Essays on Risk Premiums in Higher-O Title Financial Asset Returns

Transcription

1 Essays on Risk Premiums in Higher-O Tile Financial Asse Reurns Auhor(s) SASAKI, Hiroshi Ciaion Issue Dae Type Thesis or Disseraion Tex Version ETD URL hp://doi.org/ /27273 Righ Hiosubashi Universiy Reposiory

2 Essays on Risk Premiums in Higher-Order Momens of Financial Asse Reurns Thesis by Hiroshi Sasaki Advisor Associae Professor Hideoshi Nakagawa In Parial Fulfillmen of he Requiremens for he Degree of Docor of Philosophy, Graduae School of Inernaional Corporae Sraegy (ICS), Hiosubashi Universiy, Tokyo, Japan July 31, 2014

3 Copyrigh c 2014 Hiroshi Sasaki, All Righs Reserved. 1

4 This Thesis is Dedicaed o My Wife, Yukiko, and My Son, Yu. 2

5 Acknowledgemens I would like o give hearful hanks o Associae Prof. Hideoshi Nakagawa whose enormous suppor and insighful commens were invaluable during he course of my sudy. Wihou his encouragemen and guidance, his disseraion would no have maerialized. Special hanks also go o Prof. Kazuhiko Ohashi, Prof. Ryozo Miura, Prof. Toshiki Honda, Prof. Fumio Hayashi, Prof. Nobuhiro Nakamura, Prof. Tokuo Iwaisako, Associae Prof. Tasuyoshi Okimoo, Associae Prof. Daisuke Yokouchi, whose opinions and informaion have helped me very much hroughou he producion of his hesis. I would also like o express he deepes appreciaion o Prof. Shigeo Kusuoka, Prof. Masayuki Ikeda, Prof. Jun Sekine, Associae Prof. Masaaki Fukasawa, Associae Prof. Koichiro Takaoka, whose advice and commens given by hem have been a grea help in sudying he problems focused in his hesis. Moreover, Dr. Takanori Adachi gives me consrucive commens and warm encouragemen, and I am paricularly graeful for he assisance given by him. I have had he suppor and encouragemen of superiors and colleagues a work in my company for proceeding o he degree, and finally, I would also like o express my graiude o my family for heir moral suppor and warm encouragemens for his hesis. 3

6 Conens 1 Inroducion Moivaion for his sudy Absrac and focus for each chaper The Skewness Risk Premium in Equilibrium and Sock Reurn Predicabiliy Inroducion Model Framework Model Seup and Assumpions The Model Soluion in Equilibrium Risk Premiums in Higher-Order Momens in Equilibrium An Equiy Risk Premium Represenaion Model Implicaions Empirical Measuremens Measuremens for he Higher-Order Momens Daa Descripion Main Empirical Findings Concluding Remarks Appendix 2.A Proof of Proposiion Appendix 2.B The Risk-Free Rae An Approach o he Opion Marke Model Based on End-user Ne Demand Inroducion The Model Assumpions of a Financial Opion Marke Assumpions for Opion Prices Opimaliy and Equilibrium

7 CONTENTS Opimizaion for he Marke-maker and he End-user The Pricing Kernel in Equilibrium Concluding Remarks Appendix 3.A Expeced Dela-Hedged Gain and Loss Appendix 3.B Proof of Proposiion Appendix 3.C Proof of Proposiion Appendix 3.D Mone Carlo Simulaion Resuls Undersanding Dela-hedged Opion Reurns in Sochasic Volailiy Environmens Inroducion The Model and he Mehodology An explici represenaion for dela-hedged opion reurns Esimaion Sraegy for he Volailiy Risk Premium An Analyical Process for a Conribuion Analysis Daa and Mehodology for an Empirical Implemenaion Descripion of he OTC Currency Opion Marke and Daa Parameer Esimaion for he Heson[1993] Sochasic Volailiy Model Esimaion of he Volailiy Risk Premium Esimaion of he Expeced DHGL An Empirical Analysis Esimaion Resuls for he Model Parameers A Conribuion Analysis on he Expeced DHGL Concluding Remarks Appendix 4.A Proof of Proposiion Appendix 4.B Time Series Daa Appendix 4.C Esimaion Resuls of he Heson Model Concluding Remarks 113 5

8 Lis of Figures 2.1 The Risk Premiums in Higher-Order Momens and he Equiy Risk Premium The Facor Loading β vp,q The Facor Loading β vp,λ The Facor Loading β sp,q The Facor Loading β sp,λ The Facor Loading o V ar P :π V ar The Facor Loading o vp :π vp The Facor Loading o skp :π skp The VIX and The Curren Volailiy The Risk-Neural Skewness and The Curren Skewness The end-user ne demand δ 2 and he opion price p 2 (δ 2) in equilibrium The end-user ne demand δ 3 and he opion price p 3 (δ 3) in equilibrium The parameer c 1 in equilibrium The shape of erminal wealh for he end-user : W E T (δ ; S T ) The Disribuions of Dela-Neural Hedging Errors ɛ i T, i = 1, 2, The End-User Ne Demand δi and he Opion Prices p i (δ ) in Equilibrium The ime series of he volailiy risk premium parameer λ Risk aversion parameer γ USD-JPY WM/Reuer Closing Spo Rae USD-JPY 1M ATM Implied Volailiy(Mid Price) USD-JPY 1M ATM Mid-Bid Spread The ime-series of he esimaed parameer: k The ime-series of he esimaed parameer:ṽ The ime-series of he esimaed parameer: ρ

9 Lis of Tables 2.1 The Se of Model Parameers Summary saisics for he monhly reurns and predicor variables The Monhly and Quarerly Reurn Regressions The Univariae Regressions wih Tradiional Predicor Variables Summary saisics for he CAY The Univariae Regressions wih he CAY Summary Saisics of Hedging Error Disribuions Summary saisics on he volailiy risk premium parameer λ Conribuion Analysis for he Expeced Dela-hedged Gain Loss Summary Saisics of he DHGL for he ATM Sraddle Shor Sraegy Summary Saisics of he DHGL for he OTM Pu Shor Sraegy Relaive Conribuion Comparison of he Expeced DHGL beween Preand Pos Lehman Crisis : Based on a ime-series of overlapping resuls Relaive Conribuion Comparison of he Expeced DHGL beween Preand Pos Lehman Crisis : Based on a ime-series of non-overlapping resuls Summary Saisics for Implied Volailiies

