Aggregation of preferences for skewed asset returns

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ScienceDirect Journal of Economic Theory 154 (2014) 453–489 www.elsevier.com/locate/jet

Aggregation of preferences for skewed asset returns ✩ Fousseni Chabi-Yo a , Dietmar P.J. Leisen b,∗ , Eric Renault c a Fisher College of Business, Ohio State University, United States b Gutenberg School of Management and Economics, University of Mainz, Germany c Department of Economics, Brown University, United States

Received 4 October 2013; final version received 22 August 2014; accepted 26 September 2014 Available online 5 October 2014

Abstract This paper characterizes the equilibrium demand and risk premiums in the presence of skewness risk. We extend the classical mean-variance two-fund separation theorem to a three-fund separation theorem. The additional fund is the skewness portfolio, i.e. a portfolio that gives the optimal hedge of the squared market return; it contributes to the skewness risk premium through co-variation with the squared market return and supports a stochastic discount factor that is quadratic in the market return. When the skewness portfolio does not replicate the squared market return, a tracking error appears; this tracking error contributes to risk premiums through kurtosis and pentosis risk if and only if preferences for skewness are heterogeneous. In addition to the common powers of market returns, this tracking error shows up in stochastic discount factors as priced factors that are products of the tracking error and market returns. © 2014 Elsevier Inc. All rights reserved. JEL classification: D52; D53; G11; G12 Keywords: Skewness; Portfolio; Aggregation; Stochastic discount factor; Polynomials of pricing factors; Market return

✩

We thank Kenneth Judd for very helpful conversations. This paper has benefited from numerous comments and suggestions by the anonymous referee and associate editor; we are very grateful to both. * Corresponding author at: Faculty of Law and Economics, University of Mainz, 55099 Mainz, Germany. Fax: +49 6131 3923782. E-mail address: [email protected] (D.P.J. Leisen). http://dx.doi.org/10.1016/j.jet.2014.09.020 0022-0531/© 2014 Elsevier Inc. All rights reserved.

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1. Introduction Empirical studies documented that securities’ returns are not normally distributed; the seminal work of Harvey and Siddique [24] showed that skewness risk is an important component in the risk-premium. This renewed interest in the compensation of skewness risks and led to an active literature stream.1 This stream typically assumes that the aggregation of preferences leads to stochastic discount factors that are polynomials in the market return, conditional on the current information set. However, it is well known that the aggregation of preferences in incomplete markets leads to a representative agent where the weight of each individual is stochastic; thus, the stochastic discount factor may actually depend on individual securities due to unspanned powers of the market return. We study in detail the aggregation of preferences with skewed returns in a single-period model of investing. Our paper proceeds through two steps. Although the focus of our analysis is on aggregation of (heterogeneous) investors, our first step studies the case of a representative agent. We show that the common practice of using polynomials in the market return is warranted as long as the representative investor is myopic, i.e. as long as she only cares about investing over the next time period2 ; in line with this, we link our results to the beta pricing relationships proposed by Harvey and Siddique [24]. Our major second step studies equilibrium risk premiums and demand with heterogeneous investors. We derive individual demand and prove a three-fund separation theorem: agents hold the risk-free asset, the market portfolio and a new, so-called skewness portfolio, in proportions that reflect their preferences for variance and skewness risk. The skewness portfolio is the portfolio that provides the optimal hedge to the squared market return. We show that an asset’s skewness risk is priced as long as the co-skewness of this asset with the market portfolio as well as the aggregate skew-tolerance do not vanish. The equilibrium contribution to the risk-premium of individual stocks is driven by their covariance with the squared market return; it supports the common practice of using a stochastic discount factor (SDF) that is quadratic in the market return. A tracking error shows up in hedging the squared market return with the available securities; beyond average skew-tolerance, this leads to an additional lower order contribution to the risk premium through kurtosis due to the so-called cross-sectional variance of investor’s skewtolerances. Put differently, there may be an additional priced factor that contributes to skewness risk: the product of the market return with the tracking error in the squared market return. Another tracking error shows up in hedging the cubed market return with the available securities; beyond average skew-tolerance, this leads to an additional lower order contribution to the risk premium through pentosis (the fifth moment) due to the cross-sectional variance and the so-called cross-sectional skewness of investor’s skew-tolerances. 1 For example, Dittmar [14] and Barone-Adesi et al. [3] study how skewness risk is priced; Chung et al. [10] test whether Fama–French factors proxy for skewness and higher moments; Engle [18] links the skewness that shows up in asymmetric volatility models to systemic risk; Boyer and Vorkink [6] examine the impact of skewness preference on option prices; Smetters and Zhang [42] generalize the Sharpe ratio to rank non-normal risks in portfolio evaluation. 2 At the end of this paper (the last Subsection 5.4) we study briefly a two-period extension for the representative agent, i.e. she is no longer myopic. We argue that at least one additional priced factor shows up that is missing in polynomials of market returns: it captures intertemporal changes in the investment opportunity set. A detailed analysis of aggregation and additional priced factors in a two-period model is beyond the focus of this paper; it is carried out in Chabi-Yo et al. [8]. Further extensions to multiple periods (more than two) would be interesting as they can shed light on long-dated assets, i.e. assets with payoffs that are far in the future, see Martin [33].

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Our paper makes several contributions. First, we provide a structural analysis of individual and aggregate3 demand functions, as well as equilibrium risk premiums and associated portfolio holdings. We revisit the seminal contribution of Samuelson [39], see also the small risk analysis in Gollier [22], but depart from it in two directions: for one, we allow for a richer pattern of risk premiums in order to analyze the pricing of skewness risk; in addition, we do not intend to interpret his small noise expansion parameter as we are interested only in structural properties of risk-premiums and portfolio holdings. Thereby we confirm Samuelson [39] for myopic investors: the classical CAPM two fund separation theorem continues to hold as a first approximation; in addition we show that agents add the skewness portfolio in a second order approximation and that higher order approximations lead to lower order contributions of skewness risk through tracking errors. Second, we clarify the common practice of studying SDF that are polynomials in the market return. Most empirical studies looked at skewness extensions of the CAPM which add the squared market return as a factor; they justify this extension on ad-hoc Taylor-series expansions for agents’ utility functions that are truncated at the third-order term, see, e.g., Kraus and Litzenberger [30], Barone-Adesi [2], Dittmar [14]. We provide a rigorous foundation for this common practice and show that it is warranted in complete markets with myopic investors, that is when investors only care about one single time period and the squared market return can be hedged perfectly. When investors are not myopic or markets are incomplete, however, we show that several additional priced factors come up that depend on the cross-sectional variance and skewness of investors’ skew-tolerances. Our structural result unifies several theoretical insights. First of all, our results match previous ones about incomplete markets that investors’ heterogeneity and in particular the cross-sectional variance of investors’ characteristics may matter in equilibrium pricing, see, e.g., Constantinides and Duffie [13]. In addition, we show that aggregation of skewness risk for pricing is far more complicated than thought: the standard approach of studying polynomials of market returns (based on a postulated representative agent) misses several additional, priced factors that are products of market returns with tracking portfolios. Finally, it is well known about incomplete markets that the utility functions aggregate into a single so-called representative agent through stochastic weights, see, e.g., Magill and Quinzii [32]; our structural result therefore has the interpretation that these stochastic weights matter for pricing. Our structural analysis also provides an explanation for recent empirical evidence: Mitton and Vorkink [34] and Boyer and Vorkink [6] document for stocks and options, respectively, that more than co-skewness with the market is priced in stock returns; our unspanned factors provide a theoretical foundation for this. The outline of our paper is as follows. The next section discusses the general framework and relevant concepts. The third section studies the representative investor case, while the following sections focus on heterogeneous investors. Section 4 discusses demand and the portfolio separation theorem in the presence of skewness risk as well as the pricing of skewness risk. Finally, the first three subsections in Section 5 show for heterogeneous investors that the aggregation in incomplete markets leads to several risk factors that are unspanned by polynomial SDFs in the market return; the last Subsection 5.4 shows that dynamic pricing by a representative investor in a two-period extension leads to an additional, intertemporal priced factor. The paper concludes with Section 6. Lengthy proofs are postponed to Appendix A. The proofs of the two-period in 3 Structural properties of the market demand functions have been studied, e.g., by Hildenbrand [25] and Grandmont [23]. We focus on a portfolio choice problem.

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Subsection 5.4 follow the spirit of the proofs in Appendix A and are available as an online appendix. 2. General return framework This section introduces the general framework and the relevant concepts for understanding the market prices of higher order moments. Throughout, we consider a scale-location model for the vector R = (Ri )1≤i≤n of returns on the n risky assets of interest on a given period of time. More precisely, there exists a random vector Y = (Yi )1≤i≤n with zero mean vector under some given joint probability distribution, such that for σ > 0: R = E(R) + σ Y,

(1)

where E(R) is the expectation of R. We always assume that the vector Y admits finite moments at any required order and that the variance matrix Var(Y ) is positive and symmetric. Note that return volatilities are determined by the scale,4 parameter σ through Var(R) = σ 2 Σ,

with Σ = Var(Y ).

Since our interest focuses on (the cross section of) risk premiums, we introduce a risk free return Rf and decompose expected returns as follows: E(Ri ) = Rf + πi (σ ),

i = 1, ..., n;

here, πi (σ ) denotes the risk premium on asset i. This risk premium must obviously depend on the scale of risk σ and we will always assume that it is a smooth function of σ . In particular, when σ goes to zero, all asset volatilities go to zero and we expect risk premiums doing the same. Extending by continuity the risk premium function, we will assume throughout that: πi (0) = 0,

i = 1, ..., n.

We did not define our financial markets for σ = 0, since we would end up with (n + 1) risk-free assets. However, it makes sense to study the behavior of the risk premium function in a right neighborhood of zero, involving (by continuity extension) the value of the function πi (·) as well as of its derivatives at σ = 0. Smoothness properties of risk premiums functions will be implied by assumed smoothness properties of investors’ utility functions at stake. Let us assume for the moment that some investor with utility function u chooses her portfolio as solution of the first order conditions     E u W (σ ) (Ri − Rf ) = 0, for i = 1, ..., n. (2) These optimality conditions (2) are very general when the random variable W (σ ) stands for some wealth level produced by the portfolio choice of interest. These equations may stem from both static or dynamic portfolio optimization. We do not want to make this explicit at this stage, it is only important to assume that all randomness in W (σ ) is produced by asset volatilities and that it is erased when the scale of risk σ goes to zero. It is worth realizing that, up to required smoothness and integrability conditions, this very general setting already implies that risk premiums have a zero (right-hand) derivative at σ = 0: 4 In a different context Fama [19] solves the asset allocation problem when investors have homogeneous preferences and returns follow stable Paretian distributions that can have infinite variance. The scale parameter γ in his setup relates to the parameterization of the distribution, see p. 406 in Fama [19], whereas our scale parameter relates to return volatilities.

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Lemma 1. If u (W (0)) is a deterministic non-zero variable, we have πi (0) = 0 for all i = 1, ..., n. Lemma 1 is consistent with the general result (see, e.g., Gollier [22], p. 56) that the optimal share of wealth to invest in a risky asset is approximately proportional to the ratio of the expectation and variance of the excess return. Lemma 1 implies that risk aversion considered in this paper has a second-order nature, as defined by Segal and Spivak [41]. (First order risk aversion is beyond the scope of this paper.) Based on Lemma 1, the scale parameter σ allows us to describe the pattern of risk premiums from their series expansions: πi (σ ) = E(Ri ) − Rf =

∞ 

aij σ j .

(3)

j =2

Samuelson [39] assumes that the above expansion involves only one term, i.e. he sets aij = 0 for all j > 2; this, however, appears overly restrictive for a thorough analysis of the price of risk coming from higher order moments beyond variance. In addition, Samuelson [39] is interested in small scale parameters σ ; as he intends to motivate the use of mean-variance analysis, he also puts forward an interpretation in continuous time, where the scale parameter σ corresponds to a small interval of time t . No such restrictions are involved in our setting defined by (1) and (3). This setting is actually general, as it only maintains smoothness assumptions about investors’ preferences that are necessary to ensure that risk premiums are analytical functions of the scale parameter σ . We will actually set the focus in this paper on the first four coefficients ai2 , ai3 , ai4 and ai5 , so that we even do not really assume that risk premiums are analytical functions. There are then two interpretations of our results: either we adopt the assumption of analytical functions and describe general properties of the market; or we may consider, under less restrictive assumptions, that our results are accurate descriptions only for small σ , see Samuelson [39] and more recently Judd and Guu [26]. Since our analysis focuses on the role of investors’ preferences,5 we assume throughout that only investor’s demand (may) depend on the scale of risk σ but that the supply of financial assets is given independently of σ . In other words, a market portfolio is given by a vector ξ of shares of total wealth invested in the n risky assets. Since there is also a risk-free asset and no short sale constraints, the vector ξ can be any vector in Rn with strictly positive entries. It defines the market return RM = ξ  R,

(4)

where we denoted by ξ  the transpose of the vector ξ . Throughout this paper we will refer by the superscript  to the transpose of a vector. From the extant literature on pricing of higher order moments (see Chung et al. [10] and references therein), we expect the risk premium on any risky asset i = 1, ..., n, to be determined by the various contributions of this asset to market variance, skewness, kurtosis, pentosis, etc. To study this, we adopt: 5 Our focus is on the demand side of financial markets and should be completed when the offer of financial assets is also be seen as a function of the scale parameter σ . However, such an extension would simply complicate notation without modifying the economic interpretation of results.

