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."> 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.">
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The Dynamics of Optimal Risk Sharing

Author

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  • Patrick Bolton

    (Finance and Economics Division, Columbia University Business School)

  • Christopher Harris

    (Department of Economics, University of Cambridge)

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

  • Patrick Bolton & Christopher Harris, 2005. "The Dynamics of Optimal Risk Sharing," Economics Working Papers 0092, Institute for Advanced Study, School of Social Science, revised May 2010.
    • Handle: RePEc:ads:wpaper:0092
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    References listed on IDEAS

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    1. Veronica Rappoport & Enrichetta Ravina & Daniel Paravisini, 2010. "Risk Aversion and Wealth: Evidence from Person-to-Person Lending Portfolios," 2010 Meeting Papers 664, Society for Economic Dynamics.
    2. Luigi Guiso & Monica Paiella, 2008. "Risk Aversion, Wealth, and Background Risk," Journal of the European Economic Association, MIT Press, vol. 6(6), pages 1109-1150, December.
    3. Robert B. Barsky & F. Thomas Juster & Miles S. Kimball & Matthew D. Shapiro, 1997. "Preference Parameters and Behavioral Heterogeneity: An Experimental Approach in the Health and Retirement Study," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 112(2), pages 537-579.
    4. Pierre‐André Chiappori & Monica Paiella, 2011. "Relative Risk Aversion Is Constant: Evidence From Panel Data," Journal of the European Economic Association, European Economic Association, vol. 9(6), pages 1021-1052, December.
    5. Cvitanic Jaksa & Wan Xuhu & Zhang Jianfeng, 2008. "Principal-Agent Problems with Exit Options," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 8(1), pages 1-43, October.
    6. Dumas, Bernard, 1989. "Two-Person Dynamic Equilibrium in the Capital Market," The Review of Financial Studies, Society for Financial Studies, vol. 2(2), pages 157-188.
    7. Hui Ou-Yang, 2003. "Optimal Contracts in a Continuous-Time Delegated Portfolio Management Problem," The Review of Financial Studies, Society for Financial Studies, vol. 16(1), pages 173-208.
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    JEL classification:

    • D86 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Economics of Contract Law
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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