biometric solvency risk for portfolios of general …contracts (iv) longevity risk stochastic...

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BIOMETRIC SOLVENCY RISK FOR PORTFOLIOS OF GENERAL LIFE

CONTRACTS (IV) LONGEVITY RISK STOCHASTIC DYNAMICS

Werner Hürlimann

Wolters Kluwer Financial Services, Seefeldstrasse 69, CH-8008 Zürich

ABSTRACT

Longevity risk applies to the class of insurance contracts contingent on survival of the insured live with negative

sums at risk, as typically encountered in life annuity products. For a portfolio consisting of a single cohort of

immediate life annuities with annual benefit payments of fixed amount, the Solvency II standard approach

prescribes a target capital, which only depends on the time of valuation and not on the cohort size. In fact, our

analysis shows that the current QIS5 specification applies to the systematic risk component of longevity risk under

the assumption of a cohort of infinite size. Important elements like process risk and related cohort size are not taken

into account. Moreover, parameter risk, understood as uncertainty in the life table parameters, is not dealt with

explicitly in QIS5. The present contribution offers a comprehensive integrated model for full coverage of the

longevity risk components and compares it to the QIS5 standard.

Keywords: Solvency II, target capital, solvency ratio, value-at-risk, longevity risk, process risk, systematic risk,

parameter risk, Poisson-Gamma model, convex ordering.

1. INTRODUCTION

Within the Solvency II project, there are two ways to calculate solvency capital requirements using the standard

approach and internal models. Advanced insurance companies are likely to use internal models because they already

have them in place. An internal model can improve consistency with respect to the obligations of the insurer and

may lead to a lower (or higher) capital requirement than under the standard approach. Due to this, full acceptance of

an internal model by the supervising authority must be made before the standard formula can be replaced by the

internal model. It follows that the rationale for any internal solvency model must be rigorously funded and the made

assumptions fully disclosed. Though the standard approach and an internal model approach have certainly many

common features, there may be elements, which have not been accounted for in the standard approach. For a sound

risk management it is necessary to understand any difference between the two approaches. Companies that will or

have implemented internal models can only benefit from the insights they gain through comparison with the

standard approach. Companies that have internal models only for certain risk types or some lines of business are

allowed to use the partial internal models approach, which combines internal models with the standard approach. In

the present study, we follow the above line of thought for the longevity risk, which has recently received increased

attention (e.g. Olivieri and Pitacco [1], [2]).

Longevity risk applies to the class of insurance contracts contingent on survival of the insured live with negative

sums at risk, as typically encountered in life annuity products. For clearness and simplicity we restrict the attention

to a portfolio consisting of a single cohort of immediate life annuities with annual benefit payments of fixed amount.

A generalization via compounding to multiple cohorts with varying amounts of benefits is possible and well-known

in actuarial risk theory. For a single cohort with level benefit payments the current QIS5 standard approach

prescribes a target capital (=amount of assets required at time of valuation to meet the longevity risk), which only

depends on the time of valuation and not on the cohort size. In fact, our analysis shows that the current QIS5

specification applies to the systematic risk component of longevity risk under the assumption of a cohort of infinite

size. Important elements like process risk and related cohort size are not taken into account. It seems that the latter

risk features have been disregarded under believe that the process risk component can be offset through pooling by

increasing the population size. However, the stochastic simulation of a single cohort of annuitants through time with

a random number of deaths does not lead to a vanishing capital requirement (even for a cohort of infinitely growing

size) at least within our modeling framework. Moreover, parameter risk, understood as uncertainty in the life table

parameters, is not dealt with explicitly in QIS5. The need for internal models to capture separately or simultaneously

all risk components of longevity risk is therefore justified. Besides its own sake of interest, it leads to a better

understanding of the standard approach and its application extent. A more detailed account of our contribution

follows.

Section 2 recalls the formulas needed to evaluate the target capital and the longevity risk solvency ratio at a given

time of valuation according to the current Solvency II standard approach. In Subsection 3.1 we develop a discrete

time dynamic stochastic solvency model for longevity risk, which relies on value-at-risk. The “Longevity VaR”

criterion says that if the portfolio assets at time of valuation exceed the random present value of future cash-out

IJRRAS 16 (2) ● August 2013 Hürlimann ● Longevity Risk Stochastic Dynamics

184

flows with a high probability, then solvency at time of valuation is fulfilled. Alternatively, the initial capital

requirement at time of valuation is determined using a mean value actuarial principle, such that assets exceed

liabilities in mean value for each future time of valuation. It is important to note that for a cohort of infinitely

growing size and a random present value of future cash-out flows, which satisfies the assumptions of the central

limit theorem, both capital requirements coincide. This result suggests the use of the latter as longevity risk

minimum capital requirement (MCR) in the Solvency II framework. A more detailed stochastic analysis of the

random actuarial present value of future cash-out flows is undertaken in Subsection 3.2. Based on previous results of

the author, a simple convex ordering approximation, which is based on an independence assumption for the random

number of deaths in successive periods, is derived. Under certain circumstances, this approximation leads to

conservative approximations of longevity risk capital requirements. Based on the independence assumption, we

derive in Section 4 analytical formulas for a simple approximate evaluation of the value-at-risk capital requirement

for longevity risk under different modeling assumptions. Poisson and normal approximations to the process risk as

well as to the simultaneous process and systematic risks are considered in the Subsections 4.1 and 4.2. The

evaluation of parameter risk is discussed in Subsection 4.3. Section 5 illustrates numerically and graphically.

Finally, the obtained results are used in Section 6 to discuss a new robust approach to the longevity risk capital in

Solvency II. It is based on a separation of solvency capital into appropriate process and systematic risk components.

Finally, it is important to note that the present approach to the longevity risk is related to the previous articles

Hürlimann [3], [4], [5], but differs from them in the sense that (i) besides the process risk, a distinction is made

between the systematic and parameter risk, and (ii) the stochastic dynamics is explicitly modelled and studied.

2. SOLVENCY RATIO FOR LONGEVITY RISK: THE STANDARD APPROACH

The current standard requirements for the Solvency II life risk module have been specified in QIS5 [6], pp.147-163.

