Competitive Nash Equilibria and Two Period Fund Separation

Transcription

1 Competitive Nah Equilibria and Two Period Fund Separation Thorten Hen Intitute for Empirical Reearch in Economic Univerity of Zurich, Blümlialptrae 10, 8006 Zurich, Switzerland & Department of Finance and Management Science Norwegian School of Economic and Buine Adminitration Hellev. 30, 5045 Bergen, Norway Stefan Reimann Intitute for Empirical Reearch in Economic Univerity of Zurich, Blümlialptrae 10, 8006 Zurich, Switzerland Bodo Vogt Faculty of Economic, Univerity of Bielefeld, Univeritättr.25, Bielefeld, Germany November 28,

2 Competitive Nah Equilibria and Two Period Fund Separation Abtract We ugget a imple aet market model in which we analyze competitive and trategic behavior imultaneouly. If for competitive behavior two-fund eparation hold acro period then it alo hold for trategic behavior. In thi cae the relative price of the aet do not depend on whether agent behave trategically or competitively. Thoe agent acting trategically will however invet le in the common mutual fund. Contant relative rik averion and abence of aggregate rik are hown to be two alternative ufficient condition for two-period fund eparation. With derivative further trategic apect arie and trategic behavior i ditinct from competitive behavior even for thoe utility function leading to two-fund eparation. Keyword: trategic behavior, competitive behavior, two-fund-eparation, CAPM. JEL claification: C72, G11, D83. We like to thank Piero Gottardi, Enrico De Giorgi, and Rüdiger Frey for valuable dicuion. Financial upport by the national centre of competence in reearch Financial Valuation and Rik Management i gratefully acknowledged. The national center in reearch are managed by the Swi National Science Foundation on behalf of the federal authoritie. 2

3 1 Introduction Standard aet pricing model, a for example the Capital Aet Pricing Model, CAPM, are baed on the aumption that all market participant take price a given. Thee model give a firt intuition for the valuation of aet when portfolio conideration and diverification are important. Moreover, thee model are general in the ene that they can be applied to an arbitrary number of aet. They can however not cope with important iue that practitioner face when a lack of liquidity of market and o called lippage of aet price are a major concern. The market impact of portfolio deciion i clearly taken into account by intitutional invetor like penion fund that in mot market hold mot of the aet. Alo, many hedge fund limit their aet under management becaue running their trategie with too much capital would eliminate the potential gain from their trategie. Moreover, to benefit from portfolio diverification large invetor and hedge fund invet on many market imultaneouly. To cope with thee iue while keeping the benefit of portfolio diverification, model of imultaneou trategic interaction on a large number of aet market are needed. The idea of thi paper i to ytematically compare price taking and trategic behavior for a imple aet market with imultaneou competition on arbitrary many aet. The price of thi generality i that we limit our attention to ymmetric information model with a given complete participation on aet market. Alo in our paper initially invetor are not endowed with aet o that changing the market price doe not change the wealth of the invetor. Thee important apect hould be conidered once the difference of trategic and competitive behavior ha been undertood in our more imple etting. We conider a two period model with a finite number of tate in the econd period. A finite number of invetor are endowed with wealth that can be pent on firt period conumption and on a finite number of aet (bond and hare) delivering tate contingent payoff in the econd period. We aume that every invetor i mall on the market for firt period conumption. Firt period conumption reemble the real GDP of the world. On thi market a large number of producer, pure conumer, and invetor interact and o we aume that even large invetor are mall. Warren Buffet and George Soro, for example, are etimated to manage wealth of approximately a couple of billion USD. Thi i a huge amount a compared to the market capitalization for individual tock, while it can be neglected relative to the world GDP. Aet market can be complete or incomplete. In our model aet payoff are the only ource to finance econd period conumption. All conumer plit their wealth between firt period conumption and a portfolio of aet in order to maximize intertemporal utility. On the aet market we allow for competitively and alo for trategically acting invetor. In the firt cae the invetor take price a given while in the econd cae they take the market impact of their demand into account. One may argue that thee different type 3

4 of behavior can arie if etimating the market impact i cotly becaue it require data bae and reearch facilitie o that only ome invetor have a ufficient incentive to conider their market impact. However, thee argument are beyond our model. Throughout the paper we aume that invetor have expected utility function with homogenou belief. A it i well known, for example from Magill and Qunizii [12], the CAPM i the pecial cae of our model which i obtained with only competitively acting invetor and quadratic von-neumann-morgentern utility function. With repect to the trategic behavior the model i imilar to the famou Shapley-Shubik [17] Market Game. It will turn out that the number of aet obtained by any invetor are given by the ratio of the wealth he ha bet on that aet divided by the total wealth bet on that aet. One important difference will be that in our model aet are in fixed exogenou upply and income doe not depend on the market outcome. Hence in contrat to the Shapley Shubik model we can eaily enure that the budget retriction hold. We formulate the agent deciion in term of budget hare that are required to add up to one. Thi formulation of the invetment problem in term of wealth hare, the o called aet allocation, i tandard in finance. It allow to dicu invetment deciion baed on return, i.e. payoff per price of aet. Keeping thi convention our reult are more eaily comparable to the finance literature. The point of our paper i to analyze under which condition and in which repect trategic invetment behavior differ from competitive behavior. To tart with, we how that, a the number of invetor become large, trategic behavior tend to competitive behavior. For a general account of thi o called Cournotian foundation of competitive equilibria ee Ma-Colell [14]. To obtain thi reult we let the economy grow in a very ymmetric way. In each tep one additional identical copy, a replica, of the trategic agent i introduced. Ever ince Debreu and Scarf [6] uch limit reult for replica economie are well know in the general equilibrium literature (cf. Hildenbrand and Kirman [9]). Since in our model the upply i exogenou we increae it proportionally to the number of conumer in our economy. Beide thi tandard reult on the convergence of trategic and competitive behavior we give ufficient condition for finite economie, uch that with repect to the aet allocation problem trategic and competitive behavior become identical. We how that if a form of fund eparation hold for competitive behavior that we ugget to call two period fund eparation then trategically acting agent will form the ame portfolio of aet a competitive agent. Both type of behavior do however differ with repect to the amount of wealth inveted in the common mutual fund of aet. The trategically acting invetor take into account that their demand will let price lip to their diadvantage and hence invet le into aet a compared to invetor with identical characteritic that behave competitively. We alo give ufficient condition for two period fund eparation. One uch ufficient condition i contant relative rik averion, CRRA. An alternative condition i the cae of no-aggregate rik, NAR. 4

