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

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1 Sochasic volailiy implies fourh-degree risk dominance: Applicaions o asse pricing Chrisian Gollier Toulouse School of Economics, Universiy of Toulouse-Capiole March 17, 2017 Absrac We demonsrae ha increasing he risk surrounding he variance of fuure consumpion generaes a fourh-degree risk deerioraion in fuure consumpion, yielding an increase in is excess kurosis. Is impac on he equilibrium risk premium is hus posiive if only if he fourh derivaive of he uiliy funcion is negaive. Is impac on ineres raes is negaive only if is fifh derivaive is posiive. We also show ha he persisence of shocks o he variance of he consumpion growh rae, as assumed in long-run risk models, has no effec on he erm srucure of he variance raio which remains fla in expecaion, bu i makes he erm srucure of he annualized fourh cumulan of log consumpion increasing. I generaes erm srucures of ineres raes and risk premia ha are respecively decreasing and increasing under consan relaive risk aversion. Using recursive preferences does no qualiaively modify hese resuls, which are counerfacual. However, he persisence of shocks o he variance of changes in log consumpion is suppored by he observaion ha heir annualized 4h cumulan exhibis an increasing erm srucure over he period 1947Q1-2016Q4 in he Unied Saes. Keywords: Long-run risks, fourh-degree risk dominance, emperance, edginess, recursive uiliy, kurosis. JEL codes: D81 Acknowledgemen: I hank Jules Tinang Nzesseu for providing me wih he consumpion daa used in Secion 6. I acknowledge funding from he chair SCOR and FDIR a TSE. This paper has benefied from discussions wih Louis Eeckhoud, Chrisophe Heinzel, Nour Meddahi, and Nicolas Treich. The research leading o his resul has received he suppor from he European Research Council under he European Communiy s Sevenh Framework Programme (FP7/ ) Gran Agreemen no SolSys. Commens welcomed. 1

2 1 Inroducion This paper provides a heoreical analysis of he effec of sochasic volailiy on ineremporal welfare and asse prices. Since he seminal work by Bansal and Yaron (2004), i has been demonsraed ha inroducing sochasic volailiy in he process governing aggregae consumpion can conribue o he resoluion of he classical puzzles in finance when combined wih oher ingrediens. Our main objecive in his paper is more heoreical. Raher han showing ha a realisic calibraion of he parameers of he model can explain he observed asse prices if he model is rich enough, we focus on a single ingredien, i.e., sochasic volailiy, and we examine is consequence on asse pricing from a heoreical poin of view. Technically, he sochasic volailiy conained in he Bansal-Yaron process produces an increase in risk in he variance of log consumpion. I is ofen suggesed ha sochasic volailiy adds a new layer of risk, which implies ha risk-averse consumers should dislike i. This common wisdom is no rue. To show his, consider wo loeries. Wih loery L 1, one loses or wins one moneary uni wih equal probabiliies. Wih loery L 2, one loses or wins wo moneary unis wih probabiliy 1/8. Wha is he preferred loery? Observe ha he firs hree momens of hese wo loeries are idenical, wih a zero firs and hird momens, and a uni second momen. This implies ha all expeced-uiliy-maximizers wih a hird-degree polynomial uiliy funcion will be indifferen beween he wo loeries, independen of heir degree of risk aversion and prudence. In paricular, i is no rue ha L 2 is riskier han L 1 in he classical sense defined by Rohschild and Sigliz (1970). 1 How is his observaion relaed o sochasic volailiy? In Figure 1, we represened L 2 as wo alernaive compound loeries, L 2 and L 2. Loery L 2 compounds a sure payoff of x 0 = 0 wih probabiliy 3/4 and a zero-mean loery x 1 ( 2, 1/2; 2, 1/2) wih probabiliy 1/4. Because he variance of x 1 equals 4, L 1 differs from L 2 by he fac ha he sure variance of 1 is replaced by an uncerain variance ha is disribued as (0, 3/4; 4, 1/4). This is a mean-preserving spread in variance. This is also he case for he compound loery L 2, where he condiional variance is disribued as (0, 1/2; 2, 1/2). In oher words, deermining he preference order beween loeries L 1 and L 2 is equivalen o deermining he welfare impac of an uncerain variance. The cornersone of our analysis is our Theorem 1 which saes ha any mean-preserving spread in he disribuion of he variance of a random variable x generaes a fourh-degree risk increase of x in he sense of Ekern (1980). In oher words, i reduces (increases) Ef(x) if and only if he fourh derivaive of f is negaive (posiive). A necessary (bu no sufficien) condiion is ha he kurosis of x is increased. Thus, we conclude ha, under expeced uiliy, increasing he risk on he variance of final consumpion is disliked if and only if he fourh derivaive of he uiliy funcion is negaive. Under his condiion, loery L 1 is preferred o loery L 2. The negaiveness of he fourh derivaive of he uiliy funcion is called "emperance" 1 In fac, L 2 is obained from L 1 hrough a sequence of wo Mean-Preserving Spreads (MPS) and one Mean-Preserving Conracion (MPC). In he firs MPS, he oucome 1 of L 1 is replaced by (2, 3/4; 2, 1/4). In he second MPS, he oucome -1 of L 1 is replaced by (2, 1/4; 2, 3/4). Finally, he MPC akes he form of displacing probabiliy masses of 3/8 from respecively 2 and 2 o 0. 2

3 Figure 1: Two ways of represening loery L 2 ( 2, 1/8; 0, 3/4; 2, 1/8) as a compound loery wih sochasic variance. in expeced uiliy heory. Eeckhoud and Schlesinger (2006) have shown ha emperance is necessary and sufficien for consumers o prefer compound loery (x 1, 1/2; x 2, 1/2) over compound loery (0, 1/2; x 1 x 2, 1/2), for any pair (x 1, x 2 ) of zero-mean independen loeries. This is anoher form of aversion o a sochasic variance. In fac, he preference of L 1 over L 2 can be represened in his way, wih L 1 x 1 x 2. Gollier and Pra (1996) showed ha emperance is necessary for any zero-mean background risk o raise he aversion o any oher independen risk. Under Consan Relaive Risk Aversion (CRRA), emperance goes ogeher wih risk aversion since he successive derivaives of he uiliy funcion alernae in sign. Under Discouned Expeced Uiliy (DEU), he ineres rae is decreasing in he risk surrounding fuure consumpion if and only if i raises he expeced marginal uiliy of fuure consumpion. Since Leland (1968), Drèze and Modigliani (1972) and Kimball (1990), i is well-known ha, by Jensen s inequaliy, his is he case if and only if he represenaive agen is pruden, i.e., if he hird derivaive of he uiliy funcion is posiive. Because i increases he fourh-degree risk (kurosis), he sochasic naure of he variance of consumpion also reduces he ineres rae under DEU if and only if he fifh derivaive of he uiliy funcion is posiive. In a wo-period model wih recursive preferences à la Kreps and Poreus (1978) and Selden (1978), we characerize a necessary and sufficien condiion in he small ha combines he fourh and fifh derivaives of he risk uiliy funcion wih he difference beween risk aversion and flucuaion aversion. The long-run risks lieraure focused on he predicion of he shor-erm ineres rae and of he price of equiy. Up o our knowledge, here has been no analysis of he erm srucure of zero-coupon bond and equiy reurns, wih he excepion of Beeler and Campbell (2012) who examined ineres raes. Bansal and Yaron (2004) considered a sochasic process of log consumpion in which he variance iself is governed by an independen sochasic process. 3