A full internal model specification should take the following three risk components into account:

• process risk or insurance risk (random number of deaths or random fluctuations in the expected number of

survivals or deaths as forecasted by the life table)

• systematic risk (uncertainty in long term mortality trend or systematic deviations from the life table)

• parameter risk (uncertainty in life table parameters)

We note that the process risk, which is called unsystematic risk in Cairns et al. [7], is not covered by the standard

approach. The larger the population size the smaller is the unsystematic mortality risk, and it is believed that this risk

component can be offset through pooling. The rationale for the latter is the central limit theorem from probability

theory. Indeed, conditional on a best estimate of the mortality rate, the observed number of deaths will converge to

this value if the portfolio size is large enough. However, the stochastic simulation of a single cohort of annuitants

through time with a random number of deaths does not lead to a vanishing capital requirement (even for a cohort of

infinitely growing size) at least within our modeling framework. This result is derived rigorously in Section 3.1 and

confirmed through numerical results in Section 5. We consider a portfolio consisting of a single cohort of immediate

life annuities with annual benefit payments of fixed amount b . We assume that all the annuitants are aged s at the

time 0t of contract issue and denote by the maximum attainable age. Recall the life table and its

probabilistic interpretation:

xq : probability a life aged x will die within one year (probability of death)

xx qp 1 : probability a life aged x will survive to age 1x

xkpqpp xkxxkxk ...,,1,1,1 011 : probability a life aged x will attain age kx (survival probability)

11 kxxk qp : probability a life aged x will die within the time period xkkk ...,,1,,1 .

To simplify notation and discussion we consider a flat term structure of interest rates with annual interest rate i

and discount factor )1/(1 iv . The mathematical reserve at time st ,...,1,0 of a life annuity-due of one unit

of benefit payment payable at the beginning of each year while the annuitant survives coincides with its actuarial

present value at time t , which is defined and denoted by

)(

0

ts

ktsk

k

ts pva

. (1)

Let tA denote the amount of portfolio assets at time of valuation t and tst abV the corresponding

mathematical or actuarial reserves at the same time. The latter quantity is also called best estimate of liabilities. The

net value of assets minus liabilities at time t equals

ttt VANAV . (2)

QIS5 defines the solvency capital requirement (SCR) for longevity risk as


185

shocklongevityNAVSCR ttL , . (3)

In this formula tNAV is the change in net value of assets minus liabilities subject to a longevity shock defined as

a (permanent) 25% decrease in mortality rates at each age. Under a longevity shock the expected value of future

payments is calculated with a life table whose mortality rates are 25% lower than those in the best estimate life table.

With the shifted life table xxxx qpqq 1,)1( , 25.0,1,1 011

xkxxkxk pqpp , one obtains

)(

0, ,,

ts

ktsk

k

tststtttL pvaabVVVSCR

. (4)

Under the standard approach to Solvency II (upper index S2 in quantities) the amount of assets required at time t

to meet the longevity risk, the so-called target capital, is defined by

tLtL

S

tL RMSCRTC ,,

2

, . (5)

This formula includes a risk margin, which is currently calculated using a cost-of-capital approach with cost-of-

capital rate %6CoCi as follows:

)(

0,,

ts

kktL

k

CoCtL SCRviRM

. (6)

For comparison with internal models it is useful to consider the longevity risk solvency ratio at time t under the

Solvency II standard approach defined by the quotient

t

S

tLS

tLV

TCSR

2

,2

, . (7)

3. A DISCRETE TIME DYNAMIC STOCHASTIC SOLVENCY MODEL FOR LONGEVITY RISK

The notations and assumptions of the preceding Section remain the same. The remaining lifetimes of annuitants are

assumed to be identically distributed and, conditional on any given mortality assumption, independent of each other.

Let tN denote the random number of annuitants at contract duration time st ,...,1,0 , with 00 nN the

initial size of the cohort and 01 sN (assumption on the maximum attainable age). For congruence with the

setting made in Section 2, risks not related to the lifetime of an insured are disregarded. In particular, expenses and

related expense loadings are not taken into account.

Consider the annual cash-out flow of the cohort at time t defined by

stNb t ,...,2,1, . (8)

Then, the random present value of future cash-out flows at time t is given by

,,...,2,1,1

stNvbZtT

jjt

j

t

(9)

where 1)( tsTt represents the maximum remaining cohort lifetime at time t and use has been made of

the assumption 01 sTt NNt .

3.1. Value-at-risk solvency criterion and mean value actuarial principle

Given the initial deterministic amount of portfolio assets tA at time of valuation t , the random path of future

assets is determined by the recursive relationship

tttt TNbiAA ,...,2,1,11 , (10)

Through calculation one obtains the formula

tj

jt

j

tt TNvbAiA ,...,2,1,11

. (11)

Since tV is the best estimate of liabilities, the target capital

ttt VATC (12)

represents the amount of assets available at time t to meet the longevity risk.

In general, solvency rules suppose that the insurer is solvent if, with some (high) probability, assets meet liabilities

within a chosen time horizon. Details about the practical implementation of such a definition have been formulated

and discussed in Olivieri and Pitacco [1], [2]. For our purposes, we will adopt their solvency rule [R3], which

satisfies an attractive “asset and liability” logic (as opposed to the “deferral and matching” logic of their rule [R1]).


186

Under this rule, given an accepted (small) default probability 0 , say %5.0 to be consistent with QIS5, the

initial capital requirement tA at time t over the time horizon tT solves the probability inequality

1,...,2,1, ttt TZAP . (13)

Now, using (9) one obtains

.,...,2,1,111

t

T

jjt

jT

kkt

k

t TNvbiNvbZtt

(14)

Combining this with (11) using again (9) one gets further

ttttt TZAiZA ,...,2,1,1

, (15)

which implies the equivalence of (13) with the simpler value-at-risk condition

1tt ZAP . (16)

The derived “Longevity VaR” criterion says that if the portfolio assets tA exceed the random present value of

future cash-out flows with a probability of at least 1 , then solvency at time of valuation t is fulfilled. Let VaR

tLA , denote a minimum solution to (16). Then the target capital t

VaR

tL

VaR

tL VATC ,, meets the longevity risk. For

comparison with (7) one considers the longevity risk solvency ratio at time t defined by

t

VaR

tLVaR

tLV

TCSR

,

, . (17)

As an alternative, suppose that the initial capital requirement tA at time t is determined using a mean value

actuarial principle, such that assets exceed liabilities in mean value for each future time of valuation, that is instead

(13) one requires less stringently that

ttt TZAE ,...,2,1,0 . (18)

Equivalently, using again (3.8) one requires that

tt ZEA . (19)

Let t

mean

tL ZEA , denote the minimum solution to (19). Then the target capital t

mean

tL

mean

tL VATC ,, meets the

longevity risk according to the mean value principle. The corresponding longevity risk solvency ratio at time t

reads

t

mean

tLmean

tLV

TCSR

,

, . (20)

It is instructive to remark that for a cohort of infinitely growing size and provided the random variable tZ satisfies

the assumptions of the central limit theorem of probability theory, the solvency ratio (17) converges to (20). This

result suggests the use of mean

tLTC , as longevity risk minimum capital requirement (MCR) in the Solvency II

framework.

