انتخاب بازار و بقای استراتژی های سرمایه گذاری

The paper analyzes the process of market selection of investment strategies in an incomplete market of short-lived assets. In the model under study, asset payoffs depend on exogenous random factors. Market participants use dynamic investment strategies taking account of the available information about current and previous events. It is shown that an investor allocating wealth across the assets according to their conditional expected payoffs eventually accumulates total market wealth, provided the investor’s strategy is asymptotically distinct from the portfolio rule suggested by the Capital Asset Pricing Model (CAPM). This assumption turns out to be essentially necessary for the result.

It has long been argued that, in competitive environments, market pressures would eventually select those traders who are better adapted to the prevailing conditions. According to the standard paradigm of economic theory, agents maximize preferences or utilities. From the evolutionary point of view, what matters is not the utility level, but the chances of survival. The evolutionary principle leads to the consideration of the process of economic natural selection among the market participants, or among the strategies of behavior they adopt. This view has been put forward by Alchian, 1950, Enke, 1951 and Penrose, 1952 and pursued by many others. The purpose of the paper is to elaborate on an evolutionary approach to the study of investment strategies in financial markets. This work combines ideas from economic theory and finance. We examine the process of market selection in the framework of incomplete markets with traders using dynamic investment strategies. In the model under study, it is supposed that each trader follows a portfolio rule specifying the allocation of wealth across the available assets at any moment of time and for any history of events. We retain the key feature of economic equilibrium models where a market-clearing mechanism determines prices endogenously in every period. However, we depart from individual utility maximization. Instead, we assume that trading strategies are compared with each other in terms of their abilities to survive under market selection, rather than in the conventional terms of discounted values. We consider a market with short-lived assets that live only one time period but are identically reborn every next period. The assets are in positive supply, and their payoffs depend on the realization of exogenous states of the world described in terms of a homogeneous finite-state Markov chain. Short sales are ruled out. The focus is on the long-run dynamics of the distribution of wealth across the investors. Following Epstein and Zin, 1989 and Epstein and Zin, 1991 the model prescribes reinvestment of total wealth and thus precludes consumption. This assumption, in particular, avoids the trade-off between the rate of consumption and the evolutionary fitness of the trading rule in a market. In the case where only a complete set of Arrow securities is traded, the states of the world are independent and identically distributed, and all the traders use only simple portfolio rules (independent of time and observations), our framework reduces to the model considered by Blume and Easley (1992, Section 3). They have demonstrated the remarkable role of the portfolio rule prescribing the investor to distribute wealth between the assets according to the probability of the state in which the asset pays out—this portfolio rule is often referred to as “betting one’s beliefs.” Blume and Easley (1992) show that if a trader uses this rule, whereas all the others use different (simple) portfolio rules, then the trader will eventually accumulate total market wealth. In other words, the investor will be a single survivor in the market selection process. Apparently, the first who stated the principle underlying the rule of “betting one’s beliefs” was Kelly (1956). He showed (in a different context) that this principle leads to the maximization of the expected logarithm of the portfolio growth rate. This idea gave rise to a large body of research—see, e.g., Breiman, 1961, Thorp, 1971, Algoet and Cover, 1988 and Hakansson and Ziemba, 1995. A common feature of these papers is that they study single-agent problems with exogenous prices. Blume and Easley (1992) considered an equilibrium model with prices determined endogenously. Nonetheless, due to the completeness of the market (and the special structure of Arrow securities), their result regarding the rule of “betting one’s beliefs” can be reduced to the maximization of the expected logarithm of an appropriately defined relative growth rate. The Blume–Easley approach has been extended by Sandroni, 2000 and Blume and Easley, 2001 to expected utility maximizing investors with general utility functions. These papers focus, basically, on dynamically complete markets. As regards to incomplete markets, Sandroni (2000, Proposition 1) provides a version of the result that traders maximizing the expected logarithm of the growth rate eventually accumulate all wealth. Note, however, that with endogenous prices it is not clear that this can be achieved with a specific investment strategy like the one derived in this paper. Incomplete markets with simple trading strategies and independent identically distributed states of the world have been considered in Evstigneev et al. (2002) (see also Hens and Schenk-Hoppé, 2003 