Abstract
Estimating the ruin probability or the probability of insolvency of a financial institution constitutes
a problem, which Lundberg first addressed in considering the compound Poisson process to
represent the surplus and find an upper bound of a company bankruptcy’s probability. Although
this model is simple and unrealistic, it serves as a benchmark in risk theory.
To improve the classical collective model and make it useful for practitioners, one needs to use a
sophisticated model by considering more realistic risk processes so as to reflect the true risks undertaken
and account for the economic environment, as well as to minimise the model error, since
failing to do so can result in risk of under- or overestimation. This will lead to a capital requirement
or solvency capital requirement under/overestimation. For example, it is well known that the
dependence structure misspecification is among the causes of the global financial crisis.
This thesis aims to analyse ruin probabilities problems and their related quantities under a nonstandard
renewal risk process. To make the model more realistic, we incorporate the dependency
within the risks, the claim number process, and assume a constant force of interest.
To be more precise, we first transfer the ruin problem in the banking sector, where we analyse the
impact of default from obligors on the insolvency of a bank, via the ruin probability techniques.
By embedding the bank’s stochastic cash flows to the Sparre Andersen model and assuming a
phase-type distribution for the default arrival process, we find a lower and upper bound of the ruin
probability via the Cramér Lundburg adjustment coefficient.
Secondly, we investigate the ruin problem in those cases where risks exhibit some types of dependence
structure. To do so, we assume a mixture representation for both the inter-arrival time and
claim amounts. Furthermore, we assume that the dependency between their mixture factors gives
the dependency between the claim inter-arrival time and its corresponding claim amount. Using
Archimedean copula to model this dependency, we derive explicit closed-expression and the upper
bound of the ruin probability.
Finally, we extend our work to the tail probability framework by considering the discounted aggregate
claim with a more general dependence structure within risks. Assuming a constant interest
rate and under the heavy-tailed assumption for the claims severities, we derived a closed analytical
formula of the survival probability (asymptotically) and apply those results to investigate some risk
measures, such as the probability of ruin and the value-at-risk.