Kaiserslautern - Fachbereich Mathematik
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This thesis deals with the simulation of large insurance portfolios. On the one hand, we need to model the contracts' development and the insured collective's structure and dynamics. On the other hand, an important task is the forward projection of the given balance sheet. Questions that are interesting in this context, such as the question of the default probability up to a certain time or the question of whether interest rate promises can be kept in the long term, cannot be answered analytically without strong simplifications. Reasons for this are high dependencies between the insurer's assets and liabilities, interactions between existing and new contracts due to claims on a collective reserve, potential policy features such as a guaranteed interest rate, and individual surrender options of the insured. As a consequence, we need numerical calculations, and especially the volatile financial markets require stochastic simulations. Despite the fact that advances in technology with increasing computing capacities allow for faster computations, a contract-specific simulation of all policies is often an impossible task. This is due to the size and heterogeneity of insurance portfolios, long time horizons, and the number of necessary Monte Carlo simulations. Instead, suitable approximation techniques are required.
In this thesis, we therefore develop compression methods, where the insured collective is grouped into cohorts based on selected contract-related criteria and then only an enormously reduced number of representative contracts needs to be simulated. We also show how to efficiently integrate new contracts into the existing insurance portfolio. Our grouping schemes are flexible, can be applied to any insurance portfolio, and maintain the existing structure of the insured collective. Furthermore, we investigate the efficiency of the compression methods and their quality in approximating the real life insurance portfolio.
For the simulation of the insurance business, we introduce a stochastic asset-liability management (ALM) model. Starting with an initial insurance portfolio, our aim is the forward projection of a given balance sheet structure. We investigate conditions for a long-term stability or stationarity corresponding to the idea of a solid and healthy insurance company. Furthermore, a main result is the proof that our model satisfies the fundamental balance sheet equation at the end of every period, which is in line with the principle of double-entry bookkeeping. We analyze several strategies for investing in the capital market and for financing the due obligations. Motivated by observed weaknesses, we develop new, more sophisticated strategies. In extensive simulation studies, we illustrate the short- and long-term behavior of our ALM model and show impacts of different business forms, the predicted new business, and possible capital market crashes on the profitability and stability of a life insurer.
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.
Das zentrale Thema dieser Arbeit sind vollständig gekoppelte reflektierte Vorwärts-Rückwärts-Stochastische-Differentialgleichungen (FBSDE). Zunächst wird ein Spezialfall, die teilweise gekoppelten FBSDE, betrachtet und deren Verbindung zur Bewertung Amerikanischer Optionen aufgezeigt. Für die Lösung dieser Gleichung wird Monte-Carlo-Simulation benötigt, daher werden verschiedene Varianzreduktionsmaßnahmen erarbeitet und miteinander verglichen. Im Folgenden wird der allgemeinere Fall der vollständig gekoppelten reflektierten FBSDE behandelt. Es wird gezeigt, wie das Problem der Lösung dieser Gleichungen in ein Optimierungsproblem übertragen werden kann und infolgedessen mit numerischen Methoden aus diesem Bereich der Mathematik bearbeitet werden kann. Abschließend folgen Vergleiche der verschiedenen numerischen Ansätze mit bereits existierenden Verfahren.