A General Diversification Theorem: A Note

Abstract

Theorem four) have provided successively more general models which show that nonpositive interdependence of asset returns yields an incentive to diversify (and is free of the two-parameter restriction found in the analyses of Markowitz and Tobin). However, positive interdependence (or positive correlation in a twoparameter model) is a common relation between risky assets. If two risky assets are characterized by positive interdependence, there will still be an incentive to diversify if the investor can alter his portfolio to preserve the same mean income and reduce the riskiness of his portfolio income. It is easy to demonstrate the conditions which ensure this in the context of Tobin's model where it is well known that positive correlation is consistent with positive diversification. An analogous result has not been demonstrated for a more general portfolio model which includes a safe asset.1 Such a diversification theorem is provided in the next section. The theorem provides the conditions on the joint distribution function which suffice to show that the optimal portfolio of any risk averse investor will be positively diversified if it exists. The theorem is sufficiently robust to allow for positive interdependence of the risky assets and to characterize the extent of the positive interdependence. In two corollary results, it is shown that if the conditions in the diversification theorem are satisfied then by positively diversifying the investor is able to reduce the weight in the tails of his portfolio income distribution and reduce the variance of the portfolio income distribution.