Rare Disasters and Risk Sharing with Heterogeneous Beliefs

Transcription

1 Rare Disasers and Risk Sharing wih Heerogeneous Beliefs Hui Chen Sco Joslin Ngoc-Khanh Tran January 31, 21 Absrac Alhough he hrea of rare economic disasers can have large effec on asse prices, difficuly in inference regarding boh heir likelihood and severiy provides he poenial for disagreemens among invesors. Such disagreemens lead invesors o insure each oher agains he ypes of disasers each one fears mos. Due o he highly non-linear relaionship beween consumpion losses in a disaser and he risk premium, a small amoun of risk sharing can significanly aenuae he effec ha disaser risk has on he equiy premium. The impac of risk sharing becomes sronger when he differences in beliefs ge larger or when he minoriy wealh holders in he economy have lower risk aversion. Our model implies a non-monoonic relaionship beween he equiy premium and he size of he disaser insurance marke. I also shows ha he equiy premium can someimes become lower as marke paricipans on average become more concerned wih disaser risk. Chen: MIT Sloan School of Managemen (huichen@mi.edu). Joslin: MIT Sloan School of Managemen (joslin@mi.edu). Tran: MIT Sloan School of Managemen (khanh@mi.edu). We hank Jakub Jurek, Leonid Kogan, Monika Piazzesi, and Jiang Wang for commens. All he remaining errors are our own.

2 1 Inroducion In his paper, we demonsrae how heerogeneous beliefs abou rare disasers affec asse prices and rading aciviies. Research by Riez (1988), Longsaff and Piazzesi (24), Barro (26) and ohers show ha he hrea of rare economic disasers ha cause severe losses in oupu and consumpion can have large impac on he equiy premium. However, wih a relaively shor sample of hisorical daa, i is difficul o esimae he likelihood of disasers or he size of heir impac, which suggess ha here is likely o be large heerogeneiy in he beliefs of marke paricipans abou disasers. We show ha such disagreemens can generae srong risk sharing moives among invesors and significanly affec asse prices in he equilibrium. We sudy an exchange economy wih wo ypes of agens, whose beliefs on disasers can differ in various ways. For example, one ype of agens can be more opimisic abou disaser risk han he oher. 1 These opimiss migh believe in a lower probabiliy of disasers (e.g., once every 1 years as opposed o once every 6 years), or hey migh hink he poenial loss of aggregae endowmen during a disaser is smaller. Alernaively, boh ypes of agens could be concerned abou disasers, excep ha one hinks disasers are large and rare, while he oher hinks hey are smaller bu more frequen. We assume ha markes are complee, so ha he agens can rade coningen claims and achieve opimal risk sharing. Thus, equilibrium asse prices depend on he beliefs and he disribuion of wealh among he agens. Our main finding is ha adding a second ype of agens wih differen beliefs abou disasers can cause he equiy premium o drop subsanially, even when he new agens only have a small amoun of wealh. This resul holds wheher he disagreemen is abou he inensiy or impac of disasers. Ineresingly, his resul is sill rue even when he new agens are generally more pessimisic abou disasers, so long as hey believe he disasers he original agens fear mos are relaively less likely. When we calibrae he beliefs of one agen using inernaional daa (from Barro (26)), he oher using only consumpion daa from he US (where disasers have been relaively mild), raising he fracion of oal wealh for he second agen from o 1% lowers he equiy premium from 4.4% o 2.%. The decline in he equiy premium becomes faser when he disagreemen is larger, or when he new agens also have lower risk aversion. 1 In he sense ha he disribuion of consumpion growh under one s beliefs firs-order sochasically dominaes ha of he oher s. 1

3 The key reasons behind his resul are he following: (1) he equiy premium derives almos enirely from jump (disaser) risk, (2) high prices for jump risk induce aggressive risk sharing, and (3) here is a highly nonlinear relaionship beween risk premium and disaser risk. Firs, in our economy, as is ypically he case in sandard power uiliy models, here is very lile compensaion for Brownian risk due o he low volailiy of consumpion and moderae levels of risk aversion. Consequenly, he equiy premium derives primarily from disaser risk, and he compensaion for bearing disaser risk mus be high. For example, if here is a single ype of disaser resuling in a 4% loss o he consumpion claim and he equiy premium due o disaser risk is 4%, hen he annual premium for a disaser insurance conrac ha pays $1 when disaser srikes mus cos 1 cens or more, regardless of he acual chance of payoff. Second, he high premium for disaser risk provides a srong moivaion for risk sharing when agens have differen beliefs abou disasers. In a benchmark example of our model, he pessimiss may be willing o pay up o 13 cens per $1 of insurance coverage, even hough he payoff probabiliy is only 1.7% under heir own beliefs, or.1% under he beliefs of he opimisic agens. Such high prices induce he opimiss o underwrie insurance conracs wih noional value up o 4% of heir oal wealh, despie he risk of losing 7% of heir consumpion if a disaser srikes. Third, he disaser risk premium is highly non-linear in he size of disasers, so ha small amouns of risk sharing may have significan effecs on risk premia. Since disasers are rare, in order for hem o have large impac on he risk premium, marginal uiliy in he disaser saes needs o rise subsanially as he size of he consumpion drop increases. Wih power uiliy, his is achieved by having marginal uiliy rise wih he log disaser size a an exponenial rae. As a resul, he equiy premium is sensiive o changes in he size of individual consumpion losses during a disaser. For example, if an agen (wih γ = 4) manages o reduce her consumpion loss in a disaser from 4% o 35%, he equiy premium she demands will fall by 4%! Thus, a small amoun of risk sharing beween agens wih heerogeneous beliefs is enough o significanly lower he equiy premium hey demand. I is imporan o poin ou ha he above mechanism does no require he new agens o be globally more opimisic abou disasers han he exising ones. The criical componen in he risk sharing mechanism is he exisence of minoriy invesors who believe ha he ypes of disasers he majoriy wealh holders fear mos are relaively less likely o occur. Alhough hese minoriy 2

