What is the Impact of Stock Market Contagion on an Investor s Portfolio Choice?

Transcription

1 What s the Impact of Stock Market Contagon on an Investor s Portfolo Choce? Ncole ranger Holger Kraft Chrstoph Menerdng Ths verson: prl 29, 2008 Fnance Center Münster, Westfälsche Wlhelms-Unverstät Münster, Unverstätsstr , D Münster, Germany. Emal: Ncole.ranger@ww.un-muenster.de. Fnance Department, Goethe Unversty, D Frankfurt am Man, Germany. E-mal: holgerkraft@fnance.un-frankfurt.de. Fnance Center Münster, Westfälsche Wlhelms-Unverstät Münster, Unverstätsstr , D Münster, Germany. Emal: Chrstoph.Menerdng@ww.un-muenster.de.

2 What s the Impact of Stock Market Contagon on an Investor s Portfolo Choce? Ths verson: prl 29, 2008 bstract Stocks are exposed to the rsk of sudden downward jumps, and a crash n one stock or ndex may ncrease the rsk of a crash for other stocks or ndces. Ths may have a crucal mpact on nvestors portfolo choces, snce t reduces ther ablty to dversfy ther portfolos. llowng the economy to be n ether of two regmes, agon, we explctly take agon rsk nto account and study ts mpact on the portfolo decson of a CRR nvestor both n a complete and n an ncomplete market. We fnd that the nvestor sgnfcantly adjusts hs portfolo when agon s more lkely to occur. Capturng the tme dmenson of agon,.e. the tme dfference between the downward jump n the frst and n the second stock, s thus of frst-order mportance when analyzng portfolo decsons. n nvestor gnorng agon completely or accountng for agon whle gnorng ts tme dmenson suffers a large and economcally sgnfcant utlty loss. Ths loss s larger n a complete than n an ncomplete market, and the nvestor mght be better off f he does not trade dervatves at all. Keywords: sset llocaton, Jumps, Contagon, Model Rsk JEL: G12, G13

3 1 Introducton and Motvaton The noton of agon n fnancal markets refers to a phenomenon where losses n one asset, one asset class, or one country ncrease the rsk of subsequent losses n other assets, other asset classes, or other countres. Contagon may arse due to economc relatons, e.g. when one frm s the man customer of another frm, due to the exposure to a common macroeconomc rsk factor, or due to psychologcal reasons, when e.g. problems for one fnancal nsttuton ncrease the rsk of a bank run for other fnancal nsttutons. One example for such a stuaton s the recent subprme crss that has been threatenng the fnancal markets all over the world: When real estate prces n the US started to decrease, homeowners who had borrowed heavly aganst the equty n ther homes were suddenly realzng that they could no longer afford to keep up ther mortgage payments. n estmate from December 2007 states that subprme borrowers wll probably default on 220 bllon 450 bllon of mortgages. 1 Ths threat has had a sgnfcant effect on the markets for structured credt racts lke Collateralzed Debt Oblgatons CDOs leadng to huge losses that the banks have now started to report. ll along the way, the fear has extended nto equty markets: Fears about an end to the leveraged buy-out boom trggered heavy sellng of global equtes yesterday, leadng to the FTSE 100 s worst one-day slde for more than four years.... The FTSE 100 fell more than 200 ponts, or 3.2%, to 6.251,2; ts bggest drop snce March 2003 n the run-up to the Iraq war.... y early afternoon n New York, the Dow Jones Industral verage was down more than 300 ponts, or 2.4%. FT, July 27, 2007 In ths sort of clmate t s all about sentment, not about the numbers at all, and sentment at present s all about fear and nervousness, sad Kevn Gardner, head of global equty strategy at HSC. WSJ, July 27, 2007 or as catchly summarzed: The grevous experence of two centures of fnancal busts s that when the bankng system s n dffcultes the mess spreads. Economst, Dec 19, 2007 These examples show how losses n one part of the economy or n one country can spread out nto other parts of the economy or other countres. 1 See Economst, Dec

4 Our paper concentrates on agon effects occurrng n stock markets. We study the optmal portfolo decson of an nvestor who s exposed to these effects. The stocks n our economy follow a jump-dffuson process where the jumps are downward jumps. Contagon s bult nto the model by allowng for a dependence between these downward jumps of the stock prces. Das and Uppal 2004 study an economy where downward jumps n stock prces always happen smultaneously. The dependence between stocks s thus drven by the perfect correlaton of the jumps. We generalze ths dea and focus on the probabltes that jumps happen. More precsely, we model agon usng a Markov chan wth two states, a state and a agon state. In the state, the probablty of losses s rather low, whle t ncreases sgnfcantly when the economy enters the agon state whch s therefore much more rsky. In both states, there are occasonal downward jumps n the stock prces. Some of these jumps n the state do not only lead to a loss n one of the stocks, but also trgger a jump of the economy nto the agon state and thus ncrease the overall rskness of stocks. Subsequently, the economy can jump back nto the state, and ths knd of jumps occurs wthout a jump n stock prces. Our approach allows us to capture two stylzed facts at the same tme: Frstly, agon s not a one tme event n the sense that t occurs, leads to mmedate losses n several stocks, but has no longer-lastng mpact. Usually, the probablty for subsequent crashes remans hgher for some tme. Ths tme-dmenson of agon mples that the nvestor can adjust hs portfolo when the threat of agon becomes apparent. Secondly, agon s usually trggered by an ntal crash n a partcular market,.e. the jump nto the agon state occurs when some stock prces drop. Our paper s related to the lterature on nuous-tme portfolo choce startng wth Merton 1969, There are two approaches to deal wth agon effects n portfolo problems. One strand of the lterature models agon as jont Posson jumps. Papers n ths area nclude Das and Uppal 2004 and Kraft and Steffensen 2008 whch, however, dsregard the tme dmenson of agon. In partcular, the probablty of subsequent jumps remans the same after a jont jump has happened, snce these papers do not allow for regme shfts. These frameworks therefore do not allow to study how an nvestor hedges aganst subsequent losses gven that agon effects become apparent n the market. The second strand of the lterature are so-called regme-swtchng models. Papers n ths area nclude ng and ekaert 2002 and Gudoln and Tmmermann 2007a,b. lthough these models capture the tme dmenson of agon, regme shfts are not trggered by jumps n asset prces, but occur ndependently of crashes n the stock market. urasch, Porcha, and Trojan 2007 study a model wth stochastc correlatons between assets, but do not allow for jumps. The relevance of agon s also emprcally 2

