Master Thesis
Asset Pricing with Higher Moments, a European Perspective
by Robin IJzerman
MSc. Finance University of Groningen Faculty of Economics and Business
Author: Robin IJzerman Student Number: S1798154 Email: robinijzerman@gmail.com Phone: +31611244400
Date: June 24, 2014
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Asset Pricing with Higher Moments, a European Perspective
Robin IJzerman
ABSTRACT
This paper studies if higher-order co-moments have the ability to explain the variation in cross-sectional index returns from January 2001 to December 2012. Consistent with previous research, results suggest there is no clear relationship between risk and return. I use a conditional pricing model to examine how conclusions of such analysis depend on whether the market return in excess of the risk-free rate is positive or negative. In the conditional setting, I find robust evidence investors price exposures to beta risk. Support that investors care about risk factors beyond beta is found in the turbulent period 2007 to 2009. During this period, where returns were more negative, extreme and negative skewed than in others in the sample, systematic skewness is priced and contributes in explaining cross-sectional differences in asset returns. The systematic kurtosis factor does not appear to be priced.
Keywords: CAPM, Beta, Higher Moments, Risk Premium, Skewness, Kurtosis
JEL Classification: G100, G110, G120, G150
1. Introduction
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As a result, traditional pricing models that lack to incorporate higher moments than variance may appear to be inefficient.
By way of illustration, developed countries often have negative skewed market returns and the recent credit crunch of 2007 can be considered a reminder that fat tails do exist despite the unlikely probability understated by a normal distribution. During this period, stock returns were more negatively skewed and more extreme implicating the difficulties for conventional CAPM in explaining price differences for market risk across countries. In this regard, other systematic (undiversifiable) risk factors beyond beta could increase explanatory power to variation in cross-sectional asset returns. In fact, Hwang and Satchell (1999), Dittmar (2002), Patton (2004) and Hung et al. (2004) show that investors are particularly concerned about downside risk which is a function of systematic skewness and systematic kurtosis.
Recent literature is primary focused on the estimation of systematic risk factors of an individual asset relative to the index of the market in which the asset trades. Harvey and Siddique (1999, 2000), Hung et al. (2004) and Galagedera et al. (2003) find evidence that higher moments than beta are priced by investors in respectively the US, UK and Australian market. For an international investor this may not be the best approach and would be better served with risk estimation relative to an international index (Heston et al. 1999). Since capital markets have become more integrated, barriers for cross-border investing continue to decline and passive investing (index tracking) becomes popular, it will become increasingly important for investors to explain return variation across countries. To this end, it should be of interest for international investors whether higher moments of systematic risk are priced in terms of asset allocation to identify aggressive and defensive countries.
Fletcher (2000) and Tang (2003) find support that beta has a role in explaining cross-sectional differences in country index returns. To date, no study investigated whether in addition to beta also higher moments of systematic risk explain cross-sectional differences in international index returns. This paper contributes to the debate by extending the approach of Pettengill et al. (1995), Fletcher (2000) and Hung et al. (2004) to examine if systematic risk factors are able to explain weekly cross-sectional index returns between January 2001 and January 2014.1 In other words, do risk averse investors price exposures to
systematic variance, systematic skewness and systematic kurtosis risk?
By following the Fama and MacBeth (1973) rolling time series regression methodology, systematic risk factors are estimated of country index excess returns against European market excess returns. These estimates are used as parameters in weekly cross-sectional regressions to estimate their risk premia. Since in reality only realized asset and market returns are available, averaged realized returns are
1 This paper investigates ten European Morgan Stanley Capital International country indices since the
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used to proxy the market expected market returns. Pettengill et al. (1995) and later Isakov (1999) observed that the use of realized returns as a proxy for expected returns produce biased results due to aggregation of negative and positive market excess return periods. To better access the reliability of higher moment factors as measures of risk also an adjusted methodology is used which is conditional on whether the excess market return is positive (up market) or negative (down market).
Adopting the above procedure, I find some support for pricing models that incorporate higher moments of systematic risk. Results show that beta risk is significantly related to returns conditional on the state of the market. Hence, beta is still a useful measure of risk for investors in making optimal investment decisions. Evidence that investors care about higher moments is found in the turbulent period 2007 to 2009. During this period where returns were more negative, more extreme and negatively skewed than in other periods, systematic skewness risk is priced and contributes in explaining cross-sectional differences in index returns.In economic terms this translates to investors require additional compensation for holding assets that are more negatively skewed than the market in down markets periods, whereas investors require less compensations for holding assets that are more positively skewed than the market in up market periods. My findings suggest that pricing models incorporating higher moment risk have better explanatory power than traditional models do.
The remainder of this paper is outlined as follows. Section 2 discusses prior literature, while section 3 sets out the regression methodology used in the tests. Section 4 reports the data selection and descriptive statistics. Section 5 provides the main empirical results. The final section contains concluding comments and offers suggestions for direction in which this research might be extended. Additional test results are presented in the appendices.
2. Literature Review
This chapter will discuss prior literature regarding unconditional and conditional CAPM and higher-order pricing models.
2.1 The Conventional CAPM
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CAPM applies diversification in the MPT theory. The model states that the return on risky assets only depend on security risk, where the equity return is linearly related to the market portfolio (Fama, 1970). An individual investment has both non-systematic risk (asset risk) and systematic risk (market risk). Through diversification CAPM assumes that only market risk affect return and the investor receives no added return for bearing diversifiable risk. The systematic risk thus cannot be eliminated and is called the beta component. This beta coefficient describes the co-movement the returns of the asset with the market as a whole.
