Working paper

A general theory of risk apportionment

Christian Gollier


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.


Stochastic dominance; risk orders; prudence; temperance; concordance.;

JEL codes

  • D81: Criteria for Decision-Making under Risk and Uncertainty


Christian Gollier, A general theory of risk apportionment, TSE Working Paper, n. 19-1003, April 2019.

See also

Published in

TSE Working Paper, n. 19-1003, April 2019