Competition in Fund Management and Forward Relative Performance Criteria
aa r X i v : . [ q -f i n . P M ] N ov Competition in fund management and forward relativeperformance criteria ∗ Michail Anthropelos † , Tianran Geng ‡ and Thaleia Zariphopoulou § November 3, 2020
Abstract
In an Itˆo-diffusion market, two fund managers trade under relative performance concerns. Forboth the asset specialization and diversification settings, we analyze the passive and competitivecases. We measure the performance of the managers’ strategies via forward relative performancecriteria, leading to the respective notions of forward best-response criterion and forward Nashequilibrium. The motivation to develop such criteria comes from the need to relax various crucial,but quite stringent, existing assumptions - such as, the a priori choices of both the market modeland the investment horizon, the commonality of the latter for both managers as well as the fulla priori knowledge of the competitor’s policies for the best-response case. We focus on locallyriskless criteria and deduce the random forward equations. We solve the CRRA cases, thusalso extending the related results in the classical setting. An important by-product of the workherein is the development of forward performance criteria for investment problems in Itˆo-diffusionmarkets under the presence of correlated random endowment process for both the perfectly andthe incomplete market cases. ∗ A considerable part of this work is part of the Ph.D. Thesis (2017) of the second author ([27]) and first appearedin https://ssrn.com/abstract=2870040. The work has been presented at conferences and seminars (Oxford, BrownUniversity and National and Technical University of Athens). The authors would like to thank the participants for theirvaluable comments and suggestions. † Department of Banking and Financial Management, University of Piraeus; [email protected]. ‡ Boston Consulting Group (BCG), Southborough, MA; [email protected]. § Departments of Mathematics and IROM, The University of Texas at Austin and the Oxford-Man Institute ofQuantitative Finance; [email protected]. Introduction
Relative performance is of tantamount importance in both the mutual and hedge fund managementindustries. It impacts a variety of factors spanning from a company’s reputation and net cash inflowsto the incentive structure and promotion schedule for its managers. Such facts have been very welldocumented in the finance practice and have been extensively studied in academic research (see,among others, [17, 18, 26, 36, 58]).While classifying the various kinds patterns of relative performance among managers is rathercomplex (classification by sectors, asset riskiness, market conditions, business cycles and others),there is a prevailing dichotomy based on asset specialization or asset diversification . In the former,the competing managers specialize in distinct asset classes while, in the latter, they invest in commonones.Asset specialization stems from a variety of reasons like familiarity with a certain sector, reductionof costs to enhance knowledge of new stocks, trading costs and constraints ([15]), liquidity costs ([62]),and ambiguity aversion ([16, 46]). The above evidence has been well established in the empiricalliterature ([19, 23, 32]) and has been incorporated in several theoretical models ([2, 25, 42, 61]; seealso [7, 19, 32, 42, 43, 59, 60, 62]).In asset diversification, the motivation is mainly to increase the net money inflows from clients([10, 20]). This setting is also more suitable to model relative performance concerns against a givenbenchmark portfolio (typically, a mix of asset classes). In a different direction, asset diversificationalso occurs in delegated portfolio management, where the role of one manager is replaced by the client([55]). In a related family of models, it also appears in the so-called “catching up with the Joneses”literature (see [1, 28]).Relative performance has been also considered in terms of how each manager reacts to the per-formance of a competitor. This interaction can be passive (best-response) in that the manager takesthe competitors’ policies as given (arbitrary but fixed) and trades accordingly, without any furtherinteraction (see, among others, [17, 36, 38]). On the other hand, interaction may be also competitive ,when managers compete with each other dynamically while investing among the various accounts([18, 26]).Whether managers compete within the same or different asset classes and/or interact in a passiveor competitive manner, there are common underlying assumptions that limit the generality and ap-plicability of the existing studies. The aim herein is to revisit some of these assumptions, propose analternative approach, study the related optimization problems and develop a comparative study withprevious works. We are motivated to do so not only by theoretical and conceptual arguments but,also, by various recent empirical works that point out to strong dependencies of the observed policiesto dynamically evolving factors, a dependency that cannot be explained in traditional settings; see,for example, [31, 36], where the effects of the current (and possibly non-anticipated) phase of themarket on the managers’ behavior is discussed.The first such assumption, ubiquitous for solving all underlying expected utility problems, isthat the market model is a priori chosen for the entire duration of the investment activity (see, forexample, [9, 13]). This is, however, rather unrealistic since model error and model decay alwaysoccur. Of course, genuinely dynamic model revision may be incorporated (for example, in the contextof adaptive control), but then, intertemporal consistency is violated. We also note that even in thepopular robust case, widely used to remedy model uncertainty and ambiguity, there is a stringentunderlying assumption that the plausible family of models is itself a priori chosen. Similar restrictionsare also present in filtering, in that the associated observation process is also pre-chosen.The second assumption is related to the investment horizon choice. In all existing works, it isassumed that the horizon is i) a single one (finite or infinite), ii) a priori chosen and iii) commonacross competitors (see, for example, [9, 13, 39]). In practice, however, this is not the case. While itis customary for managers to report their performance at common standardized time intervals (e.g.quarterly, annually), they almost always have internal sub-horizons that depend on company-relatedfactors and which are themselves difficult to model. Furthermore, even if a common horizon is a priorichosen, the investment activity does not stop at this specific pre-assigned time, as managers always2oll their positions from one investment horizon to the next. One could then argue that managersapply the same (or very similar) goals for the upcoming period. This, however, is not supported byexisting empirical evidence which shows that managers always adapt their goals in a rather complexmanner, depending on realized losses and gains, new upcoming (frequently unpredictable) marketconditions and others (see, for example, [1, 6, 11, 14, 21]).The third assumption is related to managers’ interaction. It is always assumed that each managerhas full, and for the entire investment horizon, knowledge of the competitor’s policy, in that she knowsthe stochastic process that models the investment policy she competes against. This modeling inputis needed in order to solve all associated expected utility problems but as an assumption, it is quiteunrealistic. Indeed, not only the manager cannot have such foreknowing skills for her competitor,the competitor herself might not a priori know how her own strategy will be changing as the marketenfolds.Herein, we propose a new framework aiming to remedy some of the shortcomings of these threestringent assumptions. We make no specific assumptions about the market model besides a weakstructural one that the asset prices are Itˆo-diffusion processes whose coefficients adapt to the currentinformation (see (3)). We also make no specific assumption about any pre-specified investment hori-zon, allowing for each manager to invest till personalized discretionary times. Finally, we make noassumption on a priori choosing the stochastic process that yields the competitor’s policy. Rather, weallow (besides mild integrability conditions) this policy to be dynamically revealed to her competitor.For tractability, we only consider the case with two managers. The general case for N -managers aswell as the mean field game limit are left for future research .The new framework is built on extensions of the so-called forward performance criteria. Suchcriteria, introduced by one of the authors and M. Musiela, and further developed by others (see[47, 49] and [63]), are modeled as stochastic processes, say ( U ( x, t )) t ≥ , that adapt to the marketinformation, are (local) supermartingales along all admissible policies and (local) martingales alongan optimal policy. To characterize forward criteria in such markets, a stochastic PDE was proposedin [50]. Depending on the choice of its volatility and structural parametrizations, various forms of U ( x, t ) have been studied (see, among others, [5, 41, 57]). However, several questions remain open asthe underlying stochastic optimization problems are ill-posed, fully non-linear and degenerate.To build forward criteria that allow for interaction - passive or competitive - between two decisionmakers, we proceed as follows. Let us assume that each manager uses admissible policies α and β, generating wealths X α and X β .For the case of best-response, we introduce the best-response forward criterion for manager 1 asa process U ( x , x , t ; β ) such that U ( X α , X β , t ; β ) is a (local) supermartingale for each admissiblepolicy α and becomes a (local) martingale, U ( X α ∗ , X β , t ; β ) , along an optimal α ∗ . We stress that incontrast to all classical cases, the competitor’s policy process β is not pre-assumed. Rather it is beingrevealed in “real time” and, in turn, the performance criterion U ( x , x , t ; β ) adapts to it dynamically.The best-response forward criterion for manager 2 , U ( x , x , t ; α ) , is defined analogously, and withthe competitor’s policy process α also not a priori known.For the case of competitive interaction between the managers, we introduce a forward Nash equilib-rium criterion, consisting of two pairs (cid:16) U ( x , x , t ; β ) t ≥ , ( α ∗ t ) t ≥ (cid:17) and (cid:16) U ( x , x , t ; α ) t ≥ , ( β ∗ t ) t ≥ (cid:17) such that U ( X α ∗ , X β , t ; β ) and U ( X α , X β ∗ , t ; α ) are (local) supermartingales, and U ( X α ∗ , X β ∗ , t ; α ∗ )and U ( X α ∗ , X β ∗ , t ; β ∗ ) are (local) martingales.For each kind of interaction, based on best response or on competition, we analyze both the assetspecialization and the asset diversification cases. Herein, we focus on forward criteria that are locally One may consider the case of infinite competitors and build the notion of a “forward” mean-field game (MFG).However, constructing such a notion is not immediate as various concepts might not “carry over”, especially when thereis common noise and/or the forward MFG performance process has its own volatility in a general Itˆo-diffusion setting.To