4 invesors may fear oher disasers (perhaps even larger and/or more frequen ones), hey will sill be willing o share he disaser risk he majoriy wealh holders fear. Thus, heerogeneiy among agens may resul in a low equiy premium even if each would individually demand a high equiy premium when oher ypes of agens are no presen. Our model implies ha he marke risk premium can sill remain low when he majoriy of marke paricipans are concerned wih he risks of major disasers. Before a disaser srikes, he opimisic invesors gain wealh by selling disaser insurance, which gradually drives down he equiy premium. This occurs regardless of he rue likelihood of disasers; ha is, agens wih correc pessimisic beliefs abou disasers will lose wealh o agens wih incorrec beliefs in imes ouside of disasers. However, when a disaser occurs, hese opimiss will lose a large fracion of heir wealh, and heir risk sharing capaciy will be grealy reduced. As a resul, he equiy premium will jump up significanly. A number of oher ineresing resuls and predicions arise from our analysis. Firs, we show ha hisorical daa combined wih simple economic resricions can provide useful bounds on beliefs abou disasers. While sampling error pus igh resricions on he mean of consumpion growh, i leaves much more room for disagreemens abou he frequency of disasers. Second, we show ha he degree o which variaion in he disaser inensiy influences asse prices and risk premia crucially depends on he disribuion of wealh among agens wih differen beliefs. Moreover, alhough he wealh disribuion in our model is non-saionary, agens who are overly opimisic abou disasers are likely o survive and even gain wealh for long periods of ime. Third, our model predics ha he equiy premium is no necessarily increasing in invesors weighed average belief of disaser risk; he premium could become lower if he rise in he average perceived disaser risk is accompanied by larger disagreemens. Finally, similar o he link beween asse prices and he size of he marke for riskless lending in Longsaff and Wang (28), our model predics a non-monoonic relaionship beween he equiy premium and he size of he disaser insurance marke. This paper conribues o he disaser risk lieraure, which goes back o he work of Riez (1988). Barro (26, 29) has reinvigoraed his lieraure by providing inernaional evidence ha disasers have been frequen and severe enough o generae a large equiy premium. A series of recen sudies demonsrae ha disaser risk can also help mach a wide range of facs in 3

5 financial markes, including asse volailiy, reurn predicabiliy, corporae bond spreads, opion pricing, exchange raes, ec. Among hese sudies are Liu, Pan, and Wang (25), Gabaix (29), Wacher (29), Farhi and Gabaix (29), Marin (28), and ohers. The majoriy of hese sudies adop a represenaive-agen framework. The few excepions include Dieckmann and Gallmeyer (25), Baes (28), and Dieckmann (29). The paper closes o ours is Dieckmann (29), who also sudies a model of heerogeneous beliefs abou disasers under boh complee markes and incomplee markes. He only considers he case of log uiliy and consan disaser risk, where risk sharing has limied effecs on he equiy premium. Our paper builds on he lieraure of heerogeneous beliefs models. See Basak (25) for a survey. Recen developmens on heerogeneous beliefs and asse pricing include Kogan, Ross, Wang, and Weserfield (26), Buraschi and Jilsov (26), Yan (28), David (28), Dumas, Kurshev, and Uppal (29), Xiong and Yan (29), among ohers. Our main finding is relaed o he resuls of Kogan, Ross, Wang, and Weserfield (26), who show ha irraional raders can sill have large price impac when heir wealh becomes negligible. Our affine heerogeneous beliefs model provides a racable ye flexible framework, hrough he generalized ransform resuls of Chen and Joslin (29), o sudy he implicaions of general forms of heerogeneiy in beliefs abou disasers. In he special case wih consan disaser probabiliy, we derive closed form soluions for prices, risk premia, and porfolio posiions for he cases where relaive risk aversion γ > 1. We also provide explici parameer resricions for asse prices o be finie. We also compare our resuls o models of heerogeneous preferences. Among he works on his opic are Dumas (1989), Wang (1996), Chan and Kogan (22), and more recenly Longsaff and Wang (28). When agens risk aversions are differen, we show ha he effecs on he equiy premium are qualiaively similar o he case wih heerogeneous beliefs. Moreover, combining lower risk aversion wih opimisic beliefs can make he effecs of risk sharing on he equiy premium paricularly srong. The res of he paper is organized as follows. Secion 2 presens our model of heerogeneous agens and disasers. Secion 3 discusses how o bound he beliefs abou disasers. Secion 4 analyzes he effec of heerogeneous beliefs and risk sharing in a seing wih consan disaser risk. The resuls are hen exended o he seing wih ime-varying disaser risk in Secion 5. Secion 6 compares he resuls wih he model of heerogeneous risk aversion. Secion 7 concludes. 4

6 2 Model Seup We firs presen he resuls of he general model where agens have boh heerogeneous beliefs and preferences, and he disaser risk is ime-varying. Then we review he resuls for a special case wih homogeneous agens and consan disaser risk. 2.1 Disasers and Heerogeneous Agens We consider a coninuous-ime, pure exchange economy. There are wo agens (A, B), each being he represenaive of her own class. Agen A believes ha he aggregae endowmen is C = e cc +cd, where c c is he diffusion componen of log aggregae endowmen, which follows dc c = ḡd + σ c dw c, (1) where ḡ and σ c are he expeced growh rae and volailiy of consumpion wihou jumps, and W c is a sandard Brownian moion under agen A s beliefs. The erm c d jumps arrive wih sochasic inensiy λ, is a pure jump process whose dλ = κ( λ A λ )d + σ λ λ dw λ, (2) where λ A is he long-run average jump inensiy under A s beliefs, and W λ is a sandard Brownian moion independen of W c. The jumps c d have ime-invarian disribuion ν A. We summarize agen A s beliefs wih he probabiliy measure P A. Agen B believes ha he probabiliy measure is P B, which we shall suppose is equivalen o P A. 2 She may disagree abou he growh rae of consumpion wihou jumps, he likelihood of disasers or he disribuion of he severiy of disasers when hey occur. We assume ha he wo agens are aware of each ohers beliefs, bu noneheless agree o disagree. 3 Specifically, agen B s beliefs are characerized by he Radon-Nikodym derivaive η (dp B /dp A ), 2 Essenially, wo probabiliy measures are equivalen if hey agree on he se of impossible evens. 3 We do no explicily model learning abou disasers. Given he naure of disasers, such learning will likely be quie slow, and he main source of disagreemens will be he priors. 5

