Stochastic volatility implies fourth-degree risk dominance:Applications to asset pricing

Christian GollierToulouse School of Economics, University of Toulouse-Capitole

December 9, 2016

Abstract

Increasing the risk surrounding the variance of future consumption generates a fourth-degree stochastically dominated shift in distribution of consumption. Its impact on theequilibrium risk premium is thus positive only if the fourth derivative of the utility func-tion is negative. Its impact on interest rates is negative only if its fifth derivative ispositive. We also show that the persistence of shocks to the variance of the consumptiongrowth rate, as assumed in long-run risk models, has no effect on the term structure ofthe variance ratio which remains flat in expectation, but it magnifies the kurtosis of logconsumption for long time horizons. It makes the term structures of interest rates andrisk premia respectively decreasing and increasing under constant relative risk aversion.Using recursive preferences does not qualitatively modify these results, which are counter-factual. However, the persistence of shocks to the variance of changes in log consumptionis supported by the observation that the term structure of their annualized 4th cumulantis increasing over the period 1947Q1-2016Q4 in the United States.

Keywords: Long-run risk, fourth-degree stochastic dominance, temperance, recursive utility,kurtosis.

JEL codes: D81

Acknowledgement: I thank Jules Tinang Nzesseu for providing me with the consumption data usedin Section 7. I acknowledge funding from the chair SCOR and FDIR at TSE. This draft is verypreliminary. Comments welcomed.

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1 Introduction

This paper provides a theoretical analysis of the effect of stochastic volatility on intertemporalwelfare and asset prices. Since the seminal work by Bansal and Yaron (2004), there is agrowing literature that demonstrates that introducing stochastic volatility in the processgoverning aggregate consumption can solve the classical puzzle in finance when combinedwith other ingredients such as the persistence of shocks to growth and the disentanglementof the attitude towards risk and consumption fluctuations. Our main aim in this paperdiffers radically from that long-run risk literature. Rather than showing that a realisticcalibration of the parameters of the model can explain the observed asset prices if the modelis rich enough, we focus on a single ingredient, i.e., stochastic volatility, and we examineits consequence on asset pricing from a theoretical point of view. In fact, the stochasticvolatility contained in the Bansal-Yaron process produces an increase in risk in the varianceof log consumption. It is often suggested that stochastic volatility adds a new layer of risk,which implies that risk-averse consumers should dislike it. This idea may be supported bythe following simple analysis. Conditional to the variance σ2 of future consumption c, theconsumer’s welfare can be estimated by using the Arrow-Pratt approximation v(Ec−0.5Aσ2),where v is the von Neumann-Morgenstern utility function, and A is its index of absolute riskaversion. If the variance is uncertain, then, the unconditional welfare could be approximatedby Ev(Ec−0.5Aσ2). This suggests that the stochastic nature of the variance makes consumersworse off only if they are risk-averse. This common wisdom is not true.

To show this, consider two lotteries. With lottery L1, one loses or wins one monetaryunit with equal probabilities. With lottery L2, one loses or wins two monetary units withprobability 1/8. What is the preferred lottery? Observe that the first three moments ofthese two lotteries are identical, with a zero first and third moments, and a unit secondmoment. This implies that all expected-utility-maximizers with a third-degree polynomialutility function will be indifferent between the two lotteries, independent of their degree of riskaversion and prudence. In particular, it is not true that L2 is riskier than L1 in the standardsense defined by Rothschild and Stiglitz (1970).1 How is this question related to stochasticvolatility? In Figure 1, we represented L2 as two alternative compound lotteries, L′2 and L′′2.Lottery L′2 compounds a sure payoff of x0 = 0 with probability 3/4 and a zero-mean lotteryx′1 ∼ (−2, 1/2; +2, 1/2) with probability 1/4. Because the variance of x′1 equals 4, L1 differsfrom L′2 by the fact that the sure variance of 1 is replaced by an uncertain variance that isdistributed as (0, 3/4; 4, 1/4). This is a mean-preserving spread in variance. This is also thecase for the compound lottery L′′2. In other words, determining the preference order betweenlotteries L1 and L2 is equivalent to determining the welfare impact of an uncertain variance.The cornerstone of our analysis is our Lemma 1 which states that a mean-preserving spread inthe distribution of the variance of a random variable x generates a fourth-degree risk increaseof x in the sense of Ekern (1980). In other words, it reduces Ef(x) if and only if the fourthderivative of f is negative. A necessary (but not sufficient) condition is that the kurtosis of

1In fact, L2 is obtained from L1 through a sequence of two mean-preserving spreads and one mean-preserving contraction. In the first MPS, the outcome 1 of L1 is replaced by (+2, 3/4;−2, 1/4). In the secondMPS, the outcome -1 of L1 is replaced by (+2, 1/4;−2, 3/4). Finally, the MPC takes the form of displacingprobability masses of 3/8 from respectively −2 and 2 to 0.

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Figure 1: Two ways of representing lottery L2 ∼ (−2, 1/8; 0, 3/4; +2, 1/8) as a compound lottery withstochastic variance.

x is increased. Thus, we conclude that, under expected utility, increasing the risk on thevariance of final consumption is disliked if and only if the fourth derivative of the utilityfunction is negative. Under this condition, lottery L1 is preferred to lottery L2.

The negativeness of the fourth derivative of the utility function is called "temperance"in expected utility theory. Eeckhoudt and Schlesinger (2006) have shown that temperanceis necessary and sufficient for consumers to prefer compound lottery (x1, 1/2;x2, 1/2) overcompound lottery (0, 1/2;x1 + x2, 1/2), for any pair (x1, x2) of zero-mean independent lot-teries. This is another form of aversion to a stochastic variance. In fact, the preference of L1over L′′2 can be represented in this way, with L1 ∼ x1 ∼ x2. Gollier and Pratt (1996) showthat temperance is necessary for any zero-mean background risk to raise the aversion to anyother independent risk. Under Constant Relative Risk Aversion (CRRA), temperance goestogether with risk aversion since the successive derivatives of the utility function alternate insign. This is why the reference to temperance is never made in the long-run risk literaturein spite of its crucial role. Because the finance literature models the stochastic process of logconsumption, the relevant (indirect) utility function is f = u ◦ exp. Since the exponentialfunction has a positive fourth derivative, f exhibits temperance under CRRA only if relativerisk aversion is larger than unity. This means that under CRRA, the stochastic nature ofthe variance of log consumption is disliked if and only if relative risk aversion is larger thanunity.

Since Leland (1968), Drèze and Modigliani (1972) and Kimball (1990), it is well-knownthat, under Discounted Expected Utility (DEU), the interest rate is decreasing in the risksurrounding future consumption if and only if it raises the expected marginal utility of futureconsumption. By Jensen’s inequality, this is obviously the case if the representative agent isprudent, i.e., if the third derivative of the utility function is positive. Because it increasesthe fourth-degree risk (kurtosis), the stochastic nature of the variance of consumption also

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reduces the interest rate under DEU if and only if the fifth derivative of the utility functionis negative. In a two-period model with recursive preferences à la Kreps and Porteus (1978)and Selden (1978), a necessary and sufficient condition in the small is obtained that combinesthe fourth and fifth derivatives of the risk utility function with the difference between riskaversion and fluctuation aversion. This condition is always satisfied when absolute or relativerisk aversion is constant.

The long-run risk literature focused on the prediction of the short-term interest rate andof the price of equity. Up to our knowledge, there has been no analysis of the term structureof zero-coupon bond and equity returns, with the exception of Beeler and Campbell (2012)who examined interest rates. Bansal and Yaron (2004) considers a stochastic process of logconsumption in which the shocks on variance are highly persistent. This is useful to explainwhy the equity premium on financial markets is time varying. Because it raises the kurtosisof future log consumption, it reduces the interest rate and it raises the equity premium, whichis helpful to solve the financial puzzles. Since the term structure of the annualized standarddeviation of the conditional variance of log consumption is increasing in this model, so is theterm structure of its annualized kurtosis. This magnifies the effect of stochastic volatilityat higher maturities. This means that, under DEU, the term structure of interest rates andrisk premia are respectively decreasing and increasing. Both properties are contradicted byasset prices observed on financial markets. In particular, recent findings document the factthat dividend strip risk premia have a decreasing term structure (Binsbergen et al. (2012),Binsbergen and Koijen (2016), Giglio et al. (2015), Belo et al. (2015), Marfè (2016) and Giglioet al. (2016)). The traditional combination of stochastic volatility with a persistence in theshocks to the expected growth rate that prevails in the long-run risk literature will make theproblem more puzzling, since it will generate an increasing term structure of the annualizedvariance of log consumption.2 Because of prudence, this will make the term structure ofinterest rates more decreasing (Campbell (1986), Gollier (2008)). Because of risk aversion, itmakes the term structure of risk premia more increasing. And the traditional combinationof stochastic volatility with recursive preferences does not solve the problem either. This isshown in this paper by deriving analytically the term structures from the stochastic volatilitymodel extracted from Bansal and Yaron (2004). In the most recent calibration of the modelby Bansal et al. (2016), the half-life of the shocks to the variance exceeds 35 years. This hasstrong implications for this term structures. In particular, it takes many centuries of durationfor the interest rates and risk premia to converge to their asymptotic value. For example, adividend strip generated by a diversified portfolio of equity has a risk premium 1.71% for aone-month maturity, and it goes up to 3.17% when the maturity tends to infinity. However,this equity premium is only 1.86% for a 10-year maturity, and 2.28% for a 50-year maturity.