date, the proper definition of a forward MFG has not been produced, and neither the convergence of the forwardfinite game to the forward MFG. Formally, one may mimic the definitions herein (see also [27]) and the ones in [39] forthe classical case, and calculate a special solution within the CARA functions (see [54]). dU ( x , x , t ; β ) = b ( x , x , t ; β ) dt and dU ( x , x , t ; α ) = b ( x , x , t ; α ) dt , (1)for suitable adapted processes ( b ( x , x , t ; β )) t ≥ and ( b ( x , x , t ; α )) t ≥ . We choose this class be-cause, in the absence of relative concerns, locally riskless forward criteria were the first to be exten-sively analyzed not only because of their tractability but, also, for the valuable intuition in terms ofnum´eraire choice, time monotonicity of preferences, dependence on market performance, and others(see [50] for details).Throughout, we model the market having one riskless bond and two risky securities representingproxies of two asset classes. Such proxies have been consistently used in the literature (see, for example,[9, 30, 37]). We model their prices as Itˆo-diffusion processes (cf. (3)) but we stress, once more, thattheir coefficients are not a priori chosen but, rather, become known gradually, infinitesimally in time,as the market evolves.When managers invest in isolation, their wealths evolve as in (5) and in (36), (37), for the assetspecialization and diversification cases, respectively. Under relative performance, the competitor’swealth needs to be incorporated. One way to do this was proposed in [8, 9, 10, 37]), which we alsoadopt herein. Namely, we introduce the relative wealth processes ( ˜ X ,t ) t ≥ and ( ˜ X ,t ) t ≥ , with˜ X := X X θ and ˜ X := X X θ , where the competition biases θ , θ ∈ [0 ,
1] model the degree of relative performance considerations.The limiting case θ = 0 (resp. θ = 0) expresses that manager 1 (resp. 2) is not at all concernedwith the output of the opponent. The other limiting case, θ = 1 (resp. θ = 1) corresponds to thetraditional relative performance in terms of a benchmark (such as S&P500 index, collection of indexfunds, and others; see, for example, the related discussion in [9, Section 1]).The form of the relative state dynamics ˜ X and ˜ X , see (9), (11) for the asset specialization and(40), (42) for the asset diversification cases, prompts us to introduce “personalized” fictitious marketsand define the relevant forward criteria within. Informationally, the original and these virtual marketsdo not differ but the forward criteria may have different characteristics, depending on the choice ofthe modified state variables.The above choice of ˜ X and ˜ X suggests to develop criteria of the reduced scaled form, namely, U ( x , x , t ; β ) = V x x θ , t ; β ! and U ( x , x , t ; α ) = V x x θ , t ; α ! , (2)˜ x = x /x θ , ˜ x = x /x θ , for suitable processes ( V (˜ x , t ; β )) t ≥ and ( V (˜ x , t ; α )) t ≥ . Other formsof relative criteria may be introduced depending on admissibility domains, type of risk preferences,etc. (see, for example, additive cases in [13, 22, 24, 39]).In the asset specialization case, neither manager may invest in the asset class of the competitor.As a result, the market (be the original or the fictitious ones) is incomplete. Forward criteria forincomplete markets have been developed before but only when incompleteness comes exclusively fromimperfectly correlated stochastic factors affecting the stocks’ dynamics (see, among others, [4, 40, 41,57]). Herein, however, the kind of incompleteness is different, for it is generated by the specializationconstraints. These constraints alter the relative wealth processes ˜ X and ˜ X in a way that the relateddynamics may be interpreted as either including non-zero constrained allocations (cf. (9) and (11))or, alternatively, having a stream with imperfectly correlated return (cf. (14) and (16)). The formerinterpretation is closer to the original formulation herein. The latter has a different scope. It shows howthe forward criteria under asset specialization may be used to define analogous criteria for problemswith (imperfectly correlated) random endowment process (also called stochastic income stream). Thisis a new class of forward criteria, not been considered so far.We analyze both the best-response and the competition cases, and introduce the corresponding best-response forward relative performance criteria. The definitions extend the original ones in [47].4e, in turn, derive a random PDE (cf. (17)) that the (locally riskless) criterion is expected to satisfy.Its coefficients are adapted processes, and depend on both the market dynamics and the competitor’spolicies. In general, equation (17) is not tractable unless for the homothetic class, which we solve.Nevertheless, its solution is used to derive and represent the optimal policies in a stochastic feedbackform (see (18)).When dynamic competition is allowed, this naturally leads to the new concept of forward Nashequilibrium, which we introduce in Definition 5. To derive the equilibrium policies, one needs to solvea system of equations, in general intractable due to interdependent nonlinearities. The homotheticcase is solvable and we provide the relevant policies. Their form resembles the ones in the log-normalcase studied in [9], but it is now derived, under the new criteria, in the general Itˆo-diffusion setting.In the asset diversification case, both managers invest in a common market. Their relative perfor-mance concerns distort the original wealth processes (cf. (40) and (42)) which, as in the previous case,leads to two distinct personalized fictitious markets, each depending on the individual competitionparameter and the policy of the opponent. Now, however, each of these markets is complete as invest-ment is allowed in both stocks with modified risk premia. Forward criteria may, in turn, be defined asin the asset specialization case and we focus again on locally riskless ones. The completeness of themarkets enables us to extend the results of [50] and characterize both the relative performance andNash equilibrium criteria, their optimal wealth and investment policies in full generality. The specialcase of homothetic criteria is also analyzed.Conceptually, the analyses of the asset specialization and the asset diversification cases are rathersimilar, in terms of the associated fictitious markets and the optimality criteria. The fundamentaldifficulty is in their (in)completeness which affects the tractability of the problem and the form of theoptimal policies. A key difference is that the locally riskless forward criteria in the asset diversificationcase are always time-decreasing while, in the asset specialization case, they are not. We furtherelaborate on this later on.We conclude the introductory section mentioning that an underlying assumption herein - which isalso widely present in the classical literature - is that the managers have common information for boththe market and the opponent’s strategies; we refer the reader to [8, 11] for discussion of supportingarguments for this assumption. While the access to such information is much more realistic in oursetting (as it occurs in “real-time”), the fact that both managers share common access to it is, in ourview, a rather stringent requirement. As the focus herein is to develop the new, forward frameworkwith relative performance, we also adopt this assumption. We provide ideas how to relax it and futureresearch in this direction in section 4.The paper is organized as follows. In section 2 we present the asset specialization case and analyzethe forward best-response and the forward Nash equilibrium cases. In section 3, we analyze the assetdiversification case while in section 4 we conclude and comment on possible extensions. The market consists of one (locally) riskless asset and two risky securities, representing proxies oftwo distinct asset classes. The prices of the risky securities, ( S ,t ) t ≥ and ( S ,t ) t ≥ are Itˆo-diffusionssolving dS S = µ dt + σ dW and dS S = µ dt + σ dW , (3)with S , , S , >
0. The processes ( W ,t ) t ≥ , ( W ,t ) t ≥ are standard Brownian motions on a filteredprobability space (Ω , F , ( F t ) t ≥ , P ) , with correlation coefficient ρ ∈ ( − ,
1) and F t being the filtrationgenerated by ( W , W ). The market coefficients ( µ i,t ) t ≥ ,( σ i,t ) t ≥ , i = 1 , , are F t -adapted processeswith values in R and R + , respectively. The riskless asset is a money market account ( B t ) t ≥ offeringpositive interest rate ( r t ) t ≥ , an F t -adapted process.We denote this original market by M = ( B, S , S ) . The related market price of risk processes,5 λ ,t ) t ≥ and ( λ ,t ) t ≥ , are given by λ = µ − rσ and λ = µ − rσ , (4)and assumed to be bounded processes, 0 < c ≤ λ , λ ≤ C < ∞ , t ≥ , for some (possibly determin-istic) constants c, C .In this market environment, we consider two asset managers, indexed by i = 1 , . They specialize in assets S and S , respectively, in that manager 1 (resp. 2) trades between the riskless asset and S (resp. S ). However, both managers have access to the common filtration ( F t ) t ≥ (as for for examplein [11] and [12]).We denote by ( X ,t ) t ≥ , ( X ,t ) t ≥ the wealths of manager 1 and 2 and by ( α t ) t ≥ and ( β t ) t ≥ thecorresponding self-financing strategies in assets S and S . Then, (3) yields dX X = σ α ( λ dt + dW ) and dX X = σ β ( λ dt + dW ) , (5)with X i, = x i > , i = 1 ,
2; herein, X , X , α, β are expressed in discounted (by the riskless asset)units.The set of admissible policies A and A , of manager 1 and 2 , respectively, are defined for ( π t ) t ≥ =( α t ) t ≥ , ( β t ) t ≥ , and i = 1 , , A i = (cid:26) π : π t ∈ F t , E (cid:20)Z t σ i,s π s ds (cid:21) < ∞ and X i > , t > (cid:27) . (6)The wealth positivity constraint is in accordance to what is frequently observed in the asset man-agement industry (for instance, mutual funds cannot have negative wealth). The measurability ofthe individual investment policies reflects the access by both managers to the common informationgenerated by F t (see section 4 for a discussion on this assumption).We work with relative wealth processes with competition parameters θ , θ ∈ (0 ,
1] following theframework of [8, 9, 60]). Specifically, if manager 2 follows an arbitrary strategy β ∈ A generatingwealth X , the relative wealth of manager 1 , ( ˜ X ,t ) t ≥ , is defined as˜ X := X X θ , (7)with X and X solving (5). Symmetrically, the relative wealth of manager 2, ( ˜ X ,t ) t ≥ , given anarbitrary strategy α ∈ A of manager 1 generating wealth X , is defined as˜ X := X X θ . (8)As discussed in the introduction, we introduce three new modeling elements. Firstly, while wemake the structural model assumption (3), we do not pre-choose (at initial time) the processes µ i , σ i , i = 1 , . Secondly, in a similar manner, we do not assume that the competitors’ policies, α and β, are a priori chosen stochastic processes. Rather, each manager learns the market coefficients and theopponent’s strategy as time enfolds. Thirdly, there is no pre-chosen investment horizon.The biased benchmark processes ( X θ ,t ) t ≥ and ( X θ ,t ) t ≥ solve, for θ , θ ∈ (0 , ,dX θ X θ = σ θ α (cid:18)(cid:18) λ −
12 (1 − θ ) σ α (cid:19) dt + dW (cid:19) and dX θ X θ = σ θ β (cid:18)(cid:18) λ −
12 (1 − θ ) σ β (cid:19) dt + dW (cid:19) .