4 This is useful o explain why he equiy premium on financial markes is ime varying. Because i raises he kurosis of fuure log consumpion, i reduces he ineres rae and i raises he equiy premium, which is helpful o solve he financial puzzles. Moreover, he shocks on variance are highly persisen in his lieraure, wih an half-life around 35 years in he mos recen calibraion of he model by Bansal e al. (2016). This persisence implies ha he erm srucure of he annualized kurosis of log consumpion is increasing. This magnifies he effec of sochasic volailiy a higher mauriies. This means ha, under DEU, he erm srucures of ineres raes and risk premia are respecively decreasing and increasing. These heoreical predicions of his model are conradiced by asse prices observed on financial markes. In paricular, recen findings documen he fac ha dividend srip risk premia have a decreasing erm srucure (Binsbergen e al. (2012), Binsbergen and Koijen (2016), Belo e al. (2015), and Marfè (2016)). 2 The radiional combinaion of sochasic volailiy wih persisen shocks o he expeced growh rae will make he problem even more puzzling, since i will generae an increasing erm srucure of he annualized variance of log consumpion. 3 Because of prudence, his will make he erm srucure of ineres raes more decreasing (Campbell (1986), Gollier (2008)). Because of risk aversion, i makes he erm srucure of risk premia more increasing. And he radiional combinaion of sochasic volailiy wih recursive preferences does no solve he problem eiher. This is shown in his paper by deriving analyically he erm srucures from he sochasic volailiy model exraced from Bansal and Yaron (2004). Because of he high persisence of he shocks, i akes many cenuries of duraion for he ineres raes and risk premia o converge o heir asympoic value. For example, a dividend srip generaed by a diversified porfolio of equiy has a risk premium 1.71% for a one-monh mauriy, and i goes up o 3.17% when he mauriy ends o infiniy. However, his equiy premium is only 1.86% for a 10-year mauriy, and 2.28% for a 50-year mauriy. The paper is organized as follows. In Secion 2, we provide some generic resuls linking sochasic volailiy, fourh-degree sochasic dominance and kurosis. We apply hese findings in a wo-period Kreps-Poreus preferences in Secion 3. In he nex wo secions, we explore he erm srucures of asse prices under he sandard long-run risk specificaion yielding persisen shocks o he variance of log consumpion. We do ha in he DEU framework in Secion 4, and in he case of Epsein-Zin-Weil preferences in Secion 5. In Secion 6, we es wheher he erm srucure of he 4h cumulan of changes in log consumpion is upward-sloping due o he persisence of shocks o he variance. We provide some concluding remarks in he las secion. 2 These findings are for mauriies up o 10 years. For longer mauriies, Giglio e al. (2015) and Giglio e al. (2016) provide evidence for real esae asses (leasehold conracs) wih mauriies measured in decades and cenuries. 3 Beeler and Campbell (2012) show evidence of mean-reversion raher han persisence in U.S. consumpion growh in he period since On he conrary, mean-reversion makes he aggregae risk in he longer run relaively smaller and can hus explain why ineres raes and risk premia are respecively increasing and decreasing in mauriy, conrary o wha he persisence in volailiy shocks implies. 4

5 2 Sochasic volailiy and sochasic dominance This secion is devoed o he analysis of he impac of he sochasic variance of a random variable x on Ef(x), where f is a four ime differeniable real-valued funcion. The risk srucure of x is described by he following model: x = x ση (1) σ 2 = σ 2 w, (2) where (η, w) is a pair of independen random variables wih a zero mean so ha x and σ 2 are he mean of respecively x and σ 2. 4 If we assume ha Eη 2 = 1, hen σ 2 measures he variance of x condiional o σ. Finally, we also assume ha Eη 3 = 0. This assumpion guaranees ha he uncondiional hird momen of x is zero, independenly of he disribuion of σ. Consider any real-valued funcion f ha is a leas wice differeniable. We wan o characerize he impac of he sochasic variance of x on Ef(x). By he law of ieraed expecaions, we have ha [ ] Ef(x) = E E[f(x) σ 2 ] = Eh(σ 2 ), (3) where funcion h is derived from funcion f in such a way ha h(σ 2 ) equals E[f(x) σ 2 ] for all σ. An increase in risk of variance is defined as a sequence of Rohschild-Sigliz Mean-Preserving Spreads (MPS) in he disribuion of he variance σ 2 of x. From Rohschild and Sigliz (1970), i is also defined as a change in he disribuion of σ 2 ha reduces he expecaion of any concave funcion of σ 2. This implies ha, from equaion (3), an increase in risk on variance reduces Ef if and only if h is concave in σ 2. The following heorem saes he necessary and sufficien condiion for his o be rue independen of he disribuion of he zero-mean random variable η. Theorem 1. Suppose ha x = xση, where x is a consan and σ and η are wo independen random variables wih Eη = Eη 3 = 0. Any increase in risk in he variance σ 2 of x reduces (raises) Ef(x) if and only if f is concave (convex). Proof: See he appendix. This resul is relaed o he heory of sochasic dominance orders. Following Ekern (1980), a random variable undergoes a n-h degree risk deerioraion if and only if his change in disribuion reduces he expecaion of any funcion g G n of ha random variable, where G n is he se of all real-valued funcions g such ha ( 1) n g (n) 0, where g (n) denoes he n-h derivaive of g. For example, he case n = 1 corresponds o he concep of firs-order sochasic dominance, whereas he case n = 2 corresponds o he Rohschild-Sigliz s noion of an increase in risk. Theorem 1 means ha raising he uncerainy affecing he variance of 4 Noice ha σ is no he mean of σ in his model. By Jensen s inequaliy, he uncerainy affecing σ 2 has a negaive impac on he expeced value of σ. 5

