Evolutionary Portfolio Selection with Liquidity Shocks

Transcription

1 Evolutonary Portfolo Selecton wth Lqudty Shocks Enrco De Gorg Insttute for Emprcal Economc Research Unversty of Zurch Frst Draft: 17th December 2003 Ths Verson: 27th Aprl 2004 Abstract Insurance companes nvest ther wealth n fnancal markets. The wealth evoluton strongly depends on the success of ther nvestment strateges, but also on lqudty shocks whch occur durng unfavourable years, when ndemntes to be pad to the clents exceed collected prema. An nvestment strategy that does not take lqudty shocks nto account, exposes nsurance companes to the rsk of bankruptcy, when lqudty shocks and low nvestment payoffs ontly appear. Therefore, regulatory authortes mpose solvency restrctons to ensure that nsurance companes are able to face ther oblgatons wth hgh probablty. Ths paper analyses the behavour of nsurance companes n an evolutonary framework. We show that an nsurance company that merely satsfes regulatory constrants wll eventually vansh from the market. We gve a more restrctve no bankruptcy condton for the nvestment strateges and we characterze tradng strateges that are evolutonary stable,.e. able to drve out any mutaton. Keywords: nsurance, portfolo theory, evolutonary fnance. JEL Classfcaton Numbers: G11, G22, D81. SSRN Classfcaton: Captal Markets: Asset Prcng and Valuaton, Bankng and Fnancal Insttutons. I am grateful to Thorsten Hens and Klaus Schenk-Hoppé for ther valuable suggestons. Fnancal supports from Credt Susse Group, UBS AG and Swss Re through RskLab, Swtzerland and from the natonal center of competence n research Fnancal Valuaton and Rsk Management and are gratefully acknowledged. The natonal centers of competence n research are managed by the Swss Natonal Scence Foundaton on behalf of the federal authortes.

2 1 Introducton Insttutonal nvestors, lke penson plans or nsurance companes, are usually actve on asset markets that do not provde complete nsurance aganst all possble rsks. The performance of those nsttutons strongly depends on the success of ther nvestment strateges. On the other hand, penson plans and nsurance companes are also exposed to lqudty shocks, that occur when the pensons or clams to be pad out to the clents exceed collected prema. In order to ensure that nsurance companes or penson plans are able to face ther oblgatons wth hgh probablty, regulatory authortes mpose solvency constrants, that are constrants on the nvestment strateges, such that a safely nvested reserve captal exsts. A part from the regulatory constrants, nsttutonal nvestors stll face the problem of choosng the proporton of wealth to be nvested prudently, n order to be able to cover future losses and, therefore, to avod gong bankrupt, but, on the other hand, to also proft from growth opportuntes offered by fnancal markets. In ths paper we analyse the long-run performance of nsurance companes wth an evolutonary model, that s well suted to study the performance of large nsttutonal nvestors, snce they have a consderable mpact on asset prces, face relatvely small transacton costs and ther nvestment horzon s potentally nfnte. Accordng to ths approach, nvestors tradng strateges compete for the market captal and the endogenous prce process s thus a market selecton mechansm along whch some strateges gan market captal whle others lose. Analogously, nsurance companes sell nsurance contracts dependng on ther ablty of facng lqudty shocks, and prema are also endogenously determned by demand and supply. We use the theory of dynamcal systems (Arnold 1998) to derve evolutonary stable tradng stable strateges,.e. those that have the hghest exponental growth rate n a populaton where they determne asset prces. The evolutonary model that we present n ths paper, has one long-lved rsky asset and cash. Wthdrawals and savngs are the dfference between collected prema and pensons or ndemntes to be pad. Here, n partcular, we consder nsurance companes, and the prcng prncple for nsurance contracts s gven by regulatory constrants, set up to ensure that wth ther supply for nsurance contracts, nsurance companes are able to face ther oblgatons wth hgh probablty. Ths relates to the nvestment strateges,.e. the proporton of wealth safely nvested, so that, fnally, regulatory constrants represent mnmal requrements on nvestors nvestment strateges. A bankruptcy occurs when the nvestment s payoff and prema are not enough to pay the ndemntes. Moreover, borrowng and short-sellng are not allowed, so nvestors that go to bankrupt smply dsappear from the market. We establsh a no bankruptcy condton for the nvestment strateges. The no bankruptcy condton s the mnmal suffcent condton on the tradng strateges that ensures that, n the presence of any type of compettor or tradng strategy, the nvestor s able to face almost surely lqudty shocks. In fact, snce asset prces are endogenously determned, t happens that, dependng on other players strateges and wealth shares, wth any strategy that satsfes a less restrctve condton than the no bankruptcy condton, the probablty of gong bankrupt s strctly 1

3 postve. In partcular, f an nvestor s the unque survvor at some pont n tme, the no bankruptcy condton s suffcent but also necessary to avod almost surely gong bankrupt. Nevertheless, whle nvestors wth strateges satsfyng the no bankruptcy condton wll not almost surely go bankrupt, we also show that nvestors who use the smple strategy that corresponds to the no bankruptcy boundary, wll eventually dsappear from the market. Moreover, we characterze tradng strateges that are evolutonary stable, f they exst, followng the dea frst ntroduced by Hens and Schenk-Hoppé (2002a). We gve the condton on the dvdend process and lqudty shocks factor for the exstence of evolutonary stable strateges, when the state of the world follows an..d. process. We show that the condton for the exstence of evolutonary stable strateges s related to the growth rate of the tradng strateges n a neghbourhood of the strategy nvestng accordng to the no bankruptcy boundary. If ths growth rate s strctly postve, then an nvestor puttng more than the no bankruptcy boundary on the rsky assets s able to further ncrease her market share, when asset prces are domnated by the strategy correspondng to the no bankruptcy boundary. However, ths s true only as long as lqudty shocks do not force the nvestor to use all her wealth, or else she wll dsappear from the market. Therefore, whle t can happen that the growth rate of a strategy n a neghbourhood above the no bankruptcy boundary s postve, ths strategy cannot be evolutonary stable, snce t wll almost surely dsappear, because of lqudty shocks. In ths case, no evolutonary stable strateges could exst. Ths result also suggest that n the presence of lqudty shocks, evolutonary stablty should be characterzed n term of both the growth rate and the probablty of default. Ths work contrbutes to the development of the evolutonary portfolo theory, that started wth the semnal paper of Blume and Easley (1992), where an asset market model s frst ntroduced to study the market selecton mechansm and the long run evoluton of nvestors wealth and assets prces. In ther model, Blume and Easley (1992) consder dagonal securtes 1, wth no transacton costs and postve proportonal savng rates are exogenously gven. In the case of complete markets wth dagonal securtes, Blume and Easley (1992) show that there s a unque attractor of the market selecton mechansm and prces do not matter. Wth smple strateges 2 and constant, dentcal savng rates across nvestors, the unque survvor s the portfolo rule known as bettng your belefs (Breman 1961), where the proporton of wealth to be put on each asset s the probablty of the correspondng state of nature. Ths strategy can also be generated by maxmzng the expected logarthm of relatve returns, whch s know as the Kelly rule, studed n dscrete-tme by Kelly (1956), Breman (1961), Thorp (1971) and Hakansson and Zemba (1995) (for an overvew, see also Zemba 2002) and, n contnuous-tme, by Pesten and Sudderth (1985), Heath, Orey, Pesten, and Sudderth (1987) and Karatzas and Shreve (1998), among others. Hens and 1 A system of securtes s called dagonal, f for each state of nature there s exactly one asset whch has a strctly postve payoff. 2 A portfolo rule called a smple strategy, f the proporton of wealth put on each asset s constant over tme. 2

