Diversification and equilibrium in securities markets

Diversification is the strategy of investing small fractions of wealth in each of a large number of securities and thereby reducing risk. The basic argument in support of diversification is an appeal to the Law of Large Numbers. A portfolio in which an investor's wealth is invested in equal fractions in each of N securities with i.i.d. returns, has a return that converges to a riskless return as N increases indefinitely. Any risk averse investor would prefer the limit riskless return to the return of any portfolio. In this paper we use the space ha of all finitely additive signed measures on the set of natural numbers (the index set for securities) to model the set of portfolios from which an investor may choose. This portfolio space contains arbitrary finite portfolios and a nontrivial portfolio which is the limit of the sequence of portfolios having the EI action of wealth 1/N invested in each of N securities. We identify a class of portfolios which we call perfectly diversified. These portfolios are described by purely finite additive measures. In a perfectly diversified portfolio, wealth invested in each individual security is a negligible fraction of the total wealth of the portfolio. We show that the return of every perfectly diversified portfolio of securities with uncorrelated returns is riskless. It is shown that modeling of the portfolio space as the space ba makes possible a rigorous derivation of a general equilibrium version of the Arbitrage Pricing Theorem (APT). The APT provides a good illustration of the implications of diversification for equilibrium prices of securities. Journal of Economic Literature Classification Numbers: D52, G20. (C) 1997 Academic Press.