7 where η = e a+bcc I, (3) I = (bḡ + 12 )) ( λb b2 σ 2c + λ s λ A 1) ds, (4) for some consans b and λ B >, and a is a pure jump process whose jumps are coinciden wih he jumps in c d and have size ( λb dν B ) a = log λ A dν A. (5) Here, dνb dν A is a funcion of he disaser size, and reflecs he disagreemen abou he disribuion of disaser; dνb dν A will be large (small) for he ype of disasers ha agen B hinks are relaively more (less) likely han agen A. Inuiively, he Radon-Nikodym derivaive expresses he differences in beliefs beween he agens by leing agen B assign a higher probabiliy o hose saes where η is high. The erms a and bc c reflec B s poenial disagreemens regarding he likelihood of disasers and he growh rae of consumpion, respecively. For example, if b >, hen η is large in hose saes where c c is high, which is equivalen o agen B believing in a higher expeced growh rae of consumpion wihou jumps. Similarly, if B believes a disaser of a cerain size d is more likely han A does, eiher due o a higher inensiy in general or a higher probabiliy for disasers of size d condiional on a disaser occurring, hen η jumps up when such a disaser occurs. Finally, he inegral erm I ensures ha η is a maringale under measure P A. I follows from he specificaion of η in (3-5) ha, under agen B s beliefs, he expeced growh rae of consumpion wihou jumps is ḡ+bσ 2 c, a disaser occurs wih inensiy λ λ B λ A (he long run average inensiy is λ B ), and he disaser size disribuion is ν B (which is equivalen o ν A ). Now we see ha he jumps in η specified in (5) are deermined by he log likelihood raio for disasers of differen sizes under he wo agens beliefs. Wihin his seup, agen B no only can disagree wih A on he average frequency of disasers, bu also he likelihoods for disasers of differen magniude. Moreover, his seup also has he advanage of remaining wihin he affine family as X = (c c,c d,log η,λ ) follow a joinly affine process, so ha he equilibrium can be compued using he generalized ransform mehod in Chen and Joslin (29). 6

8 We assume ha he agens are infiniely lived and have consan relaive-risk aversion (CRRA) uiliy over life ime consumpion: U i (C i ) = E i [ e ρ i (Ci ) 1 γ ] i d, (6) 1 γ i for i = A, B. We also assume ha markes are complee and agens are endowed wih some fixed share of aggregae consumpion (θ A,θ B = 1 θ A ). The equilibrium allocaions can be characerized as he soluion of he following planner s problem, specified under he probabiliy measure P A, max C A, CB E A [ e ρ A (CA ) 1 γ ] A + 1 γ ζ e ρ B (CB ) 1 γb d, (7) A 1 γ B s.. C A + C B = C, (8) where ζ ζη is he sochasic Pareo weigh for agen B. The firs order condiions imply e ρ A (C A ) γ A = ζ e ρ B (C B ) γ B, (9) which ogeher wih he marke clearing condiion (8) gives he equilibrium consumpion allocaions: C A = f A (ˆζ )C, (1a) C B = (1 f A (ˆζ ))C. (1b) where ˆζ = e (ρ A ρ B ) C γ A γ B ζ, and f A is an implici funcion. The sochasic discoun facor under A s beliefs, M, is given by M = e ρ A (C A ) γ A = e ρ A f A (ˆζ ) γ A C γ A. (11) Then, we can solve for ζ hrough he life-ime budge consrain for one of he agens (see Cox and Huang (1989)), which is linked o he iniial allocaion of endowmen. Since our emphasis is on heerogeneous beliefs abou disasers, for he remainder of his secion we focus on he case where here is no disagreemen abou he disribuion of Brownian shocks, 7

9 and he wo agens have he same preferences. 4 In his case, b =, γ A = γ B = γ, ρ A = ρ B = ρ. The equilibrium consumpion share hen simplifies o f A ( ζ ) = ζ 1 γ. (12) From he definiion of a, we see ha as a disaser of size d occurs, ζ is muliplied by he likelihood raio λ A λ B dν B dν A (d). Thus, if agen B is more pessimisic abou a paricular ype of disaser (because λ B > λ A and/or dνb dν A (d) > 1), she will have a higher weigh in he planner s problem when such a disaser occurs, so ha her (relaive) consumpion increases. The equilibrium allocaions can be implemened hrough compeiive rading in a sequenialrade economy. We consider hree ypes of raded securiies: (i) a risk-free money marke accoun, (ii) a claim o aggregae consumpion, and (iii) a series (or coninuum) of disaser insurance conracs wih 1 year mauriy, which pay $1 on he mauriy dae if a disaser of size d occurs wihin he year. We can compue he insananeous riskfree rae from he sochasic discoun facor M, r = ρ + γḡ 1 2 γ2 σc 2 λ E e γ cd ) f ( ζ A e a f A ( ζ ) 1. (13) The price of he aggregae endowmen claim is P = [ E e ρτ M ] +τ C +τ dτ, (14) M which can be viewed as a porfolio of zero coupon consumpion claims. I can be shown ha he price/consumpion raio only depends on he sochasic weigh and he disaser inensiy: P = C h(λ, ζ ). (15) In he case where λ is consan, he price of he consumpion claim furher reduces o closed form soluions. Similarly, we can also compue he individual agens wealh as he prices of heir equilibrium consumpion sreams. See Appendix A for deails. 4 In Secion 6, we invesigae he case wih heerogeneous preferences. 8

