The dynamics of optimal risk sharing

"We study a dynamic-contracting problem involving risk sharing between two parties - the Proposer and the Responder - who invest in a risky asset until an exogenous but random termination time. In any time period they must invest all their wealth in the risky asset, but they can share the underlying investment and termination risk. When the project ends they consume their final accumulated wealth. The Proposer and the Responder have constant relative risk aversion R and r respectively, with R>r>0. We show that the optimal contract has three components: a non-contingent flow payment, a share in investment risk and a termination payment. We derive approximations for the optimal share in investment risk and the optimal termination payment, and we use numerical simulations to show that these approximations offer a close fit to the exact rules. The approximations take the form of a myopic benchmark plus a dynamic correction. In the case of the approximation for the optimal share in investment risk, the myopic benchmark is simply the classical formula for optimal risk sharing. This benchmark is endogenous because it depends on the wealths of the two parties. The dynamic correction is dri