Chapter 12 actuarial funding of dismissal and resignation risks

CHAPTER

12

Actuarial Funding
of Dismissal and
Resignation Risks
Werner Hürlimann
CONTENTS
12.1 I ntroduction
12.2 Dismissal and Resignation Causes of Decrement
12.2.1 Dismissal by the Employer
12.2.2 Resignation by the Employee
12.2.3 Death of the Employee
12.2.4 Survival to the Retirement Age
12.3 Asset and Liability Model for Dismissal Funding
12.4 Dynamic Stochastic Evolution of the Dismissal Fund
Random Wealth
12.5 The Probability of Insolvency: A Numerical Example
Acknowledgments 28
References 286

268
269
271
272
273
274
276
279
280

5

B

esides t he u sua l p ension benefits, the pension plan of a firm may
be forced by law in some countries to offer wage-based lump sum
payments due to death, retirement, or dismissal by the employer, but no
payment is made by the employer when the employee resigns. An actuarial risk model for funding severance payment liabilities is formulated and
studied. The yearly aggregate lump sum payments are supposed to follow a
classical collective model of risk theory with compound distributions. The
final wealth at an arbitrary time is described explicitly including formulas
267

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268 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling

for the mean and the variance. Annual initial-level premiums required for
“dismissal f unding” a re de termined a nd u seful g amma approximations
for c onfidence i ntervals o f t he w ealth a re p roposed. A spec ific numerical example illustrates the nonnegligible probability of default in case the
employee structure of a “dismissal plan” is not well balanced.
Keywords: Asset and liability management (ALM), solvency, actuarial funding, dismissal risk, resignation risk, compound distributions.

12.1 INTRODUCTION
In some countries, for example, Austria, modern social legislation stipulates
besides usual p ension benefits, fixed wage-based lump sum pa yments by
death and retirement as well as through dismissal by the employer of a firm,
so-called severance payments (see, e.g., “Abfertigung neu” 2002, Holzmann
et al. 2003, “Abfertigung neu und alt” 2005, Koman et al. 2005, Grund 2006,

“Abfindung im Arb eitsrecht” 2007). H owever, if t he co ntract t erminates
due to resignation by the employee, no lump sum payment is made by the
employer. In this situation, there are four causes of decrement, which have
a random effect on the actuarial funding of the additional liabilities in the
pension plan, referred to in this chapter as the “dismissal plan.”
We are interested in actuarial risk models that are able to describe all
random lump sum payments until retirement for the dismissal plan of a
firm. The aggregate lump sum payments in each year are supposed to follow a classical collective model of the risk theory with compound distributions. The evaluation of the mean and standard deviation of these yearly
payments requires a separate analysis of the four causes of decrement. See
Section 12.2 for further details.
Actuarial funding with dismissal payments is based on the dynamic stochastic evolution of the random wealth of the dismissal fund at a spec ific
time. The final wealth at the end of a time horizon can be described explicitly, and formulas for the mean and the variance are obtained. In particular,
given the initial capital of the dismissal fund as well as the funding capital,
which should be available at the end of a time horizon to cover all expected
future random lump sum payments until the retirement of all employees, it
is possible to determine the required annual initial level premium necessary
for dismissal funding. This is described in Section 12.3.
Section 12.4 considers the dynamic stochastic evolution of the random
wealth at an arbitrary time and proposes a useful gamma approximation
for confidence intervals of the wealth.
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Actuarial Funding of Dismissal and Resignation Risks ◾ 269

Section 12.5 is devoted to the analysis of a specific numerical example,
which illustrates the nonnegligible probability of insolvency of a dismissal
fund in case the employee structure is not well balanced.

12.2 DISMISSAL AND RESIGNATION

CAUSES OF DECREMENT
Consider the “dismissal plan” of a firm, which offers wage-based lump
sum payments by death and retirement as well as through dismissal by
the em ployer. H owever, i f t he co ntract ter minates d ue t o r esignation
by t he em ployee, n o l ump su m pa yment i s made b y t he em ployer. I n
this situation, there are four causes of decrement, which have a random
effect on the dismissal funding. They are described as follows:
• Dismissal by the employer with a probability PDx at age x
• Resignation by the employee with a probability PR x at age x
• Death of the employee with a probability PTx at age x
• Survival to the deterministic retirement age s with a probability PSsx
at age x
Survival to retirement age s of an employee aged x happens if the employee
does not die and there is neither dismissal by the employer nor resignation
by the employee. The probability of this event depends on the probabilities
that an employee aged x survives to age x + k, namely,
k −1

k

PS x = ∏ (1 − PD x + j − PR x + j − PTx + j ),
j=0

0

PS x = 1,

(12.1)

and equals PSsx = s −x PS x . Note t hat i f a n employee attains t he common

retirement age s, then retirement payment due to survival takes place and
neither d ismissal, resignation, nor death is possible. Therefore, it can be
assumed that PDx = PR x = PTx = 0 for all x ≥ s.
We consider an actuarial risk model, which describes all random lump
sum pa yments u ntil r etirement f or t he d ismissal p lan o f a firm with M
employees at the initial time of valuation t = 0. F or a l onger time horizon
H, say 25 or 30 years, and for an initial capital K0, let P be the annual initial
level premium of the dismissal fund required to reach at fixed interest rate
i the funding capital KH at time H. The latter quantity is supposed to cover
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270 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling

at time H all expected future random payments until the retirement of all
employees [see formula (12.37)]. The considered (overall) funding premium
should not be confused with the individual contributions of the employees
for t heir ben efits, wh ich ma y va ry be tween em ployees. S ince l ump su m
payments a re p roportional t o t he wa ges o f t he em ployees, i t i s a ssumed
that the annual premium increases proportionally to the wages. With an
annual wage increase of 100 · g%, the annual premium at time t reads
Pt = P ⋅ (1 + g )t −1 , t = 1,…, H .

