A general theory of risk apportionment

Suppose that the conditional distributions of ˜x (resp. ˜y) can be ranked according to the m-th (resp. n-th) risk order. Increasing their statistical concordance increases the(m, n) degree riskiness of (˜x, ˜y), i.e., it reduces expected utility for all bivariate utility functions whose sign of the (m, n) cross-derivative is (−1)m+n+1. This means in particular that this increase in concordance of risks induces a m + n degree risk increase in ˜x + ˜y. On the basis of these general results, I provide different recursive methods to generate high degrees of univariate and bivariate risk increases. In the reverse-or-translate (resp.reverse-or-spread) univariate procedure, a m degree risk increase is either reversed or translated downward (resp. spread) with equal probabilities to generate a m + 1 (resp.m + 2) degree risk increase. These results are useful for example in asset pricing theory when the trend and the volatility of consumption growth are stochastic or statistically linked

Non-differentiable transformations preserving stochastic dominance

In this paper, we solve the following problem: when does a stochastic improvement in one risk maintain itself under a non everywhere continuously differentiable transformation of this risk? Using the notion of divided differences, we show that stochastic dominance at the third (and higher) order, and sometimes at the second one, is not preserved after simple piecewise linear transformation of the initial risk. Our analysis complements the one that exists for everywhere continuously differentiable transformations