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Definition 1. For asset i = 1, ..., n, the systematic co-moments are defined as follows:       1 Cov Ri , RM − E(RM ) = Cov Yi , ξ  Y , 2 σ     2  2  1 ci,M = 3 Cov Ri , RM − E(RM ) = Cov Yi , ξ  Y , σ     3  3  1 di,M = 4 Cov Ri , RM − E(RM ) = Cov Yi , ξ  Y , σ     4  4  1 fi,M = 5 Cov Ri , RM − E(RM ) = Cov Yi , ξ  Y . σ We refer to bi,M , ci,M , di,M and fi,M as the market beta, the market co-skewness, the market co-kurtosis and the market co-pentosis, respectively. bi,M =

By comparison with standard notations, the definition of market moments and the asset comoments standardizes by the scale of risk; thus they do not depend on σ . This standardization simplifies the presentation but does not change the main messages of the paper. Note also that we set the focus on centered systematic co-moments and do not standardize by variance; this is different from, e.g., Chung et al. [10]. Our convention allows us to interpret the co-moments as shares of asset i in the aggregate corresponding moment namely respectively market variance VM , skewness SM , kurtosis KM and pentosis PM : n 

ξi bi,M =

  1 Var[RM ] = Var ξ  Y = VM , 2 σ

ξi ci,M =

 3  3  1  = SM , E RM − E(RM ) = E ξ  Y 3 σ

ξi di,M =

 4  4  1  = KM , E RM − E(RM ) = E ξ  Y 4 σ

ξi fi,M =

  5  5  1  = E ξ Y = PM . E R − E(R ) M M σ5

i=1 n  i=1 n  i=1 n  i=1

From an economic perspective, it is worth pointing out that even if the market skewness or pentosis is zero, some specific asset may have a non-zero systematic component and that this may play a role in the determination of its market price. Chung et al. [10] study asset pricing implications of a model that “does not stop at the second moment” and considers up to the fourth moment. They stress that “there is no reason to stop with the fourth moment”, i.e. even a higher order moment like pentosis may matter for asset pricing. They base their argument on the remark that “variance, skewness and kurtosis tell us something about the tails of the distribution, but they fall short of specifying the tail precisely”. In this paper, our main interest is in the price of skewness; we will argue that when considering investors with heterogeneous preferences for skewness, an asset pricing model that does not want “to stop with the second moment” must take also into account some specific pricing factors due to preference heterogeneity. To put it differently, we will argue that the simple fact that investors care for higher order moments implies that they need to track not only the market return, but also the pricing factor (RM − E(RM ))2 that shows up through market co-skewness. We will see that this has a significant pricing impact in case of market incompleteness (there is no perfect hedge

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of the squared market return) jointly with investor heterogeneity. To see this, we will study terms in the series expansion (3) as far as ai4 or ai5 and the associated pricing of market co-kurtosis and market co-pentosis. 3. Pricing kernels and moment preferences in representative agent analysis This section studies the pricing of (higher order) moments without taking into account complications due to preference heterogeneity: throughout this section, we assume that the representative investor has a known preference structure. This assumption will be questioned in the following sections through explicit discussions of aggregation issues. Ultimately, our analysis with heterogeneous investors implies that the representative investor paradigm falls short of properly characterizing the impact of preferences for skewness on financial asset prices. However, before carrying out this analysis we first recall in this section the classical picture of risk premiums assigned to systematic co-moments in the case of a representative investor. The pricing formulas in the literature that result from a Taylor expansion of the utility function of a representative agent (see, e.g., Harvey and Siddique [24] and Dittmar [14]) will find a more general interpretation in the context of our series expansions (3). 3.1. Risk premiums Let us consider a representative investor who maximizes expected utility E[u(W )] of her terminal wealth W obtained by investing her initial wealth q in a portfolio. Throughout, the portfolio is defined in terms of shares of wealth invested θ = (θi )1≤i≤n ; this gives terminal wealth:   n  W (σ ) = q Rf + θi (Ri − Rf ) . (5) i=1

The demand θ of the representative investor for risky assets solves the first order conditions (2) of her expected utility maximization where the wealth level W (σ ) is induced by the portfolio choice according to (5). Since the representative investor must hold the market portfolio, the first order conditions with θ = ξ determine the equilibrium pricing of the financial assets, i.e. the coefficients aij , j ≥ 2 of the risk premium expansion (3); these coefficients solve for all levels σ of risk:     E u W (σ ) Ri (σ ) − Rf = 0 for i = 1, ..., n,   n    ξi Ri (σ ) − Rf , subject to W (σ ) = q Rf + i=1

where Ri (σ ) − Rf = πi (σ ) + σ Yi

for i = 1, ..., n.

(6)

We are then able to prove: Theorem 1. The equilibrium risk premium πi (σ ) = E(Ri ) − Rf = ai2 =

1 bi,M , τ

ai3 = −

ρ ci,M , τ2

ai4 =

∞

j =2 aij σ

κ 1 − 3ρ di,M + VM bi,M , τ3 τ3

and ai5 = −

j

χ 3κ − ρ(1 − ρ) κ − ρ(1 − 2ρ) fi,M + VM ci,M + SM bi,M , τ4 τ4 τ4

fulfills

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where τ =− κ=

u (qRf ) , qu (qRf )

ρ=

[u (qRf )]2 u[4] (qRf ) , 6[u (qRf )]3

u (qRf )u[3] (qRf ) , 2[u (qRf )]2 χ=

[u (qRf )]3 u[5] (qRf ) . 24[u (qRf )]4

Theorem 1 puts forward the preference parameters τ , ρ, κ, χ that characterize the representative investors’ concerns for higher order moments. Following Scott and Horvath [40], we expect investors to have a negative preference for even moments and a positive preference for odd ones. Caballe and Pomansky [7] have shown that this property is displayed by utility functions exhibiting “mixed risk aversion”, that is having all odd derivatives positive and all even derivatives negative. As noted by Pratt and Zeckhauser [37], the latter property means that utility functions have completely monotone first derivatives, a property known to correspond to Laplace transforms of probability distributions on the positive real line. Eeckhoudt and Schlesinger [16] have provided some theoretical underpinnings for this property in terms of “risk apportionment”. We will then rely upon this literature to maintain throughout the following assumptions: u[k] (qRf ) ≤ 0 for k even,

and u[k] (qRf ) ≥ 0 for k odd.

Under this assumption, all the preference parameters τ , ρ, κ, χ of Theorem 1 are nonnegative; since they contain successively a higher order derivative of the representative investor’s utility function u, we call them risk tolerance, skew tolerance, kurtosis tolerance and pentosis tolerance, respectively.6 The main insight of a higher moment CAPM based on a representative investor is, that in pricing only matter systematic volatility, skewness, kurtosis, pentosis, etc.; Theorem 1 matches this intuition by displaying risk premiums proportional to market co-moments bi,M , ci,M , di,M , fi,M , etc. Up to some additional terms discussed below, the proportionality coefficients are defined by the right preference parameters. First, we find that the relative risk aversion (Rf /τ ) weights the systematic volatility bi,M of asset i. This risk premium corresponds to the traditional Sharpe–Lintner CAPM and we just reproduce here the key insight of Samuelson [39]: we match the traditional CAPM by focusing on the first term ai2 σ 2 . Second, while Samuelson [39] did not introduce higher order terms, we find here that co-skewness weighted by opposite of skew tolerance ρ (or more precisely (−ρ/τ 2 )) is important, as displayed in the next term ai3 σ 3 : this has been well known since Kraus and Litzenberger [30]. Similarly, the terms (κdi,M /τ 3 ) in ai4 σ 4 reproduce the four moment CAPM studied by Dittmar [14]. A five moments CAPM comes with the term (−χfi,M /τ 4 ) in ai5 σ 5 . Note that additional terms, not captured by a Taylor expansion along the lines of Dittmar [14], show up in the formulas for ai4 and ai5 . They imply an additional pricing impact of market beta and market co-skewness. While their economic significance will be discussed below through a 6 Skewness tolerance is related to the prudence concept of Kimball [28]; kurtosis tolerance is related to the temperance concept of Kimball [29] and to risk vulnerability of the utility function in the presence of background risks, see Gollier [22]. Eeckhoudt and Schlesinger [16] study lotteries with the property that prudence and temperance can be fully characterized by a preference relation over these lotteries. Pentosis tolerance is driven by the fifth derivative of the utility function. It is related to the concept of edginess that Lajeri-Chaherli [31] introduced to study decision making in the presence of background risks. Eeckhoudt [15] discusses the sign of various derivatives (third, fourth, fifth) of the utility function.

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study of the market risk premium, the mathematical explanation of this difference is simple. The proof of Theorem 1 shows that omitting these terms is akin to making only the series expansion of the marginal utility while forgetting that the optimality condition (whose expansion is called for) actually involves the product of marginal utility times net return. Even without considering these additional terms, the empirical evidence about the usefulness of higher order moments CAPM is mixed. However, the disappointing results may be due more to statistical issues than to the lack of economic content. One problem is the poor finite-sample behavior of sample counterparts of higher order moments. Since we know that asset returns have rather fat tails, finite-sample averages of returns to the power 3, 4, etc. are likely to be far from their hopefully asymptotic Gaussian behavior. Moreover, a proper empirical study of the CAPM should rather set the focus on conditional moments, whose estimation is even more problematic. A recent strand of literature, including Chang et al. [9] and Conrad et al. [12], tries to circumvent these difficulties by using moment estimates extracted from option prices data. While the former sets the focus on market skewness and kurtosis from daily S&P500 option data, the latter goes even further by considering options written on individual securities. Consistent with our theoretical result, Chang et al. [9] find that stocks with high exposure to innovations in market skewness exhibit low returns on average, while exposure to market kurtosis implies somewhat higher returns on average. The evidence is weaker for kurtosis but this may be explained by the well-documented large negative correlation between skewness and kurtosis innovations that makes it difficult to fully separate the effects of skewness and kurtosis. An additional difficulty is that the technique used for extracting moment estimates from option prices, stemming from the work of Bakshi and Madan [1] (that shows how to span any smooth payoff function by a portfolio of calls and puts), is able to deliver only risk-neutral moments. It is actually well-known that option-implied skewness and kurtosis have an alternative interpretation as a measure of jump risk or downside risk. To convert risk-neutral moments to their historical counterpart, one should choose one from many possible specifications of the volatility risk and jump risk premiums (see, e.g., Bates [5] and Pan [36]). However, Conrad et al. [12] do check that there is a positive and statistically significant relation between estimates of historical and risk-neutral moments. As far as practical implications of Theorem 1 are concerned, some possibly more robust conclusions can be drawn by setting the focus on the case i = M, that is on the market risk premium. The result of Theorem 1 can then be rewritten as follows: Proposition 1. Up to terms of order higher than σ 5 , the risk premium on the market is:

 5  1 1 − 3ρ 2 κ − ρ(1 − 2ρ) 3 2 aMj σ j = σ V + σ S + M M σ VM τ τ3 τ4 j =2

 ρ 3κ − ρ(1 − ρ) 2 κ χ + − 2+ σ VM σ 3 SM + 3 σ 4 KM − 4 σ 5 PM . τ τ4 τ τ Barro [4] provides a striking empirical illustration of the consequences7 of Proposition 1. Following the aforementioned intuition that skewness and excess kurtosis may be caused by 7 The risk premium formula in Barro [4] is consistent with that in our Proposition 1, ignoring the pricing impact of terms with higher order than σ 3 . To see this, we use the approximation (1 − b)−α − (1 − b)1−α ≈ b + b2 α + b3 12 α(α + 1) + O(b4 ), and the approximation log(1 + x) ≈ x, that is log E[RM ] ≈ E[RM ] − 1 and log Rf ≈ Rf − 1 and find that his Eq. (13) on p. 838 becomes log E[RM ] − log Rf ≈ E[RM − Rf ] ≈ α(Var(ut+1 ) + p(1 − qBarro )E(b2 )) +

