A model for determining early retirement incentives

Insurance:

Mathematics

and

Econnmir-\

117

6 (19X7) 117-127

North-Ho!!and

A model fordeter early retirement incentives

It is becoming common for an employer to set up an early retirement incentive plan (ERIP). ’ A lump sum coupled with an immediate pension is offered to employees who are within a few years of retirement. One motive for the establishment of the ERIt’ is to reduce the number of employees: another is to improve career prospects for junior employees. In cases where service since hire is the main determinant of salary. a primary motive mav be the desire to reduce payroll while maintaining employee numbers. For example. school systems tend to have a fairly rigid service-dependent salary grid. Hence the replacement of a senior teacher with a recently hired teacher can result in salary savings. and Indeed school boards have been active in setting up early retirement incentive plans. The practical problem is to determine the maximum lump sum early retirement incentive which

can be offered if long-term savings to the employer are still to arise from the early retirement. This paper considers the problem in a situation where the salary structure can be approximated as being dependent only on service. Several papers consider the early retirement decision from the employee’s rather than the employer’s point of view. Clark and McDermed f 1982) consider the employee’s early retirement decision through analysis of the discounted value of the future stream of employment earnings or pension payments and give particular consideration to the effect of inflation. Burkhauser (1979) discusses the decision through consideration of a lifetime utility function and analyses data to determine the impact of variables such as the employee’s marital status 2nd accumulated assets. Wolfe (1983) gives another discussion of the employee’s retirement decision i;ith consideration of the effect of the perception of the employee’s future longevity on the discounted value of Social Security benefit streams. In a subsequent paper Wolfe (1985) considers this decision from the viewpoint of the maximization of lifetime ~.~ti!itv and consideration of health as a depreciating asset. The effect of alternative employment prospects on the discounted value of future earnings or pension payments is considered by Baltz (1984). Parnes (1983 j discusses the results of surveys aiming to determine the reTson for an employee’s decision to retire early. with particular emphasis on the effect of health. None of the above papers. however. includes the employer’s cost perspective on early retirement. Meyer and Fox (I971) give useful background data on the pension benefits available on earl! retirement under private sector plans. ’ Hansen and Holden (1985) model the development of the

’ See

’ Another

Keith P. SHARP Lftlrr-ersilJ of Wurerioo.

Wuterloo.

Received

1985

Revised

13 September 29 October

The salary cmplovce are tions

Results

the case

entrant

rates

arc

and a numerical Ker.~w(I);

Early

rctircment paid

two

special

casts. termina-

is considered

the prohabilii\

is small

of an

empln~cc

pre-retirement

the cast that

be of assistance example

for

where

so high

retirement

should

the early

presented

Sccondlv.

are

reaching

The results

from

bv a less highiv

is considered

can be neglected.

termination

I..

resulting

his replacement

analysed.

Firstly.

!V.?L 3GI

1986

savings

and

Otlr., Cmudu

enough

Lvhere

of a nt\l

to he neglected.

in solving

actual

prohkr-1s.

ia given.

retirement.

Replaccmcnt

cmplo!cc.

Introduction

P/m

for

instance

Rwiew.

10. in Busitzess Nuriottul May

anonymous

February Itwmm-e,

Utzdenvriter.

8. 1982.

0167-66X7/X7/$3.50

p. 4

and

articles

198i.

June

(Life

p.

p.

in

12 and

Ettytlqrve

13. 1983.

utld Healrh

L&we/it

December

1983.

p. 19 and Itlswat~ce

in The Ediiiott ).

14.

’ 1987. Elsevier

Science

p.

Puhlishcrs

reference

is Rnrremerf

Trettd itt rbz Prrrxte

Secror.

GAO,/HRD-X5-81.

July

sector

eariy

design

of retirement

R.V. (Snrth-Hnlland)

retirement

15.

f~$vx~ .4,qe 6.i I.\ u Gr~wrr:g

CS General 1985

which

as a relevant systems

for federal

Accounting

Office.

discusses

private

consideraticn employees.