4 Schenk-Hoppé (2002a) proposed a more general settng, wth ncomplete markets, general short-lved assets that re-born each perod and constant, postve, proportonal and, dentcal savng rates across nvestors. In ther evolutonary model, the equlbrum noton refers to wealth dstrbutons that are nvarant under the market selecton process. The authors show that nvarant wealth dstrbutons are generated by a populaton, where only one nvestor (or portfolo rule) exsts (a so-called monomorphc populaton). Moreover, they ntroduce the concept of evolutonary stable portfolo rules, that s also consdered n ths paper. The man result of Hens and Schenk-Hoppé (2002a) s that, n the case of ergodc state of the world processes and wthout redundant assets, there s a unque evolutonary stable portfolo rule, whch s the one that puts on each asset the proporton of wealth correspondng to the expected relatve payoff of the asset. In Evstgneev, Hens, and Schenk-Hoppé (2003) ths result s extended to a model wth long-lved assets, under the assumpton of Markow state of the world. Introducng long-lved assets allows to take nto account the captal gans and losses due to assets prces changes. Ths wll also be of much mportance n the presence of lqudty shocks, as we wll dscuss n ths paper. Moreover, n Evstgneev, Hens, and Schenk-Hoppé (2002) t s also shown that, wth ndependent and dentcally dstrbuted state of world processes, the strategy that nvests accordng to relatve dvdends s the unque smple portfolo rule that asymptotcally gathers total wealth. A generalzaton of the results obtaned by Blume and Easley (1992). Sandron (2000), and Blume and Easley (2002) have also studed the case of long-lved assets, to nclude market prces n the evoluton of wealth shares. The man result of Blume and Easley (2002) and Sandron (2000), s that, wth complete markets, among all nfnte horzon expected utlty maxmzers, those who happen to have ratonal expectaton wll eventually domnate the market and ths result holds ndependently of nvestors rsk averson. In hs model Sandron (2000) also ncludes endogenously determned postve and proportonal savng rates. All these models assume that wthdrawals and savngs are a postve proporton of the current wealth, so that bankruptcy s excluded n ther setup. Moreover, e.g. n Evstgneev, Hens, and Schenk-Hoppé (2003), the wthdrawal rates are assumed to be dentcal among nvestors. Under these assumptons, the only crteron that matters for a tradng strategy to be evolutonary stable, s ts exponental growth rate n the presence of a mutant strategy. Ths paper shows that wth non-proportonal and maybe negatve wthdrawal rates, a second crteron has to be consdered, snce n fact, even f a strategy has the maxmal exponental growth rate n the presence of any mutant, t can dsappear because exogenously determned lqudty shock occurs. In the classcal fnance approach wth exogenously gven prce dynamcs, asset-lablty management models already assume that nvestors maxmze the nvestment s expected payoff less penaltes for bankruptcy or targets not meet (see Carno, Myers, and Zemba 1998, Carno and Zemba 1998). Lu, Longstaff, and Pan (2003) consder a prce dynamc for the rsky asset wth umps (event rsk) and take utlty functons dentcal to for strctly negatve termnal wealth, so that no portfolo rule, that has a strctly postve probablty 3

5 of gong bankrupt, wll be optmal. They obtan lower (snce they do not exclude shortsellng) and upper bounds for the proporton of wealth to be put on the rsky asset and they provde optmal portfolo weghts. Alternatvely, Browne (1997) dstngushes between the survval problem and the growth problem. He frst looks at portfolo rules that maxmze the probablty of survvng n the so-called danger-zone (where bankruptcy has strctly postve probablty to occur) and second, he consders portfolo rules that maxmze the growth rate n the safe-zone, where bankruptcy s almost surely excluded. Browne (1997) dentfes wealth-level dependent strateges, but n hs tme-contnuous setup, no optmal strategy s found for the danger-zone, and a weaker optmalty crteron s ntroduced. The optmal strategy for the safe-zone corresponds to a generalzaton of the Kelly crteron prevously dscussed. Zhao and Zemba (2000) propose a model wth a reward functon on mnmum subsstence,.e. the obectve functon to maxmze equals the sum of the expected fnal wealth and a concave ncreasng functon on the supremum over the wealth levels that are almost surely smaller than fnal wealth. In ths way, the optmal portfolo rule solves a trade-off between expected payoff and mnmum subsstence. The rest of ths paper s organzed as follows. In the next secton we present the model setup. In Secton 3 we derve the no bankruptcy condton on nvestment strateges, that ensures that lqudty shocks do not cause bankruptcy. In Secton 4 we present the man results of the paper. Secton 5 concludes. Techncal results and proofs are gven n the Appendx. 2 An evolutonary model wth bankruptcy Tme s dscrete and denoted by t = 0, 1, 2,... Uncertanty s modelled by a stochastc process (S t ) t Z wth values n some nfnte space S, endowed wth power σ-algebra 2 S. F t = σ(...,s 0,S 1,...,S t ) denotes the σ-algebra gvng all the nformaton avalable at tme t and F = σ ( t Z F t ). Let Ω = S Z be the space of sample paths (s t ) t Z, where s t, t Z s the realzaton of S t on S. Fnally, P denotes the unque probablty measure on (Ω, F) generated by (S t ) t Z. There are = 1,...,I (I 2) nvestors, wth ntal wealth w 0. There s one long-lved rsky asset and cash. Cash s rsk-less both n terms of ts return R = 1 + r 1 and prce, whch s taken as numérare. The rsky asset pays a dvdend D t (s t ) 0 at tme t, dependng on the hstory s t = (...,s 1,s 0,s 1,...,s t ) up to tme t. Moreover, at each tme t each nvestor wthdraws or collects the amount C t(s t ), also dependng on the hstory s t up to tme t. Here, we consder nsurance companes, so that C t s the dfference between the ndemntes to be pad and collected prema. Let w t be the total wealth of nvestor at tme t after clams payment and prema collecton, m t 0 and a t 0 be the unt of cash and rsky asset, respectvely, held by nvestor at tme t and q t be the prce of the rsky asset. The budget constrant at tme t of each nvestor 4