The model furthermore predicts all efficient portfolios lie on the same Security Market Line (SML). It provides a contribution to understanding the manner in which capital markets function and describes the equilibrium return on all portfolios as well as individual securities. The more general idea behind the CAPM is that these investors need to be compensated for investing their money in two ways: (a) the time value of cash, represented by the risk-free rate, and (b) risk, represented by the compensation is required for taking additional risk. The CAPM defines all efficient assets lie on the SML:
( ) ( ( ) ) (1)
Where ( ) represents the expected return of a security i, ( ) is the expected return on the market portfolio, denotes the risk-free rate and is a measure of risk for a security i. The formula attempts to describe the relationship between the security’s expected return and the systematic risk it faces for the reason that the beta defines the risk of a security and the term between brackets is considered the market risk premium. The risk premium is defined as the average additional remuneration required by investors over the risk free rate to invest in the risky market portfolio. Therefore, it reflects the judgments by risk averse investors how much risk they perceive is present and what price is attached to risk. Beta is technically defined as the covariance between the individual security return and the return of the market, divided by the market return variance. All individual securities combined together represent the whole market with an average beta of one. As described above, the model suggests that the only risk that is priced by rational investors is systematic risk.
The SML furthermore suggests an overall positive relationship between expected returns and systematic risk. Thus, when the realized risk premium is positive, there should be a positive relationship between risk and return. Over long periods of time high beta assets should on average produce higher returns, but also incur more risk (Black, 1993).
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stocks. In addition, Kraus and Litzenberger (1976) argue that the slope of the model is lower and the intercept is higher than predicted. Moreover, Fama and French (1992) found a flat relationship between return and beta using nearly 50 years of US stock returns data. Some researchers have argued in favor of measuring systematic responsiveness to several macroeconomic variables (Chen, Roll and Ross, 1986). Fama and French (1993, 1996, 1998) suggested using a size and value factor next to the beta factor as a proxy for systematic risk, where Lakonishok and Shapiro (1986) found empirical evidence that asset returns are also affected by measures of unsystematic risk.
Despite inconclusive results and difficulties testing the validity of the CAPM, the model is still widely used in corporations as a tool for financial asset valuation (Graham and Harvey, 2001; Duke University, 2008). However, as prior research indicates, when there is often no relationship between cross-sectional returns and beta observed, continued reliance on beta as a solely measure of risk appears inappropriate.
2.2 Higher Moments of Systematic Risk
The traditional pricing model described in section 2.1 predicts that beta is the only measure of risk which has to be compensated. However, this version of CAPM is only valid under the following two assumptions: (a) investors have a quadratic utility function, and (b) the return distribution is normal. Under a quadratic utility, the third and fourth order derivatives are zero and therefore an investor’s asset or portfolio choice is completely determined in terms of a preferred relation defined over the mean and variance of expected returns. In other words, an increase in the degree of risk aversion is accompanied by a growth in wealth. Conclusions that are based on this quadratic utility function seem to be rather counter intuitive.
Research by Rubinstein (1973) and Scott and Horvath (1980) already established that investors may have non-quadratic utility functions. This implies that not only the variance is sufficient to capture the shape of the return distribution and therefore all moments of the return distribution should be taken into account. According to Scott and Horvath (1980) there is no reason to stop at the first two moments (mean and variance). The mean variance theory of Markowitz stops at the second order and therefore fails to consider the possible effects of higher moments than the second. The observation that some people buy lottery tickets even though these investments have a negative expected rate of return suggests that positive skewness enhances their utility. From this example it becomes intuitive that investors who seek for maximizing their expected utility indeed may care about moments of the wealth distribution in addition to mean and variance.
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evidence that the returns from stocks and international stock indices are non-normal and rather more skewed or have an unexpected higher probability of being more extreme. A return distribution with positive skewness has a longer tail on the higher return side of the curve while a return distribution with negative skewness has a longer tail in the lower return side of the curve. This implies that with a negatively skewed distribution, there is greater downside risk than measured by the standard deviation. Furthermore, a normal return distribution has a kurtosis level of three were a distribution with kurtosis greater than three is considered leptokurtic. A leptokurtic distribution has a fatter tails and a sharper peak compared to the normal distribution and indicates a lower probability than a normally distributed variable of values near the mean and a higher probability than a normally distributed variable of extreme values.
Hung et al. (2004) therefore argue that if the market returns are not normal but instead more skewed or leptokurtic, investors are concerned about the skewness (third moment) and kurtosis (fourth moment) of their asset or portfolio returns. These so called skewness and fat tails are not favored by the risk averse investor since they indicate the chance of overpaying for an asset given its risk and return (Dittmar, 2002). This implies that when investors’ preferences contain portfolio skewness and kurtosis measures, each asset contribution to systematic skewness and systematic kurtosis may determine the asset its relative attractiveness and therefore its required return.2 Harvey and Siddique (2000) state that skewness exists in asset prices and that a pricing model incorporating a skewness factor would help to explain expected returns in assets. Consequently, systematic risk factors beyond beta could explain variation in cross-sectional stock and index returns.
Scott and Horvath (1980), Harvey and Siddique (2000) and Patton (2004) argue that the risk averse investor prefers positive asymmetry and low kurtosis and therefore requires a risk premium for negative skewness. Other things being equal, an investor prefers an asset bringing longer right tails to his portfolio distribution, but dislikes an asset making his portfolio distribution more left tailed. Thus, in the presence of negative asymmetry (negative sign of systematic skewness) in the return distribution of an asset relatively to the market, a risk premium is expected as a required compensation for taking risk. In contrast, when there is presence of positive asymmetry (positive sign of systematic skewness) investors expect a negative required risk premium for systematic skewness. Similarly, an index with positive (negative) systematic skewness reduces (increases) the risk of a portfolio to large absolute market returns, it should yield a lower (higher) expected return in equilibrium. Since a more leptokurtic return distribution adds additional dispersion in returns, its required risk premium is expected to be positive.
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2.3 Higher-Order Pricing Models
In the previous section, although theoretical, several studies highlighted the possible importance of systematic skewness and systematic kurtosis in addition to beta in explaining variation in cross-sectional stock returns. Assuming that investors prefer positive systematic skewness, Kraus and Litzenberger (1976) were the first that found empirical evidence that higher orders of systematic risk are indeed actually priced by investors. Where systematic skewness measures the symmetry of the asset its return distribution relatively to the distribution of the market, systematic kurtosis measures the degree of peak of the asset returns probability distribution in relation to the return distribution of the market. In this context, systematic skewness and kurtosis are considered non-diversifiable measures of skewness and kurtosis and therefore consistent with the assumption of portfolio theory that only systematic risk is relevant to an investor’s decision. As suggested by Kraus and Litzenberger (1976), the third and fourth moment are represented by the following equations (2) and (3).