in the

;*;nn of university faculty. They inag,c composlL... clude 30-year forecasts of salary costs under scenarios where the mandatory retirement age is changed and where an early retirement incentive program is put into effect. The calculation of the employee’s salary savings resulting from a single early retirement is not an aim of their paper. Thus Hansen and Holden do not consider the long-term cost effects of an early reurement which are the focus of the current paper. In this paper consideration is given to the calculation of the lump sum incentive which can economically be offered under an ERIP. The emphasis is on the calculation of the lump sum value of the future salary savings to the employer rather than on the caiculation of the financial effects on the empioyer’s pension plan. The permanent effect on the employer of an early retirement is influenced by the following factors. (a) Cost due to payment of lump sum early retirement incentive. (b) Savings in the short term due to replacement’s lower salary. (c) Cost in the longer term due to replacement’s climb on service-related salary scale. The replacement will already have accrued service by the time the original employee would have retired without the incentive and would have been replaced by a zero service employee. (d) Cost due to possible extra burden on pension plan. (e) Cost or saving due to possible different level of skill or productivity of replacement. There may be self selection by those offered the ERIP incentive which could result in the average level of skill of those accepting !he incentive differing from that of those rejecting the incentive. (f) Cost or savings due to decline or improvement in staff morale. The aim is to set tne lump sum incentive (a) at a level such that the overall effect of fac:ors (a)-(f) is to produce a saving to the employer. Factor (d) can be analysed through standard techniques of pension valuation [see e.g.. Anderson (198.5, Section 4.5)] and the lump sum equivalent of the extra cost to the pension plan would be subtracted from the salary savings lump sum equivalent calculated according to the models presented in this paper. Factor (e) may be difficult to quantify, and the result will depend on the extent to which it is reasonable to believe that the

salary paid is related to the employee’s value to the employer. Factor (f) is likely to be the most difficult to quantify. In this paper. factors (b) and (c) are analysed. In this paper the cost savings from an induced early retirement in a situation where salary depends only on service are found by comparing total costs with and without an early retirement. Thus we compare salary costs under two scenarios. only one of which will actually occur: (i) a scenario in which the ERIP induces an initial early retirement. but then ceases to operate. so that future retirements are at the usual age; (ii) a scenario in which no early retirement is induced , and a!! retirements. including the first, are at the usual age. The result will be to derive a figure for the salary cost saving resulting from a single induced early retirement. Thi; cost is useful to the employer since it determines the maximum lump sum incentive which should be offered. In the analysis, employee termination rates (resignations. dismissals, deaths and disablements) are of great importance. An analytic model appropriate to the case of a general termination scale has not yet been developed. In Section 2 is presented a model for the special case of zero termination rates and a general service-dependent salary scale. In Section 3 this model is specialised to the case of a linear scale. In Section 4. a model is presented for high terminaticn rates and a general salary scale. and in Section 5 this model is specialised to a linear salary scale. Section 6 presents a practical example which is based on the consulting assignment from which this paper evolved. A brief summary and a comment on future research are given in Section 7.

2. Zero terminations

and general

salary scalp

The situation is considered where an early retirement is prompted by an ERIP lump sum offer. It is assumed for this section and for Section 3 that terminations do not occur, and that after the first induced early retirement, subsequent retirements by the replacements are at the usual age. It is emphasized that the ‘usual age’ may not be the normal retirement age defined in the pension plan document.

T:12 actuarial assumptions Which remain to be determined are those concerning the interest rate at which future payments should be discounted. the rate of increase in overall salary ieveis and the service-related salary grid of the employer. In this paper the rate of increase in overall salary levels is taken, without loss of generality, to be zero. Constant dollars, that is current dollars deflated by the salary inflation rate, are used throughout. Then the real (after national average salary inflation) interest rate is set relative to the salary inflation rate. The real rate used will be an estimate of the average real rate which will prevail for the next few decades. ’ The actuary will make an assumption about the real rate appropriate to the case under consideration. probably after consideration of tables of economic statistics such as those of lbbotson and Sinquefield (1982). An additional advantage of the real rate is that it is more stable than the rates of interest and salary inflation. An assumption about the real rate of interest is routinely made by actuaries when valuing pension plans in which the benefits depend on the final salary. In this paper no judgement is made about the appropriate level to assume for the real interest rate, and consideration is give to positive. zero and negative values of the rate.

of R(r)

= S(J(Z))

-.5(x-(r)).