5 The aet pricing implication of two period fund eparation are that the ratio of the price of riky aet do not depend on whether agent behave competitively or trategically. Moreover, for the cae of no-aggregate rik the weight of any aet in the mutual fund turn out to be the expected value of it payoff relative to the total payoff of all aet. Thi coincide with o called log-optimal pricing (cf. Luenberger [11], chapter 15). It i well known that log-optimal pricing i alo obtained if all invetor acting competitively have logarithmic von-neumann Morgentern utilitie, a pecial cae of CRRA (cf. Krau and Litzenberger [10]). We how that thi i till true if one allow for trategic behavior. In the cae of the CAPM heterogeneity in market behavior matter if there i aggregate rik. We oberve that trategically and competitively acting agent do chooe ubtantially different portfolio and hence aet price differ ubtantially. On the other hand, if the market doe not exhibit aggregate rik, both invetor, even if they differ in their trategic behavior, chooe the ame portfolio. Introducing derivative lead to a new trategic apect of the model. On changing demand for the underlying aet agent can change the payoff of the derivative that are baed on the price of that underlying. Indeed in thi cae it turn out that even with logarithmic utility function equilibria depend ubtantially on the form of market behavior. There i an impreive literature on trategic competition in general equilibrium model. Thi literature ha at leat the two line originating in Gabzewicz and Vial [8] and Shapley-Shubik [17]. For a recent account ee the recent pecial iue of the Journal of Mathematical Economic, Vol. 39, No. 5-6, edited by Gaël Giraud. Our contribution in thi repect i that we highlight the importance of two-fund eparation to obtain more pecific reult. The cae under which we how that two-period fund eparation hold, CRRA and NAR, are clearly not general in the et of all theoretically poible economie but they are important cae tudied extenively in the finance literature. Ever ince Merton [15] CRRA ha become the work hore of finance. Alo Campbell and Viceira [4] (page 24) argue convincingly that only the cae of CRRA i compatible with oberved aggregate time erie of conumption and rik premia: Wealth ha grown coniderably while the rik premium remained quite table over time. The econd cae for which we can how two-period fund eparation i the cae of no aggregate rik. Ever ince Borch [2] and Malinvaud [13] alo thi cae ha been extenively tudied in the literature. It i the work hore cae for inurance theory. In the finance literature market impact ha been a eriou concern, for example, in the field of derivative (cf. Taleb [19]), when aymmetric information (cf. Brunnermeier [3]) ha been conidered and in model with endogenou market participation (cf. Pagano [16]). Only the cae of derivative eem ufficiently imilar to the model conidered here. When preenting our reult in ection 6.1 concerning derivative we 5

6 will dicu the difference of thi literature to our approach. The ret of the paper i organized a follow. The next ection give the detail of the model. Then we ugget an equilibrium concept, that we call Competitive Nah Equilibrium, CNE, in which we tudy competitive and trategic behavior imultaneouly. Having made the equilibrium notion precie we demontrate the limit theorem. Thereafter, two-period fund eparation i defined and it i hown that under tandard differentiability aumption on the utility function CNE with two period fund eparation do exit. Then we how that CRRA and NAR are ufficient condition for CNE with two period fund eparation. Baed on thi the pricing implication are derived. Alo, when preenting the general reult we give numerical example for the CAPM cae and the log-utility cae to illutrate the robutne. Finally, we conider the cae of derivative. 2 The model In the following we define the model we are concerned with. The definition i divided into mainly two part, the firt one concern the market while the econd one concern the characteritic of the agent on the market. 2.1 The market (q, A) We conider a 2 period model with period t =0and t =1of an economy with S tate and K aet k. Let u denote by S 0 := {0} S the et of tate, where for convenience =0i the tate at time 0, and S := {1,.., S} i the et of tate at time 1. Let k =0be the conumption good while K = {1,.., K} i the et of aet available at time 1. Let A R K S + be the matrix of non-negative payoff of the aet k K over tate S. We aume that there are no redundant aet, i.e. rank A = K. Aet k K are in exogenou upply which i normalized to 1, while the conumption good i in elatic upply. q R K + i the price ytem on the market A, while the price for the conumption good i normalized to The invetor i Let I = {1,.., I} be the et of invetor on the market. It i aumed that invetor have homogenou believe about tate in period 1, i.e. p i = p S + i the vector of probabilitie for tate S. An invetor i characterized by hi firt period wealth (endowment) w i R + and by hi utility U i on hi conumption 6

7 ( ) in period t =0, 1. Hi invetment trategy i denoted by λ i = λ i (w i )= λ i 0(w i ), λ i 1(w i ) R K+1 +, where λ i 0(w i ) i hi (budget) hare of invetment in the conumption good and λ i 1(w i ) i hi invetment in aet k K on A. Let λ =(λ i,i I) be the vector of invetment trategie over the invetor population I. Each invetor i i uppoed to partition all hi wealth into 0 period conumption and invetment in aet k K to obtain 1-t period conumption. Formally, hi budget contraint therefore read K k=0 λi k =1 or equivalently λ i K+1 +, i I. Note that we exclude hort ale. Thi excluion i a conequence of allowing for trategic behavior. Strategically acting agent know that they could decreae aet price below zero by going hort in aet. A an effect portfolio return would then become poitive and it would pay even more to hort the aet further. Without any hort ale contraint thi would reult in unlimited arbitrage opportunitie, ruling out the poibility of any type of equilibria. The conumption of invetor i reult from hi invetment trategie a follow. The conumption ( ) function of the i-th invetor i defined a c i : K+1 + R S+1 + by c i (λ i ):= c i 0(λ i ), c i 1(λ i ), where c i 1(λ i )= ( c i (λ i ), S ) i the conumption of the i th invetor over tate S according to hi invetment trategy λ i : c i 0(λ i ) = λ i 0 w i (1) c i (λ i ) = A k λ i kw i S. (2) q k k K Recall that all aet are in unit upply. The equilibrium price ytem q then i given by the invetment trategie by requiring q k = i I λi kw i for all aet k K. Hence market clearing price are the wealth average of the invetor trategie. Given the probabilitie p, the preference of the i-th invetor are repreented by an expected utility function U i : R S+1 + R defined by U i (c i (λ i )) = u i 0(c i 0(λ i )) + β i U1(c i i 1(λ i )), where β i i a real-valued dicount factor, 0 β i 1, and u i 0 : R + R, and U1 i : R S + R i defined by U1(c i i 1(λ i )) := p u i 1(c i (λ i )), S where u i 1 : R + R. Note that U i =(u i 0,U i 1). We arrive at [U i c i ](λ i )=U i (c i (λ i )) = u i 0(c i 0(λ i )) + β i p u i 1(c i (λ i )). (3) S We make the following tandard aumption about the utility function for any i I: u i t : R + R, t =0, 1, i twice continuouly differentiable, trictly increaing, trictly concave and 7