3.2. Convex ordering approximation

Our goal is the evaluation of the Longevity VaR criterion (16) under different stochastic mortality models of

longevity risk, as exposed in Section 4. For this a more detailed stochastic analysis of the random actuarial present

value of future cash-out flows tZ turns out to be useful. One notes that at time of valuation t the cohort size

tt nN is known. Let tD denote the random number of deaths produced by the cohort in the time period

tTtt ...,,1,,1 . Clearly, one has

tk

kttt TDnN ,...,1,1

. (21)

Inserted into (9) one obtains

t

T

tt Xbi

vnbZ

t

1

, (22)

with

kt

T

k

TkT

kktt D

i

vvDvX

tt

t

11 1

)()(

. (23)


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For interpretation and later use in Section 4, we note that )(:)(1

kktt DvD represents the discounted

cumulated number of deaths in the cohort over the time period tt, . It follows that

tT

tt DX1

)(

represents

the aggregated sum of the discounted cumulated number of deaths over all time periods tTtt ,...,1,, . Since

the random numbers of deaths between successive time periods are dependent and correlated, the analytical

evaluation of the distribution function in (23) is not straightforward. To simplify both modeling and calculations,

one often assumes independence of the deaths between successive periods. The effect of this approximation on

actuarial calculations has been studied previously by Hürlimann [8] in another context. As a general and useful

result, which will be applied in Section 4, we show below that the independence assumption increases the riskiness

of (23) in convex order. In particular, the independence assumption implies equal mean but increased variance in

(23). Under certain circumstances, this property yields conservative approximations to capital requirements. In

particular, this is the case if the value-at-risk criterion (16) is replaced by a conditional value-at-risk or expected

shortfall criterion. Indeed, it is well-known that the conditional value-at-risk measure is preserved under the convex

ordering relationship (e.g. Hürlimann [9], Theorem 1.1).

To derive the stated result it is necessary to introduce some more notations and rely on several prerequisites from the

established theory of stochastic orderings. It suffices to deal with (23) for a single annuitant aged ts and alive at

time t . Let tID denote an indicator random variable representing the death status of an annuitant during the

time period tTtt ...,,1,,1 , such that 1tID if the annuitant dies and 0tID otherwise. Given

the life table these random variables are determined by the death probabilities

ttstst TqpIDPtq ...,,1,1:)( 11 . The sum

kt

T

k

TkT

kktt ID

i

vvIDvIX

tt

t

11 1

)()(

(24)

represents the contribution of a single annuitant to (3.16). For stochastic ordering comparison, let

tt TID ...,,1,

, be independent random variables with the same distributions as the tID ’s and set

kt

T

k

TkT

ktt ID

i

vvIDvIX

tt

t

11 1

)()(

. (25)

Under this independence assumption the sum (23) is replaced by the sum denoted

tX . Clearly, the means of tX

and

tX are equal but the variances are not. We claim that tX precedes

tX in stop-loss order by equal means or

equivalently in convex order, denoted by

tslt XX , or tcxt XX . Recall that

tslt XX , if the

corresponding stop-loss transforms satisfy

)()( xXExXE tt for all x . In particular, since the means

are equal, this implies tt XVarXVar , an increased variance under the independence assumption. Since the

annuitants in the cohort are assumed to be independent and the stop-loss order is preserved under convolutions (e.g.

Kaas et al. [10], Theorem III.2.1), it suffices to show that

tslt IXIX , , which follows from the fact that

tTttt IDIDID ...,,, 21 are mutually exclusive risks. Before proceeding, some prerequisites on the last property

must be recalled.

One observes that an annuitant dies only once, which means that over the time horizon tT exactly one of the

tID ’s equals one. This is the intuitive interpretation behind the notion of mutually exclusive risk considered in

Dhaene and Denuit [11].

Definition 3.1. The non-negative risks nXX ...,,1 are called mutually exclusive, or equivalently )...,,( 1 nXX is a

mutually exclusive multivariate risk, provided 0)0,0Pr( ji XX for all ji .

The following main properties about mutually exclusive risks in the Fréchet space ),...,( 1 nn FFR of all n-

dimensional multivariate risks )...,,( 1 nXX with fixed marginal distributions )Pr()( xXxF ii are known.

Theorem 3.1. Let ),...,( 1 nnn FFRR be a Fréchet space of non-negative risks.

Property 1. The space nR contains mutually exclusive risks if, and only if, the following condition is fulfilled:


188

(ME) 1)0(11

n

iiF .

Property 2. If nR satisfies (ME) then X= nn RXX )...,,( 1 is mutually exclusive if, and only if,

0,1)(max)(1

nxFxFn

iiiX is the Fréchet lower bound distribution.

Property 3. If nR satisfies (ME) and X= nn RXX )...,,( 1 is a mutually exclusive risk, then

n

iisl

n

ii YX

11

for all

Y= nn RYY )...,,( 1 .

Proof. These properties are respectively Theorem 7, 8 and 10 in Dhaene and Denuit [11].

With these preliminaries the following results can be shown.

Proposition 3.1. For each tT...,,1 the multivariate indicator )...,,,( 21 ttt IDIDID of an annuitant with

mortality structure ))(...,),2(),1(( tqtqtq is a mutually exclusive risk.

Proof. This is shown similarly to Proposition 4.1 in Hürlimann [8].

Corollary 3.1. For all tT...,,1 the discounted cumulated number of deaths satisfy the convex order relations

)()()()(11

kkttcx

kktt DvDDvD . (26)

Similarly, the aggregated sum of the discounted cumulated number of deaths over all time periods

tTtt ,...,1,, , satisfies the convex order relation

tt T

ttcx

T

tt DXDX11

)()(

. (27)

Proof. This follows from Property 3 in Theorem 3.1 using Proposition 3.1. The derivation of (27) uses additionally

the representations (24) and (25) and the explanations made above.

4. STOCHASTIC MORTALITY MODELS: PROCESS, SYSTEMATIC AND PARAMETER RISKS Given the distinct risk components of longevity risk, it is desirable to construct stochastic mortality models, which

allow for separate or simultaneous modeling of these risk components. The notations and assumptions of the

preceding Sections are further in use.