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some jump risk, the idea is to introduce rare “disasters”. A disaster is an event with a small probability of occurrence but which has a large negative impact on output when it happens. This creates negative skewness and higher kurtosis in the distribution of the output and in the market return as well. As a consequence, the market risk premium, instead of being explained only by the variance term (σ 2 VM /τ ), is significantly increased by a skewness term (−ρσ 3 SM /τ 2 ) and possibly also a kurtosis term (κσ 4 KM /τ 3 ). This allows Barro [4] to explain in particular the equity premium puzzle. Barro [4] sets the focus on the explanation by negative skewness; one may indeed wonder whether the analysis could not be limited to terms of order up to σ 3 while the additional terms coming from the part [aM4 σ 4 + aM5 σ 5 ] could be overlooked. Besides the kurtosis and pentosis tolerances, these terms actually imply an additional impact of market variance and skewness in the market risk premium. To see that, let us first assume for simplicity that the market return is not skewed. Then, we see from Proposition 1 that the derivative of the risk premium on the market return with respect to its variance Var(RM )(= σ 2 VM ) is: M =

1 1 − 3ρ Var(RM ). +2 τ τ3

(7)

In other words, the dependence of the market risk premium on the market variance is not fully characterized by the constant positive slope (1/τ ) but is also impacted by an additional term (typically brought by the expansion term aM4 σ 4 ). We argue below that this additional term is not negligible and may even make the total M negative for sufficiently large levels of risk aversion (small τ ) and reasonable values of market volatility. This is reminiscent of the result of Rothschild and Stiglitz [38] that has been presented in a different context: in a one-safe-one-risky asset model, they show that an increase in the riskiness of the risky asset does not necessarily reduce the demand for it by risk-averse investors whose utility function is not quadratic. In Rothschild and Stiglitz [38], an increase in risk is defined by a mean-preserving spread of the return, that is conformable to the common concept of Seconddegree Stochastic Dominance (SSD). Several authors (see Gollier [21] and references therein) have addressed this puzzle by defining a tighter concept of increased riskiness that is not comparable to SSD. By contrast, we note here that when keeping the SSD concept to define an increase in risk, our result (7) actually just confirms the result of Rothschild and Stiglitz [38]: an increase in risk has no obvious comparative statics property, since an increase of market variance does not imply that investors will always request a higher expected market return in equilibrium. To make the comparison even more precise in quantitative terms, let us illustrate formula (7) in the case of an utility function with constant relative risk aversion (CRRA): u (w) = w −α ,

α > 0.

We can then easily check that: τ=

Rf , α

ρ=

1 1 + , 2 2α

α(α+1) p(1 − qBarro )E(b3 ). (Here we denoted (differently from Barro [4]) by α the CRRA risk aversion parameter and 2 by qBarro the default probability of his government bond.) Structurally, it is important to note that E[b2 ] and E[b3 ] relate

to the variance and skewness of disaster shocks, weighted by risk tolerances α and skew-tolerances α(α + 1)/2, see our CRRA calculations later in this subsection.

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so that: M = α

Var(RM ) Rf3



Rf2 Var(RM )

 − α(α + 3) .

(8)

We obviously deduce from (8) that there exists a strictly increasing function α¯ such that

Rf M > 0 ⇐⇒ α < α¯ √ , Var(RM ) with the property that α(2) ¯ = 1. In other words, the upper bound α¯ = 1 pointed out by Rothschild and Stiglitz [38] (see their statement on the top of p. 72) corresponds for us to the value:  Rf = 2 Var(RM ). This means that our analysis confirms that for plausibly large values of α, a risk averse investor may choose to hold more of the risky asset when its volatility increases. While in their framework Rothschild and Stiglitz [38] stress that this surprising fact may occur as √soon as α > 1, we slightly discount their statement by finding that it actually takes α > α(R ¯ f / Var(RM )). However, it is worth keeping in mind that we found this latter result by assuming that the market return is not skewed. If it is negatively skewed as in Barro [4], the surprising result of a negative derivative of market risk premium with respect to its variance (for given skewness) is even more likely to occur. To see that, note that from Proposition 1, this derivative is in the general case:

 1 κ − ρ(1 − 2ρ) 3κ − ρ(1 − ρ) 3 1 − 3ρ M = + 2 Var(RM ) + + σ SM . τ τ3 τ4 τ4 The two additional terms, both brought by the expansion term aM5σ 5 , are typically negative when the market skewness (σ 3 SM ) is negative. For instance, in the CRRA case described above, we have: κ=

(α + 1)(α + 2) , 6α 2

so that: 1 1 5 + > 0, + α 6α 2 6 3 1 5 3κ − ρ(1 − ρ) = + > 0. + 2α 4α 2 4 κ − ρ(1 − 2ρ) =

3.2. Quadratic SDF A convenient way to describe an asset pricing model is through the stochastic discount factor (SDF), see, e.g., Cochrane [11], i.e. through a random variable H with the property that risk premiums are given by E[Ri ] − Rf = −Rf Cov(H, Ri ),

for i = 1, . . . , n.

The literature usually considers time-series applications, where expectations are conditional on the information available at each date. In line with our earlier analysis and for simplicity we do not write out this dependence explicitly. Thus, essentially, our analysis appears to boil down to a single-period analysis but the reader should keep in mind the conditional viewpoint.

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It is well known that the Sharpe–Lintner CAPM is tantamount to an SDF that is affine w.r.t. the market return. To accommodate some widely documented departures from the CAPM, empirical asset pricing may consider higher-order polynomials in the market return as a candidate SDF. For instance a quadratic SDF should accommodate pricing of skewness, see, among others, Harvey and Siddique [24] and Dittmar [14]; up to a constant, a quadratic SDF, H (2), has the form 2 Rf H (2) = λ¯ 1 RM + λ¯ 2 RM ,

(9)

for suitable parameters λ¯ 1 , λ¯ 2 . It is well-known that the quadratic SDF leads to a linear pricing relationship for excess returns:   2 , (10) E[ri ] = λ1 Covt [ri , rM ] + λ2 Covt ri , rM where ri = Ri − Rf , rM = RM − Rf stand for excess returns on asset i and the market, respectively. It is tempting to apply the linear pricing relationship (10) to the squared market return; to 2 corresponds to a do so, let us assume for a moment and for purely illustrative reasons that rM 2 and thus r 2 may portfolio available in the market. (We will discuss in the next section that RM M 2 /η − R the not be a portfolio in incomplete markets.) We denote by η the price and by r¯M = rM f excess return on the squared market portfolio; applying the linear pricing relationship (10) to this excess return gives  2  2 2 rM rM 2 rM E − Rf = λ1 Cov rM , + λ2 Cov rM , . η η η The linear pricing relationship (10) can also be applied to the market excess return; this gives a system of two equations that can be resolved: λ1 = λ2 =

2 )E[r ] − Cov(r , r 2 )(E[r 2 ] − ηR ) Var(rM M M M f M 2 ) − (Cov(r , r 2 ))2 Var(rM ) Var(rM M M 2 ] − ηR ) − Cov(r , r 2 )E[r ] Var(rM )(E[rM f M M M 2 ) − (Cov(r , r 2 ))2 Var(rM ) Var(rM M M

, .

One may be tempted to set η = 0; this characterization of λ1 , λ2 then coincides with formulas (7b), (7c) put forward by Harvey and Siddique [24]. However, this appears to be at odds with a no-arbitrage condition, since η is the price of a strictly positive payoff and should therefore be strictly positive. Also, when markets are incomplete we will stress in the next section that the squared market returns may not be in the asset span, such that its price cannot be observed easily. Our analysis in Subsection 3.1 provides a foundation for the common practice of quadratic SDF and allows us to relate the terms λ1 , λ2 to preference parameters. Based on our analysis of ai2 σ 2 + ai3 σ 3 we can identify the parameters λ¯ 1 , λ¯ 2 in the quadratic SDF of Eq. (9); we have, up to terms higher then σ 3 :   1 ρ bi,M σ 2 − 2 ci,M σ 3 = ai2 σ 2 + ai3 σ 3 = E[Ri ] − Rf = −Rf Cov H (2), Ri . τ τ Based on Definition 1, this shows: Proposition 2. Up to risk premium terms of order higher than two in σ , the SDF is quadratic; in particular it fulfills up to a constant 2 1 ρ Rf H (2) = − RM + 2 RM − E(RM ) . (11) τ τ

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Our characterization of the SDF in Eq. (11) allows us to identify to the SDF in Eq. (9), up to a constant: 1 ρ λ¯ 1 = − − 2 2 E[rM ], τ τ

λ¯ 2 =

ρ . τ2

The price of skewness is contained in the sensitivity λ¯ 2 : when it is zero, we are back in a meanvariance pricing world, while λ¯ 2 = 0 leads to mean-variance-skewness pricing. Note that here λ¯ 2 is something like a structural invariant, only varying through the value of preference parameters computed from the derivatives of the utility function at Rf . Ultimately, estimating the size of λ¯ 2 is at the core of empirical studies, see, e.g., Harvey and Siddique [24]. Overall, we find that analyzing a myopic representative agent confirms the common practice of quadratic SDF. Ultimately, we will elaborate in the first three subsections of Section 5 that this paradigm breaks down with myopic investors: we will take up the issue in great detail with heterogeneous agents and point out that there are priced factors in incomplete markets that do not show up in complete markets. The last Subsection 5.4 shows that it breaks down also for the representative agent analysis when the single-period setup is extended to a two-period one. 4. Portfolio separation and pricing of skewness risk with heterogeneous agents The previous section studied the case of a single (representative) investor. This section discusses demand of our heterogeneous investors and the pricing of skewness risk when we focus on the first two terms ai2 σ 2 + ai3 σ 3 ; along the way we also look into the impact of market incompleteness. Throughout, we assume the economy is populated by S (heterogeneous) investors s = 1, ..., S; each investor s maximizes expected utility E[us (Ws )] of her terminal wealth Ws obtained by investing her initial wealth qs in a portfolio. Also, we assume that the portfolio of each investor s = 1, ..., S is defined in terms of shares of wealth invested θs = (θis )1≤i≤n ; this gives terminal wealth   n  W s = q s Rf + θis (Ri − Rf ) . (12) i=1

Note that this is the heterogeneous investor analogue (for each investor s = 1, ..., S) of the single period wealth dynamic that we considered in the representative investor case, Eq. (5). The demand θs of investor s for risky assets solves the first order conditions of her expected utility maximization, where the first order condition for each investor s = 1, ..., S is merely Eq. (2) written out for the utility function us and terminal wealth Ws based on demand θs of each investor. Samuelson [39] characterized the impact of higher order return moments on the demand for risky assets through a series expansion of the optimal portfolio θs = (θis )1≤i≤n : θis =

∞ 

θisj σ j .

(13)

j =0

In Section 2 we assumed that the total offer of risky assets was a fixed vector ξ = (ξi )1≤i≤n of shares of the total amount S q¯ = Ss=1 qs of invested wealth and we continue to adopt this S assumption. Note that s=1 θis qs is the aggregate demand of any asset i = 1, ..., n. Therefore,

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we have the following market clearing conditions for each risky asset i = 1, ..., n and all orders j > 0: S 

θis0 qs = ξi S q, ¯

S 

and

s=1

θisj qs = 0.

(14)

s=1

This differs from our representative investor analysis (the previous section) in several ways. There, the total offer was ξ which effectively limited us to θ = ξ , i.e. all terms of order higher than 1 had to vanish. Here, this does not have to be the case at the level of individual investors; this means that investors can use higher order terms to hedge risks stemming from powers of market return not being in the asset span (in incomplete markets); this in turn will have an impact on risk premiums that only shows up when investors are heterogeneous. The market clearing conditions (14) together with the first order conditions for each investor determine the equilibrium asset demand and pricing of financial assets; the coefficients aij , j ≥ 2 of the risk premium expansion (3) solve for all levels σ of risk:   E us (Ws )(Ri − Rf ) = 0, for i = 1, ..., n,   n    θis (σ ) · Ri (σ ) − Rf , subject to Ws (σ ) = qs Rf + i=1

where Ri (σ ) − Rf = πi (σ ) + σ Yi ,

for i = 1, ..., n.