6n turn, the relative wealth ˜ X satisfies d ˜ X ˜ X = σ α (cid:16) ˜ λ , dt + dW (cid:17) − σ θ β (cid:16) ˜ λ , dt + dW (cid:17) , (9)and ˜ X , = x /x θ , x , x > , with the processes (˜ λ , ,t ) t ≥ and (˜ λ , ,t ) t ≥ , ˜ λ , := λ − ρσ θ β and ˜ λ , := λ − σ (1 + θ ) β. (10)Symmetrically, the relative wealth ˜ X satisfies d ˜ X ˜ X = − σ θ α (cid:16) ˜ λ , dt + dW (cid:17) + σ β (cid:16) ˜ λ , dt + dW (cid:17) , (11)and ˜ X , = x x θ , x , x > , with the processes (˜ λ , ,t ) t ≥ and (˜ λ , ,t ) t ≥ ,˜ λ , := λ − σ (1 + θ ) α and ˜ λ , := λ − ρσ θ α. (12)We may now interpret the relative wealth dynamics (9) as follows. In the original market M =( B, S , S ) , manager 1 chooses (proportional) risky allocations ( α,
0) in securities S and S , due tospecialization. In the relative formulation, it is as if she invests in a “personalized” fictitious market ˜ M s := (cid:16) B, ˜ S , , ˜ S , (cid:17) with (pseudo) stocks ( ˜ S , ,t ) t ≥ , ( ˜ S , ,t ) t ≥ solving (in discounted units) d ˜ S , ˜ S , = σ (cid:16) ˜ λ , dt + dW (cid:17) and d ˜ S , ˜ S , = σ (cid:16) ˜ λ , dt + dW (cid:17) , (13)with modified Sharpe ratios ˜ λ , and ˜ λ , defined in (10). In this virtual market, the original spe-cialization constraint is not binding, as the manager may now invest in both risky securities, ˜ S , and˜ S , , with respective proportional allocations ( α, − θ β ) , with only α being controlled by manager 1.The constrained allocation − θ β depends on both managers’ characteristics, statically on the biasparameter θ (chosen by manager 1) and dynamically on β (chosen by manager 2).Alternatively, we may view (9) as wealth dynamics in market ˜ M s where manager 1 invests inthe riskless security B and chooses ratio α to allocate in the fictitious stock ˜ S , , while he receives a process of random endowment returns ( ˜ Y ,t ) t ≥ , namely, d ˜ X ˜ X = σ α (cid:16) ˜ λ , dt + dW (cid:17) + dY , (14)with dY = − σ θ β (cid:16) ˜ λ , dt + dW (cid:17) = − θ β d ˜ S , ˜ S , , with Y , = 0 . Note that Y is driven only by W and its dynamics do not depend on λ , σ , α. Analogous interpretations may be derived for manager 2 , who now invests in the “personalized”fictitious market ˜ M s := (cid:16) B, ˜ S , , ˜ S , (cid:17) with (pseudo) stocks ( ˜ S , ,t ) t ≥ and ( ˜ S , ,t ) t ≥ , solving d ˜ S , ˜ S , = σ (cid:16) ˜ λ , dt + dW (cid:17) and d ˜ S , ˜ S , = σ (cid:16) ˜ λ , dt + dW (cid:17) , (15)with modified Sharpe ratios ˜ λ , and ˜ λ , as in (12). We may then interpret (11) as the outcome ofinvesting ratio β in stock ˜ S , while maintaining (ratio) allocation − θ α in stock ˜ S , . Alternatively, d ˜ X ˜ X = σ β (cid:16) ˜ λ , dt + dW (cid:17) + dY , (16) The superscript “s” corresponds to specialization. Y ,t ) t ≥ solving dY = − σ θ α (cid:16) ˜ λ , dt + dW (cid:17) = − θ α d ˜ S , ˜ S , , with Y , = 0 . Clearly, the personalized fictitious markets ˜ M s and ˜ M s do not coincide due to the asymmetry inboth the competition parameters θ and θ , and the competitors’ allocations α and β . As the originalmarket M , the specialization constraints make both these markets incomplete. Note also that, informationally, the markets M , ˜ M s and ˜ M s do not differ but, conceptually, forward performancecriteria are developed within ˜ M s and ˜ M s . Each manager invests between the riskless asset and the stock in which she specializes. She alsocompetes with her opponent passively, in the sense that she observes and takes into account thecompetitor’s policy but without interacting with him. In contrast to all existing settings, however,the competitor’s policy is not a priori modeled, it is only taken to be a process in the admissibleset A , and is being revealed by the competitor gradually, as time moves. To model, measure andoptimize in this relative performance setting, we first introduce a suitable criterion. It extends theoriginal forward criterion, proposed by Musiela and Zariphopoulou (see [47, 48]) and further developedby them and others (see, [34, 49, 50, 51, 52, 53, 63]).Throughout, we will be working with the following set of random functions in the domain D = R + × R + . Definition 1
Let U be the set of random functions u ( z, t ) , ( z, t ) ∈ D , such that, for each t ≥ and P -a.s., the mapping z → u ( z, t ) is strictly concave and strictly increasing, and u ( z, t ) ∈ C , . Definition 2
Let policy β ∈ A . An F t -adapted process ( V (˜ x , t ; β )) t ≥ , ˜ x ≥ , is called a best-response forward relative performance criterion for manager 1 if the following conditions hold:i) For each t ≥ , V (˜ x , t ; β ) ∈ U a.s. ii) For each α ∈ A , V ( ˜ X , t ; β ) is a (local) supermartingale, where ˜ X solves (9) with α beingused.iii) There exists α ∗ ∈ A , such that V (cid:16) ˜ X ∗ , t ; β (cid:17) is a (local) martingale, where ˜ X ∗ solves (9) with α ∗ being used. Analogously, we define the best-response forward relative performance for manager 2, ( V (˜ x , t ; α )) t ≥ ,˜ x ≥ α ∈ A , requiring that V ( ˜ X , t ; α ) and V ( ˜ X ∗ , t ; α ) are, respectively, a (local) supermartin-gale for any β ∈ A and a (local) martingale for an optimal β ∗ ∈ A . The notational presence of β in V and α in V is self-evident.In the absence of competition and for Itˆo-diffusion markets, forward performance criteria havebeen constructed also as Itˆo-diffusion processes (cf. [49]). However, contrary to the classical expectedutility case, their volatility process is an “investor-specific” modeling input. For a chosen volatilityprocess, the supermartingality and martingality properties impose conditions on the drift of the for-ward criterion. Under enough regularity, these conditions lead to the forward performance SPDE (see[51]), which is a fully nonlinear infinite dimensional equation. Depending on whether the forwardprocess is path-dependent or a deterministic functional of stochastic factors, the forward volatilitycan be chosen to be path- or state-dependent (see, for example, [33, 34, 41, 52, 53, 57]). In general,the underlying problems are inherently ill-posed and extra analysis is required to identify the viableinitial conditions (see, for example, [12, 50]).As mentioned in the introduction, we will work with locally riskless (no volatility) performanceprocesses, dV (˜ x , t ; β ) = b (˜ x , t ; β ) dt and dV (˜ x , t ; α ) = b (˜ x , t ; α ) dt, for some suitably chosen F t -adapted processes ( b (˜ x , t ; β )) t ≥ and ( b (˜ x , t ; α )) t ≥ . V (˜ x , t ; β ). Similar results may be derived for manager 2 and are, thus, omitted. Throughout, it isassumed that ρ = 1 , as the case ρ = 1 is more natural for the asset diversification setting. Proposition 3
Let β ∈ A , ρ = 1 , and ˜ λ , and ˜ λ , as in (10). Consider the random PDE v t −
12 ˜ λ , v z v zz + 12 (cid:0) − ρ (cid:1) θ σ β z v zz + (cid:16) ρ ˜ λ , − ˜ λ , (cid:17) θ σ βzv z = 0 , (17) for ( z, t ) ∈ D , and assume that a solution v ( z, t ) ∈ U exists, for some admissible initial datum v ( z,
0) = V ( z, β ) . Furthermore, let the process ( α ∗ t ) t ≥ be given by α ∗ = α ∗ ( ˜ X ∗ , t ) , with the random function α ∗ ( z, t ) , ( z, t ) ∈ D , defined as α ∗ ( z, t ) = ˜ λ , σ R ( z, t ) + ρ σ σ θ β, (18) with R ( z, t ) := − v z ( z, t ) zv zz ( z, t ) , (19) and ( ˜ X ∗ ,t ) t ≥ solving (5) with the control process α ∗ being used. If ˜ X ∗ is well defined and α ∗ ∈ A ,then the process V (˜ x , t ; β ) := v (˜ x , t ) , ˜ x ≥ , is a locally riskless best-response forward relative performance criterion and the investmentstrategy α ∗ is optimal. Proof.
We first rewrite (9) as d ˜ X ˜ X = σ ˆ α (cid:16) ˜ λ , dt + dW (cid:17) + θ σ β (cid:16)(cid:16) ρ ˜ λ , − ˜ λ , (cid:17) dt − p − ρ dW ⊥ (cid:17) , (20)for W ⊥ being a standard Brownian motion orthogonal to W and the modified policy ( ˆ α t ) t ≥ , ˆ α := α − ρθ σ σ β. (21)Assuming that v ( z, t ) ∈ U , Ito’s formula yields dv ( ˜ X , t ) = v t ( ˜ X , t ) dt + (cid:18) σ ˆ α ˜ X v zz ( ˜ X , t ) + ˜ λ , ˆ α ˜ X v z ( ˜ X , t ) (cid:19) dt + (cid:18) (cid:0) − ρ (cid:1) ( σ θ β ) ˜ X v zz ( ˜ X , t ) + (cid:16) ρ ˜ λ , − ˜ λ , (cid:17) σ θ β ˜ X v z ( ˜ X , t ) (cid:19) dt + v z ( ˜ X , t ) (cid:16) σ ˜ αdW − σ θ β p − ρ dW ⊥ (cid:17) . Note that for v zz < , we have12 σ ˆ α ˜ X v zz ( ˜ X , t ) + ˜ λ , ˆ α ˜ X v z ( ˜ X , t ) −
12 ˜ λ , v z v zz , with the maximum ˆ α ∗ occurring at ˆ α ∗ = − ˜ λ , σ v z ( ˜ X ,t )˜ X v zz ( ˜ X ,t ) . The rest of the proof follows easily.
