DOI: 10.25549/usctheses-c127-621437
title: Modeling default dependency and its application to finance and actuarial science

description: This thesis studies the modeling of default dependency in the reduced-form model and its application to both finance and actuarial science. Default dependency concerns about whether credit risky assets or firms are more likely to default together or separately. In the literature of credit risk, there are several approaches to constructing default dependency models such as copula, conditional independence, and contagion. This thesis consists of three chapters, and the first two chapters are mainly devoted to how the contagion model and the conditional independence model can be applied in finance and actuarial science. The copula approach is briefly discussed in the appendix. In addition, an insurance pricing problem is independently discussed in chapter 3. ❧ The first chapter of this thesis explores the contagion model. We use an intensity-based framework to analyze and compute the correlated default probabilities, both in finance and actuarial sciences, following the idea of change of measure initiated by Collins-Dufresne et al. [11]. Our method is based on a representation theorem for joint survival probability among an arbitrary number of defaults, which works particularly effectively for certain types of correlated default models, including the counter-party risk models of Jarrow and Yu [21] and related problems such as the phenomenon of flight to quality. The results are also useful in studying the recently observed dependent mortality for married couples involving spousal bereavement. In particular we study in details a problem of pricing Universal Variable Life (UVL) insurance products. The explicit formulae for the joint-life status and last-survivor status (or equivalently, the probability distribution of first-to-default and last-to-default in a multi-firm setting) enable us to derive the explicit solution to the indifference pricing formula without using any advanced results in partial differential equations. ❧ The second chapter of this thesis examines the conditional independence model from the actuarial science point of view. Using this methodology, we investigate the indifference pricing problem for pure endowments in a stochastic model in light of the idea of separation of variables established by Ludkovski and Young [26].  Our model considers multiple mortality intensities in a heterogeneous population, including a financial market consisting of a riskless asset and d-risky tradable assets. Under exponential utility, the utility indifference price is examined for issuing n pure endowments to n heterogeneous individuals. We derive a probabilistic representation of the indifference price in a purely probabilistic way. This probabilistic representation provides a useful tool to analyze the characteristics of the utility indifference price. We show that the indifference price converges to a lower or upper bound as the insurer's risk aversion parameter approaches zero or infinity, respectively. Furthermore, we discuss a heterogeneous model with frailty where the utility indifference method can be applied to price the pure endowment contracts. ❧ The third chapter investigates the pricing problem of equity-indexed universal life insurances via the utility indifference method. This type of pricing problems has been studied by Ma and Yu [27] and Young [40] under the deterministic force of mortality. We extend such the pricing model from the deterministic mortality setting to a stochastic mortality model.