(15)

Samuelson’s key insight that we extend in this paper is the following: for any investor s, the first K coefficients θisj , j = 0, ..., K − 1, only depend on the first (K + 1) derivatives of the utility functions us , s = 1, ..., S, computed at the level of wealth qs Rf . (This level can be seen as the terminal wealth resulting from investing the initial wealth qs only in the safe asset.) This permits successive determination of demand and market clearing, starting from terms of order zero and going up one order at a time. We are then able to prove: Theorem 2. The equilibrium risk premium πi (σ ) = E(Ri ) − Rf =

∞

j =2 aij σ

j

fulfills

1 ρ¯ ai2 = bi,M , ai3 = − 2 ci,M , (16) τ¯ τ¯ where individual (τs ) and average (τ¯ ) risk tolerance coefficients as well as individual (ρs ) and average (ρ) ¯ skew tolerance coefficients are defined by τs = − ρs =

us (qs Rf ) , qs us (qs Rf )

τ¯ =

us (qs Rf )u s (qs Rf ) ,  2(us (qs Rf ))2

S 1  τ s qs , S q¯ s=1 S qs τ s ρ s ρ¯ = s=1 . S s=1 qs τs

(17)

The equilibrium demand of investor s is given through ¯ −1 τs τs (ρs − ρ) Σ c, (18) ξ, θs1 = τ¯ τ¯ 2 where θs0 = (θis0 )1≤i≤n , θs1 = (θis1 )1≤i≤n describe the zero, first order demand vector and c = (ci,M )1≤i≤n describes the vector of market co-skewness. θs0 =

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Theorem 2 has interesting implications for equilibrium risk premiums. Approximating the risk premium (E(Ri ) − Rf ) by its first two terms σ 2 ai2 + σ 3 ai3 is similar to extending the CAPM to an asset pricing model that takes into account skewness risk. Skewness risk is priced in equilibrium as soon as the properly defined average skew tolerance is non-zero and some assets display a non-zero co-skewness (while the market return may well be symmetric). Our result displays the aggregate impact on the skewness premium in case of investors heterogeneity. Heterogeneity across investors’ risk tolerances and skew tolerances τs , ρs (s = 1, ..., S) may come either from heterogeneity of their utility functions us or from heterogeneity of their invested wealth qs . In both cases we find here the well-known result that heterogeneity does not really matter for mean-variance-skewness pricing: only the average risk tolerance coefficient τ¯ , ρ¯ matters. Note in particular that for homogeneous investors (same utility function u = us , same wealth q = q¯ invested), the risk premium on variance and on skewness risk would simply be that of Theorem 1. Let us now take a look the implications of Theorem 2 for equilibrium asset demand. We have seen that the Sharpe–Lintner CAPM gives a correct picture of actual risk premiums insofar as we focus on the first term ai2 σ 2 of their series expansion. Theorem 2 shows that the same applies to the equilibrium demand for risky assets, i.e. a standard mean-variance approach is sufficient insofar as we focus on the first term θs0 of the series expansion for the equilibrium demand of investor s. This is the celebrated two-fund separation theorem, where all investors hold the same portfolio of risky assets defined by portfolio weights ξ . By analogy with the CAPM relationship, let us study an additional fund defined by portfolio weights ξ sk that lead sk to a return R = Rf + ni=1 ξisk (Ri − Rf ) with the property σ 3 aj 3 = −

  ρ¯ Cov Rj , R sk , 2 τ¯

for all j = 1, . . . , n.

(19)

Note that this equation defines ξ sk as a unique solution of a linear system of n equations with non-singular matrix Var(R). Using cM = (ci,M )i of Definition 1, Theorem 2 states that ξ sk = σ Σ −1 cM . This new portfolio plays the same role for pricing skewness risk that the market portfolio played to price (co-)variance risk. Therefore, throughout this paper we refer to it as the skewness portfolio. Not surprisingly, it also appears in investors’ equilibrium demand for risky assets, i.e. we find: σ θs1 =

¯ sk τs (ρs − ρ) ξ . τ¯ 2

(20)

Eq. (18) establishes a three-fund theorem: All investors hold the same two portfolios of risky assets (so-called mutual funds) defined by portfolio weights ξ and ξ sk , respectively. The only difference in actual holdings comes from wealth heterogeneity (ceteris paribus, the amount invested is proportional to wealth), heterogeneity in risk tolerances (the amount invested is proportional to risk tolerance at this wealth level) and, regarding the second mutual fund, heterogeneity in skew tolerances (the amount invested is proportional to relative spread between the skew tolerance of investor s and the average one). Thus, an investor with stronger or weaker preferences for skewness than the average must underdiversify with respect to mean-variance efficiency.

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Recall that the wealth weighted sum of τs ρs in relation to the wealth weighted sum of τs gives the average skew tolerance coefficient ρ. ¯ Therefore, although investors’ holdings in the skewness portfolio may be different from zero, aggregate demand of it vanishes.8 It is worth elaborating in more detail why investors will have non-zero positions in the skewness portfolio when their preferences for skewness depart from the average skewness preference. Comparing (16) and (19) shows that for all assets i = 1, ..., n:     2  Cov Ri , R sk = Cov Ri , RM − E(RM ) .

(21)

In other words, we have the following decomposition for the squared market return: 

 2  RM − E(RM ) = R sk − E R sk + T sk ,

(22)

where the so-called tracking error (T sk − E(T sk )) is uncorrelated with all traded asset returns Ri , i = 1, ..., n. Up to an additive constant, the skewness portfolio return R sk can therefore be interpreted as an affine regression of the squared (demeaned) market return on the traded asset returns. In other words, the skewness portfolio is the mutual fund that is best able to track the squared market return; thus, demand for the skewness portfolio is induced by the demand for tracking the squared market return.9 In complete markets, the squared market return can be replicated through traded assets and the tracking error disappears. In incomplete markets, however, the squared market return may not be in the asset span. The variance Var(T sk ) of the tracking error can then be interpreted as a measure of market incompleteness. The bigger this variance, the less accurate is the feasible hedge of the squared market return that a representative agent would like to hold for the sake of skewness preferences (see, e.g., Dittmar [14]). This interpretation in terms of market incompleteness will be confirmed in the next section. For further illustration, let us assume in the remainder of this section that all investors s = 1, . . . , S have utility functions us with constant relative risk aversion (CRRA) αs , i.e. us (w) = w −αs .

(23)

Note that within the CRRA utilities framework, the relative risk aversion coefficient is the key driver for all preference parameters. In particular, the corresponding risk and skew tolerance for an investor s = 1, . . . , S are τs =

Rf αs

and ρs =

αs + 1 . 2αs

(24)

The difference between heterogeneous and homogeneous investors lies in the dispersion of their risk preferences; here, this leads us to study the dispersion in investor’s risk tolerances Rf /αs . For this we define first wealth weights ωs∗ by setting for each investor qs ωs∗ = S

s=1 qs

.

(25)

8 Essentially this is due to our simplifying assumption that the offer of risky assets does not depend on the scale parameter σ . Changing this assumption may introduce higher order dependence. 9 In line with this reasoning, a careful examination of option markets would show that preferences for skewness introduce non-zero demand in asymmetric assets like options, see Judd and Leisen [27].

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Then, the average risk tolerance can be written as a wealth-weighted average of individual risk tolerances, i.e. τ¯ =

S  s=1

ωs∗

Rf . αs

(26)

We use the wealth weights to define a measure of dispersion in investors’ risk tolerances, 



2  S S S  Rf ∗ ∗ Rf ∗ Rf Var ωs − ωs = ωs∗ (τs − τ¯ )2 . = αs s αs αs s=1

s=1

(27)

s=1

The “variance” measure Var∗ captures market-wide quadratic variation (second order dispersion) in risk tolerances Rf /αs across investors. We find that ρ¯ =

(τ¯ /Rf ) + (τ¯ /Rf )2 + Var∗ ((Rf /αs )s )/Rf2 2(τ¯ /Rf )

,

(28)

i.e. the average skew-tolerance is driven by the average risk-tolerances and dispersion in investor’s risk aversion. Using Theorem 2, we can write the equilibrium risk premium as 1 Cov(Ri , RM ) τ¯   τ¯ + τ 2 /Rf + Var∗ ((Rf /αs )s )/Rf (29) Cov Ri , (RM − ERM )2 . − 3 2τ¯ In a framework, where all investors exhibit constant relative risk aversion (CRRA), this shows that the dispersion of (variation in) investor’s risk tolerance plays a significant role in explaining the price of co-skewness risk. In particular, a high dispersion of their risk tolerances amplifies the expected excess returns on individual stocks. Eq. (29) can be used to estimate the average risk tolerance τ¯ as well as the cross sectional variance of investor’s risk tolerance in the CRRA utility-based framework. E(Ri ) − Rf =

5. Unspanned SDF in polynomials of market return This section studies the priced factors in the SDF; the common theme is showing that the usual Taylor series approach based on the representative agents leads to additional priced factors that have been overlooked in the current literature. The first three subsections study, in the singleperiod setup with heterogeneous investors that forms the core of this paper, whether tracking errors in incomplete markets lead to additional priced factors. In addition, the last Subsection 5.4 shows in a two-period extension for a representative, price-setting investor that a dynamic factor shows up that has been missed so far; it captures intertemporal changes in the investment opportunity set. 5.1. The spanning issue with polynomials of market return In our analysis of heterogeneous investor so far, we approximated the risk premium in the single period case through the first two terms ai2 σ 2 + σ ai3 σ 3 of its expansion (3); in this approximation, a polynomial of degree two in the market return is a sufficient statistic to define the SDF, see Eq. (11); this is in accordance with the representative agent model of Dittmar [14].

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However, this ignores the impact of higher order terms in the expansion (3). Therefore, this section carries out a careful investigation of higher order terms aij σ j (j = 4, 5) in the expansion (3) and shows that polynomials in the market return do not capture completely the pricing impact of investors’ heterogeneity in incomplete markets. Following the logic of Harvey and Siddique [24] as well as based on the representative agent models in Dittmar [14] and Chung et al. [10], we may expect that additional terms in the expansion (3) lead to a characterization of the risk premium on asset i, i = 1, . . . , n as a linear function that consists not only of the standard market beta σ 2 bi,M and of the standard co-skewness σ 3 ci,M , but also of the co-kurtosis σ 4 di,M , the co-pentosis σ 5 fi,M , and so on, as we go further in the expansion. This would lead to SDFs represented as linear functions of RM and (RM − E(RM ))j , j = 2, 3, 4, ..., i.e. as polynomials in the market return RM . The problem with this reasoning is that it overlooks a form of market incompleteness, namely the impossibility to get a perfect hedge for powers of the market return from the payoff of linear portfolios. Let us characterize this issue for the power j = 2. As explained in Section 4 above (see Eq. (22)) there is in general a tracking error:   2  RM − E(RM ) = R sk − E R sk + T sk . By construction, the tracking error (T sk − E(T sk )) is uncorrelated with any primitive asset return, so that the squared market return captures perfectly the price of skewness risk:      σ 3 ci,M = Cov Ri , R sk − E R sk = Cov Ri , R sk , for all i = 1, ..., n. Unfortunately, this may not be true anymore when it comes to the price of kurtosis risk and that of pentosis risk. In the former case we have          σ 4 di,M = Cov Ri , RM − E(RM ) R sk − E R sk + Cov Ri , RM − E(RM ) T sk , while we find in the latter case 2   2      + Cov Ri , T sk σ 5 fi,M = Cov Ri , R sk − E R sk      + 2 Cov Ri , R sk − E R sk T sk  2    = − Cov Ri , R sk − E R sk     2   2   + 2 Cov Ri , RM − E(RM ) R sk − E R sk + Cov Ri , T sk . This is the reason why we may expect that proper pricing of higher order risks will involve pricing factors like (RM −E(RM ))(R sk −E(R sk )), (R sk −E(R sk ))2 and (RM −E(RM ))2 (R sk − E(R sk )) in order to span the otherwise unhedgeable risk brought by (RM − E(RM ))T sk and (T sk )2 respectively. In other words, in case of market incompleteness, spanning the SDF may require not just polynomials in the single variable (RM − E(RM )), but polynomials in two variables: (RM − E(RM )) and, in addition, (R sk − E(R sk )). Note that only these two variables show up because we have limited our analysis so far to the tracking error on the skewness portfolio. Higher order terms in the expansions take polynomials with additional variables corresponding to returns on additional mimicking portfolios. These additional mimicking portfolios consist of two types: first, there are mimicking portfolios of higher powers of returns. We will illustrate this remark in Subsection 5.3 below by introducing, similarly to the skewness portfolio, a kurtosis portfolio with return R kurt and a tracking error (T kurt − E(T kurt )) on it:   3  RM − E(RM ) = R kurt − E R kurt + T kurt , (30)

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where T kurt is uncorrelated with all primitive asset returns Ri , i = 1, ..., n. Second, there are mimicking portfolios for products of market returns with tracking errors. In the next subsection we will introduce, similarly to the skewness portfolio, a co-cross skewness portfolio with return R ∗sk and a tracking error (T ∗sk − E(T ∗sk )) on it:       RM − E(RM ) R sk − E R sk = R ∗sk − E R ∗sk + T ∗sk , (31) where T ∗sk is uncorrelated with all primitive asset returns Ri , i = 1, ..., n. We will then illustrate in Subsection 5.3 the pricing impact of products of the market return with this co-cross skewness portfolio. Overall, we expect pricing kernels that capture pentosis risk to be polynomials of four variables: (RM − E(RM )), (R sk − E(R sk )), R ∗sk − E(R ∗sk ) and (R kurt − E(R kurt )). Of course, market incompleteness, as manifest in the non-zero tracking errors (T sk −E(T sk )), ∗sk − E(T ∗sk )) and (T kurt − E(T kurt )) will have a pricing impact only if it matters for indi(T vidual investors in equilibrium, i.e. these investors are heterogeneous in terms of preferences for higher order moments. 5.2. Co-kurtosis pricing and unspanned co-cross skewness In this subsection, we go one step further than Subsection 3.2. We want to characterize the pricing factors which matter to describe the approximation of the risk premium E(Ri ) − Rf through the first three terms ai2 σ 2 + ai3 σ 3 + ai4 σ 4 of its expansion (3). We show that the addition of the third term σ 4 ai4 may introduce two additional pricing factors: First, we find an additional power of the market return, namely the cubic one, to go beyond quadratic SDF of Section 4, as in the representative agent model of Dittmar [14]. However, second, we also find the co-cross skewness factor (RM − E(RM ))(R sk − E(R sk )). This factor will be priced in equilibrium if and only if investors display some heterogeneity in their skew tolerances. Before doing so, we define for all investors s = 1, . . . , S her weight qs τs ωs = S . (32) s=1 qs τs We remind the definition of individual skew-tolerance ρs and define kurtosis-tolerance κs by ρs =

us (qs Rf )u s (qs Rf ) ,  2[us (qs Rf )]2

κs =

[us (qs Rf )]2 u s (qs Rf ) .  6[us (qs Rf )]3

(33)