(12.2)

Let Xt be a t ime-dependent random variable, which represents the aggregate lump sum payments in year t due to the above four causes of decrement. We assume that this random variable can be described by a random
sum of the type
Nt

Xt = ∑ Yt , j , t = 1,…, H ,

j =1

(12.3)

where
Nt co unts t he number o f em ployee w ithdrawals d ue t o a ny ca use o f
decrement
Yt,j is t he individual r andom lump s um pa yment g iven t he jth wi thdrawal occurs
Under the assumption of a collective model of risk theory, the Yt,j are independent a nd i dentically d istributed l ike a r andom va riable Yt, a nd t hey
are i ndependent f rom Nt. As shown i n Hürlimann (2007) it is a lso possible to model in a simple way a continuous range of positive dependence
between independence (the present model) and comonotone dependence.
Assuming that Nt has a mean λt = E[Nt] and a standard deviation σ Nt , the
mean µ Xt and the standard deviation σ Xt of Xt are given by (e.g., Beard
et al. 1984, Chapter 3, Bowers et al. 1986, Chapter 11, Panjer and Willmot
1992, Chapter 6, Kaas et al. 2001, Chapter 3)
µ Xt = λ t ⋅µYt , σ Xt = λ t ⋅σY2t + σ N2 t ⋅ µYt ,

(12.4)

where µYt and σYt denote the mean and the standard deviation of Yt. The
evaluation of these quantities requires a separate analysis for each of the
four causes of decrement.

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Actuarial Funding of Dismissal and Resignation Risks ◾ 271

12.2.1 Dismissal by the Employer
Let N tD be the random number of dismissals in year t, and let YtD, j ~ YtD be

the independent and identically distributed individual random lump sum
payments in year t given the jth dismissal by the employer occurs. If Ak is
the age of the employee number k at the initial time of valuation, then the
expected number of dismissals in year t equals
M

λ tD = E ⎡⎣ N tD ⎤⎦ = ∑ t −1 PS Ak ⋅ PD Ak +t −1 .
k =1

(12.5)

Consider the probability of dismissal of an employer in year t given a population of M employees at initial time defined by the ratio
ptD =

λ tD
.
M

(12.6)

Since decrement by the cause of dismissal follows a binomial distribution
with parameter ptD , the variance of the number of dismissals is given by
σ2NtD = M ⋅ ptD ⋅ (1 − ptD ) =

λ tD ⋅ ( M − λ tD )
.
M

(12.7)

Furthermore, suppose that at the initial time of valuation, it is known
that by d ismissal the kth em ployee w ill r eceive t he l ump su m pa yment
B0,k. Since the lump sum payment is wage based and the wages increase at
the rate of 100 · g%, the effective lump sum payment in year t equals B0,k
(1 + g)t−1. U nder t he a ssumption o f a co mpound d istributed m odel f or
the aggregate lump sum payments due to dismissal by the employer, that
ND

t
is XtD = ∑ j =1 YtD,j , t = 1,…, H , i t f ollows t hat t he m ean a nd t he va riance

of YtD are given by
µtD = E[YtD ] =

1
⋅ E[ XtD ],
λ tD

1
[σtD ]2 = Var[YtD ] = D ⋅(Var[XtD ] − σ2NtD ⋅[µtD ]2 ) ,
λt

(12.8)

where the mean and the variance of the aggregate lump sum payments are
obtained from

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272 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling
M

E[ XtD ] = ∑ t −1 PS Ak ⋅ PD Ak + t −1 ⋅ B0,k ⋅ (1 + g )t −1 ,
k =1

(12.9)

M

Var[XtD ] = ∑ t −1 PS Ak ⋅ PD Ak +t −1 ⋅[ B0,k ⋅ (1 + g )t −1 ]2 − E[ XtD ]2 .
k =1

12.2.2 Resignation by the Employee
The e valuation i s s imilar t o t he s ituation o f d ismissal b y t he em ployer,
with th e d ifference t hat t he f oreseen l ump su m pa yment i s r eleased to
the remaining beneficiaries of the dismissal fund. Let N tR be the random
number of resignations in year t, and let YtR, j ~ YtR be the independent and
identically d istributed i ndividual r andom l ump su m pa yments i n y ear
t g iven t he kth resignation by t he employee occ urs. Again we a ssume a
compound distributed model for the aggregate lump sum payments due
NR

to r esignation b y t he em ployee, t hat i s, XtR = ∑ j =t1 YtR,j , t = 1,…, H. The
expected number of resignations in year t equals
M