8 (INADA): for any c R +, c ui t(c) a c 0. Recently, Alo-Ferrer and Ania [1] have tudied Nah equilibria in a imilar model when agent are rik neutral. Thi cae require different technique. It turn out that all agent chooing a portfolio with weight equal to the relative expected payoff i the unique Nah equilibrium. 3 The equilibrium concept: A firt definition In a competitive equilibrium the agent take the market price ytem q a given. Thi ituation i different in the Nah equilibrium where invetor anticipate that trading alter price on the market. Invetor j, thinking trategically, know that q k ( λ i (w i )) = λ i kw i + λ j j i k wj, k K. Hence for a given wealth ditribution, the equilibrium price ytem q i anticipated to depend on the et of invetment trategie λ, i.e. q = q( λ). Conequently, any individual optimal trategy λ i depend directly on the trategie of all other trader i I ( i). On a market both type of invetor, i.e. thoe following the competitive equilibrium concept and thoe following the Nah equilibrium concept, coexit. The conumption of a competitively behaving invetor on A therefore i ( ) c i (λ i ; q) = λ i 0w i, A k λ i kw i, q given, i I C, q k k K S while the conumption of a trategically behaving invetor i on A relative to invetor {j i} yield ( ) c i (λ i ; λ ( i) )= λ i 0w i, A k λ i kw i λ i k wi +, λ ( i) =(λ j j i λj k wj k ) j i given i I N k K Note that we have partitioned the et of invetor I into the et of thoe following the competitive trategy I C and thoe following the Nah trategy, I N, i.e. I = I C I N. S Now we are in a poition to define Competitive Nah Equilibria: Definition 1 (Competitive Nah Equilibrium (CNE)) Given an economy with wealth ditribution w R I ++, a Competitive Nah Equilibrium i a pair (ˆq, ˆλ), ˆλ =(ˆλ i,i I), uch that for all invetor i I C I N the following condition are fulfilled imultaneouly [ ˆλi argmax U i c i] (λ i ) (4) λ i K+1 + ˆq k = i I ˆλ i kw i, k K, (5) 8

9 where the conumption of a competitively behaving invetor i ( ) c i (λ i ; ˆq) = λ i 0w i, A k λ i kw i, ˆq k k K S ˆq given, i I C, while the conumption of a trategically behaving invetor i relative to invetor {j i} i ( ) c i (λ i ; ˆλ ( i) )= λ i 0w i, A k λ i kw i λ i k wi + ˆλ, (ˆλ) j j i given i I N j i k wj k K 3.1 The FOC and State price Vector in CE and NE In the following we will how that under the condition made for the utility function the Firt Order Condition (FOC) i ufficient for determining the optimum. Let u therefore firt derive the Firt Order Condition for CNE. Lemma 1 Conider i I with wealth w i R +. Defining the caled nabla operator i =( i ) S, where ( i := β i u i 0 (c i 0 ) c i 0 read ) 1 c i S, the firt order condition for the optimization problem for a CNE (q, λ), λ =(λ i ) q = A i U i 1 ( ) c i 1(λ i ) N i (λ), (6) where denote the componentwie multiplication of two vector. N i (λ) ha component 1 i I C Nk(λ) i = (7) 1 λi k wi i I N j λj k w j Furthermore, the Firt Order Condition i neceary and alo ufficient for determining the maximum. Proof 1 The agent optimization problem read max[u i c i ](λ i ) ubject to the condition K k=0 λi k =1 and λ i k 0. Defining g(λ i ):= K k =0 λi k, the firt order condition (FOC) are [ U i c i] (λ i ) α K g(λ i )+ α k, R α, α k 0 k K λ i k λ i k Becaue of the INADA aumption about the utility function U i, we can exclude the cae {α =0} {α k = 0} k=0..k and hence all olution are interior. Since U i i aumed to be increaing, the FOC hold with equality and we obtain w i = β i p k=0 u i c 1(c i (λ i )) i u i c i 0 (ci 0 (λi )) 0 Denoting by i the operator for the caled partial derivative ( ) u i := β i i 0 (c i 1 0) c i 0 ( c i (λ i ) λ i k c i, ). (8) 9

10 the FOC for the k-th component in K become A traightforward calculation yield c i (λ i ) λ i k w i = i U i 1 = w i k A k = wi A k q k ( ( c i 1(λ i ) ( λ i k λ i k ) ( c i (λ i ) 1 1 λi kw i j λj k λ i k λi k q k wj δi ). ) q k (q k ) 2 λ i k ), where δ i =1if i I N and 0 if i I C. Thu, by defining the o called Nah term the Firt Order Condition take the form ( q k = A k i U1 i N i k(λ) = 1 λi kw i j λj k wj δi. k K (9) q = A i U i 1 ( c i 1(λ i )) ) N i k(λ) (10) ( ) c i 1(λ i ) N i (λ) i I, (11) where i i the vector of the caled partial derivative i defined above and denote the componentwie multiplication of two vector. Note that FOC for CE and NE only differ by a factor N i (λ). Moreover, note that for δ =1we get N i k(λ) = j i λj k wj j λj k wj. (12) It remain to be hown that thi condition i ufficient for determining the maximum. Thi follow from above becaue c i i concave in each component ince [ ] 2 c i Ak wi δ i 0 k = k q = 2 λ i k k λi k 0 k k Hence, a a compoition of concave function, [U i c i ] i componentwie concave and o the FOC i neceary and alo ufficient for determining the maximum. In the cae of a population with homogenou behavior thi reduce to the tandard definition. Corollary 1 (Competitive equilibrium) Conider a 2 period economy with I invetor I = I C and wealth w R I +, where invetor i ha an utility function U i =(u i 0,U1):R i S+1 + R a defined above. Then a competitive equilibrium i a tuple (q, λ ), λ =(λ i,i I), where q R K + and λ i K+1 + uch that q = A i U i 1(c i 1(λ i )) i I C where (13) c i (λ i )= k K A k λ i k w i j I λ j k wj (14) 10

