A joint analysis of financial and biometrical risks in life insurance [Elektronische Ressource] / vorgelegt von Marcus Christian Christiansen

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• Mathematics

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• Publié par: universitat_rostock
• Publié le: 01 janvier 2007
• Nombre de lectures: 8
• Langue: English

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A joint analysis of nancial and
biometrical risks in life insurance
Marcus C. ChristiansenA joint analysis of nancial and biometrical
risks in life insurance
Dissertation
zur
Erlangung des akademischen Grades
doctor rerum naturalium (Dr. rer. nat.)
der Mathematisch-Naturwissenschaftlichen Fakult at
der Universit at Rostock
vorgelegt von
Marcus Christian Christiansen
geboren am 13. Oktober 1978 in Michigan (Vereinigte Staaten)1. Gutachter: Prof. Dr. Hartmut Milbrodt, Universit at Rostock
2.hter: Prof. Dr. Holger Drees, Universit at Hamburg
3. Gutachter: PD Dr. Ulrich Orbanz, Towers Perrin Tillinghast
Tag der Verteidigung: 17. Januar 2007Contents
Introduction 7
1 Financial and biometrical risks in life insurance 11
1.1 Classi cation of risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Life insurance model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Expanded life insurance model . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Decomposition of risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 A sensitivity analysis approach for functionals on speci c function spaces 25
2.1 A gradient vector for functionals on L () . . . . . . . . . . . . . . . . 26p
2.2 A gradient vector for on BV . . . . . . . . . . . . . . . . 28
2.3 Comparison with concepts of the statistical literature . . . . . . . . . . 31
d2.4 An extended gradient vector for L ()-functionals . . . . . . . . . . . 33p
d2.5 Anedt vector for BV -functionals . . . . . . . . . . . . 34
3 A sensitivity analysis of life insurance contracts 37
3.1 Gradient vector with respect to interest . . . . . . . . . . . . . . . . . 40
3.2t vector with respect to a single transition . . . . . . . . . . . 47
3.3 Gradient vector with respect to interest and all transitions simultaneously 54
3.4t vector of the premium level . . . . . . . . . . . . . . . . . . . 59
3.5 Sensitivities of typical life insurance contracts . . . . . . . . . . . . . . 61
4 An uncertainty analysis of life insurance contracts 75
4.1 Probabilistic model for the technical basis . . . . . . . . . . . . . . . . 76
4.2 Decomposition of risk and approximation of its components . . . . . . 79
4.3 Uncertainties of typical life insurance contracts . . . . . . . . . . . . . 93
References 105
A Appendix 109
A.1 Representation of the present value . . . . . . . . . . . . . . . . . . . . 109
A.2 Functions of bounded variation . . . . . . . . . . . . . . . . . . . . . . 110
A.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
A.4 Some properties of product integrals . . . . . . . . . . . . . . . . . . . 116
A.5 Integration of stochastic di usions with additive noise . . . . . . . . . 120Introduction
Though the actuary of the 20th century used to model the lifespan of an insured
stochastically, he usually relied on a deterministic prognosis of interest rates and
mortality probabilities, denoted as ’actuarial assumptions’ or ’technical basis’. The
past has shown that these assumptions can vary signi cantly within a contract period.
Especially in recent years nancial markets have experienced increased volatility, and
life expectancies have risen in many developed countries with an unforeseen rate.
As changes of the technical basis can have a crucial e ect on pro ts and losses, the
actuary of the 21st century is well advised to pay attention to the nancial and the
systematic biometrical risk, in particular the longevity risk. This need is also re ected
in the International Financial Reporting Standard (IFRS) No. 4 of the International
Accounting Standards Board (IASB, 2004).
Already in 1905 Lidstone studied the e ect of interest rate and mortality rate changes
on premium values. Since then his sensitivity analysis concept has been improved by
various authors leading to a number of insights. A younger and very capable approach
of sensitivity analysis in life insurance is to study partial derivatives with respect to
the parameters of the technical basis. Classically this works for (actuarial) functionals
with an in nite-dimensional domain. However, if one wants to employ that sensitivity
analysis concept for continuous time models where the technical basis parameters are
functions on the real line, some kind of generalized gradient for functionals on func-
tion spaces is needed. This paper presents a proper approach, which will turn out to
have some similarities to concepts of robust or nonparametric statistics. Applying the
