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Article Excerpt
Abstract The aim of this paper is to test the performance of the capital asset pricing model (CAPM) in an evolutionary framework. We model an economy where a heterogeneous population of long-lived agents invest their wealth according to different portfolio rules, and prove that traders who either "believe" in CAPM and use it as a rule of thumb, or are endowed with genuine mean-variance preferences, under some very weak conditions, vanish in the long run. We show that a sufficient condition to drive CAPM or mean-variance traders' wealth shares to zero is that an investor endowed with a logarithmic utility function enters the market.
Keywords Evolution * Market selection * Wealth dynamics * Portfolio rules * CAPM
JEL Classification Numbers C61 * D81 * G11
1 Introduction
1.1 Motivation
A major part of the research in financial economics is directed towards improving our understanding of how investors make their portfolio decisions and hence of how asset prices are determined. Many capital asset-pricing models have been put forth in the literature. In particular, mean-variance analysis and the Sharpe-Lintner-Mossin CAPM (1) are widely viewed as one of the "major contributions of academic research in the postwar era" (Jagannatan and Wang (1996), p.4).
Over the past few decades a number of studies have examined the empirical performance of CAPM, (2) invariably providing strong evidence of its inability to explain (and therefore to predict) the behaviour of financial markets. Nevertheless, "[i]n spite of the lack of empirical support, the CAPM is still the preferred model for classroom use in MBA and other managerial finance courses. In a way it reminds us of cartoon characters like Wile E. Coyote who have the ability to come back to original shape after being blown to pieces or hammered out of shape"(Jagannatan and Wang (1996), p.4).
In this paper we are after Wile E. Coyote once again, but with a new device. In fact, econometricians have empirically rejected its predictions and financial theorists have criticised its restrictive assumptions, but no one to our knowledge has studied CAPM in an evolutionary framework. The focus of our paper is to fill this gap in the literature and, in particular, to test the performance of the standard version of CAPM in an evolutionary setting.
Conventional financial theory shows that, under well-known assumptions, CAPM stems from rational behaviour. However, a recent strand of literature on evolution and market behaviour stresses that rationality is neither sufficient nor a necessary condition for survival. Therefore an interesting question to ask is whether CAPM prescribes a behaviour which can be considered "fit" in an evolutionary sense.
We imagine a heterogeneous population of long-lived agents who invest according to different portfolio rules and we ask what is the asymptotic market share of those who happen to behave as prescribed by CAPM. Namely we aim at detecting the asymptotic properties of the wealth shares of traders that either "believe" in CAPM and use it as a rule of thumb for their portfolio decisions, or display genuine mean-variance behaviour. Our results suggest that there are several circumstances of economic interest in which their wealth share will converge almost surely to zero. A condition sufficient to drive CAPM traders to extinction is that an investor endowed with a logarithmic utility function enters the market.
We believe that this is an interesting result not only because it proves that CAPM is not robust in an evolutionary sense, but also because it triggers once again the debate on the normative appeal and descriptive appeal of the logarithmic utility approach as opposed to the mean-variance approach in finance. Since a seminal article by Kelly (1956), several financial economists and applied mathematicians have been debating whether maximising a logarithmic utility function is "more rational" for a rational trader. The debate originates from the dissatisfaction with the mean-variance approach which fails to single out a unique optimal portfolio. In fact, the chosen mix between the risk free asset and the market portfolio depends on each investor's degree of risk aversion. Several authors (3) have argued that a rational long run investor should maximise the expected growth rate of his wealth share and, therefore, should behave as if he were endowed with a logarithmic utility function (4) (the so called Kelly criterion). This yields a unique solution to the optimal portfolio problem. This claim has been opposed by Merton and Samuelson (1974) and Goldman (1974).
In particular, Merton and Samuelson's critique stressed the obvious contradiction which lies in arguing that rational traders should maximise a utility function which is different from their own (5). The evolutionary framework adopted in this paper suggests that maximising a logarithmic utility function might not make you happy, but will definitely keep you alive!
