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Journal of Banking & Finance 32 (2008) 1178–1187 www.elsevier.com/locate/jbf

Risk aversion and skewness preference Thierry Post *, Pim van Vliet, Haim Levy Erasmus University Rotterdam, P.O. Box 1738, 3000 DR, Rotterdam, The Netherlands Received 2 September 2005; accepted 15 February 2006 Available online 5 October 2007

Abstract Empirically, co-skewness of asset returns seems to explain a substantial part of the cross-sectional variation of mean return not explained by beta. This ﬁnding is typically interpreted in terms of a risk averse representative investor with a cubic utility function. This paper questions this interpretation. We show that the empirical tests fail to impose risk aversion and the implied utility function takes an inverse S-shape. Unfortunately, the ﬁrst-order conditions are not suﬃcient to guarantee that the market portfolio is the global maximum for this utility function, and our results suggest that the market portfolio is more likely to represent the global minimum. In addition, if we do impose risk aversion, then co-skewness has minimal explanatory power. 2007 Elsevier B.V. All rights reserved. JEL classiﬁcation: G11; G12 Keywords: Co-skewness; Asymmetry; 3M CAPM; Asset pricing

1. Introduction The traditional Mean–Variance Capital Asset Pricing Model (MV CAPM) assumes that investors construct portfolios by balancing the mean against the variance. In capital market equilibrium, the model predicts an exact linear relationship between mean and beta, i.e. standardized covariance with the market portfolio. Kraus and Litzenberger (1976), Friend and Westerﬁeld (1980), and Harvey and Siddique (2000) among others, analyze a threemoment capital asset pricing model (3M CAPM) that considers skewness in addition to mean and variance. The idea behind this model is that the return distribution of capital assets in many cases is non-symmetric and investors then prefer a positively skewed distribution to a negatively skewed one. The 3M CAPM forwards an exact linear equilibrium relationship between mean, beta and gamma, i.e. standardized co-skewness with the market portfolio. Interestingly, gamma seems to explain a substantial part of the cross-sectional variation of mean return not *

Corresponding author. Tel.: +31 104081428. E-mail address: [email protected] (T. Post).

0378-4266/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankﬁn.2006.02.008

explained by beta. Speciﬁcally, there is evidence for a substantial gamma premium, i.e. securities that make the market portfolio more negatively skewed earn positive ‘abnormal’ average returns. The estimates for the annualized premium range from 2.5% (Kraus and Litzenberger, Friend and Westerﬁeld) to 3.6% (Harvey and Siddique). In other words, the empirical evidence suggests that asset returns are suﬃciently skewed and investors’ preference for positive skewness is suﬃciently strong for co-skewness to play an important role for asset pricing. In our opinion, too little attention has been given to the economic meaning of the gamma premium. The 3M CAPM is typically motivated by a representative investor model with Mean–Variance–Skewness preferences, or a cubic utility function (or a third-order Taylor series approximation to the true utility function). Speciﬁcally, an exact linear relationship between mean, beta and gamma represents the ﬁrst-order condition for maximizing the expectation of such a utility function. In this paper, we provide theoretical and empirical arguments against this interpretation. Our arguments relate to the regularity condition of risk aversion or concavity for the utility function. We will

T. Post et al. / Journal of Banking & Finance 32 (2008) 1178–1187

demonstrate that the empirical tests generally fail to impose this condition. In fact, the estimation results severely violate risk aversion and they imply an inverse Sshaped utility function with risk seeking beyond a return level. If we explicitly impose concavity, then the maximal annualized gamma premium is only 0.4%. By contrast, if concavity is not imposed, then the annualized gamma premium is 2.9%, more than seven times as large as with concavity. However, for the inverse S-shaped utility function, the ﬁrst-order conditions are no longer suﬃcient for a global maximum. In fact, our empirical results suggest that the market portfolio is more likely to represent the global minimum. These ﬁndings lead us to question the theoretical interpretation of the gamma premium. One motivation for our analysis is the recent ﬁnding in Post (2003) that the value-weighted market portfolio is signiﬁcantly ineﬃcient in terms of second-order stochastic dominance (SSD) relative to benchmark portfolios formed on market capitalization and book-to-market equity ratio. Since the SSD criterion accounts for all concave utility functions, this ﬁnding suggests that a concave cubic utility function (3M CAPM) cannot explain what a concave quadratic utility function (MV CAPM) leaves unexplained. The remainder of this paper is structured as follows. Section 2 recaptures the representative investor model behind the 3M CAPM. Next, Section 3 gives our theoretical objections against the interpretation of the gamma premium in terms of this model. Section 4 illustrates our objections by means of an empirical application of the 3M CAPM to well-known US stock market data. Finally, Section 5 presents conclusions and suggestions for further research.