λ tR = ∑ t −1 PS Ak ⋅ PR Ak +t −1 .
k =1

(12.10)

The probability of resignation of an employer in year t given a population
of M employees at the initial time is defined by the ratio
ptR =

λ tR
.
M

(12.11)

Since decrement by the cause of resignation follows a binomial distribution
with parameter ptR , the variance of the number of resignations is given by
σ2NtR = M ⋅ ptR ⋅ (1 − ptR ) =

λ tR ⋅ ( M − λ tR )
.
M

(12.12)

The mean and the variance of YtR are given by
1
⋅ E[ XtR ],
λ tR
1
[σtR ]2 = Var[YtR ] = R ⋅ (Var[ XtR ] − σ2NtR ⋅[µ tR ]2 ),
λt
µ tR = E[YtR ] =

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(12.13)


Actuarial Funding of Dismissal and Resignation Risks ◾ 273

where the mean and the variance of the aggregate lump sum payments are
obtained from
M

E [ XtR ] = ∑ t −1 PS Ak ⋅ PR Ak + t −1 ⋅ B0,k ⋅ (1 + g )t −1 ,
k =1

M

Var[ XtR ] = ∑ t −1 PS Ak ⋅ PR Ak +t −1 ⋅[B0,k ⋅ (1 + g )t −1 ]2 − E[ XtR ]2 .

(12.14)

k =1

12.2.3 Death of the Employee
Suppose that by death of an employee the portion θ of the dismissal payment is due to its legal survivor ( θ = 1/2 in our numerical example). Let
N tT be the random number of deaths in year t, a nd let Yt T, j ~ Yt T be t he
independent a nd i dentically d istributed i ndividual r andom l ump su m
payments i n year t g iven t he jth de ath occ urs. We a ssume a co mpound
distributed model for the aggregate lump sum payments due to the death
NT

of an employee, t hat is, XtT = ∑ j =t1YtT,j , t = 1,…, H. The expected number
of deaths in year t equals
M

λ tT = ∑ t −1 PS Ak ⋅ PTAk + t −1 .
k =1

(12.15)

The probability of the death of an employer in year t given a population of
M employees at initial time is defined by the ratio
ptT =

λ tT
.
M

(12.16)

Since decrement by the cause of death follows a binomial distribution with
parameter ptT , the variance of the number of deaths is given by
σ2NtT = M ⋅ ptT ⋅ (1 − ptT ) =

λ tT ⋅ ( M − λ tT )
.
M

(12.17)

The mean and the variance of YtT are given by
1
⋅ E[ XtT ],
T
λt
1
[σtT ]2 = Var[YtT ] = T ⋅ (Var[ XtT ] − σ2NtT ⋅[µtT ]2 ) ,
λt
µtT = E[YtT ] =

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(12.18)


274 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling

where the mean and the variance of the aggregate lump sum payments are
obtained from
M

E[ XtT ] = ∑ t −1 PS Ak ⋅ PTAk +t −1 ⋅θ⋅ B0,k ⋅ (1 + g )t −1 ,
k =1

M

2

Var[ XtT ] = ∑ t −1 PS Ak ⋅ PTAk + t −1 ⋅[θ⋅ B0, k ⋅ (1 + g )t −1 ] − E[ XtT ]2 .

(12.19)

k =1

12.2.4 Survival to the Retirement Age
Let NtS be the random number of retirements in year t, and let YtS, j ~ YtS be the
independent a nd identically d istributed i ndividual random lump su m payments generated upon retirement of the jth employee in year t. Taking into
account that an employee numbered k and aged Ak attains retirement in year
t such that Ak + t − 1 = s and using the definition of the retirement probability
PSsx = s −x PS x , one obtains for the expected number of retirements in year t
M

λ tS = ∑ t −1 PS Ak ⋅ I ( Ak + t − 1 = s),
k =1

(12.20)

where I(·) is an indicator function such that I(W) = 1 if the statement W is
true and I(W) = 0 else. The probability of survival to the retirement age of
an employer in year t given a population of M employees at initial time is
defined by the ratio
ptS =

λ tS
.
M

(12.21)

Since decrement by the cause of survival to retirement follows a binomial

distribution with parameter p tS , the variance of the number of retirements
is given by
σ 2NtS = M ⋅ ptS ⋅ (1 − ptS ) =

λ tS ⋅ ( M − λ tS )
.
M

(12.22)

Furthermore, suppose that at the initial time of valuation, it is known that
at re tirement t he kth em ployee w ill r eceive t he l ump su m pa yment Ck.
Due to wages increase, the effective sum in year t equals Ck(1 + g)t−1. Again,
assume a co mpound distributed model for the aggregate lump sum payNS

ments due to retirement, that is, XtS = ∑ j =t1YtS,j , t = 1,…, H . The mean and
the variance of YtS are given by
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Actuarial Funding of Dismissal and Resignation Risks ◾ 275

1
⋅ E[ XtS ],
S
λt
1
S 2
[σt ] = Var[YtS ] = S ⋅ (Var[XtS ] − σ 2NtS ⋅[µ tS ]2 ),
λt

µ tS = E[YtS ] =

(12.23)

where the mean and the variance of the aggregate lump sum payments are
obtained from
M

E[ XtS ] = ∑ t −1 PS Ak ⋅ I ( Ak + t − 1 = s) ⋅ Ck ⋅ (1 + g )t −1 ,
k =1

M

Var[X ] = ∑ t −1 PS Ak ⋅ I ( Ak + t − 1 = s) ⋅[Ck ⋅ (1 + g ) ] − E[ X ] .
S
t

t −1 2

k =1

(12.24)

S 2
t

The abo ve p reliminaries a re u sed t o o btain t he cha racteristics (12.4) a s
follows. The expected number of employee withdrawals in year t due to all
four causes of decrement equals
λ t = λ tD + λ tR + λ tT + λ tS .

