Chapter 12 actuarial funding of dismissal and resignation risks

Actuarial Funding of Dismissal and Resignation Risks Werner Hürlimann CONTENTS 12.4 Dynamic Stochastic Evolution of the Dismissal Fund 12.5 Th e Probability of Insolvency: A Numerical

Actuarial Funding

of Dismissal and

Resignation Risks

Werner Hürlimann

CONTENTS

12.4 Dynamic Stochastic Evolution of the Dismissal Fund

12.5 Th e Probability of Insolvency: A Numerical Example 280

References 286

Besides t he u sua l p ension benefi ts, the pension plan of a fi rm may

be forced by law in some countries to off er 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 actuar-ial risk model for funding severance payment liabilities is formulated and studied Th e yearly aggregate lump sum payments are supposed to follow a classical collective model of risk theory with compound distributions Th e

fi nal wealth at an arbitrary time is described explicitly including formulas

for the mean and the variance Annual initial-level premiums required for

“dismissal funding” are determined and useful gamma approximations for c onfi dence i ntervals of t he wealth a re proposed A spec ifi c numeri-cal 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,

actuar-ial funding, dismissal risk, resignation risk, compound distributions

12.1 INTRODUCTION

In some countries, for example, Austria, modern social legislation stipulates besides usual p ension benefi ts, fi xed wage-based lump sum pa yments by death and retirement as well as through dismissal by the employer of a fi rm, 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,

“Abfi ndung 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 eff ect 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

fi rm Th e aggregate lump sum payments in each year are supposed to fol-low a classical collective model of the risk theory with compound distribu-tions Th e 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 sto-chastic evolution of the random wealth of the dismissal fund at a spec ifi c time Th e fi nal wealth at the end of a time horizon can be described explic-itly, 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 Th is 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 confi dence intervals of the wealth

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

CAUSES OF DECREMENT

Consider the “dismissal plan” of a fi rm, which off ers 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 employee, no lump su m payment is made b y t he employer I n this situation, there are four causes of decrement, which have a random eff ect on the dismissal funding Th ey are described as follows:

Dismissal by the employer with a probability PD

Resignation by the employee with a probability PR

Death of the employee with a probability PT

Survival to the deterministic retirement age

x

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 Th e probability of this event depends on the probabilities

that an employee aged x survives to age x + k, namely,

−

=

0

and equals PSs PS

x =s x− x Note that if an employee attains the common

retirement age s, then retirement payment due to survival takes place and

neither dismissal, resignation, nor death is possible Th erefore, it can be assumed that PDx = PRx = PTx = 0 for all x ≥ s.

We consider an actuarial risk model, which describes all random lump

sum payments u ntil retirement for t he d ismissal plan of a fi rm with M employees at the initial time of valuation t = 0 For a longer 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 fi xed interest rate

i the funding capital K H at time H Th e latter quantity is supposed to cover

at time H all expected future random payments until the retirement of all

employees [see formula (12.37)] Th e considered (overall) funding premium should not be confused with the individual contributions of the employees for t heir benefi ts, wh ich may va ry be tween employees Since lump su m payments a re proportional to t he wages of t he employees, it is assumed that the annual premium increases proportionally to the wages With an

annual wage increase of 100 · g%, the annual premium at time t reads

−

= ⋅ +(1 ) ,t 1 = …1, ,

t

Let X t be a time-dependent random variable, which represents the aggre-gate lump sum payments in year t due to the above four causes of

decre-ment We assume that this random variable can be described by a random sum of the type

=

t N

j

where

N t co unts t he number o f em ployee w ithdrawals d ue t o a ny ca use o f

decrement

Y t,j is t he individual r andom lump s um pa yment g iven t he jth

with-drawal occurs

Under the assumption of a collective model of risk theory, the Y t,j are

inde-pendent a nd identically d istributed like a r andom va riable Y t, a nd t hey

are independent from N t As shown in Hürlimann (2007) it is also pos-sible to model in a simple way a continuous range of positive dependence between independence (the present model) and comonotone dependence