1179

if it satisﬁes the following three regularity conditions: (RC1) non-satiation, i.e. u 0 (xjh) > 0 "x 2 D, (RC2) risk aversion, i.e. u00 (xjh) 6 0 "x 2 D, and (RC3) nonincreasing absolute risk aversion (NIARA), for which u000 (xjh) P 0 "x 2 D is a well-known necessary condition. We assume throughout the text that condition (RC1) is satisﬁed, and we analyze the role of conditions (RC2) and (RC3). Under these assumptions, the representative investor solves R the constrained optimization problem maxk2RN uðxT k j hÞdGðxÞ. The value-weighted market portfolio of risky assets, say s 2 RNþ , must represent the optimal solution to this problem. The well-known Euler equation gives the ﬁrst-order conditions for optimality: Z E½u0 ðxT sjhÞx ¼ u0 ðxT sjhÞx dGðxÞ ¼ 0: ð1Þ For cubic utility, the Euler equation implies an exact linear relationship between mean, beta and gamma (standardized co-skewness): l E½x; T

b c

ð2Þ T

E½ðx s l sÞðx lÞ 2

E½ðxT s lT sÞ E½ðxT s lT sÞ2 ðx lÞ 3

E½ðxT s lT sÞ

;

ð3Þ

:

ð4Þ

Speciﬁcally, using the Taylor series expansion u0 ðxT sjhÞ ¼ 2 u0 ðlT sjhÞ þ u00 ðxT sjhÞðxT s lT sÞ þ 12 u000 ðxT sjhÞðxT s lT sÞ , we may reformulate Eq. (2) as the following Security Market Plane:

2. Preliminaries

l ¼ q1 b þ q2 c;

The 3M CAPM is typically motivated by a single-period, portfolio-based, representative investor model that satisﬁes the following assumptions:

with the following risk premiums: q1

ð5Þ

E½u00 ðxT sjhÞE½ðxT s lT sÞ2 E½u0 ðxT sjhÞ 2

ð2h1 þ 6h2 lT sÞE½ðxT s lT sÞ

1. The investment universe consists of N risky assets and a riskless asset. The excess returns of the risky assets are denoted by x 2 RN .1 2. The excess returns are random variables with a continuous joint cumulative distribution function (CDF) G:DN ! [0, 1], with D R for the return domain. 3. The representative investor constructs a portfolio by choosing portfolio weights k 2 RN , so as to maximize the expectation of a utility function that is deﬁned over portfolio return. We represent Mean–Variance–Skewness preferences using the (standardized) two-parameter cubic utility function u(xjh) x + h1x2 + h2x3, with h (h1h2)T.2 The utility function is well behaved only

3. Theoretical objections

Throughout the text, we will use RN for an N-dimensional Euclidean space, and RNþ denotes the positive orthant. 2 This utility function is standardized such that u(0jh) = 0 and 0 u (0jh) = 1. Since utility functions are unique up to a linear transformation, this standardization does not aﬀect our results.

Empirical 3M CAPM studies typically do not estimate the utility parameters but rather directly estimate the beta and gamma premiums, and they do not report the implied utility parameters. Still, there are compelling theoretical arguments to expect that the implied utility function takes

1

¼ q2 ¼

2

1 þ 2h1 lT s þ 3h2 E½ðxT sÞ

;

ð6Þ

12 E½u000 ðxT sjhÞE½ðxT s lT sÞ2 E½u0 ðxT sjhÞ 3h2 E½ðxT s lT sÞ3 2

1 þ 2h1 lT s þ 3h2 E½ðxT sÞ

:

ð7Þ

Since the market portfolio has a beta and gamma of one, the market risk premium equals the sum of the beta premium and the gamma premium, i.e. lTs = q1 + q2.

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an inverse S-shaped form with risk aversion up to some return level and risk seeking beyond that level:

The empirical results in Section 4 further support this view. The inverse S-shape introduces two complications:

1. The regularity conditions (RC1), (RC2) and (RC3) are frequently mentioned as desirable properties in empirical 3M CAPM studies (see, e.g. Kraus and Litzenberger, 1976, p. 1086). However, the studies do not actually impose or test these conditions! The studies typically use two simple restrictions. First, the beta premium q1 should be non-negative. Using (6), this restriction can be reformulated in terms of the utility parameters. Since E[(xTs lTs)2] > 0 and E[u 0 (xTsjh)] > 0 (using (RC1)), we ﬁnd that q1 P 0 if and only if E[u00 (xTsjh)] 6 0, or, equivalently,

1. We may ask if (local) risk seeking is economically meaningful. Of course, risk aversion is not a law of nature and there are several arguments to support (local) risk seeking. For example, Markowitz (1952) argues that the willingness to purchase both insurance and lottery tickets (the Friedman-Savage puzzle) implies that marginal utility is increasing for gains. Also, ‘seemingly’ risk seeking behavior may arise if irrational investors subjectively overweigh the true probability of extremely high returns, as in, e.g., the Cumulative Prospect Theory (Kahneman and Tversky, 1992)). Still, we may object to assuming risk aversion in the theoretical section of a study and neither imposing that assumption in the empirical analysis nor reporting whether the empirical results satisfy that assumption. 2. Another problem associated with relaxing concavity is that the ﬁrst-order conditions are no longer suﬃcient for optimality. We may wrongly classify a minimum or a local maximum (which will also satisfy the ﬁrst-order conditions) as the global maximum. There exist various multivariate global optimization methods for locating the global maximum if the (known) objective function is not concave (see, e.g. Horst and Pardalos, 1995). Unfortunately, these methods generally are computationally more complex than checking the ﬁrst-order condition, and the computational burden becomes prohibitive if the problem dimensionality is high (i.e. many assets are included). In addition, we do not see a simple solution for the case where the objective function is not known but rather has to be estimated empirically. Still, it is relatively simple to test some weak necessary conditions (in addition to the ﬁrst-order conditions) for a global maximum. For example, if the market portfolio is the global maximum, then its expected utility must exceed that of all individual assets. Section 4 provides empirical evidence that this condition is severely violated and that the market portfolio does not maximize the expectation of the utility function implied by the 3M CAPM test results, even though the ﬁrst-order conditions are satisﬁed.