(12.25)

Denote by Mt the number of remaining employees in year t. Starting with
an initial number M of employees, one has M0 = M and for year t > 1 one
has Mt = Mt−1 – λt, wh ich shows t hat t he ex pected number of remaining
employees dec reases o ver t ime, a s sh ould be. The i ndividual l ump su m
payment in year t satisfies the following equation:
λ t ⋅Yt = λ tD ⋅YtD − λ tR ⋅YtR + λ tT ⋅YtT + λ tS ⋅YtS . (1

2.26)

Indeed, t he a ggregate lump su m payments i n y ear t a re t he su m of t he
payments due to dismissal by the employee, death, and retirement less the
payments due to t he resignation of employees. Under t he assumption of
independence of the different random variables, one obtains for the mean
and the variance of Yt the formulas
1 D D
(λ t ⋅µt − λ tR ⋅µtR + λ tT ⋅µtT + λ tS ⋅µtS ),
λt
2
σYt = Var[Yt ] = (σtD )2 + (σtR )2 + (σtT )2 + (σtS )2 .
µYt = E[Yt ] =

(12.27)

Moreover, the variance of the number of withdrawals in year t due to all
four causes of decrement is given by
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276 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling

σ2Nt = Var[N t ] = σ2NtD + σ2NtR + σ2NtT + σ2NtS . (1

2.28)

The cha racteristics (12.4) f ollow i mmediately b y i nserting t he f ormulas
(12.25), (12.27), and (12.28).

12.3 ASSET AND LIABILITY MODEL FOR
DISMISSAL FUNDING
Let Wt be the random wealth of the dismissal fund at time t, where t = 0 is the
initial time of valuation. The random rate of return on investment in year t is
denoted It. The wealth at time t satisfies the following recursive equation:
Wt = (Wt −1 + Pt − Xt ) ⋅ (1 + It ).

(12.29)

Taking into account (12.1), the final wealth at the time horizon H is given by
H

H

H

t =1

t =1

j =t

WH = W0 ⋅ ∏ (1 + It ) + ∑ {P (1 + g )t −1 − Xt } ⋅ ∏ (1 + I j ) .

(12.30)

It is clear that the initial wealth coincides with the initial capital, that is,
W0 = K0. For simplicity, assume that accumulated rates of return in year t
are independent and identically log-normally distributed such that
1 + It = exp(Zt ),

(12.31)

where Zt is normally distributed with mean µ and standard deviation σ.
Consider the products
H

∏ (1 + I j ) = exp(Zt, H ),
j=t

1 ≤ t ≤ H,

(12.32)

which r epresent t he acc umulated r ates o f r eturn o ver t he t ime per iod
H

[t−1,H], wh ere t he su ms Zt, H = ∑ j = t Z j a re n ormally d istributed w ith
mean and standard deviation
µ t , H = E[Zt , H ] = (H − t + 1) ⋅µ, σt , H = Var[Zt, H ] = H − t + 1 ⋅σ.

(12.33)
The mean and the variance of the final wealth are given by the following
result.

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Actuarial Funding of Dismissal and Resignation Risks ◾ 277

THEOREM 12.1
Under the simplifying assumption that the random rates of return I1,…, IH
are independent from the aggregate lump sum payments Xt, the mean of
the final wealth is given by the expression
r H − (1 + g )H H
− ∑ µ Xt ⋅ r H −t +1
r − (1 + g )
t =1

E[WH ] = K 0 ⋅ r H + P ⋅ r ⋅

(12.34)

and the variance of the final wealth by the formula
H

Var ⎡⎣WH ⎤⎦ = K 02 ⋅ r 2 H ⋅ (e H σ − 1) + K 0 ⋅ ∑ r 2 H −t +1 ⋅ (P(1 + g )t −1 − µ Xt ) ⋅ (e (H − t +1)σ − 1)
2

2

t =1

H

2 H − t +1)
+ ∑r (
⋅
t =1

+2⋅

∑

{(P(1 + g )

t −1

− µ Xt )2 ⋅ (e(

H − t +1)σ2

− 1) + σ2Xt ⋅ e(

H − t +1)σ 2

r 2 H − t − s + 2 ⋅ (P (1 + g )s −1 − µ Xs ) ⋅ (P (1 + g )t −1 − µ Xt ) ⋅ (e(

}

H − t +1)σ 2

1≤ s < t ≤ H

− 1),

(12.35)
where r = exp(µ + 12 σ2 ) is the one-year risk-free accumulated rate of return
over the time horizon [0,H].
Proof Using the notation (12.32) the expression (12.30) can be rewritten as
H

WH = W0 ⋅ exp(Z1, H ) + ∑ {P (1 + g )t −1 − Xt } ⋅ exp(Zt ,H ),

(12.36)

t =1

from which one gets without difficulty (12.34). To get t he expression for
the variance, several terms must be calculated. One has
2

2

2

Var[W0 ⋅ exp(Z1, H )] = K 02 ⋅ (e 2 H (µ+σ ) − e H (2µ+σ ) ) = K 02 ⋅ r 2 H ⋅ (e H σ − 1).

