Durational eﬁects and nonsmooth
semiMarkov models in life insurance
Dissertation
zur Erlangung des akademischen Grades
doctor rerum naturalium (Dr. rer. nat.)
der MathematischNaturwissenschaftlichen Fakult˜at
der Universit˜at Rostock
vorgelegt von
Dipl.Math. Marko Helwich,
geboren am 09. Dezember 1976 in Schwerin,
aus Rostock
Rostock, im Dezember 2007Gutachter:
Prof. Dr. Hartmut Milbrodt, Institut fur˜ Mathematik, Universit˜at Rostock
Prof. Dr. Friedrich Liese, Institut fur˜ Universit˜at Rostock
Prof. Dr. Klaus D. Schmidt, Institut fur˜ Mathematische Stochastik, TU Dresden
Tag der Verteidigung:
27. Mai 2008Great are the works of the LORD;
they are pondered by all who delight in them.
Psalm 111, verse 2Contents
1 Introduction 1
2 Modelling a single risk 13
A Classes of processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
B Some general remarks on Markov processes . . . . . . . . . . . . . . . . . . . . . 18
C Homogeneous Markovian marked point processes . . . . . . . . . . . . . . . . . . 22
D Nonsmooth semiMarkovian pure jump processes . . . . . . . . . . . . . . . . . . 31
D.1 Backward and forward equations . . . . . . . . . . . . . . . . . . . . . . . 48
D.2 A special case  Markovian pure jump processes . . . . . . . . . . . . . . . 63
3 Interest, payments and reserves 65
A Reserves in a nonrandom setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
B Relating payment streams and probabilities . . . . . . . . . . . . . . . . . . . . . 67
C Durationdepending actuarial payment functions . . . . . . . . . . . . . . . . . . 68
4 Prospective reserves and prospective losses 73
A Thiele’s integral equations of type 1 . . . . . . . . . . . . . . . . . . . . . . . . . 76
B Premiums and reserves for PKV and PHI . . . . . . . . . . . . . . . . . . . . . . 79
C Martingale representation of the prospective loss . . . . . . . . . . . . . . . . . . 100
D Thiele’s integral equations of type 2 . . . . . . . . . . . . . . . . . . . . . . . . . 103
E Solvability of integral equations for the prospective reserve . . . . . . . . . . . . . 108
F Loss variances and Hattendorﬁ’s theorem . . . . . . . . . . . . . . . . . . . . . . 110
G Integral equations for the loss variance . . . . . . . . . . . . . . . . . . . . . . . . 112
5 Retrospective reserves 117
A Diﬁerent deﬂnitions of the retrospective reserve . . . . . . . . . . . . . . . . . . . 118
B Retrospective reserves in a nonsmooth Markov setup . . . . . . . . . . . . . . . 120
Cectivees in aoth semiMarkov setup . . . . . . . . . . . . 121
A Tools and Proofs 123
1 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
vChapter 1
Introduction
The aim of this work is to model and investigate socalled durational eﬁects in life insurance.
This means, in the context of a multiple state model for a single risk, that probabilities of
transitions between certain states as well as actuarial payments are not completely determined
by the current state of the policy, but may also depend on the time elapsed since entering this
state.
The development of an insured risk is usually described by nonhomogeneous Markovian
pure jump processes with a ﬂnite state space containing all possible states of the policy. The
Markov property assures that the future development of such a process only depends on the
state which is occupied at a certain time. Further, the actuarial payments, which are often
separately considered as sojourn payments and payments due to transitions, are constructed
such that they also solely depend on the current state of the policy. Here, a generalized model
is presented that allows an appropriate implementation of durational eﬁects, such that both
transition probabilities and actuarial payments are additionally allowed to depend on the time
elapsed since entering the current state. In order to achieve this, the development of an insured
risk is modelled by nonhomogeneous semiMarkovian pure jump processes. Note that each
Markov process is also a semiMarkov process.