1.2 Related literature
This paper contributes to the literature on evolution and market behaviour, which aims at studying long run market outcomes as the result of a process akin to natural selection, where population dynamics are endogenous and emerge from the process of wealth accumulation. This literature is in its relative infancy but has already attracted considerable attention in finance. In particular, one of its most fertile fields of application has been the study of survival of noise traders (6). Moreover some recent contributions have adopted an evolutionary framework to study asymmetric information in financial markets: Sciubba (2005) has looked at the survival of uninformed versus informed traders; Mailath and Sandroni (2003) have examined the role of exclusive and/or frequent information in determing survival in financial markets. Finally, models where agents adopt heterogenous portfolio rules and 'evolve' by learning about the environment have been used to explain financial market puzzles, such as the equity premium puzzle (7).
A natural further field of application for this literature is in the selection that markets operate among competing portfolio rules. A seminal study in this direction is Blume and Easley (1992). They develop an evolutionary model of a financial market, identify conditions for survival and prove false the common belief that rational behaviour is always selected for and irrational behaviour is always selected against by market forces. In particular, they show that the fittest behaviour in a risky security market is prescribed by a logarithmic utility function. Namely, traders who follow the Kelly criterion dominate and determine equilibrium prices asymptotically. On the other hand, whenever a logarithmic utility maximiser enters the market, all other types of traders are driven to extinction unless they asymptotically behave as if they were logarithmic utility maximisers. As a result, in the long run, traders who are endowed with a logarithmic utility function will survive, as well as successful imitators.
In this paper, we adopt Blume and Easley's framework and definitions of dominance, survival and extinction of traders. However, in order to show that logarithmic traders dominate and CAPM and mean-variance traders vanish, we cannot directly apply their results. This is because of two main reasons.
In the first place, Blume and Easley (1992) results on logarithmic traders' dominance do not necessarily imply that CAPM traders would vanish. In fact, nothing in principle excludes that CAPM traders will asymptotically behave as logarithmic utility maximisers. This remark, which is certainly true for any general trading behaviour, is particularly biting for CAPM behaviour because of its imitative nature. In fact, a trader who believes in CAPM invests according to a risk-free and a market portfolio, where the most successful trading strategies are better represented. There is, therefore, some sort of imitative behaviour implicit in CAPM.
A second reason why our results do not stem from a direct application of Blume and Easley (1992), is that both CAPM and mean-variance trading rules do not satisfy a crucial boundedness assumption which Blume and Easley (1992) impose. Their main theorem, (8) in fact, requires that the amount of wealth each trader invests in each asset has a uniform strictly positive lower bound. This technical assumption is not exactly harmless since it prevents us from applying their results to many interesting economic situations in which portfolio weights do not display uniform boundedness (9) of this type. In our setting, for instance, it prevents us from comparing the relative fitness of CAPM and mean-variance behaviour as opposed to logarithmic utility maximisation.
In this paper we limit the scope of our analysis to the selection that markets operate over portfolio rules. In particular, we assume that all traders have the same (correct) beliefs over the distribution of the returns of the risky assets. Moreover, as in Blume and Easley (1992), we do not model savings and consumption decisions. Sandroni (2000) and Blume and Easley (2005) address the problem of selection of beliefs when savings decisions are endogenous. Sandroni (2000) finds that, controlling for discount rates, when markets are complete and agents' utilities satisfy Inada conditions, all traders with correct beliefs survive. (10) This implies that preferences are irrelevant for survival and all that matters are beliefs. In this paper, for simplicity, markets are assumed to be complete; nevertheless, we do not consider the results in Sandroni (2000) as a challenge for three main reasons.
In the first place, we believe that the analysis of the selection that markets operate over portfolio rules, in a behavioural sense, is interesting per se, even when savings and consumption decisions are not taken into account. Suppose, for example, that the agents under scrutiny are fund managers who have been entrusted identical initial wealth endowments and follow alternative portfolio rules by reinvesting period after period all their capital and investment proceeds; we believe that an interesting problem to address is which portfolio rule allows the fund manager to accumulate wealth at the fastest rate, even when consumption and savings decisions are taken to be exogenous.
Secondly, mean-variance preferences as defined in this paper do not satisfy Inada conditions, so that Sandroni (2000)'s results do not directly apply here.
Finally, as we argue in section 5.2, even if mean-variance traders were to survive, they would not determine asset prices asymptotically, since there would be other surviving traders. Hence CAPM, which stems from a homogeneous population of mean-variance traders would not hold, not even in the long run.
1.3 Overview
The structure of the paper is as follows. In section 2 we present the model and develop...
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