11 Corollary 2 (Nah equilibrium) Conider a 2 period economy with I invetor I = I N and wealth w R I +, where invetor i ha an utility function U i =(u i 0,U1):=R i S+1 + R with component u i 1 a defined above. Then a Nah equilibrium i a pair ( q, λ), λ =( λ i,i I), where q R + and λ i K+1 + uch that q = A i U1(c i i 1( λ i )) N i ( λ) i I N where (15) c i ( λ i )= λ A k kw i k K j I λ j k wj (16) 4 A Limit Theorem While in general CE and NE differ for mall economie, both coincide in the limit of a large economy. Let u conider a market on which a N-multiplicity of invetor act, i.e. we have N I agent. Each agent i i uppoed to have N identical replica i(1),.., i(n) having identical utility function U i,n = U i and income ditribution w i,n = w i following the trategie λ i,n. We aume that upply or equivalently ( ) payoff are caled appropriately, i.e. A (N) = f k (N)A k, where f k (N) 0 for all k K. Then for trategically acting agent N i,(n) k (λ) :=1 immediately from Theorem 1. λ i k wi N I j=1 λ j k wj k K, S for k K. The following tatement follow Corollary 3 Let λ i,n K+1 +, i =1..I, n =1..N be a Nah optimal invetment trategy for the i th invetor in a N fold replica economy a defined above. Then λ (i,n) λ,i a N provided that f k (N) 1 for all k, where λ i i the optimal competitive trategy of invetor i in the one fold replica, N N =1. Proof 2 According to equation 11 the FOC for the N replica NE economy i a follow where q k = I,N i,n=1 i,n λ k wi,n = N I N q = A (N) i U i 1(c i 1( λ i ) N i,(n) ( λ), λ i i=1 kw i uch that we have ( ) I λ i kw i = A k i U1(c i i 1( λ i ) i=1 f k (N) N i,(n) k ( λ) Finally oberve that N i,(n) k ( λ) 1 a N. Hence if N and f k (N)/N 1 the expreion reduce to the FOC of CE. Therefore, under thee condition q k q 1 and hence the claim follow. k 5 Two-Period Fund Separation In thi ection we demontrate that increaing the ize of the economy i not the only cae in which competitive and trategic behavior become imilar. Actually for any finite economy it i hown that for thi 11

12 to hold a form of two-fund eparation i deciive. Recall that imilar form of two fund eparation are known to be the bai for many important reult in finance, a for example the CAPM. We will dicu the ditinction between the two fund eparation for our paper and that of CAPM once we have defined our notion. The invetment trategy of invetor i i λ i K+1 + R K. We now repreent each invetment trategy in term of elementary invetment trategie λ k R K +, where 0 k k (λ k ) k =. 1 k = k Hence λ k i the relative invetment in the aet k K. Each K-ubet of elementary invetment trategie clearly contitute a bai for the pace of invetment trategie. Thu each invetment trategy λ i K+1 + can be written a a linear uperpoition of thee elementary invetment trategie K K λ i = λ i kλ k, λ i k [0, 1], λ i k =1 k=0 k=0 Pleae inert Figure 1 about here Two Fund Separation concern the partition of an optimal fund into two regime. Here we conider eparation of an equilibrium fund over period, i.e. the partitioning of wealth ditribution w into 0 period conumption and 1 period portfolio election on the ecurity market A. We therefore call thi eparation Two-Period Fund Separation (2pF S). Definition 2 (Two-period-Fund Separation (2pFS)) Let λ i (w i ) K+1 + be the invetment trategy of agent i on the market A in a CNE economy given hi wealth w i R +. Then 2pF S hold if and only if for all invetment trategie λ i K+1 +, there exit a unique common portfolio invetment λ K+1 + for all invetor i on the ecurity market A uch that λ i (w i ) λ 0, λ K+1 + (17) for an equilibrium trategy for all invetor i I. Since dim λ 0, λ =1, thi i equivalent to aying that each invetment trategy i uniquely repreented by a real number λ i 0(w i ) [0, 1]: λ i (w i ):=λ i 0(w i )λ 0 +(1 λ i 0(w i )) λ, where λ i 0(w i ) i the relative invetment of invetor i in 0 period conumption and λ i the unique mutual fund on A. Thi ituation i diplayed in Figure 1. In other word, under 2pF S optimal invetment 12

13 trategie only differ in relative invetment in 0 period conumption. Invetment trategie then have the following repreentation with repect to the coordinate ytem (λ 0, λ): λ i (w i )= ( ) λ i 0(w i ), (1 λ i 0(w i )) Standard two fund eparation (Ca and Stiglitz [5]) refer to eparation of invetment deciion in a rikle aet and a fund of riky aet component. In our model zero period conumption play a imilar role a the rikle aet in tandard two fund eparation ince it alo guarantee rik free payoff - however delivered one period before the other aet pay off. If in our model ome of the aet k K were rik free then, due to borrowing and aving, the different time period of the rikle payoff would not matter. Yet our model ue a lightly tronger tructure than only eparating between rikle and riky payoff. In our model additive eparability over time and the INADA condition imply that one ha to conume omething in period 0, i.e. rikfree conumption i eential and cannot be ubtituted by poibly riky conumption. (18) The main quetion i which propertie on the market tructure A and on the utility function U i permit 2pFS. Our firt tatement concern the market, the econd the utility function. We firt how in Theorem 2 that 2period fund eparation hold for any economy provided there i no aggregate rik. Ever ince Borch and Malinvaud [2, 13] thi cae ha been intenively tudied in the literature. Furthermore, a Theorem 3 how, 2pFS alo hold if utility function are CRRA. Ca and Stiglitz [5] have already found the importance of CRRA for fund eparation. In our model with only one period, CRRA i equivalent to having a ingle fund on aet. Let u conider thee cae in more detail. Theorem 2 Conider an economy without aggregate rik, i.e. k Ak = a, a R + and non zero endowment, i.e. (w i ) i R I +. Then there exit an equilibrium in which 2pFS hold, the mutual fund being λ k = S A k p k A k = 1 p A k. a Thi particular mutual fund preerve a pecial notation, λ = λ. S Proof 3 Obviouly λ k k =1. We how that, provided there i no aggregate rik, there exit an λ i 0 [0, 1] uch that λ i = λ i 0λ 0 +(1 λ i 0) λ i an CNE equilibrium. Suppoe 2pFS hold. Let ˆλ 0 := (ˆλ1 0,.., ˆλ ) I 0, and define ν i (ˆλ 0)= 1 ˆλ i 0 j (1 ˆλ j 0 )wj, then the Nah term become N i k(ˆλ 0)=1 ν i (ˆλ 0)w i k K, while conumption reduce to c i 0(ˆλ 0)=ˆλ i 0w i and c i (ˆλ ( 0)= k ) Ak ν i (ˆλ 0)w i for all S. Note that if ˆλ i 0 1, then ν i (ˆλ i 0) 0 and o c i 0, while if ˆλ i 0 0 thee quantitie remain finite. 13