2h1 þ 6h2 lT s 6 0:

ð8Þ

Second, securities that increase market skewness should earn a non-positive premium. Assuming that the market portfolio is negatively skewed (as is true in our analysis), this means that the gamma premium q2 should be non-negative or, using (7) and (RC1), h2 P 0:

ð9Þ

Since u000 (xjh) = 6h2, this condition is equivalent to the NIARA condition (RC3). However, it is easily veriﬁed that condition (8) is not equivalent to the risk aversion condition (RC2); it oﬀers only a weak necessary condition for risk aversion (the second-order derivative is restricted only at lTs), and it allows for an inverse S-shaped utility function. 2. As argued by Levy (1969), a cubic utility function cannot be concave over an unbounded range, i.e. if D ¼ R. Even the smallest possible non-zero value for h2 suﬃces to make utility convex over a range. Marginal utility u 0 (xjh) = 1 + 2h1x + 3h2x2 is a quadratic function and hence it is increasing over a range. Therefore, if we impose concavity for an unbounded range, then the cubic term h2 and the gamma premium q2 must equal zero and gamma cannot explain asset prices. 3. Of course, utility can be concave over a bounded sample range, say D = [b, b+] with 1 < b < 0 for the sample minimum and 1 > b+ > 0 for the sample maximum. However, Tsiang (1972) demonstrates that a quadratic function is likely to give a good approximation for any (continuously diﬀerentiable) concave utility function over the typical sample range, and that higher-order polynomials (including cubic functions) are unlikely to improve the ﬁt. Hence, if we impose the regularity conditions for the sample range, then the gamma premium generally will be very small for realistic return distributions, and gamma is unlikely to help explain asset prices. Using these theoretical arguments, the sizable gamma premium found in empirical studies suggests that the underlying utility function is not concave (even over the sample range), but rather takes an inverse S-shape.

4. Empirical analysis 4.1. GMM estimation In empirical applications, the CDF G(x) generally is not known. Rather, information is typically limited to a discrete set of time-series observations, say (x1 xT), with xt (x1t xNt)T, which are here assumed serially independently and identically distributed random draws from G(x). Using these observations, we can construct the following sample equivalent of the Euler equation (1): mðhÞ

T 1 X u0 ðxTt sjhÞxt ¼ 0: T t¼1

ð10Þ

T. Post et al. / Journal of Banking & Finance 32 (2008) 1178–1187

We may estimate the unknown parameters h using the generalized method of moments (GMM). The GMM estimator selects parameter estimates so that the pricing errors m(h) are as close to zero as possible, as deﬁned by the criterion function J min mðhÞT WmðhÞ; h

þ 3h2

T T 3 1 X x s ¼ 0: T t¼1 t

ð15Þ

4.2. Gamma premium bounds

T 1 X xT s 6 0: T t¼1 t

ð12Þ

In addition, we can also impose the proper risk aversion condition over the sample range, i.e. u00 (xjh) 6 0 "x 2 [b, b+], by means of a single inequality restriction. Speciﬁcally, if the NIARA condition (RC3) or (9) is satisﬁed, then the second-order derivative is a non-decreasing function and hence it suﬃces to impose the following condition for the sample maximum: 2h1 þ 6h2 bþ 6 0:

ð13Þ

Similarly, we can impose non-satiation, i.e. u 0 (xjh) P 0 "x 2 [b, b+], by means of one inequality restriction. If the risk aversion condition is satisﬁed, then marginal utility is non-increasing, and hence it suﬃces to impose the following condition: 1 þ 2h1 bþ þ 3h2 b2þ P 0:

ð14Þ

Finally, we wish to standardize the parameters to exclude extreme solutions. For this purpose, we require that the sum of the beta premium and the gamma PT premium equals the total market risk premium, i.e. T1 t¼1 xTt s ¼ q1 þ q2 . It is easily veriﬁed that this is equivalent to requiring the value-weighted average of the pricing errors to equal zero, i.e.

3

T T T 2 1 X 1 X xTt s þ 2h1 x s T t¼1 T t¼1 t

ð11Þ

where W is a weighting matrix. In this study, the weighting matrix is set equal to the inverse of the covariance matrix of the pricing errors, i.e. W (m(h)m(h)T)1, and we use the continuous-updating method, which continuously alters W as h is changed in the minimization; see, e.g. Hansen et al. (1996).3 This approach focuses on estimating the utility parameters h1 and h2 rather than the risk premiums q1 and q2. However, we can compute the implied risk premiums from the utility parameters by using Eqs. (6) and (7). The advantage of this approach is that linear inequality restrictions suﬃce to impose and test the utility conditions that are of interest here. We have already seen that the typical restrictions on the beta and gamma premiums can be formulated equivalently as linear restrictions on the utility parameters, i.e. (8) and (9). In empirical applications, we may use the following empirical equivalent of (8): 2h1 þ 6h2

T

mðhÞ s ¼

1181

Similar results are obtained for a one-steep GMM procedure that sets W equal to the identity matrix, or to the sample covariance matrix of the return observations.