6 x in he sense of Rohschild and Sigliz (1970) deerioraes random variable x in he sense of fourh-degree risk. Because f(x) = (x x) i has a linear second derivaive for i = 1, 2 and 3, an immediae consequence of Theorem 1 is ha he mean, he variance and he skewness of x is unaffeced by an increase in risk in σ 2. In he same vein, he fourh cenered momen of x is increased by i. More specifically, we have ha he excess kurosis of x equals Kur[x] = E(x x)4 (E(x x) 2 ) 2 3 = Eσ4 Eη 4 (Eσ 2 ) 2 (Eη 2 3. (4) ) 2 Because funcion σ 4 is convex in σ 2, he kurosis of x is increased by any increase in risk in σ 2. If we assume ha η is Normal, hen he above equaliy simplifies o ( ) Eσ 4 Kur[x] = 3 (Eσ 2 ) 2 1 = 3 V ar[σ2 ] (Eσ 2 ) 2 = 3σ2 w σ 4. (5) The excess kurosis of x is proporional o he variance of he condiional variance of x in ha case. Thus, he variance of σ 2 should be inerpreed as a measure of he excess kurosis of x. An exreme illusraion of his phenomenon has been proposed by Weizman (2007). Suppose ha η is N(0, 1) and ha he precision p = σ 2 has a Gamma disribuion. Then, as is well-known, he uncondiional disribuion of x is a Suden-, which has faer ails han he Normal disribuion wih he same expeced variance. The momen-generaing funcion of he Suden- is undefined, which means ha he expecaion of exp(kx) is unbounded, for all k R. This is an exreme illusraion of Theorem 1 in which moving from a sure σ 2 o a risky one wih he same mean makes he expecaion of f undefined. 5 I is useful o measure a "sochasic volailiy premium" associaed o funcion f which is defined as he sure reducion π f in x ha has he same impac on Ef as he uncerainy affecing σ 2. Technically, π f saisfies he following condiion: Ef(x ση π f ) = Ef(x ση), (6) where σ 2 is he expecaion of he uncerain variance σ 2. Using Taylor expansions for boh sides of he above equaliy, i is easy o show ha he sochasic volailiy premium saisfies he following propery when η = kɛ wih k R: π f = 1 4! ψ f (x)v ar[σ 2 ]Eη 4 O(k 5 ), (7) where ψ f (x) = f (4) (x)/f (x) is an index of concaviy of f. When η has a sandard Normal disribuion, he above approximaion simplifies o π f 0.125ψ f (x)σ 2 w. (8) In he remainder of his secion, we suppose ha funcion f is exponenial wih f(x) = exp( Ax) for some A R, in which case ψ f = A 3. In ha case, we have ha Ef(x) = χ( A, x ση), (9) 5 Gollier (2016) examined risk profiles for σ ha generae a bounded soluion. 6

7 where χ(α, y) = log (E exp (αy)) is he Cumulan-Generaing Funcion (CGF) of random variable y. 6 If η is sandard normal, his can be rewrien as follows: Using he properies of he CGF funcion, his implies ha Ef(x) = Ax χ(0.5a 2, σ 2 ). (10) Ef(x) f(x) = n=1 A 2n 2 n n! κσ2 n, (11) where κ σ2 n is he nh cumulan of σ 2. 7 The firs erm in he righ-hand side of his equaliy is 0.5A 2 σ 2, which corresponds o he sandard Arrow-Pra risk premium 0.5Aσ 2. The second erm is Aπ f = 0.125A 4 V ar[σ 2 ], which corresponds o he sochasic volailiy premium approximaed in equaion (8). Two special cases are useful o examine when f is exponenial. In line wih he lieraure on long-run risks pioneered by Bansal and Yaron (2004), suppose firs ha σ 2 has a Normal disribuion. This implies ha all cumulans of σ 2 of order larger han 2, as hey appear in equaion (11), are zero. This implies ha approximaion (8) is exac in ha case. Alhough his specificaion of he model is ubiquious in he long-run risks lieraure, i is problemaic because of he posiive probabiliy of a negaive variance. 8 An alernaive specificaion which has a more saisfacory heoreical foundaion is obained when assuming ha σ 2 has a Gamma disribuion. We show in he appendix ha his implies he following analyical characerizaion of he sochasic volailiy premium: ( π f = σ4 Aσw 2 log 1 1 ) 2 A2 σ2 w σ Aσ2. (12) Le us apply hese resuls o he case of an agen who has a consan relaive risk aversion γ and whose log consumpion nex year condiional o σ is x N(µ, σ 2 ). This agen s expeced uiliy nex period is hus given by Ef(x), where f(x) is proporional o exp((1 γ)x). The above formulas allow us o compue he sochasic volailiy premium, i.e., he sure reducion in he growh rae of consumpion ha has he same impac on welfare han he uncerainy affecing is variance, in he Normal and Gamma cases. In Figure 2, we assume γ = 10. Suppose also ha he expeced annual volailiy is σ = 3%. The dashed curve corresponds o he sochasic volailiy premium as a funcion of he sandard deviaion of he variance σ 2 when σ 2 is normally disribued. As explained earlier, his premium is measured exacly by equaion (8) wih ψ f = (γ 1) 3. The plain curve measures ha funcion in he alernaive case in which σ 2 has a Gamma disribuion wih he same firs wo cumulans. This premium 6 Marin (2013) uses he properies of he CGF funcion o derive analyical soluions for ineres raes and risk premia under Epsein-Zin-Weil preferences wih i.i.d. growh raes. 7 The nh cumulan of y is defined as κ y n = χ (n) (0, y). For example, he 4h cumulan of y is equal o µ 4 3µ 2 2, where µ n is he nh cenered momen of y. 8 Under his specificaion, he cumulans of x compued wih formula κ x n = χ (n) (0, x) are equal o x, σ 2 and 3σ 2 w respecively for he firs, second and fourh orders. All oher cumulans are zero. In fac, here is no random variable wih a well-defined cumulaive disribuion funcion having such series of cumulans. 7

8 π Gamma 0.05 Normal σw Figure 2: The relaive sochasic volailiy premium as a funcion of he degree of uncerainy σ w affecing he variance of log consumpion. The dashed curve corresponds o σ 2 N(3%, σ 2 w). The plain curve corresponds o σ 2 having a Gamma disribuion wih he same mean and variance. We assume a CRRA of 10 and η N(0, 1). is given by equaion (12) wih A = γ 1. I is easy o verify from hese wo funcions are idenical up o he fourh order of σ w, so ha he wo curves coincide for low levels of uncerainy. In he long-run risks lieraure, σ w is in he order of magniude of on an annual basis (Bansal e al. (2016)). Figure 2 hus suggess ha he Normal approximaion is accepable, a leas for small mauriies. 9 3 Asse pricing in he Kreps-Poreus model In his secion, we explore he pricing implicaions of sochasic volailiy in he simple 2- period model proposed by Kreps and Poreus (1978) and Selden (1978): W 0 = u(c 0 ) e δ u(e 1 ) (13) v(e 1 ) = Ev(c 1 ). (14) The ineremporal welfare W 0 is a discouned sum of he curren uiliy exraced from curren consumpion c 0 and of he fuure uiliy exraced from he cerainy equivalen e 1 of fuure consumpion c 1. Uiliy funcions u and v are he ime and risk aggregaors, respecively. They are assumed o be increasing and five imes differeniable. Parameer δ is he rae of pure preference for he presen. Fuure consumpion is affeced by sochasic variance srucured as follows: c 1 = c 1 ση (15) σ 2 = σ 2 w, (16) where η and w are independen and have a zero mean, wih Eη 3 = 0. 9 However, he high persisence of he volailiy shocks magnifies he variance of log consumpion a very long mauriies. 8