14 If there i no aggregate rik, i.e. k Ak = a, the conumption i independent of, i.e. c i (ˆλ 0)= aν i (ˆλ 0)w i 1, where 1 = (1,.., 1) i an S-dimenional vector and c i i contant over all tate. By defining c i (ˆλ 0)=aν i (ˆλ 0)w i, we write c i (ˆλ 0)=c i (ˆλ 0)1. Under the NAR aumption with thee definition the FOC for CNE take the form i U1(c i i (ˆλ 0)1)A k (1 ν i (ˆλ 0)w i) = λ k (1 ˆλ j 0)w j Thu we arrive at (1 ˆλ j 0)w j = j ( A k p λ k ) β i u i c 1(c i (ˆλ 0)) i u i 0 (ci 0 (ˆλ 0) c i 0 = a βi u i c 1(c i (ˆλ 0)) i u i 0 (ci 0 (ˆλ 0) c i 0 j ( 1 ν i (ˆλ 0)w i) (19) ( 1 ν i (ˆλ 0)w i) (20) It remain to be hown that a olution in ˆλ 0 exit. Therefore note that the left hand ide j (1 ˆλ j 0 )wj i poitive and finite for any ˆλ 0.Ifˆλ i 0 0 then 0 <ν i (ˆλ 0) < and the term β i ui c 1(c i (ˆλ i 0)) i poitive and finite, while u i c 0(ν i (ˆλ i 0)w i ) and hence the right hand ide tend to 0 a ˆλ i 0 0. On the other hand, 0 if ˆλ i 0 1 then ν i (ˆλ 0) 0 and therefore c i ( ˆλ 0) 0. While 0 < u i c 0(ν i (ˆλ i 0)) <, ui c 1(c i (ˆλ i 0)) 0 and hence the right hand ide tend to a ˆλ i 0 1. Since both ide are continuou in ˆλ 0, a olution exit. In the mutual fund λ the weight of any aet turn out to be the expected value of it payoff relative to the total payoff of all aet. Thi coincide with o called log-optimal pricing (cf. Long [?]). Indeed the ame mutual fund i obtained in the cae of logarithmic utility function - a pecial cae of CRRA which i covered by our Theorem 3. Pleae inert Figure 2 about here Some intuition for thi reult holding in the cae of no aggregate rik i provided by referring to efficient rik haring (cf. Borch [2] and Malinvaud [13]) a diplayed in Figure 2. Since all agent have expected utility function and belief are homogenou, in the cae of no aggregate rik efficient rik haring i obtained at fair aet price, i.e. at price that are equal the expected payoff of the aet. In thi cae every conumer receive a fraction of the aggregate payoff and hence no individual need to carry any rik. A Borch and Malinvaud have hown thi i clearly a competitive equilibrium. When agent take their market impact into account they realize that their budget et are not given by a budget line but by a curve that lie below the budget line and coincide with it only at the point of 14

15 efficient rik haring. Thi i becaue any demand different to the efficient level would turn price to the diadvantage 1 of the agent deviating from the efficient allocation. Thi intuition can be derived from a reinterpretation of the firt-order-condition q k /N i k(λ) = A k i U i 1 ( ) c i 1(λ i ) k K, i I. Writing the firt-order-condition thi way, on changing the aet allocation λ 1 on A taking ratio of any two component of the vector on the right hand ide give the change in the marginal rate of ubtitution between any two aet while the correponding ratio on the left-hand-ide give the perceived change of relative aet price. Now uppoe a competitive equilibrium i obtained in which thi firt-order-condition hold ignoring the K Nah-term. Then chooing the ame portfolio a in the competitive equilibrium i alo budget feaible in the ituation with trategic interaction. Moreover, a price are turned to your diadvantage, the perceived budget et in the cae of trategic interaction i included in the budget et keeping price a given. The firt-order-condition how that, moreover, the lope of the budget et anticipating your market impact coincide with that of the competitive budget et at thoe point where all agent chooe the ame portfolio. Thi i becaue at thee point all K Nah term identical. Hence, alo in the cae of trategic behavior, independently of the rik averion, the market outcome will be given by complete rik haring. Pleae inert Figure 3 about here CAPM and NoAggregateRik We illutrate thi theorem by conidering an economy without aggregate rik and two equally probable tate =1, 2 in which two invetor i =1, 2 with identical wealth w 1 = w 2, compete for two aet k =1, 2. Invetor can act competitively or trategically. Aet 1 ha payoff (1,α), while aet 2 ha payoff (0, 1 α) over tate 1, 2. The market tructure i given by A = 1 α, 0 α α Note that thi market ha no aggregate rik, i.e. k Ak = 1 independent of. The utility function u i := R + R conidered i of the form u i (c) =c γ 2 c2. Thi function i identical acro period and alo among conumer. Note that thi function doe not atify the INADA aumption made above. Hence thi illutration i not really covered by our previou theorem. Neverthele, we ee from Figure 4 that all implication of our theorem alo hold for thi important cae. Pleae inert Figure 4 about here 1 Recall that agent are not endowed with aet o that changing price doe not change their income. 15