Discussion:
Equation (17) is, in general, non-tractable due to the presence of the second-order lin-ear term (cid:0) − ρ (cid:1) θ σ β z v zz (the first-order term θ (cid:16) ρ ˜ λ , − ˜ λ , (cid:17) σ βzv z may be easily absorbed9ith a mere time-rescaling). Its form is random and evolves with the market and the competitor’spolicy forward in time.Equations of similar structure also arise in expected utility problems in the classical setting whenthere is random endowment and/or labor income processes. To the best of our knowledge, they arealso non-tractable and only general abstract results exist to date (see, among others, [44] and themore recent work [45]). In the forward case, an additional complication arises from the ill-posednessof the problem, for one also needs to specify the class of admissible initial conditions V (˜ x , β ) . Thisis a rather challenging question, currently investigated by the authors. On the other hand, the CRRAclass provides an example, showing that Definition 2 is not vacuous.Despite its non-tractability, equation (17) demonstrates that the best-response criterion V (˜ x , t ; β )is endogenously specified and depends on the current evolution of the market and the competitor’spolicy. Both these features are in contrast to their analogues in the classical cases.The optimal policy is constructed through the random feedback functional α ∗ ( z, t ) , which consistsof the “myopic”-type term ˜ λ , σ R ( z, t ) and the linear term ρ σ σ θ β. The first component resemblesthe one in the original forward setting but now with modified risk premium ˜ λ , . It depends on thecompetitor’s policy β through ˜ λ , and R ( z, t ) . If ρ = 0 , it may become zero if there exist time(s),say t , such that β t = λ ,t ρσ ,t θ . In general, it is difficult to provide any qualitative conclusions on how α ∗ ( z, t ) is influenced by β butat least (18) highlights its endogeneity and that it is affected by the realized market performance, thecompetitor’s policy, and the manager’s realized performance. These characteristics are the outcomeof the flexibility of the normative best-response forward criterion. We stress that empirical evidencestrongly supports such features; see, for example, [17, 29, 35], for the effects of past performance bythe manager and [36] for the impact of realized market performance. The classical model in which the(terminal) risk tolerance is exogenously chosen does not seem to capture these phenomena, as arguedin these papers.Next we note that, in general, V (˜ x , t ; β ) may not be time-monotone (albeit being locally riskless).This can be seen from equation (17) when written as (recall ρ = 1) v t + 12 (cid:0) − ρ (cid:1) θ v zz ( σ βz − c ) ( σ βz − c ) = 0 , with c , = v z θ v zz − ( ρ ˜ λ , − ˜ λ , ) ±√ ∆1 − ρ , and the process (∆ t ) t ≥ given by ∆ := ˜ λ , − ρ ˜ λ , ˜ λ , + ˜ λ , > . We easily deduce that c c < v. We recall that in the absence of competition ( θ = 0), the analogous locally riskless criterion isgiven by u ( x , R t λ ds ) , with u satisfying u t = u z u zz , ( z, t ) ∈ D . This process is always decreasingin time. The lack of time-monotonicity is one of the fundamental differences between the forwardperformance processes with and without competition, V (˜ x , t ; β ) and u ( x , R t λ ds ). We commentmore on this in the next section.If ρ = 0, then ˜ λ , = λ and equation (17) reduces to v t − λ v z v zz + 12 θ σ β z v zz − ˜ λ , θ σ βzu z = 0 . In turn, α ∗ ( z, t ) = − λ σ v z ( z,t ) zv zz ( z,t ) , with v still depending on β through the coefficients in the reducedequation above.If ρ = 0 , relative performance concerns might lead to zero allocation in ˜ S , , at time(s) t suchthat ˜ λ , ,t σ ,t R ( z, t ) + ρ σ ,t σ ,t θ β t = 0 . To provide further insights on the forward relative performance criteria and also compare them withthe ones in the classical setting, we study the case of homothetic criteria for manager 1 . We impose10o assumption on what criterion manager 2 might follow, we only use assume that she follows anarbitrary policy β ∈ A . Proposition 4
Let policy β ∈ A , ρ = 1 , and ˜ λ , and ˜ λ , as in (10). Let γ > , γ = 0 , and ( η ,t ) t ≥ be given by η = ˜ λ , + 2 (cid:16) ρ ˜ λ , − ˜ λ , (cid:17) θ σ βγ − (cid:0) − ρ (cid:1) θ σ β γ . (22) Then, the process V (˜ x , t ; β ) = ˜ x − γ − γ e − R t − γ γ η ds , (23) is a locally riskless best-response forward criterion and the investment policy α ∗ = 1 γ ˜ λ , σ + ρθ σ σ β (24) is optimal. Proof.
We look for candidate criteria of the separable form V (˜ x , t ; β ) = ˜ x − γ − γ K, where ( K t ) t ≥ is an F t -adapted process, differentiable in t with K = 1. Using equation (17), the boundednessassumption on the Sharpe ratios and the admissibility of β, we easily conclude.We may rewrite the process ( η ,t ) t ≥ as η = ( λ , − δ θ σ β ) + (cid:18) ρ (1 − γ ) + γ (1 − γ + 1 θ ) − δ (cid:19) θ σ β , (25)with ( δ ,t ) t ≥ given by δ = γ λ λ + ρ (1 − γ ) . (26)Similar expressions were derived in [9] for the special case of log-normal markets for power utilities inthe classical setting. Herein, we have analogous results for general F t -adapted processes η and δ . We stress that no solutions of form (23), (25) and (26) may be derived in the classical setting beyondthe log-normal case.The criterion V (˜ x , t ; β ) resembles its forward counterpart in the absence of relative performance( θ = 0), given by u ( x , t ) = x − γ − γ e − R t − γ γ λ ds (see [50]), which is however always time-monotone.Rewriting (24) as α ∗ = 1 γ λ σ + ρθ (cid:18) − γ (cid:19) σ σ β, (27)we see that depending on the sign of the various terms, manager 1 might invest more or less in the riskyasset under relative performance concerns. For example, for ρ > σ σ >
0, and a long competitor’sstrategy, β > , we have ρθ (cid:16) − γ (cid:17) σ σ β > γ <
0, while ρθ (cid:16) − γ (cid:17) σ σ β < < γ < . These results are also consistent with the ones in [9] but, now, for a much more flexible framework.Finally, if the market price of risk λ increases, the position on the familiar asset always increaseseven with relative performance concerns. This is consistent with the fact that when the performanceof the asset the manager invests in improves, she tends to increase her position to it. The process λ usually refers to the manager’s active-management ability (see among others [56]).Symmetric results are deduced for manager 2 if her competitor follows policy α ∈ A . Namely, for γ > , γ = 1 , and ( η ,t ) t ≥ with η := ˜ λ , + 2 (cid:16) − ˜ λ , + ρ ˜ λ , (cid:17) σ θ αγ − (cid:0) − ρ (cid:1) ( σ θ α ) γ , (28)11he process ( V (˜ x , t ; α )) t ≥ given by V (˜ x , t ; α ) = ˜ x − γ − γ e − R t − γ γ η ds , (29)is a locally riskless best-response forward criterion and the investment policy β ∗ = 1 γ ˜ λ , σ + ρθ σ σ α = 1 γ λ σ + ρθ (cid:18) − γ (cid:19) σ σ α (30)is optimal.Finally, we may construct a best-response (locally riskless) forward criterion for the limiting cases γ = 0 and/or γ = 0 . Looking for a candidate process of the additive form V (˜ x , t ; β ) = log ˜ x + K, for a suitable process ( K t ) t ≥ , equation (17) yields V (˜ x , t ; β ) = log ˜ x + Z t (cid:18)
12 ˜ λ , − (cid:16) ρ ˜ λ , − ˜ λ , (cid:17) θ σ β + 12 (cid:0) − ρ (cid:1) θ ( σ β ) (cid:19) ds, with optimal policy α ∗ = ˜ λ , σ + ρ σ σ θ β. Similar results can be produced for the case γ = 0 . The asset managers not only trade between the riskless account and the respective specialized riskyasset but, also, interact dynamically with each other. Then, the individual best-response problemslead conceptually to a pure-strategy Nash game. We call the equilibrium of this game a forward Nashequilibrium and propose the following definition for its analysis.We recall the modified risk premia ˜ λ , ( β ) , ˜ λ , ( β ) and ˜ λ , ( α ) , ˜ λ , ( α ) (cf. (10) and (12)), high-lighting their dependence on the competitor’s policies. Definition 5
A forward Nash equilibrium consists of two pairs of F t -adapted processes, (cid:16) V (˜ x , t ; β ∗ ) t ≥ , ( α ∗ t ) t ≥ (cid:17) and (( V (˜ x , t ; α ∗ )) t ≥ , ( β ∗ t ) t ≥ ) , ˜ x , ˜ x > , t ≥ , with the followingproperties:i) The processes α ∗ ∈ A and β ∗ ∈ A , ii) The processes V (˜ x , t ; β ∗ ) , V (˜ x , t ; α ∗ ) ∈ U . ii) For α ∈ A , V ( ˜ X , t ; β ∗ ) is a (local) super-martingale and V ( ˜ X ∗ ,t , t ; β ∗ ) is a (local) martingalewhere ˜ X and ˜ X ∗ solve (9) with ˜ λ , = ˜ λ , ( β ∗ ) and ˜ λ , = ˜ λ , ( β ∗ ) , and with α and α ∗ being,respectively, used.iii) For β ∈ A , V ( ˜ X , t ; α ∗ ) is a (local) super-martingale and V ( ˜ X ∗ ,t , t ; α ∗ ) is a (local) martingalewhere ˜ X and ˜ X ∗ solve (11) with ˜ λ , = ˜ λ , ( α ∗ ) and ˜ λ , = ˜ λ , ( α ∗ ) , and with β and β ∗ being,respectively, used. If, under appropriate integrability conditions, the processes (cid:16) V ( ˜ X , t ; β ∗ ) (cid:17) t ≥ and (cid:16) V ( ˜ X ∗ , t ; β ∗ ) (cid:17) t ≥ are, respectively, a true supermartingale and a true martingale then, for any α ∈ A , E h V ( ˜ X ∗ , t ; β ∗ ) i = E [ V (˜ x , ≥ E h V ( ˜ X , t ; β ∗ ) i . Analogously, E h V ( ˜ X ∗ , t ; α ∗ ) i = E [ V (˜ x , ≥ E h V ( ˜ X , t ; α ∗ ) i . In other words, no unilateral deviation in strategy by either manager will result in an increase in theexpected utility of her relative performance metric.12rom Proposition 3 and, in particular, the best-response strategy (18) and analogous results forthe optimal policy β ∗ , it follows that the candidate forward Nash equilibrium strategies should satisfythe system of equations α ∗ = ˜ λ , ( β ∗ ) σ R ∗ (cid:16) ˜ X ∗ , t ; β ∗ (cid:17) + ρθ σ σ β ∗ β ∗ = ˜ λ , ( α ∗ ) σ R ∗ (cid:16) ˜ X ∗ , t ; α ∗ (cid:17) + ρθ σ σ α ∗ , (31)where (cid:16) R ∗ ,t (cid:16) ˜ X ∗ , t ; β ∗ (cid:17)(cid:17) t ≥ and (cid:16) R ∗ ,t (cid:16) ˜ X ∗ , t ; α ∗ (cid:17)(cid:17) t ≥ are defined as R ∗ (cid:16) ˜ X ∗ , t ; β ∗ (cid:17) = − v ,z ( ˜ X ∗ , t )˜ X ∗ v ,zz ( ˜ X ∗ , t ) and R ∗ (cid:16) ˜ X ∗ , t ; α ∗ (cid:17) = − v ,z ( ˜ X ∗ , t )˜ X ∗ v ,zz ( ˜ X ∗ , t ) , with v ( z, t ) and v ( z, t ) , ( z, t ) ∈ D , solving v ,t −
12 ˜ λ , ( β ∗ ) v ,z v ,zz + 12 (cid:0) − ρ (cid:1) σ θ β ∗ z v ,zz + (cid:16) ρ ˜ λ , ( β ∗ ) − ˜ λ , ( β ∗ ) (cid:17) σ θ β ∗ zv ,z = 0 (32)and v ,t −
12 ˜ λ , ( α ∗ ) v ,z v ,zz + 12 (cid:0) − ρ (cid:1) σ θ α ∗ z v ,zz + (cid:16) − ˜ λ , ( α ∗ ) + ρ ˜ λ , ( α ∗ ) (cid:17) σ θ α ∗ zv ,z = 0 . (33)System (31) is in general non-tractable because of the highly non-linear terms R ∗ (cid:16) ˜ X ∗ , t ; β ∗ (cid:17) and R ∗ (cid:16) ˜ X ∗ , t ; α ∗ (cid:17) . For tractability and to highlight the differences between the forward approach and the classical setting,we examine the case of homothetic criteria for both managers.