10 In order for prices of he aggregae endowmen claim o be finie in he heerogeneous-agen economy, i is necessary and sufficien ha prices are finie under each agen s beliefs in a singleagen economy (see Appendix B for a proof). As we show in he appendix, finie prices require ha he following wo inequaliies hold: < κ 2 2σλ 2 (φp i (1 γ) 1), (16a) > κ λ iκ κ 2 + 2σλ 2(1 φp i (1 γ)) ρ + (1 γ)ḡ (1 γ)2 σc, 2 (16b) σ 2 λ where φ P i is he momen generaing funcion for he disribuion of jumps in endowmen ν i under measure P i. The firs inequaliy reflecs he fac ha he volailiy of he disaser inensiy canno be oo large relaive o he rae of mean reversion. I prevens he convexiy effec induced by he poenially large inensiy from dominaing he discouning. The second inequaliy reflecs he need for enough discouning o counerac he growh. The disaser insurances are priced similarly hrough he sochasic discoun facor. For he simple case of a single ype of disaser, we can compue he price of disaser insurance by considering he couning processes, N, which couns he number of disasers ha have occurred: M P DI = E [M +1 1 {N+1 >N }]. (17) In he case where λ is consan, his reduces o a simple expression. For he general case, we use he ransform analysis of Duffie, Pan, and Singleon (2). See Appendix A for deails. Finally, he risk premium for any securiy under agen i s beliefs (i = A,B) is he difference beween he expeced reurn under P i and under he risk-neural measure Q. In he case of he aggregae endowmen claim, he condiional equiy premium under agen i s beliefs is E P i [R e ] = γσc 2 + λi EP i [ R] λq EQ [ R], (18) where E m [ R] is he expeced reurn in a disaser under he measure m, λa = λ, and λ B = λ λ B λ A. The risk neural inensiy, λ Q E i [M /M ]λ i, is deermined by he expeced jump size of he sochasic discoun facor a he ime of a disaser, and is he same for boh agens under complee markes. The difference beween he las wo erms is he premium for bearing disaser risk. This 9

11 premium is large if he jump-risk premium, λ Q /λi, is large, and/or he expeced loss in reurn in a disaser is large, especially under he risk-neural measure. I immediaely follows ha he difference in equiy premium under he wo agens beliefs is E P A [R e ] E P B [R e ] = λ A E P A [ R] λ B E P B [ R], which depends on he differences in beliefs abou he disaser inensiy and he expeced reurn in a disaser. This difference will be small compared o he size of he equiy premium when he risk-neural inensiy λ Q is large relaive o he acual inensiy λ i. In he remainder of he paper, we repor he equiy premium relaive o he probabiliy measure of agen A, P A. One inerpreaion for picking P A as he reference measure is ha A has he correc beliefs, and we are sudying he impac of he incorrec beliefs of an opimis on asse prices. 2.2 Homogeneous agens and consan disaser risk When agens have he same preferences and beliefs abou disasers, and ha he disaser inensiy λ is consan, we recover he basic version of he represenaive agen disaser risk model. We now review his case before presening he resuls of he heerogeneous-agen model. The aggregae endowmen process is a special case of he process in Secion 2, where c d is now a pure jump process wih consan inensiy λ and momen generaing funcion (MGF) φ for he jump size disribuion. The sochasic discoun facor, M, is given by M = e ρ C γ. From he sochasic discoun facor we can compue he consan riskfree rae r = DM M = ρ + γḡ 1 2 γ2 σc 2 + λ(φ( γ) 1). (19) Addiionally, he sochasic discoun facor allows us o easily compue he risk neural dynamics, which faciliaes he compuaion and inerpreaion of excess reurns. Under he risk-neural measure, dc = (ḡ σ 2 c γ)d + σ cdw Q + dj Q, (2) where disasers arrive wih inensiy λ Q = φ( γ)λ and have disribuion wih momen generaing funcion φ Q (s) = φ(s γ)/φ( γ). When he riskfree rae and disaser inensiy are close o zero, he 1

12 risk-neural disaser inensiy λ Q can be approximaely viewed as he value of a one-year disaser insurance conrac ha pays $1 a + 1 when a disaser occurs beween and The risk adjusmens for he jumps are quie inuiive. If aggregae consumpion drops during a disaser, hen φ( γ) > 1 for γ >, so ha λ Q > λ, i.e. disasers occur more frequenly under he risk-neural measure. Moreover, he risk-adjused disribuion of jump size condiional on a disaser saisfies dv Q /dv = e γ c /φ( γ), which slans he probabiliies owards large negaive jumps, making severe disasers more likely. where The price of he claim o aggregae dividends is P = E [ e ρτ M ] +τ C +τ dτ = C M θ, θ = ρ (1 γ)ḡ 1 2 (1 γ)2 σ 2 c λ(φ(1 γ) 1). (21) Finally, he risk premium on he aggregae consumpion claim is ] E [R e ] = γσc [λ(φ(1) 2 + 1) λ Q (φ Q (1) 1). (22) The premium for disaser risk reflecs boh he increased likelihood of disasers under Q, λ Q (relaive o λ), and he increased severiy of losses for a given disaser under Q, φ Q (1) 1 (relaive o φ(1) 1). Imporanly, he premium rises exponenially wih he size of he consumpion drops. Thus, a small reducion in he consumpion exposure o disasers (especially he mos severe ones) can subsanially lower he equiy premium. This mechanism is key o he resuls of he heerogeneous agens model. 3 Bounds for Beliefs abou Disasers While he beliefs of individual agens abou consumpion growh and disasers are no direcly observable, hisorical consumpion daa ogeher wih simple economic resricions can provide guidance on how exreme hese beliefs could be. We define an admissible belief as one ha saisfies [ ] 5 The value of he disaser insurance is D 1 = e r +1 λ Q e λq (s ) ds. When r and λ Q are close o, D 1 λ Q. 11

13 .4 pessimism abou consumpion growh (%) Gaussian p value disaser p value p value disaser inensiy (%) Figure 1: Bounds for exreme beliefs. The graph plos he p-value for various disaser inensiies (boom axis) and mean growh raes of consumpion (op axis) based on 1 years of daa. he following condiions: (i) he belief canno be rejeced by he daa a a given significance level α (we choose α = 5%), and (ii) he price of consumpion claim under he belief is finie. In he quaniaive analysis ha follows, we will only consider beliefs ha are admissible. We firs use sampling error as a way o judge wha ypes of beliefs are plausible. Specifically, we consider wheher an agen wih a given null hypohesis abou eiher he growh rae of consumpion or he likelihood of disasers would be able o rejec he null using 1 years of hisorical daa. Figure 1 plos he p-value associaed wih differen beliefs abou he expeced growh rae ḡ in a Gaussian model (no disasers) and beliefs abou λ in a consan disaser risk model. The p-value in he Gaussian case is he probabiliy of observing he sample mean growh rae when he rue growh rae is x% lower, an amoun specified in he op axis. This p-value is compued based on he assumpion ha he volailiy of consumpion growh is σ c = 2%. I falls rapidly, reaching 1% when he expeced growh rae is jus.5% below he sample mean. Such a iny difference in beliefs abou he growh rae has essenially no impac on asse prices in sandard seings. 6 In conras, since he sampling error for disaser inensiy is significanly larger, here is more 6 Cecchei, Lam, and Mark (2) and Abel (22) discuss oher sources of disagreemen beyond sampling error, which could allow for more disagreemens and larger impac on asse pricing. Malmendier and Nagel (29) argue ha individual experiences of macro-economic oucomes can have long-erm effecs on heir preferences and beliefs. 12