Apart from computing the risk premiums that are implied by the GMM estimates of the utility parameters, we may also directly compute an upper bound to the gamma premium. Speciﬁcally, the utility parameters must satisfy the linear conditions (9,13,14) and (15). Hence, we may obtain upper and lower bounds for h2 as the solution to the following linear programming problem: h2 sup h2 h1 ;h2 s:t:

ð16Þ

2h1 þ 6h2 bþ 6 0 1 þ 2h1 bþ þ 3h2 b2þ P 0 T T T T 2 T 3 1 X 1 X 1 X xTt s þ 2h1 xt s þ 3h2 x s ¼0 T t¼1 T t¼1 T t¼1 t h1 6 0 h2 P 0

Associated with h2 is the following value for h1: , T T X X T 2 T x s x s h1 0:5 t

t¼1

1:5h2

t

t¼1

T X t¼1

3 xTt s

,

T X

2 xTt s :

ð17Þ

t¼1

Using (7), we may compute the following bound on the gamma risk premium from h2 and h1 : 3 T T P P 1 T 1 T 3h2 T xt s T xt s t¼1 t¼1 2 : ð18Þ q T T 2 P P 1 þ 2h1 T1 xTt s þ 3h2 T1 xTt s t¼1

t¼1

In a similar way, we can obtain a lower bound, say q2, by searching for the smallest admissible value for h2. Rather than an estimate that best describes the data according to 2 and q2 give theoretia speciﬁc estimation methodology, q cal bounds that depend only on the return distribution of the market portfolio and hence apply for all estimation methodologies and sets of benchmark assets. 4.3. Data In our empirical analysis, we use individual stock returns, portfolio returns and index returns. The monthly stock returns (including dividends and capital gains) are from the Center for Research in Security Prices (CRSP) at the University of Chicago. The CRSP all-share index,

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T. Post et al. / Journal of Banking & Finance 32 (2008) 1178–1187

a value-weighted average of all common stocks listed on the NYSE, AMEX, and NASDAQ markets, is a proxy for the market portfolio. Further, we use the one-month US Treasury bill and the Long-term Government total return index as the riskless asset and as a proxy for the bond market. The one-month T-Bill and bond index are obtained from Ibbotson and the size decile portfolios are from the data library of Kenneth French. We use data returns ranging from July 1963 to December 2001. Given our focus on the gamma premium, it is necessary to use benchmark portfolios with a large spread for gamma to proxy for the individual risky assets. Therefore we construct 10 portfolios formed by sorting individual stocks based on past gamma. However, ex post skewness or ex post co-skewness may not be a good predictor of future (co-)skewness due to a high sensitivity to sampling error in these measures (see, i.e. Lau et al., 1989). An alternative approach is to sort stocks based on an observable instrument variable that is highly correlated with gamma. A natural candidate for the instrument is market capitalization of equity (ME), because small cap stocks typically have relatively high upside potential compared to large caps. For this reason, this study also uses the 10 CRSP size decile portfolios, which are described in Fama and French (1992). The gamma portfolios are constructed using individual stocks which are placed into 10 decile portfolios based on their past 60-months gamma as deﬁned in Eq. (4). We select ordinary common US stocks listed on the New York

Stock Exchange (NYSE), American Stock Exchange (AMEX) and NASDAQ markets. We exclude ADRs, REITs, closed-end-funds, units of beneﬁcial interest, and foreign stocks. Hence, we only include stocks that have a CRSP share type code of 10 or 11. We require a stock to have (1) 60 months of data available (for gamma estimation) and (2) information about market capitalization at formation date (to calculate value-weighted portfolio returns). Portfolio formation takes place at December of each year. For example, to be included at December 1980 a stock must have trading information since January 1976 and a positive market capitalization for December 1980. A stock is excluded from the analysis if there is no more price information available. In that case, the delisting return or partial monthly return provided by CRSP is used as the last return observation. Table 1 shows some descriptive statistics for the excess returns of the market portfolio and the 20 benchmark portfolios. All benchmark portfolios have a negatively skewed return distribution. Interestingly, this negative skewness is not ‘diversiﬁed away’; the market portfolio is more negatively skewed than most of the individual benchmark portfolios. Apparently, investing in the market portfolio yields a relatively small reduction in downside risk (relative to the individual benchmark portfolios) at the cost of a relatively large reduction in upside potential. As expected, the small cap stocks are less negatively skewed than the large caps. Nevertheless, the small caps

Table 1 Descriptive statistics benchmark portfolios Standard deviation

Beta

Skewness

Gamma

Min

Max

Panel A: Size decile portfolios Market 0.471 ME 1 0.703 ME 2 0.697 ME 3 0.712 ME 4 0.682 ME 5 0.711 ME 6 0.596 ME 7 0.621 ME 8 0.604 ME 9 0.523 ME10 0.439

Mean

4.467 6.494 6.322 6.023 5.852 5.544 5.323 5.161 5.048 4.632 4.288

1.000 1.090 1.184 1.178 1.163 1.136 1.110 1.098 1.087 1.009 0.932

0.502 0.083 0.236 0.452 0.507 0.607 0.563 0.449 0.417 0.338 0.297

1.000 1.653 1.595 1.506 1.475 1.431 1.302 1.155 1.092 0.905 0.829

23.09 29.30 30.60 29.70 30.05 28.27 26.89 26.69 24.90 22.99 20.33

16.05 28.72 27.79 25.50 24.18 24.33 20.52 22.09 18.00 17.70 17.43

Panel B: Gamma decile portfolios Market 0.471 G1 0.536 G2 0.666 G3 0.587 G4 0.554 G5 0.484 G6 0.570 G7 0.460 G8 0.516 G9 0.522 G10 0.640