Proposition 6
Let γ , γ > with γ , γ = 1 , and assume that δ := γ γ − ρ θ θ (1 − γ )(1 − γ ) = 0 . (34) Consider the processes ( α ∗ t ) t ≥ , ( β ∗ t ) t ≥ given by α ∗ = γ λ − ρθ (1 − γ ) λ σ δ and β ∗ = γ λ − ρθ (1 − γ ) λ σ δ . (35) Let also (cid:0) η ∗ ,t (cid:1) t ≥ and (cid:0) η ∗ ,t (cid:1) t ≥ be given by (22) and (28) when β ∗ and α ∗ are, respectively, used and ( V (˜ x , t ; β ∗ )) t ≥ and ( V (˜ x , t ; α ∗ )) t ≥ defined as V (˜ x , t ; β ∗ ) = ˜ x − γ − γ e − R t − γ γ η ∗ ds , V (˜ x , t ; α ∗ ) = ˜ x − γ − γ e − R t − γ γ η ∗ ds . Then, the pair of processes (( V (˜ x , t ; β ∗ ) , α ∗ ) and ( V (˜ x , t ; α ∗ ) , β ∗ ) constitutes a forward Nashequilibrium. Proof.
From (27) and (30), we deduce that the candidate strategies ( α ∗ t ) t ≥ , ( β ∗ t ) t ≥ must solve thesystem α ∗ − (cid:16) − γ (cid:17) ρθ σ σ β ∗ = γ λ σ − (cid:16) − γ (cid:17) ρθ σ σ α ∗ + β ∗ = γ λ σ . δγ γ , with δ as in (34) and, by assumption, δ = 0, we easilydeduce (35). Furthermore, α ∗ ∈ A and β ∗ ∈ A , given the assumption on bounded λ and λ . Therest of the proof follows easily.In the special case ρ = 0 , the forward Nash equilibrium strategies simplify to, α ∗ = 1 γ λ σ and β ∗ = 1 γ λ σ , which are the optimal policies without competition. Note, however, that the associated forward Nashcriteria still depend on the other manager’s strategy through the processes η ∗ and η ∗ .Continuing the discussion in 2.1.1., we mention that the forward Nash equilibrium investmentstrategies have the same form as those of the classical setting in a log-normal market (see [9, Propo-sition 1]). Hence, we may generalize all comparative statics of [9] in the general Itˆo-diffusion marketsetting herein. In this section we impose the situation where both managers invest in the same market M =( B, S , S ) , with S , S solving (3) and without any trading constraints. This case is particularlypopular when managers aim to beat the same benchmark. The managers have relative performanceconcerns and may interact passively or competitively. As in the asset specialization case, we incorpo-rate these concerns by working with relative wealth processes with competition parameters θ , θ . Wemeasure the performance of their strategies using forward best response and forward Nash equilibriumcriteria, respectively. We define them as in Definitions 2 and 5, and we also work with locally risklessprocesses.Using (3), the (discounted by the bond) wealth processes ( X ,t ) t ≥ and ( X ,t ) t ≥ , t ≥ , satisfy dX X = σ α ( λ dt + dW ) + σ α ( λ dt + dW ) (36)and dX X = σ β ( λ dt + dW ) + σ β ( λ dt + dW ), (37)with X , = x > X , = x > , and α , α (resp. β , β ) being the fractions of wealth X (resp. X ) invested in asset classes S and S , respectively. The set A of admissible policies α = ( α , α ) and β = ( β , β ) is defined similarly to (6).For θ , θ ∈ (0 , , α = ( α , α ) and β = ( β , β ) , the biased benchmark processes (cid:16) X θ ,t (cid:17) t ≥ and (cid:16) X θ ,t (cid:17) t ≥ solve dX θ ,t X θ ,t = θ σ α ( λ dt + dW ) + θ σ α ( λ dt + dW ) + 12 θ ( θ − C ( α ) dt with the process ( C ,t ( α )) t ≥ , C ( α ) := σ α + 2 ρσ σ α α + σ α . (38)Similarly, dX θ ,t X θ ,t = θ σ β ( λ dt + dW ) + θ σ β ( λ dt + dW ) + 12 θ ( θ − C ( β ) dt, with the process ( C ,t ( β )) t ≥ , C ( β ) = σ β + 2 ρσ σ β β + σ β . (39)14irect calculations yield that the relative wealths processes ˜ X := X X θ ,t and ˜ X := X X θ ,t satisfy d ˜ X ˜ X = σ α (cid:16) ˜ λ , dt + dW (cid:17) + σ α (cid:16) ˜ λ , dt + dW (cid:17) − σ θ β ( λ dt + dW ) − σ θ β ( λ dt + dW ) + 12 θ (1 + θ ) C ( β ) dt, (40)with the processes (cid:16) ˜ λ , ,t (cid:17) t ≥ and (cid:16) ˜ λ , ,t (cid:17) t ≥ , ˜ λ , := λ − θ ( σ β + ρσ β ) and ˜ λ , := λ − θ ( ρσ β + σ β ) . (41)Similarly, d ˜ X ˜ X = σ β (cid:16) ˜ λ , dt + dW (cid:17) + σ β (cid:16) ˜ λ , dt + dW (cid:17) − σ θ α ( λ dt + dW ) − σ θ α ( λ dt + dW ) + 12 θ (1 + θ ) C ( α ) dt, (42)with the processes (cid:16) ˜ λ , ,t (cid:17) t ≥ and (cid:16) ˜ λ , ,t (cid:17) t ≥ ,˜ λ , := λ − θ ( σ α + ρσ α ) and ˜ λ , := λ − θ ( ρσ α + σ α ) . (43)As in the asset specialization case, we may interpret (40) as the wealth of a manager who invests inthe personalized fictitious market ˜ M d := (cid:16) B, ˜ S , , ˜ S , (cid:17) with (pseudo) stocks ˜ S , , ˜ S , solving d ˜ S , ˜ S , = σ (cid:16) ˜ λ , dt + dW (cid:17) and d ˜ S , ˜ S , = σ (cid:16) ˜ λ , dt + dW (cid:17) , with ˜ λ , and ˜ λ , given in (41), while receiving returns from a random endowment process ( Y ,t ) t ≥ ,d ˜ X ˜ X = α σ (cid:16) ˜ λ , dt + dW (cid:17) + α σ (cid:16) ˜ λ , dt + dW (cid:17) + dY ,t with dY = − θ σ β ( λ dt + dW ) − θ σ β ( λ dt + dW ) + 12 θ (1 + θ ) C ( β ) dt and Y , = 0 . Similarly, manager 2 invests in a personalized fictitious market ˜ M d := (cid:16) B, ˜ S , , ˜ S , (cid:17) with (pseudo)stocks ˜ S , , ˜ S , solving d ˜ S , ˜ S , = σ (cid:16) ˜ λ , dt + dW (cid:17) and d ˜ S , ˜ S , = σ (cid:16) ˜ λ , dt + dW (cid:17) , with ˜ λ , and ˜ λ , given in (43), and d ˜ X ˜ X = σ β (cid:16) ˜ λ , dt + dW (cid:17) + σ β (cid:16) ˜ λ , dt + dW (cid:17) + dY with dY = − θ σ α ( λ dt + dW ) − θ σ α ( λ dt + dW ) + 12 θ (1 + θ ) C ( α ) dt, and Y , = 0 . The personalized fictitious markets ˜ M d and ˜ M d are both complete, in contrast to their counter-parts ˜ M s and ˜ M s in the asset specialization case. This completeness makes the underlying problemstractable, as we discuss next. 15 .1 Best-response forward relative performance criteria In analogy to the asset diversification case, we apply Definition 2 to define the best-response forwardperformance criteria, denoted with a slight abuse of notation by ( V (˜ x , t ; β )) t ≥ and ( V (˜ x , t ; α )) t ≥ , ˜ x = x xθ , ˜ x = x x θ , where α, β stand for arbitrary policies of the competitors. Because of symmetry,we only analyze the quantities pertinent to manager 1 . We provide complete characterization of herrelative forward criterion, the optimal investment and the optimal wealth processes under relativeperformance concerns.We first recall two auxiliary functions, u : D → R + and h : R × R + → R + . Function u solves u ,t = 12 u ,z u ,zz , (44)with initial condition given by ( u ′ ( z, ( − := Z ∞ + z − y dν ( y ) , (45)for a finite positive Borel measure ν .The function h is defined as h ( z, t ) := ( u ,z ) ( − ( e − z + t , t ) (spatial inverse). It solves h ,t + h ,zz = 0 with h ( z,
0) = R ∞ + e yz dν ( y ), and is given by h ( z, t ) = Z ∞ + e yz − y t dν ( y ) . (46)Let also R : D → R + , R ( z, t ) := − u ,z ( z, t ) zu ,zz ( z, t ) = h ,z (cid:16) h ( − ( z, t ) , t (cid:17) h (cid:16) h ( − ( z, t ) , t (cid:17) , (47)with the latter equality following from the definition of h . The functions u , h and R were introduced in [50], and used to construct in full generality thelocally riskless forward criteria in the absence of competition ( θ = 0); we refer the reader therein fordetails, and especially for the assumptions on measure ν . Finally, we consider the processes ( A ,t ) t ≥ and ( M ,t ) t ≥ defined, for ˜ λ , , ˜ λ , as in (41), as A := 11 − ρ Z t (cid:16) ˜ λ , − ρ ˜ λ , ˜ λ , + ˜ λ , (cid:17) ds and M := Z t ˜ λ , dW + Z t ˜ λ , dW . (48)Next, we present the main result in the asset diversification case. Proposition 7
Let policy β = ( β , β ) ∈ A and C ( β ) as in (39), and define ( B ,t ) t ≥ as B := e θ (1 − θ ) R t C ( β ) ds . (49) Let the processes A and M be as in (48), u (˜ x , as in (45) with u ( z, t ) solving (44), and introducethe process ( H ,t ) t ≥ , H := h ,z (cid:16) h ( − (˜ x ,
0) + A + M , A (cid:17) h (cid:16) h ( − (˜ x ,