who examine the local dynamics of the market selection process, dealing, however, with more general strategies). There is a significant distinction between models of this type and the previous ones. Owing to market incompleteness and the endogenous mechanism of price formation, the questions studied in this, more general, framework cannot be reduced to the single-agent maximization of growth rates. Usually, the performance of a strategy in a market selection process cannot be determined only by the strategy itself, it depends on the combination of all the strategies employed by the whole group of investors (this combination determines the endogenous asset prices). The main finding in the last two papers is that investors distributing their wealth according to the expected relative payoffs dominate the market. The present paper continues the study conducted in Evstigneev et al. (2002). Our goal is to remove two simplifying assumptions that substantially reduce the scope of the models under study. Firstly, we consider general, rather than simple, investment strategies. Secondly, we abandon the assumptions of independence and identical distribution of the random variables describing the states of the world and assume instead that the sequence of these random variables is a homogeneous discrete-time Markov process. This more general set of assumptions results in a considerably enlarged scope of the theory at hand, with enhanced realism and thus potential applicability of the results. In a financial investment setting, the state of the world is a description of a large and complex set of variables characterizing investors’ information, including, among many others, business cycle indicators, central bank policy variables, various firm-level indicators and consumer indices. The complexity underlying the evolution of so many relevant state variables could not possibly be captured by a random process with independent and identically distributed values. Some serial dependence, at least of a Markov nature, must be postulated. Furthermore, restricting consideration to simple investment strategies amounts to asking each trader to commit to one and the same constant strategy for the entire duration of the process, as if the agent had no access to any relevant information throughout. This is hardly compatible with the real-life behavior of investors, who typically react with considerable frequency to a whole array of economic indices. Consequently, it seems imperative to model investors’ behavior as reflecting unfolding events and disclosed information.1 To outline our main result, let us denote by λ∗ the portfolio rule that requires a trader to allocate wealth across the assets in accordance with their relative conditional expected payoffs. (In the Markov rather than i.i.d. setting, we have to deal with conditional, rather than unconditional, expectations.) Our main result is that, in any—complete or incomplete—market for short-lived assets, a trader following the rule λ∗ eventually accumulates total market wealth, provided the trader’s strategy λ∗ is asymptotically distinct from the Capital Asset Pricing Model (CAPM) rule. The latter prescribes investing into the market portfolio.2 A trader using the CAPM rule keeps a constant fraction of market wealth. Thus the trader can neither accumulate total market wealth nor be driven out of the market. Investing into the market portfolio means mimicking the“average” portfolio (therefore the CAPM strategy does not, typically, belong to the class of simple strategies). We prove that the λ∗-trader accumulates total market wealth at exponential rate if λ∗ is bounded away from the CAPM rule for “sufficiently many” time periods. More precisely, we impose the following condition: there exists a random number κ>0, such that, almost surely, the distance between λ∗ and the CAPM rule is greater than κ in nt periods during every time-horizon of length t, where View the MathML source. Remarkably, this requirement turns out to be not only sufficient but also necessary for the λ∗-trader to be a single survivor in the market selection process, accumulating wealth at exponential rate (see Theorem 2 in Section 3). The need for such a requirement arises here due to the added complexity of strategic behavior. To the best of our knowledge, the above result has no counterparts in the related literature. Although our analysis is complete within the present framework, there are certainly many desirable extensions of the model. One can mention, for instance, long-lived assets, changes in the market structure, endogenous asset supply, and variations of the investment–consumption ratio. These generalizations are left to future research. For recent studies on evolutionary finance dealing with different (albeit related) models and questions see Brock and Hommes, 1998 and Brock et al., 2003, and references therein. While our paper stands in the tradition of analyzing the market selection hypothesis (cf. Blume and Easley, 1992, Sandroni, 2000 and Blume and Easley, 2001), this strand of literature focusses on asset pricing properties in interacting heterogenous agents models. In particular in those models changes of investment strategies driven by comparisons of realized and anticipated returns contribute to the dynamics of the price processes. The paper is organized as follows. Section 2 introduces the model. The main results are presented in Section 3. All the proofs are relegated to the Appendix.

