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Evolutionary stable stock markets.

Article Excerpt
Summary. This paper shows that a stock market is evolutionary stable if and only if stocks are evaluated by expected relative dividends. Any other market can be invaded in the sense that there is a portfolio rule that, when introduced on the market with arbitrarily small initial wealth, increases its market share at the incumbent's expense. This mutant portfolio rule changes the asset valuation in the course of time. The stochastic wealth dynamics in our evolutionary stock market model is formulated as a random dynamical system. Applying this theory, necessary and sufficient conditions are derived for the evolutionary stability of portfolio rules when relative dividend payoffs follow a stationary Markov process. These local stability conditions lead to a unique evolutionary stable portfolio rule according to which assets are evaluated by expected relative dividends (with respect to the objective probabilities).

Keywords and Phrases: Evolutionary finance, Portfolio theory, Incomplete markets.

JEL Classification Numbers: G11, D52, D81.

1 Introduction

The expected discounted dividends model is one of the cornerstones of finance. According to this model the rational and fair value of common stocks is given by the expected value of the discounted sum of future dividends paid out by the company. Indeed in the very long run the trend of stock market prices coincides with the trend of the dividends paid by the companies. Yet over shorter horizons (sometimes even for decades) stock market prices can considerably deviate from their fundamentals. This phenomenon, called excess volatility, was first pointed out by Shiller (1981). While models based on complete rationality have difficulties to cope with excess volatility, models based on adaptive behavior typically go to the other extreme and generate too irregular price dynamics. We suggest here a solution in between these two extremes.

This paper considers a stock market model with a heterogenous population of portfolio rules. In our model rationality is important on the level of the market since market selection may ultimately give pressure for selecting the rational portfolio rules. It turns out that only a rational market in which assets are evaluated by expected relative dividends is evolutionary stable. Any other market can be invaded, i.e. there are portfolio rules that will gain market wealth, and hence the valuation of assets changes. While, as in De Long et al. (1990), rational strategies clearly face the risk that there are too many irrational strategies, any set of irrational strategies is however more easily turned over by invasion of even a small fraction of slightly different strategies. That is to say, every now and then the market can be displaced from its rational valuation by a big push of irrationality but eventually the market selection pressure will lead the market back to the rational valuation because from any irrational market there exists a sequence of small and nearby innovations leading back to the rational market. This stability property may be the explanation that on long-term averages stock markets look quite rational while severe departures are possible in the short- and medium-term.

In a sense our results give support to a long-held belief by Friedman (1953), Fama (1965) and others, who argued that the market naturally selects for rational strategies which, in effect, would lead to market efficiency. However, our paper also makes clear that the mutation force has to be added to the selection argument in order to prove this conjecture. Considering only the market selection process, the economy can get stuck at any situation in which all investors use the same portfolio rule. Moreover, our paper shows which portfolio rules can successfully enter which market. For example, an irrational market can be turned over by portfolio rules that are not themselves rational portfolio rules. The rational portfolio rule actually may fail to invade an irrational market.

To make these ideas precise, we study an asset market (complete or incomplete) where a finite number of portfolio rules manage capital by iteratively reinvesting in a fixed set of long-lived assets. In every period assets pay dividends according to the realization of a stationary Markov process in discrete time. In addition to the exogenous wealth increase due to dividends, portfolio rules face endogenously determined capital gains or losses. Portfolio rules are encoded as non-negative vectors of expenditure shares for assets. The set of portfolio rules considered is not restricted to those generated by expected utility maximization. It may as well include investment rules favored by behavioral finance models. Indeed any portfolio rule that is adapted to the information filtration is allowed in our framework.

Portfolio rules compete for market capital that is given by the total value of all assets in every period in time. The endogenous price process provides a market selection mechanism along which some portfolio rules gain market capital while others lose. We give a description of the market selection process from a random dynamical systems perspective. In each period in time the evolution of the distribution of market capital, i.e. the wealth shares of the portfolio rules as percentages of total market wealth, is given by a map that depends on the exogenous process determining the asset payoffs. An equilibrium in this model is provided by a distribution of wealth shares across portfolio rules that is invariant under the market selection process. Provided there are no redundant assets, every invariant distribution of market shares is generated by a monomorphic population, i.e. all investors with strictly positive wealth use the same portfolio rule at such an equilibrium. These invariant distributions are fixed points in an appropriate space.

A portfolio rule is evolutionary stable if the state in which this rule has total market wealth is robust against the entry of new portfolio rules with sufficiently small wealth. In other words, an evolutionary stable portfolio rule drives out any mutation. Criteria for evolutionary stability as well as evolutionary instability are derived for such fixed points. The derivation is via the linearization of the local dynamics. These sufficient and necessary conditions can be used to single out one particular portfolio rule, denoted by [lambda]*, that is the unique evolutionary stable portfolio rule. The rule [lambda]* is the only one that has highest exponential growth rate at its own market prices. In a sense, when the population pursues the evolutionary stable portfolio rule, it plays the "best response against itself." Moreover, any other market with one portfolio rule can successfully be invaded by a slightly different rule, i.e. the market can be destabilized by mutant portfolio rules that are small variations of the incumbent portfolio rule. This mutant portfolio rule will then lead to a change of the asset valuation in the market.

An explicit formula for the [lambda]*-rule is given, and it is applicable to real financial markets. This [lambda]*-rule prescribes to divide one's wealth proportionally to the expected relative dividends of assets. It is therefore justified to call a financial market with [lambda]* only rational, while any other market is termed irrational.

The effect of this [lambda]*-rule on asset prices is to equalize all assets' expected relative returns--in particular asset pricing is log-optimal in the sense of Luenberger (1997, Chapter 15). It is well known that log-optimal pricing is obtained if all investors have logarithmic von Neumann-Morgenstern utilities (Kraus and Litzenberger, 1975). The portfolio rule [lambda]* could therefore be obtained as well as in an idealized market with a single representative agent having rational expectations. For a market selection model based on rational expectations see Blume and Easley (2000) and Sandroni (2000). Our paper shows that such an idealized market with rational expectations can be justified by evolutionary reasoning.

A further implication of our evolutionary stability result is that, among all proportional portfolio rules, only the [lambda]*-rule is a candidate for a globally evolutionary stable portfolio rule, i.e. convergence to the status quo when the disturbance of the market can be large in the sense that any initial distribution of wealth is permitted....

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