Balance, Growth and Diversity of Financial Markets

A financial market comprising of a certain number of distinct companies is considered, and the following statement is proved: either a specific agent will surely beat the whole market unconditionally in the long run, or (and this “or” is not exclusive) all the capital of the market will accumulate in one company. Thus, absence of any “free unbounded lunches relative to the total capital” opportunities lead to the most dramatic failure of diversity in the market: one company takes over all other until the end of time. In order to prove this, we introduce the notion of perfectly balanced markets, which is an equilibrium state in which the relative capitalization of each company is a martingale under the physical probability. Then, the weaker notion of balanced markets is discussed where the martingale property of the relative capitalizations holds only approximately, we show how these concepts relate to growth-optimality and efficiency of the market, as well as how we can infer a shadow interest rate that is implied in the economy in the absence of a bank. 0. Introduction 0.1. Discussion and results. We consider a model of a financial market that consists of d stocks of certain “distinct” companies. The distinction between companies clings on their having different risk and/or growth characteristics, and will find its mathematically precise definition later on in the text. In absence of clairvoyance, the total capital of each company is modeled as a stochastic process Si, i = 1, . . . , d. Randomness comes through a set Ω of possible outcomes — for each ω ∈ Ω we have different realizations of Si(ω). Financial agents decide to invest certain amounts of their wealth to different stocks, and via their actions the value of Si t for each time t ∈ R+ is determined. Of major importance in our discussion will be the distribution of market capital, given by the relative capitalization κi := Si/(S1 + . . . + Sd) of each company (S1 + . . . + Sd is the total market capital). In particular, the limiting, i.e., long-run, capital distribution will be investigated. For addressing this question, a probability P is introduced that weights the different outcomes of Ω (for all events in some σ-algebra F); P reflects the average subjective feeling of the financial agents, but in this average sense it is not subjective anymore: each agent’s investment decisions are fed back into the relative capitalization of the companies, and thus affects the random choice of the outcome. Via this mechanism, P becomes a real-world probability, and can also be regarded as the subjective view of a representative agent in the market, whose decisions alone reflect the cumulative decisions of all “small” agents. Date: December 3, 2008.