10 Chaper 1 Inroducion 1.1 Moivaion for his sudy I is well known ha marke prices of financial opions reflec common assessmen of he probabiliy disribuion of he underlying asse on he expiraion day, adjused for he invesors olerance for bearing risk. Recen sudies sugges ha here is a difference beween he probabiliy disribuion of he underlying asse implied by financial opion prices and ha realized in he underlying asse marke. For praciioners such as risk managers in he invesmen banks or speculaors in he hedge funds indusry, i is very imporan o undersand why his difference beween he risk neural and physical probabiliy disribuions arise from he heoreical poin of view. Wha are he heoreical deerminans of his difference? Wha role does invesor behavior play in explaning dynamic movemens in his difference? In his sudy, we invesigae risk premiums in higher-order momens, such as variance and skewness, of financial asse reurns and some recen opics relaed o hose risk premiums. In he area of financial economics, he risk premiums in higher-order momens have recenly received broad aenion due o a rich body of implicaions for financial asse pricing mechanism and asse reurn predicabiliy, and a lo of effors have been pu ino examining he exisence and he characerisics of hose risk premiums. The purpose of his sudy is o provide he way of a deeper undersanding on he risk premiums in higher-order momens examined by recen academic sudies hrough boh heoreical and empirical approaches. The concern wih risk premiums in higher-order momens has been growing for he las several years in erms of he pracical poin of view as well as he academic viewpoins. In paricular, here has been a growing ineres in he informaion of he difference 8

11 1.1. MOTIVATION FOR THIS STUDY beween opion-implied and realized disribuions, which is usually recognized as a risk premium required by he represenaive agen, in erms of he financial risk managemen or he asse pricing implicaions. This sudy especially focuses on he variance and skewness risk premiums in financial asse reurns and invesigaes heir exisence in financial markes and heir implicaions for financial asse pricing mechanism in deail. The variance risk premium is defined as he difference beween opion-implied and realized variance and he skewness risk premium is also defined as he same wih he variance risk premium, ha is, he difference beween opion-implied and realized skewness. Recen invesigaions on he risk premiums in higher-order momens of financial asse reurns have focused on hree major aspecs, ha is, empirical and heoreical aspecs on he exisence of hose risk premiums and an aspec on he asse reurn predicabiliy. Bakshi and Kapadia[2003], Low and Zhang[2005], Carr and Wu[2009], and Broadie, Chernov, and Johannes[2009] examine he exisence of he volailiy risk premiums based on empirical manners and hey show ha he risk premiums in higher-order momens of financial asse reurns significanly exis in acual financial marke. Bansal and Yaron[2004], Eraker[2008], Branger and Völker[2010], Drechsler and Yaron[2011], and Bollerslev, Sizova, and Tauchen[2012] provide general equilibrium models ha are consisen wih he exisence of he risk premiums in higher-order momens of financial asse reurns and hey sugges heoreical approaches o prove he exisence of hose risk premiums. Bali and Hovakimian[2009], Goyal and Sareo[2009], Bollerslev, Tauchen, and Zhou[2010], and Drechsler and Yaron[2011] invesigae he asse reurn predicabiliy of he risk premiums in higher-order momens of financial asse reurns empirically and hey find ha hose risk premiums have superior predicive power for fuure asse reurns. In his sudy, we aim o advance hese previous sudies wih heoreical and empirical manners in order o shed ligh on he naure of he risk premiums in higher-order momens of financial asse reurns. Specifically, we provide hree major opics on hose risk premiums. Firs, in Chaper 2, we develop he model which is consisen wih he exisence of he skewness risk premium. The variance risk premium is recognized as he compensaion for he risk induced by sochasic volailiy in financial asse price processes. I is well known ha his risk premium is essenially relaed o dela-hedged opion reurns (e.g., Bakshi and Kapadia[2003]). And moreover, in recen years, here remains an ever-increasing ineres and challenge o develop an enirely self-conained equilibrium-based explanaion for he nonzero variance risk premium and is predicabiliy for sock index reurns (e.g., Bollerslev, Tauchen, and Zhou[2009]). Thus, recen sudies have mainly focused on he 9

12 1.1. MOTIVATION FOR THIS STUDY variance risk premium as one of he risk premiums in higher-order momens. However, as far as we know, here have been few repors abou he risk premium which compensaes for uncerainy of he hird momen, ha is, he skewness, of asse reurns in financial markes. Therefore, firs of all, we begin o demonsrae ha he skewness risk premium exiss in equilibrium and capures aiudes oward economic uncerainy as well as he variance risk premium. Among recen sudies on self-conained equilibrium-based models for he nonzero variance risk premium referred above, almos all he sudies model he asse price process as condiional normal, so ha he one-sep-ahead condiional disribuion of he marke reurn is also normal and, as a resul, he skewness of ha disribuion is zero. Therefore, he models proposed by hese recen sudies can no explain he negaive risk-neural skewness, which is found by he previous sudies such as Aï-Sahalia and Lo[1998] and Aï-Sahalia, Wang, and Yared[2001]. Exending he models proposed in he previous sudies by inroducing a sochasic jump inensiy ino he consumpion growh rae process, we will provide a represenaion of he skewness risk premium based on a general equilibrium model and prove ha his skewness risk premium should be non-zero in equilibrium. Second, in Chaper 3, we invesigae he reason why he variance risk premium exiss in financial markes heoreically and empirically in more deail under a parial equilibrium seing. We consider a heoreical relaionship beween he ne demand of end-users for financial opions and he variance risk premiums in a sochasic volailiy environmen wih he demand-based pricing kernel, which is he pricing kernel relaed o he amoun of he ne demand of end-users for financial opions in equilibrium, and provide an explici explanaion of he exisence of he variance risk premium in erms of he ne opion demand of end-users. To he bes of our knowledge, his approach is he firs o provide some implicaions in he well-known empirical evidence, ha is, he negaive variance risk premium, in erms of he end-user ne demand for financial opions. Third and finally, in Chaper 4, he effec of parameer esimaion risk on he resuls of empirical sudies for he exisence and he sign of he variance risk premium is examined. Theoreical financial models ofen assume ha he economic agen who makes an opimal financial decision knows he rue parameers of he model. However, he rue parameers are rarely if ever known o he decision maker. In realiy, model parameers have o be esimaed based on hisorical informaion and, hence, he model s usefulness depends parly on how good he esimaes are. This gives rise o esimaion risk in virually all financial valuaion models. Bakshi and Kapadia[2003] and Low and Zhang[2005] find a heoreical relaionship beween dela-hedged opion reurns in financial opions and he variance risk premium. They sudy dela-hedged opion reurns in a sock index 10