( ̅̅̅̅)( ̅̅̅̅̅̅) ( ̅̅̅̅̅̅)
(2)
( ̅̅̅̅)( ̅̅̅̅̅̅) ( ̅̅̅̅̅̅)
(3)
Where, and represents respectively the third and fourth systematic co-moment of an individual
index i, while denotes the return of stock i at time t, and represents the market return. 2.3.1 Unconditional Higher-Order Pricing Models
There are basically two approaches for testing whether beta and higher moments are priced and whether it can contribute to explanatory power to the unadjusted CAPM evident in the literature. The first approach involves tests of the unconditional form of higher moment asset pricing models.
Fama and MacBeth (1973) developed a still commonly used three step approach for testing the validity of the model. Firstly, with time series regressions individual asset betas are estimated to form portfolios (high to low beta) according to these estimated betas. Secondly, in the period following, betas of the formed portfolios are estimated. In the first and second stage betas of individual assets and portfolios are estimated by the following time series regression:
( ) (2)
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(3)
To assess the effect of higher moment risk factors, the formulas (2) and (3) can be extended with systematic skewness and systematic kurtosis factors. Similar, the slope coefficients estimated with time series regressions (4) are used as parameters in cross-sectional regressions (5) to estimate their related risk premium.
( ) ( ) ( ) (4) (5)
Where are the estimated slope coefficients, is the intercept term and are risk premiums dedicated by covariance risk, systematic skewness risk and systematic kurtosis risk. In other words, the additional returns required by investors for taking extra systematic risk. The intercept should not be significant different from zero. With this methodology results are combined across different cross-sectional regressions by performing a test on the average estimated risk premia (negative and positive values) unconditional of whether the market went actually up or down.
Results of Kraus and Litzenberger (1976) showed that the inclusion of the third moment to the traditional CAPM led being the intercept value being equal to the risk free rate. Studies (Friend and Westerfield, 1980; Barone-Adesi, 1985; Lim, 1989, Harvey and Siddique, 2000; Smith, 2007) also provided evidence that systematic skewness is priced by investors’ implying that by adding the third moment, the model better explains the variation in cross-sectional returns. Moreover, according to Harvey and Siddique (2000) systematic skewness is economically important and requires a risk premium, on average, of 3.6% per year. The authors measured the third moment systematic risk by performing a cross-sectional regression on portfolio excess return on the square of the market premium. Friend and Westerfield (1980) find that the slope coefficient of systematic skewness is significantly different from zero but also is the intercept implicating that skewness can be priced however it does not explain the total risk premium.
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cube of the market premium to incorporate the fourth moment of systematic risk. Results showed only weak evidence since adjusted R2 did not improve when higher moments were added.
2.3.2 Conditional Higher-Order Pricing Models
Since the unconditional formula (5) often fails to predict positive expected return premia as a function of the market beta, systematic skewness and systematic kurtosis (Hung et al. 2004; Pettengill et al. (1995); Galagedera et al. (2003), the second approach involves tests of the conditional form of higher moment asset pricing models (including beta). In fact, investigating historical data shows that there are fluctuations in actual market realizations and that they produce data points with negative and positive realized market risk premia (Hung et al. 2004). As stated before, many empirical studies simply average risk premia estimations of all cross-sectional periods in the last stage of the Fama and MacBeth (1973) estimation method.
Since in reality only realized asset and market returns are available, Pettengill et al. (1995) argue that this unconditional version of CAPM theoretically models the expected returns while in empirical research realized returns are used as proxies for the expected returns. This implies that results are biased due to aggregation of negative and positive market excess return periods. Since realized returns on the market portfolio often fall below the returns of the risk free asset it can result in a sometimes negative ex post risk premium is these periods.
To access the reliability of both beta and higher moment factors as measures of risk an adjusted methodology is preferred, conditional on whether the excess market return is positive (up market) or negative (down market). To adjust the CAPM such that it incorporates information relating to the sign of the market realization in a specific period, Pettengill et al. (1995) refined the cross-sectional regression. To this end, a dummy is used that is one for positive and zero for negative excess market returns in the specific period to allow for the fact that an ex-post realized risk premium can be negative. According to this methodology, the conditional relationship between beta and return is cross-sectional estimated as:
( ) (6)
Where, D = 1, if ( ) 0 (i.e., when market excess returns are positive), and D = 0, if ( ) (i.e., when market excess returns are negative), and represent the estimated risk premium
for beta depending on the sign of the market excess return.
Similarly, to transform the unconditional higher moment CAPM (5) into a conditional model, the cross-sectional regression formula is adjusted in:
11 I refer to equation (7) as the four-moment conditional model.
Using this approach, Pettengill et al. (1995) provide evidence that for the period 1936 to 1990 there is strong support for positive and negative beta relationship in US stock returns conditional on the split of the sample into up and down market months. Also Crombez and Vander Vennet (2000) confirmed that the conditional CAPM is indeed able to explain Belgium cross-sectional stock returns whereas the unconditional model does not. Campbell (1996) and Hung et al. (2004) indicate that the conditional model improves the ability of explaining the relationship between beta and cross-sectional domestic asset returns.
Support for a conditional relation between the third moment and domestic stock returns is found by Harvey and Siddique (1999, 2000) for the US market, Hung et al. (2004) for the UK market and Galagedera et al. (2003) for the Australian market. Dittmar (2002) showed that the risk averse investor prefers positive skewness and normal kurtosis in returns. Evidence is also found in emerging markets by Hwang and Satchell (1999), who argue by estimation with generalized method of moments (GMM) that higher moments better explain prices than the conventional mean-variance CAPM. However, it cannot compensate for the fundamental non-stationary of emerging market returns. Patton (2004) showed that when knowledge of higher moments and asymmetric dependences is taken into account it can lead to gains that are economically and statistically significant in cases with no short-sales constraints.