If an early retirement is induced e years earlier than retirement would otherwise have occurred. and if subsequent retirements under both scenarios occur consistently p years after hire and If S( -_) is an increasing function of ;. R(t)20.

o
R(i)lO,

c-
2.2. Arlalvsis to use the foliowing notation: time since the ERIP-ir,duced r~rly retiret: ment under scenario (i), s( t ): service (i.e.. year since hire) of empioyee at time t under scenario (i), _F(t ): service of employee at time t under scenario (ii). S(Z): salary rate for person with length of service _. The salary scale S(Z) takes account of length of service (and hence. for an individual. age also) and average promorion and merit rises, but not of overall saIa;y inflation. Salaries are assumed to be paid continuously. An induced early retirement at time 0 results in a cash flow rate of saving (see Figure 1) at time t ’ . Real rate’ usually

refers

rate5 and price inflation. wed

to refer

national

to the differcncc

to the difference

average salary

between

In this paper. the term ‘real inflation.

hetwccn

intcrcst

intcrc\t rate’ is

ratcb and

(2.2)

Expression (2.2) follows becauss for 0 < f # e. the seniority under scenario (ii) is greater but for e < I # p the seniority under scenario (i) is greater. The function R(t) is periodic with a period of p years. as can be seen by considering the behaviour when the replacements for the original retirees at times 0 [scenario (i j] and e [scenario (ii)] themselves retire at times p and p + e. respectively. Furthermore. the years of positive saving are exactly balanced by the years of negative saving since the functions S(s(r)) and S( J( f)) are themselves periodic with period p. Therefore for any f,, we have

-

It is convenient

(2.1 i

‘“-‘S(_X-(I)) I f#,

dt=O.

(2.3)

In considering the maximum early retirement lump sum to offer. it is necessary to consider at time 0 the dicro!tnterl vallle of the cash flow rate of saving. taking into account factors (b) and (c) of Section 1. If the period 0 to t is considered. the discounted value of the savings is A(r,

i) =

'R(u)?

J0

du.

(2.4)

where i is the real rate of interest and (7= l/(1

+

i).

The value of A( 1. i) depends on the value chosen for t. the ‘final accounting date’. Indeed. as can be seen from Figure 1. an apparently minor change in t (e.g.. from 2p to e + Zp) can result in a large change in A( t. i). The savings resulting from a single ERIP-induced early retirement last for an indefinite period of time. Therefore it is appropriate to use as the discounted value lim, _ r A( t. i ). The use of this infinite time horizon might be criticised as being unrealistic. However. as is demonstrated in Appendices A and B it can be

interpreted for i 2 0 as the expectation of A(t, i) where the expectation is taken over the possible v-,;ues of the final accounting date. t. which is assumed to have a very diffuse distribution. The use of an infinite time horizon does not. of course. correspond well with the use of any finite planning horizon by the employer. and it will be seen that it leads to some interesting results. However. it would appear to be difficult to justify on theorpt;rrt nrfi,lnrlc ths= snv oo&n finic~ !i~_p *___ ,ICP _.,_ nf . __.v_. D’_-_.-.~ -- ---, horizon because its use would result in the neglect of some real long-term c”fects, especiaiiy in the case of a real interest rate of zero or close to zero. lim , _ r. A( I. i) exists only for i 3 0. The cases i > 0. i = 0 and i < 0 are considered separately below. Result for i > 0 If the actuary assumes effective rate of interest, tively A(%,

a strictly positive real i, the analysis is rela-

counted value of a perpetuity due of k( p. i) paid once every p years. The formula is readily capable of evaluation in practical situations. especially in the case of the linear salary scale considered in Section 3. The interpretation of the final accounting date t as having a diffuse distribution is given in Appendix A.

Resuh jar i = 0 Thn on,....--. iii&j; d&k . ‘S., “b’UU‘J

iu USCil ~cru reai (i.e.. after salary inf!ation) interest rate. The function Air. 0) has a periodic behaviour as illustrated in Figure 1, and hm, _ 3cA(r. 0) does not exist. in a practical situation, some value of the lump sum incentive must be derived. and this value depends on a chosen definition of A( x, 0). As demonstrated in Appendix B equivalent definitions for A(%. 0) are

A(%,

0) = ,li-~+ ,limL A(r, i)

straightforward: i) =

lim A(t, I -

=(i Equation

%

-r”jj’A(p. (2.5)

can

ij =

rR(~l)c~”

du

=+~JpA(u,O)

du-2

(2.6)

f 0

i)

(i>(I).

be interpreted

(2.5) as the dis-

where z is defined as the mean value of A( t. 0) over one period of length p years. This result is intuitively reasonable in view of the interpretation given in Appendix B of a random but diffuse distribution of the final accounting date t.

A’. I’. .s:w,7 ,/’ I.~url,

For

a negative

rL’,,,-L’,,I<‘I,I

C’,,,,

,,,!

