A Comparative Statics Analysis of Risk Premiums

With the notable exception of Lintner, the comparative statics of security premiums in a mean-variance context has received sparse rigorous treatment in the published literature.' Lintner examines the implications of changing the number of individuals in the market, changing present wealth, and redistributing wealth. Most of his analysis, however, is hampered by the unrealistic assumption of constant absolute aversion, failure to consider simultaneously several types of comparative statics changes, and an exogenous specification of a fixed risk-free rate of return. This essay attempts to remedy these deficiencies. In Section I, applying the mean-variance security valuation model developed formally in the Appendix, I isolate useful categories of comparative statics influences which have clear analogies in the more traditional capital theory under certainty. In Section II, premiums (ratios of one plus the expected rate of return on a portfolio to one plus the risk-free rate) are shown to depend on per capita wealth and social attitudes toward risk. For the special case of constant proportional aversion, single-period spot-rate relative premiums are constant over time, even with changes in population and changes in social wealth. In Sfction III, alternative definitions of the market price of risk and measures of are compared on the basis of their comparative statics implications. The theoretical basis for many of the contributions of this essay is provided in the Appendix, in which the general relationship between the market price of risk and an aggregation of individual measurable utility functions is derived, assuming either normal probability distributions for security returns or quadratic utility. This new theorem motivates brief comments on measures of aversion and mean-variance efficiency analysis.