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Valuation and risk management in life insurance

Fischer, Tom (2004):
Valuation and risk management in life insurance.
Darmstadt, Technische Universität, TU Darmstadt, [Online-Edition: urn:nbn:de:tuda-tuprints-4125],
[Ph.D. Thesis]

Abstract

In this thesis, several aspects of modern life insurance mathematics are considered in a discrete finite time framework. The dissertation consists of three parts. Part I: The classical Principle of Equivalence ensures that a life insurance company can accomplish that the mean balance per contract converges to zero almost surely for an increasing number of independent clients. In an axiomatic approach, this idea is adapted to the general case of stochastic financial markets. The implied minimum fair price of general life insurance products is then uniquely determined by the product of the given equivalent martingale measure of the financial market with the probability measure of the biometric state space. This minimum fair price (valuation principle) is in accordance with existing results. Part II: Assuming a product space model for biometric and financial events, there exists a rather natural principle for the decomposition of gains of life insurance contracts into a financial and a biometric part using orthogonal projections. The dissertation shows the connection between this decomposition, locally variance-optimal hedging and the so-called pooling of biometric risk contributions. For example, the mean aggregated discounted biometric risk contribution per client converges to zero almost surely for an increasing number of clients. A general solution of Buehlmann's AFIR-problem is proposed. Part III: This part proposes differentiability properties for positively homogeneous risk measures which ensure that the gradient can be applied for reasonable risk capital allocation on non-trivial portfolios. It is shown that these properties are fulfilled for a wide class of coherent risk measures based on the mean and the one-sided moments of a risky payoff. In contrast to quantile-based risk measures like Value-at-Risk, this class allows allocation in portfolios of very general distributions, e.g. discrete ones.

Item Type: Ph.D. Thesis
Erschienen: 2004
Creators: Fischer, Tom
Title: Valuation and risk management in life insurance
Language: English
Abstract:

In this thesis, several aspects of modern life insurance mathematics are considered in a discrete finite time framework. The dissertation consists of three parts. Part I: The classical Principle of Equivalence ensures that a life insurance company can accomplish that the mean balance per contract converges to zero almost surely for an increasing number of independent clients. In an axiomatic approach, this idea is adapted to the general case of stochastic financial markets. The implied minimum fair price of general life insurance products is then uniquely determined by the product of the given equivalent martingale measure of the financial market with the probability measure of the biometric state space. This minimum fair price (valuation principle) is in accordance with existing results. Part II: Assuming a product space model for biometric and financial events, there exists a rather natural principle for the decomposition of gains of life insurance contracts into a financial and a biometric part using orthogonal projections. The dissertation shows the connection between this decomposition, locally variance-optimal hedging and the so-called pooling of biometric risk contributions. For example, the mean aggregated discounted biometric risk contribution per client converges to zero almost surely for an increasing number of clients. A general solution of Buehlmann's AFIR-problem is proposed. Part III: This part proposes differentiability properties for positively homogeneous risk measures which ensure that the gradient can be applied for reasonable risk capital allocation on non-trivial portfolios. It is shown that these properties are fulfilled for a wide class of coherent risk measures based on the mean and the one-sided moments of a risky payoff. In contrast to quantile-based risk measures like Value-at-Risk, this class allows allocation in portfolios of very general distributions, e.g. discrete ones.

Place of Publication: Darmstadt
Publisher: Technische Universität
Uncontrolled Keywords: Hedging, Law of Large Numbers, Life insurance, Principle of Equivalence, Valuation, Pooling, Risk decomposition, Variance-optimal hedging, Coherent risk measures, One-sided moments, Risk capital allocation, Value-at-Risk
Divisions: 04 Department of Mathematics
Date Deposited: 17 Oct 2008 09:21
Official URL: urn:nbn:de:tuda-tuprints-4125
License: only the rights of use according to UrhG
Referees: Lehn, Prof. Dr. Jürgen and Rieder, Prof. Dr. Ulrich
Refereed / Verteidigung / mdl. Prüfung: 5 February 2004
Alternative Abstract:
Alternative abstract Language
In this thesis, several aspects of modern life insurance mathematics are considered in a discrete finite time framework. The dissertation consists of three parts. Part I: The classical Principle of Equivalence ensures that a life insurance company can accomplish that the mean balance per contract converges to zero almost surely for an increasing number of independent clients. In an axiomatic approach, this idea is adapted to the general case of stochastic financial markets. The implied minimum fair price of general life insurance products is then uniquely determined by the product of the given equivalent martingale measure of the financial market with the probability measure of the biometric state space. This minimum fair price (valuation principle) is in accordance with existing results. Part II: Assuming a product space model for biometric and financial events, there exists a rather natural principle for the decomposition of gains of life insurance contracts into a financial and a biometric part using orthogonal projections. The dissertation shows the connection between this decomposition, locally variance-optimal hedging and the so-called pooling of biometric risk contributions. For example, the mean aggregated discounted biometric risk contribution per client converges to zero almost surely for an increasing number of clients. A general solution of Buehlmann's AFIR-problem is proposed. Part III: This part proposes differentiability properties for positively homogeneous risk measures which ensure that the gradient can be applied for reasonable risk capital allocation on non-trivial portfolios. It is shown that these properties are fulfilled for a wide class of coherent risk measures based on the mean and the one-sided moments of a risky payoff. In contrast to quantile-based risk measures like Value-at-Risk, this class allows allocation in portfolios of very general distributions, e.g. discrete ones.English
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