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.