Vigilant Measures of Risk and the Demand for Contingent Claims

35 Pages Posted: 23 Jul 2012 Last revised: 28 Jan 2015

Date Written: August 12, 2013


I examine a class of utility maximization problems with a not necessarily law-invariant utility, and with a not necessarily law-invariant risk measure constraint. The objective function is a Lebesgue integral of some function U with respect to some probability measure P, and the constraint set contains some risk measure constraint which is not necessarily P-law-invariant. This introduces some heterogeneity in the perception of uncertainty. The primitive U is a function of some given underlying random variable X and of a contingent claim Y on X. Many problems in economic theory and financial theory can be formulated in this manner, when a heterogeneity in the perception of uncertainty is introduced. Under a consistency requirement on the risk measure that I call Vigilance, supermodularity of the primitive U is sufficient for the existence of optimal contingent claims, and for these optimal claims to be comonotonic with the underlying random variable X. Vigilance is satisfied by a large class of risk measures, including all distortion risk measures. In the latter case, I give a quantile characterization of optimal solutions, which can be useful in many applications. As an illustration, I consider a problem of demand for insurance indemnity schedules, where the insurer has ambiguous beliefs about the realizations of the insurable loss and is ambiguity-averse in the sense of Schmeidler. In this case, I give an explicit characterization of the optimal indemnity schedule for the insured and I show how the result naturally extends the classical result of Arrow on the optimality of the deductible indemnity schedule. Arrow’s result is then obtained as a special case.

Keywords: Utility Maximization, Contingent Claims, Optimal Insurance, Deductible, Ambiguity, Choquet Integral, Distorted Probabilities, Quantiles, Monotone Likelihood Ratio

JEL Classification: C02, D89, G11, G22

Suggested Citation

Ghossoub, Mario, Vigilant Measures of Risk and the Demand for Contingent Claims (August 12, 2013). Insurance: Mathematics and Economics, Vol. 61, No. 1, 2015. Available at SSRN: or

Mario Ghossoub (Contact Author)

University of Waterloo ( email )

Dept. of Statistics & Actuarial Science
200 University Ave. W.
Waterloo, Ontario N2L 3G1


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