published_or_final_version / abstract / Statistics and Actuarial Science / Master / Master of Philosophy

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

In this dissertation the theory of solvency assessment for long-term insurers is reviewed and how it evolved. Key international bodies and standards are identified and selected jurisdictions’ solvency frameworks are reviewed. The South African framework required by legislation introduced in 1998 is compared to these standards. Solvency capital requirements, valuation methods and risk management standards are the key areas considered. The financial results of a model office according to the South African requirements are compared to the financial results modelled stochastically according to the identified international standards. It is shown that the South African framework does not meet international standards. The key problem areas are the prescribed nature of the solvency capital requirement, the onerous treatment of policy cancellations and the treatment of new business. The role of actuaries in solvency assessment is also investigated. South Africa’s statutory-actuary role is compared with similar international roles. It is concluded that although similar international roles, notably the appointed actuary of the UK, have evolved the role of the statutory actuary has remained the same.

Cheung, Oi Lam Eunice. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 43-45). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Risk Theory --- p.1 / Chapter 1.2 --- Hybrid Numerical Scheme --- p.3 / Chapter 2 --- The Model --- p.5 / Chapter 2.1 --- Model --- p.5 / Chapter 2.2 --- Integro-Differential Equations --- p.8 / Chapter 2.3 --- Explicit Formulas and Asymptotic Properties --- p.13 / Chapter 3 --- Numerical Method --- p.16 / Chapter 3.1 --- From Integro-Differential Equations to Integral Equations --- p.17 / Chapter 3.2 --- Prom Integral Equations to Linear Equations --- p.19 / Chapter 3.3 --- Boundary Conditions --- p.20 / Chapter 3.4 --- Importance Sampling --- p.23 / Chapter 4 --- Numerical Study --- p.27 / Chapter 4.1 --- Exponential Claims with Equal Means --- p.28 / Chapter 4.1.1 --- Importance Sampling --- p.28 / Chapter 4.1.2 --- System of Linear Equations --- p.31 / Chapter 4.2 --- Exponential Claims with Unequal Means --- p.32 / Chapter 5 --- Conclusion --- p.40 / Bibliography --- p.43

Leung, Kit Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 47-48). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Classical Model --- p.1 / Chapter 1.2 --- Diffusion-perturbed model --- p.3 / Chapter 1.3 --- Hybrid computational scheme --- p.5 / Chapter 2 --- Integro-differential Equations --- p.7 / Chapter 2.1 --- Integro-differential equation of Chiu and Yin (2003) --- p.7 / Chapter 2.2 --- Integro-differential equations for ψs(u) and ψd(u) --- p.16 / Chapter 3 --- Numerical Method --- p.17 / Chapter 3.1 --- Trapezoidal approximation --- p.18 / Chapter 3.2 --- Boundary Conditions --- p.19 / Chapter 4 --- Importance Sampling --- p.22 / Chapter 4.1 --- Simulation Recipe --- p.25 / Chapter 4.2 --- Discussion --- p.26 / Chapter 5 --- Numerical Examples --- p.28 / Chapter 5.1 --- Probabilities of ruin: Oscillation and claim --- p.28 / Chapter 5.2 --- Comparison with the asymptotic results --- p.32 / Chapter 5.2.1 --- Ruin Probability --- p.38 / Chapter 5.2.2 --- Surplus before ruin --- p.40 / Chapter 5.2.3 --- Deficit after ruin --- p.42 / Chapter 6 --- Conclusion --- p.45 / References --- p.47

Graduation date: 2004

This thesis explores the feasibility of the application of statistical regression theory to the curve fitting of mortality data. Equations derived from Makeham's first law were used. These include:1x = ksxgcXlog lx=a+hx+bcx color pX = A + BcxThe 1941 CSO and 1958 CSO mortality tables were used for initial study.Extending this work, pure raw mortality data in conjunction with a modified version of the last equation above was employed. Results were quite interesting.

published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

Considerably much work has been done on Backward Stochastic Differential Equations (BSDEs) in continuous-time with deterministic terminal horizon or stopping times. Various new models have been introduced in these years in order to generalize BSDEs to solve new practical financial problems. One strand is focused on discrete-time models. Backward Stochastic Difference Equations (also called BSDEs if no ambiguity) on discrete-time finite-state space were introduced by Cohen and Elliott (2010a). The associated theory required only weak assumptions. In the first topic of this thesis, properties of non-linear expectations defined using the discrete-time finite-state BSDEs were studied. A converse comparison theorem was established. Properties of risk measures defined by non-linear expectations, especially the representation theorems, were given. Then the theory of BSDEs was applied to optimal design of dynamic risk measures. Another strand is about a general random terminal time, which is not necessarily a stopping time. The motivation of this model is a financial problem of hedging of defaultable contingent claims, where BSDEs with stopping times are not applicable. In the second topic of this thesis, discrete-time finite-state BSDEs under progressively enlarged filtration were considered. Martingale representation theorem, existence and uniqueness theorem and comparison theorem were established. Application to nonlinear expectations was also explored. Using the theory of BSDEs, the explicit solution for optimal design of dynamic default risk measures was obtained. In recent work on continuous-time BSDEs under progressively enlarged filtration, the reference filtration is generated by Brownian motions. In order to deal with cases with jumps, in the third topic of this thesis, a general reference filtration with predictable representation property and an initial time with immersion property were considered. The martingale representation theorem for square-integrable martingales under progressively enlarged filtration was established. Then the existence and uniqueness theorem of BSDEs under enlarged filtration using Lipschitz continuity of the driver was proved. Conditions for a comparison theorem were also presented. Finally applications to nonlinear expectations and hedging of defaultable contingent claims on Brownian-Poisson setting were explored. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

published_or_final_version / abstract / toc / Statistics and Actuarial Science / Master / Master of Philosophy