13 1.2. ABSTRACT AND FOCUS FOR EACH CHAPTER opion marke and currency opion markes, respecively, and relying on a heoreical relaionship, hey provide evidence ha he variance risk premiums are no zero and negaive because of non-zero expeced dela-hedged opion reurns wih hedging-based empirical ess on he variance risk premium. We show ha he effec of parameer esimaion risk on opion prices quoed in acual financial opion markes is significan, and herefore, sugges ha he sandard hedging-based empirical ess on he exisence of he variance risk premiums, which is explored and examined by, for example, Bakshi and Kapadia[2003] and Low and Zhang[2005], may be unreliable because of ha significan effec of parameer esimaion risk on opion prices. These analyses in his sudy lead o he conclusion ha he uncerainy of he variance and skewness of financial asse reurns are consisenly priced in equilibrium and we also find ha hese prices of he risks have broad implicaions for financial asse pricing mechanism, which is one of he mos imporan problems o be solved in he financial economics. 1.2 Absrac and focus for each chaper Le us provide he absrac and focus for each chaper in his sudy in he following. Chaper 2 In his chaper, we sudy risk premiums in higher-order momens of financial asse reurns in a general equilibrium seing. To he bes of our knowledge, he firs aemp o demonsrae he exisence of he volailiy risk premium based on a general equilibrium marke model is made by Eraker[2008]. Eraker[2008] develop an equilibrium explanaion for he volailiy risk premium based on he long-run risks (LRR) model 1 which emphasizes he role of long-run risks, ha is, low-frequency movemens in consumpion growh raes and volailiy, in accouning for a wide range of asse pricing puzzles. The LRR model feaures an Epsein and Zin[1989] uiliy funcion wih an invesor preference for early resoluion of uncerainy and conains (i) a persisen expeced consumpion growh componen and (ii) long-run variaion in consumpion volailiy. On he basis of he LRR model, Eraker[2008] sudies he volailiy risk premium hrough he framework of a general equilibrium model. The recen sudies on he risk premiums in higher-order momens focus mainly on he risk premium of he second momen, ha is, he volailiy or he variance. Conversely, 1 The long-run risks model is pioneered by Bansal and Yaron[2004], which is a sylized self-conained general equilibrium model incorporaing he effecs of ime-varying economic uncerainy. 11

14 1.2. ABSTRACT AND FOCUS FOR EACH CHAPTER as far as we know, here are few repors abou he risk premium which compensaes for uncerainy of he hird momen, ha is, he skewness, of asse reurns. In his chaper, we demonsrae ha he skewness risk premium, defined by he difference beween wo expeced values of he skewness under he risk-neural and physical probabiliy measures, respecively, also capures aiudes oward economic uncerainy as well as he variance risk premium. Exending he long-run risks (LRR) model proposed by Bansal and Yaron[2004] by inroducing a sochasic jump inensiy for jumps in he LRR facor and he variance of consumpion growh rae, we provide an explici represenaion for he skewness risk premium, as well as he volailiy risk premium, in equilibrium. On he basis of he represenaion for he skewness risk premium, we propose a possible reason for he empirical fac of ime-varying and negaive risk-neural skewness. Moreover, we also provide an equiy risk premium represenaion of a linear facor pricing model wih he variance and skewness risk premiums. The empirical resuls prove ha he skewness risk premium, as well as he variance risk premium, has superior predicive power for fuure aggregae sock marke index reurns. Compared wih he variance risk premium, he resuls show ha he skewness risk premium plays an independen and essenial role for predicing he marke index reurns. Chaper 3 In his chaper, we sudy financial opion prices in erms of demand pressure effecs based on he preferences of he represenaive marke-maker and he represenaive enduser under a parial equilibrium seing. We assume ha here are wo ypes of agens in he opion marke: marke-makers and end-users. Marke-makers play a key role in providing liquidiy o end-users by aking he oher side of end-user ne demand. We examine a demand-based opion marke model in which marke-makers hedge heir opion posiions based on he dela-neural hedging sraegy wih fuures or forwards in order o conrol he risks induced by aking he oher side of end-user ne demand. If we can assume a dynamically complee financial marke ha can be governed by sandard marke models such as he Black-Scholes-Meron model[1973], he no-arbirage heory deermines derivaive prices uniquely and independenly of invesor demand for opions because marke-makers can hedge heir opion posiions perfecly hrough coninuous ime rading wih he underlying asses and cash. However, he assumpion of a dynamically complee marke would no apply o he acual financial marke. Under he assumpion of marke incompleeness induced by addiional risk facors such as sochasic volailiy and/or jumps, marke-makers canno hedge heir opion posiions perfecly 12

15 1.2. ABSTRACT AND FOCUS FOR EACH CHAPTER and are exposed o he risk of significan losses in he processes of making markes and managing heir opions porfolios. Assuming an incomplee marke governed by a sochasic volailiy facor in underlying asse price processes, we demonsrae ha he demand pressure for an opion conrac direcly impacs raded opion prices due o he covariance of he unhedgeable pars of a demanded opion and he oher raded opions. Whereas Gârleanu e al.[2009] also provide a resul which is similar o he fac saed above, bu hey assume ha he aggregae opion demand of end-users is provided exogeneously and independen of he preferences of marke-makers and end-users. In conras, as menioned by Green and Figlewski[1999], we assume ha he preference of marke-makers o he background risks induced by he ne demand of end-users affecs opion prices direcly and sudy he supply-demand balance for opion conracs by considering he preferences of boh marke-makers and end-users. Considering each of opimizaion problems for he represenaive marke-maker and he represenaive end-user independenly, we derive he equilibrium demand pressures for raded opion conracs and provide an explici represenaion for he pricing kernel in equilibrium as a funcion of he equilibrium demand pressures. Moreover, we provide some ineresing implicaions in he exisence of he variance risk premium and he shape of he implied risk aversion funcion wih he pricing kernel derived above. Chaper 4 In developing risk managemen sraegies for financial opion porfolios in incomplee markes, i is necessary o specify he risk facors in he markes and selec an opion pricing model which is consisen wih hose specified risks. In paricular, for he praciioners i is essenial o consider he maers menioned above for heir risk managemen processes. For his reason, in his chaper, we sudy he feaures on empirical opion prices and dela-hedged opion reurns in a sochasic volailiy environmen. A rich body of sudies on empirical opion prices and dela-hedged opion reurns in financial opion markes has developed in recen years wih some sylized empirical analyses. In paricular, Bakshi and Kapadia[2003] and Low and Zhang[2005] sudy dela-hedged opion reurns in a sock index opion marke and currency opion markes, respecively, and hey provide evidence of he exisence of he negaive sochasic variance risk premiums based on non-zero expeced dela-hedged opion reurns. While we also explore an empirical sudy on he exisence of he variance risk premiums in his chaper, our sudy is differen from ha explored by Bakshi and Kapadia[2003] in ha we explicily consider he effec of parameer esimaion risk on financial 13