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278 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling

For 1 ≤ t ≤ H one has
Var ⎡⎣(P (1 + g )t −1 − Xt ) ⋅ exp(Zt, H )⎤⎦
= Var ⎡ E ⎡⎣(P (1 + g )t −1 − Xt ) ⋅ exp(Zt, H ) Zt, H ⎤⎦ ⎤
⎣
⎦
+ E ⎡ Var ⎡⎣(P (1 + g )t −1 − Xt ) ⋅ exp(Zt, H ) Zt, H ⎤⎦ ⎤
⎣
⎦
= (P (1 + g )t −1 − µ Xt )2 ⋅ Var [exp(Zt , H )] + σ2Xt ⋅ E [exp(2 ⋅ Zt , H )]
= (P (1 + g )t −1 − µ Xt )2 ⋅ (e 2( H −t +1)(µ+σ ) − e ( H −t +1)(2µ+σ
2

2

)

) + σ2X

t

⋅ e 2(H −t +1)(µ+σ

2

)

t −1
= r 2(H −t +1) ⋅{(P (1 + g ) − µ Xt )2 ⋅ (e( H −t +1)σ − 1) + σ2Xt ⋅ e( H −t +1)σ }.

2

2

For 1 ≤ s < t ≤ H one has
Cov ⎡⎣(P (1 + g )s −1 − X s ) ⋅ exp(Z s , H ), (P (1 + g )t −1 − Xt ) ⋅ exp(Zt, H )⎤⎦
= (P (1 + g )

s −1

− µ X s ) ⋅ (P (1 + g )t −1 − µ Xt ) ⋅ Cov [exp(Zs ,H ),exp(Zt, H )] ,

where for the covariance term one gets
Cov[ exp(Z s , H ),exp(Zt , H )] = Cov ⎡⎣ E ⎡⎣exp(Z s , H ) Z s , H − Zt, H ⎤⎦ ,
E ⎡⎣exp(Zt, H ) Z s , H − Zt, H ⎤⎦ ⎤⎦
+ E ⎡⎣Cov ⎡⎣exp(Z s , H ),exp(Zt, H ) Z s , H − Zt, H ⎤⎦ ⎤⎦
= E [exp(Z s , H − Zt, H )]⋅ Var [exp(Zt , H )]
=e

(t − s )(µ+ 2 σ

1 2

)

2
2
⋅ (e 2( H −t +1)(µ+σ ) − e ( H −t +1)(2µ+σ ) )

= r 2 H −t − s + 2 ⋅ (e( H −t +1)σ − 1) .

2

In a similar way, for 1 ≤ t ≤ H one has
Cov [W0 ⋅ exp(Z1, H ), (P (1 + g )t −1 − Xt ) ⋅ exp(Zt, H )]
= K 0 ⋅ (P (1 + g )t −1 − µ Xt ) ⋅ Cov [exp(Z1, H ),exp(Zt , H )]
2

= K 0 ⋅ (P (1 + g )t −1 − µ Xt ) ⋅ r 2 H −t +1 ⋅ (e ( H −t +1)σ − 1).

© 2010 by Taylor and Francis Group, LLC


Actuarial Funding of Dismissal and Resignation Risks ◾ 279

Gathering a ll ter ms t ogether a nd su mming a ppropriately o ne o btains
finally (12.35).
Note t hat t he r isk-free r ate of re turn r must be realized in order to
guarantee w ith c ertainty t he ex pected fi nal wealth. We a re n ow i nterested in the determination of the required premium for dismissal funding with dismissal payments. Let K0 be the initial capital of the dismissal
fund. Suppose that at time H the funding capital KH should be available
in order to cover all expected future random lump sum payments until
the retirement of all employees. If Hmax denotes the maximum time horizon at which all employees from the initial population of M employees
have be en r etired w ith c ertainty, t hen t he r equired f unding c apital i s
given by
KH =

H max

∑ rH

t =H

max −t

⋅µ Xt ,

(12.37)

where r is a fi xed one-year guaranteed accumulated rate of return. Setting
this quantity equal to the expected final wealth, that is, E[WH] = KH, one
sees that by fixed r and with the formula (12.34) this equation can be solved
for the required annual initial level premium P.

12.4 DYNAMIC STOCHASTIC EVOLUTION OF THE
DISMISSAL FUND RANDOM WEALTH
The dynamic stochastic evolution of the random wealth at time t is determined by the recursive equation (12.29). Similar to (12.30), one obtains the
explicit expression
t

t

t

j =1

j =1

k= j

Wt = W0 ⋅ ∏ (1 + I j ) + ∑ {P (1 + g ) j −1 − Xt } ⋅ ∏ (1 + I k ).

