Optimal Risk Exchanges

Abstract

The determination of optimal rules for sharing risks and constructing reinsurance treaties has important practical and theoretical interest. Medolaghi, de Finetti, and Ottaviani developed the first linear reciprocal reinsurance treaties based upon minimizing individual and aggregate variance of risk. Borch then used the economic concept of utility to justify choosing Pareto-optimal forms of risk exchange; in many cases, this leads to familiar linear quota-sharing of total pooled losses, or to stop-loss arrangements. However, this approach does not give a unique, risk-sharing agreement, and may lead to substantial fixed side payments. Gerber showed how to constrain a Pareto-optimal risk exchange to avoid invasion of reserves.To these ideas, the authors have added the actuarial concept of long-run fairness to each participant in the risk exchange; the result is a unique, Pareto-optimal risk pool, with “quota-sharing-by-layers” of the total losses. There are many interesting special cases, especially when all individual utility functions are of exponential form, giving linear quota-sharing-by-layers. Algorithms and numerical examples are given.

The composite iteration algorithm for finding efficient and financially fair risk-sharing rules

“…This will be done in a series of lemmas below. We also demonstrate that the approach via nonlinear Perron-Frobenius theory leads to a proof of existence and uniqueness of Pareto efficient and financially fair solutions, independent from the approach via reformulation as an optimization problem (Gale and Sobel, 1979;Bühlmann and Jewell, 1979). …”

“…The proof of uniqueness in these papers is based on the construction of a "social welfare function", which is such that it reaches its optimum on the set of financially fair allocations at a Pareto efficient point. Bühlmann and Jewell (1979) note that essentially the same technique can be applied as well to their formulation of the problem. Sobel (1981) gives a proof of uniqueness that avoids the introduction of the social welfare function, in order to accommodate a generalization in which agents use private valuation functionals.…”

“…A model similar to those proposed by Gale (1977) and by Bühlmann and Jewell (1979), but using more general preference specifications, was developed contemporaneously and independently by Balasko (1979). Balasko was motivated by developments in general equilibrium theory, in particular fixed-price equilibria as studied by Drèze (1975) and Benassy (1975).…”

“…However, it would be of interest to make more use of the particular structure of PEFF problems. An early discussion of specialized methods is given by Bühlmann and Jewell (1979). They discuss in particular the case of two agents, for which a line search suffices, and the case of exponential utility.…”

Multi-Period Risk Sharing Under Financial Fairness

“…It is given exogenously. It can then be shown that it is possible to find such a solution that is PE while the FF constraints are satisfied at the same time; moreover, the PEFF solution is unique; see Bühlman and Jewell [14], Gale [19] and Pazdera et al [33]. The y's are increasing functions of X which can be solved numerically.…”