6 s gven by The wealth of nvestor evolves as follows 3 w t = m t + q t a t. (1) w t+1 = (1 + r)m t + (D t+1 + q t+1 )a t C t+1. (2) We say that nvestor goes bankrupt durng perod (t,t + 1] (or smply perod t + 1) ff w t+1 0. In ths case she uses all her wealth to pay the ndemntes and vanshes from the market,.e. we arbtrarly wrte m s = a s = 0 for all s t + 1 (and thus we also set w s = 0 for all s t + 1). Note that the nvestor s wealth at tme t + 1 also depends on the prce q t+1 of the rsky asset, whch s determned at equlbrum by nvestors demand for the rsky asset and supply. Thus, tme t + 1 nvestors strateges may cause a bankruptcy. Let I t = { w t > 0} be the set of nvestors, who survve perod t. Obvously, I t I t 1 and thus m t = a t = 0 for all I t 1. Investor s sad to be the unque survvor at tme t f and only f I t = {}. The next perod amount C t+1 s determned by the followng. At tme t nvestor can decde to sell δ t 0 nsurance contracts on one sngle future stochastc clam X t+1 0 (whch s dentcal for all nvestors). The premum P t+1 of each contract s F t -measurable (depends only on nformaton avalable up to tme t), determned by the market clearng condton on the nsurance market at tme t, and s pad at tme t + 1 by the buyer of the nsurance contract, who s supposed to be external to the economy ust defned,.e. buyers of nsurance contracts do not partcpate to the fnancal market. The amount collected or wthdrawn by nvestor at tme t + 1 s then gven by the dfference C t+1 = δ t (X t+1 P t+1 ) between clams and prema. We suppose that the nsurance market s regulated and solvency constrants are mposed. Each nvestor should be able to meet her oblgaton, n a way that condtonng on the current hstory s t only a proporton α t > 0 of her current wealth wll be affected wth a small probablty ǫ t > 0,.e. where for all t and I t P [ δ t (X t+1 P t+1 ) > α t w t s t] = ǫ t (3) α t (0,α) and ǫ t ǫ. (4) 3 To be formally correct, the wealth evoluton of equaton (2) should be replaced by w t+1 = [ (1 + r)m t + (D t+1 + q t+1 )a t C t+1] +, where for x R, x + = max(0,x). We prefer to keep the notaton smpler and snce we are essentally lookng at strateges that survve n the long run, the wealth evoluton of those strateges s correctly gven by equaton (2). 5

7 The parameters α and ǫ are exogenously gven by regulatory authortes. Equaton (3) defnes the prcng rule for nsurance contracts and s called quntle prncple and t has been dscussed n Schneper (1993) and Embrechts (1996). Moreover, t corresponds to the proportonal value-at-rsk constrants studed by Leppold, Vann, and Troan (2003) wth tme ndependent proportonal factors, and sameness between nvestors n ther general equlbrum consderaton. The parameters α t and ǫ t are fxed and can be nterpreted as the loss acceptablty of nvestor and, n the general settng of the model, we assume that they can vary between nvestors. Other smplfyng assumptons wll be ntroduced later. For a gven premum P t+1 and parameters α t and ǫ t, equaton (3) serves to compute the number δ t of nsurance contracts that nvestor can sell, n order to satsfy the solvency constrant. The premum P t+1 s determned endogenously when the nsurance market clears. We should bear n mnd that for an nvestor, gong bankrupt means vanshng from the market and thus should be avoded! They can further decrease ther nsurance rsk by choosng a smaller α or a smaller ǫ. As we wll see below, an nvestor wth a small α, who s a safer nvestor wth respect to mnmal solvency requrement, s also forced to reduce her exposure to the nsurance market, losng n ths way growth opportuntes when clams are less than prema. The amount α t w t represents the techncal reserve or the proporton of current wealth to be nvested prudently by nvestor to make the rsky nsurance busness acceptable n the future (see Norberg and Sundt 1985). If the amount δ t (X t+1 P t+1 ) s strctly greater than the techncal reserves, we say that nvestor faces a lqudty shock. From equaton (3), nvestor faces lqudty shocks wth probablty ǫ t durng perod t + 1. In our settng, the solvency constrant essentally mposes that m t α t R w t, for all, or equvalently m t w t α t R, I t. (5) Let µ t = E [ X t F t 1] and σt 2 = Var(X t F t 1 ) be the condtonal expectaton and the condtonal varance of X t, gven F t, respectvely, and let F t be the condtonal cumulatve dstrbuton functon of Y t = Xt µt σ t,.e. F t (y) = P [ Y t y F t 1]. Moreover, Ft 1 denotes the generalzed nverse of F t. To avod the premum P t+1 fully coverng the nsurance rsk, we mpose the followng restrctons Assumpton 1 (Insurance market). For t Z and I t, let (α t,ǫ t) and ( α t, ǫ t) be two possble choces for the loss acceptablty parameters of nvestor. Let α t = α t. Then for all prema P t+1 δ t > δ t ǫ t > ǫ t. Ths assumpton says that for gven techncal reserves, the probablty of havng lqudty shocks strctly ncreases wth the number of nsurance contracts sold. If ths s not satsfed, 6

8 then t would be possble to cover addtonal nsurance rsk only through collected prema, whch s not a far prcng rule. Snce δ t = 0 solves equaton (3) wth α t = 0 and ǫ t = 0, Assumpton 1 also mples that an nsurance company wthout techncal reserves that sells a strctly postve number of contracts, faces lqudty shocks wth a strctly postve probablty. Assumpton 1 ndrectly mposes restrctons on equlbrum prema, as shown n the followng lemma. Lemma 1. If Assumpton 1 holds, then for all t Z and I t : Proof. Let us suppose that P t+1 < µ t+1 + σ t+1 F 1 t+1(1 ǫ t). P t+1 µ t+1 + σ t+1 F 1 t+1(1 ǫ t). for some t and I t. Then and thus for all δ > 0 P t+1 µ t+1 σ t+1 F 1 t+1(1 ǫ t) P [ δ (X t+1 P t+1 ) > 0 ] = P [ X t+1 P t+1 > 0 ] = P [ Y t+1 > P t+1 µ t+1 ] ǫ σ t t+1 ndependently from δ. Ths contradcts Assumpton 1, snce the last nequalty shows that the probablty of lqudty shocks s n fact ndependent of the number of contracts, wth fxed techncal reserves. From equaton (3) and Lemma 1, we obtan [ ] δt µt+1 + σ t+1 Ft+1(1 1 ǫ t) P t+1 = α t wt, (6) or δ t = α t w t µ t+1 + σ t+1 F 1 t+1(1 ǫ t) P t+1. (7) Lemma 1 ensures that δ t 0. Equaton (7) says that nvestor supply for nsurance contracts s proportonal to her techncal reserve and decreases wth ncreasng probablty ǫ t. For a fxed supply of nsurance contracts, an nvestor can therefore decrease her techncal reserve by decreasng her lqudty shock probablty ǫ t. Naturally, the solvency constrants (3) and (4) do not take nto account the magntude of a lqudty shock! Ths s a well know crtque of quntle constrants (see e.g. Artzner, Delbaen, Eber, and Heath 1997). We assume that demand for nsurance contracts s normalzed to 1,.e. δ t = 1 for all t. It follows: P t+1 = µ t+1 + σ t+1 δt Ft+1(1 1 ǫ t) αt wt. (8) 7