Consider the random dynamical system (9) describing the evolution of the relative market shares rti(st) of the investors i=1,2,…,I. Note that if rt=(rti) is a strictly positive vector, then, as is easily seen from (9), (10) and (5), rt+1 is a strictly positive vector as well. Thus rt=rt(st) is a random process with values in the relative interior Δ+I of the unit simplex View the MathML source The initial state r0=(r01,…,r0I)∈Δ+I, from which this process starts, is fixed (r0i=w0i/∑w0j). We will analyze the above random dynamical system under the following assumptions. (A1) The functions equation(13) View the MathML source take on strictly positive values for eachs∈S. (A2) For everys∈S, the functionsR1(·,s),…,RK(·,s) restricted to the set Π(s)={σ∈S:p(σ|s)>0} are linearly independent. According to (A1), the conditional expectation equation(14) View the MathML source of the relative payoff Rk(st+1,st) of every asset k given st=s is strictly positive at each state s. Assumption (A2) means the absence of conditionally redundant assets. The term “conditionally” refers to the fact that the functions Rk(·,s), k=1,…,K, are linearly independent on the set Π(s)—the support of the conditional distribution p(σ|s). In what follows, we will restrict attention to those investment strategies λ=(λk,t) that satisfy the following additional assumption. (B) The coordinatesλk,t(st) of the vectorsλt(st) are bounded away from zero by a strictly positive non-random constantρ (that might depend on the strategyλ, but not onk,tandst), i.e. infi,k,t,stλik,t(st)>ρ>0. In (5), we included in the definition of a strategy the condition λk,t>0 (such strategies are sometimes termed completely mixed). Assumption (B) contains the additional requirement of uniform strict positivity of λk,t. A key role in our analysis will be played by the strategy View the MathML source defined according to the formula equation(15) View the MathML source where View the MathML source is the conditional expectation of Rk(st+1,st) given st=s (see (13) and (14)). This is the direct analog of the strategy of “betting one’s beliefs,” which takes on, in the case of independent identically distributed variables st, the form (12). Note that View the MathML source does not explicitly depend on t, and, furthermore, View the MathML source is a function of only the current state st of the process (st), rather than the whole history st of it. This implies, by virtue of (A1) and in view of the finiteness of S, that the strategy λ∗ satisfies condition (B). To proceed further, we need to describe a recursive method of constructing strategies based on (Markovian) decision rules. Suppose one of the traders, say 1, has a privilege of making her investment decision at time t with full information about the current market structure rt and the actions λt2(st), λt3(st),…,λtI(st) that have just been undertaken by all the other traders 2,3,…,I. Formally, the decision of investor 1 is specified by a function View the MathML source taking values in Δ+K. Suppose such functions—decision rules—are given for all t=0,1,2,…. Furthermore, suppose investors 2,…,I have chosen some strategies λt2,…,λtI (t=0,1,2,…). Then we can construct a strategy λt1(st), t=0,1,…, of investor 1 by using the formula equation(16) View the MathML source where rt=rt(st) and λtj=λtj(st), j=2,…,I. Let us consider a particular decision rule f=(f1,…,fK) (which does not explicitly depend on t) defined by equation(17) View the MathML source where r=(r1,…,rI)∈Δ+I, lj=(l1j,…,lKj)∈Δ+K, and so the vector f=(f1,…,fK) belongs to Δ+K. Note that the vector f is a convex combination of the vectors l2,…,lI with weights rj(1−r1)−1. This implies, in particular, the following: if the coordinates lkj of the vectors lj are bounded away from 0 by a constant ρ>0, then the coordinates fk of f are bounded away from 0 by the same constant. Consequently, if the strategies λt2,…,λtI satisfy condition (B), the strategy (16) satisfies condition (B) as well. In what follows, we will use the notation f=(fk) for the specific decision rule described in (17). The decision rule (17) has a number of remarkable properties. First of all, observe the following. Suppose investor 1 employs the strategy λt1(st) defined by (16) in terms of the decision rule (17). Then we have equation(18) View the MathML source which, in view of (9) and (10), yields View the MathML source Thus, if investor 1 uses the strategy generated by the decision rule (17), then, regardless of what strategies are used by the others, the relative market share of this investor remains constant over time. This observation leads to the following conclusion. If one of the traders 2,…,I uses the strategy λ∗, she cannot be a single survivor, as long as trader 1 uses the strategy (16), (17) and, consequently, keeps a constant positive market share rt1=r01 for all t. Further, we can see that the portfolio of investor 1, who uses the strategy λt1 defined in terms of the decision rule (17), is given by View the MathML source for all k=1,2,…,K (see (7) and (18)). Thus the vector ht1= (h1,t1,…,hK,t1) turns out to be proportional to the market portfolio, i.e. the vector View the MathML source whose components indicate the amounts of assets k=1,2,…,K traded at the market. According to the