinds of portfolios that best respond to differences in coskewness. We begin by presenting the results of Fama and French to provide a benchmark. Since our period only starts in July 1968 the values are understandably different, however, we provide the exact Fama and French results in Table A2 of the Appendix.
The regression equation presented by Fama and French (2015) is the following:
To the five factor regression we proceed to add the coskewness based factors one at a time in order to create three new six-factor regressions:
From analyzing the intercepts and the values as well as their respective t-values (see Panel B of Table 4) we may observe that the factor that represents the returns of the ex-ante formed, most positive coskewness loaded portfolio, has the strongest effect when a relatively “extreme” portfolio is at hand. By extreme we refer to it either being part of the smallest or highest capitalization quintile as well as lowest or highest book-to-market quintile. The results seem to point that should be included in the asset pricing formula if the purpose is to better represent small capitalized companies, as for 2 of the 5 quintiles the coefficients for are highly significant.
Comparing the intercepts of the six-factor model with the five-factor model of Fama and French (Panel A of Table 4) we can observe that for the smallest size quintile, the absolute value of 4 of the 5 t-values corresponding to the regression intercepts have decreased. Abiding by the reasoning of Fama and French (2015), such an effect is indicative of the better performance of the six-factor model in terms of pricing small stocks.
Table 4
Panel A: Five-factor intercepts: (1968-2013)
a t(a)
Size/BM Low 2 3 4 High Low 2 3 4 High
Small -0.34 0.11 0.01 0.11 0.10 -3.64 1.65 0.18 1.81 1.52 2 -0.10 -0.08 0.01 0.01 -0.04 -1.50 -1.40 0.14 0.13 -0.62 3 0.06 -0.01 -0.08 -0.06 0.05 0.93 -0.17 -1.12 -0.88 0.65 4 0.20 -0.23 -0.15 0.00 -0.11 2.93 -3.09 -1.94 0.00 -1.27 Big 0.09 -0.06 -0.11 -0.17 -0.08 1.79 -1.03 -1.49 -2.56 -0.79
Panel B: Six-factor coefficients: (1968-2013)
a t(a)
Size/BM Low 2 3 4 High Low 2 3 4 High
Small -0.33 0.08 -0.06 0.04 0.11 -3.20 1.10 -0.97 0.57 1.49 2 -0.05 -0.09 -0.01 0.01 -0.08 -0.67 -1.48 -0.13 0.10 -1.16 3 0.07 -0.03 -0.11 -0.06 -0.05 0.95 -0.34 -1.43 -0.85 -0.59 4 0.10 -0.23 -0.18 -0.04 -0.17 1.43 -2.86 -2.16 -0.51 -1.74 Big -0.02 -0.04 -0.01 -0.11 -0.19 -0.42 -0.59 -0.17 -1.49 -1.71 t( )
Size/BM Low 2 3 4 High Low 2 3 4 High
Small -0.03 0.07 0.15 0.15 -0.01 -0.41 1.10 2.96 2.90 -0.25 2 -0.11 0.02 0.03 0.00 0.09 -1.88 0.49 0.68 0.06 1.53 3 -0.01 0.03 0.06 0.01 0.22 -0.24 0.48 1.04 0.09 3.13 4 0.20 0.00 0.07 0.09 0.11 3.40 0.00 0.97 1.34 1.47 Big 0.23 -0.05 -0.21 -0.14 0.23 5.52 -0.96 -3.17 -2.31 2.56
Nevertheless, the troublesome microcap portfolio in the lowest B/M quintile remains a problem. Fama and French show that the RMW and CMA negative factor loadings absorb around 40% of its three factor intercept (from -0.49, t=-5.18 to -0.29, t=-3.31). Since our period can only begin in July 1968 due to the estimation of the conditional coskewness measure, we note the five-factor model performs worse in terms of pricing the troublesome portfolio for the period 1968-2013 (as can be seen from Panel A of Table 4, the five factor intercept is almost 15% higher, t=-3.64). The addition of the positive skewness factor only provides a small
improvement in this respect (t=-3.20), this is also indicated by the factor loading of which is also not significant. The results discussed in this section seem to indicate that even though the
six-factor positive coskewness model performs better overall, its contribution to explaining the average returns of small stocks with low B/M is weak at best.