We use these and the weights ωs to extend earlier definitions of aggregate characteristics of preferences as follows: S S S S  qs τs κs  s=1 qs τs ρs ρ¯ = S = ω s ρs , κ¯ = s=1 = ωs κs , (34) S s=1 qs τs s=1 qs τs s=1 s=1 S S qs τs (ρs − ρ) ¯ 2  Var(ρ) = s=1 S = ωs (ρs − ρ) ¯ 2. (35) q τ s=1 s s s=1 (Note that weights ω in Eq. (32) differ from the earlier wealth-weights that we introduced in Eq. (25) to study the CRRA case. Consequently, the weights in calculating the variance term differs from the previous one and we denote it here differently.) We are now in a position to characterize the approximation E(Ri ) − Rf ≈ ai2 σ 2 + ai3 σ 3 + ai4 σ 4 ; it leads us to complete Theorem 2 by

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Theorem 3. With heterogeneous investors s = 1, . . . , S we complete the characterization of the j equilibrium risk premium πi (σ ) = E(Ri ) − Rf = ∞ j =2 aij σ in Theorem 2 by  1 − 3ρ¯    κ¯  di,M σ 4 + VM σ 2 bi,M σ 2 3 3 τ¯ τ¯      Var(ρ) Cov Ri , RM − E(RM ) R sk − E R sk , −2 3 τ¯ where τ¯ is defined in Eq. (17) and ρ, ¯ κ, ¯ Var(ρ) are defined in Eqs. (34), (35). σ 4 ai4 =

(36)

Comparing the description of ai4 in Theorem 3 to that in Theorem 1, i.e. with the fourth order risk premium term in the case of a representative agent, we see that the covariance of individual asset returns i with the co-cross skewness factor (RM − E(RM ))(R sk − E(R sk )) enters into pricing. We will discuss below that this covariance may not vanish. Before doing so, let us consider again, purely for illustrative purposes, a particular preference parameterization. At the end of Section 4 we studied mean-variance-skewness risk premiums and demand an economy, where all investors exhibit constant relative risk aversion (CRRA). To illustrate the impact of co-kurtosis on risk premiums, let us assume for a moment a similar economy, i.e. all investors s = 1, . . . , S have utility functions us (w) = w −αs with (constant) risk aversion parameter αs . We recall that we defined initially relative wealth weights ωs∗ for each investor, see Eq. (25); we then used these to rewrite the average risk tolerance (Eq. (26)) and ultimately defined the variation (dispersion) in investor’s risk tolerances Rf /αs , see Eq. (27). Analogously we define the third order dispersion (“skewness”) across investor’s risk tolerance by setting 



3  S S S  Rf ∗ ∗ Rf ∗ Rf Skew ωs − ωs = ωs∗ (τs − τ¯ )3 . = αs s=1,...,S αs αs s=1

s=1

s=1

Now let us look at the approximation in Theorem 3 that includes co-kurtosis within the CRRA framework. Note that within the CRRA utilities framework, the relative risk aversion coefficient is the key driver of all our skew tolerance terms, i.e. of ρs , κs , χs . At the end of Section 4 we related the average skew-tolerance to the average risk tolerance and the variance Var∗ in investors’ risk aversion. A similar characterization holds here and to see this, we compute κs =

(αs + 1)(αs + 2) , 6αs2

such that τ¯ + 2τ¯ 3 /Rf2 + 3τ¯ 2 /Rf + (6τ¯ + 3Rf )/Rf2 Var∗ (( αfs )s ) + 2/Rf2 Skew∗ (( αfs )s ) R

κ¯ =

R

. 6τ¯ This shows that the average kurtosis coefficient κ¯ is determined by the average risk tolerance τ¯ , the cross sectional variance of investors’ risk tolerances Var∗ ((Rf /αs )s ), and the cross-sectional skewness of investors’ risk tolerances Skew∗ ((Rf /αs )s ). If the distribution of investor risk tolerances is determined by the first four cross sectional moments of investor risk tolerances, the expected excess return in Theorem 3 is a useful guideline for empirical frameworks that aim to recover the cross sectional distribution of investor preferences in the CRRA framework.

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The previous lines elaborated on the link between CRRA preferences and the average kurtosis tolerance κ. ¯ Let us now return to our general setup (general preferences). The approximation (36) in Theorem 3 puts forward the following SDF, up to a constant: Proposition 3. Up to risk premium terms of order higher than four in σ , the SDF is up to a constant: H (3) = H (3, 1) + H (3, 2), where

 1 1 − 3ρ¯ (−Rf )H (3, 1) = Var(R ) + M RM τ¯ τ¯ 3 2 3 ρ¯  κ¯  − 2 RM − E(RM ) + 3 RM − E(RM ) , τ¯ τ¯   sk  Var(ρ)  (−Rf )H (3, 2) = −2 RM − E(RM ) R − E R sk . 3 τ¯

(37) (38)

Here, H (3, 1) is similar to the classical cubic SDF provided by representative agent models, see, e.g., Dittmar [14]. As usual, the three pricing factors RM , (RM − E(RM ))2 and (RM − E(RM ))3 have prices proportional to risk aversion, skew tolerance and kurtosis tolerance, respectively. Note that adding the expansion term σ 4 a4 has led us to adding the correction term (1 − 3ρ) ¯ Var(RM )/τ¯ 3 to the standard weight (1/τ¯ ) of the market return; note also, however, that this correction term is of lower order in small noise expansions since Var(RM ) is proportional to σ 2 . More important, by comparison with the classical SDF that is a polynomial in the market return, our setting may add one pricing factor (RM − E(RM ))(R sk − E(R sk )). It is worth analyzing what may make it necessary to account explicitly for this additional factor within the SDF, beyond the common cubic market return (RM − E(RM ))3 . First, we note that by definition of the tracking error      3 RM − E(RM ) R sk − E R sk = RM − E(RM ) ⇔ T sk = 0. But, even more important we have:        3 Cov RM − E(RM ) R sk − E R sk , Ri = Cov RM − E(RM ) , Ri       ⇔ Cov RM − E(RM ) T sk , Ri = 0 ⇔ E RM − E(RM ) T sk Ri = 0. In other words, the tracking error, albeit uncorrelated with all asset returns, may become correlated when multiplied by the market return. The resulting additional pricing factor has a price proportional to the cross-sectional variance Var(ρ) of skew tolerances across investors. In particular, this additional factor is priced in equilibrium if and only if investors are heterogeneous in terms of skew tolerances. We call co-cross skewness of asset i the risk quantity Cov[(RM − E(RM ))(R sk − E(R sk )), Ri ] that shows up in the risk premium formula (36). The importance of the cross-sectional variance Var(ρ) in H (3, 2) should not come as a surprise: it matches a similar result of Constantinides and Duffie [13], where market incompleteness (here: tracking error (T sk − E(T sk ))) gives rise to an additional pricing factor with a price proportional to the variance of the distribution of heterogeneity. As far as market incompleteness is concerned, it is worth noting that non-zero individual kurtosis tolerances will not lead the agents to invest directly in the cubic market return but only

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in its best tracking portfolio with return R kurt corresponding to the affine regression defined in (30): R

kurt

= Rf +

n 

ξikurt (Ri − Rf ),

  where ξ kurt = ξikurt 1≤i≤n = σ 2 Σ −1 dM .

i=1

Here, dM = (di,M )i is the vector of market co-kurtosis of asset i introduced in Definition 1, ξ kurt defines the so-called kurtosis portfolio that is hold in equilibrium by agent s insofar as her kurtosis tolerance differs from the average. Similarly, due to market incompleteness the agent cannot invest into (RM − E(RM ))(R sk − E(R sk )) but only in the return R ∗sk of the best tracking portfolio ξ ∗sk , corresponding to its affine regression defined in (31): R ∗sk = Rf +

n 

ξi∗sk (Ri − Rf ),

  where ξ ∗sk = σ 2 Σ −1 c ξ, ξ sk ,

i=1

where we used the vector of co-cross skewness c(ξ, ξ sk ) = (ci (ξ, ξ sk ))1≤i≤n with Definition 2. The co-cross skewness of asset i is defined as            1 ci ξ, ξ sk = 4 Cov RM − E(RM ) R sk − E R sk , Ri = Cov ξ  Y ξ sk, Y , Yi . σ Similarly to (20), we show in Appendix A that the equilibrium portfolio of investor s involves a third order term σ 2 θs2 such that:    τs  σ 2 θs2 = − 3 (κs − κ)ξ ¯ kurt − 2 ρs (ρs − ρ) ¯ − Var(ρ) ξ ∗sk − 3(ρs − ρ) ¯ Var(RM )ξ . τ¯ (39) Note that, by contrast with (20), this expansion term involves two departures from the mean variance portfolio: Most important, the kurtosis portfolio leads to a portfolio adjustment when the kurtosis tolerance of investor s is different from average. In addition to that, the co-cross skewness portfolio ξ ∗sk is reflected in the portfolio holdings of investor s when the term ρs (ρs − ρ) ¯ differs from its cross-sectional average Var(ρ). 5.3. Co-pentosis pricing and unspanned squared skewness portfolio return In this subsection, we go one step further than in the former one. We want to characterize the pricing factors which matter to describe the approximation of the risk premium (E(Ri ) − Rf ) through the first four terms σ 2 (ai2 + σ ai3 + σ 2 ai4 + σ 3 ai5 ) of its expansion (3). We show that the addition of the fourth term σ 5 ai5 may lead to introduce five additional pricing factors: Most important, we find an additional power of the market return, namely the quartic one, to go beyond cubic pricing kernels of the previous subsection as in the representative agent model of Dittmar [14]. However, we also find the squared skewness portfolio factor (R sk − E(R sk ))2 (second), the co-cross skewness portfolio (RM − E(RM ))(R ∗sk − E(R ∗sk )) (third), the cocross-kurtosis portfolio (RM − E(RM ))(R kurt − E(R kurt )) (fourth) and the co-cross quadratic market-skewness portfolio (RM − E(RM ))2 (R sk − E(R sk )) (fifth additional factor). These factors will be priced in equilibrium when investors display heterogeneous skew tolerances (for a non-zero price of the squared skewness portfolio factor), and when these skew tolerances display

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some cross sectional correlation with kurtosis tolerances (for a non-zero price of the co-crosskurtosis and the co-cross quadratic market-skewness portfolios). Here we extend the definitions of aggregate characteristics of preferences introduced earlier by introducing A1 (ρ, κ), A2 (ρ, κ) as follows: A1 (ρ, κ) =

−ρ¯ + 2ρ¯ 2 − 4 Var(ρ) + κ¯ , τ¯ 4

A2 (ρ, κ) =

−ρ¯ + ρ¯ 2 − 8 Var(ρ) + 3κ¯ . (40) τ¯ 4

The approximation E(Ri ) − Rf ≈ ai2 σ 2 + ai3 σ 3 + ai4 σ 4 + ai5 σ 5 leads to complete Theorem 2 by . . , S we complete the characterization of the Theorem 4. With heterogeneous investors s = 1, . j equilibrium risk premium πi (σ ) = E(Ri ) − Rf = ∞ j =2 aij σ in Theorems 2, 3 by       τ¯ 4 σ 5 ai5 = A1 (ρ, κ) SM σ 3 bi,M σ 2 + A2 (ρ, κ) VM σ 2 ci,M σ 3  2   − Var(ρ) Cov R sk − E R sk , Ri − χ¯ fi,M σ 5        − 4 Skew(ρ) + ρ¯ Var(ρ) Cov RM − E(RM ) R ∗sk − E R ∗sk , Ri 