9 σ t+1 δ t F 1 t+1(1 ǫ t) α t w t s the so-called loadng factor and s supposed to be strctly postve. In fact, t s well known from the run theory, that f P t+1 µ t+1,.e. f the premum at tme t s less or equal to the condtonal expectaton of next perod clams gven all nformaton avalable at tme t, then for any value for the ntal wealth (wthout fnancal market) the probablty of gong bankrupt s equal one (see Feller 1971, page 396). From the last equaton, we see that the premum of the nsurance contract ncreases wth ncreasng condtonal varance, as one would expect, and decreases when the weghted wealth I =1 α t w t ncreases. Moreover, a safer nvestor, wth a smaller α or a smaller ǫ than a rsker nvestor, contrbutes to an ncrease of the premum, from whch all nvestors beneft. Ths behavour has also been descrbed by Ceccarell (2002). Equatons (7) and (8) can be solved for δ t and P t+1 : they provde a unque soluton wth a strctly postve premum (ths wll become clear for the specal case consdered below; however, we gve a general proof of the exstence and unqueness of a soluton n the Appendx 6.1). Snce the goal of ths paper s to analyse nvestors long-run wealth evoluton wth respect to ther nvestment strateges on fnancal markets, we assume that ther profles on nsurance markets are dentcal, meanng that they possess the same loss acceptablty parameters. Here, we do not address the queston of nvestors strateges (choce of the loss acceptablty parameters) on the nsurance market. It s not clear whether an nvestor who has hgher loss acceptablty, wll growth faster or not. In fact, whle t s true by equaton (7) that hgher loss acceptablty means greater lqudty shocks (for both the probablty and the amount), t must also be sad that nvestors who sell a larger number of contracts beneft from growth opportuntes when prema are greater than clams. Moreover, less techncal reserves means a smaller exposure to lqudty shocks (as dscussed above), but also less restrctve constrants for the nvestment strateges, meanng that those nvestors can put less money nto the rsk-free asset and proft from growth opportuntes on the fnancal market. We address these ssues n other works. Here, as n Leppold, Vann, and Troan (2003) we make the followng assumpton. Assumpton 2 (Loss acceptablty). Investors loss acceptablty s constant over tme and dentcal for all nvestors,.e. α t = α (0,α) and ǫ t = ǫ for all t and = 1,...,I. Note that by equatons (7) and (8), when all nvestors possess the same ǫ t, δ t does not depend on ǫ t anymore and therefore the magntude of lqudty shocks s mnmzed for all nvestors f ǫ t = ǫ. Moreover, by Assumpton 2 and equatons (7) and (8) t follows P t+1 = µ t+1 + σ t+1 F 1 t+1(1 ǫ) α w t, (9) δ t = wt, (10) w t 8

10 and therefore nvestor s supply for nsurance contracts corresponds to her relatve wealth. Now, we ntroduce a precse structure for the clam X t+1. In partcular, we assume that the total clam X t+1 s proportonal to the aggregate wealth avalable at tme t, meanng that the amount of nsured clams ncreases or decreases dependng on the aggregate success of the nvestors (a smlar assumpton wll be also made for the dvdend process). Ths assumpton also prevents a shock from destroyng the economy. The proportonal factor s supposed to be ndependent of the hstory up to tme t and can be nterpreted as the lqudty shock factor for the economy. Mathematcally we have X t+1 = η t+1 W t, (11) where η t+1 [0, 1] s ndependent of F t and W t = I t w t = I =1 w t s the aggregate wealth avalable n the economy at tme t. From equaton (11) t follows that µ t+1 = W t E [ η t+1 ] and σ 2 t+1 = W 2 t Var(η t+1 ). Moreover, η t+1 G t+1 where F t+1 (y) = G t+1 ( y W t ), y. Thus P t+1 = ( µ(η t+1 ) + σ(η t+1 )G 1 t+1(1 ǫ) α ) W t. Therefore, the premum P t+1 s strctly postve for all t, f the loadng factor (σ(η t+1 )G 1 t+1(1 ǫ) α)w t s greater than zero for all t. Moreover, for the sake of smplcty, we make the followng assumpton: Assumpton 3 (Lqudty shocks). Lqudty shocks (η t ) t 1 are ndependent and dentcally dstrbuted,.e. G t = G for all t, η t η G, where G s a contnuous cumulatve dstrbuton functon. Let µ = E [ η ] and σ 2 = Var(η), then by Assumptons 2 and 3, P t+1 = µw t + σ G 1 (1 ǫ)w t α W t = (β α)w t, (12) C t+1 = (η t+1 β + α) w t, (13) where β = µ + σ G 1 (1 ǫ). As dscussed above for the general case, we mpose that the loadng factor (σ G 1 (1 ǫ) α) W t s strctly postve,.e. α < mn{α,σ G 1 (1 ǫ)}. Then β α > β σ G 1 (1 ǫ) = µ > 0 and thus P t+1 > 0 for all t. We now turn our attenton to the fnancal market. We suppose that the rsky asset s n fxed supply, normalzed to one. Instead, the supply of cash s exogenously gven by cumulated dvdends and collected prema less wthdrawals. The market clearng condtons are I a t = a t = 1 (14) I t =1 M t = R I t m t 1 + D t I t a t 1 q t 9 ( 1 I t a t 1 ) C t (15)

11 where M t = I t m t and C t = I t C t. Note that I t m t 1 M t 1 = I t 1 m t 1 snce I t I t 1. Moreover, f no bankruptcy occurs durng perod t, then I t = I t 1 and the usual equaton for M t follows,.e. M t = R M t 1 + D t C t. Note that M t 0 for all t. In fact f for some t, M t < 0, then there exsts at least one nvestor, say I t, wth m t < 0. But snce borrowng s not allowed, nvestor s forced nto bankruptcy durng perod t, a contradcton to I t. To be consstent wth Assumpton 3, and n order to avod that dvdends become very small as compared to nsurance shocks, we make the followng assumpton for the dvdend process: Assumpton 4 (Dvdend process). () For each t, () D t = d t W t 1, for some process (d t ) t>0, wth d t d H ndependently and dentcally dstrbuted wth cumulatve dstrbuton functon H on [0, 1]. P [ d > 0 ] = 1 H(0) (0, 1),.e. at each tme dvdends have strctly postve probablty of beng zero and of beng strctly postve. Ths assumpton, together wth Assumpton 3, solves the dffculty encountered by Hens and Schenk-Hoppé (2002b), where the rate of return on the long-lved asset eventually domnates that of the numérare, so that the strategy that nvests only n long-lved asset s able to drve out any other strategy. Hens and Schenk-Hoppé (2002b) suggest to base evolutonary fnance model on Lucas (1978), where assets payoffs are n term of a sngle pershable consumpton good. In ths way, the consumpton rate s at least as the growth rate of the total payoff of the market. In our model, also wthout relayng on Lucas (1978), the prcng rule for nsurance contracts (that also determnes C t ) and, Assumpton 3 and 4, ensure that the rate of consumpton ncreases proportonally to the growth rate of the total payoff. Moreover, as wll dscuss later, f assets payoffs were n term of pershable consumpton goods, t would not be possble to fnd a tradng strategy that preserves the wealth (the reserve captal n the nsurance busness) and have postve growth rate. Let λ t [0, 1] be the proporton of wealth nvested n the rsky asset by nvestor I t at tme t. We have a t = λ t w t q t and m t = (1 λ t)w t. We call the sequence (λ t) {t>0 It} the tradng strategy of nvestor and λ t the strategy of nvestor at tme t. We use the conventon that λ t = 0 f / I t. Note that λ t s a random varable,.e. t depends on the state of the world up to tme t, s t. Other assumptons on the 10