The paper is organized as follows. In Section 2, we provide some generic results linkingstochastic volatility, fourth-degree stochastic dominance and kurtosis. We apply these find-ings in a two-period Kreps-Porteus preferences in which the variance of future consumption

2Beeler and Campbell (2012) show evidence of mean-reversion rather than persistence in U.S. consumptiongrowth in the period since 1930. Mean-reversion makes the aggregate risk in the longer run relatively smallerand can thus explain why interest rates and risk premia are respectively increasing and decreasing in maturity,contrary to what the persistence in volatility shocks implies.

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is uncertain in Section 3. Section 4 is devoted to the same model, but it is log consumption,rather than consumption, whose volatility is stochastic. In the next two sections, we explorethe term structures of asset prices under the standard long-run risk specification yieldingpersistent shocks to the variance of log consumption, in the DEU case in Section 5, and inthe case of Epstein-Zin-Weil preferences in Section 6. In Section 7, we test whether the termstructure of the 4th cumulant of changes in log consumption is upward-sloping due to thepersistence of shocks to the variance. We provide some concluding remarks in the last section.

2 Stochastic volatility and stochastic dominance

This section is devoted to the analysis of the impact of the stochastic variance of a randomvariable x on Ef(x), where f is a four time differentiable real-valued function. The riskstructure of x is described by the following model:

x = x+ ση (1)σ2 = σ2 + w, (2)

where (η, w) is a pair of independent random variables with a zero mean so that x and σ2

are the mean of respectively x and σ2.3 If we assume that Eη2 = 1, then σ2 measures thevariance of x conditional to σ. Finally, we also assume that

Eη3 = 0. (3)

This assumption guarantees that the unconditional third moment of x is zero, independentlyof the distribution of σ. Moreover, the unconditional fourth moment of x is equal to

E(x− x)4 = E (ση)4 = Eσ4Eη4. (4)

By Jensen’s inequality, we know that Eσ4 is larger than (Eσ2)2. This implies that theuncertainty affecting variance σ2 has no effect on the first three moments of x, but it raisesits fourth centered moment, i.e., its kurtosis.

We now characterize the impact of the stochastic variance of x on Ef(x). By the law ofiterated expectations, we have that

Ef(x) = E[E[f(x) | σ2]

]= Eh(σ2), (5)

where function h is derived from function f in such a way that h(σ2) equals E[f(x) | σ2]for all σ. An increase in risk of variance is defined as a sequence of Rothschild-StiglitzMean-Preserving Spreads (MPS) in the distribution of the variance σ2 of x. From Rothschildand Stiglitz (1970), it is also defined as a change in the distribution of σ2 that reduces theexpectation of any concave function of σ2. This implies that, from equation (5), an increasein risk on variance reduces Ef if and only if h is concave in σ2. The following proposition

3Notice that σ is not the mean of σ in this model. By Jensen’s inequality, the uncertainty affecting σ2 hasa negative impact on the expected value of σ.

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states the necessary and sufficient condition for this to be true independent of the distributionof the zero-mean random variable η. The proof of this cornerstone lemma is relegated to theappendix.

Lemma 1. Suppose that x = x+ση, where x is a constant and σ and η are two independentrandom variables satisfying condition (3). Any increase in risk in the variance σ2 of x reducesEf(x) if and only if f ′′ is concave.

Proof: See the appendix.

This result is related to the theory of stochastic dominance orders. Following Ekern(1980), a random variable undergoes a n-th degree risk deterioration if and only if this changein distribution reduces the expectation of any function g ∈ Gn of that random variable, whereGn is the set of all real-valued functions g such that (−1)ng(n) ≤ 0.4 For example, the casen = 1 corresponds to the concept of first-order stochastic dominance, whereas the case n = 2corresponds to the Rothschild-Stiglitz’s notion of an increase in risk. Ekern (1980) showsa n-th degree risk deterioration requires the n − 1 moments of the random variable to beunaffected. Lemma 1 means that raising the uncertainty affecting the variance of x in thesense of Rothschild and Stiglitz (1970) deteriorates random variable x in the sense of fourth-degree risk.

An extreme illustration of this phenomenon has been proposed by Weitzman (2007).Suppose that η is N(0, 1) and that the precision p = σ−2 has a Gamma distribution. Then,as is well-known, the unconditional distribution of x is a Student-t, which has fatter tailsthan the Normal distribution with the same expected variance. The moments and cumulantsof the Student-t are undefined, which means that the expectation of exp(kx) is unbounded,for all k ∈ R. This is an extreme illustration of Lemma 1 in which moving from a sure σ2 toa risky one with the same mean pushes the expectation of f to plus or minus infinity. Gollier(2016) examined risk profiles for σ that generate a bounded solution.

It is useful to measure a "stochastic volatility premium" associated to function f whichis defined as the sure reduction πf in x that has the same impact on Ef as the uncertaintyaffecting σ2. Technically, πf satisfies the following condition:

Ef(x+ ση − πf ) = Ef(x+ ση), (6)where σ2 is the expectation of the uncertain variance σ2. The following proposition providesan estimation of the stochastic volatility premium for welfare in the small.

Lemma 2. Suppose that η is small in the sense that η = kε, with k small and Eε = 0 =Eε3 = 0. Then, the stochastic volatility premium associated to f defined by equation (6)satisfies the following property:

πf = 14!ψf (x)σ2

wEη4 +O(k5), (7)

where ψf (x) = −f ′′′′(x)/f ′(x) is an index of concavity of f ′′.4g(n) denotes the n-th derivative of g.

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Proof: See the appendix.

This proposition implies that the concavity of f ′′ is also necessary and sufficient in thesmall. It is noteworthy that when η has a standard Normal distribution, tEη4 is equal to 3,which implies that the approximation simplifies in that case to

πf ' 0.125ψf (x)σ2w. (8)

In the appendix, we examine two analytical solutions for the stochastic volatility premium,when σ2 is Normal or Gamma. We assume that f(x) = exp(−Ax) for all x. Notice that inthis case, the index ψf (x) of concavity of f ′′ is a constant equaling A3.

Lemma 3. Suppose that x is distributed as x = x+ση, where σ and η are independent, η isN(0, 1), and σ2 has mean σ2 and variance σ2

w. Suppose also that f(x) = exp(−Ax) for allx. If the stochastic variance v is normally distributed, then the stochastic volatility premiumequals

πf = 0.125A3σ2w. (9)

If σ2 has a gamma distribution, we have that

πf = − σ4

Aσ2w

log(

1− 12A

2σ2w

σ2

)− 1

2Aσ2. (10)

Proof: See the appendix.

Although using a Normal distribution for the variance is common practice in the Long-Run Risk literature since Bansal and Yaron (2004), it is not satisfactory because it implies anegative variance with a positive probability. In fact, when η ∼ N(0, 1) and σ2 ∼ N(σ2, σ2

w),variable x is a pseudo-random variable. The Cumulant Generating Function (CGF) χ of a(pseudo-) random variable is defined as χ(k, x) = log (E exp(kx)) for all k ∈ R.5 Using thelaw of iterated expectations, we have that

χ(k, x) = log(E exp(kx+ 0.5k2σ2)

)= kx+ 1

2k2σ2 + 1

8k4σ2w. (11)

This implies that the pseudo-random variable x has the following cumulants:6 Its mean andvariance are respectively equal to κx1 = x and κx2 = σ2, its skewness is zero and its fourthcumulant is equal to7

κx4 = 3σ2w. (12)

All other cumulants are zero. The fact that the excess kurtosis of x defined by equations(1)-(2) is proportional to the variance of the conditional variance σ2 of x is a key result when

5Martin (2013) uses the properties of the CGF function to derive analytical solutions for interest rates andrisk premia under Epstein-Zin-Weil preferences with i.i.d. growth rates.

6The nth cumulant of x is defined as κxn = χ(n)(0, x). For example, the fourth cumulant κx4 equalsE(x− Ex)4 − 3(κx2)2. Using a Taylor expansion implies that χ(k, x) =

∑+∞n=1 κ

xnk

n/n!.7This implies that the excess kurtosis of x equals 3σ2

w/σ4.

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a dynamic setting will be examined latter on in this paper. For example, in the exponentialcase, we obtain that

E exp (−Ax) = f(−A−1χ(−A, x)

)= exp

(−Ax+ 1

2A2σ2 + 1

4!A4κx4

). (13)

3 Stochastic variance of consumptionIn this section, we explore the pricing implications of stochastic volatility in the simple 2-period model proposed by Kreps and Porteus (1978):

W0 = u(c0) + e−δu(e1) (14)v(e1) = Ev(c1). (15)

The intertemporal welfareW0 is a discounted sum of the current utility extracted from currentconsumption c0 and of the future utility extracted from the certainty equivalent e1 of futureconsumption c1. Utility functions u and v are the time and risk aggregators, respectively.They are assumed to be increasing and five times differentiable. Parameter δ is the rateof pure preference for the present. Future consumption is affected by stochastic variancestructured as follows:

c1 = c1 + ση (16)σ2 = σ2 + w, (17)

where η and w are independent and have a zero mean, with Eη3 = 0.