9 An immediae applicaion of Theorem 1 is ha sochasic volailiy reduces welfare if and only if v (4) is negaive. Following Eeckhoud and Schlesinger (2006), his condiion is referred o as "emperance". Gollier and Pra (1996) showed ha emperance is necessary for any zero-mean background risk o raises he aversion o any oher independen risk. Eeckhoud and Schlesinger (2006) showed ha emperance is necessary and sufficien for an individual o prefer a compound loery yielding eiher x 1 or x 2 over anoher compound loery yielding eiher 0 or x 1 x 2, where x 1 and x 2 are wo zero-mean independen loeries. We now urn o he analysis of asse prices. As in Campbell (1986), Abel (1999) and Marin (2013) for example, le P (φ) denoe he price oday of an asse ha generaes a payoff disribued as c φ 1 in he fuure. The risk-free asse corresponds o φ = 0, and a claim on aggregae consumpion corresponds o φ = 1. In a Lucas ree economy wih a represenaive agen whose preferences are represened by equaions (13) and (14), and where c 0 and c 1 are he fruis endowmen a dae 0 and 1, he equilibrium price P (φ) is characerized by he following pricing equaion: P (φ) = e δ Ecφ 1 v (c 1 ) u (e 1 ) v (e 1 ) u (c 0 ). (17) The coninuously compounded expeced reurn r(φ) of he asse is equal o he logarihm of Ec φ 1 /P (φ). For example, he ineres rae rf is equal o log(p (0)). The risk premium of asse φ is he difference beween he expeced reurn of ha asse and of he risk-free asse. I is equal o ( ) Ec φ 1 π(φ) = log Ev (c 1 ) Ec φ. (18) 1 v (c 1 ) We firs examine he impac of sochasic volailiy on he risk-free rae. We obain a clearcu resul in he special case of Discouned Expeced Uiliy (DEU) where u and v are idenical. In he DEU case, equaion (17) direcly implies ha he price P (0) of a riskfree asse is increased by sochasic volailiy if and only if i raises Ev (c 1 ). The following proposiion is hus anoher immediae applicaion of Theorem 1. Proposiion 1. Suppose u v. Any increase in risk in he variance of fuure consumpion reduces he ineres rae if and only if v is convex. This is he consequence of he fac ha increasing risk surrounding he variance of fuure consumpion generaes a 4h-degree risk deerioraion of consumpion. By definiion, his raises Ev if he fourh derivaive of v is posiive. This is in line wih earlier resuls by Eeckhoud and Schlesinger (2008) who demonsraed ha, in he DEU model, any nh-degree increase in fuure income risk increases opimal savings if and only if sgn[v (n1) ] = ( 1) n. Noice ha condiion v (5) 0 is referred o as "edginess" by Lajeri-Chaherli (2004) and Eeckhoud and Schlesinger (2008). Deck and Schlesinger (2014) and Deck and Schlesinger (2016) esed he sign of up o he fifh derivaive of he uiliy funcion in he laboraory, showing some evidence of edginess Theoreical resuls relaing asse prices o he fifh derivaive of he uiliy funcion are scarce. Gollier (2001) showed ha wealh inequaliy reduces he equilibrium ineres rae if v /v is concave. 9

10 Wih recursive preferences, he impac of sochasic volailiy on he ineres rae is also affeced by he marginal rae of ransformaion u (e 1 )/v (e 1 ). This rae ells us how a marginal increase in risk uiliy v(e 1 ) generaed by an increase in e 1 ranslaes ino an increase in emporal uiliy u(e 1 ). Because sochasic volailiy affecs he cerainy equivalen consumpion e 1, i also affecs his marginal rae of ransformaion. I is increasing in e 1 if and only if v (c)/v (c) is larger han u (c)/u (c) for all c. If v (4) is negaive, we know ha e 1 is reduced by sochasic volailiy. This implies ha sochasic volailiy reduces he marginal rae of ransformaion u (e 1 )/v (e 1 ) when v is more concave han u. This counerbalances he direc effec of sochasic volailiy on Ev (c 1 ) when v (5) is posiive, implying an ambiguous effec. This is summarized in he following proposiion, which also characerizes he risk premium π(1) on aggregae consumpion. Proposiion 2. Suppose ha he risk on he variance of fuure consumpion is small in he sense ha η is disribued as kɛ wih k small. Any increase in risk in he variance of fuure consumpion reduces he ineres rae if and only if ( v v (5) (c 1 ) v (4) (c 1 ) (c 1 ) v (c 1 ) ) u (c 1 ) u 0. (19) (c 1 ) I raises he risk premium π(1) on aggregae consumpion if and only if v (4) (c 1 ) is negaive. Proof: See he appendix. If we assume ha v (4) is negaive, hen condiion (20) can be rewrien as follows: v (5) (c 1 ) v (4) (c 1 ) v (c 1 ) v u (c 1 ) (c 1 ) u (c 1 ). (20) Because u is concave, a sufficien condiion for an increase in risk in he variance of consumpion o reduce he ineres rae is ha he index of edginess v (5) /v (4) be larger han he index of risk aversion v /v. 11 This condiion, which is necessary and sufficien in he small when u is linear, is saisfied for example when v is a power or an exponenial funcion. Noice also ha hese resuls hold only in he small. As already shown by Gollier (1995) and Abel (2002), an increase in consumpion risk does no necessarily raise he risk premium a equilibrium under risk aversion. Similarly, a 4h-degree increase in consumpion risk does no necessarily raise i eiher under condiion v (4) 0. In he long-run risks lieraure, he uncerainy affecs he variance of log consumpion raher han consumpion iself. This means ha equaions (15) and (16) are replaced by he 11 This condiion is parallel o he condiion by Kimball and Weil (2009) who showed ha an increase in fuure income risk raises savings and hus reduces he ineres rae a equilibrium if he index of prudence v (3) /v (2) is larger han he index of risk aversion v /v. This condiion is equivalen decreasing absolue risk aversion. Bosian and Heinzel (2016) examine he impac of a nh-degree risk affecing fuure income on opimal saving in he large wihin he Kreps-Poreus framework. 10