4.467 6.322 5.195 4.708 4.569 4.371 4.578 4.767 4.507 4.998 5.636

1.000 1.238 1.051 0.955 0.934 0.917 0.964 1.009 0.956 1.053 1.111

0.502 0.088 0.081 0.281 0.154 0.145 0.134 0.464 0.632 0.637 0.565

1.000 1.153 0.827 0.744 0.876 0.771 0.781 1.042 1.062 1.232 1.351

23.09 22.54 17.56 16.54 17.78 20.37 20.97 25.46 27.30 28.42 32.12

16.05 27.48 21.76 17.28 24.64 18.53 19.56 18.08 14.32 16.59 16.97

This table includes monthly excess returns for the market portfolio (CRSP all-share index), the 10 CRSP size decile portfolios (Panel A) and the 10 gamma decile portfolios (Panel B). Descriptive statistics are computed for the full sample from July 1963 to December 2001. Excess returns are computed from the raw return observations by subtracting the return on the one-month US Treasury bill from Ibbotson. The size decile portfolios and the market portfolio are obtained from the data library of Kenneth French. The gamma portfolios are constructed using the 2002 CRSP stock database.

T. Post et al. / Journal of Banking & Finance 32 (2008) 1178–1187

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Table 2 Estimation results three-moment CAPM Risk aversion

h1

h2

q1

q2

J

Yes No

0.0114 (0.009) 0.0093 (0.051)

0.0002 (0.471) 0.0020 (0.141)

0.439 0.231

0.031 0.239

0.011 (0.745) 0.009 (0.837)

have above-unity gammas and hence increasing the weight of these stocks relative to their market weights makes the portfolio more negatively skewed, while the large caps have below-unity gammas, meaning that these stocks reduce the negative skew. This surprising ﬁnding – the ranking of skewness and gamma being opposite – presumably reﬂects that large negative returns of small caps occur during general market downturns, while large positive returns are less correlated with general market movements. At ﬁrst glance, the distribution of gamma across the benchmark portfolios suggests that skewness may help explain asset prices. For example, the ‘high yield’ small cap portfolio ME1 (with a Treynor ratio of 0.64) has a high gamma of 1.65 and hence seems unfavorable for skewness seekers, while the ‘low yield’ big cap portfolio ME10 (with a Treynor ratio of 0.23) has a low gamma of 0.83. However, as we will explain below, we must be cautious when interpreting this pattern as evidence for eﬃciency of the market portfolio for skewness-loving investors. Panel B shows the descriptives for gamma decile portfolios, formed by sorting the individual stocks based on past 60-month gamma. Past gamma does not predict future returns witness the ﬂat pattern in returns across the 10 gamma portfolios (correlation is 1%). Further, the gamma spread is substantially smaller for these portfolios than for the size decile portfolios and the predictive power of past gamma is substantially weaker than that of size. Most notably, the G1 portfolio, which includes the stocks with the lowest past gammas, has a relatively high, above-unity gamma of 1.15. As discussed above, these results presumably reﬂect the high sensitivity to sampling error of gamma and they explain why we will focus on size decile portfolios rather than gamma decile portfolios in this section. We will apply the GMM method (see Section 4.1) to this data set. We may ask if the results are sensitive to the estimation methodology (GMM), the set of benchmark assets (the 10 size deciles), the sample period (July 1963–December 2001), the return interval (monthly returns) or the composition of the market portfolio (all-equity). To answer this question, we also compute the empirical upper bound (see Section 4.1). Again, this bound applies for all estimation methodologies and for all benchmark portfolios. In addition, we also compute the bound for various 10-year subsamples, for other return frequencies (daily, quarterly, and yearly returns) and for market portfolios constructed as

Utility

We estimate the unknown parameters h1 and h2 of the cubic utility function u(xjh) x + h1x2 + h2x3 by means of GMM. The orthogonality conditions are given by the Euler equation (1). We analyze the eﬃciency of the Fama and French market portfolio relative to the 10 size portfolios over the sample period July 1963–December 2001. We compare the model that imposes the condition of risk aversion over the sample range, i.e. (13), with the model without this restriction. The table shows the GMM estimates for the parameters (p-values within brackets), the associated estimated for the monthly risk premiums q1 and q2, and the J-statistics (p-values within brackets).

100

(ii)

80 60 40

(i)

20 0 -40

-30

-20

Losses

-10

-20 -40

0

10

20

30

40

Gains

-60 -80 -100

Fig. 1. Optimal cubic utility function. The ﬁgure shows the cubic utility function u(xjh) x + h1x2 + h2x3 for (i) the case with risk aversion condition (13) and (ii) the case without the risk aversion condition. The values for h1 and h2 are obtained by means of GMM estimation of the Euler equation (10) for the Fama and French market portfolio and the 10 size portfolios, using data from July 1963 to December 2001; see Table 2.

mixtures of the CRSP all-share index and the Ibbotson US long-term government bond index. 4.4. GMM results We estimate the Euler equation for two diﬀerent models. The ﬁrst model imposes no restrictions apart from the standardization (15). The second model adds the condition of risk aversion over the sample range, i.e. (13). Table 2 reports the estimated utility parameters for both models and the associated risk premiums, as computed from (6) and (7). Further, the table reports the J-statistics and the associated p-values.4 For illustration, Fig. 1 shows the estimated utility function of both models. In addition, Fig. 2 displays the Security Market Plane for both models. For the model that imposes risk aversion, the estimate for the cubic parameter h2 is approximately zero, and the 4 We may use the J-statistic to test if the Euler equation holds (i.e. the pricing errors m(h) are equal to zero). Speciﬁcally, under the null that the (N 2) overidentifying restrictions are satisﬁed, the J-statistic times the number of regression observations, i.e. JT, asymptotically obeys a chisquared distribution with degrees of freedom equal to the number of overidentifying restrictions, i.e. (N 2) degrees of freedom.

