## The Dynamics of Optimal Risk Sharing

#### Christopher HarrisUniversity of Cambridge

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 static benchmark plus a dynamic correction. In the case of the
approximation for the optimal share in investment risk, the static 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 driven by counterparty risk. If both parties are fairly risk tolerant,
in the sense that 2 > R > r, then the Proposer takes on more risk than she
would under the static benchmark. If both parties are fairly risk averse, in
the sense that R > r > 2, then the Proposer takes on less risk than she
would under the static benchmark. In the mixed case, in which R > 2 > r,
the Proposer takes on more risk when the Responder’s share in total wealth
is low and less risk when the Responder’s share in total wealth is high. In
the case of the approximation for the optimal termination payment, the static
benchmark is zero. The dynamic correction tells us, among other things, that:
(i) if the asset has a high return then, following termination, the Responder
compensates the Proposer for the loss of a valuable investment opportunity;
and (ii) if the asset has a low return then, prior to termination, the Responder
compensates the Proposer for the low returns obtained. Finally, we exploit our
representation of the optimal contract to derive simple and easily interpretable
sufficient conditions for the existence of an optimal contract.

Joint work with Patrick Bolton