4.1. Poisson and normal approximation to the process risk

The process risk is a traditional risk in life insurance, which describes the random fluctuations in the life table under

the following assumption. Whenever insured lives are independent and identically distributed and if the portfolio

size is large enough then the ratio of observed deaths to portfolio size is close to the mortality rate with high

probability. Given a fixed time of valuation st ,...,1,0 , a known cohort size tt nN , and a life table for the

mortality of annuitants, the random number of deaths during a given time period has a (conditional) binomial

distribution such that

tktstsktkt TkqpnBinD ...,,1,,~ 11 . (28)

For each tT...,,1 consider the discounted cumulated number of deaths )()(1

kktt DvD with mean

)(tm and variance )(2 ts . Using the first part of Corollary 3.1, we replace it by its simpler convex order upper

bound approximation )()(1

k

ktt DvD with the same mean but an increased variance )()( 22 tt ss . From

the binomial distribution properties and the independence assumption one gets immediately the following formulas

.1)(,)(1

1111

22

111

kktstskktstsktt

kktstsktt qpqpvnsqpvnm (29)

Furthermore, if the portfolio size is large enough, the probabilities of death are low and the product

11 ktstskt qpn is stable, then the Poisson approximation


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tktstsktkt TkqpnPoD ...,,1,~ 11 , (30)

which is widely used in actuarial risk theory, can be made. In this situation )(~)( tt mPoD has the mean

)(tm and the simple Poisson variance

)()(2 tPot mvs . (31)

Next, let us calculate the mean t and variance )( 2

t of the aggregated sum

tT

tt DX1

)(

of discounted

cumulated deaths over the whole remaining time horizon.

Proposition 4.1. The mean and variance of the sum

tX are determined as follows:

tT

tt m1

)(

(32)

1

1 1

2

1

22 )(2)()(t tt T

k

kT

t

kT

tt svs

(Binomial model) (33)

1

1 1

2

1

22 )(2)()(t tt T

k

kT

Pot

kT

PotPot svs

(Poisson approximation) (34)

Proof. The mean formula (32) is straightforward. To obtain the variance, set t

m

tt TmDmX ,...,1,)()(1

, and

proceed by induction on m to obtain a formula for the variance of )(mX t

denoted )(2 mt . We will make

repeatedly use of the following recursive relationship

1

1)()1( ttt DvDvD . (35)

First of all, one has )1()1( 22

tt s . For 2m one has

)2(,)1(2)2()1()2(2

ttttt DDCovDVarDVar .

Using (35) and noting that )1(tD and

2tD are independent one gets

)1()1(,)1()2(,)1( 2

2

tttttt DVarvDvDvDCovDDCov ,

which implies that )1(2)1()1()2( 2222

tttt svss .

By induction assumption we suppose that the following formulas hold

1

1 1

2

1

22 )(2)()(m

k

km

t

km

tt svsm

, (36)

1

1

2 )()(),1(m

kt

km

tt ksvmDmXCov , (37)

and show their validity for the index 1m . One has

)1(),(2)1()()1()()1( 222 mDmXCovmsmmDmXVarm ttttttt .

Using (35) and noting that )1( mX t and )(mDt are independent from

1mtD one gets


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,)()(,)()1(

)(,)()1()1(),(

1

21

1

1

m

kt

km

ttt

mt

m

ttttt

ksvmDvmDmXCov

DvmDvmDmXCovmDmXCov

which shows (37) for the index 1m . Inserted in the preceding expression and using the induction assumption one

obtains (36) for the index 1m . Finally, setting tTm in (36) shows (33) and (34) with the variances in (29) and

(30) respectively.

To evaluate the Longevity VaR criterion (16) we finally approximate tX in (22) by a normally distributed random

variable with mean (32) and variance given by (33) respectively (34), called “normal approximation” respectively

“Poisson approximation” for use in Section 5. For sufficiently large portfolios this approximation is justified. The

above stochastic mortality models yield the following approximate analytical capital requirements for the process

risk:

tt

T

t

VaR

tLi

vnbA

t

)(1 1

, (normal approximation) (38)

Pott

T

tPo

VaR

tLi

vnbA

t

)()(1

)( 1

, (Poisson approximation) (39)

Finally, we note that the MCR defined by the mean value actuarial principle in (19) is determined by the exact

formula

t

T

tt

mean

tLi

vnbZEA

t

1

, . (40)

For a cohort of infinitely growing size, our approximations satisfy the central limit theorem and accordingly the

solvency ratios from (38) and (39) converge to the one from (40).

4.2. Poisson and normal approximation to the simultaneous process and systematic risk In contrast to the assumption in Section 4.1, in case the ratio of observed deaths to portfolio size is not close to the

mortality rate, even for large portfolio sizes, systematic risk exists. In this situation, the mortality rate is uncertain

and assumed to be random. Stochastic models with “fluctuating basic probabilities” are widespread in actuarial

science and have been pioneered by Ammeter [12], [13]. Usually, one assumes a Bayesian Poisson-Gamma model

such that the number of deaths is conditional Poisson distributed with a Gamma distributed random mortality, which

results in a negative binomial distribution for the unconditional distribution of the number of deaths. This model is

well-known in non-life insurance, where it is used to describe the number of claims for a heterogeneous pool of risks

in a static environment (e.g. Bühlmann [14]). It has also been used in life insurance by Hürlimann [15], which has

introduced the so-called “linear multivariate Poisson Gamma model” of risk theory. Alternatively, one can assume

that the annual number of deaths is conditional binomially distributed with a Beta distribution for the random

mortality rate, which yields a negative hyper-geometric unconditional distribution, also called Binomial-Beta or

Polyà-Eggenberger model (e.g. Panjer and Willmot [16]). We note that the latter model has been adopted by

Marocco and Pitacco [17] to describe the annual number of deaths in a portfolio of life annuities. From a

mathematical point of view, the Poisson-Gamma choice has several advantages, in particular the generalization via

compounding to multiple cohorts with varying amounts of benefits (e.g. Hürlimann [15]).

For the purpose of modeling simultaneously the process and the systematic risks, we consider the Poisson-Gamma

model with time-dependence of the type introduced in Olivieri and Pitacco [2]. This probability model is able to up-

date its parameters to experience, which should be quite attractive for the design of dynamic information based

internal risk management systems. Given is a fixed time of valuation st ,...,1,0 , a known cohort size

tt nN , and a life table for the mortality of annuitants. For the first time period 1, tt , we assume that there is no

experience available and that the random number of deaths is conditional Poisson distributed such that

ts

tststtq

GammaQQnPoD

,~,~1 . (41)

It follows that the unconditional distribution of the number of deaths in the first time period is negative binomially

distributed such that

tst

tqn

NBD

1

1

11 ,

1,~ . (42)


191

In contrast to the expected number of deaths tst qn , as predicted by the life table and the process risk models of

Section 4.1, one has here

tstt qnDE

1 . (43)