As described above, tests of the higher order CAPM were generally conducted in a single country on domestic stocks against a local market index. For an international investor this may not be the best approach and he would be better served with systematic risk estimation relative to international index (Heston et al. 1999). Since capital markets have become more integrated and barriers for cross-border investing continue to decline it is becoming increasingly important for investors to explain return variation across countries. In this context, expected returns on assets or country indices are proportional to the expected return on the world market portfolio instead to a local market index.
However, international pricing models require additional assumptions to be valid (Solnik, 1974; Adler and Dumas, 1983). It is only appropriate when capital markets around the world are integrated and purchasing power parity holds, providing exchange rates are constant. Yet, if there are deviations from purchasing power parity, investors want to hedge against foreign exchange risk. Consequently, exchange risk should be priced and exchange risk factors must then be included in the model. With respect to these assumptions, the CAPM can be written similar as equation (4) where Rm represents the world market portfolio and βi the beta of the local market index i relative to the world market index.
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relationship between beta and return in down market months. This study is followed up by Tang (2003) by testing for more robust results. The paper provides evidence for the conditional relationship for both monthly and weekly returns for two different proxies of the world portfolio. Therefore, both Fletcher (2000) and Tang (2003) indicate that beta in a useful tool for the international investor in explaining cross-sectional differences in country index returns and therefore making optimal investment decisions.
While a number of the papers discussed have tested higher order pricing models for domestic data (especially in the US), there has been no empirical study conducted regarding the explanation of international index returns. From previous studies we can learn there is evidence that in addition to beta higher moments can be priced. However, the strength of the results seems to depend on the specific market investigated and often only the third moment adds additional explanation in the cross-sectional return relationship between stocks. Moreover, to date there is no study of the international CAPM incorporating higher moments of systematic risk.
2.4 Hypotheses
Formerly conducted unconditional tests often provided inconclusive or weak results for the pricing of higher moment risk factors. Although this study differentiates itself with more recent data and international focus, based on results of prior studies it is to be expected that conditional relationship tests are preferred. Therefore, the conditional pricing model is examined in this study and is expected to support the relationship between systematic risks and return. Nevertheless, the unconditional relationship between systematic risk and return is also examined for purpose of comparison.
The conditional model relies on the fact that there is investor’s uncertainty over the sign of the realized risk premium or otherwise the risk free asset would never have been held in the first place. Ex-ante investors always anticipate a positive expected risk premium in line with CAPM but when the realized risk premium is negative ex-post, stocks with higher betas are expected to have lower returns. Thus, high beta stocks will be more sensitive to the positive (negative) excess market return and therefore have higher (lower) returns than low beta stocks in up (down) markets. It is expected that in periods were the excess market return is positive (negative) there should be a positive (negative) relationship between beta and return. From this it follows that I can formulate the following hypothesis on the conditional relationship between beta and returns:
H1: There is a conditional relationship between beta risk and cross-sectional index returns.
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As stated before, Scott and Horvath (1980), Harvey and Siddique (2000) and Patton (2004) argue that the risk averse investor prefers positive asymmetry and therefore requires a risk premium for negative skewness. Hence, in the presence of negative asymmetry (negative sign of systematic skewness) in the return distribution of an asset relatively to the market, a risk premium is expected as a required compensation for taking risk. Contrary, when there is presence of positive asymmetry (positive sign of systematic skewness) investors expect a negative required risk premium for systematic skewness. It seems reasonable to expect that during up markets periods the market skewness is positive and that during down market periods the market skewness is negative. Therefore, the risk premium for systematic skewness is expected to have the opposite sign compared to the skewness of the market distribution. This leads to the formulation of the following testable hypothesis on the conditional relationship between systematic skewness and returns:
H2: There is a conditional relationship between systematic skewness risk and cross-sectional index returns.
Hence, I expect a negative (positive) and significant risk premium for systematic skewness in periods were the excess market return is positive (negative).
Studies by Arrow (1971) and Ingersoll (1975) show that risk averse investors have a negative preference towards kurtosis since a more leptokurtic return distribution adds additional dispersion in returns. This implies that investors require a higher expected rate of return when the level of kurtosis of an index relative to the market index increases. Consequently, indices with high kurtosis will be more sensitive to the negative excess market return and therefore have lower expected returns than indices with low kurtosis in down markets. From this it follows that I can formulate the following hypothesis on the conditional relationship between systematic kurtosis and returns:
H3: There is a conditional relationship between systematic kurtosis risk and cross-sectional index returns.
Hence, I expect a positive (negative) and significant risk premium for systematic kurtosis in periods were the excess market return is positive (negative).
3. Methodology
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January 2001 to December 2012, producing a total of 626 data points of weekly returns of each individual index.
3.1 Adjusted Fama and MacBeth (1973) Procedure
To preserve comparability between this study and other studies from Fletcher (2000), Pettengill et al. (1995) and Hung et al. (2004), the choice of the regression methodology is mainly drawn from previous empirical literature on domestic and international asset pricing. A number of asset pricing models rely on time series regressions only where often portfolios are formed and excess returns are regressed on the market risk premium and mimicking portfolio returns. This approach has been criticized in literature for the reason that there is no guidance whether the risk factors are actually priced (Lewellen et al. 2010). Given the this issue with a solely time series approach, I apply the Fama and Macbeth (1973) approach to test whether systematic risk factors are priced and whether models incorporating higher systematic risk factors better explain cross-sectional index returns. This methodology, in addition to the time series regressions, also tests the statistical significance of the average of slope coefficients estimated from weekly cross-sectional regressions. In effect, to test the conditional relationship between risk and return both estimations of time series and cross-sectional regressions are required.