11 following specificaion: x = log ( ) c1 c 0 = µ ση (21) σ 2 = σ 2 w, (22) where η and w are independen and have a zero mean, wih Eη 3 = 0. In his framework, alhough he uncerainy surrounding σ does no affec he firs hree momens of x, i increases all momens of c 1, since funcion f(x) = exp(nx) has a posiive fourh derivaive for all n N 0. The problem is much simplified if one assumes ha he uiliy funcion v exhibis consan relaive risk aversion γ 0, i.e., v(c) = c 1 γ /(1 γ), so ha Ev(c 1 ) is equal o Ef(x) wih exp ( (γ 1)x) f(x) =. (23) γ 1 This implies ha funcion f exhibis consan absolue risk aversion A = γ 1, yielding f (4) (x) = (γ 1) 3 exp( (γ 1)x). Theorem 1 implies ha he sochasic volailiy of he consumpion growh rae reduces welfare in he Kreps-Poreus model wih consan relaive risk aversion if and only if relaive risk aversion is larger han uniy. When η is sandard Normal, he relaive sochasic volailiy premium is approximaely equal o 0.125(γ 1) 3 σ 2 w. This approximaion is exac when he variance of log consumpion is normally disribued. We now urn o he analysis of he impac of he uncerain variance of log consumpion on he risk-free rae and he risk premium. Following Epsein and Zin (1989) and many ohers afer hem, we hereafer assume ha v(c) = c 1 γ /(1 γ) and u(c) = c 1 ρ /(1 ρ). Parameer ρ is he relaive aversion o consumpion flucuaions over ime. I is he inverse of he elasiciy of ineremporal subsiuion. The following proposiion is a direc consequence of using equaion (8) (which is exac under our specificaion) o esimae he expecaions ha appear in equaions (17) and (18). Proposiion 3. Suppose ha log(c 1 /c 0 ) is disribued as µ ση where σ and η are independen, η has a sandard Normal disribuion, and σ 2 is N(σ 2, σw). 2 Suppose also ha relaive risk aversion γ and he relaive aversion o flucuaions ρ are consan. The ineres rae r f and he risk premium π(φ) associaed o an asse whose fuure payoff is c φ 1 saisfy he following condiions: r f = δ ρµ 1 ( ) γ 2 (γ ρ)(γ 1) σ 2 1 ( γ 4 (γ ρ)(γ 1) 3) σw (24) π(φ) = φγσ 2 1 ( 2 φγ γ 2 3 ) 2 φγ φ2 σw. 2 (25) These resuls are reminiscen of earlier resuls by Marin (2013) who characerized he ineres rae and risk premia wih power funcions for u and v when changes in log consumpion are i.i.d. bu no Normal. The coefficiens of he las erm in equaions (24) and (25) correspond o hose obained by Marin for he impac of he 4h cumulan of x on he ineres rae and he risk premia. This is he consequence of he fac ha he 4h cumulan of x equals 3σ 2 w in his framework. 11

12 Increasing he risk on he variance of log consumpion always raises he risk premium associaed o any asse wih φ > 0 since he las erm in equaion (25) has he same sign as φ. In paricular, i increases he risk premium associaed o a claim on aggregae consumpion (φ = 1). Increasing he risk on he variance of log consumpion reduces he ineres rae if γ 4 is larger han (γ ρ)(γ 1) 3. The inuiion of his resul is similar o he one of condiion (20): The increased kurosis of log consumpion raises Ev (c 1 ) proporionally o γ 4. This "precauionary effec" ends o reduce he ineres rae. I also reduces he cerainy equivalen growh rae proporionally o (γ 1) 3. This implies in urn a reducion in he marginal rae of ransformaion u (e 1 )/v (e 1 ) proporionally o (γ ρ)(γ 1) 3. This "income effec" ends o raise he ineres rae when γ is larger han ρ. Globally, he sochasic volailiy affecing log consumpion reduces he ineres rae if and only if he precauionary effec dominaes he income effec. This is he case for example when boh γ and ρ are larger han uniy. When ρ equals 1, i requires ha γ be larger han 1/2. We now explore he impac of sochasic volailiy on wealh, which is here defined as he equilibrium price a dae 0 of a claim on he fuure aggregae consumpion c 1. I is equal o Ec 1 discouned a he risk-adjused discoun rae r f π(1). Using Proposiion 3, we obain ha ( W ealh = c 0 exp δ (ρ 1)µ 1 2 (γ 1)(ρ 1)σ2 1 ) 8 (γ 1)3 (ρ 1)σw 2. (26) As is well-known, in he DEU model (ρ = γ), an increase in uncerainy affecing growh, as measured by σ 2, raises wealh in he economy. We obain he same resul for an increase in he uncerainy affecing volailiy, as measured by σ 2 w. One of he benefis of he recursive uiliy model is o reverse he sign of hese impacs when ρ 1 γ. In his case, an increase in risk on growh or on volailiy reduces wealh in he economy. 4 Long-run sochasic volailiy under Discouned Expeced Uiliy In he remainder of his paper, we characerize he erm srucures of he sochasic volailiy premia for ineremporal welfare, ineres raes and risk premia in he conex examined by Bansal and Yaron (2004) in which he variance of he growh rae of consumpion follows an auoregressive process of order 1: ( ) c1 log = µ σ η 1, (27) wih c σ 2 1 = σ 2 ν ( σ 2 σ 2) σ w w 1, (28) where σ 2 is he uncondiional variance, ν [0, 1[ is he coefficien of persisence of shocks on variance, σ w is he sandard deviaion of hese shocks, and he wo shocks η and w are assumed o be i.i.d. sandard Normal. 12 This sochasic process has hree ineresing 12 Bansal and Yaron (2004) consider a more general model in which he expeced growh rae µ also follows an auoregressive process. In his paper, we focus on he impac of he uncerain variance of growh. 12