14 room for heerogeneous beliefs abou disasers. We compue he p-value in his case assuming ha no disasers have occurred over he las 1 years (he disasers we consider laer in he paper are significanly more severe han hose observed in he US hisory). Then he p-value is he probabiliy of observing no disasers assuming he rue inensiy is consisen wih he agen s beliefs (boom axis). Figure 1 shows ha λ = 3% corresponds o a p-value of 5%. In he homogeneous-agen model, wih a relaive risk aversion γ = 4 and assuming a 4% drop of consumpion in a disaser, he equiy premium will rise from essenially o 8% when he disaser inensiy rises from o 3%, which demonsraes how powerful he differen beliefs abou λ can be for asse pricing. If we were o assume here was one disaser in he las 1 years (he Grea Depression), hen even higher values of λ will become admissible. For disasers ha rarely happen, consumpion daa provide lile informaion abou he size of heir poenial impac. However, we can sill obain resricions on he beliefs abou he jump size disribuion (and disaser inensiy) via he requiremen ha prices of he aggregae consumpion claim are finie. These condiions are given by (16a 16b) for he general case. If λ is consan, he condiions simplify o ρ (1 γ)ḡ 1 2 (1 γ)2 σc 2 λ(φ(1 γ) 1) >, which provides a bound on he momen generaing funcion of he disaser size disribuion for given preference parameers and disaser inensiy. The raher loose bounds on he likelihood of disasers and he disribuion of disaser size derived from sampling error and economic resricions highligh he relevance of heerogeneous beliefs abou disasers. Nex, we invesigae how such heerogeneous beliefs affec asse pricing. 4 Heerogeneous Beliefs: Consan Disaser Risk To cleanly demonsrae he effecs of heerogeneous beliefs and he risk-sharing mechanism, we firs keep he risk of rare disasers consan, i.e., λ = λ. We sar wih wo special examples of disagreemens abou rare disasers, one where agens disagree abou he frequency of disasers, he oher where hey disagree abou he size of disasers. We hen examine wha happens o asse prices when wo agens, boh believers of disaser risk, coexis in an economy. Finally, we calibrae our 13

15 .7.6 A. Equiy premium λ A = 1.7% λ A = 2.5% 8 7 B. Jump-risk premium λ A = 1.7% λ A = 2.5% E A [Re ].3.2 λ Q /λ A Agen B (opimis) wealh share Agen B (opimis) wealh share Figure 2: Disagreemen abou he frequency of disasers. Panel A plos he equiy premium under he pessimis s beliefs as a funcion of he wealh share of he opimis. Panel B plos he jump-risk premium for he pessimis. We consider wo ses of beliefs for he pessimis: λ A = 1.7% and λ A = 2.5%. model by exracing wo ses of beliefs abou disaser risk from he US and inernaional experiences of economic disasers. 4.1 Disagreemen abou he Frequency of Disasers In he firs example, we assume ha he disaser size is deerminisic, c d = d, and he wo agens only disagree abou he frequency of disasers (λ). We se d =.51 so ha he MGF φ( γ) in his model maches he calibraion of Barro (26) for γ = 4. I implies ha aggregae consumpion falls by 4% when a disaser occurs. 7 Agen A (pessimis) believes ha disasers occur wih inensiy λ A = 1.7% (once every 6 years), which is also aken from Barro (26). The remaining parameers are ḡ = 2.5%, σ c = 2%, and ρ = 3%. Agen B (opimis) believes ha disasers are much less likely, λ B =.1% (once every 1 years), bu she agrees wih A on he size of disasers as well as he Brownian risk in consumpion. She also has he same preferences as agen A. Figure 2 shows he condiional equiy premium and he jump-risk premium under he pessimis s beliefs. If all he wealh is owned by he pessimis, he equiy premium is 4.7%, and he riskfree 7 This value is higher han he average disaser size in Barro (26) due o he fac ha larger bu more rare jumps can have big impac on he MGF, especially when γ is large. 14

16 rae is 1.3%. Since he opimis assigns very low probabiliies o disasers, if she has all he wealh, he equiy premium is only.43% under her own beliefs, or.21% under he pessimis s beliefs. Thus, i is no surprising o see he premium fall when he opimis owns more wealh. However, he speed a which he premium declines in Panel A is impressive. When he opimisic agen owns 1% of he oal wealh, he equiy premium has fallen from 4.7% o 2.7%. When he wealh of he opimis reaches 2%, he equiy premium falls o jus 1.7%. We can derive he condiional equiy premium as a special case of (18), where he assumpion of consan disaser size helps simplify he expression: ( ) ( [R e ] = γσc 2 + λ Q λa λ A 1 h( ζ e a d)e d h( ζ ) E P A 1 ). (23) The firs erm γσc 2 is he sandard compensaion for bearing Brownian risk. Heerogeneiy has no effec on his erm since he agens agree abou he brownian risk. Given he value of risk aversion and consumpion volailiy, his erm has negligible effec on he premium. The second erm reflecs he compensaion for disaser risk. I can be furher decomposed ino hree facors: (i) he consan disaser inensiy λ A, (ii) he jump-risk premium λ Q /λa, and (iii) he reurn of he consumpion claim in a disaser. How does he wealh disribuion affec he jump-risk premium? From he definiion of he sochasic discoun facor M and he risk-neural inensiy λ Q, i is easy o show λ Q /λa = e γ ca, where c A is he jump size of he equilibrium log consumpion for agen A in a disaser, which could be very differen from he jump size in aggregae endowmen due o rading. Wihou rading, as is he case when agen A has all he wealh, c A = d, which generaes a jump-risk premium of 7.7. We have shown earlier ha λ Q is approximaely he premium of a one-year disaser insurance. Thus, wihou any risk sharing, he pessimis will be willing o pay an annual premium of 13 cens for $1 of proecion agains a disaser even ha occurs wih probabiliy 1.7%. Since he opimis views disasers as very unlikely evens, she is willing o rade away heir claims in he fuure disaser saes in exchange for higher consumpion in normal imes. For example, she will find selling an $1 disaser insurance and collecing a 13 cens premium a lucraive 15