12 process defnng the tradng strategy (λ t) t 0 wll be ntroduced later. Here, we ust mpose the followng restrcton on the strateges at tme t, (λ t) It, to prevent the prce of the rsky asset from becomng zero. Assumpton 5 (Investors strateges). For each t such that I t > 1, there exsts, I t wth (1 λ t)λ t > 0. Assumpton 5 essentally states that f more than one nvestor survves perod t, then there exsts at least one survvor wth a strctly postve proporton of her wealth nvested n the rsky asset and one survvor wth a strctly postve proporton of her wealth nvested n the rsk-free asset. Naturally, when a survvor has a mxed strategy 4 λ t (0, 1), then Assumpton 5 s obvously satsfed wth =. If I t = 1, then t mght occur that the unque survvor uses a strategy nvestng all her wealth n the rsk-free asset. The strategy λ t = 1 s excluded by the solvency constrant. In fact, the solvency condton stated by equaton (5) s equvalent to 1 λ t α R λ t 1 α R =: λ (0, 1), (16).e., for each nvestor, the proporton of wealth nvested n the rsky asset s bounded from above by λ. It seems to be a natural restrcton for an nsurance company (or a penson fund), as shown e.g. n Davs (2001, Tables 5 and 6) for lfe nsurances and penson funds of several countres. Let λ t = (λ 1 t,...,λ I t), then the market clearng condton for the rsky asset (14) mples q t = λ tw t. Note that for / I t, wt = 0 by assumpton and thus λ tw t = I t λ t wt. We rewrte equaton (2) as follows [ ] wt+1 = R (1 λ t) + (d t+1 W t + q t+1 ) λ t (η t+1 β + α) w q t. (17) t 3 The no bankruptcy condton Before dscussng the long run wealth evoluton of nvestment strateges, we gve condtons for avodng bankruptcy. In fact, a necessary condton for long-term survval s not to go bankrupt and strateges that do not almost surely exclude bankruptcy are avoded by nvestors wth long-term horzon. Therefore, as n Browne (1997) and Lu, Longstaff, and Pan (2003), we dstngush between the condtons on the strateges to avod gong bankrupt and 4 A strategy (λ t) t s called a mxed strategy, ff t assgns a strctly postve percentage to every asset, for all t. In our settng, a mxed strategy s characterzed by λ t (0,1) for all t (see Evstgneev, Hens, and Schenk-Hoppé 2002). 11

13 then, gven that nvestors satsfy those condtons, we analyses the long-term wealth evoluton. In our settng, analogously to Lu, Longstaff, and Pan (2003), we obtan upper bounds for the λ t s (a lower bound s gven by the no short sale restrcton). We wll show below that an nvestor wth a strategy that does not prevent bankruptcy at each perod, has a strctly postve probablty of vanshng from the market, even f she s the unque survvor. Moreover, f an nvestor uses a smple strategy that does not prevent bankruptcy, she has probablty 1 of vanshng from the market, even f at some pont n tme she s the unque survvor and thus domnates assets prces. In partcular, an nvestor holdng only the rsky asset (.e. λ t = 1) becomes extnct wth probablty 1. Ths result shows that Theorem 1 n Hens and Schenk-Hoppé (2002b) does not hold when bankruptcy can occur. We frst consder the case I t = 1 for some t > 0,.e. I t = {} for some {1,...,I}. We restrct ourself to strateges λ t > 0. If λ t = 0, as s clearly excluded snce R > sup supp(η) β + α 5! The prce of the rsky asset at tme t s gven by q t = λ t w t and the aggregate wealth at tme t s W t = w t: from equaton (17) t follows mmedately that I t+1 R (1 λ t) (η t+1 β + α) + d t+1 > 0. Let η = nf supp(η), η = sup supp(η) and d = sup supp(d) and K be the contnuous multvarate cumulatve dstrbuton of (η,d) on [η,η] [0,d],.e. Moreover, let K(z) = P [ d η z ] = η η functon of d η. Then K(x,y) = P [ η x, d y ]. x+z 0 dk(x, y) be the cumulatve dstrbuton P [ I t+1 ] = P [ dt+1 η t+1 > R (1 λ t) β + α ] = 1 K( R (1 λ t) β + α) and thus P [ I t+1 ] = 1 λ t R + k + β α R =: λ, (18) where k = nf supp( K). We call ths latter equaton the no bankruptcy condton. Note that λ = λ + β + k R. Therefore, the solvency constrant (16) s a stronger condton on the strateges than the no bankruptcy condton (18), f β > k,.e. f hgher shocks (greater than β) on the 5 supp(η) denotes the support of η. 12

14 nsurance market and small dvdends (less the η β) n the fnancal market do not occur smultaneously, whch s not a realstc assumpton. Ths s due to the fact that the solvency constrant does not care about dvdends, and thus does not take nto consderaton the (postve) correlaton between shocks and dvdends, such that hgher shocks wll have a smaller mpact on the wealth evoluton snce they correspond to hgher dvdends. If β < k (whch s the most common case, as for example when nsurance shocks and dvdends are consdered ndependent), the no bankruptcy condton (18) s stronger than the solvency constrant and thus nvestors ust care about the no bankruptcy condton (18). In ths case, the solvency constrant (16) does not elmnate bankruptcy! In the sequel we make the followng assumpton on the ont dstrbuton of (η,d): Assumpton 6 (Shocks and dvdends ont dstrbuton). For all δ 1 > 0 and δ 2 > 0, P [ η > η δ 1, d δ 2 ] > 0,.e., bg shocks and very small dvdends have strctly postve probablty to ontly occur. Assumpton 6 mples the followng Lemma on the dstrbuton of d η. Lemma 2. For all δ > 0, P [ d η η + δ ] > 0 and thus k = η,.e. maxmal shocks and zero dvdends have strctly postve probablty to ontly occur. Proof. P [ d η η + δ ] = P [ d η η + δ ] = P [ ] [ ] d η η + δ η η > δ 1 d P η η > δ1 0<δ 1 <δ P [ ] [ ] d δ 1 + δ η η > δ 1 d P η η > δ1 0<δ 1 <δ P [ ] d δ 1 + δ, η > η δ 1 = 0<δ 1 <δ P [ ] d P [ ] η η > δ 1 η > η δ 1 }{{} >0 > 0 Thus, K( η + δ) > 0 for all δ > 0,.e. k = η. Under Assumpton 6, the strategy λ corresponds to R η+β α and s a stronger condton on R the strateges than the solvency constrant, snce obvously β < η. From now on, we take λ = R η + β α. R 13