16 1.2. ABSTRACT AND FOCUS FOR EACH CHAPTER opion prices. Theoreical models ofen assume ha he economic agen who makes an opimal financial decision knows he rue parameers of he model. Bu he rue parameers are rarely if ever known o he decision maker. In realiy, model parameers have o be esimaed based on hisorical informaion and, hence, he model s usefulness depends parly on how good he esimaes are. This gives rise o esimaion risk in virually all opion valuaion models. Considering he effec of parameer esimaion risk on financial opion prices, we provide a novel represenaion of dela-hedged opion reurns in a sochasic volailiy environmen. The represenaion of dela-hedged opion reurns provided in his chaper consiss of wo erms; volailiy risk premium and parameer esimaion risk. Based on he represenaion for dela-hedged opion reurns, we explore an empirical simulaion. Examining he dela-hedged opion reurns of he USD-JPY currency opions wih a hisorical simulaion from Ocober 2003 o June 2010, we find ha he dela-hedged opion reurns for OTM pu opions are srongly affeced by parameer esimaion risk as well as he volailiy risk premium, especially in he pos-lehman shock period. In paricular, we find ha approximaely 13 % of he value of he OTM currency opion premium is generaed by he exisence of parameer esimaion risk in he pos-lehman crisis period, and his effec induced by parameer esimaion risk on opion prices is more significan han he effec of he volailiy risk premium. One of he mos imporan implicaions of his chaper is ha he sign and he level of he expeced dela-hedged opion reurns do no necessarily explain he exisence of volailiy risk premiums. Chaper 5 Conclusion. 14

17 Chaper 2 The Skewness Risk Premium in Equilibrium and Sock Reurn Predicabiliy 2.1 Inroducion The concern wih he informaion conen in opion-implied disribuions has been growing for he las several years. In paricular, here has been a growing ineres in he informaion of he difference beween opion-implied and realized disribuions, which is usually recognized as a risk premium required by he represenaive agen, in erms of he financial risk managemen and asse pricing implicaions. In his chaper we invesigae he risk premiums in higher order momens, especially in he skewness, of financial asse reurns under a general equilibrium seing. In his sudy, each of he risk premiums in higher order momens of financial asse reurns is defined by he difference beween wo expeced values of he momen under he risk-neural and physical probabiliy measures, respecively. In recen years, here remains an ever-increasing ineres and challenge o develop an enirely self-conained equilibrium-based explanaion for he nonzero volailiy (or variance) risk premium 1 and is predicabiliy for sock index reurns. To he bes of our knowledge, he firs aemp o demonsrae he exisence of he volailiy risk premium based on a general equilibrium marke model is made by Eraker[2008]. Eraker[2008] develop an equilibrium explanaion for he volailiy risk premium based on he long-run 1 The volailiy (variance) risk premium is defined by he difference beween wo expeced values of he volailiy (variance) under he risk-neural and physical probabiliy measures, respecively. 15

18 2.1. INTRODUCTION risks (LRR) model 2 which emphasizes he role of long-run risks, ha is, low-frequency movemens in consumpion growh raes and volailiy, in accouning for a wide range of asse pricing puzzles. The LRR model feaures an Epsein and Zin[1989] uiliy funcion wih an invesor preference for early resoluion of uncerainy and conains (i) a persisen expeced consumpion growh componen and (ii) long-run variaion in consumpion volailiy. On he basis of he LRR model, Eraker[2008] sudies he volailiy risk premium hrough he framework of a general equilibrium model. In addiion o he developmen of an enirely self-conained equilibrium-based explanaion for he risk premiums in higher order momens, several academic sudies relaed o hose risk premiums are provided in recen years. For example, moivaed by fruiful implicaions from he LRR model pioneered by Bansal and Yaron[2004], Bollerslev, Tauchen, and Zhou[2009] invesigae he sock reurn predicabiliy of he variance risk premium in erms of a general equilibrium seing based on he LRR model framework. They show ha he difference beween implied and realized variaion, or he variance risk premium, is able o explain a nonrivial fracion of he ime-series variaion in pos aggregae sock marke reurns, wih high (low) premia predicing high (low) fuure reurns. The magniude of he predicabiliy is paricularly srong a he inermediae quarerly reurn horizon, where i dominaes ha afforded by oher popular predicor variables, such as he P/E raio, he defaul spread, and he consumpion-wealh raio. Drechsler and Yaron[2011] also show he predicabiliy of he variance risk premium for sock index reurns based on an exended LRR model wih jumps in uncerainy and he long-run componen of cash-flows. They demonsrae ha a risk aversion greaer han one and a preference for early resoluion of uncerainy correcly signs he variance risk premium and he coefficien from a predicive regression of reurns on he variance risk premium. All of he sudies cied above focus only on he variance risk premium required by a represenaive invesor due o he sochasic naure of asse reurn variance. Conversely, as far as we know, here are few repors abou he risk premium which compensaes for uncerainy of he hird momen, ha is, he skewness, of asse reurns. In his chaper, we demonsrae ha he skewness risk premium, defined by he difference beween wo expeced values of he skewness under he risk-neural and physical probabiliy measures, respecively, also capures aiudes oward economic uncerainy as well as he variance risk premium. Among recen sudies on self-conained equilibrium-based models for he nonzero variance risk premium referred above, all of he sudies excep for Drechsler and 2 The long-run risks model is pioneered by Bansal and Yaron[2004], which is a sylized self-conained general equilibrium model incorporaing he effecs of ime-varying economic uncerainy. 16