16 In order to tudy the cae of AGGREGATE RISK conider the market A given a A = 2 α, 0 α 1, 0 1 α while all other pecification are the ame a in the example above, ee Figure 4. One obervation in hi cae i that conumer with identical characteritic [U i,w i ] chooe the ame portfolio if market behavior among conumer i homogenou. Both for the economy in which both agent behave competitively and alo for the cae of trategic behavior the ame portfolio i choen. On the other hand if we conider an economy with identical conumer characteritic but with different market behavior, then in the preence of aggregate rik the portfolio differ. The intuition for thi obervation i the following: The Nah equilibrium we have computed i a ymmetric Nah equilibrium, i.e. a ituation in which identical agent chooe identical trategie. Thi ymmetry i alo true in the competitive equilibria. Moreover, the available total payoff are independent of the market behavior we conider. Hence, ince there are no redundant aet, with identical conumer characteritic the portfolio choice in ymmetric Nah equilibria coincide with thoe in the competitive equilibrium. But till competitive and Nah equilibria differ with repect to the money inveted in the mutual fund. On the other hand, if we mix competitive with trategic behavior in one market, then the trategically acting agent will invet le in the aet and will conume more today o that he evaluate hi portfolio of aet at a different econd period wealth level. Hence, if relative rik averion depend on the wealth level, a it doe in the cae of quadratic utilitie, then both agent will chooe different portfolio even though they have identical characteritic [U i,w i ]. Pleae inert Figure 5 about here Thi ugget that if on the other hand the portfolio choice i independent from the wealth level, a it i in the cae of contant relative rik averion, then all invetor hould hold the ame mutual fund. The next theorem tate that even with aggregate rik 2pFS hold, if all invetor have identical relative rik averion. 16

17 Theorem 3 Suppoe there are no redundant aet, i.e. rank A = K. Moreover, aume all invetor have identical econd period relative rik averion, i.e. u i 1 = u 1 for all i I, where u 1 : R + R i defined by c η, 0 <η<1 u 1(c) = c>0 ln(c) Then in every CNE (ˆq, ˆλ) 2pFS hold, i.e. there exit a common mutual fund λ K+1 + with K k=0 λ k =1 uch that The mutual i of the form ˆλ i λ 0, λ K+1 + i I. λ k =1/µ A k p ( k Ak where µ>0 i a normalization contant o that K k=1 λ k =1. ) 1 η, (21) Proof 4 Part I: Recall that λ = ( λ ) i i the vector of invetment trategie on the market. Conider i I two invetor i, j I with identical utility function U i, U j : R S+1 + R. Note that i U1(c i i 1) i homogenou of degree ν { 1,η 1} for all i I. Both ee the ame price ytem q. Hence, ince rank A = K, the aociated linear map i injective and o the pre-image of q i unique. It thu follow that i U1(c i i 1(λ i )) N i (λ) = j U j 1 (cj 1 (λj )) N j (λ) and therefore i U1(c i i 1(λ i )) j U j 1 (cj 1 (λj )). Since i U1(c i i 1(λ i )) and j U j 1 (cj (λ j )) are homogenou of the ame degree ν and c i 1, c j 1 are homogenou of degree1inλ i, λ j, it follow that λ i λ j for any pair i, j. Hence all invetment trategie {λ i } are co-linear and are in the ame ubpace, i.e. for every pair (i, j) there exit a real valued calar 0 l(i, j) 1 uch that c j = l(i, j)c i. Particularly if λ ˆ i i an CNE invetment trategy, then there exit ome factor l>0 uch that λ := l ˆλ i and λ k k =1i the unique mutual fund which pan the correponding um pace λ 0, λ. Part II: Recall that under 2pFS, with the notation from the proof of Theorem 2, the FOC read A k i U1 i ( c i 1(ˆλ 0) )(1 ν i (ˆλ 0)w i) = λ k (1 ˆλ j 0)w j, where c i (ˆλ ( 0)= k ) Ak ν i (λ 0)w i i the conumption in tate. Aume that ˆλ i 0 > 0 and define K(ˆλ i 0):= ( β i 1. u i 0(c0)) i Note that K(ˆλi 0 ) 0 a ˆλ i 0 0, while it i poitive and finite for λ i 0 > 0. Then the c i 0 j 17

18 FOC equivalently read ( ( ηk(ˆλ i 0) p A k ηk(ˆλ i 0) ( k A k i U1 i A k A k p ( k Ak ( c i (ˆλ 0) )(1 ν i (ˆλ 0)w i) = λ (1 ˆλ i 0)w i k ) η 1 ) ( 1 ν i (ˆλ 0)w i) (ν i (ˆλ 0)w i ) η 1 = λ k (1 ˆλ i 0)w i ν i (ˆλ i 0 )wi (22) ν i (ˆλ i 0 )wi (23) ) 1 η ) ( 1 ν i (ˆλ 0)w i) (ν i (ˆλ 0)w i ) η = λ k (1 ˆλ i 0)w i, (24) Inerting (24), we obtain ( η/µk(ˆλ i 0) 1 ν i (ˆλ 0)w i) (ν i (ˆλ 0)w i ) η = (1 ˆλ i 0)w i (25) ( η/µ 1 ν i (ˆλ 0)w i) ( ) (ν i (ˆλ 0)w i ) η = (1 ˆλ i 0)w i β i u i 0(c i 0) (26) c i 0 The right-hand-ide i trictly increaing in λ i 0 and tend to 0 if λ i 0 1, while it tend to + if λ i 0 0. To dicu the behavior of the right-hand-ide, recall that ν i (λ i 0) 0 if λ i 0 1, while it remain poitive and finite for λ i 0 < 1. Put x := ν i (λ i 0)w i, then x R +. The real valued function f(x) =(1 x)x η, d 0 <η<1 i concave, f(x) x=0 =+, and ha two root (0, 1). Hence there exit two olution of dx equation (26), a trivial one ˆλ i 0 =1and a non trivial one 0 < ˆλ i 0 < 1. Note that the general mutual fund, ee equation 21, include thoe for the cae of NAR, log-utilitie, and alo of rik-neutrality: Indeed for η = 1 we get that λ k i the expected relative payoff of aet k, while for η =0we get that λ k i the relative expected payoff of aet k. Log utility function on a market with aggregate rik For illutration we conider the ame etting a defined above, i.e. a market with aggregate rik but now conider the cae that both invetor have identical CRRA utility function, particularly logarithmic one, i.e. u i 1(c) = ln(c). The extended market tructure thu i A = 2 α, 0 α α For implicity we aume that tate 1 and 2 are equally probable, i.e. p 1 = p 2 =1/2, and wealth ditribution i w 1 = w 2. Pleae inert Figure 6 about here Theorem 3 tate an intereting property of CNE. However it doe not etablih the exitence of CNE with thee propertie. The next propoition how that uch an invetment trategy in fact etablihe a CNE. 18