Assuming that N t has a mean λt = E[N t] and a standard deviation σN t, the mean µX t and the standard deviation σ of X X t t 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 = λ ⋅µX t t Y t, σ = λ ⋅σ + σ ⋅ µX t t Y2t N2t Y t, (12.4) where µY t and σY t denote the mean and the standard deviation of Y t Th e evaluation of these quantities requires a separate analysis for each of the four causes of decrement

12.2.1 Dismissal by the Employer

Let D

t

N be the random number of dismissals in year t, and let D D

, ~

the independent and identically distributed individual random lump sum

payments in year t given the jth dismissal by the employer occurs If A k is

the age of the employee number k at the initial time of valuation, then the expected number of dismissals in year t equals

=

1 PS PDk k

M

k

Consider the probability of dismissal of an employer in year t given a pop-ulation of M employees at initial time defi ned by the ratio

λ

t

p

Since decrement by the cause of dismissal follows a binomial distribution with parameter D

t

p , the variance of the number of dismissals is given by

t

Furthermore, suppose that at the initial time of valuation, it is known

that by dismissal the kth employee w ill receive t he lump su m payment

B 0,k Since the lump sum payment is wage based and the wages increase at

the rate of 100 · g%, the eff ective lump sum payment in year t equals B 0,k

(1 + g) t−1 Under t he a ssumption of a co mpound d istributed m odel for the aggregate lump sum payments due to dismissal by the employer, that

,

t

N

X =∑ = Y t= … H, it follows t hat t he mean a nd t he va riance

of D

t

Y are given by

λ

D

D

1

1

t

t

(12.8)

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

=

−

=

∑

∑

1

1

M

t

k

M

t

k

(12.9)

12.2.2 Resignation by the Employee

Th e e valuation i s similar to t he situation of d ismissal by t he employer, with the d iff erence t hat t he foreseen lump su m payment i s r eleased to the remaining benefi ciaries of the dismissal fund Let R

t

N be the random number of resignations in year t, and let R R

, ~

Y Y be the independent and

identically d istributed i ndividual r andom l ump su m pa yments i n y ear

t given the kth resignation by the employee occurs Again we assume a

compound distributed model for the aggregate lump sum payments due

to r esignation b y t he em ployee, t hat i s, R =∑ =R R = …

,

t N

expected number of resignations in year t equals

=

1 PS PRk k

M

Th e probability of resignation of an employer in year t given a population

of M employees at the initial time is defi ned by the ratio

λ

t

p

Since decrement by the cause of resignation follows a binomial distribution with parameter R

t

p , the variance of the number of resignations is given by

t

Th e mean and the variance of R

t

Y are given by

R

R

R

1

1

t

t

λ

λ

(12.13)

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

−

=

−

=

∑

∑

1

1

M

t

k

M

t

k

12.2.3 Death of the Employee

Suppose that by death of an employee the portion θ of the dismissal pay-ment is due to its legal survivor ( θ = 1/2 in our numerical example) Let

T

t

N be the random number of deaths in year t, and let T T

, ~

Y Y be t he

independent a nd i dentically d istributed i ndividual r andom l ump su m

payments in year t given the jth death occurs We assume a co mpound

distributed model for the aggregate lump sum payments due to the death

of an employee, that is, T =∑ =T T = …

,

t N

of deaths in year t equals

=

1 PS PTk k

M

Th e probability of the death of an employer in year t given a population of

M employees at initial time is defi ned by the ratio

λ

t

p

Since decrement by the cause of death follows a binomial distribution with parameter T

t

p , the variance of the number of deaths is given by

t

Th e mean and the variance of T

t

Y are given by

λ

T

T

1

1

t

t

(12.18)