5 documented n ae, Karoly, and Stulz 2003 and oyson, Stahel, and Stulz In the paper, we address the followng ponts: Frstly, we solve for the optmal stock demand n the and n the agon state both n a complete and n an ncomplete market. We show that there s a hedgng demand for those jumps that lead to a dfferent state. The sgn of ths hedgng demand depends on the nvestment opportuntes n both states and on the rsk averson of the nvestor relatve to the log nvestor. We then analyze whether and how the nvestor adjusts hs portfolo when the economy enters or leaves the agon state. These portfolo revsons turn out to be sgnfcant, and they are the larger the more the and agon state dffer. Whether the nvestor ncreases or decreases hs holdngs of the rsky assets depends on the changes n the market prces of rsk and on whether the market s complete or ncomplete. Secondly, we analyze the utlty loss an nvestor suffers from f he gnores agon or f he gnores the tme dmenson of agon. We show that the utlty loss due to model ms-specfcaton can be sgnfcant. Ths s partcularly true when the market s complete and the nvestor uses dervatves. In ths case, an nvestor wth a rather low rsk averson of 1.5 mght lose more than 20% a year when he bases hs decson on an ncorrect model. If the and agon state dffer sgnfcantly, the utlty loss s largest f the nvestor gnores agon completely. For smaller dfferences, the utlty losses are largest f he only gnores the tme dmenson of agon. Ths latter model also results n the largest losses f the market s ncomplete, even f these losses are much smaller than n a complete market, where the nvestor does not only suffer from basng hs portfolo decson on an ncorrect model, but also from mplementng hs seemngly optmal strategy usng an ncorrect prcng model for the dervatves. The utlty loss from ths second mstake mght even be so large that t more than offsets the utlty gan from havng access to dervatves, resultng n a stuaton where the nvestor s better off f he does not trade dervatves at all. The remander of the paper s structured as follows. In Secton 2, we present the model and the portfolo plannng problem. The optmal portfolos both n complete and ncomplete markets are derved n Secton 3. In Secton 4, we analyze two benchmark models where the nvestor ether completely gnores agon or just ts tme dmenson. Secton 5 provdes some numercal examples, dscusses the mpact of model ms-specfcaton, and provdes some robustness checks. Secton 6 concludes. 3

6 2 Model Setup 2.1 The Economy We consder an economy wth two stocks and. The nterest rate r s assumed to be constant. The stocks are drven by jump-dffuson processes, the dynamcs of stock {, } are ds t S t = µ Zt dt + σ Zt dw t k Zt L Zt,k dn k t. W and W are two rownan motons wth correlaton ρ Zt whch capture normal stock prce movements. Sudden large changes n the stock prces are drven by the Posson processes N k, and the loss f a jump happens s gven by L Zt,k, where we assume that the loss szes are constant. Note that n our notaton L > 0 corresponds to a loss. The dynamcs of the stock prces depend on some state of the economy Z. We nterpret these states as and agon states and assume that these states manly dffer wth respect to the jump ntenstes. Whle the jump ntenstes are rather low n a state, they ncrease sgnfcantly f the economy enters a agon state. In a agon state, the probablty that there wll be several downward jumps n stock prces n a gven tme nterval s thus much larger than n a state. Formally, agon s modeled usng a Markov chan. In general, the Markov chan jumps from state to state j j wth ntensty λ,j, and the process N j counts the number of jumps nto state j. The current state s denoted by Zt. We use a Markov chan wth eght states k { 1, 2, 1, 2, 1, 2, 1, 2 } whch s llustrated n Fgure 1. The frst subscrpt of the state ndcates the stock n whch the last jump took place, the second subscrpt s due to the techncal reason that there cannot be any jumps from a state nto tself. 2 In the states, there may be a jump n any of the two stocks, and ths jump may but needs not trgger agon. The ntensty of a jump n stock that does not trgger agon s λ,, and the correspondng loss n stock s L, the loss n the other stock s zero. When such a jump takes place, the Markov chan goes from.1 to 2 or from.2 to 1. The ntensty of a jump n stock that trggers agon s λ,, the loss of stock for such a jump s L,, and the Markov chan goes from.j to j. If the economy s n a agon state, the ntensty for a loss n stock s λ,, and the correspondng loss sze 2 If there are, e.g., several successve jumps n stock n the state, then the Markov chan changes between the states 1 and s 2. Wthout these dfferent states, the Markov chan would have to jump from the unque state nto the state, whch s not possble techncally. 4

7 s L,. If such a jump happens, the Markov chan goes from.1 to 2 or from.2 to 1. Eventually, the economy wll go back to the state. The ntensty for ths event s λ,, and we assume that ths event does not nduce any losses n the stocks,.e. L, 0 {, }. The Markov chan goes from j to j. The ntenstes for all other jumps are equal to zero. The Markov chan has four agon states and four states. We assume that the model parameters concde n all state and n all agon states, respectvely. The future behavor of the stock prces thus only depends on whether the economy s n a state or n a agon state, but not on whch specfc or agon state s realzed. Ths mples that optmal portfolos, ndrect utltes, and other economc quanttes we are nterested n also depend only on whether we are n a or agon state. The use of four agon and four states thus does not have any economc mplcatons, but s for techncal reasons only. Fnally, we have to specfy the drft and the rsk prema of the stocks. The drft of stock s equal to µ Zt = r + φ Zt + k Zt L Zt,k λ Zt,k where the last term s the compensator for the jump processes. In general, the rsk premum on the stock s φ Zt = σ Zt η Zt + L Zt,k λ Zt,k η Zt,k k Zt where η j s the premum for dffuson rsk W when the economy s n state j, and η j,k s the premum for jumps from j to k. The ntensty for a jump from j to k under the rsk neutral measure s thus 1 + η j,k tmes the ntensty under true measure. Wth our defnton of the Markov chan, the rsk premum depends only on whether the economy s n one of the or n one of the agon states. The rsk prema on stock are φ = σ φ = σ η η + L, + L, λ, η, λ, η,. + L, λ, η, esdes the stocks and the money market account, the nvestor mght also have access to dervatves. We assume that there are ether no dervatves at all, or enough dervatves to complete the market. The exposure of the dervatves to the rsk factors can be calculated usng Ito s lemma. 5