15 Let us now consder a sngle survvor wth a smple strategy λ > λ. Then at each perod she wll have a strctly postve probablty of gong bankrupt and therefore P [ t I t ] = 0, meanng that she wll vansh almost surely from the market. We state these results n the followng Lemma. Lemma 3. Let I t = {} for some t and {1,...,I},.e. nvestor s the unque survvor at tme t. The followng holds: () If λ t > λ, then nvestor has strctly postve probablty of gong bankrupt durng perod t + 1. () If λ s > λ for all s t, then nvestor wll almost surely eventually vansh from the market. In partcular, f nvestor uses a smple strategy λ > λ, then she wll eventually almost surely vansh from the market almost surely. Let us now consder the case I t > 1. Wthout loss of generalty we set I t = {1, 2}: f I t = { 1,..., n } wth n = I t > 2, then we can stll reduce the orgnal settng to a 2-nvestors settng by defnng a new nvestor wth strategy ξ s [0, 1] at tme s {t,t+1} and wealth w s, where ξ s = n l=2 λ l s w l s n l=2 w l s, w s = The prce of the rsky asset at tme s {t,t + 1} s then gven by q s = λ 1 s w 1 t + ξ s w s. Thus let us assume that I t = {1, 2}. Then from the wealth evoluton (17) t follows mmedately that for = 1, 2 where. I t+1 R (1 λ λ t t) + d t+1 W t + w t+1 λ λ t t+1 (η t+1 β + α) > 0, (19) q t q t n l=2 w l s. Proof. () Suppose that I t+1. Then w t+1 > 0 and by equaton (17) w t+1 = R (1 λ t)w t + d t+1 W t λ t w t q t + and thus w t+1 ( ) 1 λ λ t wt t+1 q t + ( ) wt+1 1 λ 1 t+1 + wt+1 2 λ 2 λ t wt t+1 (η t+1 β + α) w q t, t = R (1 λ t)wt λ t wt + d t+1 W t q t +w t+1 λ λ t wt t+1 (η t+1 β + α) w q t, t 14

16 ( ) where. Snce λ t+1 1 (solvency restrcton), then 1 λ λ t w t t+1 q t > 0, and thus from wt+1 > 0 t follows that R (1 λ t)wt λ t wt + d t+1 W t + w t+1 λ λ t wt t+1 (η t+1 β + α) wt > 0. q t q t Snce I t, then w t > 0 and therefore dvdng the last nequalty by w t we obtan () Suppose now that R (1 λ λ t t) + d t+1 W t + w t+1 λ λ t t+1 (η t+1 β + α) > 0. q t q t R (1 λ λ t t) + d t+1 W t + w t+1 λ λ t t+1 (η t+1 β + α) > 0, q t q t where. Then for I t, [ ] wt R (1 λ λ t t) + d t+1 W t + w t+1 λ λ t t+1 (η t+1 β + α) > 0, q t q t and thus w t+1 > 0, snce w t+1 = w t R (1 λ λ t) + d t+1 W t t q t + w t+1 λ λ t t+1 q t (η t+1 β + α) 1 λ λ t w t t+1 q t and 1 λ λ t w t t+1 q t > 0 by equaton (16). + The necessary and suffcent condton (19) for avodng bankruptcy for nvestor also depends on other nvestors wealths and strateges, through the term w t+1 λ λ t t+1. Speculatng on other nvestors behavour, nvestor could essentally put less wealth on the wt rsk-free asset than allowed under the no bankruptcy condton (18). Whle ths would mply a strctly postve probablty of gong bankrupt when nvestor domnates assets prces, the no bankruptcy condton s not necessary for avodng almost surely bankruptcy n the presence of compettors, when they sgnfcantly nvest n the rsky asset. However, the no bankruptcy condton s the mnmal condton on nvestment strateges that almost surely elmnates bankruptcy n the presence of each type of compettor. In fact, an nvestor who systematcally volates the no bankruptcy condton (18), wll eventually dsappear from the market wth probablty one, f her opponents are nvestng all ther wealth on the rsk-free asset,.e. an nvestment strategy that systematcally volates the no bankruptcy condton 15

17 s almost surely drven out by the rsk-free strategy. Thus the no bankruptcy condton s the mnmal condton that ensures that each nvestor wll not go bankrupt wth probablty 1, regardless from other nvestors behavour. In the sequel, because of the long horzon perspectve consdered here, and followng the approach of Lu, Longstaff, and Pan (2003), we use the no bankruptcy condton to ensure that nvestors almost surely do not face bankruptcy. In ther settng, bankruptcy s penalzed wth mnus nfnty utlty, so that no optmal strategy wll allow fnal negatve wealth wth strctly postve probablty. 4 The man results From the prevous secton, t s clear that an nvestor who uses a strategy that does not almost surely elmnate bankruptcy, wll eventually dsappear from the market, also f at some pont n tme she s the unque survvor. Thus, lookng at the long-run evoluton of nvestors wealths, a strategy that does not prevent bankruptcy would not be ft, as t s defned n Blume and Easley (1992). Nether can t be evolutonary stable, as defned n Hens and Schenk-Hoppé (2002a) and Evstgneev, Hens, and Schenk-Hoppé (2003), snce t wll also dsappear almost surely f t domnates asset prces, as shown n Lemma 3. Moreover, as long as dvdends and lqudty shocks are not postve correlated, the solvency restrcton s not enough to avod bankruptcy. In fact, under the solvency restrcton, an nvestor faces lqudty shocks wth strctly postve probablty and f her nvestment strategy provdes small payoff (n partcular, dvdends are small), then the lqudty shock destroys her wealth. Followng the dscusson of the prevous secton, we consder the case where the followng assumpton holds and we study the long run evoluton of nvestors, who are safe enough on ther nvestment postons to be almost surely able to face lqudty shocks. Assumpton 7 (The no bankruptcy condton). For all I t and all t Z, λ t [0,λ]. Followng Hens and Schenk-Hoppé (2002b), we rewrte the wealth dynamcs. We defne and By Assumpton 7, we have B t = λ t w t λ t w t, A t = R (1 λ t)w t + B t d t+1 W t (η t+1 β + α) w t. w t+1 = A t + B t λ t+1 w t+1 16

18 or (I B t λ t+1)w t+1 = A t where A t = (A 1 t,...,a I t), B t = (B 1 t,...,b I t ) and I s the dentty on R I. Note that for / I t, A t = B t = 0. The nverse of I B t λ t+1 s gven by I + (1 λ t+1 B t ) 1 B t λ t+1, provded that λ t+1 B t 1 (see Horn and Johnson 1985, Sec ). It can be easly checked that λ t+1 B t < 1 f there exts an nvestor I t+1 wth λ t < 1 and λ t+1 > 0 and ths s stll the case when I t > 1, by Assumptons 5 and 7. If I t = {} for some, then nvestor s already the unque survvor and the wealth evoluton s easly obtaned. Therefore, n the sequel we only consder the case I t > 1. Under the assumpton of no default durng perod t + 1, the wealth evoluton can then be wrtten as and the -th component s gven by w t+1 = [ w t (1 λ t+1)λ t w t w t+1 = ( I B t λ t+1) 1 At ( = I + B tλ ) t+1 1 λ A t, (20) t+1b t d t+1 W t λ t + [ R(λ λ t) + (η η t+1 ) ] ( λ tw t + (λ t λ t)λ t+1 w t )]. Ths result s explctly derved n the Appendx 6.2. We use that The prce at tme t + 1 follows: λ = R η + β α R R + β α = R λ + η. q t+1 = λ t+1 A t + λ t+1 B t λ t+1 A t 1 λ t+1 B t = λ t+1 A t 1 λ t+1 B t. Let r t = w t W t, and ζ t = λ t λ for = 1,...,I and t Z. The vector r t = (r 1 t,...,r I t ) s the vector of wealth shares,.e. r t I 1 = {r R + r t = 1}. By Assumpton 7, ζ t [0, 1] and ζ t = 1 ff λ t = λ. We obtan wt+1 rt W t = (1 λζ t+1)ζ t r t [ d t+1 ζ t + [ R λ (1 ζ t) + (η η t+1 ) ] ( ζ tr t + λ (ζ t ζ t )ζ t+1 r t )]. 17

