Modelling multivariate dependence structures in insurance and credit risk via copulas

This PhD thesis seeks to offer a new framework that accommodates dependency in pricing an insurance portfolio following the renewal risk model, corporate bonds, as well as credit default swaps (CDS). This will be achieved by combining the approach and methodology of actuarial science with stochastic processes and probability theories, as well as employing a hint of the integral calculus used in the electromagnetic and viscoelasticity fields. This thesis is a collection of three papers, which are presented in Chapters 2, 3 and 4. While Chapters 3 and 4 can be read in conjunction with each other, Chapter 2 can be read in isolation because it presents a completely different perspective of insurance to the financial perspective taken in the other two articles (Chapter 3 and 4). Nevertheless, the three papers share the same scope, which is the use of copula to capture the dependency between variables. In total, four copulas are explored: the Farlie-Gumbel-Morgenstern (FGM) copula, Gumbel copula, Gaussian copula and Student-t copula. However, only three copulas are compared in each working paper. The first article in Chapter 2 models a continuous time renewal risk process, and uses copulas to capture the dependence structure between the claims inter-arrival time and discounted claims size. The second and third articles work under the framework of a reduced form model and use various copulas to capture the dependence structure between the jump sizes of the intensity processes, each of which is represented by a jump diffusion process. Taking the insurance perspective, the first article - titled Neumann Series on the Recursive Moments of Copula-Dependent Aggregate Discounted Claims - studies the recursive moments of aggregate discounted claims, where the dependence between the interclaim time and the subsequent claim size is considered. Using the general expression for the mth order moment proposed in [1] which takes the form of the Volterra Integral Equation (VIE), we used the method of successive approximation to derive the Neumann series of the recursive moments. We then compute the first two moments of aggregate discounted claims, i.e. its mean and variance, based on the Neumann series expression where the dependence structure is captured by the FGM copula, Gaussian copula and Gumbel copula, with exponential marginal distributions. Insurance premium calculations with their figures are also illustrated. The second work – titled A multivariate jump diffusion process for counterparty risk in CDS rates – considers counterparty risk in CDS rates. To do so, it uses a multivariate jump diffusion process for obligors’ default intensity, where jumps (i.e. magnitude of contribution of primary events to default intensities) occur simultaneously and their sizes are dependent. For these simultaneous jumps and their sizes, a homogeneous Poisson process and three copulas, which are Farlie-Gumbel-Morgenstern (FGM), Gaussian and Student-t copulas are used. This project applies copula-dependent default intensities of multivariate Cox process to derive the joint Laplace transform that provides us with joint survival/default probability and other relevant joint probabilities. For that purpose, the piecewise deterministic Markov process (PDMP) theory developed in [2] and the martingale methodology in [3] are used. The survival/default probability is computed using the three copulas and exponential marginal distributions, and the results are applied to calculate CDS rates, assuming deterministic rate of interest and recovery rate. Sensitivity analysis for the CDS rates were also conducted by changing the relevant parameters and providing their figures. The final article – titled Jump diffusion model with copula dependence structure in defaultable bond pricing – studies the pricing of a defaultable bond under various copulas. For that purpose, it used a bivariate jump diffusion process for a bond issuer’s default intensity and the short rate of interest. We assume the jumps (i.e. magnitude of contribution of primary events to default intensities) occur simultaneously and their sizes are dependent. For these simultaneous jumps and their sizes, a homogeneous Poisson process and three copulas – FGM copula, Gaussian copula and Student-t copula are used, respectively. The joint Laplace transform for the variables’ integrated processes is derived to provide the expression for defaultable bond price, using copula-dependent jump sizes. Once again, we apply the piecewise deterministic Markov process (PDMP) theory developed in [2] and the martingale methodology in [3]. Zero coupon defaultable bond prices and their yield are computed using the three copulas and exponential marginal distributions. The model is then used to calibrate zero coupon bonds on one-day basis as well as for an extended period of one year. Calibration results show that the Student-t copula provides the best fit relative to the other two copulas.