Table 5
Six-factor intercepts: (1968-2013)
a t(a)
Size/BM Low 2 3 4 High Low 2 3 4 High
Small -0.35 0.11 0.01 0.11 0.10 -3.73 1.59 0.22 1.85 1.49 2 -0.11 -0.08 0.01 0.01 -0.04 -1.65 -1.45 0.13 0.10 -0.56 3 0.07 -0.01 -0.08 -0.06 0.06 1.00 -0.16 -1.12 -0.87 0.76 4 0.21 -0.24 -0.15 0.00 -0.11 3.07 -3.24 -1.93 0.06 -1.22 Big 0.10 -0.07 -0.13 -0.19 -0.06 2.11 -1.15 -1.73 -2.82 -0.61 t( )
Size/BM Low 2 3 4 High Low 2 3 4 High
Small 0.05 0.02 -0.02 -0.02 0.01 1.47 0.83 -0.70 -0.65 0.31 2 0.06 0.02 0.00 0.01 -0.02 2.26 0.90 0.15 0.51 -0.75 3 -0.03 0.00 0.00 -0.01 -0.05 -1.09 -0.06 0.02 -0.21 -1.66 4 -0.06 0.06 0.00 -0.03 -0.02 -2.13 2.16 0.03 -0.93 -0.68 Big -0.08 0.04 0.10 0.09 -0.11 -4.50 1.82 3.40 3.60 -2.67
As defined before, SKS represents the excess return on a hedge portfolio, the return of which represents the difference between the returns of the and portfolios.
Before we delve into the results presented in Table 5, we must first recall the work of Harvey and Siddique (2000) which have developed the method of construction for our factors. In their paper several left-hand-side portfolios are tested, some of which being the exact 25 Size-B/M portfolios that concern this section. Nevertheless, even though in the end they have concluded that the factor should be priced in, they did not find sufficient evidence when the 25 Size-B/M sort was concerned. Moreover, HS (2000) provide little comparison value since most of their presented statistics begin in July 1963 even though the method they have employed and developed implies that you forgo the first 60 months in order to estimate the direct conditional coskewness values.
The SKS factor performs worse in terms of pricing the 25 Size-B/M portfolios, however it is a bit more significant when it comes to pricing troublesome microcap portfolio in the lowest B/M quintile nevertheless the coefficient is still not significant. For this particular time period, SKS seems to add some value in the lowest B/M quintile as well as the highest size quintile.
The factor which performs the best in the paper of Harvey and Siddique, does not fare as well for this period and thus we will refrain from providing a detailed analysis as most
coefficients are very close to zero. It is however the most significant from our 3 proposed factors in the regression of the most problematic portfolio, the lowest quintile in both Size and B/M in which has a coefficient of 0.10 and a t-statistic of 1.88. This however does not make up for its otherwise poor performance. To sum up, this section has been dedicated to dissecting the
information that is responsible for the reductions in the GRS statistic exhibited in the previous section. Furthermore, a great deal of attention has been allocated to answering the main research question, that is investigating the particular performance of the 3 new factors in explaining the average returns of the lowest capitalized growth stock portfolio as this has proved problematic in the work of Fama and French (2015). In this regard, the best performing factor has is ,
however none of the factors analyzed had a significant effect at the 5% level. In the following section we go one step further and provide some insight into the relations between factors. 5.4 Factor interactions
The purpose of this section is to establish how the factors in our models relate to each other. As we have already observed, the summary statistics for this period appear rather counterintuitive, especially when the skewness based factors are concerned. Thus, this section will provide a snapshot of how the factors in the model relate to each other in hope that this will aid in understanding the phenomenon that prompted the unexpected returns for both the SKS and the portfolios. First of all we will provide a correlation matrix (Panel A of Table 6) followed by a panel that uses five-factors of the model to explain the average returns of the fifth (Panel B of Table 6). This is employed throughout literature as well as by FF (2015) to determine whether factors should be included in the asset pricing model, or whether they are already captured by other factors. The exact same method has been employed by Fama and French when they concluded that HML is a redundant factor in the five-factor model.