0) + A + M , A (cid:17) (50)= R ∞ + e yh ( − (˜ x , yd ˜ ν ,t ( y ) R ∞ + e yh ( − (˜ x , d ˜ ν ,t ( y ) , ith d ˜ ν ,t ( y ) = e y (1 − y ) A + yM dν ,t ( y ) . (51) The following assertions hold:i) The process ( V (˜ x , t ; β )) t ≥ , given by V (˜ x , t ; β ) = u (cid:18) ˜ x B , A (cid:19) , (52) with V (˜ x , β ) = u (˜ x , is the unique locally riskless best-response forward criterion with suchinitial condition. For each β ∈ A and ˜ x > , V (˜ x , t ; β ) is time-decreasing.ii) The optimal wealth process (cid:16) ˜ X ∗ ,t (cid:17) t ≥ is given by ˜ X ∗ = B h (cid:16) h ( − (˜ x ,
0) + A + M , A (cid:17) (53)= B Z ∞ + e yh ( − (˜ x , d ˜ ν ,t ( y ) , with ˜ ν ,t as in (51). iii) Let α ∗ ( z, t ) = ( α ∗ ( z, t ) , α ∗ ( z, t )) , ( z, t ) ∈ D , be defined as α ∗ ( z, t ) = ˜ λ , − ρ ˜ λ , (1 − ρ ) σ B R (cid:18) zB , A (cid:19) + θ β (54) and α ∗ ( z, t ) = − ρ ˜ λ , + ˜ λ , (1 − ρ ) σ B R (cid:18) zB , A (cid:19) + θ β . Then, the optimal investment processes (cid:0) α ∗ ,t (cid:1) t ≥ , (cid:0) α ∗ ,t (cid:1) t ≥ are given in the feedback form, α ∗ = α ∗ (cid:16) ˜ X ∗ , A (cid:17) and α ∗ = α ∗ (cid:16) ˜ X ∗ , A (cid:17) , (55) and in closed form, α ∗ = ˜ λ , − ρ ˜ λ , (1 − ρ ) σ H + θ β (56) and α ∗ = − ρ ˜ λ , + ˜ λ , (1 − ρ ) σ H + θ β , with H as in (50). Proof.
Let ˆ α := α − θ β , ˆ α := α − θ β . Then, the state dynamics (40) can be written as d ˜ X ˜ X = ˆ α σ (cid:16) ˜ λ , dt + dW (cid:17) + ˆ α σ (cid:16) ˜ λ , dt + dW (cid:17) + 12 θ (1 − θ ) C ( β ) dt. Defining the auxiliary process ( ˆ X ,t ) t ≥ by ˆ X = ˜ X B , we have that d ˆ X ˆ X = ˆ α σ (cid:16) ˜ λ , dt + dW (cid:17) + ˆ α σ (cid:16) ˜ λ , dt + dW (cid:17) , (57)17ith ˆ X , = ˜ X , = ˜ x . We are, then, in the complete market framework of [50, Section 3] andwe deduce that if u : D → R + solves (44) and satisfies (45), then the process u (cid:16) ˆ X , A (cid:17) is asupermartingale for any (cid:16) (ˆ α ,t ) t ≥ , ( ˆ α ,t ) t ≥ (cid:17) and becomes a martingale for (cid:0) ˆ α ∗ ,t (cid:1) t ≥ , (cid:0) ˆ α ∗ ,t (cid:1) t ≥ givenby ˆ α ∗ = − ˜ λ , − ρ ˜ λ , (1 − ρ ) σ ˆ R ∗ and ˆ α ∗ = − ρ ˜ λ , + ˜ λ , (1 − ρ ) σ ˆ R ∗ , with (cid:16) ˆ R ∗ t (cid:17) t ≥ = − u ,z ( ˆ X ∗ ,A ) ˆ X ∗ u ,zz ( ˆ X ∗ ,A ) , where ˆ X ∗ solves (57) with ( ˆ α ∗ , ˆ α ∗ ) being used. Following theanalysis in [50], we deduce that the optimal process (cid:16) ˆ X ∗ ,t (cid:17) t ≥ is given in closed form by ˆ X ∗ = h (cid:16) h ( − (ˆ x ,
0) + A + M , A (cid:17) and (53) follows. Furthermore, from the definition of h we deducethat ˆ α ∗ = ˜ λ , − ρ ˜ λ , (1 − ρ ) σ h ,z (cid:16) h ( − (ˆ x ,
0) + A + M , A (cid:17) h (cid:16) h ( − (ˆ x ,
0) + A + M , A (cid:17) and, similarly, ˆ α ∗ = ˜ λ , − ρ ˜ λ , (1 − ρ ) σ h ,z (cid:16) h ( − (ˆ x ,
0) + A + M , A (cid:17) h (cid:16) h ( − (ˆ x ,
0) + A + M , A (cid:17) . We easily deduce that ˆ α ∗ , ˆ α ∗ ∈ A as well as the rest of the assertions for the optimal wealth andoptimal policies.To establish the time monotonicity of V (˜ x , t ; β ), observe that, for each β ∈ A and ˜ x > ,ddt V (˜ x , t ; β ) = − θ (1 − θ ) C ( β ) B ( β ) u ,x (cid:18) ˜ x ˜ B , A (cid:19) + ˜ λ , − ρ ˜ λ , ˜ λ , + ˜ λ , − ρ u ,t (cid:18) ˜ x B , A (cid:19) < θ < , C > , u ,x > u ,t < . Remark 8
We note that the above best-response forward performance differs from the one introducedin [49] given by u ( x/Y t , Z t ) , where ( Y t ) t ≥ is a traded benchmark and ( Z t ) t ≥ a “market-view” process.This process is not locally riskless and its state variable is the individual wealth, and not the relativeone. Similar results may be derived for manager 2 . Let manager 1 follow an arbitrary policy, say α = ( α , α ) ∈ A . If we choose V (˜ x , α ) = R ∞ + ˜ x − y dν ( y ) , for a suitable positive Borel measure ν , we deduce that the unique locally riskless best-response forward criterion is given by V (˜ x , t ; α ) = u (cid:18) ˜ x B , A (cid:19) , with u solving (44) with u ( z,
0) = V (˜ x , α ) , ( B ,t ) t ≥ = e θ (1 − θ ) R t C ( α ) ds , ( A ,t ) t ≥ := (cid:0) − ρ (cid:1) R t (cid:16) ˜ λ , − ρ ˜ λ , ˜ λ , + ˜ λ , (cid:17) ds with ˜ λ , , ˜ λ , and C ( α ) as in (43) and (38).Furthermore, if ( M ,t ) t ≥ := R t ˜ λ , dW + R t ˜ λ , dW , h ( z, t ) := u ( − ,z ( e − z + t , t ) and ( H ) t ≥ defined as H := h ,z (cid:16) h ( − (˜ x ,
0) + A + M , A (cid:17) h (cid:16) h ( − (˜ x ,
0) + A + M , A (cid:17) , (58)then, the optimal wealth (cid:16) ˜ X ∗ ,t (cid:17) t ≥ is given by ˜ X ∗ = B h (cid:16) h ( − (˜ x ,
0) + A + M , A (cid:17) and thepolicies (cid:0) β ∗ ,t (cid:1) t ≥ , (cid:0) β ∗ ,t (cid:1) t ≥ , with β ∗ = ˜ λ , − ρ ˜ λ , (1 − ρ ) σ H + θ α , β ∗ = − ρ ˜ λ , + ˜ λ , (1 − ρ ) σ H + θ α (59)18re optimal.Replacing ˜ λ , , ˜ λ , , ˜ λ , ˜ λ , in (41) and (43), yields the simplified forms (recall that α = ( α , α )and β = ( β , β )) in the original market dynamics, a ∗ = λ − ρλ (1 − ρ ) σ H (˜ x , β ) + (1 − H (˜ x , β )) θ β a ∗ = − ρλ + λ (1 − ρ ) σ H (˜ x , β ) + (1 − H (˜ x , β )) θ β , (60)and β ∗ = λ − ρλ (1 − ρ ) σ H (˜ x , α ) + (1 − H (˜ x , α )) θ α ,β ∗ = − ρλ + λ (1 − ρ ) σ H (˜ x , α ) + (1 − H (˜ x , α )) θ α . (61) Discussion:
The best-response forward criterion (rewritten with more explicit notation) is given bythe locally riskless process u (cid:16) ˜ x B ( β,ρ,θ ) , A ( λ, σ, β ; ρ, θ ) (cid:17) . The process B ( β, ρ, θ ) > β, the correlation ρ , and the competition parameter θ . It is increasingin θ , when θ ∈ (cid:0) , (cid:1) , and decreasing when θ ∈ (cid:0) , (cid:1) , with maximum discounting at θ = . The discounting vanishes at the limiting values θ = 0 ,
1. For ρ = 1 , the process C ( β , β ; ρ ) > β , β ) and achieves a global minimum at (0 , A ( λ, σ, β, θ ) isnon-decreasing in time and represents a stochastic time change. Furthermore, its time derivative isconvex in the competition parameter θ . The case when manager 2 uses policies (cid:0) β ,t (cid:1) t ≥ , (cid:0) β ,t (cid:1) t ≥ with β = − λ − ρλ (1 − ρ ) σ , β = − λ − ρλ (1 − ρ ) σ requires special attention. Therein, the modified risk premia vanish at all times, ˜ λ , = ˜ λ , = 0 , and thus the ”personalized” fictitious market M d becomes worthless . In turn, A = M = 0 , and α ∗ = θ β and α ∗ = θ β . Therefore, the optimal risky strategy is to simply follow fraction θ of thisspecific competitor’s strategy. This yields ˜ X ∗ = B ˜ x , with B := e θ (1 − θ ) ( − ρ ) R t ( λ + λ ) ds and,thus, (cid:16) V (cid:16) ˜ X ∗ , t ; β (cid:17)(cid:17) t ≥ = u (˜ x , not change with time (provided all quantitiesare expressed in discounted units).The optimal policy is given both via a feedback and in closed form (cf. (54) and (56)). Thefeedback control depends on wealth only through the random function R ( z, t ) , which is the relativerisk tolerance associated with u ( z, t ) . Using the results in [50], we deduce that R ( z, t ) , and thus α ( z, t ) and α ( z, t ) , are decreasing in time and non-increasing in z. Let the measure in (45) be a Dirac, ν ( dy ) = δ γ , γ > . Then, h ( z, t ) = e zγ − (cid:16) γ (cid:17) t and H (˜ x , β ) = γ (cf. (46) and (50)). Criterion (52) becomes V (˜ x , t ; β ) = 11 − γ (cid:18) ˜ x B (cid:19) − γ e −
12 1 − γ γ A , and, this is the unique locally riskless homothetic criterion associated with γ . The optimal policiesand optimal wealth processes are given by (56) and (53), α ∗ = 1 γ λ − ρλ (1 − ρ ) σ + (cid:18) − γ (cid:19) θ β ,α ∗ = 1 γ − ρλ + λ (1 − ρ ) σ + (cid:18) − γ (cid:19) θ β X ∗ = ˜ x e γ (cid:16) − γ (cid:17) A + γ M B . We recall that there is no assumption for the preferences ofmanager 2, only that she follows an arbitrary policy β ∈ A . Similarly, let manager 1 follow policy α = ( α , α ) . If ν ( dy ) = δ γ , for γ > , γ = 1 , the uniquelocally riskless best-response forward criterion for manager 2 with initial condition V (˜ x , α ) = ˜ x − γ − γ is given by V (˜ x , t ; α ) = 11 − γ (cid:18) ˜ x B (cid:19) − γ e −
