The game-theoretic capital asset pricing model

Using Shafer and Vovk’s game-theoretic framework for probability, we derive a capital asset pricing model from an efficient market hypothesis, with no assumptions about the beliefs or preferences of investors. Our efficient market hypothesis says that a speculator with limited means cannot beat a particular index by a substantial factor. The model we derive says that the difference between the average returns of a portfolio and the index should approximate the difference between the portfolio’s covariance with the index and the index’s variance. This leads to interesting new ways to evaluate the past performance of portfolios and funds. The established general theory of capital asset pricing combines stochastic models for asset returns with a rich tapestry of economic ideas: no arbitrage, general equilibrium, and marginal utilities for current and future consumption (Campbell 2000, Cochrane 2001). Twenty years of work have demonstrated the power and flexibility of the combination; many different stochastic models and many different models for investors’ marginal utility can be adopted, estimated, or predicted. There is little consensus, however, concerning the empirical validity of these different instantiations of the general theory. While this can be attributed in part to the very richness of the theory, which allows data on prices and returns to be looked at in different ways, it is also related to the theory’s deep ambivalence about the meaning of its stochastic models. On the one hand, these models are hypotheses about the behavior of returns, to be compared with the empirical distribution of returns. On the other hand, they are hypotheses about investors’ beliefs, which can be combined with hypotheses about investors’ preferences in order to determine or predict asset prices. These two roles do not necessarily mesh. Investors can be mistaken about the future, and the fact that a stochastic model fits past asset returns does not go very far towards demonstrating that investors were using it to make their decisions. One can try to achieve clarity within the established theory by making parsimonious assumptions about the stochastic process driving asset returns and about the marginal utility of investors. But this does not alleviate the problems arising from the multiple meanings of stochasticity. A more effective application of the principle of parsimony requires a deconstruction of stochasticity. One needs something more modest than the assumption that asset returns are generated by a stochastic process. In this article, we propose finding this something more modest in the game-theoretic framework recently advanced by Shafer and Vovk (2001). In this framework, limited opportunities to bet can be interpreted without any assumption of stochasticity. Shafer and Vovk show that the framework is adequate for the classical limit theorems of probability (the law of large numbers, the law of the iterated logarithm, and the central limit theorem), and that it can be used to make the theory and practice of option pricing more purely game-theoretic. In this article, we apply the framework to capital asset pricing.