As in Campbell (1986), Abel (1999) and Martin (2013) for example, let P (φ) denote theprice today of an asset that generates a payoff distributed as cφ1 in the future. The risk-freeasset corresponds to φ = 0, and a claim on aggregate consumption corresponds to φ = 1.In a Lucas tree economy with a representative agent whose preferences are represented byequations (14) and (15), and where c0 and c1 are the fruit endowment at date 0 and 1, theequilibrium price P (φ) is characterized by the following pricing equation:

P (φ) = e−δEcφ1v

′(c1)v′(e1)

u′(e1)u′(c0) . (18)

The continuously compounded expected return r(φ) of the asset is equal to the logarithm ofEcφ1/P (φ).

We first examine how future welfare measured by the certainty equivalent consumptione1 is affected by the stochastic nature of volatility. The following proposition is a directconsequence of Lemma 1.

Proposition 1. Any increase in risk in the variance of future consumption reduces futurewelfare measured by e1 if and only if v′′ is concave.

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Following Eeckhoudt and Schlesinger (2006), condition u′′′′ ≤ 0 is referred to as "temper-ance". Gollier and Pratt (1996) showed that temperance is necessary for any zero-mean back-ground risk to raises the aversion to any other independent risk. Eeckhoudt and Schlesinger(2006) showed that temperance is necessary and sufficient for an individual to prefer a 50-50 compound lottery yielding either x1 or x2 over another 50-50 compound lottery yieldingeither 0 or x1 + x2, where x1 and x2 are two zero-mean independent lotteries.

Let us now turn to the analysis of the risk-free rate. We obtain a clearcut result in thespecial case of Discounted Expected Utility (DEU) where u and v are identical. In the DEUcase, equation (41) directly implies that the price P (0) of a risk-free asset is increased bystochastic volatility if and only if it raises Eu′(c1). The following proposition is thus anotherimmediate application of Lemma 1.

Proposition 2. Suppose u ≡ v. Any increase in risk in the variance of future consumptionreduces the interest rate if and only if v′′′ is convex.

Thus, the effect of the stochastic nature of the variance of future consumption on interestrates is negative under DEU if and only if the fifth derivative of the utility function ispositive. This is because this uncertainty has no effect on the first three moments of futureconsumption, but it raises its kurtosis, as shown in Section 2. This raises the expectedmarginal utility only if the fourth derivative of marginal utility is positive. In that case,stochastic volatility has the same qualitative effect on the interest rate as an increase in riskon future consumption, which is negative under prudence (v′′′ ≥ 0).

In the recursive generalization of the DEU model examined in this section, the impact ofstochastic volatility on the interest rate is also affected by the marginal rate of transformationu′(e1)/v′(e1). This rate tells us how a marginal increase in risk utility v(e1) generated byan increase in e1 translates into an increase in temporal utility u(e1). Because stochasticvolatility affects the certainty equivalent consumption e1, it also affects this marginal rate oftransformation. It is increasing in e1 if and only if −v′′(c)/v′(c) is larger than −u′′(c)/u′(c)for all c, i.e., if and only if v is more concave than u. This observation combined with Lemma1 yields the following result. For example, when v′′ is concave, stochastic volatility reducese1, and it raises u′(e1)/v′(e1) if v is less concave than u. From equation (41), this tends toraise the price of the risk-free asset, thereby reinforcing the effect described in Proposition 2when v′′′ is convex.

Proposition 3. Any increase in risk in the variance of future consumption reduces the in-terest rate if v′′′ is convex and one of the following conditions is satisfied:

• v′′ is concave and v is less concave than u;

• v′′ is convex and v is more concave than u.

If the risk is small in the sense that η is distributed as kε with k small, then the necessaryand sufficient condition is:

v(5)(c1) + v(4)(c1)(−v′′(c1)v′(c1) −

−u′′(c1)u′(c1)

)≥ 0. (19)

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Proof: See the appendix.

The sufficient conditions described in this proposition are not very satisfactory becauseit is generally assumed that v′′ is concave and that v is more concave than u. It is thereforeuseful to examine the necessary and sufficient condition in the small. If we assume that v′′ isconcave, then, this condition can be rewritten as follows:

v(5)(c1)−v(4)(c1)

≥ −v′′(c1)

v′(c1) −−u′′(c1)u′(c1) . (20)

The left-hand side of this inequality is usually referred to as the fourth-order risk aversioncoefficient. It measures the degree of convexity of v′′′. This inequality tells us that thisfourth-order risk aversion must be larger than the difference in the degrees of concavity of vand u. This condition is always satisfied when v exhibits constant absolute or relative riskaversion.

We now turn to the analysis of the impact of stochastic volatility on the risk premium ofasset φ, which is the difference between the expected return of that asset and of the risk-freeasset. it is equal to

π(φ) = log(Ecφ1Ev

′(c1)Ecφ1v

′(c1)

). (21)

How does the stochastic nature of the variance of consumption affect this risk premium?Using Lemma 1 does not provide here a clearcut result since the stochastic volatility affectssimultaneously the three expectations of the right-hand side of the above equation. This iswhy we limit our analysis in this section to the case of small risk.

Proposition 4. Suppose that risk is small in the sense that η is distributed as kε with ksmall. Any increase in risk in the variance of future consumption reduces the risk premiumassociated to a claim on aggregate consumption if and only if u′′ is concave.

Proof: See the appendix.

This proposition tells us that stochastic volatility can contribute to explain the equitypremium puzzle in the Kreps-Porteus model only if the fourth derivative of the risk utilityfunction v of the representative agent is negative. This is in line with the observation thatthe uncertainty affecting the variance of future consumption yields a fourth-degree risk dete-rioration, which corresponds in the small to an increase in the kurtosis of the distribution ofconsumption.

4 Stochastic variance of the growth rate of consumptionThe current trend in the literature is to make the variance of the growth rate of consumptionstochastic. In the two-period model examined in the previous section, this means replacing

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equations (16) and (17) by the following specification:

log(c1c0

)= µ+ ση (22)

σ2 = σ2 + w, (23)

where η and w are independent and have a zero mean, with Eη3 = 0. In this context,following Ross (2004), our earlier results can be reinterpreted by defining a utility functionfv on the change in log consumption x = log(c1/c0) as follows:

fv(x) = v(c0 expx). (24)

For example, under which condition on v does an increase in risk in the variance σ2 of thegrowth rate of consumption reduces expected utility Ev(c1) = Efv(x)? From Proposition1, we know that this is the case if and only if f ′′v is concave in x. From definition (24) andnormalizing c0 to unity, this is the case iff

f ′′′′v (x) = e4xv′′′′(ex) + 6e3xv′′′(ex) + 7e2xv′′(ex) + exv′(ex) (25)

is negative for all x ∈ R. This condition is more complex than in the case in which it isthe variance of consumption which is stochastic. Remember than the stochastic nature ofthe variance of consumption does not affect the first three moments of consumption, so thatthe first three derivatives of the utility function are irrelevant to sign its effect on welfare.But when it is the variance of the logarithm of consumption that is stochastic, the first threemoments of consumption are affected. This implies that risk aversion (u′′) and prudence (u′′′)matter to determine whether expected utility is deteriorated. Notice in particular that

Ecn1 = E exp(n(µ+ ση)). (26)

Because function hn(z) = exp(nz) has a second derivative that is convex in z for all n ∈ N0,Lemma 1 implies that all moments of consumption are increased by an increase in risk inthe variance of log consumption. The fact that the variance of consumption is increasedmakes it more likely that this form of stochastic volatility reduces welfare, because of riskaversion. This is made apparent by the the presence of the third term in the sum appearingin equation (25). But it also makes consumption positively skewed, an effect that goes inopposite direction under prudence (u′′′ ≥ 0). Equation (25) tells us how to weigh thesedifferent effects to determine whether increasing the risk on the variance of log consumptionis desirable or not.

The problem is much simplified if one assumes that the utility function v exhibits constantrelative risk aversion γ ≥ 0, i.e., v(c) = c1−γ/(1− γ), so that

fv(x) = −exp (−(γ − 1)x)γ − 1 . (27)

This implies that function f exhibits constant absolute risk aversion A = γ − 1, yieldingf ′′′′v (x) = −(γ − 1)3 exp(−(γ − 1)x). The index ψfv of concavity of f ′′v is (γ − 1)3 in thiscase, which is positive if γ is larger than unity. Lemma 1 thus implies that the stochastic

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volatility of the consumption growth rate reduces welfare in the Kreps-Porteus model withconstant relative risk aversion if and only if relative risk aversion is larger than unity. Theimpact on welfare can be measured by the relative stochastic volatility premium π, which isthe sure reduction in the growth rate of consumption that yields the same effect on intertem-poral welfare than the uncertainty affecting the variance of the growth rate of consumption.Technically, it is implicitly defined by the following equation, with v = Ev:

Ev (c0 exp(µ+ ση − π)) = Ev (c0 exp(µ+ ση)) . (28)

Proposition 5. Suppose that log(c1/c0) is distributed as µ+ ση where σ and η are indepen-dent and η has a standard Normal distribution. Suppose also that relative risk aversion is aconstant γ. The uncertainty affecting the variance of log consumption reduces the certaintyequivalent growth rate of consumption if and only if γ is larger than unity. The relativestochastic volatility premium π defined by (28) is approximately equal to

π ' 0.125(γ − 1)3σ2w. (29)

This approximation is exact when the variance of log consumption is normally distributed, orwhen v is logarithmic.