Panel B of Table 6 presents the coefficients and their respective t-values from regressions of certain factors on the other five. The intercepts of these regressions represent average returns unexplained by the other factors and thus if these are significantly different from zero we can safely conclude they add some additional information to the equation which has not already been captured by other factors.
Table 6
Panel A. Correlations between factors: July 1968 – December 2013
SMB HML RMW CMA SKS 1.00 0.27 -0.32 -0.22 -0.40 0.93 0.97 0.04 SMB 0.27 1.00 -0.13 -0.38 -0.09 0.26 0.22 0.10 HML -0.32 -0.13 1.00 0.12 0.71 -0.20 -0.30 0.15 RMW -0.22 -0.38 0.12 1.00 -0.05 -0.10 -0.23 0.21 CMA -0.40 -0.09 0.71 -0.05 1.00 -0.31 -0.35 0.02 0.93 0.26 -0.20 -0.10 -0.31 1.00 0.85 0.38 0.97 0.22 -0.30 -0.23 -0.35 0.85 1.00 -0.16 SKS 0.04 0.10 0.15 0.21 0.02 0.38 -0.16 1.00
Panel B. Using five factors in regressions to explain the returns of the fifth: Jul 1968 – Dec 2013 Intercept SMB HML RMW CMA SKS (1) -0.42 -0.03 -0.10 -0.29 -0.20 0.84 0.89 t-stat -5.94 -1.33 -3.14 -8.80 -4.11 57.80 (2) -0.39 0.10 0.06 0.02 -0.22 0.95 0.94 t-stat -7.82 6.18 2.77 0.65 -6.44 84.48 (3) 0.85 0.23 0.00 -0.42 -0.91 0.13 0.25 t-stat 4.72 3.74 0.02 -4.70 -7.33 1.83 (4) 0.65 1.03 0.08 0.12 0.28 0.08 0.88 t-stat 8.53 57.80 3.11 3.51 7.60 1.52 (5) 0.48 0.98 -0.09 -0.06 -0.04 0.16 0.94 t-stat 9.64 84.48 -5.22 -2.79 -1.86 4.61 (6) SKS 0.17 0.05 0.17 0.19 0.32 -0.08 0.10 t-stat 1.53 1.83 4.47 3.66 6.05 -1.02
Accordingly, from Panel B we observe that the intercepts of two ( of the three conditional coskewness based factors are significantly different than zero and therefore their effects are not completely captured by other factors. Since the are both basically just parts of the market return, it is expected that their correlation coefficient with the market will be rather high; this is also reflected in the R-squared of their respective regressions in Panel B. The SKS factor’s intercept is not significantly different than zero and thus we may assert that over the entire 45 year period (Jul 1968 – Dec 2013) it is mostly captured by the other Fama and French factors. This is not surprising due to the observed changes described in the previous sections which the factor has experienced since the period analyzed by Harvey and Siddique (2000).
The slopes in Panel B of Table 6 may often seem counterintuitive, with a most striking example being the positive intercepts of regressions (5) and (6), as the basic rationale would point towards two values of opposite signs. As striking as these results may be, they are the result of interactions between the factors that would require a very comprehensive analysis into their exact dynamics in order to gain a proper understanding. FF (2015) observe similar
dynamics for their five factors and conclude that multivariate regression slopes are often counterintuitive and fail to line up with univariate characteristics. As an example they offer a regression of the HML factor on the other four factors of their model. They note that the RMW slopes are strongly positive when controlling for other variables, indicating robust profitability for value stocks, even though, unconditionally value stocks tend to exhibit low profitability. Since Fama and French are not suspicious of these dynamics, we will also refrain from further speculation as to their causes. Nevertheless, further research is needed in this area as very valuable insights may be consequently revealed.