13 feaures. Firs, he volailiy σ is sochasic. Second, he shocks o volailiy exhibi some persisence. Third, he volailiy is known one period in advance. Noice also ha an asse φ ha generaes a cash flow D = c φ is governed by he following sochasic process: ( ) D1 log = µ D φσ η 1, (29) D wih µ D = φµ. 13 As in Bansal and Yaron (2004), he sochasic volailiy of dividend growh is proporional o he sochasic volailiy of consumpion growh. Under equaions (27) and (28), he variance of log-consumpion periods ahead is as follows: [ v = V ar log ( c c 0 ) w 1,...w 1, σ 0 ] = σ 2 1 ν 1 ν (σ 2 0 σ 2) σ w τ=1 1 ν τ 1 ν w τ. (30) The las erm in his equaliy characerizes he sochasic naure of he volailiy of long-erm growh. I is normally disribued. This means ha he sochasic process (27)-(28) yields he same sochasic srucure of log consumpion as he one described by equaions (21)- (22), wih mauriy-specific parameers. The following lemma characerizes he naure of he sochasic variance of log-consumpion a differen mauriies in his long-run risk conex. Lemma 1. Suppose ha log-consumpion is governed by process (27)-(28). Then, for any mauriy N 0, x 0, = log(c /c 0 ) condiional o σ 0 is disribued as µ v ε, where v and ε are independen, ε is N(0, 1), and v is normally disribued wih annualized mean E 0 [v ] = σ 2 1 ν ( σ0 2 σ 2) (31) (1 ν) and annualized variance V ar 0 [v ] = 1 3 κ x 0, 4 = σ2 w (1 ν) 2 ( ν (1 ν) 1 ) ν2 (1 ν 2, (32) ) where κ x 0, 4 is he fourh cumulan of log consumpion x 0,. Noice ha E 0 v measures he uncondiional variance of log(c /c 0 ). If one divides his measure by σ 0, one obains he "variance raio" ha has already been examined by Cochrane (1988), Beeler and Campbell (2012) and many ohers. Equaion (31) ells us ha his variance raio has a fla erm srucure in expecaion, i.e., when σ 0 equals σ. This is consisen wih he recen findings obained by Marfè (2016) who used poswar U.S. oupu. 14 As explained earlier, sochasic volailiy does no increase he risk as measured by he variance 13 This consrain on he expeced growh of dividends is irrelevan for our analysis. I affecs he price of he asse, bu no is risk premium. 14 In fac, Marfè (2016) obained erm srucures of he variance raio for oupu, salary and dividend ha are respecively fla, increasing and decreasing. He convincingly argues ha his comes from he fac ha firms provide shor-erm insurance o heir employees agains he ransiory flucuaions of heir labor produciviy, in line wih he heory of implici labor conrac. 13

14 of log consumpion. Similarly, i does no affec is skewness. Turning o equaion (32), he annualized variance of v goes from 0 for a one-period mauriy up o σ 2 w/(1 ν) 2 for very large mauriies. This upward sloping erm srucure is due o he combinaion of he fac ha he variance is known one period in advance and of he persisence of shocks o volailiy. These wo feaures of he sochasic process of growh magnify he long-erm risk on variance. As noiced in Secion 2, in his Gaussian framework, he fourh cumulan of x 0, is equal o hree imes he variance of v. I implies ha he annualized fourh cumulan of log consumpion has an increasing erm srucure oo. We es his hypohesis in Secion 6. As a benchmark, consider a Lucas ree economy wih a represenaive agen ha maximizes he discouned expeced uiliy of he flow of aggregae consumpion: [ ] W 0 = E exp( δ)v(c ). (33) =0 We hereafer assume ha W 0 is bounded. Using he same approach as in he previous secion adaped o he case wih u(c) = v(c) = c 1 γ /(1 γ) and wih he mauriy-varying parameers of he sochasic volailiy process described in Lemma 1, we obain he following proposiion. Proposiion 4. Suppose ha he growh process is governed by equaions (27)-(28). Under he DEU model wih consan relaive risk aversion γ, he erm srucures a dae 0 of ineres raes and he risk premia saisfy he following condiions: r f = δ γµ 1 E 2 γ2 0 [v ] π (φ) = φγ E 0 [v ] 1 ( 2 φγ where E 0 [v ] and V ar 0 [v ] are given by Lemma γ4 V ar 0 [v ] (34) γ 2 3 ) V 2 φγ ar0 [v ] φ2, (35) The firs wo erms in he righ-hand side of equaion (34) correspond o he Ramsey rule (Ramsey (1928)). The hird erm measures he impac of he risk affecing economic growh on he ineres rae in he absence of sochasic volailiy. Is erm srucure is fla when he curren variance σ 2 0 equals is hisorical mean σ2. This is because sochasic volailiy does no increase he long-run risk measured by he annualized uncondiional variance, which is equal o σ 2 a all mauriies in ha case. The las erm measures he sochasic volailiy premium for ineres raes. Because he volailiy of he growh rae is known one period in advance in he Bansal-Yaron model, his premium is zero for a one-period mauriy. Because of he persisence of he shocks o volailiy, he annualized variance of he condiional variance of log consumpion defined as V ar 0 (v ) / is increasing wih mauriy, hereby magnifying he kurosis of he disan log consumpion. This makes he erm srucure of ineres raes decreasing in expecaion. The firs erm in he righ-hand side of equaion (35) is he classical CCAPM risk premium, which is he produc of hree elemens: he CCAPM bea of he asse, he relaive 14

15 r f r f 1 π (1) π 1 (1) π (3) π 1 (3) γ = ρ = γ = ρ = γ = ρ = Table 1: Expeced erm spreads of ineres raes, risk premia on aggregae consumpion and equiy premia under neural expecaions (σ 0 = σ). As in Bansal e al. (2016), we assume σ w = , and ν = on a monhly basis. Spreads are expressed in percens per year. risk aversion of he represenaive agen, and he expeced annualized variance of he growh rae unil mauriy. This laer elemen has a fla erm srucure when he curren variance σ0 2 is equal o is hisorical mean σ2. An imporan feaure of his model is ha shorerm risk premia are ime-varying, in parallel o changes in expecaion abou he shor-erm volailiy. However, he long-erm risk premium of any asse φ is fixed. The second erm measures he impac of sochasic volailiy. I is always posiive and proporional o he annualized variance of he condiional variance of log(c /c 0 ). Because his elemen has an increasing erm srucure, we can conclude ha, on average, risk premia have an increasing erm srucure due o he persisence of he shocks o volailiy. This magnifies he kurosis of he disribuion of long erm log consumpion, ogeher wih he conribuion of any risky asse o his kurosis. This explain why he erm srucure of risk premia mus be increasing in expecaion in his model wih sochasic volailiy. This raises new concern abou he empirical es of his model. Indeed, i has recenly been shown ha he erm srucure of equiy premia is decreasing (Binsbergen e al. (2012), Binsbergen and Koijen (2016), Giglio e al. (2015), and Giglio e al. (2016)). Le us now quanify he impac of sochasic volailiy on asse prices under he DEU model. We focus on he erm spreads. Because volailiy is known one period in advance, sochasic volailiy has no impac on he shor-erm ineres rae and risk premium. Bu i reduces he long-erm ineres rae and i raises he long-erm risk premium. Bansal e al. (2016) assumed γ = 9.67, σ w = , and ν = on a monhly basis. This yields a erm spread of ineres raes of 2.30% on an annual basis. This number provides an upper bound of he impac of sochasic volailiy on ineres rae for finie mauriies. Bansal e al. (2016) also calibraed he elasiciy of he payoff of equiy o aggregae consumpion a φ = 3. This yields a erm spread of annualized equiy premia of 1.80%. I should be sressed ha hese erm spreads are very sensiive o relaive risk aversion. This is paricularly he case for he ineres rae whose erm spread is proporional o γ 4. In Table 1, we documen his high sensiiviy. To sum up, he sochasic volailiy of he growh rae raises he kurosis of fuure log consumpion. Because he 4h and 5h derivaives of v are respecively negaive and posiive, his ends o reduce ineres raes and o raise equiy premia in he DEU framework. The persisence of shocks o volailiy magnifies hese effecs for longer mauriies, hereby making 15

