Longevity risks [Elektronische Ressource] : modelling and financial engineering / vorgelegt von Shaohui Wang

universitat_ulm

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Publié par	universitat_ulm
Publié le	01 janvier 2008
Nombre de lectures	21
Langue	English
Poids de l'ouvrage	2 Mo

Extrait

Universität Ulm
Fakultät für Mathematik und
Wirtschaftswissenschaften
Longevity Risks:
Modelling and
Financial Engineering
Dissertation
zur Erlangung des Doktorgrades
Dr. rer. nat.
der Fakutät für Mathematik und Wirtschaftswissenschaften
der Universität Ulm
vorgelegt von
Dipl.-Math. oec. Shaohui Wang
aus
Baixiang, China
Ulm, im Mai 2008ii
Amtierender Dekan
Prof. Dr. Frank Stehling
Gutachter
1. Gutachter: Prof. Dr. Rüdiger Kiesel
2. Gutachter: Prof. Dr. Hans-Joachim Zwisler
Tag der Promotion
07. Juli 2008Acknowledgements
This thesis would not have been possible without the ﬁnancial support by German
Science Foundation Research Training Group: modelling, analysis and simulation in
economy mathematics.
I take the opportunity to thank my academic advisor Professor Dr.Rüdiger Kiesel
who opened me the door to the inter-discipline research of longevity risks from ﬁnan-
cial mathematical perspectives and who supported my studies and stays in every phase.
I highly appreciate his strategical advises to my work and his encouragement, which
builded up an ideal research environment beyond my expectation.
I would also like to thank Professor Dr. Hans-Joachim Zwiesler who introduced
the actuarial science to me and was friendly available as second supervisor to my thesis.
Further, I thank my parents for their ﬁnancial and emotional supports for my stud-
ies and stays in Germany. I thank my young brother and his wife to take care of our
parents since I was not at home when they needed us.
Last but not least, my thanks go to my wife Jiao Xueli for her caring support and
encouragement during the writing of this thesis.Contents
Contents ii
List of Figures v
List of Tables vi
1 Introduction 1
1.1 Motivation to Systematic Mortality Risks . . . . . . . . . . . . . . . . . 1
1.2 Objective of the Thesis and Contribution . . . . . . . . . . . . . . . . . 2
1.3 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Mortality Risks 9
2.1 Life Contingencies in Classical Actuarial Perspectives . . . . . . . . . . 9
2.1.1 Deterministic Mortality Rate . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Evaluation of Insurance Contracts: Principle of Equivalence . 12
2.1.3 Net Reserves and Thiele’s Diﬀerential Equation . . . . . . . . . 16
2.2 Life Contingencies on Systematic Mortality Risks . . . . . . . . . . . . 18
2.2.1 Stochastic Mortality Rate . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Systematic Mortality Risks . . . . . . . . . . . . . . . . . . . . . . 24
2.2.3 Analysis of JPMorgan LifeMetrics . . . . . . . . . . . . . . . . . 26
2.2.4 Concrete Models and Fittings with JPMorgan LifeMetrics . . 31
iiCONTENTS iii
3 Alternative Approach to No-Arbitrage Hypothesis 42
3.1 Motivation: non-tradable Bank Account . . . . . . . . . . . . . . . . . . 42
3.2 Alternative Deﬁnition of No-Arbitrage Opportunity . . . . . . . . . . . 43
3.3 Hedging of Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Financial Market involving Survivor Bonds 54
4.1 Arbitrage-free Financial Market: Lévy Finance . . . . . . . . . . . . . . 54
4.1.1 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1.2 Default-free Zero-coupon Bonds Market . . . . . . . . . . . . . 55
4.1.3 Arbitrage-free Risky Securities Market . . . . . . . . . . . . . . 60
4.2 Calibration of The Risk-neutral Measure with Spot Prices . . . . . . . . 64
4.2.1 Calibration of the Term Structure Model . . . . . . . . . . . . . 65
4.2.2 Calibration of Equity Model . . . . . . . . . . . . . . . . . . . . . 68
4.3 Arbitrage-free Survivor Bonds Market . . . . . . . . . . . . . . . . . . . . 69
4.3.1 Term Structure of Survivor Bonds . . . . . . . . . . . . . . . . . 70
4.3.2 Evaluation of Contingent Claims related to Longevity Risks . . 79
4.4 Application to Unit-linked Insurance Contracts . . . . . . . . . . . . . . 88
4.4.1 Risk-neutral Evaluation of Unit-linked Contract . . . . . . . . . 89
4.4.2 Market Reserve and Thiele’s Diﬀerential Equation for Unit-
linked Pure Endowment Contract . . . . . . . . . . . . . . . . . 91
5 Longevity Derivatives 97
5.1 Longevity Risk Mangement . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1.1 Analysis of the Maximum of Future Lifetimes . . . . . . . . . . 98
5.1.2 Pure Longevity Derivatives . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Risk-neutral Valuation and Hedging of Longevity Derivatives . . . . . . 100
5.2.1 Risk-neutral Valuation of Longevity Derivatives . . . . . . . . . 100