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T. Post et al. / Journal of Banking & Finance 32 (2008) 1178–1187

(i) without risk aversion

(ii) with risk aversion

0.80

0.80

0.60

0.60

0.40

0.40

0.20 2.0

Mean

1.00

Mean

1.00

0.20 2.0

0.00

1.5 1.0

2.00

1.0

1.50

0.5 Gamma

0.00

1.5

1.00

0.0

2.00 1.00

0.0

0.50 0.00

1.50

0.5 Gamma

Beta

0.50 0.00

Beta

Fig. 2. Security market plane. The ﬁgure shows the Security Market Plane as deﬁned in (5): l ¼ q1 b þ q2 c for (i) case with risk aversion and (ii) the case without risk aversion. The risk premiums q1 and q2 are obtained by means of GMM estimation of the Euler equation; see Table 2. The ﬁtted values are: (i) l = 0.439b + 0.031c and (ii) l = 0.231b + 0.239c.

cubic utility function eﬀectively takes a quadratic form. This illustrates the argument by Tsiang (1972) that a cubic utility function is unlikely to improve the ﬁt relative to a quadratic utility function if we maintain the assumption of risk aversion. The small value for h2 implies an annualized gamma premium of only 0.4%, or about 7% of the total market premium. The J-statistic is not signiﬁcantly diﬀerent from zero and we cannot reject the ﬁrst-order conditions. Since the ﬁrst-order conditions suﬃce for a global maximum for concave utility functions, this implies that we cannot reject the null that the market portfolio is the global maximum. For the model without the risk aversion condition, the estimated utility function takes an inverse S-shaped form with an inﬂection point around zero. Interestingly, the estimated utility parameters satisfy conditions (8) and (9) and the beta premium and gamma premium (now 2.8% and 2.9% per annum, respectively) are both positive. These ﬁndings illustrate the point that the typical 3M CAPM restrictions on the risk premiums do not suﬃce to guarantee risk aversion and that the empirical results of 3M CAPM studies may violate this condition. Again, the J-statistic is not signiﬁcantly diﬀerent from zero and hence the market portfolio does not signiﬁcantly violate the ﬁrstorder conditions for the estimated utility functions. However, the ﬁrst-order conditions are not suﬃcient to guarantee a global maximum for inverse S-shaped utility functions, and the market portfolio may represent a minimum or a local maximum for expected utility. Table 3 displays the sample mean of utility for the market portfolio and the benchmark portfolios. Interestingly, mean utility for the market portfolio is less than that for each of the 10 benchmark portfolios. This suggests that the market portfolio is more likely to represent the global minimum (!) than the global maximum of expected utility. Roughly speaking, investors who prefer positive skewness assign a relatively high weight to upside potential,

Table 3 Sample mean utility of Fama and French portfolios

Market ME 1 ME 2 ME 3 ME 4 ME 5 ME 6 ME 7 ME 8 ME 9 ME10

Concave utility function (i)

Inverse S-shaped utility function (ii)

0.237 0.234 0.242 0.289 0.280 0.347 0.262 0.311 0.309 0.276 0.228

0.252 0.441 0.371 0.331 0.300 0.348 0.264 0.348 0.351 0.323 0.269

The table shows the sample mean of the cubic utility function u(xjh) x + h1x2 + h2x3 for the Fama and French market and the 10 size portfolios over the period July 1963–December 2001. The utility parameters h1 and h2 are taken from the GMM estimation results for (i) the model with the risk aversion condition and (ii) the model without the risk aversion condition; see Table 2.

and the reduction in downside risk associated with holding the market portfolio does not suﬃciently compensate them for the reduction in upside potential. Rather than holding a well-diversiﬁed portfolio, such investors will hold a less diversiﬁed portfolio with more upside potential.5 Since investors (both individual and institutional) actually hold highly undiversiﬁed portfolios (see, e.g. Levy, 1978), actual investors may indeed exhibit skewness preference (although there are several alternative explanations for ‘underdiversiﬁcation’). However, a representative investor with 5 Simkowitz and Beedles (1978, p. 939) already made this point: ‘Many investors hold less than perfectly diversiﬁed portfolios, a phenomenon in contradiction with frequently oﬀered advice. [. . .] If positive skewness is a desirable characteristic of return distributions, then the fact that the simple act of diversiﬁcation destroys skew is a likely explanation of observed behavior.’

T. Post et al. / Journal of Banking & Finance 32 (2008) 1178–1187

Fig. 3. Utility restrictions in (h1, h2)-space. The ﬁgure shows the contours of the restrictions (9), (13), (14) and (15) in (h1, h2)-space, for the full sample period from July 1963 to December 2001. The thick dotted line represents the admissible set of solutions that satisfy the restrictions. The optimal solution is denoted by ðh1 ; h2 Þ.