To model a systematic deviation from the life table expectation, as encountered in longevity risk assessments, one

assumes that the quotient / is less than one. Suppose that at time 1t , the number of deaths 1td observed in

the cohort over the first time period is available, and let 11 ttt dnn be the up-dated cohort size. A calculation

shows that the posterior distribution of tsQ conditional on the information 11 tt dD is Gamma distributed

ts

tstttts

q

qndGammadQ

,~ 11 , (44)

which reveals that the initial structural systematic risk parameters are up-dated as follows:

tstt qnd ,, 1 . (45)

Passing to the second time period 2,1 tt , we assume similarly to the first period that

1

1111112 ,~,~ts

tsttttststtt

q

qndGammadQQnPodD

. (46)

This implies an unconditional negative binomial distribution of the type

11

2

2

2112 ,

1,~

tst

tstttt

qn

qndNBdD

. (47)

Iterating the above Bayesian scheme, one generalizes as follows. At time 2,1 t , having observed the

annual number of deaths 121 ...,,, ttt ddd , the up-dated cohort size for the next time period tt ,1 is

obtained from the recursion ,121 ktktkt dnn ,...,3,2k . The corresponding unconditional negative

binomial distribution of the number of deaths is then given by

11

1

1111

1121 ,

1,~,...,,

tst

kktskt

kkttttt

qn

qn

dNBdddD . (48)

It is interesting to look at the expected number of deaths given the available information, which is given by and

compares with the expected number from the life table as follows:

.,...,, 11111

111

1

1

121

tststttst

kktskt

kkt

tttt qpnDEqn

qn

d

dddDE (49)

If mortality experience is consistent with what is expected, the quotient of both expected values remains constant

over time. On the other side, if experience is worse than expected, the same quotient will decrease over time.

Clearly, such a systematic deviation must have an impact on the capital requirement for longevity risk.

Paraphrasing Section 4.1, it remains to find formulas for the mean )(tm and variance )(2 ts of the random

variable )()(1

k

ktt DvD for each tT...,,1 . Similarly to (29) one obtains the following formulas

211

21

111

111

1

1

2

22

2121

)(

))((

)(

)(

,,...,,)(

ktst

kktskt

kktskt

kkt

tst

tst

t

kktttkttstt

qn

qn

qnd

qnqn

vs

dddDEqnvm

. (50)


192

In the present Bayesian framework, it is also possible to use a Poisson approximation. Indeed, if the portfolio size is

large enough, the probabilities of death are low and the conditional expected values in (49) remain stable, then the

Poisson approximation

tktttktkttstt TkdddDEPoDqnPoD ...,,2,,...,,~,~ 1211 , (51)

can be made. In this situation )(~)( tt mPoD has the mean )(tm and the simple Poisson variance

)()(2 tPot mvs . (52)

Moreover, the mean t and variance )( 2

t of the aggregated sum

tX are obtained from the formulas in

Proposition 4.1 with the changed values in (50) and (52). Finally, inserting the latter quantities into the formulas

(38)-(40), one obtains analytical capital requirements for the longevity risk, which take both the process and the

systematic risks into account.

4.3. Longevity life tables and parameter risk To handle mortality risks a life table is required. It is constructed from a probabilistic mortality model, which should

allow for (i) random fluctuations (process risk), (ii) systematic deviations from the life table with respect to both age

and calendar year (systematic risk), and (iii) deviations due to the uncertainty in the model parameters (parameter

risk). Adequate methods used to represent the process or/and the systematic risks have been presented in the

previous two Subsections. We assume that our probabilistic mortality models depend on a set of parameters. A

simple and practical way to account for parameter risk consists to assign alternative hypotheses about these

parameters and mix them with a weight structure (e.g. Pitacco [18], Section 4.5, Olivieri and Pitacco [1]).

Let us start with one of the most sophisticated and best existing parametric model used to generate mortality rates in

life tables, which is the law of Heligman and Pollard [19] (see also Ibrahim [20], Jones [21]). Pitacco [22] points out

that Thiele proposed in 1867 a similar structure to model the force of mortality instead of the mortality rate. The

Heligman-Pollard life table is based on the analytical function with eight non-negative parameters

HGFEDCBA ,,,,,,, defined by

x

FxBx

x

x HGEDAp

q C

2)( )ln(exp . (53)

The model contains three terms, each of which represent a distinct mortality component. The first term is a rapidly

decreasing exponential function, which reflects the fall in mortality at early childhood ages less than 10 years. The

second term is similar to a log-normal density function describing the middle life mortality. The last term is a

Gompertz [23] exponential function, which reflects the rise in mortality at the adult and old ages. Since the focus is

on longevity risk modeling, we will assume that beyond the age of retirement one has

sxHGp

q x

x

x , . (54)

One notes that the parameter G describes the level of aging mortality and H the rate of increase of aging

mortality itself. A plot of the mortality rates xq against age x takes a logistic shape. Such functions can be

estimated very simply and efficiently, as shown by Sweeting [24]. Further interesting findings on Gompertz’s law

are found in Willemse and Kopelaar [25]. Following Pitacco [18] (see also Olivieri and Pitacco [1] for a refinement)

consider three alternative parameter sets ),(),,(),,( hhmm HGHGHG for the law (54) representing respectively

low, median and high mortality in the sense of longer, as expected and shorter expected future lifetimes. Denote by h

x

m

xx qqq ,, the life tables generated by these parameter sets. Then one assigns probabilities or weights

1,,0 hm to the assumed alternatives such that 1 hm . The resulting weighted life table is

defined by

sxqqqq h

xh

m

xmx

w

x , . (55)

The weighted life table is used to assess the parameter risk in conjunction with the models in Section 4.1 for the

process risk and the models in Section 4.2 for the simultaneous process and systematic risks.

5. A NUMERICAL CASE STUDY We discuss the impact of the considered longevity risk internal models on the current Solvency II standard

approach. In particular, we calculate solvency ratios for different modeling assumptions and compare them gaining

new insight into longevity risk. The alternative life tables from Section 4.3 have been generated using the following

sets of parameters


193

low mortality median mortality high mortality

G 2.5E-06 2.0E-06 1.5E-06

H 1.12 1.13 1.14

The median mortality should correspond to a best estimate of the life table. Our median parameters are

approximately equal to 130.1,06005.2 HEG , which are the best estimates for the cohort of Italian males

born in 1955 mentioned in Olivieri and Pitacco [1], Table 1. In a first step, it appears instructive to compare

expected future lifetimes under the low, median, high and weighted mortality assumptions for different times of

valuation 25,20,15,10,5,0t given the fixed age of retirement 65s . With an assumed maximum attainable

age 120 the “curtate-expectation-of-life” is defined by (e.g. Bowers et al. [26], Chapter 3, (3.5.5))

stpets

ktskts

,...,2,1,0,)(

1. (56)

Our choice of weights for the weighted life table in (55) is 25.0,5.0,25.0 hm . A calculation yields the

values summarized in Table 1 below. As a result we observe that the expected future lifetimes under the weighted

mortality are below but quite close to those values under the median mortality. Since the former is the life table used

to assess the parameter risk, its effect on solvency capital requirements should be intuitively small. This observation

will be confirmed quantitatively later on. Whether this also holds under more general settings has not been analyzed.