1’1

I “\

produced bvi the induced early retirement is zero. This. hovvever. is not the case. Graph (4) illustrates

real interest rate. the hmit of (2.4) as t --$ -K is not defined. Corresponding!y. the series summed in deriving (A.2) cannot be summed for i < 0. intuitively this is because A( -x;.

discount rate and a linear salary scale. A(I. 0) can be interpreted as the total salary savings accu-

i) corresponds to the discounted value of a perpetuity due. where payments of ia( p. i) are made

mulated at zero interest to time f. The linear salary scale is given by

at the start of every p years. For i) -C 0. and the value of the perpetuity tive interest

rate

is (negatively)

der the assumption

of zero

negative

real

interest

rates

incentive

plan

is feasible.

infinjte.

termination no

early

i < 0. A( p. at a nega-

the

behaviour

o<:+m. 1711z I p.

un-

S(z)

rates

and

where Scale

=H.

terminations

and linear salary scale

As an illustration of the application of the formulas of Section 2. the case of a linear salary scale is considered and the functions S( s( r)). S( J(I)), R(t) and A(t, 0) are graphed. It is emphasised that any salary scale can be handled by the methods of Section 2. Graphs (1) and (2) of Figure 1 show the salary outgo S( x( I)) and S(_r( t )) under the assumptions that retirement is or is not taken early. The service-dependent salary grid used in Figure 1 is linear for the first m years of employment and then flat until retirement. It is common in the Canadian public sector for a salary grid to exist whereby salary depends on service alone and in many cases salary varies linearly with service. In the jurisdiction relevant to the example given in Section 6. the salartes of schoolteachers are determined from a grid and depend on the level of academic qualification of the teacher and on the teacher’s length of service. Every year all the salaries in the grid are increased to an extent determined largely by the increase in average salaries in the community at large. In addition. each teacher moves to a higher level on the grid. consistent with the teacher’s increased period of service. Graph (3) illustrates the cash flow savings R( t ) produced by an early retirement. The negative values are a result of effect (c) described in Section 1. The years of negative cash flow will. unless we introduce a non-zero discount rate. give a negative savings precisely sufficient to offset the salary savings produced in the first c years after an induced early retirement. At first sight it may appear from Graph (3) that the overall saving

of a zero

(3.la)

C = ( H - L )/ttz. (3.la)

We define

assumes

continuous

also scale (3.lb)

are assumed 3. Zero

0) for the case

S(:)=L+X-.

Thus

retirement

of A(r.

to be given

salary

where

increases.

salary

at annual

increases

intervals:

S*(:)=L+int(:)C.

Os:sm.

S*(z)=H.

tnI I 5 p.

(3.lb)

where C = ( H - L)/ttz. Int(:) = integer part of :. We will use an asterix to denote functions derived using the alternative salary scale (3.lb). The function A([. 0) is the appropriate maximum to the lump sum early retirement incentive which should be offered if there is to be an overall sav-ing to the employer. Hovvever. t corresponds to a ‘final accounting date’. It can be seen from Graph (4) that as t increases. A( 1. 0) alternates between being positive and being zero. As indicated in Section 2. the limit does not e-cist. At a positive real rate of interest i. the present value at time zero of the salary savings is A( -x. i ). given by (2.5). For the speciai case of the continuous linear salary scale (3 1 a) vve have. provided that the plateau of the salary before early retirement. ’ A(rc.

i) = (1 - rJ)‘A(

scale

is reached

p. i)

=(l-r~“)~‘JP(S(?.(~))-~(.~(~)))(.’di 0 = (1 -I*~))‘[

Hii,, -C(%j,&-

HZ,

K. P. Sharp / Ear&~reriremenr iwerltir7e.r

122

A*(m,

i) =

z

Gpx?!~.

(32b)

If the real interest rate / = 0, it is shown in Appendix C that (2.6) gives the same result as does substituting i = 0 into (3.2a): em?

‘4(m, 0) = -

2p

(3.3a)

.

0) =

em(n2 +

1)C

2P

.

The result for 0 I t I e During the period 0 I t I e, under scenario the ERIP offer has been rejected and the cumbent continues to work until I = e, when retires at the usual age. Thus we assume scenario

Hence the expectation after time t is E[X(t)IX(O)=O]

(4.1)

x = t.

3

=

of X(t)

(1 -

for integer

t just

s,[l - (1 - 4’1 4

The discounted value at time 0 of the saving over the period 0 I I I e is a random variable. Consistent with definition (2.4) define

We now consider the situation where termination rates are high. In particular we assume that termination rates are so high that the probability a new entrant will remain in service until retirement is neglible. Thus the only retirement will be that of the original senior employee. We assume for the purpose of maintaining tractability that terminations are at the end of each year, and that all new entrants commence employment at the same point on the salary grid. The termination rate is taken as q independent of age and service. This is not a severe restriction. For reasonably high q, we are considering mainly younger employees with low service. It is necessary to consider two periods. First we consider the period 0 I t I e, and secondly we consider the period t > e.

up,&

osxst-1.