Since country MSCI indices are well diversified portfolios, the first step of the estimation method, which is the formation of portfolio of individual stocks, can be avoided. This may yield an advantage for the reason that when sorting individual assets or countries on a beta factor creates an advantage in estimating the dependence on this factor but disadvantage the other factors in the model specification (Hung, 2004). In other words, when assets are grouped on the basis of the first risk factor, other uncorrelated risk factors in the model may be averaged and therefore have the possibility to be similar across portfolios.
3.1.1 Rolling Time Series Regressions
The whole sample period is equally divided into four separate sub periods. The first period includes the estimation period (window) of country systematic risk factors. Factors are estimated by rolling time series regressions of country excess index returns against European market excess returns. The countries risk factors in 2004 are estimated from the period 2001-2003. Similarly, the country risk factors in 2005 are estimated from the period 2002-2004, and so on. Therefore, the regressions generate risk estimates for the next period based on the previous 3 years of weekly observations.3 Overall, with 626 weekly return observations and a 3 year rolling window, 470 time-series observations are obtained of each individual index.
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Whereas Fletcher (2000) estimated the coefficients over the whole sample by a single time series regression, the advantage of the methodology used in this paper is that I do not have to assume that systematic risk factors are constant over time and the use of non-overlapping periods for estimating risk factors would minimize the errors-in-variables problem. However, Shanken (1992) points out that the Fama and MacBeth (1973) standard errors still tend to overstate the precision of the parameters. Conversely, later research by Jagannathan and Wang (1998) dispute that the assumption of Shanken is rather strong, and when not hold, the standard errors may not be too high.
Similar to Hung (2004), I adjust the time series regression initiated by Fama and MacBeth (1973) by including higher systematic risk factors than beta to estimate the coefficients for beta, systematic skewness and systematic kurtosis (recall equation 4).
( ) ( ) ( ) (4)
Where, is the realized excess return of a MSCI country index, i, in period t. is the sensitivity
of the country index, i, to the European market index, , represents the market risk
premium which equals the realized European market return minus the risk free rate. measures the relative skewness of country index, i, to that of the market index. , measures the relative kurtosis of
country index, i, to that of the market index. , is a random error term. The regression residuals are
tested for autocorrelation and heteroscedasticity at the 5% significance level. In the presence of heteroscedasticity, I use White’s (1980) heteroscedasticity consistent covariance estimates to prevent coefficients estimates to become inefficient. When autocorrelation is detected in the residuals at a 1-month lag by the Breusch-Godfrey test (Breusch, 1979; Godfrey, 1978) I use the Newey and West (1987) heteroscedasticity and autocorrelation consistent covariance estimates to prevent unbiased but inefficient estimates.
3.1.2 Weekly Cross-Sectional Regressions
In the remaining three periods, previous estimates for beta, systematic skewness and systematic kurtosis are used to estimate their risk premia. Under the assumptions of CAPM, risk factors estimated in the estimation period are a good proxy for the risk factors in the test period.
To test the unconditional relationship between systematic risk factors and returns, the following cross-sectional regression is estimated (recall equation 5):
(5)
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( ) ( ) ( ) (7)
Where, D = 1, if ( ) 0 (i.e., when market excess returns are positive), and D = 0, if (
) (i.e., when market excess returns are negative). are the estimated slope coefficients of
the time series regressions, and where the coefficient is the intercept term and are risk premia. All of the tests in this procedure are predictive in the sense that the risk coefficients are estimated from the data of a period prior to the period of the index returns on which the regressions are run. The above cross-sectional regression is estimated for each week in the test period 2004 to 2012 by estimating either
or depending on the sign of the European market excess return resulting in a time series
for every There are a total of 470 weekly cross-sectional regressions and results for coefficient
estimates.
As stated in the literature review, many authors find that after simple averaging the , risk factors have only marginal significance. Using the conditional formula, the are only averaged across different cross-sectional regressions depending on the sign of the dummy. On the average values of t-tests are performed.4If the differences in expected returns can be explained by the systematic risk factors, the mean of the estimated coefficients for should be significantly different from zero. I evaluate the models on the significance of the factor coefficients at a 5% significance level. Adjusted R squared is the average of the adjusted R squared from each week regression. The issue of adopting the results of conditional higher moment pricing models to test the longer term risk return tradeoff is not addressed in this article.
In line with my hypothesis on the relationship between beta and return, I expect the sign of to be positive and significant in periods of positive market excess returns. Since is estimated in periods of
negative excess market returns, I expect the sign to be negative and significant. Furthermore, in line with my second hypothesis is defined as the risk premium for systematic skewness and is estimated in periods of positive market excess returns, I expect the sign to be negative and significant. It is therefore expected that the premia for systematic skewness is inversely related with the sign of the market.Since is estimated in periods of negative market excess returns, the sign of is expected to be positive and significant. Finally, in line with my third hypothesis on the relationship between systematic kurtosis and return, is estimated in periods of positive market excess returns and the expected sign associated with it is expected to be positive and significant. is estimated in periods of negative excess market returns, I
therefore expect the sign to be negative and significant.
If the null hypothesis of no relationship is rejected in both states of the market, a conditional relationship between the systematic risk factor and return is present. Such findings would imply that
4
Standard errors are computed following usual expressions:
√
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investors price exposures to that specific risk factor and the factor contributes in explaining variation in cross-sectional returns.
3.2 Robustness
3.3.1 Explanatory Power of Competing Regression Models
In addition to estimating risk premia, I examine whether adding higher moment systematic risk factors increases explanatory power relatively towards the unadjusted pricing model. To this end, for purpose of comparison, three conditional pricing models are evaluated. Where model A can be seen as the unadjusted (conditional) CAPM, model B and C add respectively systematic skewness and systematic kurtosis to the pricing model. Following previous research (Hung et al. 2004; Galagedera et al. 2003), the adjusted R2 is compared among the different models for evaluating their explanatory power. A model with a high adjusted R2 value indicates that the model has high explanatory power and that useful factors are included. It is expected that adding higher moments of systematic risk increases the explanatory power of the model.