Table 1. Expected future lifetimes under different mortality assumptions

time low mortality median mortality high mortality weighted mortality

0 25.012 21.128 18.247 20.928

5 20.660 16.921 14.172 16.722

10 16.594 13.074 10.530 12.882

15 12.897 9.687 7.431 9.512

20 9.655 6.847 4.955 6.700

25 6.934 4.601 3.115 4.494

The assessment of the possible combinations of longevity risk components in our proposed internal models is based

on the following modeling assumptions:

• process risk: median mortality + risk models of Section 4.1

• process and parameter risks: weighted mortality + risk models of Section 4.1

• process and systematic risks: median mortality + risk models of Section 4.2

• process, systematic and parameter risks: weighted mortality + risk models of Section 4.2

We assume 1b (unit benefit payments) and %3i (annual interest rate). The evaluation of longevity risk

according to the standard approach includes the following quantities:

• reserves at time t : formula (1)

• capital requirement SCR at time t under a longevity shock 25.0 : formula (4)

• risk margin RM at time t : formula (6)

• market value MV of liabilities at time t : sum of reserves and RM at time t

• target capital TC at time t : sum of SCR and RM at time t

• solvency ratio SR at time t : formula (7)

As argued in Section 2 a longevity shock takes only the systematic risk into account. Parameter risk can be assessed

using the weighted mortality table. Table 2 summarizes results for the cohort size 100tn and confirms the little

influence of parameter risk.


194

Table 2. Standard Solvency II longevity risk assessment without and with parameter risk

quantities reserves SCR RM MV TC SR

without parameter risk

t = 0 1588.4 112.4 113.0 1701.4 225.4 14.2%

t = 5 1356.7 118.5 103.0 1459.7 221.5 16.3%

t = 10 1121.3 120.7 88.0 1209.2 208.7 18.6%

t = 15 893.1 117.8 69.6 962.7 187.4 21.0%

t = 20 684.7 109.4 50.6 735.3 160.0 23.4%

t = 25 507.4 96.5 34.0 541.4 130.4 25.7%

with parameter risk

t = 0 1578.4 112.5 112.4 1690.8 224.8 14.2%

t = 5 1345.4 118.6 102.2 1447.6 220.8 16.4%

t = 10 1109.0 120.7 87.1 1196.1 207.8 18.7%

t = 15 880.6 117.6 68.6 949.3 186.2 21.1%

t = 20 673.3 109.1 49.7 723.1 158.8 23.6%

t = 25 498.5 96.1 33.3 531.8 129.4 26.0%

Let us now consider internal models. Given a life table, the implementation of the model of Section 4.1 is

straightforward. The implementation of the model of Section 4.2 for the simultaneous measurement of both the

process and systematic risk is less trivial. A specification of the annual number of deaths and the remaining cohort

sizes beyond the valuation time is required. A simple way, which is consistent with the Solvency II standard

approach, consists to assume that future mortality deviates systematically from the life table according to the

prescribed longevity shock 25.0 , that is one sets

1,...,2,1,)1(, 111 tktsktktktktkt Tkqnddnn . (57)

This is consistent with 11 tt dDE if in (43) one has )1( . Assume further that 100 , which

implies a coefficient of variation for 1tD equal to 10%. While the standard solvency ratio does not depend on the

cohort size, this is the case for stochastic risk models, which take into account the process risk. As seen in Sections

4.1 and 4.2, the relevant quantities to assess capital requirements are the mean t and variance )( 2

t of the

aggregated sum

tT

tt DX1

)(

of discounted cumulated deaths over the whole remaining time horizon. The

Table 3 illustrates evaluation for a cohort size 100tn .


195

Table 3. Mean and variance of aggregated sum tX for different modeling assumptions

cohort size

100 process process process process + par.

quantities / time + parameter + systematic + systematic

mean

t = 0 1079.1 1087.4 980.8 989.4

t = 5 1196.3 1205.8 1092.7 1102.6

t = 10 1303.2 1313.8 1197.9 1208.8

t = 15 1388.2 1399.3 1285.5 1297.1

t = 20 1438.1 1449.0 1342.8 1354.3

t = 25 1440.4 1450.4 1356.3 1366.8

standard deviation

t = 0 121.6 122.3 114.5 115.3

t = 5 141.0 141.9 134.4 135.4

t = 10 159.3 160.2 154.0 155.0

t = 15 173.8 174.7 171.0 172.2

t = 20 181.6 182.4 182.8 184.0

t = 25 180.2 180.8 187.1 188.1

standard deviation

t = 0 123.0 123.8 113.3 114.1

t = 5 143.1 144.0 132.8 133.8

t = 10 162.1 163.1 151.8 152.8

t = 15 177.8 178.8 168.0 169.1

t = 20 187.3 188.2 178.7 179.7

t = 25 188.5 189.3 181.2 182.1

longevity risk components

t

t

Pot )(

tn

According to the formulas (38) and (39), the target capital t

VaR

tL

VaR

tL VATC ,, depends besides the cohort size tn

also on the mean t and the standard deviations

t (normal approximation) or Pot )( (Poisson

approximation). The solvency ratio (17) depends additionally on the actuarial reserve. The Table 3 shows

immediately that for the fixed cohort size 100tn the target capital and solvency ratio must increase with the

valuation time independently of the risk components. By fixed valuation time, it must also increase when adding the

systematic risk either to the process risk or to the simultaneous process and parameter risk components. A somewhat

different and less clear behavior is observed when adding the parameter risk either to the process risk or to the

simultaneous process and systematic risk components. While by fixed valuation time the target capital decreases,

due to a slight increase of the mean and standard deviation, the corresponding solvency ratio still increases, due to a

decrease of actuarial reserves (as seen in Table 2), which offsets the decrease in target capital. Anywhere, the effect

of the parameter risk is rather small.

On the other hand, the cohort size tn has a dramatic effect on the solvency ratio. It reduces it from very high levels

for small cohort sizes to the minimum level prescribed by the MCR formula (20). The Tables 4 and 5 display

solvency ratios for the Poisson and normal approximations by increasing cohort size and valuation time for all

considered risk components of the longevity risk, and compare these with the standard Solvency II approach.