17 rade. Such a rade helps reduce he pessimis s consumpion loss in a disaser c A, which in urn lowers he jump-risk premium. However, he opimis s capaciy for underwriing such insurance is limied by her wealh, as she needs o ensure ha her consumpion/wealh is posiive in all fuure saes, including when a disaser occurs (no maer how unlikely such an even is). In fac, she says away from his limi imposed by he wealh consrain because he more disaser insurance she sells, he more her consumpion falls in he disaser saes, which makes her less willing o ake on addiional disaser risk. The more wealh he opimis has, he more disaser insurance she is able o sell wihou making her consumpion oo risky when a disaser srikes. The above mechanism can subsanially reduce he disaser risk exposure of he pessimis in equilibrium. Panel B of Figure 2 shows ha he jump-risk premium falls rapidly. When he opimis owns 2% of oal wealh, he jump-risk premium drops o 4.2. According o equaion (23), such a drop in he jump-risk premium alone will cause he equiy premium o fall by more han half o 2.2%, which accouns for he majoriy of he change in he premium (from 4.7% o 1.7%). Besides he jump-risk premium, he equiy premium also depends on he reurn of he consumpion claim in a disaser, which in urn is deermined by he consumpion loss and changes in he price-consumpion raio. Following a disaser, he riskfree rae drops as he wealh share of he pessimis rises. Wih CRRA uiliy, he lower ineres rae effec can dominae ha of he rise in he risk premium, leading o a higher price-consumpion raio. 8 Since a higher price-consumpion raio parially offses he drop in aggregae consumpion, i makes he reurn less sensiive o disasers, which will conribue o he drop in equiy premium. However, our decomposiion above shows ha he reducion of he jump-risk premium (due o reduced disaser risk exposure) is he main reason behind he fall in premium. Can we counerac he effec of he opimisic agen and resore he high equiy premium by making he pessimis even more pessimisic abou disasers? The dash-lines in Figure 2 plo he resuls when agen A believes ha λ = 2.5% (everyhing else equal), which according o Figure 1 is sill admissible (wih p-value of 8%). The resuls are sriking. While he equiy premium becomes significanly higher (6.8%) when he pessimis owns all he wealh in he economy, i falls o 4.1% wih jus 2% of oal wealh allocaed o he opimis (already lower han he previous case wih λ A = 1.7%), and is below 1% when he wealh of he opimis exceeds 8.5%. As he wealh share 8 Wacher (29) also finds a posiive relaion beween he price-consumpion raio and he equiy premium in a represenaive agen rare disaser model wih ime-varying disaser probabiliies and CRRA uiliy. 16

18 of he opimis grows higher, he premium can even become negaive. The decline in he jump-risk premium is sill he main reason behind he lower equiy premium. For example, when he opimis has 1% of oal wealh, he jump-risk premium falls o 4., which will drive he premium down o 3.1% (6% of he oal fall). The reason ha he equiy premium and he jump-risk premium decline faser is ha he amoun of risk sharing becomes larger as he beliefs of he wo agens become more differen, which can quickly dominae he heighened fear of he pessimis. This comparaive saic exercise has an imporan implicaion. When holding he average belief consan (weighed by wealh share), larger disagreemen beween he wo agens can drive he equiy premium lower. Thus, he equiy premium may no be increasing in he average belief of disaser risk in he marke. To beer examine he risk sharing mechanism beween agens, we compue heir porfolio posiions in he aggregae consumpion claim, disaser insurance, and he money marke accoun. Calculaing hese porfolio posiions amouns o finding a replicaing porfolio ha maches he exposure o Brownian shocks and jumps in he individual agens wealh processes. Appendix A provides he deails. The firs hing o noice is ha each agen will hold a consan proporion of he consumpion claim. Inuiively, his is because hey agree on he brownian risk and share i proporionally. Disagreemen over disaser risk is resolved hrough rading in he disaser insurance marke, which is financed by he money marke accoun. We plo he noional value of he disaser insurance sold by he opimis as a fracion of her oal wealh in Panel A of Figure 3. The dash-line is he maximum amoun of disaser insurance (as a fracion of her wealh) he opimis can sell subjec o her budge consrain. When he opimis has very lile wealh, he noional value of he disaser insurance she sells is abou 35% of her wealh. This value iniially rises and hen falls as he opimis gains more wealh. The reason is ha when he opimis has lile wealh, he pessimis has grea demand for disaser insurance and is willing o pay a high premium, which induces he opimis o sell more insurance relaive o her wealh. As he opimis ges more wealh, risk sharing improves, and he premium on he disaser insurance falls, so ha he opimis becomes less aggressive in underwriing he insurance. By comparing he acual amoun of rading o is limi, we can judge wheher he risk sharing in equilibrium is oo exreme. A is peak, he amoun of disaser insurance sold by he opimis is abou half of he maximum amoun ha she can underwrie while sill keeping her wealh posiive 17

19 Pc of opimis wealh A. Disaser insurance 2 opimal maximum feasible Pc of oal wealh B. Disaser insurance marke opimal maximum feasible C B /C C. Opimis consumpion share opimis no rade C i /C i (%) D. Consumpion change in disaser 4 opimis 2 pessimis aggregae Agen B (opimis) wealh share Agen B (opimis) wealh share Figure 3: Risk sharing. Panel A and B plo he oal noional value of disaser insurance relaive o he wealh of he opimis and oal wealh in he economy. Panel C plos he consumpion share for he opimis in equilibrium. Panel D compares he wo agens consumpion drops in a disaser wih ha of he aggregae endowmen. These resuls are for he case λ A = 1.7%. wih probabiliy 1, which migh appear reasonable. The cavea is ha, in realiy, underwriers of disaser insurance will likely be required o collaeralize heir promises o pay in he disaser saes. According o he model, all he wealh is from he claim on fuure endowmen income, which may no be used as collaeral (jus as labor income canno be used as collaeral). We will revisi he issue of marke incompleeness laer. Panel B plos he size of he disaser insurance marke (he oal noional value normalized by oal wealh). Naurally, he size of his marke is zero when eiher agen has all he wealh, and he marke is he bigges when wealh is closer o be evenly disribued. A is peak, he noional value of he disaser insurance marke is abou 16% of he oal wealh of he economy. Noice ha 18