8 2.2 The Investor We consder a representatve nvestor wth CRR-utlty uc = c1 γ 1 γ where γ > 0 denotes hs relatve rsk averson. The plannng horzon of the nvestor s T, and he derves utlty from termnal wealth only. The ndrect utlty at tme t and n state j s defned as G j t, X t = max {E ux T Zt = j} X T j t,x t where j t, X t s the set of all wealth levels at T that meet the budget restrcton and can be fnanced wth a current wealth level X t. More detals on ths set wll be gven later on. 3 sset llocaton 3.1 Complete Market In a complete market, the nvestor can choose the optmal exposures to the rsk factors frst, and then mplement these exposures by some approprate tradng strategy, as explaned e.g. n Lu and Pan We follow ths ansatz, and the budget restrcton for the nvestor s dxt Xt = rdt + θ Zt t dw t + η Zt dt + k Zt,λ Zt,k 0 + θ Zt t dw t + η Zt dt θ Zt,k t dn k t λ Zt,k dt η Zt,k λ Zt,k dt where θ j t s the exposure of wealth to dffuson rsk W n state j and θ j,k s the exposure to a jump from state j to state k. In the state, we have to choose the four exposures to jumps n stock and stock that do not nduce agon, and we denote these exposures by θ, θ,. In the agon state, we have to choose the three exposures to jumps n stock, jumps n stock, and jumps back from the agon to the state. These exposures are denoted by θ, and θ,. The portfolo plannng problem of the nvestor s G j t, X t = subject to the budget restrcton 1. max E ux T Zt = j {θ j s,θj s,θj,k s,t s<t } 6 1

9 Proposton 1 Contagon, Complete Market In an economy wth agon, the optmal exposures to the rsk factors are θ, θ, θ, θ j = ηj ρj η j γ1 ρ j 2 θ j = ηj ρj η j γ1 ρ j 2 = 1 + η, 1 γ 1 θ, = 1 + η, 1 γ 1 = 1 + η, 1 f γ f 1 θ, = 1 + η, 1 f γ f 1 = 1 + η, 1 γ 1 θ, = 1 + η, 1 γ 1 θ, = 1 + η, 1 γ f 1. f The ndrect utlty functon of the nvestor s G j t = x1 γ f j t γ 1 γ where f t f t = exp { C, C, C, C, T t } 1 1 wth C, = 1 γ γ C, = C, = 1 γ γ r + η 1 γ η 2 + η 2 2ρ η 2γ1 ρ η, 1 1 γ η, 1 λ, η, 1 + η, 1 1 γ r + η 1 1 γ λ, + λ, + λ, η, 1 + η, 2 + η 2 2ρ η 2γ1 ρ η, 1 λ, + 1 γ η, 1 λ, 1 γ η, 1 1 γ λ, C, = 1 + η, 1 1 γ λ,. η η, 1 + η, η, γ 1 1 γ λ, λ, 1 γ 1 γ λ, λ, 1 + η, 1 λ, 1 γ 1 1 γ λ, 7

10 The proof s gven n ppendx.1. Followng Merton 1971, the optmal exposures can be decomposed nto a speculatve demand and a hedgng demand. The demand for dffuson rsk s purely speculatve, snce dffuson rsk does not have any mpact on the nvestment opportunty set. It depends on the rsk prema and the correlatons only. The optmal exposure to jump rsk s more nvolved. The speculatve demand for a jump from state old to state new where the two states mght concde s gven by 1 + η old,new 1 γ 1. If the market prce of jump rsk η old,new s postve, jumps are more lkely under the rskneutral measure than under the true measure, and the optmal exposure to ths knd of jumps s negatve. In lne wth ntuton, t ncreases n absolute terms n the rsk premum, and t decreases n absolute terms n rsk averson. The second part of the demand for jump rsk s the hedgng demand, whch s gven by 1 + η old,new 1 γ f new f 1. old It dffers from zero only f the old and new state are not equal,.e. f the economy changes from to agon or vce versa. In ths case, the nvestor takes changes n the nvestment opportunty set nto account, where hs reacton to these changes depends on whether he s more or less rsk-averse than the log-nvestor, as explaned e.g. n Km and Omberg 1996, Lu and Pan 2003 or Lu, Longstaff, and Pan For f new > f old, the nduced hedgng demand s postve. If γ > 1, f new > f old mples that nvestment opportuntes are worse n the new state than n the old state. The nvestor s more rskaverse than the log nvestor, he cares about hedgng, and he wants to have more wealth n those states of the world where nvestment opportuntes are bad. Ths results n a postve hedgng demand. If γ < 1, f new > f old mples that nvestment opportuntes are better n the new state than n the old state. The nvestor s less rsk-averse than the log nvestor and he speculates on changes n the nvestment opportunty set. He thus wants to have more wealth n the good new state, and the nduced hedgng demand s postve. To assess how good the nvestment opportuntes n state j are, we rely on the certanty equvalent return CER. It s defned by xe CERj t,xt t G j t, x = 1 γ The CER gves the determnstc return on wealth that would result n the same ndrect utlty as the optmal nvestment n the rsky assets. 8 1 γ.

11 When the economy changes from the state to the agon state or vce versa, the ndrect utlty of the nvestor changes due to two reasons. Frst, hs wealth changes, where the loss or gan depends on hs exposure towards the jump. Second, the nvestment opportunty set and thus the CER changes. Consder e.g. the case where the optmal exposure to a jump from the to the agon state s negatve. If the nvestment opportuntes are worse n the agon state, the nvestor wll be worse off after the jump. If, on the other hand, the nvestment opportuntes are better n the agon state, the overall mpact on the ndrect utlty depends on the trade-off between the lower wealth and the hgher CER. 3.2 Incomplete Market If the nvestor can only trade n the two stocks and n the money market account, the market s ncomplete. The budget restrcton becomes dxt Xt = π Zt tds t S t + πzt tds t S t + 1 π Zt t πzt t rdt where π j t s the proporton of wealth nvested n stock =, at tme t and n state j. The optmal portfolo strategy s gven n Proposton 2 Contagon, Incomplete Market In an economy wth agon where only the two stocks and the money market account are traded, the ndrect utlty of the nvestor n state j {, } s G j t, x = x1 γ 1 γ f j t where f j solves the ordnary dfferental equatons 0 = f t 0 = f t + 1 γ r + π µ 0.5γ1 γ π + λ, 1 π + λ, 1 π σ + 1 γ r + π µ 0.5γ1 γ π 1 π + λ, r + π µ r f π σ 2 + 2π L 1 γ f f + λ, L 1 γ f f + λ, σ + λ, f f. π σ σ ρ f 1 π L 1 γ 1 f 1 π L 1 γ 1 f r + π µ r f π σ 2 + 2π L 1 γ 1 f + λ, π σ σ ρ f 1 π L 1 γ 1 f 9

