Gains from Diversification

By now it has been firmly established that, under a variety of circumstances, risk averters should follow a policy of diversifying their investments. Several different theorems dealing with the conditions for diversification have recently appeared in the literature; for example [2], [3], [6]. The most general of these results says that if n assets are independently distributed, have equal means, and positive finite variances, the optimal portfolio for each risk averter includes some positive amount of each asset. The attractiveness of this theorem derives from the fact that optimality follows despite only weak conditions imposed on the distributions. The disadvantage is that it cannot identify any specific diversified portfolio which all risk averters would prefer to specialized ones. For unanimous ranking of portfolios stronger conditions are required, such as those used in the diversification theorems given in [2], [3]. For example, when two assets are identically distributed, any mixture of the two assets is preferred to a specialized portfolio by every risk averter. And in the case where the joint distribution of the individual assets is symmetric, the portfolio with an equal amount of each asset is the optimal portfolio for every risk averter. In this paper we consider the case where risk averters unanimously judge a particular asset to be superior to all other assets. Somewhat surprisingly, it turns out that risk averters will unanimously prefer certain diversified portfolios to specializing in the superior asset. This result holds if and only if the independently distributed prospects have equal means and the same range.