19 Propoition 1 Let λ i λ 0, λ. Then there exit a real valued coefficient 0 λ i 0 1 uch that λ i = λ i 0λ 0 +(1 λ i 0) λ i a CNE invetment trategy for invetor i. Proof 5 Uing notation a in the proof of Theorem 2 and by defining function F and G k by F(λ 0):= A i U i 1(( k Ak )ν i (λ 0)w i ) and G k (λ 0):= λ k j (1 λj 0 )wj, the FOC read F k (λ 0)(1 ν i (λ 0)w i )=G k (λ 0). Note that 0 < G k (λ 0) < for any given λ 0 while if λ i 0 0, then F k (λ 0) 0 and if λ i 0 1 then F k (λ 0). Since both function are continuou in λ 0, a olution exit. The next theorem how that under 2pFS agent acting trategically invet le in the mutual fund than thoe acting competitively. A a conequence of thi the utility level of the agent in a market in which every agent behave trategically i higher than the utility level in a competitive market. Note that thi tatement doe not conflict with the firt welfare theorem, i.e. with the claim that competitive equilibria are Pareto-efficient. In our model from a central planning perpective the agent are trictly better off conuming almot all their wealth today and betting only very little on the aet market. Thi i becaue the aet are in fixed upply while the firt period conumption good i in infinitely elatic upply. Theorem 4 Let λ i (w i )=λ i 0 (w i )λ 0 +(1 λ 0(w i )) λ be a CE. Then λ i (w i )= λ i 0(w i )λ 0 + ( 1 λ i 0(w i ) ) λ i a NE for ome λ i 0(w i ) λ i 0 (w i ). Proof 6 Conider an economy with given wealth ditribution w =(w i,i=1,..., I) and aume that λ(w i ) λ 0, λ i a CE. We how that there exit λ i 0 uch that λ(w i )= λ i 0λ 0 +(1 λ i 0) λ i a NE. For the ake of implicity let λ 0 =(λ i 0):= ( λ i 0(w i ),i I ) be the vector of 0 period invetment of agent i. Then define the following function F i (λ 0):=A i U1(c i i 1(λ 0)) q. In fact F λ k(λ i 0) > 0 ince c i (λ 0) 0 if λ i i and hence i u i 1(c i ) + according to the INADA aumption on u i 1. The FOC for CE then take the form Fk(λ i 0)=0. Let λ 0 uch that for given w =(w i ), F i k(λ 0)=0for all k. Finally define G i (λ 0):=A i U i 1(c i 1) N i (λ 0) q(λ 0). Let λ # 0 be uch that q(λ# 0 )=q. Then ince Nk(λ i # 0 ) 1, we have Gi k(λ # 0 ) Fi k(λ # 0 ). Hence it follow that λ 0 implicitly defined by G i ( λ 0)=0fulfill λ 0 λ 0, or equivalently λ 0(w i ) λ 0(w i ). 19

20 Hence under two-period fund eparation, thinking trategically, i.e. taking into account that price lip away on increaing order, doe matter for the hare of wealth inveted in the mutual fund, however it doe not affect the portfolio allocation within the group of aet. 6 Aet pricing implication From Corollary 1 above it i clear that CE and NE price are the ame in the limit of infinitely large market with homogenou invetor population, i.e. for I. What about price on market in which invetor act trategically and other do not. The quetion i whether thinking trategically matter for aet price on mall market. The next tatement how that relative aet price are independent of the compoition of market participant a long a two-period Fund Separation hold. Particularly if 2pFS hold, then relative aet price in a pure competitive and a pure Nah invetor population are identical to thoe in combined Competitive Nah economie. Corollary 4 If 2pFS hold, relative price are independent of the compoition of the agent population. Proof 7 According to the market clearing condition, under 2pFS price fulfill ˆq k = ˆλ i i I kw i for all k K. By Theorem 2 we have 2pFS in the CNE economy with the unique mutual fund λ. Hence we have for the price of aet k K ˆq k = λ k (1 λ i 0)w i uch that ˆq k ˆqj i in fact independent of the partitioning of I. i Recall the two example mentioned above. We in fact oberve that for our repective condition relative price are identical in the different regime. By (C/C) we denote a regime in which both invetor have competitive behavior, in a (N/N) regime both invetor behave trategically while in the (C/N) regime invetor 1 act competitively while invetor 2 behave trategically. The following table give the relative price on a market with aggregate rik and a market without, when both invetor have CAPM preference and follow different trategie. A above the market i A = 2 α, 0 α α Note that, a mentioned above, the identity of price in homogenou (C / C) and (N / N) economie i a reult of the ymmetry of the etting! 20

21 α CAPM - NAR CAPM - AR (C/C) (N/N) (C/N) (C/C) (N/N) (C/N) Derivative One field in finance which ha taken market impact a a eriou concern i the field of derivative in which lippage and liquidity hole have been taken into account when hedging a contingent claim. A nice intuitive account of thee effect for managing derivative i given in Taleb [19], chapter 4. For a more rigorou analyi along thee line ee Frey and Stremme [7] and Schönbucher and Willmot [18] who have adjuted the famou Black and Schole formula for lippage of price. Thi literature alo recognize that lippage ha ome upide: Many large trader ue their buying power to prop up the market in which they accumulate poition Taleb [19], page 69. To the bet of our knowledge, the pro and con of the market impact have not been balanced ytematically by thi literature. Moreover, it i quetionable to conider one ided trategic interaction in which only one party i allowed to act trategically while the ret of the market i paive. Introducing derivative lead to a new trategic apect of the model conidered here. On changing demand for the underlying aet agent can change the payoff of the derivative aet that are baed on the price of that underlying. Indeed in thi cae it turn out that even with logarithmic utility function equilibria depend ubtantially on the form of market behavior! We illutrate thi apect by the following imple model of a look-back option. The payoff matrix i given 21