12 and where the optmal portfolo weghts solve µ µ L λ, L λ, r γσ 2 π r γσ 2 π µ 1 π L µ 1 π L γπ γπ r γσ 2 π γ f γπ σ σ ρ f L λ, 1 π L γ = 0 4 r γσ 2 π σ σ γ f γπ σ σ ρ f L λ, 1 π L γ = 0 5 ρ L λ, 1 π L γ = 0 6 σ σ ρ L λ, 1 π L γ = 0. 7 The proof s gven n ppendx.2. Equatons 2, 3, 4 and 5 form a system of so-called dfferental-algebrac equatons whch can only be solved numercally. s compared to the complete market, the nvestor can n general no longer acheve the optmal exposures, snce he s restrcted to the package of exposures offered by the two stocks, as e.g. ponted out n Lu and Pan s we wll show n some numercal examples n Secton 5, hs exposure to some rsk factors wll thus be too hgh, whle the exposure to some other rsk factors wll be too low. The exposure to jumps from the agon to the state plays a specal role. Snce the exposure of both stocks to ths jump s assumed to be zero, the nvestor has no exposure to ths jump at all, and he cannot even approxmately mplement hs hedgng demand. The ndrect utlty of the nvestor s lower n the ncomplete market than n the complete market. The sze of the utlty loss due to market ncompleteness can be measured by the dfference n the certanty equvalent returns. 4 enchmark Cases We consder two benchmark cases. In the frst case no agon, the nvestor gnores agon completely. The stocks jump ndependently of each other, and the jump ntenstes are constant over tme. In the second case jont jumps, studed e.g. by Das and Uppal 2004, the nvestor takes agon nto account by assumng that stock prce jumps can only happen smultaneously. Our model s n between these extreme cases n two respects. Frst, we assume that some jumps are normal jumps whch do not trgger agon, whle some other jumps 10

13 nduce agon. Second, we allow for a tme dmenson of agon. If the economy enters nto the agon state, the nvestor can adjust hs portfolo and take a smaller or larger poston n the rsky assets. In the benchmark model wth jont jumps, on the other hand, the jumps happen smultaneously, and the nvestor cannot react to the event of agon any more. 4.1 No Contagon: Independent Downward Jumps In the frst benchmark case, there s no agon at all, and downward jumps n the stocks happen ndependently of each other. The dynamcs of stock are ds t S t = r + φ + L λ }{{} dt + σ dw t L dn t. µ The Wener processes W and W are correlated wth correlaton ρ. N s a Posson process wth ntensty λ. The rsk premum on the stock s φ = σ η dff + L λ η jump where η dff s the premum for dffuson rsk and η jump s the premum for jumps. In a complete market, the nvestor can agan choose the exposures to the rsk factors. The budget restrcton becomes dxt Xt = rdt + θ dff t dw t + η dff dt + θ dff t dw t + η dff dt + θ jump t dn t λ dt η jump λ dt + θ jump t dn t λ dt η jump λ dt where θ dff s the exposure to dffuson rsk W, and θ jump stock. The optmal portfolo s gven n s the exposure to jumps n Proposton 3 No Contagon, Complete Market If there are no agon effects n the market, the optmal exposures to the rsk factors are θ dff θ jump = ηdff ρη dff γ1 ρ 2 θ dff = ηdff ρη dff γ1 ρ 2 = 1 + η jump 1 γ 1 θ jump = 1 + η jump 1 γ 1. The ndrect utlty functon of the nvestor s Gt, x = x1 γ 1 γ exp {γ Cnc,c T t} 11

14 where C nc,c = 1 γ γ r + ηdff 2 + η dff η jump 2 2ρη dff ηdff 2γ1 ρ 2 λ η jump η jump 1 1 γ λ η jump The proof s gven n ppendx γ λ. λ 1 1 γ λ + λ The nvestment opportunty set s constant. There s thus speculatve demand only. oth for dffuson rsk and for jump rsk, ths speculatve demand has the same structure as n the agon model dscussed n Secton 3 and s drven by the rsk prema and the dffuson correlaton only. The certanty equvalent return s gven by γ 1 γ Cnc,c. It captures how good the nvestment opportuntes are. In a complete market, t does not depend on asset specfc parameters lke stock prce volatltes and loss szes, but only on economy-wde varables lke the rsk prema and the jump ntenstes. Obvously, the certanty equvalent return s ncreasng n the rsk prema. Furthermore, t s ncreasng n the jump ntenstes λ and λ, whch s formally shown n ppendx.2. To get the ntuton, note that the rsk premum the nvestor earns on hs optmal portfolo s ncreasng n the optmal exposure to jumps.e. the loss n case of a jump, the market prces of jump rsk, and the jump ntenstes.e. the overall amount of jump rsk n the market. The CER s thus ncreasng n these three varables, too. In the ncomplete market, the nvestor chooses the optmal weghts of the two stocks, whch are gven n the next proposton. Proposton 4 No Contagon, Incomplete Market If there are no agon effects n the market and only the money market account and the two stocks are traded, the ndrect utlty of the nvestor s gven by Gt, x = x1 γ 1 γ expcnc,c T t where C nc,c = 1 γ r + π µ r + π µ r γ 2 π2 σ 2 + πσ π π σ σ ρ +λ 1 π L 1 γ 1 + λ 1 π L 1 γ 1 12

15 and where the optmal portfolo weghts are gven as the unque soluton of µ r γσ 2 π γπ σ σ ρ L λ 1 π L γ = 0 µ r γσ 2 π γπ σ σ ρ L λ 1 π L γ = 0. The proof s gven n ppendx.3. Just as n our agon model, the nvestor can n general no longer acheve the optmal exposures as compared to the complete market, snce he s restrcted to the package of exposures offered by the two stocks, as also ponted out by Lu and Pan 2003 n a model wth jump rsk, but one stock only. gan, hs exposure to some rsk factors wll be too hgh, whle the exposure to some other rsk factors wll be too low. Snce the nvestment opportunty set s constant, the nvestor does not need to mplement a hedgng demand n the ncomplete market, ether. The ndrect utlty of the nvestor s lower n the ncomplete market than n the complete market. The sze of the utlty loss due to market ncompleteness can be measured by the dfference n the certanty equvalent returns. 4.2 Jont Downward Jumps In the second benchmark case, the nvestor takes agon nto account by assumng that stock prce jumps always happen smultaneously. The dynamcs for stock are ds t S t = r + φ + L λ }{{ jont dt + σ } dw t L dn jont t µ and the rsk premum on the stock s φ = σ η dff + L λ jont η jump jont. We want the behavor of the ndvdual stocks to be the same n both benchmark cases, so that only the jont behavor dffers. Consequently, we assume that the parameters for the ndvdual stocks are the same as n Secton 4.1, and we set λ jont = λ = λ and η jump jont = ηjump = η jump. In the complete market, the soluton to the portfolo plannng problem s gven n the next proposton. 13