19 Let θ t+1 be defned by θ t+1 = ζ tr t d t ( ) R λ (1 ζt k ) + (η η t+1 ) k r k t ( ζ tr t + λ (ζ k t ζ t )ζ t+1 r t ). Then The rato θ t+1 (1 λ ζ t+1 ) ζ t r t W t+1 = θ t+1 (1 λζ t+1)ζ t r t W t. s the growth rate of the economy. From the wealth evoluton of equaton (20), we obtan the evoluton of wealth shares: rt+1 = r t θ [ t+1 d t+1 ζt + [ R λ (1 ζt) + (η η t+1 ) ] ( ζ tr t + λ (ζ t ζ t )ζ t+1r t )]. From ths last equaton t follows drectly that rt+1 = 0 f rt = 0 and therefore also, rt+1 = 1 f rt = 1. Wthout any addtonal assumpton on the dvdend process, the lqudty shock factor and the nvestment strateges we are now able to prove that a tradng strategy that corresponds to the no bankruptcy boundary λ, s almost surely drven out by any strategy that s bounded away from λ. From equaton (21) t follows that for,k I t, ( ) rt+1 r = t rt+1 k rt k ( d t+1 ζt + [Rλ(1 ζt) + (η η t+1 )] ζ T t r t + λ ) (ζ t ζ t )ζt+1r t d t+1 ζt k + [ Rλ(1 ζt k ) + (η η t+1 ) ]( ζ T t r t + λ ). k (ζk t ζ t )ζt+1r t Let us now suppose that only two nvestors exst. The frst nvestor s usng a smple strategy correspondng to the no bankruptcy boundary,.e. λ 1 t = λ for all t. The second nvestor s usng a strategy whch s bounded away from the no bankruptcy condton, as well as from the strategy puttng the wealth only on the rsk-free asset,.e. δ < λ 2 t < λ δ for all t > 0 and for some δ > 0. Usng the notaton ntroduced above, we have ζt 1 = 1 for all t and ζt 2 (δ, 1 δ) for all t and δ = δ > 0. We obtan the followng result. λ Theorem 1. Under Assumptons 3-7, and gven an nvestor wth ζ 1 t = 1 for all t > 0 and an nvestor wth ζ 2 t (δ, 1 δ) for all t > 0 and some δ > 0, the nvestor wth the smple 18 (21)

20 strategy correspondng to the no bankruptcy boundary, wll almost surely vansh from the market. Proof. On {d t+1 η t+1 > η} (by Assumptons 4 and Assumpton 6, ths set has probablty one) we have ( ) rt+1 2 r 2 = t dt+1 ζt 2 + [R λ (1 ζt 2 ) + (η η t+1 )] ( ) ζ T t r t (1 ζt 2 )λrt 1 rt+1 1 rt 1 d t+1 + (η η t+1 ) ( ) ζ T t r t + (1 ζt 2 )λζt+1 2 rt 2 ( ) r 1 = t r 2 t d t+1ζt 2 + [R λ(1 ζt 2 ) + (η η t+1 )] [1 (1 ζt 2 )λ (1 ζt 2 )(1 λ)rt 2 ] d t+1 + (η η t+1 ) [ ] 1 (1 ζt 2 )(1 λζt+1)r 2 t 2 ( ) ( r 1 t dt+1 ζt 2 + [1 (1 ζt 2 )λ] [R λ (1 ζt 2 ) + (η η t+1 )] mn, rt 2 d t+1 + (η η t+1 ) ) ζt 2 d t+1 + [R λ (1 ζt 2 ) + (η η t+1 )] d t+1 + (η η t+1 ) [ ] ζt 2 (1 λζt+1) 2 + λ ζt+1 2 ( ) ( r 1 t dt+1 ζt 2 + [1 (1 ζt 2 )λ] [R λ (1 ζt 2 ) + (η η t+1 )] mn, rt 2 d t+1 + (η η t+1 ) ) ζt 2 d t+1 + [R λ (1 ζt 2 ) + (η η t+1 )] d t+1 + η η t+1 ( ) r 1 t R λ (1 δ)δ > 0. d t+1 + η η t+1 r 2 t By the frst nequalty, we use that r dζ + [R λ (1 ζ) + (η η)] [1 (1 ζ)λ (1 ζ) (1 λ)r] [ d + (η η) 1 (1 ζ) (1 λ ζ)r ] s strctly ncreasng, strctly decreasng or flat as a functon of r, dependng on the parameters d, η, ζ, ζ, λ and R. Thus, the mnmum of the functon s attaned for r = 1 or r = 0. By the second nequalty, we use that [ ] ζ 2 t (1 λζt+1) 2 + λζt+1 2 < 1, for all ζ 2 t,ζ 2 t+1 [0, 1]. Iteratvely, we obtan log r2 t+1 r 1 t+1 t+1 ( ) R λ (1 δ)δ log d τ + η η τ τ= log r2 0. r0 1

21 Let ǫ < R (1 δ)δ λ, then log r2 t+1 r 1 t+1 C t+1 1 {dτ+η η τ ǫ} + log r2 0 r0 1 τ=1 where C = log R (1 δ) δ λ ǫ lm t by Assumpton 6. Thus > 0 by defnton of ǫ. By the Theorem, ( t+1 ) 1 C lm 1 {dτ+η η t t + 1 τ ǫ} + log r2 0 r0 1 1 t + 1 log r2 t+1 r 1 t+1 r 2 t 1 r 2 t = r2 t r 1 t τ=1 = C K(d η η + ǫ) = γ > 0, exp(t γ) as t,.e r 2 t 1 almost surely. The theorem states that, whle beng at the boundary of the no-bankruptcy condton means that bankruptcy s excluded wth probablty one, the market selecton mechansm stll forces such an nvestor to vansh from the market, f other nvestors are usng strateges that are bounded away from λ. Therefore, the tradng strategy λ t = λ cannot be evolutonary stable as defned by Evstgneev, Hens, and Schenk-Hoppé (2003). In fact, even f ths strategy possesses almost the entre wealth, an nvestment strategy that s bounded away from λ s able to drastcally perturb the dstrbuton of wealth shares and to drve out λ t. We next ask the queston about nvestment strateges that are evolutonary stable, referrng to Hens and Schenk-Hoppé (2002a) and Evstgneev, Hens, and Schenk-Hoppé (2003). The evoluton of wealth shares from equaton (21) can be wrtten as follows. For = 1,...,I let f (r t,t) = r t θ t+1 [ d t+1 ζ t + ( ) Rλ(1 ζt) + (η η t+1 ) ( ζ tr t + λ (ζ t ζ t )ζ t+1 r t )]. (22) Then or r t+1 = f (r t,t) r t+1 = f(r t,t), (23) where f = (f 1,...,f I ). Although t does not appear explctly n the defnton of f t, the functon f t also depends on the state of the world s t+1 up to tme t + 1, through nvestors strateges at tme t + 1, the dvdend d t+1 and the lqudty shock factor η t+1. We make the followng addtonal assumpton on the tradng strateges to make f ndependent from t and, therefore, the market selecton mechansm statonary, also because Assumptons 3 and 4 on (η t ) t Z and (d t ) t Z, respectvely. 20