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the risk aversion condition, and they again imply a reverse S-shaped utility function. Unfortunately, we cannot present the results for the case with the risk aversion condition, for the simple reason that we cannot impose this conditions on the risk premiums in a straightforward manner. In fact, this limitation provides our motivation for estimating the utility parameters rather than the risk premiums. Post and Levy (2005) also investigate the possibility that the market portfolio is optimal for inverse S-shaped utility functions, using the new criterion of ‘‘Markowitz stochastic dominance’’ (MSD). However, like the empirical tests of the 3M CAPM, their analysis is based on the ﬁrst-order conditions and hence cannot distinguish between a global maximum and other stationary points. Indeed, our analysis suggests that the market portfolio is the global minimum for MSD investors and hence casts serious doubt on the ability of risk seeking to explain asset prices. 4.5. Theoretical bounds on gamma premium

skewness preference is unlikely to explain asset prices, as such an investor would have to invest in the market portfolio. We focus on estimating the utility parameters here, and we may ask if directly estimating the risk premiums gives comparable results. In our case, OLS estimation of Eq. (5) gives an estimated beta premium of q1 = 0.318 (0.000) and a gamma premium of q2 = 0.217 (0.001), and R2 equals 92%. These results are very similar to our results without

We may ask if our results are speciﬁc for the estimation methodology (GMM), the set of benchmark portfolios (ME deciles portfolios), the sample period (July 1963– December 2001), the return interval (monthly returns), or composition of the market index (equity only). To answer 2 and this question, we compute the theoretical bounds q q2. Again, these bounds apply for all estimation methodologies and sets of benchmark assets. We compute the

Table 4 Bounds gamma premium Period

Mean

Panel A: 120 months subsamples 1963–1973 0.253 1973–1983 0.373 1983–1993 0.587 1991–2001 0.676 Return interval

Mean

Standard deviation 3.69 5.14 4.49 4.19

Skewness

b+

q2

2 q

% MRP

0.34 0.03 1.08 0.89

8.98 16.05 12.42 7.94

0.000 0.002 0.000 0.000

0.017 0.000 0.099 0.099

7 0 17 20

q2

2 q

% MRP

0.001 0.031 0.119 0.842

4 7 8 15

Standard deviation

Skewness

b+

Panel B: Return interval Daily 0.021 Monthly 0.471 Quarterly 1.479 Yearly 5.679

0.87 4.47 8.48 15.97

0.933 0.502 0.448 0.559

8.64 16.05 22.84 29.97

% Bonds

Standard deviation

Skewness

b+

q2

2 q

% MRP

0.502 0.358 0.138 0.137 0.357 0.452

16.05 13.72 11.38 9.98 11.98 13.98

0.000 0.000 0.000 0.005 0.007 0.006

0.031 0.022 0.013 0.003 0.000 0.000

7 6 4 1 1 1

Mean

Panel C: Mixed market portfolio 0 0.471 20 0.402 40 0.334 60 0.265 80 0.196 100 0.128

4.47 3.78 3.21 2.81 2.69 2.86

0.000 0.000 0.000 0.000

2 for varying sample periods, return interval and market index compositions. Panel A We estimate the theoretical bounds for the gamma premium q2 and q shows the estimated bounds for monthly excess returns on the CRSP-all share index for the 10-year subsamples July 1963–June 1973, July 1973–June 1983, July 1983–June 1991, and January 1991–December 2001. Panel B shows the estimated bounds for the daily, monthly, quarterly and yearly CRSP-all share index. Panel C includes the results for mixed portfolios varying from all-equity (0% bonds) to all-bonds (100% bonds). The table displays the sample descriptive statistics: market risk premium (mean), standard deviation, skewness and maximum return. Further, the theoretical bounds are in the columns 2 , where the ﬁnal column gives the absolute value of the bound on the gamma premium as a percentage of the market risk premium labeled q2 and q (% MRP).

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bounds for the full sample period from July 1963 to December 2001, as well as four 10-year subsamples: July 1963–June 1973, July 1973–June 1983, July 1983–June 1993, and January 1991–December 2001. In addition, we consider daily, monthly, quarterly and yearly returns and market portfolios that mix stocks and bonds, ranging from an all-equity index to an all-bonds index. For the full sample of monthly returns, Fig. 3 shows the contours of the restrictions (9,13,14) and (15) in (h1, h2)space. The admissible set, i.e. the parameter values that satisfy the restrictions, is very small. The maximum for h2 is achieved at h2 ¼ 0:0002. At the maximum, the risk aversion condition (13) is binding, and relaxing this condition will increase the gamma premium. The minimum is equal to 2 is exactly equal to the gamma premium q2 = 0. Further, q implied by the GMM parameter estimates (see Table 2). This reﬂects the ﬁnding that the model without the risk aversion condition yields an inverse S-shaped utility function and that the risk aversion condition is violated. Our bounds to the maximal annualized gamma premium are thus only 0.4%, which represents a small proportion (7%) of the total market risk premium. This maximum gamma premium value is much lower than reported in other empirical studies. Table 4 summarizes the results for the diﬀerent samples, return intervals and market portfolio compositions. Previous 3M CAPM studies report values for the gamma premium ranging between 40% and 60% of the total market risk premium, much higher than the values we ﬁnd in the most opportune sample, return interval or market portfolio. The gamma bound, measured as a percentage of the market risk premium, is maximal for the sample 1991– 2001 (20%), for yearly returns (15%) and for the all-equity index (7%). For other subsamples, return intervals and mixed portfolios, the maximum gamma premium is substantially lower. As expected, the high gamma bounds coincide with high levels of market skewness and/or limited maximum return (b+). For example, yearly returns are characterized by high negative skewness (0.559) and the sample 1991–2001 by a low maximum return (7.94%). Of course, the explanatory power of gamma also depends on the speciﬁc set of benchmark portfolios and the diﬀerences in co-skewness between these portfolios. Still, the low values for the bounds suggest that we need unrealistically large diﬀerences in co-skewness to obtain substantial explanatory power. Hence, our conclusion from the GMM results seems to apply also for other estimation methodologies, sets of benchmark assets, sample periods and return intervals; gamma is unlikely to help explain asset prices. 5. Conclusions The usual 3M CAPM tests fail to impose the standard regularity condition of concavity or risk aversion. We present theoretical and empirical evidence that the empirical results severely violate this condition. If we impose risk aversion, then gamma has minimal explanatory power