16 Proposton 5 Jont Downward Jumps, Complete Market If there are jont downward jumps, the optmal exposures to the rsk factors are θ dff θ jump jont = ηdff ρη dff γ1 ρ 2 = 1 + ηjump jont 1 γ 1. θ dff = ηdff ρη dff γ1 ρ 2 The ndrect utlty functon of the nvestor s where C jj,c = 1 γ γ Gt, x = x1 γ 1 γ exp { γ C jj,c T t } r + ηdff 2 + η dff 2 2ρη dff ηdff 2γ1 ρ η jump jont λjont 1 1 γ λ jont η jump 1 1 γ jont λ jont. The optmal exposures depend on the market prces of rsk and on the correlaton only. Wth dentcal parameters for the behavor of the ndvdual stocks, they are thus the same as n the case of ndependent jumps. If a jump happens, the nvestor loses exactly the same amount of money, no matter whether he assumes ndependent jumps or jont jumps. What dffers, however, s the optmal portfolo held by the nvestor. If there are jont jumps, the portfolo that s optmal wth ndependent jumps would have a jump rsk exposure that s twce as hgh as optmal. Wth jont jumps, the nvestor s thus more conservatve. The CER s lower wth jont jumps than wth ndependent jumps. To get the ntuton, note that the market prces of rsk are dentcal, whle the average number of jumps s twce as large n the case of ndependent jumps as n the case of jont jumps. Snce the CER ncreases n the jump ntensty and thus n the average number of jumps, t s ndeed smaller wth jont jumps. In the ncomplete market, the nvestor s agan restrcted to the package of exposures offered by the stocks. The optmal portfolo s gven n the next proposton. Proposton 6 Jont Downward Jumps, Incomplete Market If there are jont downward jumps and only the money market account and the two stocks are traded, the ndrect utlty of the nvestor s gven by Gt, x = x1 γ 1 γ exp{cjj,c T t} 14

17 where C jj,c = 1 γ r + π µ r + π µ r γ 2 π2 σ 2 + πσ π π σ σ ρ +λ jont 1 π L π L 1 γ 1 and where the optmal portfolo weghts are gven as the unque solutons of µ r γσ 2 π γπ σ σ ρ L λ jont 1 π L π L γ = 0 µ r γσ 2 π γπ σ σ ρ L λ jont 1 π L π L γ = 0. Just as n the model setup wthout agon, the nvestment opportunty set s constant and the nvestor does not have a hedgng demand n the ncomplete market, ether. 5 Numercal Results 5.1 Parameter Choce and Model Calbraton We consder a CRR-nvestor wth a relatve rsk averson of γ = 3 and a plannng horzon of 20 years. The nterest rate s set to r = The two stocks are assumed to follow dentcal processes. We choose the parameters such that they approxmately ft the behavor of the S&P500 over the last 25 years, where we rely on the parameter estmates of Eraker, Johannes, and Polson 2003 and roade, Chernov, and Johannes Snce we want to focus on the mpact of agon, whch s reflected n the dfference between the jump ntenstes n the and n the agon state, all other parameters are assumed to be equal n both states. The dffuson volatlty σ s set to 5, and the Wener processes drvng the stock prce dynamcs are correlated wth ρ = 0.5. oth these parameters do not depend on the current state. The jump ntensty n the benchmark models s set to 1.5, and we calbrate the jump ntenstes n our agon model such that the average long-run jump ntensty s equal to 1.5, too. More detals on ths step of the calbraton wll be gven below. The loss n case of a jump n one of the stocks s assumed to be constant and set equal to, whch s slghtly hgher than the estmate provded n models that also nclude stochastc volatlty. Remember that the loss for a jump back from the agon to the state s equal to zero. The market prce for dffuson rsk s assumed to be equal to 0.35 n both states. Jumps from the agon state back to the state are not prced. For the other market 15

18 prces of jump rsk, we consder two extreme cases. In the frst case parametrzaton 1, we assume that they are dentcal n all states. Ths mples a rather hgh drft of the stocks n the agon state. In the second case parametrzaton 2, we assume that the expected excess returns of the stocks are equal n both states, whch results n larger market prces of rsk n the state and lower ones n the agon state. We calbrate the market prces of jump rsk such that the average expected excess return of the stocks s equal to 8.25% for both parametrzatons. The two benchmark models wthout agon and wth jont jumps are calbrated such that the behavor of the stock prces n the benchmark model s as smlar as possble to the behavor n our model. Therefore, we set the local moments n the benchmark models equal to the long run averages of the local moments n our model. The statonary probablty of the and agon state s p = λ, p = λ, +λ, +λ, λ, +λ,, λ, +λ, +λ, and we know from the ergodc theorem for Markov chans 3 that 1 lm t t t where g s some state-dependent functon. 0 gzsds = gp + gp Frstly, we want the stocks to have the same rsk n the agon model and n the benchmark models. We thus equate the varance of the stock, whch gves 2 λ, σ 2 + L 2 λ = p σ σ + p + p p L, λ, 2 + L, σ η + 2 λ, + L, + L, σ η L, λ, η, λ, η, L, λ, η, We also equate the jump ntensty for those jumps that result n a loss and the average 3 See, e.g., rémaud

19 jump sze λ = p λ, L = p λ, + λ, + p λ, λ, + p L, + λ, L, + 8 λ, λ, + λ, L, Secondly, we want the stocks to have the same expected excess returns. Snce the nvestor mght deal dfferently wth jump and dffuson rsk, we also equate the rsk prema earned on stock dffuson rsk and stock jump rsk. Ths gves two addtonal restrctons σ η dff = p σ L λ η jump = p η + p σ η L, λ, η, + p L, λ, η,. + L, λ, η, The jump ntenstes and the loss szes n the benchmark models are dentcal for both parameterzatons. Ths also holds for the jump rsk prema, whch concde wth the state-ndependent jump rsk prema of parametrzaton 1. The dffuson volatlty n the benchmark models s dentcal to that n our model wth parametrzaton 2. It s slghtly larger and depends on the jump ntenstes f the market prces of rsk are equal parametrzaton 1, whch accounts for the slghtly larger varance n ths case. s a consequence, the market prce for dffuson rsk s slghtly lower n ths case. The dfferent jump ntenstes n our model are chosen such that the average number of jumps per year, whch follows from Equaton 8, s equal to the benchmark value of 1.5. Snce we want to focus on agon, we explctly rol for ts severeness and thus for the wedge drven between the two states. The dfference between the jump ntenstes n the and agon state s captured by ξ 1: λ, = ξ λ, + λ, {, }. The condtonal probablty that a loss n a stock actually trggers agon s gven by the parameter α: λ, = α λ, + λ, {, }, and the average tme the economy stays n the agon state depends on ψ: λ, = ψ λ, 17 + λ,.