16 he erm srucures of ineres raes and equiy premia respecively decreasing and increasing on average. 5 Long-run sochasic volailiy under Recursive Uiliy The use of recursive preferences is helpful o solve he equiy premium puzzle and he risk-free rae puzzle. Can i also solve he puzzle of he decreasing erm srucure of risk premia? In he Epsein-Zin-Weil model, welfare V is obained by backward inducion: ( V 1 ρ = (1 β)c 1 ρ β log V = (1 β) log c β log E V 1 γ 1 ( ) 1 ρ 1 γ E V 1 γ 1 ) 1 1 γ if ρ 1 (36) if ρ = 1. (37) where parameers γ and ρ are he indices of relaive aversion o risk and o consumpion flucuaions, respecively. Parameer β = exp( δ) is a discoun facor. The equilibrium price a dae for an asse ha generaes a single payoff D τ = c φ τ a dae τ > saisfies he following condiion: [ ] S τ P,τ = E D τ. (38) S The one-period-ahead sochasic discoun facor S τ1 /S τ o be used a dae τ o value a payoff occurring a dae τ 1 is: S τ1 S τ = β ( cτ1 c τ ) ( γ ( ) ) γ ρ 1 γ Z ρ γ 1 γ cτ1 τ1 E τ Z τ1, (39) c τ where Z τ = V τ /c τ is he fuure expeced uiliy per uni of curren consumpion. Proposiion 5 describes an approximaion of he erm srucures of ineres raes and risk premia wih recursive preferences and sochasic volailiy. 15 These resuls are obained by assuming ha he log(z ) is linear in he sae variable σ 2, which is he case when he EIS ρ 1 is equal o one. We show in he appendix ha he coefficien b = d log(z ) dσ 2 (40) σ 2 =σ 2 saisfies he following condiion: ( ) ( ( 1 b = 2 (1 γ) bν β exp (1 ρ) µ 1 2 (1 γ)σ2 1 )) 2 (1 γ)b2 σw 2. (41) When ρ equals uniy, b equals 0.5β(1 γ)/(1 βν). Noice ha b is negaive when γ is larger han uniy, which implies ha he ineremporal welfare Z is decreasing in he curren volailiy of he growh rae of consumpion. 15 As done in his lieraure, we will hereafer use equaliies o refer o he approximaions obained when using his linearizaion. 16

17 Proposiion 5. Suppose ha he growh process is governed by equaions (27)-(28). Under he recursive uiliy model (36)-(37), he erm srucures a dae 0 of ineres raes and he risk premia are approximaed by he following equaions: r f = δ ρµ 1 ( ) γ 2 E0 [v ] (γ ρ)(γ 1) 1 ( ) 2 γ 2 V ar 0 [v ] (γ ρ)(γ 1) (γ ρ)(1 ρ)b2 σw 2 1 ( γ ρ ( ) γ 2 (γ 1)(γ ρ) b 1 1 ) ν σw 2 (42) 2 1 ν (1 ν) π (φ) = φγ E 0 [v ] 1 ( 2 φ γ(γ ρ γρ) 1 ) V 2 φ(2γ2 γ ρ γρ) γφ 2 ar0 [v ] ( ) 1 2 σ2 w(γ ρ)φb φ 2γ 1 ν 1 1 ν (1 ν), (43) where E 0 [v ] and V ar 0 [v ] are given by Lemma 1, and b solves equaion (41). These approximaions are exac when ρ is equal o uniy. Proof: See he appendix. A direc consequence is ha, in expecaion (σ 0 = σ), he shor-erm ineres rae simplifies o 16 r f 1 = δ ρµ 1 2 ( ) γ 2 (γ ρ)(γ 1) σ (γ ρ)(1 ρ)b2 σw. 2 (44) Alhough he variance of growh is known one period in advance, he sochasic naure of fuure volailiy reduces he shor-erm ineres rae when he represenaive agen has a Preference for an Early Resoluion of Uncerainy (PERU), i.e., when γ is larger han ρ, and ρ is smaller han uniy. The represenaive agen will observe a dae 1 he volailiy σ 1 ha will prevail in he second period, and his addiional dae-1 risk has an impac on he willingness o raise savings oday for dae-1 consumpion. The erm srucure of ineres raes in his long-run risk model wih recursive preferences combines feaures already discussed in he DEU model and a new elemen coming from PERU. Indeed, he firs line in equaion (42) is symmeric o he equaion (34), wih adaped coefficiens for E 0 [v ] and V ar[v ] o accoun for he discrepancy beween γ and ρ. The second line in equaion (42) is new compared o he DEU framework. We have seen above ha i ends o reduce he shor-erm ineres rae. I is easy o check ha he las erm in his equaion has a decreasing erm srucure under PERU and γ 1 (so ha b is negaive). This means ha PERU canno reverse he endency of he erm srucure of ineres raes o be decreasing. This resul parallels Beeler and Campbell (2012) who showed ha when he persisence of shocks o he growh rae of consumpion is added o he model as in Bansal 16 Alhough hey do no compue he erm srucure of ineres raes, Bansal e al. (2016) characerize he shor-erm ineres rae r f 1, which corresponds o equaion (44) wih heir consan of log-linearizaion κ1 equaling 2b(1 βν)/(1 γ). 17