20 he model generaes a non-monoonic relaion beween he size of he disaser insurance marke and he equiy premium. The premium is high when here is a lo of demand for disaser insurance bu lile supply, and is low when he opposie is rue. In eiher case, he size of he disaser insurance marke will be small. Panel C plos he equilibrium consumpion share of he opimis for differen wealh disribuions. The 45-degree line corresponds o he case of no rading. The opimis s consumpion share is above he 45-degree line, especially when her wealh is small, suggesing ha she is consuming a larger share of oal consumpion han her endowmen in he non-disaser saes. However, he price for geing more o consume in normal imes is more exposure o he fall in consumpion when disaser srikes, which is eviden in Panel D. A sign of how aggressive he opimis is in being agains disaser risk is ha, when she has lile wealh, she will suffer a 7% loss in consumpion in he even of a disaser (compared o 4% drop in aggregae consumpion). As for he pessimis, he less wealh she possesses, he more disaser insurance she buys relaive o her wealh. This will gradually lower her disaser risk exposure, and can evenually urn he disaser insurance ino a speculaive posiion her consumpion can jump up as high as 2% in a disaser. This overinsurance explains why he equiy premium under he pessimis s beliefs can urn negaive when he opimis has mos of he wealh. If we make agen A s beliefs more pessimisic (e.g. λ A = 2.5%), she will pay more for disaser insurance, which presens a beer rading opporuniy for agen B. Naurally, he amoun of disaser insurance sold (boh relaive o he wealh of he opimisic agen and o oal wealh in he economy) becomes higher han he case of milder pessimism, and he equilibrium consumpion shares will become more nonlinear. As a resul, he pessimis s consumpion loss in a disaser will be reduced a a faser rae (especially near he lef boundary), which acceleraes he fall in he equiy premium. A final quesion for his example is wheher he effec of risk sharing on he equiy premium becomes sronger or weaker as he size of disaser increases. On he one hand, for larger disasers, he equiy premium becomes more sensiive o changes in he size of consumpion drops, which means he premium will decline more for he same amoun of risk sharing beween he agens. On he oher hand, he opimis will be increasingly relucan o ake on exra losses in he disaser sae because her marginal uiliy rises exponenially in he (log) size of consumpion losses. To 19

21 .5.4 A. Equiy premium more disagreemen less disagreemen 8 7 B. Jump-risk premium more disagreemen less disagreemen.3 6 E A [Re ].2 λ Q /λ A Agen B (opimis) wealh share Agen B (opimis) wealh share Figure 4: Disagreemen abou he size of disasers. The lef panel plos he equiy premium under he pessimis s beliefs. The righ panel plos he jump-risk premium for he pessimis. In he case wih more disagreemen, he pessimis (opimis) assigns 99% probabiliy o he big (small) disaser, condiional on a disaser occurring. Wih less disagreemen, he probabiliy assigned o big (small) disaser drops o 9%. sudy he ne effecs, we increase he size of disasers, bu keep he risk premium for he pessimis in he single-agen and he relaive difference in beliefs unchanged (by lowering λ A and keeping λ B /λ A fixed). Our resuls (no repored) show ha he second effec dominaes. The decline in equiy premium becomes closer o linear as d ges larger (in absolue value), and he amoun of risk sharing becomes smaller. 4.2 Disagreemen abou he Size of Disasers The second example we sudy is on disagreemen abou he disribuion of disaser size. For simpliciy, we assume ha he drop in aggregae consumpion in a disaser follows a binomial disribuion, wih he possible drops being 1% and 4%. Boh agens agree on he inensiy of a disaser (λ = 1.7%). Agen A (pessimis) assigns a 99% probabiliy o a 4% drop in aggregae consumpion, hus having essenially he same beliefs as in he previous example. On he conrary, agen B (opimis) only assigns 1% probabiliy o a 4% drop, bu 99% probabiliy o a 1% drop. The res of he parameer values are he same as in he firs example. Figure 4 (solid lines) plos he condiional equiy premium and jump-risk premium under he 2

22 pessimis s beliefs. When he pessimis has all he wealh, he equiy premium is 4.6% (almos he same as in he firs example). Again, he equiy premium falls rapidly as we sars o shif wealh o he opimis. The premium falls by almos half o 2.4% when he opimis owns jus 5% of oal wealh, and becomes 1.4% when he opimis s share of oal wealh grows o 1%. Similarly, he jump-risk premium falls from 7.6 o 4.5 wih he opimis s wealh share reaching 1%, which by iself will lower he premium o 2.4%. These resuls show ha, in erms of asse pricing, inroducing an agen who disagrees abou he severiy of disasers is similar o having one who disagrees abou he frequency of disasers. Even hough he wo agens agree on he inensiy of disasers in general, hey acually srongly disagree abou he inensiy of disasers of a specific magniude. For example, under A s beliefs, he inensiy of a big disaser is 1.7% 99% = 1.68%, which is 99 imes he inensiy of such a disaser under B s beliefs. The opposie is rue for small disasers. Thus, B will aggressively insure A agains big disasers, while A insures B agains small disasers. For agen A, he effec of he reducion in consumpion loss in a big disaser dominaes ha of he increased loss in a small disaser, which drives down he equiy premium exponenially. Such rading can also become speculaive when B has mos of he wealh: agen A will ake on so much loss in a small disaser ha he jump-risk premium rises up again. Naurally, we expec ha he agens will be less aggressive in rading disaser insurances when here is less disagreemen on he size of disasers, and ha he effec of risk sharing on he risk premium will become smaller. The case of less disagreemen in Figure 4 confirms his inuiion. In his case, we assume ha he wo agens assign 9% probabiliy (as opposed o 99%) o one of he wo disaser sizes. While he equiy premium sill falls rapidly near he lef boundary, he pace is slower han in he previous case. Similarly, we see a slower decline in he jump-risk premium. 4.3 When Two Pessimiss Mee The examples we have considered so far have one common feaure: he new agen we are bringing ino he economy has more opimisic beliefs abou disaser risk, in he sense ha he disribuion of consumpion growh under her beliefs firs-order sochasically dominaes ha of he oher s, and ha he equiy premium is significanly lower when she owns all he wealh. However, he key o generaing aggressive risk sharing is no ha he new agen demands a lower equiy premium, 21