(12.38)

Applying the same approach as in the proof of Theorem 12.1, one sees
that th e m ean an d th e v ariance o f th e r andom w ealth a t t ime t ar e
given by

E [W ] = W0 ⋅ r t + P ⋅ r ⋅

© 2010 by Taylor and Francis Group, LLC

r t − (1 + g )t H
− ∑ µ X ⋅ r H − t +1 ,
r − (1 + g ) t =1 t

(12.39)


280 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling
t

Var[Wt ] = K 02 ⋅ r 2t ⋅ (et σ − 1) + K 0 ⋅ ∑ r 2t − j +1 ⋅ (P (1 + g ) j −1 − µ X j ) ⋅ (e(t − j +1)σ − 1)
2

2

j =1

t

+ ∑ r 2(t − j +1) ⋅ {(P (1 + g ) j −1 − µ X j )2 ⋅ (e(t − j +1)σ − 1) + σ2X j ⋅ e(t − j +1)σ

2

j =1

+ 2⋅

∑

1≤i < j ≤ t

2

}

r 2t −i − j + 2 ⋅ (P(1 + g )i −1 − µ Xi ) ⋅ (P(1 + g ) j −1 − µ X j ) ⋅ (e(t − j +1)σ − 1).
2

(12.40)

Let k[Wt] be t he coefficient of va riation of t he wealth at t ime t. To e stimate a quantile of the random wealth at time t, we suppose that the wealth
is a pproximately g amma d istributed. This i s a p ractical a pproximation
under the reasonable assumptions of gamma-distributed aggregate lump
sum pa yments a nd i ndependent i dentically l og-normally d istributed
accumulated rates of return. Then, the α -quantile of the wealth at time t,
QW−1t (α) = inf x : P (Wt ≥ x ) ≥ α , is

{

}

⎛ 1 ⎞
⋅ k[Wt ]2 ⋅ E[Wt ] ,
QW−1t (α) = Γ α−1 ⎜
2⎟
⎝ k[Wt ] ⎠

(12.41)

where Γ α−1 (β) is the α -quantile of a standard gamma distribution Γ(β,1).
The α-confidence interval of the wealth at time t contains all possible realizations of the wealth in the interval ⎡⎣QW−1t (1 − α), QW−1t (α)⎤⎦ .

12.5 THE PROBABILITY OF INSOLVENCY:
A NUMERICAL EXAMPLE
The features of the present approach will be illustrated at a concrete example, which is not based on a real-life firm. The considered situation is chosen to exemplify what could happen in case a d ismissal fund is not well
balanced in its employee structure.
Suppose that the employee structure of the dismissal fund by age, term
of ser vice, a nd wage is g iven as in Table 12.1. Each age class is assumed
to be represented by 50 employees, of which half is male and half female.
Therefore the dismissal fund has a total of M = 1000 employees. The total
wages is equal to 61,500,000. The wage-based lump sum payment is evaluated u sing Table 1 2.2 u nder t he a ssumption t hat t he wa ges i ncrease b y

© 2010 by Taylor and Francis Group, LLC


Actuarial Funding of Dismissal and Resignation Risks ◾ 281
TABLE 12.1 Employee Structure by Age,
Term of Service, and Wage
Age

Term of Service

20
30
30
30
35
35
35
35
40
40
40
40
50
50
50
50
60
60
65
65

Wage

0
10
5
0
15
10

5
0
15
10
5
0
20
15
10
5
25
20
30
25

30,000
50,000
40,000
30,000
60,000
50,000
40,000
30,000
70,000
60,000
50,000
40,000
90,000
80,000
70,000

60,000
100,000
90,000
100,000
90,000

TABLE 12.2 Term of Service and Lump
Sum Payment
Term of Service
(in Years)
3
5
10
15
20
25

© 2010 by Taylor and Francis Group, LLC

Number of Monthly
Wage Payments
2
3
4
6
9
12


282 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling

TABLE 12.3

TABLE 12.4

Probabilities of Decrement in %

Age x

PDx

PRx

PTxm

PTxf

20
35
50
65

2
1
0.5
0

5
2.5
1
0

0.2
0.15
0.5
2

0.05
0.1
0.2
1

Development of Lump Sum Payments

Year

Expected
Lump Sum
Payments

Expected Future
Lump Sum
Payments

1
2
3
4
5
6
7

8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23

9,429,819
−52,479
−33,653
−31,354
−9,584
10,889,573
−105,101
−85,453
−64,532
−42,279
−41,507
−8,093
27,364

64,955
104,777
22,131,057
−42,264
−12,823
18,422
51,553
90,193
132,906
178,102

50,964,483
43,196,050
44,978,470
46,812,608
48,717,721
50,676,397
41,378,297
43,142,734
44,957,314
46,822,721
48,739,600
50,732,351
52,770,062
54,852,406
56,978,949
59,149,139
38,498,805
40,082,712
41,699,357

43,348,172
45,028,484
46,735,823
48,467,033

Year

Expected
Lump Sum
Payments

Expected
Future Lump
Sum
Payments

24
25
26
27
28
29
30
31
32
33
34
35
36
37

38
39
40
41
42
43
44
45

225,889
276,384
23,142,266
143,521
178,203
214,836
253,508
21,921,214
93,753
112,660
132,617
153,670
16,890,209
10,869
14,926
19,217
23,753
28,543
33,599
38,933
44,556