22 a A = 1 0, α q 1 where q 1 i the price of aet 1 determined in the firt period. I.e. the econd aet pay the price of the firt aet if tate 2 occur. Again tate 1 and 2 are equally probable, both invetor are identical, i.e. have the ame endowment w 1 = w 2 and have the ame logarithmic utility function. Invetor can act competitively or trategically. Hence there are three ituation: Both act competitively, both act trategically, one invetor act competitively while the other invetor act trategically. The imulation, Figure 7 how that the fund choen by the invetor differ ignificantly if both follow different trategie. Pleae inert Figure 7 about here 7 Concluion and Outlook We have uggeted a imple aet market model in which we analyzed competitive and trategic behavior imultaneouly. We have hown that if for competitive behavior two-fund eparation hold acro period then it alo hold for trategic behavior. In thi cae the relative price of the aet do not depend on whether agent behave trategically or competitively. Thoe agent acting trategically will however invet le in the common mutual fund. Contant relative rik averion and abence of aggregate rik have been hown to be two alternative ufficient condition for two-period fund eparation. With derivative further trategic apect arie and trategic behavior i ditinct from competitive behavior even for thoe utility function leading to two-fund eparation. Thee reult are firt tep in building a new capital aet market model in which trategic interaction play ome role. Further reearch may endogeneize wealth by giving agent endowment in term of aet. Moreover, the model hould be extended to multiple period. Reference [1] Alo-Ferrer, Carlo and Ana Ania (2003): The Stock Market Game and the Kelly Nah Equilibrium ; Department of Economic Dicuion Paper, Univerity of Vienna, forthcoming in Journal of Mathematical Economic. [2] Borch, Karl(1962): Equilibrium in a Reinurance Market ; Econometrica; Volume 30: [3] Brunnermeier, Marku-K (2001): Aet pricing under aymmetric information: Bubble, crahe, technical analyi, and herding ; Oxford and New York: Oxford Univerity Pre. 22

23 [4] Campbell, John and Lui Viceira (2002): Strategic Aet Allocation, Oxford Univerity Pre. [5] Ca, David and Joeph E. Stiglitz (1970): The Structure of Invetor Preference and Aet Return, and Separability in Portfolio Allocation: A Contribution to the Pure Theory of Mutual Fund, Journal of Economic Theory 2: [6] Debreu, Gerard and Herbert Scarf (1962): A Limit Theorem on the Core of an Economy ; International Economic Review 4: [7] Frey,-Rüdiger and Stremme,-Alexander (1997): Market Volatility and Feedback Effect from Dynamic Hedging ; Mathematical Finance. October 1997; 7(4): [8] Gabzewicz, Jean-Jacque and Jean-Paul Vial (1972): Oligopoly à la Cournout in general equilibrium analyi ; Journal of Economic Theory; Vol 4: [9] Hildenbrand, Werner and Alan Kirman (1988): Equilibrium Analyi, North Holland: Amterdam. [10] Krau, Alan and Robert H. Litzenberger (1975): Market Equilibrium in a Multiperiod State Preference Model with Logarithmic Utility ; The Journal of Finance, Vol. 30, pp [11] Luenberger, David G. (1997): Invetment Science, Oxford Univerity Pre. [12] Magill, Michael and Martine Quinzii: Theory of Incomplete Market,1995, MIT Pre. [13] Malinvaud, Edmond: The Allocation of Individual Rik in Large Market, Journal of Economic Theory, 1972, Vol. 4, pp [14] Ma-Colell, Andreu (1982): The Cournotian foundation of Walraian equilibrium theory: en expoition of recent theory, chapter 7 in Advance in econometric; ed. by Werner Hildenbrand, Cambridge Univerity Pre,Cambridge. [15] Merton, Robert C. (1971): Optimum Conumption and Portfolio Rule in a Continuou-Time Model, Journal of Economic Theory 3: [16] Pagano, Marco (1998): Trading Volume and Aet Liquidity, Quarterly Journal of Economic 104(2), [17] Shapley, Lloyd and Martin Shubik (1977): Trade Uing one Commodity a a Mean of Payment ; Journal of Political Economy 85: [18] Schönbucher, Philipp and Paul Willmot (2000): The Feedback Effect of Hedging in Illiquid Market SIAM Journal of Applied Mathematic, Vol. 61 (1), : [19] Taleb, Naim (1996): Dynamic Hedging: Managing Vanilla and Exotic Option, John Wiley and Son, New York. 23

24 λ 1 λ 0 λ i (w i ) λ λ 2 Figure 1: The implex 3 + of invetment trategie λ =(λ 0, λ 1 ) over period 0 and 1 on a market A R diplayed in R

25 k Ak 2 i =2 i =1 p 1 p 2 k Ak 1 Figure 2: Complete Rik Sharing in Competitive and in Nah Equilibrium Figure 3: Mutual fund for log utility function on a market with aggregate rik depending on the market parameter α a defined in the example. 25

26 Figure 4: Fund choen by the two CAPM invetor on a market WITHOUT aggregate rik. Fund of invetor coincide if both have the ame market behavior (dot). The olid line how the common mutual fund choen by BOTH invetor even if they act according to different trategie, particularly invetor 1 act competitively and invetor 2 act trategically. Thi figure hould be compared with the analogou etting for a market WITH aggregate rik 26

27 Figure 5: Fund choen by the two CAPM invetor on the market WITH AGGRGATE RISK. Due to the ymmetry of the ituation fund of invetor coincide if both have the ame market behavior (dot), while they chooe different fund on the aet market, diplayed by line (dahed for the competitive invetor and dotted for the Nah invetor), if they follow different trategie, i.e. one of them i acting trategically while the other behavior competitively. 27

28 Figure 6: Fund election of invetor with log utility function on a market with aggregate rik. Dot repreent mutual fund choen if both invetor follow the ame trategy, while the line indicate mutual fund choen if one act trategically while the other competitively. Even if both invetor follow different trategie they chooe the ame fund on the aet market. Figure 7: Selection of fund in a mall economy with derivative. Becaue of ymmetry both invetor act identically in a C/C economy or in a N/N economy, while in a C/N economy both invetor clearly behave differently. 28