2 Cov[(RM − E(RM ))(R kurt − E(R kurt )), Ri ] . (41) + Cov(ρ, κ) + 3 Cov[(RM − E(RM ))2 (R sk − E(R sk )), Ri ] Comparing this to Theorem 1, i.e. with the fifth order risk premium term in the representative agent case, we see additional pricing factors due to the cross-sectional distribution of the preference parameters and to the cross-sectional correlation of preference parameters. The cross-sectional distributions as well as the cross-sectional correlation of investors’ preferences are known to be empirically relevant to understand individual investors’ portfolio choice, see, e.g., Noussair et al. [35]. Regarding the impact of heterogeneity on the price of different risk factors, several remarks are in order. First, not surprisingly, the quartic market return has a non-zero price insofar as the investors display in average a non-zero pentosis tolerance: S S  [us (qs Rf )]3 u[5] s (qs Rf ) s=1 qs τs χs χs = , χ ¯ = = ωs χs , (42) S 24[us (qs Rf )]4 s=1 qs τs s=1 where ω denotes the weights introduced earlier in Eq. (32). Second, the cross-sectional variance Var(ρ) of skew-tolerances scales not only the price of the co-cross-skewness portfolio (as seen in the previous subsection) but also the price of the squared skewness portfolio. Third, the cross-sectional covariance Cov(ρ, κ) between skew tolerances and kurtosis tolerances scales the price of both the co-cross-kurtosis and the co-cross quadratic market-skewness portfolios. Here, analogous to the previous definitions of cross sectional averages and variances, this covariance is defined by Cov(ρ, κ) =

S 

ωs (ρs − ρ)(κ ¯ s − κ). ¯

(43)

s=1

Fourth, the cross-sectional skewness of skew tolerances Skew(ρ) =

S  s=1

ωs (ρs − ρ) ¯ 3

(44)

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scales the product of the co-cross skewness portfolio R ∗sk − E[R ∗sk ] with the market return. Finally, the quantities A1 (ρ, κ) and A2 (ρ, κ) come to modify at higher orders the prices of the market return and the skewness portfolio respectively. Adding σ 5 ai5 of Eq. (41) to the approximation (36) leads us to the following SDF, up to a constant: Proposition 4. Up to risk premium terms of order higher than five in σ , the SDF is, up to a constant: H (4) = H (4, 1) + H (3, 2) + H (4, 2) + H (4, 3) + H (4, 4) + H (4, 5), where



 1 1 − 3ρ¯ A1 (ρ, κ) (−Rf )H (4, 1) = Var(RM ) + Skew(RM ) RM + τ¯ τ¯ 3 τ¯ 4

  2 ρ¯ A2 (ρ, κ) − 2− Var(RM ) RM − E(RM ) 4 τ¯ τ¯ 3 χ¯  4 κ¯  + 3 RM − E(RM ) − 3 RM − E(RM ) . τ¯ τ¯ The additional pricing factor H (3, 2) is defined in (38). The additional pricing factors H (4, j ) (j = 2, 3, 4, 5) are defined as    Cov(ρ, κ)  RM − E(RM ) R kurt − E R kurt , τ¯ 4    Skew(ρ) + ρ¯ Var(ρ)  RM − E(RM ) R ∗sk − E R ∗sk , (−Rf )H (4, 3) = −4 4 τ¯  2   Cov(ρ, κ)  (−Rf )H (4, 4) = 3 RM − E(RM ) R sk − E R sk , τ¯ 4 2  Var(ρ)  sk (−Rf )H (4, 5) = − R − E R sk . 4 τ¯ (−Rf )H (4, 2) = 2

(45) (46) (47) (48)

Note that H (4, 1) is a fourth order polynomial in the market return that corresponds to the classical quartic SDF provided by representative agent models, see, e.g., Dittmar [14]. As usual, the four pricing factors RM , (RM − E(RM ))2 , (RM − E(RM ))3 and (RM − E(RM ))4 have prices proportional to risk aversion, skew tolerance, kurtosis tolerance and pentosis tolerance, respectively. In comparison to the cubic SDF (37), note that adding the expansion term σ 5a3 has led us to adding two correction terms. First, a further correction by A1 (ρ, κ)/τ¯ 4 SM to the standard weight (1/τ¯ ) of the market return; note this correction is in addition to the earlier term (1 − 3ρ)V ¯ M /τ¯ 3 that showed up in (37) as a correction to the quadratic SDF of (11). Second, a correction −A2 (ρ, κ)/τ¯ 4 VM to the standard weight (ρ/ ¯ τ¯ 2 ) of the squared market return. Note that both correction terms are of lower order in small noise expansions since VM is proportional to σ 2 and SM is proportional to σ 3 . Similar to H (3, 2) these pricing factors embody a form of market incompleteness, here the impossibility to hedge the squared market return, the cubic market return as well as the cross product between market return and the return on the skewness portfolio. Ultimately, the SDF will be a polynomial of all these factors at various powers; note that only the powers that appear

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477

through H (3, 2) contribute to fourth moments (kurtosis pricing), while only the terms in H (4, j ) (j = 2, 3, 4, 5) lead to fifth moments (pentosis pricing). The additional factors H (3, 2) and H (4, j ) (j = 2, 3, 4, 5), defined in Eqs. (38), (45)–(48) above, will contribute to pricing whenever there is some heterogeneity in preferences for higher order moments (Var(ρ), Cov(ρ, κ) and Skew(ρ)). In particular, note that even in a world with heterogeneous risk averse investors that have only non-vanishing skew tolerances, i.e. investors with κs = χs = 0 for all s = 1, . . . , S, the additional factors H (3, 2), H (4, 3) and H (4, 5) will be priced as long as there is heterogeneity of preferences for higher order moments with Var(ρ) = 0 or Skew(ρ) = 0. Overall so far, this section shed more light on how heterogeneity of investors’ preferences do have an impact on the spanning of the pricing kernel. The main message is threefold: (i) In a world without investor heterogeneity, the only (possibly) relevant pricing factors are the powers of the market return (RM − E(RM ))j , j = 1, 2, 3, 4, . . . as in the representative agent model of Dittmar [14]. (ii) In a world with heterogeneity, the pricing factors defined by powers of the market return still show up with a non-zero price if and only if there is a non-zero aggregated preference for the corresponding moment. (iii) However, in a world with heterogeneity of preferences, additional pricing factors must be considered based on tracking errors as explained above. The equilibrium prices of associated risks can be interpreted as measures of heterogeneity of preferences for associated higher order moments. 5.4. Co-skewness pricing and unspanned intertemporal hedge portfolio return So far, our paper studied exclusively a setup with a single investment period. In a dynamic framework this means that all investors are assumed myopic, i.e. they ignore hedging changes in her investment opportunity set over time. Here, we consider an extension to two investment periods; the so-called intertemporal hedge demand may then affect equilibrium risk premiums; however, the typical Taylor series approach starts immediately with the assumption that they do not matter for the representative investor. In line with the main theme of the previous subsections in this section, our goal is showing that the usual analysis of a myopic investor ignores a priced factor based on the intertemporal hedge portfolio. Throughout we focus on the representative investor in a setup with two time periods. A detailed analysis of dynamic pricing by the representative investor as well as of the dynamic equilibrium with heterogeneous investors is beyond the focus of this paper; we take up these issues in a companion paper Chabi-Yo et al. [8]. Here, we will borrow heavily from the extant literature as well as from our results in the previous (sub-)sections of this paper. Our simplified setup in this subsection assumes three dates: 0, 1, 2; we consider a representative investor who maximizes expected utility E[u(W )] derived from the terminal (date 2) wealth W that she obtains by investing some initial (date 0) wealth q¯ in a portfolio. The difference to the previous subsection is that we take into account that her terminal wealth W is the result of a dynamic investment strategy: the investment at date 0 leads to date 1 wealth q; further investment leads to date 2 wealth W . Throughout, we consider a scale-location model for the vector of returns that is defined recursively over consecutive periods: R(1) = (Ri,1 )1≤i≤n = E(R(1) ) + σ Y(1) ,

with E(Y(1) ) = 0,

(49)

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and R(2) = (Ri,2 )1≤i≤n = E(R(2) |Y(1) ) + σ Y(2) ,

with E(Y(2) |Y(1) ) = 0.

(50)

As before, we assume that the supply of risky assets is independent of σ : we assume that the date 0 and date 1 supply are both given by a vector ξ of shares of total (date 0) wealth q¯ and q, respectively, invested in the n risky assets. These define the market returns RM,1 and RM,2 over both periods by setting RM,1 = ξ  E(R(1) ) + σ ξ  Y(1) ,

RM,2 = ξ  E(R(2) |Y(1) ) + σ ξ  Y(2) .

In line with the extant literature on dynamic optimization we treat this as a problem to be solved backwards in time; this leads us to study at date 0, that the representative investor faces the problem of maximizing the expectation of her so-called (date 1) indirect utility functions J , evaluated at date 1 wealth q = q(σ ). Formally, this looks similar to our single-period optimization problems considered so far; however, there are delicate differences: the indirect utility function J is a function of the risk premiums over the last time period; the latter are a consequence of date 1 wealth, which is itself depends on the investment strategy over the first time period. Put differently, the investment strategy over the first time period affects not only the date 1 wealth but also the date 1 indirect utility function. Ultimately, the standard Taylor series approach is unable to capture this dependence and this is the origin of the additional pricing factor that we document below. To see how this enters, we note that the first non-zero term in the (conditional) date 1 equilibrium risk premium for each asset i = 1, ..., n in the characterization πi,1 (σ ) = E(Ri,2 |Y(1) ) − 1 j Rf = ∞ j =2 aij σ is 1 ai2 =

1 b1 , τ1 (qR ¯ M,1 ) i,M

where τ1 (w) = −

u (wRf ) wu (wRf )

(51)

and the function τ1 (w) describes the date 1 risk tolerance. This risk tolerance is analogous to the one that we used in the single period case; however, in line with our prior analysis, we need to evaluate this at date 1 wealth; seen from date 0, this function will be evaluated at qR ¯ M and so it is a random variable. Our approach allows us to view this through a series expansion in the scale parameter and therefore we introduce the date 0 preference parameter ∂τ1 (qR ¯ f ). γ0 = q¯ (52) ∂w Recall that τ1 /Rf is equal to the common (relative) risk tolerance function, evaluated at wRf . Therefore, unless preferences are described through the so-called constant relative risk aversion, the function τ1 will not be constant and wealth changes over the first period will affect the investor’s risk tolerance at date 1. The term γ0 describes the first order change in the risk tolerance function τ1 due to changes in investor’s wealth. Therefore, we call this preference parameter the wealth tolerance. Our focus in this subsection is on characterizing the impact on the SDF; therefore we refrain from presenting the risk premiums. The online appendix characterizes these in detail and derives from there the following characterization of the SDF. Proposition 5. Up to risk premium terms of order higher than two in σ and up to a constant, the date 0 one-period SDF is H˜ (2) = H˜ (2, 1) + H˜ (2, 2),

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where 1 ρ0 Rf H˜ (2, 1) = Rf H (2) = − RM,1 + 2 (RM,1 − ERM,1 )2 , τ0 τ0 Rf H˜ (2, 2) = −

1 − ρ 0 + γ0 Var[RM,2 |R1 ]. τ02

Compared to the one-period SDF in the static model, there is an additional pricing factor which is the volatility risk factor Var[RM,2 |R1 ]. The price of risk of this factor is determined not only by the risk preferences (risk and skew tolerance), but also through a wealth effect via the wealth tolerance γ0 . Our message in this subsection is that a two period analysis may be at odds with the common quadratic SDF assumption, except if one maintains two rather restrictive assumptions about the volatility risk factor Var[RM2 |R1 ]. These assumptions are statistical in nature and not underpinned by any economic theory: First, we need to assume that an aggregated volatility model is optimal:     (53) Var RM2 |R1 = Var RM2 |RM1 . Second, even under this assumption we only get the quadratic SDF specification by assuming an ARCH(1) specification (Engle [17]) for this aggregated volatility model:   2 Var RM2 |RM1 =  + ςRM1 . Even an asymmetric ARCH (Glosten et al. [20]), where the sign of RM1 would matter (to accommodate an aggregate leverage effect) would invalidate the naive quadratic SDF. Note in addition that the aggregation assumption (53) is all about the cross section of risky assets. We always assume that investors are homogeneous regarding their expectation of future returns. Aggregation of heterogeneous beliefs and/or asymmetric information is beyond the scope of this paper. 6. Conclusion This paper showed a three-fund separation theorem for investors with mean-varianceskewness preferences: agents hold the risk-free asset, the market portfolio and a skewness portfolio. In complete markets, i.e. when the skewness portfolio is the portfolio that replicates the squared market return, the skewness risk premium is driven by co-variation with the squared market such that pricing is characterized by a stochastic discount factor that is a quadratic polynomial in the market return. In incomplete markets, however, the skewness portfolio is the projection of the squared market return on the available assets leading to a tracking error. In world with heterogeneity of preferences, cross moments of the tracking error and the market return at various orders contribute to pricing in addition to the skewness portfolio; in stochastic discount factors this shows up as priced factors in addition to factors that are powers of market returns. Empirical studies of skewness risk based on stochastic discount factors that are polynomials in the market return miss all these additional factors. In addition to this, even with homogeneous preferences, an additional factor shows up that prices (intertemporal) changes in the market return variance; this factor relates to the market volatility (leverage feedback) effect documented in empirical studies.