The assumption that the variance be normally distributed is problematic since there is apositive probability that this variance be negative. This is not the case if we assume that σ2

has a Gamma distribution. In that case, Lemma 3 implies the following solution:

π = − σ4

(γ − 1)σ2w

log(

1− 12(γ − 1)2σ

2w

σ2

)− 1

2(γ − 1)σ2. (30)

In Figure 2, we compare the relative stochastic volatility premia prevailing in the Normal andGamma frameworks, assuming γ = 10 and ση = 1. Suppose also that the expected annualvolatility is σ = 3%. The dashed curve corresponds to the stochastic volatility premiumas a function of the standard deviation of the variance σ2 when σ2 is normally distributed.The plain curve measures that function in the alternative case in which σ2 has a Gammadistribution. It is easy to verify from equations (29) and (30) that the two functions areidentical up to the fourth order of σw, so that the two curves coincide for low levels ofuncertainty.

We now turn to the analysis of the impact of the uncertain variance of log consumptionon the risk-free rate and the risk premium. Again, because an increase in risk in the varianceof log consumption affects all four moments of the distribution of consumption, the first fivederivatives of function v play a role in the determination of asset prices. This is why we againlimit the analysis to the case of power functions. Following Epstein and Zin (1989) and manyothers after them, we hereafter assume that v(c) = c1−γ/(1 − γ) and u(c) = c1−ρ/(1 − ρ).Parameter ρ is the relative aversion to consumption fluctuations over time. It is the inverse ofthe elasticity of intertemporal substitution. The following proposition is a direct consequenceof using equation (8) several times to estimate the expectations that appear in equations (41)and (21).

12

Normal

Gamma

0.005 0.010 0.015 0.020 0.025 0.030σw

0.05

0.10

0.15

π

Figure 2: The relative stochastic volatility premium as a function of the degree of uncertainty σw affecting thevariance of log consumption. The dashed curve corresponds to σ2 ∼ N(3%, σ2

w). The plain curve correspondsto σ2 having a Gamma distribution with the same mean and variance. We assume a CRRA of 10 andη ∼ N(0, 1).

Proposition 6. Suppose that log(c1/c0) is distributed as µ+ ση where σ and η are indepen-dent, η has a standard Normal distribution, and σ2 is N(σ2, σ2

w). Suppose also that relativerisk aversion γ and the relative aversion to fluctuations ρ are constant. The interest raterf and the risk premium π(φ) associated to an asset whose future payoff is cφ1 satisfy thefollowing conditions:

rf = δ + ρµ− 12(γ2 − (γ − ρ)(γ − 1)

)σ2 − 1

8(γ4 − (γ − ρ)(γ − 1)3

)σ2w (31)

π(φ) = φγσ2 + 12φγ

(γ2 − 3

2φγ + φ2)σ2w. (32)

Let us first focus on the interest rate. Increasing the risk on the variance of log con-sumption reduces the interest rate if γ4 is larger than (γ − ρ)(γ − 1)3. Under DEU, theimpact of stochastic volatility on future expected marginal utility Ev′(c1) = E exp(−γx).From Lemma 3, we know that this reduces marginal expected utility exactly as an equivalentreduction in the expected growth rate that is proportional to γ3. So it reduces the interestrate proportionally to γ4. We refer to this effect as a "precautionary effect": It explains howthe increased risk on the variance raises precautionary saving through the increased expec-tation of future marginal utility. But when u and v differ, we should also take account ofthe fact that stochastic volatility affects the marginal rate of transformation u′(e1)/v′(e1)appearing in equation (41). We know from Lemma 2 that stochastic volatility reduces thecertainty equivalent growth rate proportionally to (γ−1)3. This reduces the marginal rate oftransformation – and the price of the risk-free bond – proportionally to exp((γ− ρ)(γ− 1)3).This raises in turn the interest rate proportionally to (γ−ρ)(γ−1)3. This is an income effect.Globally, the stochastic volatility affecting log consumption reduces the interest rate if andonly if the precautionary effect dominates the income effect. This is the case for examplewhen both γ and ρ are larger than unity. When ρ equals 1, it requires that γ be larger than1/2.

13

From equation (21), the effect of stochastic volatility on the risk premium associated toasset φ depends upon how it affects log(E exp(φx)), log(E exp(−γx)) and log(E exp((φ −γ)x)). From equation (13), these effects are proportional to φ4, γ4, and (φ−γ)4, respectively.Equation (32) is then the direct consequence that8

φ4 + γ4 − (φ− γ)4 = 4φγ[γ2 − 3

2φγ + φ2].

Because the bracketed term in this equation is always positive, the impact of stochasticvolatility on the risk premium associated to any asset φ has the same sign as φ. In particular,it increases the risk premium associated to a claim on aggregate consumption (φ = 1).

We now explore the impact of stochastic volatility on wealth, which is here defined as theequilibrium price at date 0 of a claim on the future aggregate consumption c1. It is equal toEc1 discounted at the risk-adjusted discount rate rf + π(1). Using Proposition 6, we obtainthat

Wealth = c0 exp(−δ − (ρ− 1)µ+ 1

2(γ − 1)(ρ− 1)σ2 + 18(γ − 1)3(ρ− 1)σ2

w

). (33)

As is well-known, in the DEU model (ρ = γ), an increase in uncertainty affecting growth, asmeasured by σ2, raises wealth in the economy. We obtain the same result for an increase inthe uncertainty affecting volatility, as measured by σ2

w. One of the benefits of the recursiveutility model is to reverse these impacts when ρ ≤ 1 ≤ γ. In this case, an increase in risk ongrowth or on volatility reduces wealth in the economy.

As a final remark related to this two-period framework, observe that from equation (12),we have that the 4th cumulant of log consumption is proportional to σ2

w. Stochastic volatilityinduces a positive excess kurtosis, and the last term in equations (31), (32) and (33) can bereinterpreted as the effect of the kurtosis of log consumption on asset prices, as analyzed byMartin (2013).

5 Long-run stochastic volatility under Discounted ExpectedUtility

In the remainder of this paper, we characterize the term structures of the stochastic volatilitypremia for intertemporal welfare, interest rates and risk premia in the context examined byBansal and Yaron (2004) in which the variance of the growth rate of consumption follows anautoregressive process of order 1:

log(ct+1ct

)= µ+ σtηt+1, (34)

8Martin (2013) shows that an increase in the excess kurtosis of the growth rate of consumption raises therisk premium if and only if φ4 +γ4− (φ−γ)4 is positive. This is in line with Proposition 6 since the stochasticvolatility of changes in log consumption increases the kurtosis of log consumption.

14

withσ2t+1 = σ2 + ν

(σ2t − σ2

)+ σwwt+1, (35)

where σ2 is the unconditional variance, ν ∈ [0, 1[ is the coefficient of persistence of shockson variance, σw is the standard deviation of these shocks, and the two shocks η and ware assumed to be i.i.d. standard Normal.9 This stochastic process has three interestingfeatures. First, the volatility σt is stochastic. Second, the shocks to volatility exhibit somepersistence. Third, the volatility is known one period in advance. Notice also that an assetφ that generates a cash flow Dt = cφt is governed by the following stochastic process:

log(Dt+1Dt

)= µD + φσtηt+1, (36)

with µD = φµ.10 As in Bansal and Yaron (2004), the stochastic volatility of dividend growthis proportional to the stochastic volatility of consumption growth.

Under equations (34) and (35), the variance of log-consumption t periods ahead is asfollows:

vt = V ar

[log

(ctc0

)∣∣∣∣w1, ...wt−1, σ0

]= tσ2 + 1− νt

1− ν(σ2

0 − σ2)

+ σw

t∑τ=1

1− νt−τ

1− ν wτ . (37)

The last term in this equality characterizes the stochastic nature of the volatility of long-termgrowth. It is normally distributed. This means that the stochastic process (34)-(35) yieldsthe same stochastic structure of log consumption as the one described by equations (22)-(23), with maturity-specific parameters. The following lemma characterizes the nature of thestochastic variance of log-consumption at different maturities in this long-run risk context.

Lemma 4. Suppose that log-consumption is governed by process (34)-(35). Then, for anymaturity t ∈ N0, x0,t = log(ct/c0) conditional to σ0 is distributed as µt+√vtε, where vt andε are independent, ε is N(0, 1), and vt is normally distributed with annualized mean

E0 [vt]t

= σ2 + 1− νt

t(1− ν)(σ2

0 − σ2)

(38)

and annualized variance

V ar0 [vt]t

= σ2w

(1− ν)2

(1− 2 1− νt

t(1− ν) + 1− ν2t

t(1− ν2)

)(39)

Notice that E0vt measures the unconditional variance of log(ct/c0). If one divides thismeasure by tσ0, one obtains the "variance ratio" that has already been examined by Cochrane(1988), Beeler and Campbell (2012) and many others. Equation (38) tells us that this varianceratio has a flat term structure in expectation, i.e., when σ0 equals σ. This is consistent

9Bansal and Yaron (2004) consider a more general model in which the expected growth rate µ also followsan autoregressive process. In this paper, we focus on the impact of the uncertain variance of growth.