12 1 − γ γ A , and the optimal policies and optimal wealth by β ∗ = 1 γ λ − ρλ (1 − ρ ) σ + (cid:18) − γ (cid:19) θ a , β ∗ = 1 γ − ρλ + λ (1 − ρ ) σ + (cid:18) − γ (cid:19) θ a (62)and ˜ X ∗ ,t = ˜ x e γ (cid:16) − γ (cid:17) A + γ M B . The forward Nash equilibrium is defined as in Definition 5. To find the equilibrium strategies ( α ∗ t ) t ≥ ,( β ∗ t ) t ≥ one then needs to solve the non-linear system (cf. (60) and (61)), a ∗ = λ − ρλ (1 − ρ ) σ H (˜ x , β ∗ ) + (1 − H (˜ x , β ∗ )) θ β ∗ a ∗ = − ρλ + λ (1 − ρ ) σ H (˜ x , β ∗ ) + (1 − H (˜ x , β ∗ )) θ β ∗ β ∗ = λ − ρλ (1 − ρ ) σ H (˜ x , α ∗ ) + (1 − H (˜ x , α ∗ )) θ α ∗ β ∗ = − ρλ + λ (1 − ρ ) σ H (˜ x , α ∗ ) + (1 − H (˜ x , α ∗ )) θ α ∗ . (63)The system is in general difficult to solve unless for special cases, one of which is examined next. We derive explicit solutions when both managers have homothetic forward criteria using (62) and(63).
Proposition 9
Let γ , γ > with γ , γ = 1 , and assume that γ γ − θ θ (1 − γ )(1 − γ ) = 0 . Then,the Nash equilibrium strategies ( α ∗ t ) t ≥ , ( β ∗ t ) t ≥ are given as α ∗ = c α λ − ρλ σ (1 − ρ ) and α ∗ = c α λ − ρλ σ (1 − ρ ) (64) β ∗ = c β λ − ρλ σ (1 − ρ ) and β ∗ = c β λ − ρλ σ (1 − ρ ) (65) where the constants c α and c β are defined as c α := γ + θ ( γ − γ γ − θ θ (1 − γ )(1 − γ ) , c β := γ + θ ( γ − γ γ − θ θ (1 − γ )(1 − γ ) . Proof.
Taking into account (62), we get that system (63) becomes linear. Assumption γ γ − θ θ (1 − γ )(1 − γ ) = 0 guarantees that the determinant is different than zero and hence the system admitsa unique solution. Simple calculations imply (64) and (65) and standing assumptions on λ and λ yield the admissibility of the equilibrium investment strategies.Similarly to the asset specialization setting, the Nash equilibrium strategies (64) and (65) havethe same form as the ones in the log-normal market and backward utility maximization criteria (see[9, Proposition 2]). All conclusions in [9] hold for the general Itˆo-diffusion setting we assume herein.20 Conclusions and extensions
We have studied portfolio allocations of two fund managers when they incorporate relative perfor-mance concerns. We have looked at the asset specialization and asset diversification settings in anItˆo-diffusion market. For both these cases, we have considered the best response and the Nash equi-libria. We studied these issues in a new framework we introduce herein that is based on forwardperformance criteria. These criteria allow for “real-time” updating of both the model coefficients andthe competitor’s policies as well as for flexible horizons. Thus, we considerably generalize the existingwork on the subject by allowing i) a considerably more general market model, ii) no a priori mod-eling of the competitor’s policy and iii) flexible investment horizons. Next, we discuss some possibleextensions. i) Multi-frequencies:
In all cases herein, we have assumed that model selection, trading and rela-tive/competitive performance valuation are all aligned and, furthermore, that they all occur continu-ously in time. In reality, however, these three fundamental attributes are not synchronized. A morerealistic scenario would allow trading to take place more frequently than model selection, and relativeperformance evaluation to occur less frequently than trading. Note that the most extreme case is inthe classical expected utility problem in which the terminal utility is specified only once, at initialtime, with no further risk preference adjustment.With regards to the relative frequency of trading and model selection, it is more realistic to assumethat the model is selected for some trading period ahead, say a week, and that within this week, tradingtakes place in discrete or continuous time. When relative performance is involved, the distinct scalesof time evolution are more critical, for each fund manager typically announces her performance atdiscrete times and not continuously. ii) Information about market and competitors:
Information availability and acquisition for boththe market and the competitor’s behavior and performance are of tantamount importance. In theexisting literature it is assumed that both managers have full access to both the market(s) and riskpreferences. While we relax the requirement that neither the model dynamics nor the competitor’sinput (risk preferences, chosen policy and investment horizon) need to be a priori modeled, we doassume that any information - acquired in real time - about them is available to both managers,together with their relative bias parameters. These assumptions are partially supported by existingresults; see, for example, [37], where it is argued that managers acquire such information from therealized, and publicly available, returns of their piers.However, several “under-specification” issues remain open, especially in terms of the manager’srisk preferences, specialized knowledge and past performance. For example, it might be more realisticto assume that at the end of each relative evaluation period, each fund manager receives informationabout the performance of the other and, right after, formulates a view about the possible upcomingperformance till the end of the next evaluation period. This will partially address the absence ofcomplete information under asset specialization. In this case, injecting personal views could lead to aforward Black-Litterman type criterion under competition. iii) Beyond locally riskless and reduced form relative performance/competition criteria:
Herein, weworked with criteria that are, from the one hand, locally riskless processes and, from the other, ofthe “homogeneous” scaling (7) and (8). In general, relative performance concerns might be modeled,at the level of the criterion, by arbitrary F t -adapted processes, say C ( x , t ) and C ( x , t ) . Theseprocesses might then model, in a more refined way, the competition dependence on past performanceof the competitors, market conditions and time in a more realistic way.In a different direction, the forward criteria might have volatility, which would capture uncertaintyabout the model dynamics and/or the competitor’s beliefs and policies. We will then work withcriteria of the form dU ( x , x , t ) = b ,t ( x , C ( x , t ) , t ) dt + a , ,t ( x , C ( x , t ) , t ) dW ,t + α , ,t ( x , C ( x , t ) , t ) dW ,t and dU ( x , x , t ) = b ,t ( C ( x , t ) , x , t ) dt + a , ,t ( C ( x , t ) , x ) dW ,t + a , ,t ( C ( x , t ) , x , t ) dW ,t , a , ,t , a , ,t ) t ≥ and ( a , ,t , a , ,t ) t ≥ being adapted and manager-specific inputprocesses. Proceeding as in [51] we would then obtain a stochastic PDE (rather than a random one)with coefficients depending on the evolving market dynamics and the competitor’s policies. As inthe absence of relative concerns, these equations will be ill-posed and degenerate with little, if any,tractability. In turn, the systems related to the forward Nash equilibria (cf. (31) and (63)) would besystems of such infinite dimensional equations.In a different direction, relative forward criteria may be modeled as discrete or a combination ofdiscrete and continuous-time processes for different, possible nested, time regimes, associated withdistinct frequencies as discussed above. For discrete processes, predictability is a natural assumption(see [3] for a binomial model and adaptive market parameter selection). References [1] A. B. Abel. Asset prices under habit formation and catching up with the joneses.
The AmericanEconomic Review , 80(2):38–42, 1990.[2] A. G. Ahearne, W. L. Griever, and F. E. Warnock. Information costs and home bias: An analysisof US holdings of foreign equities.
Journal of International Economics , 62(2):313–336, March2004.[3] B. Angoshtari, T. Zariphopoulou, and X. Y. Zhou. Predictable forward performance processes:The binomial case.