Table A3 of the Appendix presents the results of several regressions of the most problematic portfolio sort in FF (2015) with alternating regressors. Aside of a more detailed picture of how the discussed factors enter the pricing equation for this particular stock cluster, the tables also allow the reader to observe how parameters alternate between negative and positive values when other factors are added.
6. Discussion
This section is aimed at providing some mediation between the expectations we had at the onset of this academic endeavor and the observations and results that we have obtained through our analysis. Moreover, since several anomalous effects have been observed especially in relation to the sheer passing of time, we deem it of paramount importance that we provide several plausible explanations for said anomalies, which may also pave the way for further research.
The first interesting effect which we observed was the curious reversal of the premium required by the factor representing a hedge portfolio aimed at capturing the coskewness risk (SKS), as developed by Harvey and Siddique (2000). The return on the factor was negative over the six years following the period analyzed by HS (2000) and stood 70% lower for the rest of the 20 years which our analysis added. This finding was not entirely surprising as several papers have hinted to this being a possibility. Our findings are consistent with those of Smith (2007)
which has established that the price of risk varies greatly over time and that investors exhibit a very strong preference for positive skewness when the market is positively skewed. We posit that this coupled with abnormal “skill-alpha” seeking behavior of mutual funds could be the reason behind the decrease in the SKS premium.
Our proposition was reinforced by our findings when evaluating the relative performance of our three proposed models. The GRS statistic and other model performance metrics seem to indicate that both the SKS and factor have lost a significant portion of their predictive power which is now better represented in the positive coskewness loaded factor, This is consistent with the notion that investors are relatively indifferent to taking on coskewness risk in times negatively skewed market as the late 90s were relatively negatively skewed (see Fig A1).
The proposed model that augments the Fama-French five-factor model with the factor performs significantly better than its five-factor competitor leading to 28.4% decrease in the GRS statistic for the 25 Size-B/M sorted portfolios, with similar improvements for various other portfolio sorts. Furthermore this model is also the only one that is no longer strongly rejected by the GRS statistic for the Size-B/M sort.
In spite of the outstanding overall performance of the asset pricing model, the factor, adds very little explanation power when the troublesome, smallest-capitalized extreme growth portfolio is concerned. When the portfolio is regressed on a sole skewness factor, their effect is highly significant however after adding the Fama and French five factors, and SKS lose their potency. The troublesome portfolio has persistently caused problems for asset pricing frameworks throughout the decades, and our model makes no exception, thus providing a negative answer to the main question of our research.
We conclude this section by addressing several observations that have transpired during some of our robustness checks. For robustness other left-hand-side (LHS) portfolio sorts were analyzed. It is important to note that for different portfolio sorts, the Fama French five-factor model performs much worse with GRS statistics of 5.65 (for 25 Size-variance sorted) or 3.56 (for 10 portfolios sorted on Momentum as described by Carhart (1997)). Moreover a very distressing observation is that for the Industry sorted portfolios about which Lewellen et al (2010) argue are a better way to assess model performance, the GRS statistic is smallest with only the excess market return as the dependent variable, thus making the CAPM the best performing model if a GRS statistic evaluation method is used (see Table A4 of the Appendix).
This only goes to prove that we have still ways to go before developing a proper understanding of asset pricing.
7. Conclusion
The aim of this paper has been the augmentation of the Fama and French (2015) five-factor model with a five-factor that reflects conditional coskewness in order to improve the
performance of the model when the pricing of small stocks is concerned. In order to achieve such a feat we have recreated the three skewness-based factors of Harvey and Siddique (2000) which were later added to the five-factor model. To provide a comprehensive analysis of the models, their relative performance has also been thoroughly evaluated following the latest methodology employed by some of the leading authorities in asset pricing, Eugene Fama and Ken French. Due to some drastic changes in the return behavior of the HS (2000) factors during the 20 years which have passed since their analysis, the expectations regarding the performance of the factors were rather reserved. In the end some interesting results have come to light. Alas the most troublesome portfolio from the Fama and French (2015) paper has persisted as a problem (only marginal improvement) thus rejecting the main hypothesis of our research. Nevertheless, the overall performance of one of the three proposed models has been significantly improved.