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Appendix A. Proofs Proof of Lemma 1. The first order conditions (2) can be rewritten as        πi (σ )E u W (σ ) + σ E u W (σ ) Yi = 0. Differentiating w.r.t. σ gives        ∂E[u (W (σ ))Yi ] ∂E[u (W (σ ))] πi (σ )E u W (σ ) + πi (σ ) + E u W (σ ) Yi + σ = 0. ∂σ ∂σ Knowing that πi (0) = 0 we find for σ = 0 that     πi (0)u W (0) + u W (0) E(Yi ) = 0. We conclude πi (0) = 0, since u (W (0)) = 0 and E(Yi ) = 0.

2

For further derivations in this appendix, we introduce the following concepts: Definition A-1. We define the co-skewness, co-kurtosis, and co-pentosis of asset i with a portfolio θ by   2    3    4  di (θ ) = Cov Yi , θ  Y , fi (θ ) = Cov Yi , θ  Y , ci (θ ) = Cov Yi , θ  Y , and the co-cross-skewness and the co-cross-kurtosis with portfolios θ , θ˜ by       2   ci (θ, θ˜ ) = Cov Yi , θ  Y θ˜  Y , di (θ, θ˜ ) = Cov Yi , θ  Y θ˜  Y . In addition we define the third moment of a portfolio θ by  3  Sk(θ ) = E θ  Y . We will also write these as vectors as follows: c(θ ) = (ci (θ ))1≤i≤n , d(θ ) = (di (θ ))1≤i≤n , ˜ = (di (θ, θ˜ ))1≤i≤n . Finally, for θ = ξ , f (θ ) = (fi (θ ))1≤i≤n , c(θ, θ˜ ) = (ci (θ, θ˜ ))1≤i≤n , d(θ, θ) we consider the following market quantities: c = c(ξ ), ξ0sk

=Σ

−1

d = d(ξ ), f = f (ξ ), Sk = Sk(ξ ),   ∗sk −1 sk ξ0kurt = Σ −1 d. c, ξ0 = Σ c ξ, ξ0 ,

Throughout this appendix we assume for sake of expositional simplicity that the utility function us is analytical for every investor s = 1, . . . , S; however, our proofs go through as long as their utility functions are sufficiently differentiable to study the lower order expansion terms of the first order conditions. Lemma A-1 (Series expansion product marginal utility with net returns). Fix an investor s = 1, . . . , S. Denote for any asset i = 1, . . . , n by      his (σ ) = us Ws θs (σ ) · Ri (σ ) − Rf the product of the investor’s marginal utility with the asset’s net return. We assume this is an analytical function of the scale parameter σ , i.e. we write his (σ ) =

∞  j =0

hisj σ j .

F. Chabi-Yo et al. / Journal of Economic Theory 154 (2014) 453–489

481

We have his0 = 0 his1 = us (qs Rf )Yi , his2 = us (qs Rf )ai2

 + us (qs Rf )qs

his3 = us (qs Rf )ai3 + us (qs Rf )qs  +

 us (qs Rf )qs

n 

k=1  n 

 θks0 Yk Yi ,  θks0 Yk ai2

k=1

  n 2  n   1  2 (θks1 Yk + θks0 ak2 ) + us (qs Rf )qs θks0 Yk Yi , 2 k=1

that

k=1

 his4 = us (qs Rf )ai4

+ us (qs Rf )qs 

n n   (θks0 Yk )ai3 + (θks1 Yk + θks0 ak2 )ai2 k=1 2

n 



k=1

1 + u (qs Rf )qs2 θks0 Yk ai2 2 s k=1 ⎧ ⎫ n θks1 ak2 + θks0 ak3 )) ⎨ +us (qs Rf )qs ( k=1 (θks2 Yk + ⎬ n n 2 + +u s (qs Rf )qs ( k=1 θks0 Yk )( k=1 θks1 Yk + θks0 ak2 ) Yi , ⎩ 1  ⎭ + 6 us (qs Rf )qs3 ( nk=1 θks0 Yk )3 and that



his5 = us (qs Rf )ai5 + us (qs Rf )qs 

n 

 θks0 Yk ai4

k=1

  n 2  n   1  2 + (θks1 Yk + θks0 ak2 ) + us (qs Rf )qs θks0 Yk ai3 2 k=1 k=1 ⎧  ⎫ n ⎨ us (qs Rf )qs ( k=1 (θ ⎬ ks2 Yk + θks1 a k2 + θks0 ak3 )) n n 2 + u + s (qs Rf )qs ( k=1 θks0 Yk )( k=1 θks1 Yk + θks0 ak2 ) ai2 ⎩ ⎭ n 3 3 + 16 u s (qs Rf )qs ( k=1 θks0 Yk ) ⎫ ⎧  us (qs Rf )qs ( nk=1 (θ ks3 Yk + θks2 a k2 + θks1 ak3 + θks0 ak4 )) ⎪ ⎪ ⎪ ⎪ n n ⎪ ⎪  2 ⎪ ⎪ θ Y )( θ Y + θ a + θ a ) ks0 k ks2 k ks1 k2 ks0 k3 ⎬ ⎨ + us (qs Rf )qs ( k=1 k=1 n 1  2 2 + 2 us (qs Rf )qs ( k=1 (θks1 Yk + θks0 ak2 )) Yi . + ⎪ ⎪ ⎪ ⎪ + 1 u (qs Rf )q 3 ( n θks0 Yk )2 ( n θks1 Yk + θks0 ak2 ) ⎪ ⎪ s k=1 k=1 ⎪ ⎪ 2 s ⎭ ⎩ 1 (5) us (qs Rf )qs4 ( nk=1 θks0 Yk )4 + 24 

us (qs Rf )qs

Proof of Lemma A-1. We denote r(σ ) = (Ri (σ ) − Rf )1≤i≤n the vector of net risky returns and find by definition: ri (σ ) = σ Yi +

∞  j =2

aij σ j ,

i = 1, ..., n

(A-1)

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We compute the series coefficients hisj by multiplying this expansion with the series expansion of us (Ws (θ (σ ))). We first write a Taylor series around qs Rf :   us Ws (θ )  n   n 2   1   θks rk + u (qs Rf )qs2 θks rk = us (qs Rf ) + us (qs Rf )qs 2 s k=1 k=1  n 3  n 4   1  1 (5) 3  4  + us (qs Rf )qs θks rk + us (qs Rf )qs θks rk + . . . (A-2) 6 24 k=1

k=1

Now, we deduce from (A-1) the series expansion of portfolio returns: n 

θks (σ )rk = σ

k=1

n 

θks0 Yk + σ 2

k=1

+ σ3 + σ4

n  (θks1 Yk + θks0 ak2 ) k=1

n 

(θks2 Yk + θks1 ak2 + θks0 ak3 )

k=1 n 

(θks3 Yk + θks2 ak2 + θks1 ak3 + θks0 ak4 ) + . . .

(A-3)

k=1

Jointly, (A-2) and (A-3) imply:   us Ws (θ ) = us (qs Rf ) + σ us (qs Rf )qs 

n  k=1

 θks0 Yk 

 n 2   1  2 (θks1 Yk + θks0 ak2 ) + us (qs Rf )qs θks0 Yk +σ 2 k=1 k=1 ⎧ ⎫ n θks1 ak2 + θks0 ak3 )) ⎨ +us (qs Rf )qs ( k=1 (θks2 Yk + ⎬ (qs Rf )qs2 ( nk=1 θks0 Yk )( nk=1 θks1 Yk + θks0 ak2 ) + σ 3 +u ⎩ 1 s  ⎭ + 6 us (qs Rf )qs3 ( nk=1 θks0 Yk )3 ⎧  ⎫ Yk + θks2 a us (qs Rf )qs ( nk=1 (θ k2 + θks1 ak3 + θks0 ak4 )) ⎪ ⎪ ks3 ⎪ ⎪ n n ⎪ ⎪  2 ⎪ ⎨ + us (qs Rf )qs ( ⎬ k=1 θks0 Yk )( k=1 θks2 Yk + θks1 ak2 + θks0 ak3 ) ⎪ n 1 2 + 2 u θks0 ak2 ))2 + σ4 + ... s (qs Rf )qs ( k=1 (θks1 Yk + ⎪ ⎪ 1  3( n θ 2( n θ ⎪ ⎪ + u (q R )q Y ) Y + θ a ) ⎪ ⎪ s f ks0 k ks1 k ks0 k2 s s k=1 k=1 ⎪ ⎪ 2 ⎩ ⎭ 1 (5) + 24 us (qs Rf )qs4 ( nk=1 θks0 Yk )4 2





us (qs Rf )qs

n 

Multiplying with the expansion (A-1) of ri and collecting terms, this now implies the statement. 2 Proof of Theorem 1. The representative investor economy can be interpreted as an economy with a single investor S = 1, utility function us = u, demand θ = ξ , i.e. θis0 = ξi , θisj = 0, j = 1, 2, . . . , and h = u (W (σ ))(R(σ ) − Rf ) for the application of Lemma A-1. In this representative investor economy, the first-order conditions read for asset i = 1, . . . , n: E(hij ) = 0,

∀j = 1, 2, 3, ...

(A-4)

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483

Our goal is to characterize the coefficients aij , j = 2, 3, . . . implied by these equations. Lemma A-1 describes the first five terms in the expansion of hi (σ ) for each i = 1, ..., n. Recall that E(Yi ) = 0 for i = 1, ..., n and θ = ξ in this representative investor economy. Up to order five, the informative optimality conditions are then as follows: First, E(hi2 ) = 0 gives for i = 1, ..., n: 0 = u (qRf )ai2 + u (qRf )q

n 

ξk Cov(Yk , Yi ),

k=1

i.e. ai2 = −

 1  u (qRf )q Cov Yi , ξ  Y = bi,M . u (qRf ) τ

Second, E(hi3 ) = 0 gives for i = 1, ..., n:    2  1 2 , 0 = u (qRf )ai3 + u[3] s (qRf )q Cov Yi , ξ Y 2 i.e. ai3 = −

1 u[3] (qRf )q 2 ρ ci,M = − 2 ci,M . 2 u (qRf ) τ

Third, E(hi4 ) = 0 gives for i = 1, ..., n:  n     1   0 = u (qRf )ai4 + u (qRf )q ξk ak2 ai2 + u[3] (qRf )q 2 Var ξ  Y ai2 2 k=1  n    1   3   + u[3] (qRf )q 2 ξk ak2 Cov Yi , ξ  Y + u[4] (qRf )q 3 Cov Yi , ξ  Y , 6 k=1

i.e. ai4 =

    κ 1 ρ ρ di,M + ξ  a2 ai2 − 2 Var ξ  Y ai2 − 2 2 ξ  a2 bi,M . τ τ3 τ τ

However, we have seen above that ξ  a2 = we conclude:

ξ  bM τ

=

VM τ

. Using this and the formula ai2 =

κ 1 di,M + 3 (1 − 3ρ)VM bi,M . 3 τ τ Finally, fourth, E(hi5 ) = 0 gives for i = 1, ..., n: ai4 =

    1 0 = u (qRf )ai5 + u (qRf )q ξ  a2 ai3 + u[3] (qRf )q 2 Var ξ  Y ai3 2     3 1 [4]  + u (qRf )q ξ a3 ai2 + u (qRf )q 3 E ξ  Y ai2 6   1     2  [3] 2 + u (qRf )q Cov ξ Y, Yi ξ  a3 + u[4] (qRf )q 3 Cov ξ  Y , Yi ξ  a2 2    1 [5] 4 + u (qRf )q 4 Cov ξ  Y , Yi , 24

bi,M τ

,

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that is χ VM ρ fi,M + 2 ai3 − 2 VM ai3 4 τ τ τ   bi,M VM κ 1    bi,M κ 2ρ + ξ a3 + 3 SM − 2 bi,M ξ  a3 + 3 3 ci,M . τ τ τ τ τ τ τ

ai5 = −



As before we have used here the known identities ξ  a2 = ξ τbM = VτM and ai2 = remember that similarly ξ  a3 = − τρ2 SM and ai3 = − τρ2 ci,M , we end up with ai5 = −

χ 3κ − ρ(1 − ρ) κ − ρ(1 − 2ρ) fi,M + VM ci,M + SM bi,M . 4 4 τ τ τ4

bi,M τ

. If we now

2

Lemma A-2. Fix an investor s = 1, . . . , S. We use the concepts of risk tolerance τs and skew tolerance ρs introduced in Eq. (17), as well as kurtosis tolerance κs introduced in Eq. (33) and pentosis tolerance χs introduced in Eq. (42). The first four expansion terms of demand for investor s are characterized by θs0 = τs Σ −1 a2   ρs −1 c(θs0 ) 2 + a3 θs1 = τs Σ τs

(A-5) (A-6)

as well as       qs ρ s qs Σθs2 = qs τs a4 − qs θs0 a2 a2 + Var θs0 Y a2 τs qs ρ s qs ρ s  qs κs +2 c(θs0 , θs1 ) + 2 θ a2 Σθs0 − 2 d(θs0 ), τs τs s0 τs          qs ρ s  Var θs0 Y a3 − qs θs1 a2 + θs0 a3 a2 qs Σθs3 = qs τs a5 − qs θs0 a2 a3 + τs  qs κs qs ρs   θs0 Σθs1 a2 − 2 Sk(θs0 )a2 + 2 τs τs     qs ρ s   c(θs0 , θs2 ) + θs1 a2 + θs0 +2 a3 (Σθs0 ) τs     q s ρs  + c(θs1 ) + 2 θs0 a2 (Σθs1 ) τs     qs χs qs κs  − 3 2 d(θs0 , θs1 ) + θs0 a2 c(θs0 ) + 3 f (θs0 ). τs τs

(A-7)

(A-8)

Proof of Lemma A-2. The first order conditions (15) read for asset i = 1, ..., n: E(hisj ) = 0,

∀j = 1, 2, 3, . . .