SIAM Journal on Control and Optimization , 58(1):327–347, 2020.[4] M. Anthropelos. Forward exponential performances: Pricing and optimal risk sharing.
SIAMJournal on Financial Mathematics , 5(1):626–655, 2014.[5] L. Avanesyan, M. Shkolnikov, and R. Sircar. Construction of forward performance processesin stochastic factor models and an extension of Widder’s theorem. To appear in Finance andStochastics, available at https://arxiv.org/pdf/1805.04535.pdf, 2018.[6] B. M. Barber, X. Huang, and T. Odean. Which Factors Matter to Investors? Evidence fromMutual Fund Flows.
The Review of Financial Studies , 29(10):2600–2642, 2016.[7] N. Barberis and M. Huang. Stocks as lotteries: The implications of probability weighting forsecurity prices.
The American Economic Review , 98(5):2066–2100, 2008.[8] S. Basak and D. Makarov. Strategic asset allocation in money management.
Journal of Finance ,69(1):179–217, 02 2014.[9] S. Basak and D. Makarov. Competition among portfolio managers and asset specialization. ParisDecember 2014 Finance Meeting EUROFIDAI - AFFI Paper, September 2015.[10] S. Basak, A. Pavlova, and A. Shapiro. Optimal asset allocation and risk shifting in moneymanagement.
The Review of Financial Studies , 20(5):1583–1621, 2007.[11] J. B. Berk and J. H. van Binsbergen. Measuring skill in the mutual fund industry.
Journal ofFinancial Economics , 118(1):1–20, 2015.[12] F. Berrier, C. Rogers, and M. Tehranchi. A characterization of forward utility functions. Pre-rprint, May 2009.[13] J. Bielagk, A. Lionnet, and G. D. Reis. Equilibrium pricing under relative performance concerns.
SIAM Journal on Financial Mathematics , 8(1):435–482, 2017.[14] A. Bodnaruk and A. Simonov. Loss-averse preferences, performance, and career success of insti-tutional investors.
The Review of Financial Studies , 29(11):3140–3176, 2016.2215] M. J. Brennan. The optimal number of securities in a risky asset portfolio when there are fixedcosts of transacting: Theory and some empirical results.
Journal of Financial and QuantitativeAnalysis , 10(03):483–496, September 1975.[16] H. H. Cao, T. Wang, and H. H. Zhang. Model uncertainty, limited market participation, andasset prices.
The Review of Financial Studies , 18(4):1219–1251, 2005.[17] H.-l. Chen and G. G. Pennacchi. Does prior performance affect a mutual fund’s choice of risk? the-ory and further empirical evidence.
Journal of Financial and Quantitative Analysis , 44(04):745–775, August 2009.[18] J. Chevalier and G. Ellison. Career concerns of mutual fund managers.
The Quarterly Journalof Economics , 114(2):389–432, 1999.[19] J. D. Coval and T. J. Moskowitz. Home bias at home: Local equity preference in domesticportfolios.
The Journal of Finance , 54(6):2045–2073, 1999.[20] D. Cuoco and R. Kaniel. Equilibrium prices in the presence of delegated portfolio management.
Journal of Financial Economics , 101(2):264–296, 2011.[21] X. Dong, S. Feng, and R. Sadka. Liquidity risk and mutual fund performance.
ManagementScience , 65(3):1020–1041, 2019.[22] G.-E. Espinosa and N. Touzi. Optimal investement under relative performance concerns.
Math-ematical Finance , 25(2):221–257, 2015.[23] M. A. Ferreira, A. Keswani, A. F. Miguel, and S. B. Ramos. The Determinants of Mutual FundPerformance: A Cross-Country Study.
Review of Finance , 17(2):483–525, 2013.[24] C. Frei and G. D. Reis. A financial market with interacting investors: does an equilibrium exist?
Mathematics and Financial Economics , 4:161–182, 2011.[25] K. R. French and J. M. Poterba. Investor diversification and international equity markets.
Amer-ican Economic Review , 81(2):222–226, May 1991.[26] S. Gallaher, R. Kaniel, and L. T. Starks. Madison avenue meets wall street: Mutual fund families,competition and advertising. Working paper, January 2006.[27] T. Geng. Essays on forward portfolio theory and financial time series modeling.
Ph.D. Thesis,University of Texas at Austin , 2017.[28] J.-P. G´omez, R. Priestley, and F. Zapatero. Implications of keeping-up-with-the-Joneses behaviorfor the equilibrium cross section of stock returns: International evidence.
The Journal of Finance ,64(6):2703–2737, 2009.[29] A. P. Goriaev, F. Palomino, and A. Prat. Mutual fund tournament: Risk taking incentivesinduced by ranking objectives. May 2001.[30] Z. He and A. Krishnamurthy. Intermediary asset pricing.
American Economic Review ,103(2):732–770, April 2013.[31] J. Huang, C. Sialm, and H. Zhang. Risk Shifting and Mutual Fund Performance.
The Review ofFinancial Studies , 24(8):2575–2616, 2011.[32] M. Kacperczyk, C. Sialm, and L. Zheng. On the industry concentration of actively managedequity mutual funds.
The Journal of Finance , 60(4):1983–2011, 2005.[33] S. K¨allblad, J. Ob l´oj, and T. Zariphopoulou. Dynamically consistent investment under modeluncertainty: The robust forward criteria.
Finance and Stochastics , 22:879–918, 2018.2334] N. E. Karoui and M. Mrad. An exact connection between tow solvable SDEs and a nonlinearutility stochastic PDE.
SIAM Journal on Financial Mathematics , 4(1):697–736, 2014.[35] A. Kempf and S. Ruenzi. Tournaments in mutual-fund families.
Review of Financial Studies ,21(2):1013–1036, April 2008.[36] A. Kempf, S. Ruenzi, and T. Thiele. Employment risk, compensation incentives, and managerialrisk taking: Evidence from the mutual fund industry.
Journal of Financial Economics , 92(1):92–108, April 2009.[37] R. S. J. Koijen. The cross-section of managerial ability, incentives, and risk preferences.
TheJournal of Finance , 69(3):1051–1098, 2014.[38] J. L. Koski and J. Pontiff. How are derivatives used? Evidence from the mutual fund industry.
Journal of Finance , 54(2):791–816, 1999.[39] D. Lacker and T. Zariphopoulou. Mean field and N-agent games for optimal investment underrelative performance criteria.
Mathematical Finance , 29(4):1003–1038, 2019.[40] T. Leung, R. Sircar, and T. Zariphopoulou. Forward indifference valuation of American options.
Stochastics , 84(5-6):741–770, 2012.[41] G. Liang and T. Zariphopoulou. Representation of homothetic forward performance processesin stochastic factor models via ergodic and infinite horizon BSDE.
SIAM Journal on FinancialMathematics , 8(1):344–372, 2017.[42] R. C. Merton. A simple model of capital market equilibrium with incomplete information.
TheJournal of Finance , 42(3):483–510, 1987.[43] T. Mitton and K. Vorkink. Equilibrium underdiversification and the preference for skewness.
TheReview of Financial Studies , 20(4):1255–1288, 2007.[44] O. Mostovyi. Optimal investment with intermediate consumption and random endowment.
Math-ematical Finance , 27(1):96–114, 2017.[45] O. Mostovyi and M. Sˆırbu. Optimal investment and consumption with labor income in incompletemarkets.
Annals of Applied Probability , 30(2):747–787, 04 2020.[46] S. Mukerji and J.-M. Tallon. Ambiguity aversion and incompleteness of financial markets.
TheReview of Economic Studies , 68(4):883–904, 2001.[47] M. Musiela and T. Zariphopoulou. Investments and forward utilities. Technical report, Universityof Texas at Austin, 2006.[48] M. Musiela and T. Zariphopoulou.
Derivative pricing, investment management and the termstructure of exponential utilities: The case of binomial model . Princeton University Press, 2009.[49] M. Musiela and T. Zariphopoulou. Portfolio choice under dynamic investment performance cri-teria.
Quantitative Finance , 9(2):161–170, 2009.[50] M. Musiela and T. Zariphopoulou. Portfolio choice under space-time monotone performancecriteria.
SIAM Journal on Financial Mathematics , 1(1):326–365, 2010.[51] M. Musiela and T. Zariphopoulou. Stochastic partial differential equations and portfolio choice.
Contemporary Quantitative Finance , pages 195–215, 2010.[52] S. Nadtochiy and M. Tehranchi. Optimal investment for all time horizons and martin boundaryof space-time diffusions.
Mathematical Finance , 27(2):438–470, 2017.2453] S. Nadtochiy and T. Zariphopoulou. A class of homothetic forward investment performanceprocess with non-zero volatility.
Inspired by Finance: The Musiela Festschrift , pages 475–505,2014.[54] G. D. Reis and V. Platonov. Forward utilities and mean-field games under relative performanceconcerns, 2020. Sumbitted for publication, available at https://arxiv.org/abs/2005.09461.[55] K. Ron, S. Tompaidis, and T. Zhou. Impact of managerial commitment on risk taking withdynamic fund flows. Preprint, April 2016.[56] W. F. Sharpe. Mutual fund performance.
The Journal of Business , 39:119–119, 1965.[57] M. Shkolnikov, R. Sircar, and T. Zariphopoulou. Asymptotic analysis of forward performanceprocesses in incomplete markets and their ill-posed hjb equations.
SIAM Journal on FinancialMathematics , 7(1):588–618, 2016.[58] E. R. Sirri and P. Tufano. Costly search and mutual fund flows.
The Journal of Finance ,53(5):1589–1622, 1998.[59] R. Uppal and T. Wang. Model misspecification and underdiversification.
The Journal of Finance ,58(6):2465–2486, 2003.[60] J. H. van Binsbergen, M. W. Brandt, and R. S. J. Koijen. Optimal decentralized investmentmanagement.
The Journal of Finance , 63(4):1849–1895, 2008.[61] S. Van Nieuwerburch and L. Veldkamp. Information immobility and the home bias puzzle.
TheJournal of Finance , 64(3), 2009.[62] W. Wagner. Systemic liquidation risk and the diversity-diversification trade-off.
The Journal ofFinance , 66(4):1141–1175, 2011.[63] G. ˇZitkovi´c. A dual characterization of self-generation and exponential forward performances.