23 .6 A. Equiy premium 12 B. Jump-risk premium.4 1 E A [Re ].2 λ Q /λ A Consumpion change (%) C. Small disaser agen A agen B aggregae Agen B wealh share Consumpion change (%) D. Big disaser agen A agen B aggregae Agen B wealh share Figure 5: When Two Pessimiss Mee. Panel A and B plo he equiy premium and jump-risk premium under agen A s beliefs. Panel C and D plo he individual consumpion changes in small and big disasers. bu ha she is willing o insure he majoriy wealh holders agains he ypes of disasers ha hey fear mos. In order o highligh his insigh, le s consider he following example, where boh agens believe ha disaser risk accouns for he majoriy of he equiy premium. The key difference in heir beliefs is ha one agen believes ha disasers are rare bu big, while he oher hinks disasers are more frequen bu less severe. Specifically, we assume ha disasers can cause aggregae consumpion drops of a 3% or 4%. Agen A believes ha λ A = 1.7%, and assigns 99% probabiliy o he bigger disaser. B believes ha λ B = 4.2%, and assigns 99% probabiliy o he smaller disaser. By hemselves, he wo agens boh demand high equiy premium. We have chosen λ B so 22

24 ha, under he beliefs of agen A, he equiy premium is 4.6% wheher A or B has all he wealh. However, hey have significan disagreemen on he exac magniude of he disaser. For example, agen A believes ha he inensiy of he big disaser is 1.68%, while B believes ha he inensiy is only.4%. Such disagreemen generaes a lo of demand for risk sharing. As we see in Panel A of Figure 5, he condiional equiy premium falls rapidly as he wealh share of agen B moves away from he wo boundaries. In fac, he premium will be below 2% when B owns beween 9% and 99% of oal wealh. In Panel B, he jump risk premium also falls by half from 7.6 and 1 on he wo boundaries when B s wealh share moves from % and 1% o 25% and 91%, respecively. To ge more informaion on he risk sharing mechanism, in Panel C and D we examine he equilibrium consumpion changes for he individual agens during a small or big disaser. Since agen A assigns a low probabiliy o he small disaser, she insures agen B agains his ype of disasers. As a resul, her consumpion loss in such a disaser exceeds ha of he aggregae endowmen (-3%), and i increases wih he wealh share of agen B. When B has almos all he wealh in he economy, agen A sells so much small disaser insurance o B ha her own consumpion can fall by as much as 82% when such a disaser occurs. As a resul, agen B is able o reduce her risk exposure o small disasers significanly. In fac, her consumpion acually jumps up in a small disaser when she owns less han 75% of oal wealh, someimes by over 1% (when her wealh share is small). The opposie is rue in Panel D. As agen B insures A agains big disasers, she experiences bigger consumpion losses in such a disaser han he aggregae endowmen (-4%). The equilibrium consumpion changes of he wo agens are less exreme compared o he case of small disasers, which is due o wo reasons. Firs, he relaive disagreemen on big disasers is smaller han on small disasers. Second, he insurance agains larger disasers is more expensive, so ha agen A s abiliy o purchase disaser insurance is more consrained by her wealh. We can ake he insigh from his example one sep furher. Suppose he new agen added ino he economy is even more pessimisic abou disaser risk han he majoriy wealh holder. The new agen assigns higher probabiliies o more severe disasers, so ha she would demand a higher equiy premium on her own. However, he equiy premium will sill decline rapidly when we allocae a small amoun of wealh o he new agen, because despie her pessimism, she will be able o insure he old agen agains he smaller disasers. 23

25 4.4 Calibraing Disagreemen: Is he US Special? Having considered a series of special examples of heerogeneous beliefs, we now exend he analysis o a more realisic model of beliefs on disasers. The way we calibrae he beliefs of he wo ypes of agens is as follows. Agen A believes ha he US is no differen from he res of he world in is disaser risk exposure. Hence her beliefs are calibraed using cross-counry consumpion daa. Agen B, on he oher hand, believes ha he US is special. She forms her beliefs on disaser risk using only he US consumpion daa. An imporan conribuion of Barro (26) is o provide deailed accouns of he major consumpion declines cross 35 counries in he wenieh cenury. Raher han direcly using he empirical disribuion from Barro (26), we esimae a runcaed Gamma disribuion for he log jump size from Barro s daa using maximum likelihood (MLE). 9 Our esimaion is based on he assumpion ha all he disasers in he sample were independen, and ha he consumpion declines occurred insanly. 1 We also bound he jump size beween 5% and 75%. In comparison, he smalles and larges declines in per capial GDP in Barro s sample are 15% and 64%, respecively. The disaser inensiy under A s beliefs is sill λ A = 1.7%. The remaining parameers are: he mean growh rae and volailiy of consumpion wihou a disaser, ḡ = 2.5% and σ c = 2%, which are consisen wih he US consumpion daa pos WWII. As for agen B, we assume ha she agrees wih he values of ḡ and σ c, bu we esimae he runcaed Gamma disribuion of disaser size using MLE from annual per-capia consumpion daa in he US Over he sample of 119 years, here are hree years where consumpion falls by over 5%. Thus, we se λ B = 3/119 = 2.5%. Alernaively, we can also joinly esimae λ B and he jump size disribuion. Panel A of Figure 6 plos he probabiliy densiy funcions of he log jump size disribuions for he wo agens, which are very differen from each oher. The solid line is he disribuion fied o he inernaional daa on disasers. The average log drop is.36, which is equivalen o 3% drop 9 The runcaed Gamma disribuion has PDF f(d; α, β d min, d max) = f(d; α, β)/(f(d max; α, β) F(d min; α, β)), where f(x; α, β) and F(x;α, β) are he PDF and CDF of he sandard Gamma disribuion wih shape parameer α and scale parameer β. 1 These assumpions are debaable. For example, many of he major declines cross European counries are in WWI and WWII. Moreover, many of he declines spanned several years. See Donaldson and Mehra (28) for more discussions on he issue of observaion frequency. 11 The daa is aken from Rober Shiller s web sie hp:// shiller/daa.hm 24