A pure jump process (X ) with ﬂnite state space S can also be described by using at t‚0
marked point process with space of marks S. The marked point process ((T ;Z )) thatm m m2N0
appertains to a pure jump process  which is either Markovian or semiMarkovian  is a Markov
chain. Hence, in order to develop a model that covers both the Markov as well as the semi
Markov approach, our theory mostly relies on the Markov property of the appertaining marked
pointprocess((T ;Z )) . Forreasonstobeexplainedlater, thisprocessisfurtherassumedm m m2N0
to be homogeneous. The Markov property of ((T ;Z )) implies the Markov property ofm m m2N0
the bivariate process ((X ;U )) . For each t ‚ 0, the process ((X ;U )) records both thet t t‚0 t t t‚0
current state of the policy, X , and the time spent in state X up to time t since the latestt t
transition to that state, U . Thus, for our approach, the pure jump process (X ) is nott t t‚0
necessarily Markovian, but the bivariate process ((X ;U )) is a Markov process. In contrastt t t‚0
to the pure jump process (X ) , however, this process does not have a ﬂnite state space. Thet t‚0
state space of ((X ;U )) is given byS£[0;1):t t t‚0
Before giving a survey of the literature that is concerned with durational eﬁects in life
insurance,andafterwardspresentingtheoutlineofthisthesis,threeexamplesfromlifeinsurance
are discussed for which the probabilities of certain transitions actually depend on the time
elapsed since the current state was entered. Doing so, the importance of durational eﬁects with
respect to the development of a single policy shall be pointed out. The insurance products
being considered are German private health insurance (PKV: Private Krankenversicherung),
permanent disability insurance (also referred to as permanent health insurance (PHI)), and
longterm care insurance (LTC).
12 CHAPTER 1. INTRODUCTION
Germanprivatehealthinsuranceformsthecapitalfundedpartofthehealthinsurancesystem
iny. Hence, the following issues are a distinctively German matter of interest. Since
policyholders currently lose their ageing provision (i.e. the prospective reserve) if they switch
insurance companies, there is, on the one hand, a lack of competition. This is being discussed in
German politics at the present time. On the other hand, an interesting actuarial issue is raised,
namely that withdrawal probabilities decrease with the time of being insured. The reason for
this is that the prospective reserve  and with that the loss due to withdrawal  increases over
the ﬂrst years. Hence, the withdrawal probabilities do not only depend on the attained age
of an insured, but also on the previous duration of the contract. Figure 1 gives an impression
of this eﬁect. For a real existing PKV portfolio with annual withdrawal rates structured by
technical age and time elapsed since entry into the portfolio, mere agedepending withdrawal
rates w ;x 2 fx ;:::;x g are compared with age and durationdepwalx MIN MAX
rates w ;x 2 fx ;:::;x g;u 2 f0;:::;x¡x g; in the case of 36year old insured ofx;u MIN MAX MIN
both gender and previous contract durations u=0;:::;15.
–––– w36
++++ w36;u
u 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 u0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 1: Age and durationdepending annual withdrawal rates compared with mere age
depending withdrawal rates with respect to the previous contract duration u, for men (left)
and women (right) at the age of x=y =36
Currently, the decrement of a PKV portfolio is modelled by using a Markov approach with
a mere agedepending actuarial basis. However, ﬂgure 1 shows that in doing so, the actual
withdrawal rates of insured with a short previous contract duration are underestimated and thewal rates of policyholders being insured for a longer time are overestimated. The latter
is a very crucial point, because the withdrawal rates are used to calculate the reserve that is
left by the withdrawing insured and put to increase the reserve of the remaining insured. In a
certain sense, this is like giving ﬂnancial support to the remaining insured, resulting in lower
net premiums for them. Consequently, if the actual withdrawal rates are lower than the ones
used for actuarial calculations, the resulting net premiums might be too low. Hence, the risk
of the portfolio is not really covered. For this reason, a calculation relying on an age and
durationdepending actuarial basis could help to avoid losses for the insurance company, or at
least to prevent from unintended shifting of risks. Yet, according to the obligatory regulations
concerning the actuarial modelling for German private health insurance  i.e. the act concern
ing the health insurance calculation (KalV: Kalkulationsverordnung [1996]) and the German3
Insurance Supervisory Law (VAG: Versicherungsaufsichtsgesetz [2004])  it is not permitted to
basethecalculationonanageanddurationdependingactuarialbasis. Forifthiswerethecase,
premiumswouldalsodependonpreviouscontractdurations,basicallyinthefollowingway: The
longer the previous contract duration, the higher the premium. This, however, is forbidden in
order to prevent premiums for longterm insured becoming more expensive than premiums for
new entries at the same age. Incidentally, in this case, insured could withdraw their contracts
and afterwards enter a new contract at the same company with lower premiums, provided that
they pass the health examination. An appropriate solution of this dilemma  on the one hand,
the actual withdrawal rates depend on both age and previous contract duration, and on the
other hand, it is not permitted to use a model that takes this into account  will be introduced
lateron(seeexample 4.13). Further, wewill sketchthe changeof the presentsituation for PKV
modelling caused by the regulations for improving the competition for statutory health insur
ance (GKVWSG:GesetzlicheKrankenversicherungWettbewerbsst˜arkungsgesetz [2007]), and
we will clarify whether or not our approach remains appropriate.