19 2.1. INTRODUCTION Yaron[2011] model he processes of boh he variance of consumpion growh rae and he LRR facor as condiional normal, so ha he one-sep-ahead condiional disribuion of he marke reurn is also normal and, as a resul, he skewness of ha disribuion is zero. Therefore, he models proposed by hose sudies can no explain he negaive risk-neural skewness, which is found by he previous sudies such as hose by Aï-Sahalia and Lo[1998] and Aï-Sahalia, Wang, and Yared[2001]. They documen several empirical feaures of he sae price densiy for he S & P500 index opion marke over ime, including he erm srucures of mean reurns, volailiy, skewness, and kurosis, ha are implied by opion-implied disribuions. In paricular, They show ha he nonparameric sae price densiies are negaively skewed, have faer ails and he amoun of skewness and kurosis boh increase wih mauriy. We show ha jump componens in he LRR facor and/or he variance of consumpion growh rae can explain he nonzero (or negaive) skewness of he one-sep-ahead asse reurn disribuion. To he bes of our knowledge, Drechsler and Yaron[2011] is he firs paper ha indicaes an imporan role for ransien non-gaussian shocks (jumps) o fundamenals such as he LRR-facor and he variance of consumpion growh rae for undersanding how percepions of economic uncerainy and cash-flow risk manifes hemselves in asse prices. However, he assumpion of an affine srucure on he jump inensiy process λ, ha is, λ = l 0 + l 1 σ 2 where l 0, l 1 > 0 and σ 2 is he variance of consumpion growh rae, in Drechsler and Yaron[2011] can no explain an empirical fac on a simulaneous relaion beween monhly sock reurns and monhly changes of he opion-implied skewness: r m,+1 = ISkew +1, (2.46) ( 3.46) r m,+1 = V IX ISkew +1, (2.1) (3.33) ( 16.00) ( 3.94) where r m,+1 is he monhly reurn of he S & P500 Toal Reurn Index from ime o + 1, V IX +1 is he monhly change of implied volailiy calculaed wih he CBOE s VIX from ime o + 1, and ISkew +1 is he monhly change of implied skewness calculaed wih he CBOE s Skew Index from ime o + 1. These resuls are obained based on he monhly daa from Jan-1990 o Aug Under he assumpion on he jump inensiy process in Drechsler and Yaron[2011], however, we can confirm ha he regression parameers o ISkew +1 in he above regression models should be posiive. In his chaper, we propose an exension of he LRR models developed by Bansal and Yaron[2004] and Drechsler and Yaron[2011]. Our model conains a rich se of ransien 17

20 2.1. INTRODUCTION dynamics and can quaniaively accoun for he ime variaion and asse reurn predicabiliy of he skewness premium as well as he variance risk premium. In paricular, we inroduce a sochasic jump inensiy for ransien jumps o fundamenals such as he LRR facor and he variance of consumpion growh rae, and show ha his addiional inroducion of a sochasic jump inensiy enables our model o capure he various empirical aspecs of he sock index reurns and is opion implied momens including he resul of (2.1). Chrisoffersen e al.[2012] find very srong suppor for ime-varying jump inensiies for S & P500 index reurns, and hey show ha, compared o he risk premium on dynamic volailiy, he risk premium on he dynamic jump inensiy has a much larger impac on opion prices. We find ha he exisence of he negaive skewness and he skewness risk premium have a close relaionship wih he exisence of he jumps and he jump risk premium, respecively. This chaper also shows ha he skewness of asse reurn disribuion and he skewness risk premium, which compensaes for he sochasic naure of he skewness, are boh ime-varying due o he sochasic naure of he jump inensiy for ransien jumps in boh he LRR facor and he variance of consumpion growh rae. Providing an equiy risk premium represenaion of a linear facor pricing model wih ime-varying variance and skewness risk premiums, we find ha hose risk premiums can explain a nonrivial fracion of he ime series variaion in he aggregae sock marke reurns and show an empirical evidence in which he skewness risk premium, as well as he variance risk premium, has superior predicive power for fuure aggregae sock marke index reurns. Compared wih he variance risk premium, he resuls show ha he skewness risk premium plays an independen and essenial role for predicing he marke index reurns. The remainder of his chaper is organized as follows. Secion 2 oulines he basic heoreical model wih jumps in consumpion growh rae and is volailiy, shows how equilibrium is derived for our model economy, and highlighs is key feaures. In paricular, we provide an equiy risk premium represenaion of a linear facor pricing model wih ime-varying variance and skewness risk premiums. Secion 3 provides he implicaions from a calibraed version of he heoreical equiy risk premium represenaion of a linear facor pricing model derived in Secion 2 o help guide and inerpre our subsequen empirical reduced form predicabiliy regressions. Secion 4 describes he daa used for examining he equiy risk premium represenaion empirically and discusses he resuls from he predicive regressions on he sock reurns o he variance and he skewness risk premiums wih hisorical daa. Secion 5 provides concluding remarks. 18

21 2.2. MODEL FRAMEWORK 2.2 Model Framework Model Seup and Assumpions The underlying environmen is a discree ime endowmen economy. The represenaive agen s preferences on he consumpion sream are of he Epsein and Zin[1989] form, allowing for he separaion of risk aversion and he ineremporal elasiciy of subsiuion (IES). Thus, he agen maximizes his lifeime uiliy, which is defined recursively as V = [(1 δ)c 1 γ θ ( ) 1 ] θ + δ E [V 1 γ θ 1 γ +1 ], (2.2) where C is consumpion a ime, 0 < δ < 1 reflecs he agen s ime preference, γ is he coefficien of risk aversion, θ = 1 γ, and ψ is he ineremporal elasiciy of subsiuion 1 1 ψ (IES). This preference srucure collapses o a sandard CRRA uiliy represenaion if γ = 1, ha is, θ = 1, and in his case, only innovaions o consumpion are priced. In ψ he following, based on he resul provided by Bansal and Yaron[2004] we assume ha boh γ and ψ are larger han one. I hen holds ha γ > 1, which implies θ < 0. ψ Wih his choice, he invesor has a preference for early resoluion of uncerainy. Then, no only consumpion risk is priced, bu sae variables carry risk premia, oo. parameer resricions also ensure ha he signs of he risk premia are in line wih economic inuiion, and ha a worsening of economic condiions leads o a decrease in asse prices. Uiliy maximizaion is subjec o he budge consrain: W +1 = (W C )R c,+1, where W is he wealh of he agen and R c, is he reurn on all invesed wealh. As shown in Epsein and Zin[1989], for any asse j, he firs-order condiion yields he following Euler condiion: The ] E [exp(m +1 + r j,+1 ) = 1, (2.3) where r j,+1 is he log of he gross reurn on asse j and m +1 is he log of he ineremporal marginal rae of subsiuion (IMRS), which is given by m +1 = θ log δ θ ψ c +1 + (θ ( 1)r c,+1. Here, r c,+1 is log R c,+1 and c +1 is he change in log C, ha is, log C+1 C ). We model consumpion and dividend growh raes, g +1 log( C +1 C ) and g d,+1 log( D +1 D ) where D is dividend a ime, respecively, as conaining a small persisen 19