50,482

50,220,489
51,994,383
53,786,720
31,870,232
32,995,779
34,130,279
35,272,061
36,419,295
15,078,004
15,583,621
16,089,800
16,595,471
17,099,473
217,634
215,036
208,114
196,453
179,608
157,108
128,449
93,097
50,482

Note: The required premium to fund the future payments is e valuated for a time ho rizon
between H = 25 and 30 years. The initial capital is set at K0 = 10 million to avoid insolvency in t he first year because the expected lump sum pa yments for this period are
9.43 million according to Table 12.4. The funding capital at time H is set at KH = 35.272
million, which corresponds to the expected future lump sum payments at time H = 30.
Table 12.5 disp lays t he s ensitivity o f t he required p remium dep ending o n t he time

horizon and the variation of the expected aggregate lump sum pa yments (mean ±
multiple of the standard deviation). In these tables, µt, σt stand for µ Xt , σ Xt.

© 2010 by Taylor and Francis Group, LLC


Actuarial Funding of Dismissal and Resignation Risks ◾ 283

100 · g = 3% per y ear. The u sed probabilities of w ithdrawal for t he four
causes of decrement are summarized in Table 12.3, where linear interpolation is applied for ages between two values. These probabilities are only
rough values, but correspond qualitatively to real-life data. A d istinction
is made between male and female probabilities of death.
Following the formulas of Section 12.2, it is now possible to calculate the
expected aggregate lump sum payments for an arbitrary year t (formulas
(12.3), (12.19), and (12.22)) as well as the expected aggregate future lump
sum payments (formula (12.37) for an arbitrary time horizon H ≤ Hmax =
45). The a ssumed g uaranteed r ate o f r eturn i s se t a t 4 %. The obtained
results are summarized in Table 12.4. One notes that until every employee
attains w ith certainty t he retirement age of 65 years, t here is expected a
total of 75 dismissals by the employer, 170 resignations by the employee,
121 deaths, and t he remaining 634 employees are expected to attain t he
retirement age.
The dynamic stochastic development of the random wealth is displayed
in Table 12.5. The calculation is done with a volatility σ = 2% and a logarithmic r ate o f r eturn µ = ln(1.04) − 12 σ2 = 3.902%. The skew em ployee
structure implies a 5 % probability of default in t he 6t h year, a nd a s ituation close to i nsolvency i n t he first a nd 16th year w ith a p robability of
1%. Also, there is a nonnegligible probability that the overall goal at time
H = 30 will not be attained.
Traditionally, t he l ife i nsurance sec tor ha s se t a nnual p remiums a t a
constant level. It i s i nteresting to compare t his situation w ith t he above
one. To do calculations one has to replace the wage increase factor 1 + g

by a fac tor of one in the relevant formulas. Tables 12.5 and 12.6 are then
being replaced by Tables 12.7 and 12.8.

TABLE 12.5

Sensitivity of the Required Premium

H

mt – 2st

mt – st

mt

mt + st

mt + 2st

25
26
27
28
29
30

1,129,202
1,374,935
1,309,253
1,249,018

1,193,612
1,142,511

1,326,777
1,600,598
1,528,853
1,463,125
1,402,754
1,347,173

1,524,352
1,826,262
1,748,452
1,677,231
1,611,895
1,551,836

1,721,926
2,051,925
1,968,052
1,891,338
1,821,037
1,756,499

1,919,501
2,277,589
2,187,652
2,105,444
2,030,178
1,961,161

© 2010 by Taylor and Francis Group, LLC


284 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling
TABLE 12.6

Mean

Coeffici ent
of
Variation

2,206,897
4,012,079
5,919,757
7,952,719
10,097,264
1,046,963
3,125,238
5,324,024
7,648,551
10,104,250
12,720,547
15,471,813
18,363,276
21,400,334
24,588,562
5,070,224
7,906,838

10,903,994
14,068,569
17,407,696
20,925,103
24,626,232
28,518,472
32,609,476
36,907,161
17,694,659
21,733,726
26,002,703
30,511,889
35,272,000

0.530
0.307
0.219
0.171
0.142
1.941
0.684
0.422
0.309
0.246
0.205
0.177
0.157
0.141
0.129
0.784

0.529
0.404
0.329
0.280
0.245
0.219
0.199
0.183
0.170
0.393
0.337
0.296
0.265
0.241

Time
1
2
3
4
5
6
7
8
9
10
11
12
13
14

15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

Dynamic Stochastic Development of the Random Wealth

TABLE 12.7

95% Confidence
Interval
692,822
2,223,832
3,961,998
5,851,223
7,859,784
34
602,880

2,239,834
4,220,898
6,397,179
8,755,150
11,255,590
13,896,046
16,677,694
19,603,453
677,515
2,487,624
4,803,000
7,409,025
10,239,056
13,266,041
16,479,967
19,879,381
23,466,319
27,244,777
7,997,966
11,251,124
14,769,885
18,539,418
22,555,892