10This constraint on the expected growth of dividends is irrelevant for our analysis. It affects the price ofthe asset, but not its risk premium.

15

with the recent findings obtained by Marfè (2016) who used postwar U.S. output.11 Asexplained earlier, stochastic volatility does not increase the risk as measured by the varianceof log consumption. Similarly, it does not affect its skewness. Turning to equation (39),the annualized variance of vt goes from 0 for a one-period maturity up to σ2

w/(1 − ν)2 forvery large maturities. This upward sloping term structure is due to the combination of thefact that the variance is known one period in advance and of the persistence of shocks tovolatility. These two features of the stochastic process of growth magnify the long-term riskon variance. As noticed in Section 2, in this Gaussian framework, the fourth cumulant of x0,tis equal to three times the variance of vt. It implies that the annualized fourth cumulant oflog consumption has an increasing term structure.

As a benchmark, consider a Lucas tree economy with a representative agent that maxi-mizes the discounted expected utility of the flow of aggregate consumption:

W0 = E

[+∞∑t=0

exp(−δt)v(ct)]. (40)

We hereafter assume that W0 is bounded. In that context, the equilibrium price at date 0 ofan asset that delivers a single payoff cφt at date t is equal to

Pt(φ) = e−δtE0c

φt v′(ct)

v′(c0) . (41)

The interest rate rft is equal to t−1 log(Pt(0)) and the risk premium πt(φ) associated to anasset delivering cφt at date t is equal to −t−1 log(E0c

φt /Pt(φ)) minus the interest rate. They are

generally maturity-dependent. Using the same approach as in the previous section adaptedto the case with u(c) = v(c) = c1−γ/(1− γ) and with the maturity-varying parameters of thestochastic volatility process described in Lemma 4, we obtain the following proposition.

Proposition 7. Suppose that the growth process is governed by equations (34)-(35). Underthe DEU model with constant relative risk aversion γ, the term structures at date 0 of interestrates and the risk premia satisfy the following conditions:

rft = δ + γµ− 12γ

2E0 [vt]t− 1

8γ4V ar0 [vt]

t(42)

πt(φ) = φγE0 [vt]t

+ 12φγ

(γ2 − 3

2φγ + φ2)V ar0 [vt]

t, (43)

where E0[vt] and V ar0[vt] are given by Lemma 4.

The first two terms in the right-hand side of equation (42) correspond to the Ramsey rule(Ramsey (1928)). The third term measures the impact of the risk affecting economic growth

11In fact, Marfè (2016) obtained term structures of the variance ratio for output, salary and dividend thatare respectively flat, increasing and decreasing. He convincingly argues that this comes from the fact that firmsprovide short-term insurance to their employees against the transitory fluctuations of their labor productivity,in line with the theory of implicit labor contract.

16

on the interest rate in the absence of stochastic volatility. Its term structure is flat when thecurrent variance σ2

0 equals its historical mean σ2. This is because stochastic volatility does notincrease the long-run risk measured by the annualized unconditional variance, which is equalto σ2 at all maturities in that case. The last term measures the stochastic volatility premiumfor interest rates. Because the volatility of the growth rate is known one period in advancein the Bansal-Yaron model, this premium is zero for a one-period maturity. Because of thepersistence of the shocks to volatility, the annualized variance of the conditional varianceof log consumption defined as V ar0 (vt) /t is increasing with maturity, thereby magnifyingthe kurtosis of the distant log consumption. This makes the term structure of interest ratesdecreasing in expectation.

The first term in the right-hand side of equation (43) is the classical CCAPM risk pre-mium, which is the product of three elements: the CCAPM beta of the asset, the relative riskaversion of the representative agent, and the expected annualized variance of the growth rateuntil maturity t. This latter element has a flat term structure when the current variance σ2

0is equal to its historical mean σ2. An important feature of this model is that short-term riskpremia are time-varying, in parallel to changes in expectation about the short-term volatility.However, the long-term risk premium of any asset φ is fixed. The second term measures theimpact of stochastic volatility. It is always positive and proportional to the annualized vari-ance of the conditional variance of log(ct/c0). Because this element has an increasing termstructure, we can conclude that, on average, risk premia have an increasing term structuredue to the persistence of the shocks to volatility. This magnifies the kurtosis of the distri-bution of long term log consumption, together with the contribution of any risky asset tothis kurtosis. Because CRRA utility functions exhibit an aversion to kurtosis (v′′′′ < 0), thisexplains the result. Notice that the conclusion that the stochastic volatility component ofthe long-run risk model tends to make the term structure of risk premia increasing raises anew concern about the empirical test of this model. Indeed, it has recently been shown thatthe term structure of equity premia is decreasing (Binsbergen et al. (2012), Binsbergen andKoijen (2016), Giglio et al. (2015), and Giglio et al. (2016)).

Let us now quantify the impact of stochastic volatility on asset prices under the DEUmodel. Because volatility is known one period in advance, stochastic volatility has no impacton the short-term interest rate and risk premium. But it reduces the long-term interestrate and it raises the long-term risk premium. Bansal et al. (2016) assumed γ = 9.67,σw = 2.12× 10−6, and ν = 0.9984 on a monthly basis. This yields a term spread of interestrates of −2.30% on an annual basis. This number provides an upper bound of the impact ofstochastic volatility on interest rate for finite maturities. Bansal et al. (2016) also calibratedthe elasticity of the payoff of equity to aggregate consumption at φ = 3. This yields a termspread of annualized equity premia of 1.80%. It should be stressed that these term spreadsare very sensitive to relative risk aversion. This is particularly the case for the interest ratewhose term spread is proportional to γ4. In Table 1, we document this high sensitivity.

To sum up, the stochastic volatility of the growth rate raises the kurtosis of future logconsumption. Because the 4th and 5th derivatives of v are respectively negative and positive,this tends to reduce interest rates and to raise equity premia. The persistence of shocks to

17

rf+∞ − rf1 π+∞(1)− π1(1) π+∞(3)− π1(3)

γ = ρ = 1 −2.63× 10−4 5.27× 10−4 1.74× 10−2

γ = ρ = 10 −2.63 0.91 2.02γ = ρ = 20 −42.13 7.82 20.16

Table 1: Expected term spreads of interest rates, risk premia on aggregate consumption andequity premia under neutral expectations (σ0 = σ). As in Bansal et al. (2016), we assumeσw = 2.12× 10−6, and ν = 0.9984 on a monthly basis. Spreads are expressed in percents peryear.

volatility magnifies these effects for longer maturities, thereby making the term structures ofinterest rates and equity premia respectively decreasing and increasing on average.

6 Long-run stochastic volatility under Recursive Utility

In the Epstein-Zin-Weil model, welfare Vt is obtained by backward induction:

V 1−ρt = (1− β)C1−ρ

t + β(EtV

1−γt+1

) 1−ρ1−γ if ρ 6= 1 (44)

log Vt = (1− β) logCt + β log(EtV

1−γt+1

) 11−γ if ρ = 1. (45)

where parameters γ and ρ are the indices of relative aversion to risk and to consumptionfluctuations, respectively. Parameter β = exp(−δ) is a discount factor. The equilibriumprice at date t for an asset that generates a single payoff Dτ = cφτ at date τ > t satisfies thefollowing condition:

Pt = Et

[Dτ

SτSt

]. (46)

The one-period-ahead stochastic discount factor St+1/St to be used at date t to value a payoffoccurring at date t+ 1 is:

Sτ+1Sτ

= β

(Cτ+1Cτ

)−γZρ−γτ+1

(Eτ

(Cτ+1Cτ

Zτ+1

)1−γ) γ−ρ

1−γ

, (47)

where Zτ = Vτ/Cτ is the future expected utility per unit of current consumption. Proposition8 describes an approximation of the term structures of interest rates and risk premia withrecursive preferences and stochastic volatility.12 These results are obtained by assuming thatthe log(Zt) is linear in the state variable σ2

t , which is the case when the EIS is equal to one.We show in the appendix that the coefficient

b = d log(Zt)dσ2

t

∣∣∣∣σ2t=σ2

(48)

12As done in this literature, we will hereafter use equalities to refer to the approximations obtained whenusing this linearization.

18

satisfies the following condition:

b =(1

2(1− γ) + bν

)β exp

((1− ρ)

(µ+ 1

2(1− γ)σ2 + 12(1− γ)b2σ2

w

)). (49)

Notice that b is negative when γ is larger than unity, which implies that the intertemporalwelfare Z is decreasing in the current volatility of the growth rate of consumption, as alreadyobserved in the two-period model in Proposition 5.