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

1

2

1

M

t

k

M

t

k

−

=

−

=

∑

12.2.4 Survival to the Retirement Age

Let S

t

N be the random number of retirements in year t, and let S S

, ~

independent and identically distributed individual random lump sum

pay-ments generated upon retirement of the jth employee in year t Taking into account that an employee numbered k and aged A k attains retirement in year

t such that A k + t − 1 = s and using the defi nition of the retirement probability

x =s x− x , one obtains for the expected number of retirements in year t

−

=

1

M

where I(·) is an indicator function such that I(W) = 1 if the statement W is true and I(W) = 0 else Th e probability of survival to the retirement age of

an employer in year t given a population of M employees at initial time is

defi ned by the ratio

λ

= S

t

p

Since decrement by the cause of survival to retirement follows a binomial distribution with parameter S

t

p , the variance of the number of retirements

is given by

t

N

M

Furthermore, suppose that at the initial time of valuation, it is known that

at re tirement t he kth employee w ill receive t he lump su m payment C k

Due to wages increase, the eff ective sum in year t equals C k (1 + g) t−1 Again, assume a compound distributed model for the aggregate lump sum pay-ments due to retirement, that is, S S S

,

t N

X =∑ =Y t= … ThH e mean and the variance of S

t

Y are given by

S

S

1

1

t

t

λ

λ

(12.23)

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

−

−

=

−

−

=

∑

∑

1 1

1 1

k

k

M

t

k

M

t

k

(12.24)

Th e above preliminaries a re u sed to obtain t he cha racteristics (12.4) a s

follows Th e expected number of employee withdrawals in year t due to all

four causes of decrement equals

λ = λ + λ + λ + λt Dt Rt tT St (12.25)

Denote by M t 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 M t = M t−1 – λt, which shows that the expected number of remaining employees decreases over t ime, as should be Th e i ndividual lump su m

payment in year t satisfi es the following equation:

λ ⋅ = λ ⋅t Y t tD Y tD− λ ⋅Rt Y tR+ λ ⋅Tt Y tT+ λ ⋅St Y (1 tS 2.26)

Indeed, t he aggregate lump sum payments in year t are t he sum of t he

payments due to dismissal by the employee, death, and retirement less the payments due to the resignation of employees Under the assumption of independence of the diff erent random variables, one obtains for the mean

and the variance of Y t the formulas

1

Var[ ] ( ) ( ) ( ) ( )

t

t

t

E Y Y

λ

(12.27)

Moreover, the variance of the number of withdrawals in year t due to all

four causes of decrement is given by

D R T S

2t Var[ ] 2t 2t 2t 2t

Th e cha racteristics (12.4) follow i mmediately by i nserting t he formulas (12.25), (12.27), and (12.28)

DISMISSAL FUNDING

Let W t be the random wealth of the dismissal fund at time t, where t = 0 is the initial time of valuation Th e random rate of return on investment in year t is denoted I t Th e wealth at time t satisfi es the following recursive equation:

−

=( 1+ − ) (1⋅ + )

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

0

t

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

+ =

where Z t is normally distributed with mean µ and standard deviation σ Consider the products

,

H

=

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

[t−1,H], wh ere t he su ms Z t H, =∑H j t= Z j a re n ormally d istributed w ith mean and standard deviation

(12.33)

Th e mean and the variance of the fi nal wealth are given by the following result

THEOREM 12.1

Under the simplifying assumption that the random rates of return I1,…, I H are independent from the aggregate lump sum payments X t, the mean of the fi nal wealth is given by the expression

− +

=

− +

0

1

(1 )

t

and the variance of the fi nal wealth by the formula

( )

2

2 2 2 1 1 ( 1)

1

2 1 1 2 1 2 1 1

1

1

t

H

t H

t

H t

s t H

σ − + − − + σ

=

− + − − + σ − + σ

=

− + σ

− − + − −

≤ < ≤

⎡ ⎤

⎣ ⎦

∑

∑

∑

(12.35)

where = µ + σ1 2

2

r 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

rewrit-ten as

−

=

1

t

from which one gets without diffi culty (12.34) To get the expression for the variance, several terms must be calculated One has

H