(A-9)

Lemma A-1 characterized the first five terms in the expansion of his for each i = 1, ..., n; we study here the expansion of the first order conditions E(hisj ) = 0 for j = 1, ..., 5 and all i = 1, ..., n. Recall that E(Yi ) = 0 for i = 1, ..., n by definition, such that E(his1 ) = 0 for i = 1, ..., n. Up to order five, the informative first order conditions are then as follows: First, E(his2) = 0 for

F. Chabi-Yo et al. / Journal of Economic Theory 154 (2014) 453–489

485

i = 1, ..., n gives: 0 = us (qs Rf )ai2 + us (qs Rf )qs

n 

θks0 Cov(Yk , Yi ),

i.e. in vector-matrix notation

k=1

0 = us (qs Rf )a2 + us (qs Rf )qs Σθs0 ,

or equivalently θs0 = τs Σ −1 a2 .

(A-10)

This proves Eq. (A-5). Second, the condition E(his3 ) = 0 gives for i = 1, ..., n: 0 = us (qs Rf )ai3 + us (qs Rf )qs 1 + u (qs Rf )qs2 Cov 2 s



n 

θks1 Cov(Yk , Yi )

k=1 n 

θks0 Yk

2

(A-11)

 (A-12)

, Yi .

k=1

Using the definition of co-skewness with a portfolio that we introduced at the beginning of this appendix, this reads in vector-matrix notation: 1 0 = us (qs Rf )a3 + us (qs Rf )qs Σθs1 + u (qs Rf )qs2 c(θs0 ) 2 s u (qs Rf )qs −1 u (qs Rf ) −1 Σ a3 , Σ c(θs0 ) −  s θs1 = − s  2us (qs Rf ) us (qs Rf )qs

i.e.

(A-13) (A-14)

and proves Eq. (A-6). Third, the condition E(his4 ) = 0 gives for i = 1, ..., n:       1 2 0 = us (qs Rf )ai4 + us (qs Rf )qs θs0 a2 ai2 + u s (qs Rf )qs Var θs0 Y ai2 2           + us (qs Rf )qs Cov θs2 Y, Yi + u (q R )qs2 Cov θs0 Y θs1 Y + θs0 a2 , Yi s f s   3  1 3 + u s (qs Rf )qs Cov θs0 Y , Yi . 6 Using the definitions of co-cross-skewness and co-kurtosis with a portfolio that we introduced at the beginning of this appendix, this reads in vector-matrix notation:       1 2 0 = us (qs Rf )a4 + us (qs Rf )qs θs0 a2 a2 + u s (qs Rf )qs Var θs0 Y a2 2   2  2  + us (qs Rf )qs Σθs2 + u (q R )q c(θ s f s s0 , θs1 ) + us (qs Rf )qs θs0 a2 Σθs0 s 1 + u (qs Rf )qs3 d(θs0 ). 6 s This proves the representation in (A-7). Finally, fourth, the condition E(his5 ) = 0 gives for i = 1, ..., n:       1 2 0 = us (qs Rf )ai5 + us (qs Rf )qs θs0 a2 ai3 + u s (qs Rf )qs Var θs0 Y ai3 2        2  + us (qs Rf )qs θs1 a2 + θs0 a3 ai2 + u (q s Rf )qs Cov θs0 Y, θs1 Y ai2 s    3    1 3 ai2 + us (qs Rf )qs Cov θs3 + u Y, Yi s (qs Rf )qs E θs0 Y 6       2   + u s (qs Rf )qs Cov θs0 Y θs2 Y + θs1 a2 + θs0 a3 , Yi

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F. Chabi-Yo et al. / Journal of Economic Theory 154 (2014) 453–489

  2  1 2  + u s (qs Rf )qs Cov θs1 Y + θs0 a2 , Yi 2     2   1 3  θs1 Y + θs0 a2 , Yi + u s (qs Rf )qs Cov θs0 Y 2   4  1 4 + u[5] s (qs Rf )qs Cov θs0 Y , Yi . 24 Using the definition of co-pentosis with a portfolio that we introduced at the beginning of this appendix, this reads in vector-matrix notation:       1 2 0 = us (qs Rf )a5 + us (qs Rf )qs θs0 a2 a3 + u s (qs Rf )qs Var θs0 Y a3 2    1    + us (qs Rf )qs θs1 a2 + θs0 a3 a2 + us (qs Rf )qs3 Sk(θs0 )a2 6   2   θ a + u (q R )q Σθ + u (q Rf )qs Σθs3 s f s1 2 s s s s0 s       2  + us (qs Rf )qs c(θs0 , θs2 ) + θs1 a2 + θs0 a3 (Σθs0 )      1 + u (qs Rf )qs2 c(θs1 ) + 2 θs0 a2 (Σθs1 ) 2 s      1 [5] 1 3 u (qs Rf )qs4 f (θs0 ). + u s (qs Rf )qs d(θs0 , θs1 ) + θs0 a2 c(θs0 ) + 2 24 s This gives Eq. (A-8). 2 Proof of Theorem 2. We use the market clearing conditions of Eq. (14) (zero order and first order for j = 1), together with the representations of Lemma A-2 to find the first and second order equilibrium allocations. Plugging in the value of θs0 given by (A-5), the first market clearing condition gives S 

qs τs Σ −1 a2 = S qξ, ¯

s=1

i.e. a2 =

1 b, τ¯

θs0 =

τs ξ. τ¯

(A-15)

Plugging in the value of θs1 given by (A-6), the second market clearing condition (j = 1) gives   S  ρs −1 qs τ s Σ c(θs0 ) 2 + a3 = 0. τs s=1

Eq. (A-15) implies c(θs0 ) = τs2 /τ¯ 2 c. We deduce ρ¯ c, τ¯ 2 This ends the proof. a3 = −

θs1 = 2

τs τs (ρs − ρ)Σ ¯ −1 c = 2 (ρs − ρ)ξ ¯ 0sk . τ¯ 2 τ¯

Proof of Theorem 3. Using the representation of θs2 given in (A-7), the third market clearing condition of Eq. (14) (j = 2) gives:

S S         qs ρs 0 = a4 qs τ s − Var θs0 Y a2 qs θs0 a2 a2 − τs s=1

−

S  s=1

s=1

−2

qs ρ s    qs κs qs ρ s θs0 a2 Σθs0 + 2 d(θs0 ) . c(θs0 , θs1 ) − 2 τs τs τs

F. Chabi-Yo et al. / Journal of Economic Theory 154 (2014) 453–489

Noting that θs0 = 0 = a4

S 

τs τ¯ ξ , a2

qs τ s −

s=1

−

= τ1¯ b = τ1¯ Σξ , d(θs0 ) =

τs3 d, τ¯ 3

487

we get

S   τs  qs 2 ξ  Σξ a2 τ¯ s=1

S 

−3

s=1

 qs κs τs  qs ρs τs qs ρs τs    sk ξ + Σξ a − 2 (ρ − ρ)c ¯ ξ, ξ d . 2 s 0 τ¯ 2 τ¯ 3 τ¯ 3

This shows  κ¯ 1 − 3ρ¯    Var(ρ)  ξ Σξ b − 2 c ξ, ξ0sk + 3 d, 3 3 τ¯ τ¯ τ¯ where κ¯ and Var(ρ) have been introduced in Eqs. (34), (35). Based on (A-7) we then find that    τs τs  τs θs2 = 3 3 (ρs − ρ) ¯ ξ  Σξ ξ + 2 3 ρs (ρs − ρ) ¯ − Var(ρ) ξ0∗sk − 3 (κs − κ)ξ ¯ 0kurt . 2 τ¯ τ¯ τ¯ a4 =

Proof of Theorem 4. Using the representation of θs3 given in (A-8), the fourth market clearing condition (j = 3) gives: 0 = a5

S 

qs τ s +

s=1

+

S 

S  s=1

−

s=1

         qs ρ s  −qs θs0 a2 a3 + Var θs0 Y a3 − qs θs1 a2 + θs0 a3 a2 τs

 qs κs qs ρ s   θs0 Σθs1 a2 Sk(θs0 )a2 + 2 τs τs2

S      qs ρs      qs ρ s   c(θs0 , θs2 ) + θs1 c(θs1 ) + 2 θs0 + 2 a2 + θs0 a3 (Σθs0 ) + a2 (Σθs1 ) τs τs s=1     qs χs qs κs  − 3 2 d(θs0 , θs1 ) + θs0 a2 c(θs0 ) + 3 f (θs0 ). τs τs

Recalling that θs0 = ξ  c = Sk, we get 0 = a5

S 

qs τ s +

s=1

+

S  s=1

τs τ¯ ξ ,

τs (ρ τ¯ 2 s

− ρ)Σ ¯ −1 c, f (θs0 ) =

τs4 f, τ¯ 4

S  qs τ s  s=1

−

a2 = τ1¯ Σξ , a3 = − τ¯ρ¯2 c, θs1 =

τ¯ 4

 ¯ qs τs ρs    qs τs (ρs − 2ρ) ξ  Σξ ρc ξ Σξ ρc ¯ − ¯ − (Sk · b) τ¯ 4 τ¯ 4

  ¯    qs τs κs qs τs ρs (ρs − ρ) qs τs ρs ξ c b+6 (Sk · b) + 2 (ρs − ρ) ¯ ξ  Σξ c 4 4 4 τ¯ τ¯ τ¯

S       qs τ s ρ s qs τ s ρ s  + ρs (ρs − ρ) 4 ¯ − Var(ρ) c ξ, ξ0∗sk − 2 (κs − κ)c ¯ ξ, ξ0kurt 4 4 τ¯ τ¯ s=1

+

S      qs τ s ρ s qs τ s ρ s q s τs 2 (ρs − 2ρ)(Sk ¯ · b) + 4 (ρs − ρ) ¯ 2 c ξ0sk + 2 (ρs − ρ) ¯ ξ  Σξ c 4 4 τ¯ τ¯ τ¯ s=1

+

S  s=1

−3

   qs τs κs  qs τs κs qs τs χs (ρs − ρ)d ¯ ξ, ξ0sk − 3 4 ξ  Σξ c + f, 4 τ¯ τ¯ τ¯ 4

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that is −ρ¯ + ρ¯ 2 − 8 Var(ρ) + 3κ¯    −ρ¯ + 2ρ¯ 2 − 4 Var(ρ) + κ¯ ξ Σξ c + (Sk · b) τ¯ 4 τ¯ 4  Cov(ρ, κ)      Skew(ρ) + ρ¯ Var(ρ)  2c ξ, ξ0kurt + 3d ξ, ξ0sk c ξ, ξ0∗sk + −4 4 4 τ¯ τ¯ Var(ρ)  sk  χ¯ c ξ0 − 4 f, − τ¯ 4 τ¯ where χ, ¯ Cov(ρ, κ), Skew(ρ) are as introduced in Eqs. (42)–(44). In terms of higher moments of returns, one may want to reread this as:    3      2  Cov Y, ξ  Y + A2 (ρ, κ) Var ξ  Y Cov ξ  Y , Y σ 5 a5 = A1 (ρ, κ)E ξ  Y     Skew(ρ) + ρ¯ Var(ρ) Cov ξ  Y ξ0∗sk, Y , Y −4 4 τ¯  sk, 2  χ¯   4  Var(ρ) Cov ξ , Y − Cov ξ Y ,Y − 0 τ¯ 4 τ¯ 4      2    Cov(ρ, κ)  2 Cov ξ  Y ξ0kurt, Y , Y + 3 Cov ξ  Y ξ0sk, Y , Y , + 4 τ¯ where A1 (ρ, κ), A2 (ρ, κ) have been defined in Eq. (40). 2 a5 =

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