20 Gven ξ, α, and ψ and the average jump ntensty of 1.5, all other jump ntenstes can be calculated. In the base case calbraton, we set ξ = 4, α = 0.5 and ψ = The resultng parameters are gven n Table 1. Table 2 shows the resultng condtonal equty rsk prema and varances of stock returns for both parameterzatons and n the benchmark models as well as ther decomposton nto dffuson and jump components. Several other combnatons of the parameters we have consdered n robustness checks are gven n Table 3, where we use ξ 1, 10, α 0.2, 0.5 and ψ 0.2, 2/ Optmal Exposures and Optmal Portfolos Table 4 gves the soluton to the portfolo plannng problem for the base-case parameters from Table 1 both for the complete and the ncomplete market. We dscuss the case of complete markets frst, where the nvestor can acheve any desred payoff profle. The demand for dffuson rsk s drven by the speculatve component only. It s dentcal n the and n the agon state and for both parametrzatons, because the market prces of dffuson rsk are dentcal by assumpton. In the benchmark models, the optmal demand s the same for parametrzaton 2 and slghtly lower for parametrzaton 1, whch can be attrbuted to the lower market prce of dffuson rsk n ths case. The demand for jump rsk can be decomposed nto a speculatve component and for those jumps that change the state a hedgng component. The speculatve demand depends on the market prces of jump rsk. Snce jumps from the agon state back to the state are not prced by assumpton, the speculatve demand s zero. For the other jumps, whch all lead to a loss n stock prces, jump rsk s prced, and there s a negatve speculatve demand. If the market prces of rsk are dentcal n all states parametrzaton 1, the speculatve demand does not depend on the state and concdes wth the speculatve demand n the two benchmark models. If equty rsk prema are constant parametrzaton 2, on the other hand, the market prce of rsk s lower n the agon state than n the state, and consequently, the speculatve demand s lower n absolute terms n the agon state, too. The nvestor s thus more aggressve n the state, and less aggressve n the agon state, as compared to parametrzaton 1. The market prce of rsk n the benchmark models s n between the market prces of rsk n the and the agon state, and the speculatve demand s n between those from the agon model, too. The sgn of the hedgng demand depends on whch of the two states s the better one. The rght panel of Fgure 2 shows the certanty equvalent returns n both states. If the market prces of rsk are constant parametrzaton 1, the nvestment opportunty set 18

21 s better n the agon state where jumps happen more often than n the state. Gven that γ > 1, the hedgng demand for jumps from the to the better agon state s negatve, whch mples that the nvestor takes a more aggressve poston n jump rsk n the state. In the agon state, on the other hand, hs optmal exposure to jumps back to the worse state s postve. If the expected returns are equal parametrzaton 2, the state s better than the agon state. Then, the hedgng demand n the better state s postve, reducng the demand for jump rsk n ths state, whle the hedgng demand n the worse agon state s negatve. The optmal exposures are dfferent n the and n the agon state, and the nvestor wll adjust hs portfolo when the state of the economy changes. He thus profts from the tme dmenson of agon captured n our model. Hs tradng desre due to agon s much more pronounced for the case of equal equty rsk prema parametrzaton 2, where tradng s nduced by changes n the market prces of rsk and n the hedgng demand, than for the case of dentcal market prces of rsk, where tradng s nduced by changes n the hedgng demand only. If the market s ncomplete, the nvestor cannot mplement the overall optmal exposures. s can be seen n Table 4, the realzed exposures wll be somewhere n between the optmal exposures from the complete case. For the gven parameters, the exposure to dffuson rsk s too hgh, and the exposure to jump rsk s too low n absolute terms both n our model and n the benchmark models. The poston n rsky assets s larger n the state n whch nvestment opportuntes are better, that s n the state n case of equal equty rsk prema and n the agon state n case of equal market prces of rsk. In the benchmark models, the nvestor does not dstngush between and agon states. If he gnores agon completely, the optmal poston n stocks s somewhere n between the optmal postons n the and n the agon state. If the nvestor assumes that there are jont jumps, he s more conservatve and reduces hs optmal poston n stocks sgnfcantly. The certanty equvalent returns n our model and n the two benchmark models are shown n the left panel Fgure 2. s expected, the utlty loss due to market ncompleteness s largest n our agon model snce the nvestor fals to mplement the optmal myopc demand as well as the ntertemporal hedgng demand, whereas a hedgng demand does not exst n both benchmark models. In the benchmark models, the utlty loss s larger n case of no agon than n case of jont jumps for our parametrzatons n partcular, for our choce of the relaton between dffuson rsk and jump rsk n stocks. To get the ntuton, note that the actual exposures n the ncomplete market are much closer to the optmal exposures n the complete market for the case of jont jumps, so that the utlty 19

