Standardized Variables and Optimal Risky Investment

This paper studies the optimal risky investment problem with fewer restrictions on utilities, and more structure on risks, than does the current literature. It uses discrete random variables defined on a common domain, hereafter called standardized variables, to obtain new results without important loss of generality. The optimal amount of investment in a single risky asset does not always decrease as risk increases in the Rothschild-Stiglitz ([1970, 1971]; hereafter RS) sense. However, by using standardized variables to define wealth dependent measures of risk and return, the paper finds necessary and sufficient conditions on risks such that an increase in risk does cause decreasing optimal risky investment. The paper thus complements the RS results. For investment in two risky assets, the paper uses standardized variables to find conditions on risks such that the riskier asset's demand to decrease (increase) as the Arrow-Pratt absolute risk aversion index increases (decreases), and thereby complements Ross' [1981] results.