Proposition 8. Suppose that the growth process is governed by equations (34)-(35). Underthe recursive utility model (44), the term structures at date 0 of interest rates and the riskpremia satisfy the following conditions:

rft = δ + ρµ− 12(γ2 − (γ − ρ)(γ − 1)

) E0 [vt]t− 1

8(γ2 − (γ − ρ)(γ − 1)

)2 V ar0 [vt]t

−12(γ − ρ)(1− ρ)b2σ2

w + 12γ − ρ1− ν

(γ2 − (γ − 1)(γ − ρ)

)b

(1− 1− νt

t(1− ν)

)σ2w (50)

πt(φ) = φγE0 [vt]t

+ 12φ(γ(γ − ρ+ γρ)− 1

2φ(2γ2 + γ − ρ+ γρ) + γφ2)V ar0 [vt]

t

+12σ

2w(γ − ρ)φbφ− 2γ

1− ν

(1− 1− νt

t(1− ν)

)(51)

where E0[vt] and V ar0[vt] are given by Lemma 4, and scalar b solves equation (49). Theseapproximations are exact when ρ is equal to unity.

Proof: See the appendix.

A direct consequence is that the short-term interest rate simplifies to

rf1 = δ + ρµ− 12(γ2 − (γ − ρ)(γ − 1)

)σ2

0 −12(γ − ρ)(1− ρ)b2σ2

w. (52)

Remember that in the Bansal-Yaron’s stochastic volatility process, the variance of growthis known for the upcoming period, so this result cannot be directly compared to equation(50). In particular, the last term in this equation disappears due to this early resolution ofuncertainty. However, it is replaced by the last term of equation (52), which is negative whenρ is smaller than unity and the representative agent has a Preference for an Early Resolutionof Uncertainty (PERU), i.e., when γ is larger than ρ. The representative agent will observe atdate 1 the volatility σ1 that will prevail in the next period. Seen from date 0, this additionaldate-1 risk raises precautionary savings when (σ − ρ)(1 − ρ) is positive. As an increase inrisk on date-1 consumption, it reduces the short-term interest rate.

The term structure of interest rates in this long-run risk model with recursive preferencescombines features already discussed in the DEU model and a new element coming fromPERU. Indeed, the first line in equation (50) is symmetric to the equation (42), with adaptedcoefficients for E0[vt] and V ar[vt] to account for the discrepancy between γ and ρ. The

19

β 0.999 discount factorγ 9.67 relative risk aversionρ−1 2.18 elasticity of intertemporal substitutionµ 0.0016 expected growth rate of consumptionσ0 = σ 0.007 current and expected volatilityσw 2.12× 10−6 standard deviation of shocks to volatilityν 0.9984 coefficient of persistence

Table 2: Calibration parameter based on monthly data extracted from Bansal et al. (2016).

second line in equation (50) is new compared to the DEU framework. We have seen abovethat it tends to reduce the short-term interest rate. It is easy to check that the last termin this equation has a decreasing structure under PERU and γ ≥ 1 (so that b is negative).This means that PERU cannot reverse the tendency of the term structure of interest ratesto be decreasing. Beeler and Campbell (2012) showed that when the persistence of shocks tothe growth rate of consumption is added to the model as in Bansal et al. (2012), then realinterest rates are negative for maturities exceeding 10 years.

The short-term risk premium π1(φ) = φγσ20 is not affected by stochastic volatility. This

is because the volatility for the first period is known at date 0. The middle term in the right-hand side of equation (51) is similar to the last term of equation (43) in the DEU model. Thepersistence of shocks to volatility magnifies the long-term kurtosis of log consumption, therebytending to make the term structure of risk premia increasing. The last term of equation (51)takes account of PERU in the recursive utility model. As long as φ is smaller than 2γ, thisnew term has an increasing term structure too. We can thus conclude that PERU cannotreverse the increasing nature of the term structure of risk premia already observed underDEU.

It may be interesting to quantify these effects. Consider again the calibration extractedfrom Bansal et al. (2016), as summarized in Table 2. Figure 3 describes the term structuresunder this calibration, respectively for the interest rates (rrt ), the risk premia on aggregateconsumption (πt(1)), and the equity premia (πt(3)). The benchmark case without stochasticvolatility is obtained by selecting σw = 0. It is represented by the dashed lines in this figure.As in the DEU framework, although shocks to volatility are small, their high persistencehas a strong impact on the pricing of long-dated assets. It reduces the long interest rate byaround 0.4%, and it raises the long equity premium by almost 1.5%. Notice that because thehalf-life of the shocks to the variance of log consumption is around 36 years, it takes manycenturies for these term structures to converge to these asymptotic values.

A comparative static analysis is summarized in Table 3. Long-term rates are much moreaffected by the unilateral change of a parameter than short-term rates. For example, consideran increase in the persistence parameter ν from its benchmark value of 0.9984 to 0.999.Because Bansal et al. (2016) estimated ν with a standard error of 0.0007, this change in ν

20

100 200 300 400 500t

1.2

1.3

1.4

1.5

1.6

1.7

rtf

100 200 300 400 500t

0.6

0.7

0.8

0.9

1.0

1.1

1.2

πt(1)

100 200 300 400 500t

1.8

2.0

2.2

2.4

2.6

2.8

3.0

πt(3)

Figure 3: The term structures of interest rates (top), risk premia on aggregate consumption (φ = 1, middle)and equity premia (φ = 3, bottom) with recursive utility under the stochastic process (34)-(35). The calibrationof the parameters is described in Table 2. The dashed curves are obtained by imposing σw = 0 (no stochasticvolatility). Rates are in percent per year and durations are in years.

21

rf1 rf+∞ π1(3) π+∞(3)Benchmark 1.61 1.08 1.71 3.17γ = 5 1.87 1.80 0.88 1.04ρ = 1 2.58 2.04 1.71 2.76σw = 4× 10−6 1.46 -0.32 1.71 6.57ν = 0.999 1.54 0.35 1.71 4.91Fixed volatility (σw = 0) 1.68 1.68 1.71 1.71

Table 3: Comparative static analysis based on the benchmark described in Table 2, withγ = 9.67, ρ = 1/2.18, σw = 2.12 × 10−6, and ν = 0.9984. The first column describes themodified parameter, everything else held unchanged. Rates are in percent per year.

cannot be excluded by their data.13 This unilateral change has no effect on the risk premiumon short-term equity, but it raises its term spread from 1.46% to 3.20%.

7 The term structure of the 4th cumulant of log consumptionThe long-run risk specification (34)-(35) of stochastic volatility generates an increasing termstructure of the uncertainty affecting the conditional variance of the growth rate. This doesnot change the annualized variance whose term structure remains flat. But it makes theterm structure of the annualized fourth cumulant of log consumption increasing. This is thedriving force behind the shapes of the term structures described in the previous two section.It is useful to test whether consumption data generates an increasing term structure of theannualized fourth cumulant of log consumption. As Bansal et al. (2016) and many others,our consumption data are extracted from the NIPA Tables 2.33, 2.34 et 7.1 of the Bureauof Economic Analysis. We use quarterly data of per-capita real consumption expenditureon nondurables and services from 1947Q1 to 2016Q4. In Figure 4, we used plain circles torepresent the term structure of the empirical annualized fourth moment of log consumption,which is computed as follows for each maturity t ∈ {1, ..., 40}:

Kx0,t4t

= 1t

(278− t)−1278−t∑τ=1

(xτ,τ+t − xt)4 − 3(

(278− t)−1278−t∑τ=1

(xτ,τ+t − xt)2)2 , (53)

where xt is the mean of the series(xτ,τ+t

)τ=1,...,278−t. The increasing term structure for

Kx0,t4 /t observed in the data suggests the persistence of shocks to the variance and is sup-

portive of decreasing interest rates and increasing risk premia. This is compatible with thelong-run risk model (34)-(35) with a positive coefficient of persistence. The quarterly cali-bration proposed by Bansal et al. (2016) is also described in Figure 4, with ν = 0.9978 and

13In fact, ν = 0.999 is the calibration used by Bansal et al. (2012). Notice that using the calibrationwith ν = 0.987 as in Bansal and Yaron (2004) would yield the following equilibrium prices: rf1 = 1.68%,rf+∞ = 1.67%, π1(3) = 1.71% and π+∞(3) = 1.73%. As noticed by Beeler and Campbell (2012), the stochasticnature of volatility has very little effect on asset prices in this initial calibration of the long-run risk model.

22

+ + + + + + ++

++

++

++

++

++

++

++

++

+

+

+

+

+

+

+

+

+

+

+

+

+

+

0 10 20 30 40t

2.×10-9

4.×10-9

6.×10-9

8.×10-9κ4x0,t /t

Figure 4: The term structure of the annualized 4th cumulant of log consumption κx0,t

4 /t, where the maturityt is measured in quarters. The bullets are obtained for maturities t ∈ {1, ..., 40} by measuring the empirical4th cumulant of the series

(xτ,τ+t

)τ=1,...,278−t

, using quarterly real U.S. consumption from 1947Q1 to 2016Q4.The squares correspond to the theoretical term structure of the 4th cumulant using equation (54) with thecalibration ν = 0.9978 and σw = 6.07× 10−6 proposed by Bansal et al. (2016). The crosses correspond to analternative calibration with ν = 0.9826 and σw = 2.96× 10−6.

