Fischer, Tom (2004):
Valuation and risk management in life insurance.
Darmstadt, Technische Universität, TU Darmstadt, [OnlineEdition: urn:nbn:de:tudatuprints4125],
[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 varianceoptimal hedging and the socalled 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 AFIRproblem 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 nontrivial portfolios. It is shown that these properties are fulfilled for a wide class of coherent risk measures based on the mean and the onesided moments of a risky payoff. In contrast to quantilebased risk measures like ValueatRisk, 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 varianceoptimal hedging and the socalled 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 AFIRproblem 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 nontrivial portfolios. It is shown that these properties are fulfilled for a wide class of coherent risk measures based on the mean and the onesided moments of a risky payoff. In contrast to quantilebased risk measures like ValueatRisk, 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, Varianceoptimal hedging, Coherent risk measures, Onesided moments, Risk capital allocation, ValueatRisk 
Divisions: 
04 Department of Mathematics 
Date Deposited: 
17 Oct 2008 09:21 
Official URL: 
urn:nbn:de:tudatuprints4125 
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 varianceoptimal hedging and the socalled 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 AFIRproblem 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 nontrivial portfolios. It is shown that these properties are fulfilled for a wide class of coherent risk measures based on the mean and the onesided moments of a risky payoff. In contrast to quantilebased risk measures like ValueatRisk, this class allows allocation in portfolios of very general distributions, e.g. discrete ones.  English 

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