A. Two-moment conditional model
( ) (8)
B. Three-moment conditional model
( ) ( ) (9) C. Four-moment conditional model
( ) ( ) ( ) (10)
3.3.2 Parameter Stability over Time
To evaluate whether the results for the test sample period of 2004 to 2012 are robust or whether they depend on the period examined, I developed a sub period analysis by partitioning using three different periods of approximately equal size: 2004 to 2006, 2007 to 2009, and 2010 to2012. Applying equation (8), (9) and (10) to each of these sub periods tests whether periodical inconsistencies are observed. The entire adjusted Fama and MacBeth (1973) procedure is conducted separately for each sub period.
4. Data
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are collected from Thomson Datastream in local (Euro) currency and are transformed into arithmetic returns.5 Since the return influence of dividends over longer investment horizons cannot be neglected, total return indices where dividends are added back are generated to provide a sufficient measure of the return that would accrue to a holder of the index during time t. Weekly return data is used for the reason that compared to monthly data, it is typically less normally distributed (Chung et al. 2006) and provides a larger number of observations. Therefore, in terms of revealing the existence of any nonlinear return dependencies, the use of weekly data may provide advantages. Moreover, Galagedera et al. (2003) showed that the kurtosis and skewness of the return distribution becomes more prominent in high frequency data.
To examine an international CAPM strong assumptions concerning purchasing power parity and exchange risk should hold (Adler and Dumas, 1983). Since previous studies provide evidence of the pricing of exchange risk, my sample only includes European countries (and market portfolio) that had identical currency during the whole sample period.
The use of MSCI indices for individual countries and as a proxy for the market portfolio over countries large cap indices is preferred since they are live indices reflecting the performance of all funds that are constituent at each point in time. The historical performance of discontinued funds affects the performance of the index. Therefore, there is no impact of survivorship bias present in the sample period.
The weekly return on a 3 month German T bill is chosen as a proxy for the risk free rate for the reasons of its lowest standard deviation and highest level of liquidity among European countries. Weekly total return data for the countries included in my sample is available from the beginning of 2001 resulting in a selection of observations of returns in the period 2001-2012 containing 53.5% up market weeks (Appendix A). During this period, there were no missing or omitted observations.
To provide the return characteristics of the individual countries, table 1 includes summary statistics of the 10 MSCI country indices, the European market index and the risk free rate. The table reports the mean weekly return, maximum and minimum returns, standard deviation, skewness and excess kurtosis. The results reported here are for the whole sample period (see Appendix B for specific sub periods).
As reported in table 1, the average weekly stock return varies between -0.07% and 0.14% for the various stock markets. Although Austria has the highest mean return, it does not have the highest standard deviation. Remarkably, Portugal has lower standard deviation than the diversified European market index. It is clear that for the stock market indexes, the minimum ranges from -17.29% to -31.39%, while the maximum varies between 9.85% and 19.03%. As proxy for the risk free rate, the average performance of the German T Bill is better than some of the individual country indices while having an extremely low
5
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standard deviation of 0.027%. As expected, this is mostly explained by the relative high return during periods of crisis as a safety haven as we can see from the descriptive statistics of the sub periods presented in appendix B (sub period 2001-2003 and 2007-2009). The skewness is negative for all country indices and ranges from -1.31 to -0.37. Also, the distributions of the stock indices appear to differ in the tails since excess kurtosis can be as high as 10.91 with the minimum being 2.10.
Table 1 - Summary Statistics of Return Indices Based on Sample Period
The total sample consists of 626 country return index observations of ten European countries that had the euro currency in the period 2001-2012. This table presents the characteristics of the MSCI return indices of the individual countries, European market and risk free rate. The return figures are in weekly percentages.
Return Index Mean Max Min SD Skewness Excess Kurtosis Obs 'N'
France 0.043 13.263 -21.713 3.148 -0.648 4.617 626 Finland -0.036 13.210 -23.504 4.448 -0.511 2.106 626 Italy -0.006 20.258 -22.208 3.358 -0.698 6.644 626 Belgium 0.067 12.481 -19.534 3.178 -1.052 6.150 626 Austria 0.143 19.034 -30.262 3.679 -1.311 10.915 626 Portugal 0.002 9.852 -17.297 2.663 -0.766 4.554 626 Ireland -0.070 18.594 -31.393 3.963 -1.026 8.780 626 Spain 0.111 12.752 -21.225 3.414 -0.525 3.531 626 Germany 0.091 16.929 -21.224 3.459 -0.376 4.184 626 Netherlands 0.051 16.632 -25.690 3.276 -0.879 7.342 626 Europe 0.057 13.333 -21.600 2.841 -0.708 7.296 626 German T Bill 0.048 0.102 0.002 0.027 0.199 -1.042 626
The regression diagnostic tests (as presented in Appendix B) show that normality of the return distribution of all stock indices is rejected and that nine out of the 10 indices have significant heteroscedasticity in the residuals at the 5% level. Furthermore, there are five indices with significant autocorrelation in the residuals at an eight week lag. This suggests that the standard errors may require correction for the effects of heteroscedasticity and autocorrelation using the method of Newey and West (1987) consistent covariance estimates to prevent unbiased but inefficient estimates (however, correction effects on significance are only marginal).
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meaning that there is sufficient variance left over to estimate the effect of the independent variables. Still, the standard errors could be potentially larger than they would be otherwise, providing less significant results.
When results in the next section produce insignificant factor estimates but the adjusted R squared is high, multicollinearity could still be an issue. Evaluating different sub periods and higher order models therefore could provide more evidence (see section 3.3.2). Multicollinearity in itself does not reduce the predictive power or reliability of the model as a whole, at least within the sample data themselves; it only affects calculations regarding whether individual factors are priced. Hence, the effect (explanatory power, adjusted R squared) of the entire set of co-moments can still be measured whether they matter to the risk averse investor.
5. Empirical Results
This section reports the main results from regression analysis and is structured as follows: estimation results of the unconditional relationship between risk and return are provided, followed by a focus on the conditional relationship. The remainder offers findings from robustness tests of the conditional model.