22 2.2. MODEL FRAMEWORK predicable componen x, which deermines he condiional expecaion of consumpion growh, x +1 = ρ x x + ϕ e σ e +1 + J x,+1, g +1 = µ g + x + ϕ η σ η +1, g d,+1 = µ d + ρ d x + ϕ ζ σ ζ +1, (2.4) where ϕ e, ϕ η, ϕ ζ, ρ x, ρ d > 0, µ g, µ d R, e, η, and ζ are muually independen i.i.d.n(0, 1) processes, and J x,+1 is a compound-poisson process represened by J x,+1 N x +1 j=1 ɛj x where N x +1 is he Poisson couning process for ha jump componen whose he inensiy process is λ x,+1 l x λ +1, l x > 0, and ɛ j x i.i.d. N(0, δ 2 x), δ x > 0, is he size of he jump ha occurs upon he N x +1. Furhermore, we also model he dynamics of he volailiy as follows: σ 2 +1 = µ σ + ρ σ σ 2 + q w +1 + J σ 2,+1, q +1 = µ q + ρ q q + ϕ ξ q ξ +1, (2.5) where he parameers saisfy µ σ > 0, µ q > 0, ρ σ < 1, ρ q < 1, ϕ ξ > 0, and w and ξ are muually independen i.i.d.n(0, 1) processes and are independen of each of e, η, and ξ. J σ 2,+1 is a compound-poisson process, which is represened by J σ 2,+1 N σ j=1 ɛj σ where 2 N+1 σ2 is he Poisson couning process for ha jump componen whose he inensiy process is λ σ 2,+1 l σ 2λ +1, l σ 2 > 0, and ɛ j σ i.i.d. N(0, δ 2 2 σ ), δ 2 σ 2 > 0, is he size of he jump ha occurs upon he N σ2 +1. We assume ha N x +1 and N σ2 +1 are muually independen and ɛ j x and ɛ j σ 2 are oo. The sochasic variance process σ 2 represens ime-varying economic uncerainy in consumpion growh wih he variance-of-variance process q in effec inducing an addiional source of emporal variaion in ha same process. We also model he variance-of-variance process q in he same fashion as Bollerslev e al.[2009]. Imporanly, we inroduce he jump inensiy dynamics in he economy which is represened by he following discree-ime sochasic process, λ +1 = µ λ + ρ λ λ + ϕ u q (ρξ ρ 2 u +1 ), (2.6) where µ λ > 0, ρ λ < 1, ρ 1, and u is an i.i.d.n(0, 1) process, which is independen of each of e, η, ζ, w, and ξ. One of he noable feaures of our model seup is his inroducion for he jump inensiy process (2.6). Chrisoffersen e al.[2012] also find very srong suppor for imevarying jump inensiies for S & P500 index reurns, and hey show ha, compared o he risk premium on dynamic volailiy, he risk premium on he dynamic jump inensiy has

23 2.2. MODEL FRAMEWORK a much larger impac on opion prices. In he previous sudies, Drechsler and Yaron[2011] is he firs paper ha inroduces ransien jumps o fundamenals such as he LRR-facor x and he variance of consumpion growh rae σ 2. However, i assumes ha he jump inensiy process λ is represened by an affine srucure of λ = l 0 + l 1 σ 2 where l 0, l 1 > 0. As menioned in he inroducion of his chaper, such assumpion for he jump inensiy process can no explain he empirical fac of regression (2.1). We exend he LRR models of Bansal and Yaron[2004] and Drechsler and Yaron[2011] so as o inroduce a sochasic jump inensiy of (2.6) ino he economy. As shown in he following, his inroducion enables our model o have a consisency wih he empirical fac shown in (2.1) and plays a key role in describing he characerisics of asse reurn disribuions The Model Soluion in Equilibrium We disinguish beween he unobservable reurn on a claim o aggregae consumpion, R c,+1, and he observable reurn on he marke porfolio, R m,+1 : he laer is he reurn on he aggregae dividend claim. Solving our model numerically, we demonsrae he mechanisms working in our model via approximae analyical soluions in he same fashion as he previous sudies such as hose by Bansal and Yaron[2004], Bollerslev e al.[2009], Drechsler and Yaron[2011], ec. To derive hese soluions for our model, we use he sandard approximaion uilized in Campbell and Shiller[1988], r c,+1 = κ 0 + κ 1 v +1 v + g +1, (2.7) where lowercase leers refer o logs, so ha r c,+1 = log(r c,+1 ) is he coninuous reurn, v = log( P C ) is he log price-consumpion raio of he asse ha pays he consumpion endowmen, {C +i } i=1, and κ 0 and κ 1 are approximaing consans ha boh depend only on he average level of v 3. Analogously, r m,+l and v m,+1 correspond o he marke reurn and is log price-dividend raio and he similar approximaion presened below can also be derived: r m,+1 = κ 0,m + κ 1,m v m,+1 v m, + g d,+1. (2.8) The sandard soluion mehod for finding he equilibrium in a model like he one defined above hen consiss in conjecuring soluions for v and v m, as an affine funcion of he sae variables, x, σ 2, q, and λ, v = A 0 + A x x + A σ σ 2 + A q q + A λ λ, (2.9) 3 Noe ha κ 1 = exp( v) 1+exp( v) and his value is approximaely (cf) Bansal and Yaron[2004]), which is also consisen wih magniudes used in Campbell and Shiller[1988]. 21

24 2.2. MODEL FRAMEWORK v m, = A 0,m + A x,m x + A σ,m σ 2 + A q,m q + A λ,m λ, (2.10) respecively, solving for he coefficiens A 0, A x, A σ, A q, and A λ in v and for he coefficiens A 0,m, A x,m, A σ,m, A q,m, and A λ,m in v m,. Subsiuing (2.9) for (2.7), we have a emporal represenaion for r c,+1 wih he sae variables, x, σ 2, q, and λ, and furhermore, subsiuing his r c,+1 for he Euler equaion (2.3), we can derive an ideniy wih hose sae variables. Solving he ideniy in he same manner as Bansal and Yaron[2004], Bollerslev e al.[2009], Drechsler and Yaron[2011], ec., we can derive he equilibrium soluions for he four parameers as follows: A x = A σ = 1 2 γ 1 θ(κ 1 ρ x 1), (1 γ) 2 ϕ 2 η + θ 2 κ 2 1A 2 xϕ 2 e, θ(κ 1 ρ σ 1) A λ = 2 exp( 1 2 θ2 κ 2 1A 2 xδx) 2 exp( 1 2 θ2 κ 2 1A 2 σδσ) 2, θ(κ 1 ρ λ 1) A q is a soluion of he quadraic equaion presened below: θa q (κ 1 ρ q 1) + θ2 κ 2 [ ] 1 A 2 σ + A 2 2 qϕ 2 ξ + 2A q A λ ϕ ξ ϕ u ρ + A 2 λϕ 2 u = 0. Considering he expressions of (2.11), he following proposiion can be proven easily: Proposiion 1 If γ > 1 and ψ > 1, hen, A x > 0, A σ < 0, A q < 0, and A λ < 0. (2.11) The above proposiion suggess ha if he IES and risk aversion are higher han 1, a rise in each of he sae variables of σ 2, q, and λ lowers he price-consumpion raio. Having solved for r c,+1 wih he four parameers derived above, we can subsiue i (and c +1 = g +1 ) ino m +1 o obain an expression for he condiional innovaion o he log pricing kernel a ime + 1: m +1 E [m +1 ] = θ log δ θ ψ c +1 + (θ 1)r c,+1 E [ θ log δ θ ψ c +1 + (θ 1)r c,+1 ] = ( θ ψ + θ 1 ) ϕ η σ η +1 + (θ 1)κ 1 A x ϕ e σ e +1 + (θ 1)κ 1 A σ q w +1 + (θ 1)κ 1 (A q ϕ ξ + A λ ϕ u ρ) q ξ +1 + (θ 1)κ 1 A λ ϕ u 1 ρ 2 q u +1 + (θ 1)κ 1 A x (J x,+1 E [J x,+1 ]) + (θ 1)κ 1 A σ (J σ 2,+1 E [J σ 2,+1]) = Λ (G z +1 + J +1 E [J +1 ]), (2.12) 22