4,413,146
6,227,286
8,198,931
10,319,397
12,566,069
4,976,769

7,257,567
9,474,570
11,900,489
14,501,908
17,292,620
20,239,259
23,343,486
26,608,985
30,040,568
12,852,005
15,799,055
19,006,586
22,451,506
26,120,528
30,006,842
34,110,296
38,435,301
42,988,043
47,775,880
30,476,084
34,999,255
39,809,538
44,909,763
50,306,448

99% Confidence
Interval
399,041
1,710,377
3,327,517

5,128,543
7,063,043
0
264,374
1,503,764
3,239,545
5,237,385
7,447,177
9,815,828
12,332,805
14,994,805
17,802,014
239,218
1,434,460
3,300,388
5,560,046
8,099,782
10,867,754
13,840,080
17,006,984
20,365,200
23,915,138
5,567,451
8,373,491
11,492,782
14,891,006
18,552,088

5,785,244
7,414,207

9,341,901
11,462,095
13,729,412
9,859,078
10,073,121
11,883,455
14,181,928
16,753,370
19,560,732
22,551,984
25,719,470
29,062,113
32,582,106
18,457,196
20,705,055
23,652,256
27,016,366
30,698,009
34,654,113
38,866,622
43,330,197
48,045,416
53,016,289
37,753,115
42,261,068
47,148,505
52,389,783
57,975,579

Sensitivity of the Required Premium

H

mt − 2st

mt − st

mt

mt + s t

mt + 2st

25
26
27
28
29
30

1,129,202
1,374,935
1,309,253
1,249,018
1,193,612
1,142,511

1,326,777
1,600,598
1,528,853

1,463,125
1,402,754
1,347,173

1,524,352
1,826,262
1,748,452
1,677,231
1,611,895
1,551,836

1,721,926
2,051,925
1,968,052
1,891,338
1,821,037
1,756,499

1,919,501
2,277,589
2,187,652
2,105,444
2,030,178
1,961,161

© 2010 by Taylor and Francis Group, LLC


Actuarial Funding of Dismissal and Resignation Risks ◾ 285
TABLE 12.8 Dynamic Stochastic Development of the Random Wealth

for a Level Premium
Time
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28

29
30

Mean

Coeffici ent
of Variation

2,941,511
5,462,274
8,064,287
10,767,991
13,557,202
5,122,858
7,785,600
10,534,419
13,371,433
16,298,785
19,342,428
22,473,065
25,692,053
29,000,706
32,400,289
13,028,525
15,942,145
18,941,690
22,028,723
25,204,781
28,467,695
31,816,704

35,252,670
38,776,376
42,388,516
22,364,624
25,458,471
28,640,002
31,910,696
35,272,000

0.397
0.225
0.160
0.126
0.105
0.395
0.273
0.212
0.176
0.152
0.134
0.121
0.112
0.104
0.098
0.304
0.261
0.231
0.209
0.192
0.179

0.169
0.160
0.153
0.147
0.310
0.286
0.267
0.252
0.240

95% Confidence
Interval
1,318,252
3,612,278
6,065,441
8,637,159
11,297,847
2,303,596
4,644,543
7,143,772
9,753,547
12,459,791
15,280,195
18,183,285
21,168,118
24,234,430
27,382,281
7,270,643
9,762,520
12,352,045

15,029,295
17,789,267
20,625,684
23,534,829
26,515,276
29,565,925
32,685,827
12,311,133
14,772,358
17,311,308
19,923,914
22,607,494

5,086,515
7,624,833
10,297,587
13,093,124
15,987,332
8,844,611
11,585,772
14,463,928
17,458,432
20,562,819
23,800,275
27,138,816
30,578,994
34,121,858
37,768,730
20,145,833
23,354,574

26,681,485
30,123,576
33,680,084
37,347,836
41,125,765
45,014,849
49,016,294
53,131,449
34,845,671
38,503,688
42,287,094
46,196,695
50,233,935

99% Confidence
Interval
913,730
3,017,245
5,368,605
7,864,281
10,458,998
1,599,411
3,698,604
6,035,767
8,516,055
11,107,927
13,820,489
16,619,029
19,500,096
22,461,838

25,503,221
5,610,061
7,869,927
10,251,288
12,733,256
15,304,421
17,954,575
20,677,303
23,469,223
26,327,767
29,250,836
9,437,490
11,611,309
13,870,655
16,206,918
18,614,135

6,310,620
8,714,272
11,365,858
14,174,014
17,096,979
10,966,572
13,572,662
16,424,579
19,438,873
22,588,170
25,887,037
29,299,116
32,822,585

36,457,189
40,203,525
23,950,965
27,194,971
30,601,159
34,153,059
37,842,703
41,662,938
45,610,332
49,684,428
53,885,625
58,214,874
41,547,960
45,388,811
49,390,007
53,547,485
57,859,685

ACKNOWLEDGMENTS
I a m deeply g rateful to t he referees of t he fi rst International Actuarial
Association L ife C olloquium f or t heir de tailed co mments a nd h elpful
suggestions on a first version of this contribution. Special thanks go to
B. Sundt for corrections in formula (12.35).

© 2010 by Taylor and Francis Group, LLC


286 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling

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© 2010 by Taylor and Francis Group, LLC