22 loss s smaller n ths case. In absolute numbers, however, the jont jumps model gves the lowest utlty both n an ncomplete and n a complete market, snce the average number of jumps s cut n half compared to the other models. Comparng both parametrzatons equal market prces of rsk and equal equty rsk prema, the utlty loss caused by market ncompleteness s roughly equal. The overall sze of the utlty loss s thus determned rather by a certan model specfcaton than by the assumptons on the market prces of rsk. The utlty dfferences between the and the agon state, however, are more pronounced wth equal market prces of rsk parametrzaton 1 due to the extreme changes of the condtonal equty rsk premum. Robustness checks show that the results do not change qualtatvely when we vary ξ, α and ψ,.e. the overall sze of agon, the rsk of enterng the agon state, and the duraton of the agon state. In lne wth ntuton, a larger dfference between the and agon state,.e. a larger value of ξ, leads to more extreme results. Ths effect s most pronounced for equal market prces of rsk parametrzaton 1, where the nvestment opportuntes n the agon state become very attractve and can nduce the nvestor to take a hghly levered poston n stocks. Wth equal equty rsk prema parametrzaton 2, ths effect s much smaller. For both parametrzatons, the utlty of the nvestor s ncreasng n ξ n a complete market, but s not monotonous n an ncomplete market. Therefore, utlty losses due to market ncompleteness also tend to ncrease n ξ. The probablty α of enterng the agon state does not have much mpact on the results. On the other hand, the smaller ψ,.e. the longer the economy stays n the agon state once t has entered ths state, the more extreme the portfolo weghts, exposures and utlty functons. 5.3 Model Ms-Specfcaton If the nvestor reles on a benchmark model nstead of the true model from Secton 2.1, the nvestor wll not hold the optmal portfolo. In ths secton, we analyze the utlty loss he suffers from due to ths suboptmal behavor Incomplete Market In the ncomplete market, the nvestor can only nvest nto the two stocks and nto the money market account. In case of model ms-specfcaton, he ncorrectly uses one of the benchmark models to determne the optmal portfolo. For both these models, the optmal 20

23 portfolo weghts are constant over tme. The ndrect utlty derved from ths strategy s gven n the next proposton. Proposton 7 Model Ms-Specfcaton, Incomplete Market In an economy wth agon where only the two stocks and the money market account are traded and for an nvestor who uses the portfolo weghts π, π, the ndrect utlty n state j {, } s where f j s gven by f t where f t = exp Ĉ, = 1 γ r + π µ λ, Ĉ, = λ, Ĉ, = λ, G j t, x = x1 γ 1 γ f j t { Ĉ, Ĉ, Ĉ, Ĉ, r + π µ r T t 0.5γ1 γ π σ 2 + π σ π π σ λ, + λ, 1 π L 1 γ 1 + λ, Ĉ, = 1 γ r + π µ 1 π L 1 γ 1 1 π L 1 γ + λ, 1 π L 1 γ r + π µ r 0.5γ1 γ π σ 2 + π σ π π σ λ, + λ, 1 π L 1 γ 1 + λ, The proof s gven n ppendx C.1. } 1 1 σ ρ σ ρ 1 π L 1 γ 1. The upper panels of Fgure 3 and 4 show the certanty equvalent returns n case of model ms-specfcaton for equal market prces of rsk and equal equty rsk prema, respectvely. For the base case parametrzaton, the nvestor looses up to 20 bass ponts a year f he reles on an ncorrect model. The losses are larger for equal market prces of rsk parametrzaton 1 than for equal equty rsk prema parametrzaton 2, snce the dfferences n the optmal portfolos between the states whch the nvestor fals to pck up are larger n the frst case. Surprsngly, the nvestor s slghtly worse off f he assumes jont jumps and thus only gnores the tme dmenson of agon than f he gnores agon completely. nd agan, the results,.e. the utlty losses, ncrease n the dfference between the and agon state as measured by ξ. 21

24 5.3.2 Complete Market Next, we analyze the mpact of model ms-specfcaton when the market s complete. To determne whether enough dervatves are traded for market completeness, the nvestor reles on the benchmark model. In the case of ndependent jumps, four rsky assets are needed, whle n the case of jont jumps, three rsky assets are enough. We assume that the nvestor uses the two stocks, an TM-call on stock wth a tme to maturty of 3 months, and f needed an dentcal TM-call on stock. These short-term TM-optons are usually among the most lqud racts. Note however that the choce of racts wll have an mpact on the utlty loss due to model ms-specfcaton. The analyss of model ms-specfcaton s more complcated than n case of an ncomplete market. In the frst step, the nvestor determnes the seemngly optmal exposures n the benchmark model. In the second step, he uses the rsky assets and ther rsk exposure to mplement these seemngly optmal exposures, where he ncorrectly determnes the senstvtes of the dervatves n the benchmark model. Gven the seemngly optmal portfolo, we but not the nvestor can then use the senstvtes from the true model to determne the realzed exposure. Gven these realzed exposures θ, whch are agan constant over tme, we can then fnally calculate the realzed ndrect utlty. Proposton 8 Model Ms-Specfcaton, Complete Market In a complete market wth agon effects, the utlty obtaned by an nvestor who uses the ncorrect rsk factor exposures θ s gven by Ĝ j t, x = x1 γ 1 γ f j t where j {, } and { f t = exp Ĉ, Ĉ, f t Ĉ, Ĉ, T t }

25 wth Ĉ, = 1 γ r + 0.5γ1 γ + λ, λ, Ĉ, = λ, 1 + θ η +, θ λ, θ, λ, θ θ η, 1 + θ 1 γ 1 λ, θ, 1 + η, 1 + η, 2 + θ 2 + 2ρ θ + λ, 1 γ + λ, 1 + Ĉ, = λ, 1 + θ, 1 γ Ĉ, = 1 γ r + θ η + θ η θ, λ, 1 + η, θ, λ, 1 + η, 0.5γ1 γ θ 2 + θ 2 + 2ρ θ + λ,, 1 + θ 1 γ 1 + λ, λ, The proof s gven n ppendx C.2., θ, θ θ λ, λ, 1 + η, 1 + η,, 1 + θ 1 γ 1, θ 1 γ θ, θ λ, 1 + η,, 1 + θ 1 γ 1 The lower panels of Fgure 3 and 4 show the certanty equvalent returns when the correct model s used and when one of the benchmark models s used to determne the seemngly optmal portfolo. The CER losses are hghly economcally sgnfcant, and they are much hgher than n the ncomplete market, snce the nvestor now makes an addtonal mstake. To set up the optmal portfolo, he has to convert the optmal exposures nto portfolo weghts. Whle the exposures of the stocks are model ndependent, the exposures of the dervatves depend on the model, and an nvestor usng an ncorrect model for portfolo plannng wll use the same ncorrect model for prcng dervatves, too. s can be seen from the fgures, the mstakes n calculatng the exposures and n prcng the dervatves do not cancel each other, but rather add up. Fgure 5 compares the utlty losses for dfferent values of ξ, where we assume equal equty rsk prema n both states. The results for equal market prces of rsk not shown here are qualtatvely smlar. s can be seen from the graphs, the dfference between the and agon state has a very large mpact on the utlty losses. They are already far from neglgble for a rather low value of ξ = 2, and ncrease to around 10%-15% 23