Appendix D graphically presents coefficients of the systematic risk factors estimated by rolling time series regressions of country excess index returns against European market excess returns. I use these coefficients as parameters in weekly cross-sectional regressions. From these cross-sectional regressions, I show average time series of factor loadings for beta, systematic skewness and systematic kurtosis factors in table 2 and 3.
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Table 2 - Test of an Unconditional Risk and Return Relationship
Estimate 0.0035 -0.0028 0.0001 0.0000 T-Statistic 1.6486 -1.1805 0.2932 -0.5944 P-Value 0.099*** 0.2384 0.7695 0.5525 N 470 Adjusted R² 0.1395
*, **, *** indicates significance at the 1%, 5% and 10% level, respectively. For tests of risk premia in the cross-section, the Fama-MacBeth (1973) methodology is applied. All reported slope coefficients and adjusted R² are mean values across 470 cross-section weeks (entire sample). The t-statistics (two-tail) are the mean divided by the standard error of the slope coefficient.
5.2 Estimation of the Conditional Relationship between Systematic Risk Factors and Return
In this section, I estimate risk premia for the systematic risk factors of the four moment conditional pricing model through equation (7). In this model, excess market returns are assumed to have asymmetric effects on the model parameters, depending on whether the excess market return is positive or negative.The evidence in table 3 is consistent with the predictions of the Pettengill et al. (1995) model. Examination of the estimated regression beta coefficients and , a risk premium with a mean value of respectively 0.0123 and -0.0219 rejects the null hypothesis of no conditional relationship between beta and returns at the 1% level. Therefore, results provide strong support for a significant positive cross-sectional relationship between international stock returns and beta in up market weeks and a significant negative relationship between return and beta in down market weeks. The significant conditional sign of beta indicates that investors that face systematic risk do price this risk. This evidence of the relevancy of beta in international stock returns is in line with previous research by Fletcher (2000) and Tang et al. (2003). It provides additional support that high beta stock indices outperform (underperform) low beta stock indices when the market excess return is positive (negative). Hence, international investors should note this finding in making their optimal investment decisions.
Although systematic skewness and systematic kurtosis as explanatory variables, as indicated by regression coefficients , and , , have the expected sign of the market price postulated in up and down markets, they appear not to be priced. Contrary to domestic evidence, the null hypothesis of no conditional relationship between higher order systematic risk factors beyond beta cannot be rejected at the 5% level. This implies that investors, conditional on the state of the market, do not require a higher expected rate of return when the level of positive skewness (kurtosis) of a stock index decreases (increases) relatively to the market.
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As we can see in table 3, the constant ( ) in the up market rejects the null hypothesis that the mean is equal to zero. Economically speaking, portfolio returns are influenced by more than the systematic risk factors. Earlier studies provide possible explanations by including size, value and momentum factors in regression equation (Fama and French, 1993, 1996, 1998).
The explanatory power of the four moment model is slightly higher in the down market (where adjusted R squared is 0.15%) than in the up market (where adjusted R squared is 0.13%). In other words, the model conditional on down markets marginally better explains the variation in cross sectional international stock returns. To examine the sensitivity of the results reported in table 2, the next section investigates other conditional pricing models and separate sub periods.
Table 3 - Estimates of Risk Premia for Up Markets and Down Markets
Up Market Down Market
0 1 3 5 0 2 4 6 Estimate 0.0060 0.0123 -0.0003 0.0000 0.0003 -0.0219 0.0005 -0.0001 T-Statistic 2.3263 4.4599 -0.7587 0.5982 0.0814 -6.0837 1.1016 -1.4076 P-Value 0.021** 0.000* 0.4488 0.5503 0.9352 0.000* 0.2719 0.1608 N 263 207 Adjusted R² 0.1298 0.1519
*, **, *** indicates significance at the 1%, 5% and 10% level, respectively. For tests of risk premia in the cross-section, the Fama and MacBeth (1973) methodology is applied. All reported slope coefficients and adjusted R² are mean values across N cross-section weeks. The t-statistics (two-tail) are the mean divided by the standard error of the slope coefficient. During January 2004 and December 2012, there are 263 weeks when excess market returns were positive and 207 weeks when excess market returns were negative.
5.3 Tests of Explanatory Power among Competing Pricing Models
For comparison purpose I investigate three (A, B and C) conditional pricing models as presented by equations (8), (9) and (10) over the whole sample period. Following previous research (Hung et al. 2004; Galagedera et al. 2003), the adjusted R2 is compared among the different models for evaluating their respective explanatory power. A model with a high adjusted R2 value indicates that the model has high explanatory power and that useful factors are included.
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Table 4 - Competing Conditional Pricing Models
Up Market Down Market
Model A Estimate 0.0056 0.0126 -0.0031 -0.0188 T-Statistic 2.332** 4.945* -0.9705 -5.651* Adjusted R² 0.0902 B Estimate 0.006 0.0126 -0.0001 -0.0022 -0.0198 0.0002 T-Statistic 2.416** 4.979* -0.3024 -0.6285 -5.656* 0.4603 Adjusted R² 0.1101 C Estimate 0.0060 0.0123 -0.0003 0.0000 0.0003 -0.0219 0.0005 -0.0001 T-Statistic 2.326* 4.459* -0.6315 0.5982 0.0814 -6.084* 1.1016 -1.4076 Adjusted R² 0.1409
*, **, *** indicates significance at the 1%, 5% and 10% level, respectively. For tests of risk premia in the cross-section, the Fama-MacBeth (1973) methodology is applied. Model A, B and C refers to regression equation (8), (9) and (10) respectively. Therefore, for each individual model both time series and cross sectional regressions are estimated. All reported slope coefficients and adjusted R² are mean values across 470 cross-section weeks. The t-statistics (two-tail) are the mean divided by the standard error of the slope coefficient.
Evidence for the pricing of beta risk ( , ) by investors is robust in all of the conditional models at the 1% significance level contributing in explaining cross sectional international stock returns. This furthermore suggests consistency of premia estimates in the up and down market for beta when other systematic risk factors are also included in the pricing model. The significant conditional sign of beta indicates that investors that face systematic risk do price this risk.