The results of this paper build on the work of Fama and French (2015), Harvey and Siddique (2000), Smith (2007) and Kostakis (2007). Furthermore, our analysis has also revealed several aspect that would certainly be worthy of further research. First of all, our paper proposes several explanations for the reversal of SKS premium in the second part of the 1990s however our explanations are only mere hypotheses which should be adequately researched. Secondly, in section 5.4 we have encountered several counterintuitive regression slopes and so we would argue that further investigation into the dynamics of multivariate regression slopes with alternating factors could yield valuable insights for asset pricing.
Appendix
Fig A1
Market skeweness as retrieved from Smith (2007): 1963-1998
Fig A2
Table A1
CAPM and other various model GRS-statistics for two types of portfolio sorts.
a. 25 Size-B/M sorted portfolios GRS b. 10 Size sorted portfolios GRS CAPM 4.22 CAPM 2.09 CAPM + SMB 4.18 CAPM + SMB 2.37 SKS 4.49 SKS 1.26 3-factor + 3.67 3-factor + 2.29 3-factor + SKS 3.70 3-factor + SKS 2.41 3-factor + 2.31 3-factor + 1.97 Table A2
Fama and French (2015) reproduction of results for 25 Size-B/M sorted portfolios. Panel A: Three-factor intercepts: (1963-2013)
a t(a)
Size/BM Low 2 3 4 High Low 2 3 4 High
Small -0.48 0.01 0.02 0.15 0.14 -5.14 0.12 0.37 2.76 2.26 2 -0.17 -0.04 0.12 0.08 -0.02 -2.82 -0.80 2.16 1.49 -0.40 3 -0.06 0.06 0.03 0.05 0.12 -0.98 0.83 0.43 0.82 1.61 4 0.13 -0.10 -0.06 0.07 -0.08 2.14 -1.37 -0.77 1.01 -1.01 Big 0.17 0.03 -0.07 -0.12 -0.18 3.52 0.46 -1.04 -1.90 -1.94
Panel B: Five-factor intercepts: (1963-2013)
a t(a)
Size/BM Low 2 3 4 High Low 2 3 4 High
Small -0.29 0.10 0.01 0.12 0.12 -3.27 1.60 0.14 2.03 1.94 2 -0.12 -0.11 0.04 0.00 -0.04 -1.84 -1.94 0.80 0.06 -0.66 3 0.02 -0.01 -0.06 -0.03 0.04 0.38 -0.20 -0.99 -0.40 0.59 4 0.17 -0.23 -0.14 0.04 -0.10 2.59 -3.22 -1.98 0.65 -1.18 Big 0.11 -0.10 -0.11 -0.15 -0.10 2.48 -1.66 -1.55 -2.42 -0.99
Table A3 Linear regressions of the smallest capitalized, extreme growth portfolio of the 25 Size-B/M sorted
portfolios on various combinations of regressors; July 1968-December 2013.
Int. SMB HML RMW CMA 5F+SKS -0.35 1.03 1.21 -0.44 -0.51 -0.09 0.05 0.94 t -3.73 46.74 37.28 -10.07 -11.00 -1.40 1.47 3F+SKS -0.52 1.08 1.34 -0.48 -0.05 0.92 -5.21 46.56 40.30 -13.61 -1.33 SKS 0.02 0.09 0.00 0.06 0.65 5F+ -0.33 1.07 1.21 -0.43 -0.50 -0.09 -0.03 0.94 -3.20 12.84 37.15 -9.93 -10.96 -1.36 -0.41 -0.57 1.02 1.34 -0.49 0.06 0.92 -5.18 11.51 39.75 -14.05 0.67 -1.24 1.35 0.57 -5.32 27.06 Table A4
GRS statistics of various asset pricing models for 32 Industry sorted portfolios; July 1968-December 2013
GRS Industry CAPM 1.28 Three-factor 2.20 Five-factor 2.41 s- 2.69 sks 2.33 s+ 2.36 3fact s- 2.26
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