well-known Tobin mutual fund theorem (Magill and Quinzii, 1996, Proposition 16.15), portfolios having this structure result from the mean-variance optimization in the Capital Asset Pricing Model (CAPM). Therefore it is natural to term the decision rule (17) the CAPM decision rule and the strategy generated by it the CAPM strategy. The CAPM decision rule plays a key role in the formulation of the main results below. The notions we have just described pertain to investor 1. We can introduce analogous notions for any m∈{1,2,…,I}. To this end, consider the vector function View the MathML source of r=(r1,…,rI)∈Δ+I and lj=(l1j,…,lKj)∈Δ+K. This function specifies the CAPM decision rule for trader m. Given strategies λtj(st) of all the other traders j∈{1,2,…,I}\{m}, the CAPM strategy of m is defined by λtm=fm(rt,λt1,…,λtm−1,λtm+1,…,λtI). Those properties we discussed for m=1, extend to an arbitrary m. In Theorem 1 below, we describe a condition sufficient for the strategy (15) to be a single survivor. We consider the dynamical system (9), assuming that the investors i∈{1,2,…,I} use some strategies λi=(λti) satisfying requirement (B). We define View the MathML source where f is the CAPM decision rule (17). The symbol |·| denotes the sum of the absolute values of the coordinates of a finite-dimensional vector. Theorem 1. Let investor 1 use the strategyλ1=λ∗defined by (15). Let the following condition be fulfilled:(C) With probability 1, we have equation(19) View the MathML source Then investor 1 is a single survivor, and, moreover, equation(20) View the MathML source almost surely. Property (20) means that the relative market share of investor 1 tends to one at an exponential rate, whereas the relative market shares of all the other investors vanish at such rates, and so the strategy λ∗ dominates the others exponentially. Condition (C) can be restated as follows: there exists a strictly positive random variable κ, such that, almost surely, equation(21) View the MathML source for all t large enough. The last inequality requires that the actions View the MathML source prescribed by the strategy λ∗ should differ by not less than κ>0 from the actions View the MathML source prescribed by the CAPM decision rule. Here, we do not assume that there is at least one investor who indeed employs the CAPM rule; we need it only as an indicator, a proper deviation of which from λ∗ guarantees λ∗ to be a single survivor. In concrete instances, it might not be easy to verify condition (C) directly. Therefore we provide another hypothesis, (C1), which is stronger than (C) but can conveniently be checked in various examples. (C1) There exists a strictly positive random variableκsuch that, with probability 1, the distance between the vectorView the MathML sourceand the convex hull of the vectorsView the MathML sourceis not less thanκfor all t large enough. Clearly (C1) implies (C) because ζt=f(rt,λt2,…,λtI) is a convex combination of λt2,…,λtI. Condition (C), which is sufficient for investor 1 to be a single survivor, turns out to be close to a necessary one. The theorem below provides a version of hypothesis (C) that is necessary and sufficient for the conclusion of Theorem 1 to hold. Theorem 2. Investor 1 using the strategy (15) is a single survivor in the market selection process, and, moreover, dominates the others exponentially, if and only if the following condition is fulfilled:(C2) There exists a random variableκ>0 such that equation(22) View the MathML source with probability 1. The symbol # in the above formula stands for the number of elements in a finite set. Observe that (C2) follows from (C). Indeed, (C) is equivalent to the existence of a random variable κ for which, almost surely, inequality (21) is fulfilled for all t large enough. In this case, the limit in (22) is equal to 1. The limit in (22) may be thought of as a density (in the set of natural numbers) of those natural numbers t for which inequality (21) holds. Hypothesis (C2) only requires this density to be strictly positive, whereas (C) says that (21) should hold from some t on. Let us return to Theorem 1. From this theorem, it follows immediately that if the relation equation(23) View the MathML source holds with positive probability, then, with positive probability, there exists a (random) sequence tk, such that equation(24) View the MathML source Can we make a stronger statement about convergence in (24) if we strengthen (23) appropriately? A result along these lines is provided by the next theorem. Theorem 3. Let the following condition be satisfied:(D1) There exists a random variable 0<γ<1 such thatView the MathML sourceand View the MathML source a.s. for allt.Then we have View the MathML source We will actually prove Theorem 3 under a weaker assumption: (D2) The expectations View the MathML source do not converge to −∞. Clearly (D1) is stronger than both (D2) and (23), but (D2) does not necessarily imply (23). Condition (D1) holds, for example, if one of the investors i=2,I uses the CAPM strategy (and so her relative market share remains constant). Then, as Theorem 3 asserts, the difference between the budget shares of investor 1 prescribed by the strategy λ∗ and the budget shares prescribed by the CAPM decision rule converges a.s. to zero.