5.2.2 Hedging Strategies for Longevity Derivatives . . . . . . . . . . . 104
5.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106CONTENTS iv
A Technical Backgrounds 114
A.1 Calculus with Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . 114
A.2 Calculus with Lévy Processes . . . . . . . . . . . . . . . . . . . . . . . . . 124
Bibliography 128List of Figures
2.1 QQnorm plot of standardized residuals . . . . . . . . . . . . . . . . . . . 28
2.2 Empirical density of standardized residuals. . . . . . . . . . . . . . . . . 30
2.3 Model W1 with M2: generalized Hyperbolic distribution . . . . . . . . 36
2.4 Model W1 with M2: Hyperbolic distribution . . . . . . . . . . . . . . . 37
2.5 Model W1 with M2: Normal-inverse Gaussian distribution . . . . . . . 38
2.6 Model W1 with M2: Variance Gamma distribution . . . . . . . . . . . . 39
vList of Tables
2.1 Table of skewness and kurtosis . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Table of normality tests I . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Table of normality tests II . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Fitted parameters of model W1 with M2 . . . . . . . . . . . . . . . . . . 40
2.5 Fitted parameters of model W2 with M2 . . . . . . . . . . . . . . . . . . 40
2.6 Fitted parameters of model W3 with M2 . . . . . . . . . . . . . . . . . . 40
2.7 Fitted parameters of model W4 with M2 . . . . . . . . . . . . . . . . . . 41
viChapter 1
Introduction
1.1 Motivation to Systematic Mortality Risks
Let us observe a given population that satisﬁes some homogeneity conditions, for ex-
ample, the people are of the same age, sex, and similar health situations. How many
will survive to a speciﬁc date in the future, say one year? If we know the one-year sur-
vival probability of this population, we can estimate the number of survivors in one
year by the expected number of this population, that is, just multiplying the survival
probability and the number of initial livings. For a suﬃcient large population, such a
prediction works pretty well. The life insurance industry (or the classical actuarial sci-
ence) lays itself principally on this idea to calculate the mortality risks in the insurance
contracts. In eﬀect, by selling suﬃcient many policies, an insurance company bears
theoretically no mortality risks in its business.
But if we are not sure of the survival probability, how are the things going to be?
Suppose there are two possible values of the one year survival probability, we will then
get two estimators by calculating the expected number of survivors for the population.
Therefore no matter how many policies an insurance company sells,the mortality risks
can not be diversiﬁed. The insurance companies can not sit on their own chairs to run
the business, they need to call on outside help, for example, the help coming from the
capital market to share the risks.
Such non-diversiﬁable mortality risks are called systematic mortality risks. In par-
ticular, they are called longevity risks when survival risks are concerned.
1CHAPTER 1. INTRODUCTION 2
1.2 Objective of the Thesis and Contribution
Chapter 2 : Mortality Risks
Life Contingencies in Classical Actuarial Perspectives Under the assump-
tion of a deterministic mortality rate, we review some main concepts in classical actu-
arial science.
We investigate the survival probabilities and the deterministic force of mortality un-
der the framework of a single jump process associated with each underlying individual.
Under the assumption that all individuals have independent future lifetimes, we can
approximate the number of future livings by the number of expected future livings.
Furthermore, the independent assumption of future lifetimes allows us to evaluate
the insurance contracts by the principle of equivalence. We explain this principle under
the framework of the standard-deviation principle. In particular, we investigate the
individual safety loading. In the current setting the individual safety loading converges to
zero, which shows that the principle of equivalence is suitable to evaluate the mortality
risks involved in the insurance contracts.
Finally, we state the classical time-continuous version of Thiele’s diﬀerential equation
for the net reserves of pure endowment contract.
Life Contingencies on Systematic Mortality Risks Empirical research such
as Currie et al. [2004] shows that the number of future livings can not be precisely
estimated by the expected future livings where the calculation uses a deterministic
mortality rate.
We generalize the concept of conditional independence of future lifetimes (seeNor-
berg [1989]) into a dynamic setting.No