generalized sensitivity analysis concept to typical life insurance contracts will show
that not only a change of interest rates but also of mortality or disability rates can
have a signi cant in uence on pro ts or losses.
Though a sensitivity analysis is a helpful approach for studying risks, it does not
take into account the diversity of the volatilities of the technical basis parameters,
neither in respect it di erent rates (e.g., interest rate, mortality rate, etc.) nor in
respect to time (e.g., interest rate at di erent time points). For an exhaustive risk
study it is therefore inevitable to model the technical basis stochastically.
Regarding the nancial risk, the thriving development of nancial theory during
the last decades has inspired many actuaries to model capital pro ts in life insurances
stochastically, too. It proved to be fruitful to adopt techniques and insights of nancial
mathematics for actuarial tasks such as valuation or pricing. Today the literature
o ers quite a number of life insurance models with stochastic interest rates.
In contrast, the systematic biometrical risks were for a long time widely ignored.
Hoem (1988, p. 192), for example, recommended to cover the systematic mortality risk8 Introduction
by a generous security loading on the interest rate. Lately, the unforeseen increase
of life expectancies directed the attention to the systematic mortality risk. In the
meantime, the literature o ers several life insurance models with stochastic mortality
rates.
For quantifying and comparing the nancial and biometrical risks, the overall un-
certainty of actuarial quantities such as present bene t/premium value has to be
decomposed with regard to its di erent sources. Though the literature o ers sev-
eral approaches for studying nancial and biometrical risks, it is up to now lacking
a concept that allows for (a) quantifying nancial (interest rate) risk, unsystematic
biometrical risk, systematic mortality risk, systematic disability risk, et cetera simul-
taneously (b) with risk measures which are comparable to each other (c) for a wide
variety of life insurance contract types. This paper wants to ll that gap, presenting
some uncertainty analysis concept.
Applying that concept to typical life insurance contracts will show that even though
the nancial (interest rate) risk is largely predominant over the systematic biometrical
risks, the latter are in many cases of signi cant size.
The structure of this paper is as follows:
After a short overview over the types of risk considered here, chapter 1 introduces
the life insurance model of Milbrodt and Helbig (1999), which is one of the most
general modeling frameworks of individual contracts in life insurance and includes
both discrete time and continuous time approaches. Section 1.3 expands this model
on a very general level to a stochastic technical basis, which will be further speci ed
in chapter 4. Section 1.4 addresses the task of decomposing the overall randomness
to its di erent sources.
Modeling the compounding factor and the transition probabilities as functions on
the real line, the prospective reserve and the premium level have as mappings of the
technical basis a function space as domain. For this reason it is not possible to perform
a classical sensitivity analysis on them by just calculating their partial derivatives.
Chapter 2 presents a new concept for a sensitivity analysis of functionals on speci c
function spaces by introducing a kind of generalized gradient vector. The latter has
some similarities with concepts in robust statistics and nonparametric statistics; a
comparison is given in section 2.3.
Using the tools of chapter 2, chapter 3 performs a sensitivity analysis on the actuar-
ial functionals ’prospective reserve’ and ’premium level’ as mappings of the technical
basis. Several realistic examples in section 3.5 exemplify the capability of the intro-
duced concept. An empirical study of the basic life insurance contract types ’annuity
insurance’, ’pure endowment insurance’, ’temporary life insurance’, ’disability insur-
ance’, and their combinations yields valuable hints for risk management.
Based on the preliminary work of section 1.3, in chapter 4 the technical basis is
modeled stochastically by assuming that the interest rate and the transition inten-
sities are linear combinations of di usion processes. Section 4.2 enhances the risk
decomposition of section 1.4, and allows one to separate the nancial risk, the un-
systematic biometrical risk, and the systematic biometrical risks such as systematic
mortality risk or systematic disability risk. In section 4.3, the approach of section9
4.2 is applied to examples of of typical life insurance contracts. An empirical study
leads to the following insights: Contrary to the statement of Hoem (1988, p. 192), the
systematic mortality risk can be of great impo