25 2.2. MODEL FRAMEWORK where ( Λ γ (1 θ)κ 1 A x (1 θ)κ 1 A σ (1 θ)κ 1 A q (1 θ)κ 1 A λ 0), ϕ η σ ϕ e σ G 0 0 q ϕ ξ q 0 0, ρϕ u q ϕ u 1 ρ 2 q 0 (2.13) ϕ ζ σ ( ) z +1 η +1 e +1 w +1 ξ +1 u +1 ζ +1, ( J +1 0 J x,+1 J σ 2, ), ( E [J +1 ] 0 E [J x,+1 ] E [J σ 2,+1] 0 0 0). Λ can be inerpreed as he price of risk for Gaussian shocks and also he sensiiviy of he IMRS o he jump shocks. From he expression of Λ, one can see ha he prices of risks are deermined by he A coefficiens, ha is, A x, A σ, A q, and A λ. The expression of Λ also shows ha he signs of he risk prices depend on he signs of he A coefficiens and (1 θ). In paricular, when γ = 1, θ = 1, and we are in he case of consan relaive risk ψ aversion (CRRA) preferences, i is clear ha only he ransien shock o consumpion z c,+1 η +1 is priced, and prices do no separaely reflec he risk of shocks o x (longrun risk), σ 2 (volailiy-relaed risk), q (variance-of-variance-relaed risk), and λ (jump inensiy-relaed risk). In he discussion and calibraions explored below, we especially focus on he case in which he agen s risk aversion γ and he IES ψ are boh greaer han 1, which implies ha A x > 0, A σ < 0, A q < 0, and A λ < 0 by he proposiion provided above. Thus, posiive shocks o long-run growh decrease he IMRS, while posiive shocks o he levels of he oher sae variables, σ 2, q, and λ, increase he IMRS. Noe ha in his case, since (1 θ) > 0, each of he A coefficiens has he same sign as he corresponding price of risk. To sudy he risk premiums in higher-order momens of he marke reurns, we firs need o solve for he marke reurn. A share in he marke is modeled as a claim o a dividend wih growh process given by g d,. To solve for he price of a marke share, we proceed along he same lines as for he consumpion claim and solve for v m,+1, he log price-dividend raio of he marke, by using he he conjecure (2.10), Campbell and 23

26 2.2. MODEL FRAMEWORK Shiller[1988]-approximaion (2.8), and he Euler equaion (2.3) 4. Wih he equilibrium soluions for he parameers of A x,m, A σ,m, A q,m, and A λ,m in (2.10), we can obain an expression for r m,+1 in erms of he sae variables and is innovaions by subsiuing he expression for v m,(+1) ino (2.8): where r m,+1 = κ 0,m + κ 1,m A 0,m + κ 1,m A σ,m µ d + κ 1,m A q,m µ g + κ 1,m A λ,m µ λ A 0,m + µ d + (κ 1,m A x,m ρ x A x,m + ρ d )x + (κ 1,m A σ,m ρ σ A σ,m )σ 2 + (κ 1,m A q,m ρ q A q,m )q + (κ 1,m A λ,m ρ λ A λ,m )λ + κ 1,m A x,m ϕ e σ e +1 + κ 1,m A σ,m q w +1 + κ 1,m (A q,m ϕ ξ + A λ,m ϕ u ρ) q ξ +1 + κ 1,m A λ,m ϕ u 1 ρ 2 q u +1 + ϕ ζ σ ζ +1 + κ 1,m A x,m J x,+1 + κ 1,m A σ,m J σ 2,+1 = r 0 +(B rf A m)y + B rg z +1 + B rj +1, (2.14) r 0 κ 0,m + (κ 1,m 1)A 0,m + (κ 1,m A σ,m + 1)µ d + κ 1,m A q,m µ g + κ 1,m A λ,m µ λ, B r κ 1,m A m + e d, A x,m 0 0 ρ x A m A σ,m A, e d 0 q,m 0, F 0 0 ρ σ ρ q 0 0, Y A λ,m ρ λ ρ d g x 2 σ q λ g d,. (2.15) Risk Premiums in Higher-Order Momens in Equilibrium Before proceeding o invesigaing he risk premiums in higher-order momens in equilibrium, we need o provide some furher explanaion on he jump dynamics and he feaures of he pricing kernel inroduced above. 4 Because he daails of he four parameers, A x,m, A σm, A q,m, and A λ,m, are insignifican and do no affec he discussion explored in he following a all, for simpliciy, we express he parameers, A x,m, A σm, A q,m, and A λ,m, as hey are and do no show explici represenaions of hose parameers in his sudy. 24