22 Assumpton 8 (Statonary tradng strateges). The tradng strateges are statonary,.e. for all t Z and all I t λ t(s t ) = λ (s t ). The market selecton process (23) generates a random dynamcal system (see Arnold 1998) on the smplex I 1. Gven a vector of ntal wealth shares r I 1 and t > 0, the map φ(t,ω,r) = f(s t, ) f(s t 1, ) f(s 1,r), (24) on N Ω I 1 gves the nvestors wealth shares at tme t, f the state of the world s ω = (s t ) t Z, and φ(0,ω,r) = r. In the sequel we characterze vectors of wealth shares that are nvarant under φ. We ntroduce the followng defnton. Defnton 1 (Fxed pont). The vector of relatve wealth shares r I 1 s called a determnstc fxed pont of φ, f and only f φ(t,r, ) = r almost surely for all t. The dstrbuton of market shares r s sad to be nvarant under the market selecton process (23). Clearly, r s a determnstc fxed pont of φ f and only f f(r) = φ(1,r, ) = r almost surely. Therefore, the vectors of wealth shares r = e for = 1,...,I are determnstc fxed ponts of φ, where { 1 f = e, =. 0 else The followng lemma shows that e are the unque determnstc fxed ponts of φ. Ths result also holds f Assumpton 8 s not satsfed. Lemma 4. Let r be a determnstc fxed pont of φ. Then r = e for some = 1,...,I. Proof. Let assume that r = rt+1 = rt (0, 1). Then ( ) ( θ t+1 = d t+1 ζt + R λ (1 ζt) + (η η t+1 ) ζ tr t + λ (ζ t ζ t )ζ t+1 r t ), or equvalently d t+1 ζt k rt k + + k k ( ) R λ (1 ζt k ) + (η η t+1 ) r k t ( ζ tr t + λ (ζ k t ζ t )ζ t+1 r t ) = d t+1 ζt (1 rt) + ( )( +(1 rt) R λ (1 ζt) + (η η t+1 ) ζ tr t + λ (ζ t ζ t )ζ t+1 r t ). (25) 21

23 Snce 1 r t = k rk t, the rght-hand sde of equaton (25) corresponds to d t+1 ζt rt k + k k ( ) ( R λ (1 ζt) + (η η t+1 ) rt k ζ tr t + λ (ζ t ζ t )ζ t+1 r t ) and thus equaton (25) s equvalent to 0 = d t+1 t k (ζ ζt k )rt k + ζ tr t R λ (ζt k ζt)r t k k + ( ) R λ (1 ζt) + (η η t+1 ) rt k λ (ζt ζ t )ζ t+1 r t k + ( ) R λ (1 ζt k ) + (η η t+1 ) rt k λ (ζ t ζt k )ζ t+1 r t k = d t+1 (ζt ζt k )rt k + λ (η η t+1 ) (ζt ζt k )rt k ζ t+1 r t k k +R λ 2 (ζt ζt k )rt k (1 + ζ t )ζ t+1 r t k Rλ 2 (ζt ζt k )(ζt + ζt k )rt k ζt+1r t R λ (ζt ζt k )rt k k k ζ t rt. Let us frst suppose that k (ζ t ζt k )rt k = 0. Then ζ = ξ t, where ξ t = the last equaton s equvalent to (ζt ζt k )ζt k rt k = 0 k k ζk t rk t. Moreover, 1 rt and thus ξ 2 t = k (ζ k t ) 2 r k t. Ths last equaton mples ζ k t = 0 for all k, or r k t = 0 for k. In the frst case we have a contradcton to Assumpton 5. In the second case we have a contradcton to r t (0, 1). Let us now suppose that k (ζ t ζ k t )r k t 0. Wthout loss of generalty, we take k (ζ t 22

24 ζ k t )r k t > 0 (the same argument can also be used for the case k (ζ t ζ k t )r k t < 0). Then 0 = d t+1 + λ (η η t+1 ) ζ t+1 r t R λ ( R λ 2 (ζ l (ζ t ζ t) 2 (ζ t)r l t l t k ) 2 k l (1 λζ t+1)ζ t r t (ζ t ζ l t)r l t ) r k t ζ t+1 r t = d t+1 + λ (η η t+1 ) ζ t+1 r t R λ (1 λζ t+1)ζ t r t R λ 2 ζ k l (ζ t ζt)r l t l t (1 ζt k )rt k ζ t+1 rt. Snce r t+1 = r t (0, 1), the set k { } ζ t+1 r t = 0 has probablty zero by Assumpton 5. Thus ζ t+1 r t > 0 almost surely. Let δ > 0, then by Assumptons 4 and 6, the set {s t+1 d t+1 (s t+1 ) = 0, η η t+1 (s t+1 ) < δ} has strctly postve probablty ndependently from s t. Thus (1 λ) ζ t r λ t + ζ k l (ζ t ζt)r l t l t (1 ζt k )rt k < δ. Snce ths s true for all δ > 0, ζ t = 0 for all nvestors wth strctly postve wealth share at tme t, a contradcton to Assumpton 5 or r t = r t+1 (0, 1). Therefore, r t = 0 or r t = 1. The Lemma mples that we can restrct ourselves to monomorphc populatons of nvestors (all nvestors wth a strctly postve market share possess the same tradng strategy), to analyse nvarant wealth share dstrbutons. In partcular, we are lookng at determnstc fxed ponts that are stable, such that a small perturbaton of the vector of wealth shares does not change the long-run evoluton. Snce nvarant wealth share dstrbutons correspond to monomorphc populatons, the stablty of nvestment strateges s related to the stablty of the assocated fxed pont. Therefore, we consder a populaton of tradng strateges wth an ncumbent strategy λ (wth market share r t) and a dstnct mutant strategy λ (wth market share r t = 1 r t). Defnton 2 (Evolutonary stable strateges). A tradng strategy λ s called evolutonary stable f, for all strateges λ, there s a random varable ǫ > 0 such that lm t φ (t,ω,r) = 1 for all r 1 ǫ(ω). If the choce of mutant strateges s restrcted to those tradng strateges that are a local mutaton of λ,.e. there exsts a random varable δ(ω) > 0 wth λ (ω) λ (ω) < δ(ω) almost surely, then λ s called locally evolutonary stable. k 23