The Dynamics of Optimal Risk Sharing

65 Pages Posted: 21 Jun 2010  

Patrick Bolton

Columbia Business School - Department of Economics; Centre for Economic Policy Research (CEPR); National Bureau of Economic Research (NBER); European Corporate Governance Institute (ECGI)

Christopher Harris

University of Cambridge - Department of Applied Economics

Date Written: June 2010

Abstract

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 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 myopic 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 myopic 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 myopic 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.

Suggested Citation

Bolton, Patrick and Harris, Christopher, The Dynamics of Optimal Risk Sharing (June 2010). NBER Working Paper No. w16094. Available at SSRN: https://ssrn.com/abstract=1626584

Patrick Bolton (Contact Author)

Columbia Business School - Department of Economics ( email )

420 West 118th Street
New York, NY 10027
United States

HOME PAGE: http://www0.gsb.columbia.edu/faculty/pbolton/

Centre for Economic Policy Research (CEPR)

77 Bastwick Street
London, EC1V 3PZ
United Kingdom

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

European Corporate Governance Institute (ECGI)

c/o ECARES ULB CP 114
B-1050 Brussels
Belgium

HOME PAGE: http://www.ecgi.org

Christopher J. Harris

University of Cambridge - Department of Applied Economics ( email )

Sidgwick Avenue
Cambridge, CB3 9DE
United Kingdom
+44 1223 330152 (Phone)
+44 1223 330152 (Fax)

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