The numerical figures show that the parameter risk can be neglected throughout. One observes some small

differences between the Poisson and the normal approximation for all risk components of longevity risk. Compared

to the normal approximation, the Poisson approximation slightly overestimates solvency ratios without the

systematic risk component while it underestimates them with the systematic risk component. Furthermore, the latter

effects increase over time. To gain further insight, it is instructive to display some graphs. We begin by comparing

the time dependence of solvency ratios by varying cohort sizes for the normal approximation. Figure 1 for the

process risk reveals an important gap between the standard approach and the lower bound. For small cohort sizes

and late valuation times, the standard approach prescribes insufficient solvency ratios. In fact, as already explained,

it does not take into account the process risk. On the other side, solvency ratios of cohort sizes exceeding 10’000

annuitants are clearly below those of the standard approach and tend more and more to the lower bound as expected

from the central limit theorem. Since the standard approach only takes into account the systematic risk, the made


196

comparison is more or less worthless. However, Figure 2 for the simultaneous process and systematic risks reveals

both a similar feature and a new instructive finding. Firstly, below 1’000 annuitants the standard approach

underestimates solvency ratios, especially for late valuation times. By increasing valuation time this underestimation

remains even for infinitely growing cohort sizes. The gap between the standard approach and the lower bound

decreases by increasing valuation time and almost coincides with it. The last effect is displayed in Figure 3.

Table 4. Comparison of solvency ratios for the Poisson approximation

0 5 10 15 20 25

Solvency II Standard Approach

without parameter risk 14.2% 16.3% 18.6% 21.0% 23.4% 25.7%

with parameter risk 14.2% 16.4% 18.7% 21.1% 23.6% 26.0%

Solvency II Internal Models

process risk

nt = 100 21.7% 30.2% 41.9% 57.9% 79.1% 105.8%

nt = 500 10.7% 15.2% 21.4% 29.6% 40.3% 53.1%

nt = 1000 8.1% 11.7% 16.5% 22.9% 31.1% 40.5%

nt = 10000 3.8% 5.8% 8.5% 11.8% 15.9% 19.9%

nt = 100000 2.5% 4.0% 5.9% 8.3% 11.1% 13.3%

nt = ∞ 1.8% 3.1% 4.7% 6.7% 8.8% 10.3%

process + parameter risks

nt = 100 22.1% 30.7% 42.7% 59.1% 80.9% 105.8%

nt = 500 11.0% 15.6% 21.9% 30.3% 41.2% 53.9%

nt = 1000 8.3% 12.0% 16.9% 23.5% 31.8% 41.2%

nt = 10000 4.0% 6.0% 8.7% 12.2% 16.2% 20.0%

nt = 100000 2.6% 4.1% 6.1% 8.6% 11.3% 13.3%

nt = ∞ 2.0% 3.3% 4.9% 7.0% 9.0% 10.3%

process + systematic risks

nt = 100 26.3% 35.9% 48.9% 66.5% 89.8% 118.6%

nt = 500 16.2% 22.0% 29.7% 39.9% 52.8% 68.0%

nt = 1000 13.8% 18.7% 25.1% 33.5% 44.0% 55.9%

nt = 10000 9.9% 13.3% 17.6% 23.1% 29.5% 36.1%

nt = 100000 8.6% 11.5% 15.2% 19.7% 24.9% 29.8%

nt = ∞ 8.0% 10.7% 14.1% 18.2% 22.7% 26.9%

process + parameter + systematic risks

nt = 100 26.7% 36.5% 49.8% 67.9% 91.6% 120.9%

nt = 500 16.5% 22.4% 30.3% 40.7% 53.8% 69.1%

nt = 1000 14.0% 19.0% 25.6% 34.2% 44.8% 56.8%

nt = 10000 10.0% 13.5% 18.0% 23.5% 30.0% 36.4%

nt = 100000 8.7% 11.7% 15.5% 20.1% 25.3% 30.0%

nt = ∞ 8.2% 10.9% 14.4% 18.6% 23.1% 27.0%

Time of Valuation

Table 5. Comparison of solvency ratios for the normal approximation


197

0 5 10 15 20 25

Solvency II Standard Approach



Solvency II Internal Models

process risk

nt = 100 21.5% 29.9% 41.3% 56.8% 77.1% 101.8%

nt = 500 10.6% 15.1% 21.1% 29.1% 39.4% 51.2%

nt = 1000 8.1% 11.6% 16.3% 22.6% 30.4% 39.2%

nt = 10000 3.8% 5.8% 8.4% 11.7% 15.7% 19.4%

nt = 100000 2.5% 4.0% 5.9% 8.3% 11.0% 13.2%

nt = ∞ 1.8% 3.1% 4.7% 6.7% 8.8% 10.3%

process + parameter risks

nt = 100 21.9% 30.4% 42.2% 58.1% 78.8% 101.8%

nt = 500 10.9% 15.4% 21.6% 29.8% 40.2% 52.0%

nt = 1000 8.3% 11.9% 16.7% 23.1% 31.1% 39.8%

nt = 10000 3.9% 6.0% 8.7% 12.1% 16.0% 19.6%

nt = 100000 2.6% 4.1% 6.1% 8.6% 11.2% 13.2%

nt = ∞ 2.0% 3.3% 4.9% 7.0% 9.0% 10.3%

process + systematic risks

nt = 100 26.6% 36.3% 49.5% 67.5% 91.5% 121.8%

nt = 500 16.5% 22.4% 30.3% 40.9% 54.6% 71.3%

nt = 1000 14.0% 19.0% 25.7% 34.5% 45.6% 58.9%

nt = 10000 10.0% 13.4% 17.9% 23.5% 30.2% 37.4%

nt = 100000 8.6% 11.6% 15.3% 19.9% 25.1% 30.2%

nt = ∞ 8.0% 10.7% 14.1% 18.2% 22.7% 26.9%

process + parameter + systematic risks

nt = 100 27.0% 36.9% 50.4% 68.9% 93.5% 124.2%

nt = 500 16.7% 22.7% 30.9% 41.7% 55.7% 72.5%

nt = 1000 14.3% 19.4% 26.2% 35.2% 46.5% 59.8%

nt = 10000 10.1% 13.7% 18.2% 24.0% 30.8% 37.8%

nt = 100000 8.8% 11.8% 15.6% 20.3% 25.5% 30.4%

nt = ∞ 8.2% 10.9% 14.4% 18.6% 23.1% 27.0%

Time of Valuation

Figure 1. Time evolution of solvency ratios (process risk)


198

Figure 2. Time evolution of solvency ratios (process and systematic risk)


199

Figure 3. Solvency ratios at different valuation times (process and systematic risks)

6. A ROBUST APPROACH TO THE LONGEVITY RISK CAPITAL IN SOLVENCY II

Since the Solvency II standard approach only takes into account the systematic risk component of longevity risk, it

appears useful to separate the process risk component from the systematic risk component in the capital model of