σw = 6.07× 10−6. This is computed as follows:

κx0,t4t

= 3σ2w

(1− ν)2

(1− 2 1− νt

t(1− ν) + 1− ν2t

t(1− ν2)

). (54)

This figure shows that this calibration tends to generate too much kurtosis for maturitiesexceeding one year. This is why we also calibrated the term structure of the annualized 4thcumulant with parameters ν = 0.9826 and σw = 2.96× 10−6 that better fit the data. In fact,these parameter values minimize the sum of the square of the differences between Kx0,t

4 /t andκx0,t4 /t for maturities between 1 and 40 quarters. This calibration exhibits a smaller degree

of persistence of shocks to the variance, and a smaller standard deviation for these shocks.

8 Concluding remarks

With power utility functions, risk aversion, prudence, temperance and higher degree riskaversions are all summarized by one parameter, i.e., the degree of relative risk aversion.The classical asset pricing theory, which heavily relies on this specification of the utilityfunction, has therefore developed arguments on the sole concept of risk aversion. We findthat problematic for the development of this theory. For example, the negative impact ofrisk on the interest rate is exclusively generated by the notion of prudence (v′′′ ≥ 0), whichis orthogonal to the concept of risk aversion. In the same fashion, we show in this paperthat stochastic volatility also reduces the interest rate, but this is exclusively generated bythe negativity of the fifth derivative of v, which is independent – at least in the small –

23

of risk aversion, prudence and temperance (v(4) ≤ 0). In other words, departing from thepower utility function would in theory allows us to disentangle these psychological traits ofour preferences under risk. This agenda of research is in line with recent findings that areincompatible with constant relative risk aversion (see for example Ogaki and Zhang (2001),Guiso and Paiella (2008), and Deck and Schlesinger (2014)). This is also in line with thetrend of behavioral finance in which the utility function is distorted by additive formation,or a reference point for example.

One of the important innovations of this paper is to show that the stochastic volatilitypremia are proportional to the third or fourth order risk aversions, depending upon whetherone examines its impact on risk premia or on interest rates. In the special case of powerutilities, this simplifies to them being proportional to the third or fourth power of the coeffi-cient of relative risk aversion. This implies that the impact of stochastic volatility is highlysensitive to the choice of this parameter. This magnifies the challenge of estimating riskaversion correctly.

Finally, we have shown that stochastic volatility generates a flat term structure of thevariance ratio in expectation. This is in striking contrast with the result that the termstructure of risk premia is decreasing. The explanation of this seemingly inconsistent resultscomes from the fact that the aggregate risk cannot be measured only by annualized varianceof future log consumption alone. Outside the Gaussian world, the annualized kurtosis matterstoo, and the persistence of shocks to the variance of log consumption makes its term structureincreasing. This explains the increasing nature of the term structure of risk premia. Thekurtosis ratio should be used in parallel of the variance ratio to interpret the term structures.

24

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Gollier, C. (2008): “Discounting with fat-tailed economic growth,” Journal of Risk andUncertainty, 37, 171–186.

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26

Appendix

Appendix 1: Proof of Lemma 1

We have to prove that function h is concave, with

h(σ2) = E [f(x+ ση)|σ] . (55)

It is easy to verify that

h′′(σ2) = σ4

4 E[(σ2η2f ′′(x+ ση)− σηf ′(x+ ση)

)∣∣∣σ] .For each σ, define random variable yσ in such a way that yσ/σ be distributed as η. Thisimplies that Eyσ = 0 and Ey3

σ = 0. Using this change of variables, the above equation canbe rewritten as follows:

h′′(σ2) = σ4

4 E[y2σf′′(x+ yσ)− yσf ′(x+ yσ)

].

The concavity of h requires the right-hand side of this equality to be non-positive for allrandom variable yσ such that Eyσ = Ey3

σ = 0. This condition is summarized as follows:

Ey = 0 and Ey3 = 0 ⇒ E[y2f ′′(x+ y)− yf ′(x+ y)

]≤ 0. (56)

We will use the following lemma, which is a direct consequence of Theorem 3 in Gollier andKimball (1996).

Lemma 5. Consider three functions (f1, f2, f3) from R to R such that f1(0) = f2(0) =f3(0) = 0. The following two conditions are equivalent:

• For any random variables y such that Ef1(y) = 0 and Ef2(y) = 0, we have thatEf3(y) ≤ 0.

• There exists a pair (λ1, λ2) ∈ R2 such that f3(y) ≤ λ1f1(y) + λ2f2(y) for all y ∈ R.

Applying this lemma to condition (56) makes it equivalent to requiring that there existsa pair (λ1, λ2) ∈ R2 such that

H(y) = y2f ′′(x+ y)− yf ′(x+ y)− λ1y − λ2y3 ≤ 0 (57)

for all y. Notice that H(0) = 0 and H ′(0) = −f ′(x)− φ1. This implies that λ1 = −f ′(0) is anecessary condition for H to be non-positive. We also have that

H ′′(y) = 3yf ′′′(x+ y) + y2f ′′′′(x+ y)− 6λ2y,

which implies that H ′′(0) also vanishes. We also get

H ′′′(y) = 3f ′′′(x+ y) + 5yf ′′′′(x+ y) + y2f (5)(x+ y)− 6λ2.

27

This implies that H ′′′(0) = 3f ′′′(x)− 6λ2. A necessary condition for H to be non-positive isthus that λ2 = 0.5f ′′′(x). Because H ′′′′(0) = 8f ′′′′(x), a necessary condition for an increasein risk in the variance v of x to reduces Ef(x) is that f ′′′′ be non-positive, or that f ′′ beconcave.

We now show that this condition is also sufficient. Using the conditions on λ1 and λ2, wecan rewrite H in such a way that for all y, H(y) = yK(y) with

K(y) = yf ′′(x+ y)− f ′(x+ y) + f ′(0)− 0.5y2f ′′′(0). (58)

We would be done if we could show that H is uniformly non-positive. This requires K(y) tohave a sign opposite to y. Because K(0) = 0 a sufficient condition for this to be true is thatfunction K be decreasing. Observe now that

K ′(y) = y[f ′′′(y)− f ′′′(0)

]is negative when f ′′′ is decreasing, i.e., if f ′′ is concave. This demonstrates that this conditionis necessary and sufficient for an increase in risk in variance to reduce Ef . �

Appendix 2: Proof of Lemma 2

Function πf (k) is implicitly defined as follows:

Ef(x+ σkε− πf ) = Ef(x+ σkε), (59)

Observe that πf (0) = 0 and

E[(σε− π′f (k))f ′(x+ σkε− πf )

]= E

[σ2εf ′(x+ σkε)

].

This equality estimated at k = 0 implies that π′f (0) = 0, since Eε = 0. Fully differentiatingthis equality implies that

E[(σε− π′f (k))2f ′′(x+ σkε− πf )

]− π′′f (k)E

[f ′(x+ σkε− πf )

]= E

[σ2ε2f ′′(x+ σkε)

].

Estimating this equality at k = 0 yields

σ2σ2ε f′′(x)− π′′f (0)f ′(x) = σ2σ2

ε f′′(x),

where σ2ε is the variance of ε. This implies that π′′f (0) = 0. Fully differentiating again yields

E[(σε− π′f (k))3f ′′′(x+ σkε− πf )

]− 2π′′f (k)E

[f ′′(x+ σkε− πf )

]−π′′′f (k)E

[f ′(x+ σkε− πf )

]− π′′f (k)E

[(σε− π′f (k))f ′′(x+ σkε− πf )

]= E

[σ3ε3f ′′(x+ σkε)

].

Because the the third moment of ε is zero, the above equation implies that π′′′f (0) = 0. Finally,fully differentiating one more time and estimating the equality at k = 0 yields

28

σ4E[ε4]f ′′′′(x)− π′′′′f (0)f ′(x) = E

[σ4]E[ε4]f ′′′′(x).

This implies that

π′′′′f (0) = −f′′′′(x)f ′(x) V ar(σ

2)Eε4. (60)

Using a fourth-order Taylor approximation of πf (k) around k = 0 yields equation (7). �For further exploration, it is useful to explore the properties of the following function:

Hf (k) = log (Ef(x+ σkε))− log (Ef(x+ σkε)) (61)= log (Ef(x+ σkε− πf (k)))− log (Ef(x+ σkε))

Function Hf measures the relative increase in the expectation of f which is due to thestochastic nature of the variance of x. Using the fact that the first three derivatives offunction πf evaluated at k = 0 are zero together with the assumption that Eε = Eε3 = 0, itis straightforward to show that

Hf (k) = − 14!π′′′′f (0)f

′(x)f(x) +O(k5).

Using equation (60), this implies that

Hf (k) = k4

4!f ′′′′(x)f(x) V ar(σ2)Eε4 +O(k5). (62)

Appendix 3: Proof of Lemma 3

Suppose first that σ2 is N(σ2, σ2w). This proposition is based on the property that when

x ∼ N(µ, σ2), then E exp(kx) = exp(kµ+ 0.5k2σ2). Using this property twice yields

E [exp(−A(x+ ση)] = E[exp(−Ax+ 0.5A2σ2σ2

η)]

= exp(−Ax+ 0.5A2σ2σ2η + 0.125A4σ2

wσ4η).

