A general theory of risk apportionment

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