Section 4.2. Without loss of generality we restrict the discussion to the normal approximation. Consider the

following decomposition of the capital requirement for the simultaneous process and systematic risks:

t

T

t

s

tLt

p

tL

s

tL

p

tL

sp

tLi

vnbAbAAAA

t

1

,)1(, )(

,

1)(

,

)(

,

)(

,

),(

, , (58)

where

tt , are determined by the formulas (32), (33), with )(tm and )(2 ts as in (49) and (50). The

remark following (40) suggests to identify mean

tL

s

tL AA ,

)(

, with the systematic risk component. This implies the target

capital decomposition

t

s

tL

s

tL

s

tL

p

tLt

sp

tL

sp

tL VATCTCAVATC )(

,

)(

,

)(

,

)(

,

),(

,

),(

, , , (59)

which implies the solvency ratio decomposition

t

s

tLs

tL

t

p

tLp

tL

s

tL

p

tL

sp

tLV

TCSR

V

ASRSRSRSR

)(

,)(

,

)(

,)(

,

)(

,

)(

,

),(

, ,, . (60)

In virtue of the central limit theorem, and as confirmed by our numerical examples in Table 5, the systematic

solvency ratio )(

,

s

tLSR is a lower bound to the overall solvency ratio ),(

,

sp

tLSR and the process solvency ratio )(

,

p

tLSR

vanishes as the cohort size grows to infinity. Therefore, the process risk can be diversified away in this internal

model, as should be. To achieve consistency with the current QIS5 approach and for a more robust approach to

Solvency II, the solvency ratio 2

,

S

tLSR in (7), which represents the systematic risk component, should be set equal to

the systematic risk component )(

,

s

tLSR in (60). This leads to the following equation

)(

,

2

,

s

tL

S

tL TCTC , (61)

which can be solved in at least two ways. A similar analysis can be made if one includes additionally the parameter

risk component of longevity risk.

6.1. Solving for the cost-of-capital rate

In case the longevity shock remains fixed at the current level 25.0 , the only way to achieve equality in (61) is

through adjustment of the cost-of-capital rate. With the formulas of Section 2, one obtains the following time

dependent “robust” cost-of-capital rate

)(

0,

,

)(

,

, ts

kktL

k

tL

s

tLrob

tCoC

SCRv

SCRTCi

. (62)


200

Table 6 illustrates this robust procedure for the parameter assumptions of Section 5. One observes that the cost-of-

capital rate for a 25-year time of valuation slightly exceeds the 6% rate assumed in the standard approach. This is

due to the fact that the defined robust solvency ratio slightly exceeds the QIS5 solvency ratio over this time horizon.

Table 6. Robust cost-of-capital rate and solvency ratios

Robust cost-of-capital rate 0 5 10 15 20 25



Solvency ratios 0 5 10 15 20 25


nt = 100 26.6% 36.3% 49.5% 67.5% 91.5% 121.8%

nt = 500 16.5% 22.4% 30.3% 40.9% 54.6% 71.3%

nt = 1000 14.0% 19.0% 25.7% 34.5% 45.6% 58.9%

nt = 10000 10.0% 13.4% 17.9% 23.5% 30.2% 37.4%

nt = 100000 8.6% 11.6% 15.3% 19.9% 25.1% 30.2%

Robust Standard Approach 8.0% 10.7% 14.1% 18.2% 22.7% 26.9%

QIS4 Standard Approach 14.2% 16.3% 18.6% 21.0% 23.4% 25.7%

with parameter risk

nt = 100 27.0% 36.9% 50.4% 68.9% 93.5% 124.2%

nt = 500 16.7% 22.7% 30.9% 41.7% 55.7% 72.5%

nt = 1000 14.3% 19.4% 26.2% 35.2% 46.5% 59.8%

nt = 10000 10.1% 13.7% 18.2% 24.0% 30.8% 37.8%

nt = 100000 8.8% 11.8% 15.6% 20.3% 25.5% 30.4%



Time of Valuation

Robust cost-of-capital rate 0 5 10 15 20 25



Solvency ratios 0 5 10 15 20 25




with parameter risk



Time of Valuation

6.2. Solving for the longevity shock

If the cost-of-capital rate remains fixed at the level %6CoCi , the only way to achieve equality in (61) is through

adjustment of the longevity shock level. The time dependent “robust” longevity shock rob

t is the unique solution

of the following equation

t

T

t

ts

ktkt

k

CoCti

vnbVVviV

t

1

)()(

0

. (63)

It must be emphasized that the mean

t , as defined in (32), (49) and (50), that is

2

1211

,...,,)(,)(k

ktttkttstt

T

tt dddDEqnvmmt

, (64)

depends via the calibration choice (57) on the longevity shock specification. A calculation leads to the results shown

in Table 7. Since the effect of parameter risk is negligible the corresponding results are not shown. Similarly to the

observation made about Table 6, we note that for a 25-year time of valuation the robust longevity shock slightly

exceeds the 25% shock assumed in the standard approach, which also leads to a slightly higher systematic risk

solvency rate than in the current QIS5 standard approach. With this robust procedure the Figure 2 transforms to the

Figure 4, which is clearly more risk consistent.


201

Table 7. Robust longevity shocks and solvency ratios without parameter risk

Solvency Ratio of Longevity Risk 0 5 10 15 20 25

robust longevity shock 0.064770 0.099725 0.140515 0.187598 0.236791 0.273925

Poisson approximation

nt = 100 22.8% 32.2% 45.6% 64.1% 89.2% 120.2%

nt = 500 12.0% 17.7% 25.7% 37.0% 52.0% 69.7%

nt = 1000 9.5% 14.3% 21.0% 30.5% 43.2% 57.8%

nt = 10000 5.2% 8.5% 13.3% 19.9% 28.6% 38.0%

nt = 100000 3.9% 6.7% 10.8% 16.5% 24.0% 31.7%

Normal approximation

nt = 100 23.0% 32.6% 46.1% 65.1% 90.9% 123.4%

nt = 500 12.3% 18.1% 26.3% 38.0% 53.8% 73.0%

nt = 1000 9.7% 14.6% 21.6% 31.5% 44.8% 60.7%

nt = 10000 5.4% 8.7% 13.6% 20.4% 29.4% 39.3%

nt = 100000 3.9% 6.8% 10.9% 16.7% 24.3% 32.2%



Time of Valuation

Figure 4. Time evolution of solvency ratios (process and systematic risk)


202

7. REFERENCES [1]. A. Olivieri and E. Pitacco, Solvency requirements for life annuities allowing for mortality risks: internal

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