5.4 Tests for Robustness over Sub Periods
To evaluate whether the tests results of the whole sample period of 2004-2012 are robust or whether they depend on the period examined, I estimate a sub period analysis by partitioning using three different periods of approximately equal size: 2004-2006, 2007-2009, and 2010-2012. Applying equation (8), (9) and (10) to each of these sub periods tests whether periodical inconsistencies are observed. The entire adjusted Fama and MacBeth (1973) procedure is conducted separately for each sub period.
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Table 5 - Estimates of Conditional Risk Premia in Sub-sample Periods
Test Period Up Market Down Market
2004-2006 A Estimate 0.0097 0.0026 -0.0031 -0.0188 T-Statistic 7.409* 1.678*** -0.9705 -5.651* N 100 57 B Estimate 0.0094 0.0029 0.0001 -0.0011 -0.0106 -0.0004 T-Statistic 6.809* 1.904* 0.2157 -0.5252 -5.436* -0.8035 N 100 57 C Estimate 0.0110 0.0020 0.0002 0.0000 -0.0036 -0.0088 -0.0005 0.0000 T-Statistic 6.430* 1.2554 0.7047 0.9012 -1.3332 -4.034* -1.5122 -1.4374 N 100 57 2007-2009 A Estimate 0.0022 0.0223 -0.0007 -0.0109 T-Statistic 0.3736 3.624* -0.3276 -5.308* N 79 77 B Estimate 0.0034 0.0210 -0.0010 -0.0008 -0.0290 0.0016 T-Statistic 0.5896 3.453* -0.9518 -0.1059 -3.857* 1.747*** N 79 77 C Estimate 0.0031 0.0201 -0.0018 0.0002 0.0068 -0.0356 0.0017 -0.0002 T-Statistic 0.5083 2.971* -1.805*** 1.0932 0.8955 -4.650* 1.697*** -1.6418 N 79 77 2009-2012 A Estimate 0.0041 0.0154 -0.0048 -0.0165 T-Statistic 0.8293 3.128* -0.8822 -2.963* N 84 73 B Estimate 0.0038 0.0161 0.0005 -0.0044 -0.0171 0.0000 T-Statistic 0.7797 3.279* 0.6771 -0.8103 -3.095* -0.0230 N 84 73 C Estimate 0.0028 0.0171 0.0005 -0.0001 -0.0035 -0.0176 0.0001 0.0000 T-Statistic 0.5298 3.258* 0.6720 -0.6944 -0.6489 -3.160* 0.1540 -0.2682 N 84 73
*, **, *** indicates significance at the 1%, 5% and 10% level, respectively. For tests of risk premia in the cross-section in each specific sub period and model, the Fama-MacBeth (1973) methodology is applied. Model A, B and C refers to regression equation (8), (9) and (10) respectively. Therefore, for each individual model both time series and cross sectional regressions are estimated. All reported slope coefficients are mean values across N cross-section weeks. The t-statistics (two-tail) are the mean divided by the standard error of the slope coefficient.
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positive skewness, they seem to require a risk premium when facing systematic skewness risk in these periods.
Tests of systematic skewness risk premia estimates reveal that in sub periods 2004-2006 and 2010-2012, where returns where less extreme and negative skewed, a conditional relationship between systematic skewness risk and return is not supported. Similar to Hung et al. (2004) I reason that if asset returns are heavily skewed (as in the turbulence period of 2007-2009), investors are concerned about the skewness of their asset or portfolio returns. This implies that when investor’s preferences contain skewness measures each asset contribution to systematic skewness may determine the asset its relative attractiveness and therefore its required return.
This study finds additional international evidence for the results of earlier research which are purely based on domestic markets (Harvey and Siddique, 2000; Galagedera et al. 2003; Hung et al. 2004) that systematic skewness can be important to asset pricing since it characterize a more true distribution of asset returns. By contrast, the hypothesis of no conditional relationship between systematic kurtosis and return cannot be rejected in each model and sub period at the 5% significance level indicating that investors do not price this risk.
Furthermore, the results in Table 5 of the two, three and four moment model inform that when excess market returns are positive (negative), a significant positive (negative) relationship exists between beta risk and international index return for each sub period suggesting that the outcomes for beta are (with a single exception for the four moment model in 2004-2006) consistent in all sub periods.
6. Conclusion
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To date, no study investigated whether in addition to beta also higher moments of systematic risk explain cross-sectional differences in international index returns. Therefore for European index returns, this study made a first attempt to demonstrate whether investors price higher moments of systematic risk. Solely in the turbulent period 2007-2009, systematic skewness is priced and contributes in explaining variation in cross sectional international stock returns. During this turbulent period, returns were more negative, more extreme and negatively skewed than in others in the sample. Similar to Hung et al. (2004) I argue that if asset returns are heavily skewed, investors may be concerned about the skewness of their asset or portfolio returns. If investor’s preferences contain skewness measures each asset contribution to systematic skewness it determines the asset its relative attractiveness and therefore its required return. It tells us that investors, conditional on the down (up) market state, do require a higher (lower) compensation when the level of skewness of a stock index decreases (increases) relatively to the market index. As expected, the sign of the systematic skewness premium is inversely related to the state of the market.
Further, this analysis shows that adding higher moment risk factors to the traditional CAPM increases explanatory power. Therefore, investors should note that models incorporating higher moments marginally better explain the variation in cross sectional international stock returns. The systematic kurtosis does not appear to be priced.
In line with previous studies, evidence of the pricing of moments of systematic risk factors beyond beta seems still dependent on the market, model and period investigated. Given this, it makes it difficult to load these results with economic meaning in making optimal decisions in portfolio management over time. Investors should therefore be aware that higher moment results are bound to the recent financial crisis and usability during next crisises is yet to be awaited. Since this study is limited to European indices and there is room left to incorporate other risk factors, global follow up research including currency risk premia and size and value factors may be interesting. The issue of adopting the results of conditional higher moment pricing models to test the average long run risk return tradeoff is not addressed in this article and is a topic for future research.
Acknowledgements
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