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. / published_or_final_version / Mathematics / Master / Master of Philosophy
| Identifer | oai:union.ndltd.org:HKU/oai:hub.hku.hk:10722/174483 |
| Date | January 2012 |
| Creators | Sung, Ka-chun, Joseph., 宋家俊. |
| Contributors | Yung, SP |
| Publisher | The University of Hong Kong (Pokfulam, Hong Kong) |
| Source Sets | Hong Kong University Theses |
| Language | English |
| Detected Language | English |
| Type | PG_Thesis |
| Source | http://hub.hku.hk/bib/B47753031 |
| Rights | The author retains all proprietary rights, (such as patent rights) and the right to use in future works., Creative Commons: Attribution 3.0 Hong Kong License |
| Relation | HKU Theses Online (HKUTO) |
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