Optimal Insurance Coverage: Reply

Abstract

The introduction to my original article notes that the results are explicit, albeit approximate, expressions for the amount of coverage (p. 704, emphasis added). Previously existing exact results sometimes offer meagre guidance for real-life decisions. Schlesinger is wary of approximations and cites a numerical example to support his position. Certainly most of us would prefer a better approximation; but this requires knowing either the utility function itself (in which case there would be no need for an approximation) or, at least, higher order terms in the Taylor expansionterms that relate to preference for skewness and higher moments. In this instance, however, 'multivariate' measures of risk aversion would have to be assessed (i.e., in terms of variance, skewness and higher moments) and the expressions would become much more complicated. The original article must be read in proper context. Consider, for instance, the following propositions: if a consumer becomes more risk averse, additional will be purchased, and [if absolute risk aversion decreases with wealth], an individual with less wealth . . . will purchase more insurance (Schlesinger pp. 469, 471). Such statements are very nearly tautological and cannot be put to operational use; they do not indicate by how much the insured amount increases for a given increase in wealth. One cannot even specify the order of magnitude. Does the insured amount increase by one hundred dollars, or by a million dollars? Theorems which simply restate the obvious do not add much to the understanding of complex phenomena. My original paper permits approximate calculations, thereby adding insight and operational value. Table 1, which applies my formula to Schlesinger's example, contrasts the approximations with the qualitative results, as well as with the exact numerical solutions. Note that in the qualitative approach no distinction can be drawn between the magnitudes of change for case 2 ($753) and for case 3 ($378,088). Hence, for example, no aid is given to companies who would like to know