Optimal reinsurance: a contemporary perspective

Abstract

In recent years, general risk measures have played an important role in risk management

in both finance and insurance industry. As a consequence, there is an

increasing number of research on optimal reinsurance problems using risk measures

as yard sticks beyond the classical expected utility framework.

In this thesis, the stop-loss reinsurance is first shown to be an optimal contract

under law-invariant convex risk measures via a new simple geometric argument.

This similar approach is then used to tackle the same optimal reinsurance problem

under Value at Risk and Conditional Tail Expectation; it is interesting to note

that, instead of stop-loss reinsurances, insurance layers serve as the optimal solution

in these cases. These two results hint that law-invariant convex risk measure

may be better and more robust to expected larger claims than Value at Risk and

Conditional Tail Expectation even though they are more commonly used.

In addition, the problem of optimal reinsurance design for a basket of n insurable

risks is studied. Without assuming any particular dependence structure, a

minimax optimal reinsurance decision formulation for the problem has been successfully

proposed. To solve it, the least favorable dependence structure is first

identified, and then the stop-loss reinsurances are shown to minimize a general

law-invariant convex risk measure of the total retained risk. Sufficient condition

for ordering the optimal deductibles are also obtained.

Next, a Principal-Agent model is adopted to describe a monopolistic reinsurance

market with adverse selection. Under the asymmetry of information, the reinsurer

(the principal) aims to maximize the average profit by selling a tailor-made reinsurance

to every insurer (agent) from a (huge) family with hidden characteristics.

In regard to Basel Capital Accord, each insurer uses Value at Risk as the risk assessment,

and also takes the right to choose different risk tolerances. By utilizing

the special features of insurance layers, their optimality as the first-best strategy

over all feasible reinsurances is proved. Also, the same optimal reinsurance

screening problem is studied under other subclass of reinsurances: (i) deductible

contracts; (ii) quota-share reinsurances; and (iii) reinsurance contracts with convex

indemnity, with the aid of indirect utility functions. In particular, the optimal

indirect utility function is shown to be of the stop-loss form under both classes

(i) and (ii); while on the other hand, its non-stop-loss nature under class (iii) is

revealed.

Lastly, a class of nonzero-sum stochastic differential reinsurance games between

two insurance companies is studied. Each insurance company is assumed to maximize

the difference of the opponent’s terminal surplus from that of its own by

properly arranging its reinsurance schedule. The surplus process of each insurance

company is modeled by a mixed regime-switching Cramer-Lundberg approximation.

It is a diffusion risk process with coefficients being modulated by both

a continuous-time finite-state Markov Chain and another diffusion process; and

correlations among these surplus processes are allowed. In contrast to the traditional

HJB approach, BSDE method is used and an explicit Nash equilibrium is

derived.