for asset prices. If we do not impose risk aversion, then the implied utility function takes an inverse S-shape with risk aversion up to some return level and risk seeking beyond that level. Unfortunately, the ﬁrst-order conditions are not suﬃcient to guarantee that the market portfolio is the global maximum for the expectation of this utility function. In fact, our empirical results suggest that the market portfolio is more likely to represent the global minimum. This may reﬂect the empirical fact that stocks generally are more strongly correlated in falling markets than in rising markets. Consequently, investing in the market portfolio yields a relatively small reduction in downside risk at the cost of a relatively large reduction in upside potential. For investors with skewness preference, the small reduction in downside risk does not suﬃciently compensate for the large reduction in upside potential. Put diﬀerently, skewness preference cannot explain why the market portfolio is mean–variance ineﬃcient, because the market portfolio has a relatively high negative skewness relative to less diversiﬁed portfolios. Our results lead us to believe that the usual theoretical interpretation of the gamma premium is not valid. We do not deny the statistical association between mean, beta and gamma, and results of, e.g. the thorough empirical study by Harvey and Siddique (2000) remain fascinating. However, we do call into question the economic interpretation that is typically given to this association. Speciﬁcally, the large gamma premium is unlikely to represent the price that investors are willing to pay for assets that increase the skewness of the market portfolio; if investors are risk averse, then the gamma premium will be very small, and if investors are risk seeking, then they will not hold the market portfolio (and gamma is not an appropriate risk measure). Rather, the large gamma premium suggests that gamma serves as a proxy for omitted variables that do explain asset prices. In a follow-up study, Poti (2005) conﬁrms that for the typical stock the co-skewness premium must be substantially smaller than the covariance premium if utility is required to be well-behaved. Furthermore, he shows that a large gamma premium implies a very volatile pricing kernel and unduly high maximal Sharpe ratios. Further research on asset pricing models with heterogeneous investors and skewness preference, e.g. along the lines of Levy (1978), may help to identify these omitted variables. This paper focuses on the 3M CAPM and the economic meaning of the gamma premium. More generally, we may question the economic meaning of many asset pricing studies. Typically, such studies impose strong parametric structure that does not follow from economic principles. In addition, conditions that do have economic meaning, like non-satiation and risk aversion, frequently are not imposed and many studies do no not report whether the results satisfy these conditions. An alternative to explicitly imposing the regularity conditions, is using asset pricing models that rely on well-behaved utility functions. One example is the mean–semivariance

T. Post et al. / Journal of Banking & Finance 32 (2008) 1178–1187

CAPM by Hogan and Warren (1974), Hadar and Russell (1969), and Bawa and Lindenberg (1977), which replaces variance with semivariance and replaces the regular market beta with a downside market beta that emphasizes the comovements with the market portfolio in a falling market. This model basically assumes a utility function that is quadratic for losses and linear for gains and avoids violations of non-satiation. Interestingly, Price et al. (1982) show that the regular beta underestimates the risk for low-beta stocks and overestimates the risk for high-beta stocks, and Post and van Vliet (2005) empirically show that the MS CAPM better explains the returns of portfolios sorted both on beta and downside beta than the standard MV CAPM does. Another example is the model used by Rubinstein (1976), Breeden and Litzenberger (1978) and Leland (2003) based on an exponential utility function, which is a canonical example of a well-behaved utility function. Acknowledgement We thank Marien de Gelder for programming assistance. We appreciate the comments by two anonymous referees, Winfried Hallerbach, Jan van der Meulen, Nico van der Sar and Jaap Spronk, as well as participants at the 30th and 32th meeting of the EURO Working Group on Financial Modeling, FMA European conference 2003 Dublin, and seminars at Catholic University of Leuven, Erasmus University Rotterdam, Ghent University and Maastricht University. Financial support by Tinbergen Institute, Erasmus Research Institute of Management and Erasmus Center of Financial Research is gratefully acknowledged. Any remaining errors are our own. References Bawa, Vijay, Lindenberg, Eric B., 1977. Capital market equilibrium in a mean-lower partial moment framework. Journal of Financial Economics 5, 189–200. Breeden, Douglas T., Litzenberger, Robert H., 1978. Prices of statecontingent claims implicit in option prices. Journal of Business 51, 621–652. Fama, Eugene F., French, Kenneth R., 1992. The cross-section of expected stock returns. Journal of Finance 47, 427–465.

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