Figure 2 presents the durational eﬁects in the situation of the PKV portfolio on the level of
probabilities of remaining in the portfolio. The pure jump process used to model the decrement
of a PKV portfolio allows the states active, withdrawal and dead. Thus, at discrete times, the of remaining p and p can, under certain assumptions, be calculated with the¢ x ¢ x;u
aid of both annual withdrawal rates and annual mortality rates (cf. example 2.38).
–––– pk 31
++++ pk 31;0
ƒ ƒ ƒƒ pk 31;10
¢ ¢ ¢ ¢ pk 31;15
0 10 20 30 40 50 60 70 k 0 10 20 30 40 50 60 70 k
Figure 2: Comparison of probabilities of remaining kyears in the portfolio for a model with a
mereagedependingactuarialbasiswithcorrespondingprobabilitiesforamodelwithanageand
durationdepending actuarial basis and previous contract durations u = 0;10;15, for men (left)
and women (right) with attained age x=y =31
We turn to the second example for which durational eﬁects play an important role, the
permanent disability insurance. Since the pressure on existing social welfare systems increases
for diﬁerent reasons, the care for disabled and elderly persons must be funded more privately.
Therefore, insurance products to cover the risk of ﬂnancial losses or even the risk of a ﬂnancial
ruin due to disability become more important. We refer to insurance products of this kind as
permanent health insurance (PHI). This usage originates from the British one. A permanent
health insurance policy should not be mistaken for a private health insurance policy. While the
intention of the latter is basically the absorbtion of costs due to health services, a PHI contract
provides an insured with an income if the insured is prevented from working by disability due4 CHAPTER 1. INTRODUCTION
to sickness or injury.
PHIpoliciesareusuallyalsomodelledbymultiplestatemodelswithstatespaceS :=fa;i;dg.
The three possible states are referred to as a»active, i»invalid and d»dead. In cases where
recovery is implemented in the model, the set of possible transitions  generally being a subset
2off(y;z)2S ;y = zg  is given by J :=f(a;i);(a;d);(i;a);(i;d)g: Figure 3 illustrates the set
of possible states and the corresponding transitions for the model of a PHI policy.
(a;i)

a ? i
(i;a)
@ ¡
@ ¡
@ ¡(a;d) (i;d)
@ ¡
@ ¡@R ¡“
d
Figure 3: Set of states and set of transitions for the PHI model
One can easily imagine that especially for transitions from the state invalid the correspond
ing probabilities depend not only on the information that a person is disabled at a certain time
or at a certain age (both of which are equivalent), but also on the time elapsed since disable
ment. It is a widespread opinion that for disabled insured both the probability of recovering
and the probability of dying decrease with increasing duration of disability. Segerer [1993], for
example, investigated mortality and recovery rates relying on a data base coming from rein
surance portfolios of several German life insurance companies. He came to the conclusion that
mortality during the ﬂrst years of disability is signiﬂcantly higher than the mortality used for
the life insurance premium calculation. For higher ages and longer durations of disability, the
mortality of disabled persons approaches the normal mortality of insured persons. Regarding
the recovery rates, Segerer stated that with increasing age and duration of disability, recovery
becomes less probable.
Recovery and mortality rates for disabled insured that depend on both age and time elapsed
since disablement are provided by socalled selectandultimate tables. Such tables generally
contain annual rates that depend on two variables (cf. Bowers et al. [1997], section 3.8). One
variable, [x]; is the age at selection (e.g. onset of disability), and the second variable, t; is the
duration since selection. Thus, a twodimensional array is generated. The dependence on age is
recorded along the columns, and the dependence on time since selection is recorded along the
iirows. For example, q is understood as the annual mortality rate of an disabled insured[x¡t]+t
with attained age x who became invalid t years ago at the age of [x¡t]: The impact of the
time since selection on the annual rates often diminishes following selection, such that beyond a
certain period, the dependence on the time since selection can be neglected. Consequently, it is
economical to construct selectandultimate tables by truncation of the twodimensional array
after the ﬂrst (r+1) columns, for example by means of
ii ii ii ii iiq ; q ;:::; q =: q = q ; t‚r:x[x] [x¡1]+1 [x¡r]+r [x¡t]+t
The number r is referred to as select period. Figure 4 sketches the structure of a selectand
ultimate table for recovery rates r with a select period of 5 years.⁄
Toprovidenumericalexamplesforourresults, theactuarialbasisforcalculationsconcerning
PHIcontractsisformedbytheGermanselectandultimatetablesDAVSRT1997RIM(forthe
recovery of disabled insured) and DAVSST TI 1997 M (for the mortality of disabled insured).
6