Kaiserslautern - Fachbereich Mathematik
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The new international capital standard for credit institutions (“Basel II”) allows banks to use internal rating systems in order to determine the risk weights that are relevant for the calculation of capital charge. Therefore, it is necessary to develop a system that enfolds the main practices and methods existing in the context of credit rating. The aim of this thesis is to give a suggestion of setting up a credit rating system, where the main techniques used in practice are analyzed, presenting some alternatives and considering the problems that can arise from a statistical point of view. Finally, we will set up some guidelines on how to accomplish the challenge of credit scoring. The judgement of the quality of a credit with respect to the probability of default is called credit rating. A method based on a multi-dimensional criterion seems to be natural, due to the numerous effects that can influence this rating. However, owing to governmental rules, the tendency is that typically one-dimensional criteria will be required in the future as a measure for the credit worthiness or for the quality of a credit. The problem as described above can be resolved via transformation of a multi-dimensional data set into a one-dimensional one while keeping some monotonicity properties and also keeping the loss of information (due to the loss of dimensionality) at a minimum level.
Life insurance companies are asked by the Solvency II regime to retain capital requirements against economically adverse developments. This ensures that they are continuously able to meet their payment obligations towards the policyholders. When relying on an internal model approach, an insurer's solvency capital requirement is defined as the 99.5% value-at-risk of its full loss probability distribution over the coming year. In the introductory part of this thesis, we provide the actuarial modeling tools and risk aggregation methods by which the companies can accomplish the derivations of these forecasts. Since the industry still lacks the computational capacities to fully simulate these distributions, the insurers have to refer to suitable approximation techniques such as the least-squares Monte Carlo (LSMC) method. The key idea of LSMC is to run only a few wisely selected simulations and to process their output further to obtain a risk-dependent proxy function of the loss. We dedicate the first part of this thesis to establishing a theoretical framework of the LSMC method. We start with how LSMC for calculating capital requirements is related to its original use in American option pricing. Then we decompose LSMC into four steps. In the first one, the Monte Carlo simulation setting is defined. The second and third steps serve the calibration and validation of the proxy function, and the fourth step yields the loss distribution forecast by evaluating the proxy model. When guiding through the steps, we address practical challenges and propose an adaptive calibration algorithm. We complete with a slightly disguised real-world application. The second part builds upon the first one by taking up the LSMC framework and diving deeper into its calibration step. After a literature review and a basic recapitulation, various adaptive machine learning approaches relying on least-squares regression and model selection criteria are presented as solutions to the proxy modeling task. The studied approaches range from ordinary and generalized least-squares regression variants over GLM and GAM methods to MARS and kernel regression routines. We justify the combinability of the regression ingredients mathematically and compare their approximation quality in slightly altered real-world experiments. Thereby, we perform sensitivity analyses, discuss numerical stability and run comprehensive out-of-sample tests. The scope of the analyzed regression variants extends to other high-dimensional variable selection applications. Life insurance contracts with early exercise features can be priced by LSMC as well due to their analogies to American options. In the third part of this thesis, equity-linked contracts with American-style surrender options and minimum interest rate guarantees payable upon contract termination are valued. We allow randomness and jumps in the movements of the interest rate, stochastic volatility, stock market and mortality. For the simultaneous valuation of numerous insurance contracts, a hybrid probability measure and an additional regression function are introduced. Furthermore, an efficient seed-related simulation procedure accounting for the forward discretization bias and a validation concept are proposed. An extensive numerical example rounds off the last part.