18 β discoun facor γ 9.67 relaive risk aversion ρ elasiciy of ineremporal subsiuion µ expeced growh rae of consumpion σ 0 = σ curren and expeced volailiy σ w sandard deviaion of shocks o volailiy ν coefficien of persisence Table 2: Calibraion parameer based on monhly daa exraced from Bansal e al. (2016). e al. (2012), hen real ineres raes have a decreasing erm srucure, and are negaive for mauriies exceeding 10 years. The shor-erm risk premium π 1 (φ) = φγσ0 2 is no affeced by sochasic volailiy. This is because he volailiy for he firs period is known a dae 0. The middle erm in he righhand side of equaion (43) is similar o he las erm of equaion (35) in he DEU model. The persisence of shocks o volailiy magnifies he long-erm kurosis of log consumpion, hereby ending o make he erm srucure of risk premia increasing. The las erm of equaion (43) akes accoun of PERU in he recursive uiliy model. As long as φ is smaller han 2γ, his new erm has an increasing erm srucure oo. We can hus conclude ha PERU canno reverse he increasing naure of he erm srucure of risk premia already observed under DEU. Consider again he calibraion exraced from Bansal e al. (2016), as summarized in Table 2. Figure 3 describes he erm srucures under his calibraion, respecively for he ineres raes (r f ), he risk premia on aggregae consumpion (π (1)), and he equiy premia (π (3)). The benchmark case wihou sochasic volailiy is obained by selecing σ w = 0. I is represened by he dashed lines in his figure. As in he DEU framework, alhough shocks o volailiy are small, heir high persisence has a srong impac on he pricing of longdaed asses. I reduces he long ineres rae by around 0.4%, and i raises he long equiy premium by almos 1.5%. Noice ha because he half-life of he shocks o he variance of log consumpion is around 36 years, i akes many cenuries for hese erm srucures o converge o hese asympoic values. A comparaive saic analysis is summarized in Table 3. Long-erm raes are much more affeced by he unilaeral change of a parameer han shor-erm raes. For example, consider an increase in he persisence parameer ν from is benchmark value of o Because Bansal e al. (2016) esimaed ν wih a sandard error of , his change in ν canno be excluded by heir daa. 17 This unilaeral change has no effec on he risk premium 17 In fac, ν = is he calibraion used by Bansal e al. (2012). Noice ha using he calibraion wih ν = as in Bansal and Yaron (2004) would yield he following equilibrium prices: r f 1 = 1.68%, r f = 1.67%, π1(3) = 1.71% and π (3) = 1.73%. As noiced by Beeler and Campbell (2012), he sochasic naure of volailiy has very lile effec on asse prices in his iniial calibraion of he long-run risk model. 18

19 r f π(1) π(3) Figure 3: The erm srucures of ineres raes (op), risk premia on aggregae consumpion (φ = 1, middle) and equiy premia (φ = 3, boom) wih recursive uiliy under he sochasic process (27)-(28). The calibraion of he parameers is described in Table 2. The dashed curves are obained by imposing σ w = 0 (no sochasic volailiy). Raes are in percen per year and duraions are in years. 19

20 r f 1 r f π 1 (3) π (3) Benchmark γ = ρ = σ w = ν = Fixed volailiy (σ w = 0) Table 3: Comparaive saic analysis based on he benchmark described in Table 2, wih γ = 9.67, ρ = 1/2.18, σ w = , and ν = The firs column describes he modified parameer, everyhing else held unchanged. Raes are in percen per year. on shor-erm equiy, bu i raises is erm spread from 1.46% o 3.20%. 6 The erm srucure of he 4h cumulan of log consumpion The long-run risk specificaion (27)-(28) of sochasic volailiy generaes an increasing erm srucure of he uncerainy affecing he condiional variance of he growh rae. This does no change he annualized variance whose erm srucure remains fla. Bu i makes he erm srucure of he annualized fourh cumulan of log consumpion increasing, as expressed in equaion (32). This is he driving force behind he shapes of he erm srucures described in he previous wo secions. I is useful o es wheher consumpion daa generaes an increasing erm srucure of he annualized fourh cumulan of log consumpion. As Bansal e al. (2016) and many ohers, our consumpion daa are exraced from he NIPA Tables 2.33, 2.34 e 7.1 of he Bureau of Economic Analysis. We use quarerly daa of per-capia real consumpion expendiure on nondurables and services from 1947Q1 o 2016Q4. In Figure 4, we used plain circles o represen he erm srucure of he empirical annualized fourh momen of log consumpion, which is compued as follows for each mauriy {1,..., 40}: K x 0, 4 = 1 ( ) 278 (278 ) 1 (x τ,τ x ) (278 ) 1 (x τ,τ x ) 2, (45) τ=1 where x is he mean of he series ( x τ,τ. The increasing erm srucure for )τ=1,...,278 K x 0, 4 / observed in he daa suggess he persisence of shocks o he variance and is supporive of decreasing ineres raes and increasing risk premia. This is compaible wih he long-run risk model (27)-(28) wih a posiive coefficien of persisence. The quarerly calibraion proposed by Bansal e al. (2016) is also described in Figure 4, wih ν = and σ w = This is compued as follows: κ x 0, 4 = 3σ2 w (1 ν) 2 ( τ= ν (1 ν) 1 ν2 (1 ν 2 ) ). (46) This figure shows ha his calibraion ends o generae oo much kurosis for mauriies exceeding one year. This is why we also calibraed he erm srucure of he annualized 4h 20

21 x0, κ 4 / Figure 4: The erm srucure of he annualized 4h cumulan of log consumpion κ x 0, 4 /, where he mauriy is measured in quarers. The bulles are obained for mauriies {1,..., 40} by measuring he empirical 4h cumulan of he series ( x τ,τ, using quarerly real U.S. consumpion from 1947Q1 o 2016Q4. )τ=1,...,278 The squares correspond o he heoreical erm srucure of he 4h cumulan using equaion (46) wih he calibraion ν = and σ w = proposed by Bansal e al. (2016). The crosses correspond o an alernaive calibraion wih ν = and σ w = cumulan wih parameers ν = and σ w = ha beer fi he daa. In fac, hese parameer values minimize he sum of he square of he differences beween K x 0, 4 / and κ x 0, 4 / for mauriies beween 1 and 40 quarers. This calibraion exhibis a smaller degree of persisence of shocks o he variance, and a smaller sandard deviaion for hese shocks. 7 Concluding remarks Wih power uiliy funcions, risk aversion, prudence, emperance, edginess and higher degree risk aiudes are all summarized by one parameer usually referred o as risk aversion. The classical asse pricing heory, which heavily relies on his isoelasic specificaion, has herefore developed argumens based on he sole concep of risk aversion. We find ha problemaic for he developmen of his heory. For example, he negaive impac of risk as measured by he variance on he ineres rae is exclusively linked by he noion of prudence (v 0), which is orhogonal o he concep of risk aversion (v 0). In he same fashion, his paper demonsraes ha sochasic volailiy also reduces he ineres rae if and only if he fifh derivaive of he risk uiliy funcion v is posiive. This condiion is someimes referred o as "edginess". In oher words, deparing from he power uiliy funcion would in heory allows us o disenangle hese psychological rais of our preferences under risk. This agenda of research is in line wih recen findings ha are incompaible wih consan relaive risk aversion (see for example Ogaki and Zhang (2001), Guiso and Paiella (2008), and Deck and Schlesinger (2014)). This is also in line wih he rend of behavioral finance in which he uiliy funcion is disored by addiive habi formaion, by background risks, or by a reference 21