26 a year for ξ = 10. For ths hgh level of ξ, the CER can even become negatve, and the nvestor would be better off f he just nvested hs wealth at the rsk-free rate only, gnorng all rsky assets. The graphs also show that the utlty losses n the and agon state are approxmately equal. If the nvestor reles on one of the benchmark models, the parameters of ths model and thus also hs seemngly optmal portfolo represents knd of an average between the two states. The dstance to the truly optmal portfolos s thus approxmately equal for both states, and ths also holds for the utlty losses. Dfferent from the ncomplete market, t now depends on ξ,.e. on the severeness of agon, whch of the two benchmark models leads to the smaller utlty loss. For low values of ξ, for whch the dfferences between the and agon state are rather moderate, the nvestor s stll better off f he gnores agon completely. For hgher values of ξ, however, he s sgnfcantly better off f he just gnores the tme dmenson of agon. To get the ntuton, note that the nvestor uses two optons n case of ndependent jumps when he gnores agon, but only one opton n case of jont jumps when he only gnores the tme dmenson of agon. Snce the use of dervatves s the man reason for the hgh utlty losses, he s better off the less dervatves he adds to hs portfolo,.e. f he reles on the model wth jont jumps. In ths model, however, he s too conservatve and does not take advantage of the jump rsk prema offered n the market. The trade-off between these two arguments depends on the absolute sze of the poston n dervatves. The larger ths poston, the larger the relatve advantage of the model wth jont jumps. Snce the poston n dervatves ncreases n the dfference between the and agon state, the nvestor s ndeed better off f he uses the model wth jont jumps for hgh values of ξ. Ths effect s more pronounced for equal market prces of rsk parametrzaton 1 than for equal equty rsk prema parametrzaton 2, snce the dfferences n the optmal exposures are larger n the frst case. n nvestor who reles on the correct model s better off n the complete market. In case of model ms-specfcaton, ths may no longer be true, as can be seen n Fgure 3 and 4. Whle an nvestor who ncorrectly bases hs decsons on a model wth jont jumps s stll better off n the complete market, an nvestor gnorng any agon mght be better off n the ncomplete market. In ths case, the utlty gan from havng access to dervatves and thus more payoff patterns s more than offset by the utlty loss from usng the ncorrect senstvtes and mplementng the seemngly optmal strategy n the wrong way. We also dd a robustness check wth respect to α and ψ, whch govern the rsk of enterng the agon state and the average tme the economy stays n the agon state. s already seen above, the mpact of these two parameters s rather small, and the 24

27 qualtatve results do not change. 5.4 Robustness Checks In the precedng sectons, we have shown that agon has a substantal effect on optmal exposures, optmal portfolo weghts, and the nvestor s expected utlty. Furthermore, an nvestor who uses an ncorrect model mght suffer large utlty losses n partcular n a complete market where he also uses dervatves. Whle we have already dscussed the senstvty of our results wth respect to the severeness of agon, we now do some addtonal robustness checks wth respect to the rsk averson, the sze of the losses, and the dffuson correlaton between the stocks Relatve Rsk verson The results up to now have been based on a relatve rsk averson of γ = 3. We have redone the analyss for values of γ between 1.5 and 10. In lne wth ntuton, the results become less extreme the hgher the rsk averson and the less the nvestor therefore nvests n rsky assets. The qualtatve results, however, do not change. Whle the utlty losses due to model ms-specfcaton decrease n γ, they are stll hghly economcally sgnfcant even for a hgh rsk averson of γ = 10. The nvestor s much more conservatve n ths case. Nevertheless, the loss n CER can well exceed 8% n the complete market and s thus far from neglgble Loss Sze In a second step, we have changed the loss sze from L = to the more moderate value of L = Ths has no mpact on the results n the complete market, whch are ndependent of the exact losses n the stocks, but depend on only on the ntensty of jumps and ther market prces of rsk. In the ncomplete market, however, the smaller loss sze decreases the utlty of the nvestor, snce the package offered by stock fts the optmal exposure now even worse. Consequently, the utlty loss due to market ncompleteness ncreases. The mpact of the loss sze on the losses due to model ms-specfcaton s mxed. Whle the utlty loss n the ncomplete market and n case the jont jumps model s used decreases wth the lower loss sze, the opposte s true n a complete market and n case 25

28 the nvestor reles on a model wth no agon at all. Overall, however, the results do not change qualtatvely when we change the loss sze Dffuson Correlaton s an addtonal robustness check, we consder dfferent values for the dffuson correlaton parameter ρ, whch was set to ρ = 0.5 n our base case. We redo the analyss for ρ = 0 and ρ = 0.5. When ρ decreases, the overall level of rsk decreases n all economes, whle the market prces of rsk and the equty rsk prema on ndvdual stocks, respectvely, stay the same. Consequently, the utlty of the nvestor ncreases both n a complete and n an ncomplete market and for all models. The ncrease n utlty s smallest n the benchmark model wth jont jumps, where the stocks are also correlated due to jont jumps and where the decrease n dffuson correlaton thus s of second-order mportance. The utlty loss due to market ncompleteness s smallest for ρ = 0.5 n our agon model. Ths can be explaned by the fact that the package offered by the stocks s closest to the overall optmal exposure n ths case. The result s specfc to the parameters used and wll not hold n general. The utlty losses due to model ms-specfcaton n an ncomplete market may also depend on ρ. Whle there s hardly any mpact f the nvestor gnores agon completely, the utlty loss ncreases to around 2% for ρ = 0.5 f the nvestor reles on a model wth jont jumps. Fgure 6 shows the CER n case of model ms-specfcaton n a complete market, where we assume equal market prces of rsk parametrzaton 1. The results are qualtatvely smlar for equal equty rsk prema parametrzaton 2. s can be seen from the graphs, t depends on ρ whether the nvestor s better off f he gnores agon completely or f he just gnores the tme dmenson of agon. To get the ntuton, remember that the model wth jont jumps leads to a portfolo that s too conservatve, but reduces the mpact of calculatng the ncorrect senstvtes. For ρ = 0.5, the optmal portfolo ncludes only a small poston n dervatves, so that the model wth jont jumps performs worse than the model wth no agon at all. For ρ = 0.5, on the other hand, the nvestor s better off f he uses the model wth jont jumps, snce the poston n dervatves s now sgnfcantly larger. gan, the utlty loss due to model ms-specfcaton may exceed the utlty gan due to market completeness f the dfferences between the and the agon state are large enough. Ths agan suggests that the nvestor may be better off f he does not use dervatves at all nstead of usng them the wrong way. 26