27 2.2. MODEL FRAMEWORK To handle he jumps, we inroduce some noaion. ψ k (u k ) = E[exp(u k ɛ k )] (k is x or σ 2 ) denoes he momen-generaing funcion (mgf) of he jump size ɛ k. The mgf for he jump componen of k, E[exp(u k J k,+1 )], hen equals exp(ψ,k (u k )), where Ψ,k (u k ) = λ k, (ψ k (u k ) 1). Ψ,k is called he cumulan-generaing funcion (cgf) of J k,+1 and is a very helpful ool for calculaing asse pricing momens. The reason is ha is n-h derivaive evaluaed a 0 equals he n-h cenral momen of J k,+1. Regarding he feaures of he pricing kernel, we can show wha described below in line wih Drechsler and Yaron[2011]. Le us se he Radon-Nikodym derivaive dq = M +1, dp E [M +1 ] where P is he physical probabiliy measure and Q is he risk-neural probabiliy measure M in our economy. From (2.12), we have +1 E [M +1 exp( Λ (G ] z +1 + J +1 )). Since z +1 and J +1 are independen, we can rea heir measure ransformaions beween P and Q separaely. As a consequence, Drechsler and Yaron[2011] show ha z +1 Q N( G Λ, I), (2.16) where I is he ideniy marix in R 6 6. Tha is o say ha, under Q, z +1 is sill a vecor of independen normals wih uni variances, bu wih a shif in he mean. For he case of J +1, we could also proceed by ransforming he probabiliy densiy funcion direcly. As guided in Drechsler and Yaron[2011], Proposiion (9.6) in Con and Tankov[2004] shows ha under Q, he J +1,k are sill compound Poisson processes, bu wih cgf given by ( Ψ Q,k (u ψk (u k Λ k ) ) k) = λ k, ψ k ( Λ k ) 1, (2.17) ψ k ( Λ k ) where k = x or k = σ 2 and Λ x denoes he price of risk for he LRR-facor x, ha is, (1 θ)κ 1 A x, and Λ σ 2 denoes he price of risk for he variance of consumpion growh rae, ha is, (1 θ)κ 1 A σ. (see (2.13)) In he following discussion, we use he facs menioned above o calculae he higher-order momens of he marke reurn and o invesigae he risk premiums in he momens. The Variance Risk Premium in Equilibrium According o Bollerslev e al.[2009] and Drechsler and Yaron[2011], he variance risk premium in equilibrium, vp, is defined by vp E Q [Var Q +1(r m,+2 )] E P [Var P +1(r m,+2 )], (2.18) where Var P +1 (Var Q +1) is he variance operaor under he physical (risk-neural) probabiliy measure a ime + 1. From (2.14), he condiional variance of he marke reurn 25

28 2.2. MODEL FRAMEWORK r m,+2 a ime + 1 under P can be obained as follows: where Var P +1(r m,+2 ) = BrG +1 G +1B r + Br 2 (i)var P +1(J i,+2 ) i = BrG +1 G +1B r + Br 2 Ψ (2) +1(0), B r = κ 1,m A m + e d ( (2.15)) ( B r (1) B r (2) B r (3) B r (4) B r (5) B r (6)) R 6, ( Br 2 Br 2 (1) Br 2 (2) Br 2 (3) Br 2 (4) Br 2 (5) Br (6)) 2 R 6, ( Ψ (2) +1(0) 0 Ψ (2) +1,x(0) Ψ (2) +1,σ (0) 0 0 0) R 6, 2 (2.19) and Ψ (2) +1,x(0) and Ψ (2) +1,σ 2 (0) are respecively he second derivaive of he cgf (cumulangeneraing funcion) for J x,+1 and J σ 2,+1 evaluaed a 0, ha is, Ψ (2) +1,x(0) 2 u 2 Ψ +1,x(u) u=0 = 2 u 2 λ x,+1(ψ x (u) 1) u=0, Ψ (2) +1,σ(0) 2 u 2 Ψ +1,σ 2(u) u=0= 2 u 2 λ σ 2,+1(ψ σ 2(u) 1) u=0. Thus he expression of (2.19) is rearranged o he following represenaion, Var P +1(r m,+2 ) = BrG +1 G +1B r + Br 2 Ψ (2) +1(0) ( ) = Br(H σ 2σ H q q +1 )B r + Br 2 diag ψ (2) (0) Π +1, (2.20) where ϕ 2 η ϕ 2 e H σ , H q ϕ 2 ξ ρϕ ξ ϕ u 0, ρϕ ξ ϕ u ϕ 2 u ϕ 2 ζ ψ (2) x (0) λ x,+1 ( ) diag ψ (2) (0) 0 0 ψ (2) σ (0) , Π +1 λ σ 2,

29 2.2. MODEL FRAMEWORK Under he risk-neural probabiliy measure Q, he condiional variance of he marke reurn r m,+2 a ime +1 also can be obained in he same manner demonsraed above. As a consequence, we can show he following proposiion based on he definiion of he variance risk premium (2.18). Proposiion 2 (The Variance Risk Premium in Equilibrium) In equilibrium, he variance risk premium a ime, vp, is linear o he variance-of-variance, q, and he jump inensiy, λ, and he represenaion of i is provided as follows: vp = β vp,c + β vp,q q + β vp,λ λ, (2.21) where [ ] β vp,c l x Br 2 (2)(ψ x (2) ( Λ x ) ψ x (2) (0)) + l σ 2Br 2 (3)(ψ (2) σ ( Λ 2 σ 2) ψ (2) σ (0)) µ 2 λ, ] β vp,q Br [Λ σ 2H σ 2 + ϕ ξ (ϕ ξ Λ q + ρϕ u Λ λ )H q B r ϕ u (ρϕ ξ Λ q + ϕ u Λ λ )(l x Br 2 (2)ψ x (2) ( Λ x ) + l σ 2Br 2 (3)ψ (2) σ ( Λ 2 σ 2)), β vp,λ BrH σ 2B r ψ (1) σ ( Λ 2 σ 2) [ ] + l x Br 2 (2)(ψ x (2) ( Λ x ) ψ x (2) (0)) + l σ 2Br 2 (3)(ψ (2) σ ( Λ 2 σ 2) ψ (2) σ (0)) ρ 2 λ. Proof See he Appendix. A number of ineresing implicaions arise from he expression (2.21). In paricular, any emporal variaion in he endogenously generaed variance risk premium is solely due o he variance-of-variance q and he jump inensiy λ. Moreover, provided ha θ < 0, Λ x > 0, and Λ σ 2 < 0, as would be implied by γ > 1 and ψ > 1, he facor loading o he jump inensiy, ha is, β vp,λ, is guaraneed o be posiive, bu ha o he variance-of-variance, ha is, β vp,q, can be boh posiive and negaive in general. However, if he correlaion beween he dynamics of he variance-of-variance and ha of he jump inensiy, ha is, ρ, is posiive, hen β vp,q is also guaraneed o be posiive due o he facs ha Λ q < 0 and Λ λ < 0. The Skewness Risk Premium in Equilibrium On he basis of he same manner used o derive he expression (2.20) in he previous subsecion, we can also derive he represenaions for he skewness of he marke reurn under P and Q, respecively, as follows: Skew P (r m,+1 ) = B 3 r diag(ψ (3) (0))Π, Skew Q (r m,+1 ) = B 3 r diag(ψ (3) ( Λ))Π, (2.22) 27