Risk Sharing with a Time Preference

Classic risk sharing results determine the optimal share of each member in a group that faces a present deal by maximizing the sum of expected utilities of the group members. For decision-makers with exponential utility functions, this formulation is equivalent to maximizing the sum of certain equivalents of the group members. This paper investigates the effects of time preference and different (but constant) risk tolerances among the group members on the individual shares when the payoff is received at a future time period. The analysis first defines several concepts: (i) a group future risk tolerance to be used for valuing the certain equivalent of future payoffs, (ii) a group time preference compounding factor that takes into account the time preference of individuals in the group, and (iii) a group present risk tolerance with time preference by which the partnership should operate for the discounted value of future deals. The results show that if the individuals in a group have the same time preference, then the classic risk sharing results still apply. However, when individuals have different time preferences, then the optimal shares of the individuals are modified by two components; the first depends on the ratio of the individual time preference compounding factor to the group time preference compounding factor, and the second depends on the surety of the deal multiplied by the group future risk tolerance. Several examples illustrate the results.