Similarly, we have that

E [exp(−A(x+ ση − πf )] = exp(−Ax+ 0.5A2σ2σ2η +Aπ). (63)

Combining these two results implies equation (9).

Suppose alternatively that σ2 has gamma distribution Γ(a, b), so that its density functionis

f(v; a, b) = va−1 e−v/b

baΓ(a) , for all v > 0,

29

where a and b are two positive constants. This implies that the mean variance is σ2 = ab,and its variance is σ2

w = ab2. It implies that

E [exp(−A(x+ ση)] = E[exp(−Ax+ 0.5A2σ2σ2

η)]

=∫ +∞

0exp(−Ax+ 0.5A2vσ2

η)f(v; a, b)dv

= exp(−Ax)baΓ(a)

∫ +∞

0va−1 exp

(v(0.5A2σ2

η − b−1))dv

= exp (−Ax)(

1− 12A

2σ2ηb

)−a. (64)

The last equality is a consequence of the observation that14∫ +∞

0va−1e−v/kdv = kaΓ(a).

Combining equations (6), (63) and (64) yields

exp(0.5A2σ2σ2η +Aπf ) =

(1− 1

2A2σ2ηb

)−a.

Using the fact that a = σ4/σ2w and b = σ2

w/σ2, this equation is equivalent to (10). �

Appendix 4: Proof of Proposition 3

We limit our proof to the demonstration of the necessary and sufficient condition (20) in thesmall. By equation (62), we have that

log(Ev′(c1)

)= log

(Ev′(c1 + ση)

)+ k4

4!v(5)(c1)v′(c1) V ar(σ2)Eε4 +O(k5). (65)

Define function φ such that φ(e) = u′(e)/v′(e) for all e, and define e?1(k) as the certaintyequivalent of c1 + σkη. From equation (7), we have that

log(u′(e1)v′(e1)

)= log (φ(e1))

= log (φ(e?1)) + φ′(c1)φ(c1)

k

4!v(4)(c1)v′(c1) V ar(σ2)Eε4 +O(k5)

= log (φ(e?1)) + v′(c1)u′(c1)

u′′(c1)v′(c1)− u′(c1)v′′(c1)(v′(c1))2

k

4!v(4)(c1)v′(c1) V ar(σ2)Eε4 +O(k5)

= log (φ(e?1)) +(u′′(c1)u′(c1) −

v′′(c1)v′(c1)

)k

4!v(4)(c1)v′(c1) V ar(σ2)Eε4 +O(k5). (66)

From equation (41) and the above two equations, we see that the increase in log price due tostochastic volatility is equal to

k4

4!v(5)(c1)v′(c1) V ar(σ2)Eε4 +

(u′′(c1)u′(c1) −

v′′(c1)v′(c1)

)k

4!v(4)(c1)v′(c1) V ar(σ2)Eε4 +O(k5).

14This is a direct consequence that the integral of the density f must be equal to 1.

30

If k is small enough, this is positive if and only if equation (20) is satisfied.

Appendix 5: Proof of Proposition 4

We have to prove that the following inequality holds for small k if and only if u′′′′ is negative:

log(E(c1 + σkε)Eu′(c1 + σkε)E(c1 + σkε)u′(c1 + σkε)

)≥ log

(E(c1 + σkε)Eu′(c1 + σkε)E(c1 + σkε)u′(c1 + σkε)

)(67)

Notice thatE(c1 + σkε) = E(c1 + σkε)

because the expectation of ε is zero. This implies that we can rewrite the above inequalityas follows:

log(Eu′(c1 + σkε)

)− log

(Eu′(c1 + σkε)

)≥ log

(E(c1 + σkε)u′(c1 + σkε)

)− log

(E(c1 + σkε)u′(c1 + σkε)

). (68)

Let Hf (k) be defined by equation (61), and let function f1 and f2 be respectively defined byf1(c) = u′(c) and f2(c) = cu′(c). This implies that the above inequality can be rewritten asfollows:

Hf1(k) ≥ Hf2(k). (69)

Using equation (62) twice, this inequality holds for k small if and only if

f(4)1 (c1)f1(c1) ≥

f(4)2 (c1)f2(c1) . (70)

This can be rewritten asu(5)(c1)u′(c1) ≥

4u(4)(c1) + c1u(5)(c1)

c1u′(c1) .

This is true if and only if u(4) is negative. �

Appendix 6: Proof of Proposition 8

Let us define the growth rate of consumption as xt,τ = log cτ/ct and the annualized log returnis rt,τ = (τ − t)−1 log(cφτ /Pt), τ > t. Let us also define the log SDF st = logSt, and the logfuture normalized utility zt = logZt. This allows us to rewrite the pricing equations (46) and(47) as follows:

0 = χ0 (tr0,t + st − s0) , (71)

sτ+1 − sτ = −δ − γxτ,τ+1 + (ρ− γ)zτ+1 + γ − ρ1− γ χτ ((1− γ)(xτ,τ+1 + zτ+1)) , (72)

where operator χt is defined as

χτ (x) = log (Eτ exp(x)) . (73)

31

One can obtain zτ by backward induction from equations (44)-(45) which are rewritten asfollows:

zτ =

(1− ρ)−1 log(1− β + β exp

(1−ρ1−γχτ ((1− γ)(xτ,τ+1 + zτ+1))

))if ρ 6= 1

β1−γχτ ((1− γ)(xτ,τ+1 + zτ+1)) if ρ = 1,

(74)

Suppose that the growth process is governed by equations (34)-(35). In case ρ = 1, it iseasy to check that the guess solution

zt = at + bσ2t (75)

satisfies equation (74) with

b = β(1− γ)2(1− βν) . (76)

When ρ 6 1, the linearization (75) is a first-order approximation of the exact link between ztand the state variable σ2

t around the steady state with σ2t = σ2 if b satisfies the following

condition:

b =(1

2(1− γ) + bν

)β exp

((1− ρ)

(µ+ 1

2(1− γ)σ2 + 12(1− γ)b2σ2

w

)). (77)

Using this linearization, equation (72) can be rewritten as

sτ+1 − sτ = −δ − ρµ+ 12(γ − ρ)(1− γ)

(σ2τ + b2σ2

w

)− γστητ+1 − (γ − ρ)bσwwτ+1. (78)

By definition of r0,t, we have that

tr0,t = tE0r0,t + φ(x0,t − E0x0,t) = tE0r0,t + φt−1∑τ=0

στητ+1. (79)

Combining this with equation (78), one can rewrite the pricing equation (71) as follows:

0 = χ0 (Q) , (80)

with

Q = tE0r0,t + 12(φ− γ)2

t−1∑τ=0

σ2τ − δt− ρµt

+ 12(γ − ρ)(1− γ)

(t−1∑τ=0

σ2τ + b2σ2

wt

)− (γ − ρ)bσw

t−1∑τ=0

wτ+1. (81)

This is rewritten as

Q = tE0r0,t − δt− ρµt+ 12(γ − ρ)(1− γ)b2σ2

wt− (γ − ρ)bσwt−1∑τ=0

wτ+1

+ 12((φ− γ)2 + (γ − ρ)(1− γ)

) t−1∑τ=0

σ2τ . (82)

32

Using equation (37), this can be rewritten as follows:

Q = tE0r0,t − δt− ρµt+ 12(γ − ρ)(1− γ)b2σ2

wt+ 12((φ− γ)2 + (γ − ρ)(1− γ)

)E0vt

+ σw

t∑τ=1

(12((φ− γ)2 + (γ − ρ)(1− γ)

) 1− νt−τ

1− ν − (γ − ρ)b)wτ . (83)

Because Q is normally distributed, the pricing equation (80) implies that

0 = tE0r0,t − δt− ρµt+ 12(γ − ρ)(1− γ)b2σ2

wt+ 12((φ− γ)2 + (γ − ρ)(1− γ)

)E0vt

+ 12σ

2w

t∑τ=1

(12((φ− γ)2 + (γ − ρ)(1− γ)

) 1− νt−τ

1− ν − (γ − ρ)b)2

. (84)

The return of the strategy in which one purchases at price P0 at date 0 an asset that deliverspayoff cφt at date t is R0,t = cφt /P0. The expected return expressed as an annualized rate is

t−1 log (E0R0,t) = E[r0,t] + 12φ

2E0vtt

+ 18φ

4V ar[vt]t

. (85)

Combining the above two equations implies that

t−1 log (E0R0,t) = δ + ρµ− 12((φ− γ)2 − (γ − ρ)(γ − 1)− φ2

) E0vtt

− 18

(((φ− γ)2 − (γ − ρ)(γ − 1)

)2− φ4

)V ar[vt]

t− 1

2(γ − ρ)(1− ρ)b2σ2w

+ 12σ2wb(γ − ρ)

1− ν((φ− γ)2 − (γ − ρ)(γ − 1)

)(1− 1− νt

t(1− ν)

). (86)

The interest rate is obtained by replacing φ by zero, thereby generating equation 50). Therisk premium is obtained by subtracting the interest rate from the expected return describedabove, which yields equation (51). This concludes the proof. �

33