(4.2)

4. I. Introduction

Cl”ll.3

\

(3.3b)

4. High termination rate and general salary scale

terminat;nnc L-1 . ..‘I‘

P(X(t)=xlX(O)=O)

=(i~wai‘~.

where (3.3a) has the same interpretation as (2.6). The interpretation of (3.3a) as the average value of A(t. 0) can be intuitively verified by considering the average accumulation of the difference between Graph (1) and Graph (2). For the case of annual increases. the corresponding result from (2.6) or (3.2b) is A*(m,

service X(0) = 0 where X(t) is a randoin variable representing the service under scenario (i). It is possible that the young new entrant will be replaced several times during the period 0 I t I e. The probability distribution for positive integer t of X(t), the service just after time t, is given by

(ii)

d-tiring

this

A(e,

i)=ju”[s(y(t))-S(X(t))]z~‘di

We require the expectation of A(e, i) which in general is difficult to obtain. A useful special case is given in Section 5. The result for t > e The person in scenario (ii) who rejected the ERIP incentive retires at time t = e. After that date, under scenarios (i) and (ii) there are terminations at random intervals. -We represent the random variables for the lengths of service just after integer time ! by X(t) and Y( t ). respectively. The expected discounted value at integer time t, of the future salary savings conditional on the time r0 lengths of service is EV( xg. JrJ- i)

xX(t,)

(ii) inhe no

period.

Under scenario (i) the offer has been accepted and a young new entrant has started work with initial

(4.3)

=x0,

Y(t,)

=_v(J

.

(4.4)

1

The expectation is taken over all future possible paths for the le rgths of service X( t ) and Y( t ) for t 2 t,. The function El/(x,, _vo, i) is a deterministic rather than a stochastic function. Further anaiysis requires that a specific salary scale be chosen and this is done in Section 5.

5. i;igir iwwinztion

rate and linear salary scale

For the annual salary scale (3.1 b) the corresponding expression is

5. I. Assurnptimw EA*(e. We assume that the termination rate q is high enough that new entrants have a negligible probability of reaching the plateau of the salary scales (3.la) and (3.lb) and the result (5.3) is corresponding derived assuming that the salary scale increases without limit. To the extent that y will actually be less than 1. this assumption introduces an approximation into the deriv.ation. The high termination rate leads to a short expected period of service. so a linear salary scale should be a good approximation to the relevant region of a non-hnear salary scale. The result for 0 < t < e The assumption of a high termination rate applies only to new entrants. It is assumed. consistent with the two-scenario model, that the existing employee ha ving refused the eariy retirement incentive does not terminate until his re:rrement e years later. Therefore his length of service does not vary stochastically. and his salary is given by H. Then using the salary scale (3.la). (4.3) becomes A(e.

i)=~~[H-L-CX(u)]~~~dli.

(5.1)

Defining the expectation of A(e. i) over all possible paths for X(U) by EA(e.i) and using (4.2) for integer t we have

(1-q) q

i)=(H-E)Z,--CIIr;p 1 -

..

ffc-

(1

-

q)(‘l.?,

l-(l-q)r.

1

(5.2b)

.

The result for t :>e For the salary scales (3.la) and (3.1 b). Ci

E t’( -x0, J(, 9 i)=t?;,-X(J)

all

(5.3)

+ 4)

where 6 = ln(l + i) is the force of interest. A derivation is given in Appendix D. For i = 0 the derivation can be modified to give EV(x,,

J,.O)=(_Q,-x,,)$.

(5.4)

This has an interpretation as the product of the initial salary difference (.r-i,- ,xo)C and the expected time to the first termination under a given scenario. l/q. The result for the whole period t > 0 To obtain the expectation EA(x. if for the entire future salary savings under the continuous salary scale (3.la). we may add the expectation (5.2a) to the expectation of (5.3) discounted to time 0 over all possible values of sO_ Here _Y”is taken as the length of service under scenario (i) just ai:cr time e i X( e ), m the notation of Section 41. We note that the corresponding length of service J just after time e under scenario (ii) is zero. Use (4.2) at t = 2 to obtain EA(x.

i) = EA(e.

i) +r”E[EV(X(e).

0. i)]

e-1

-CE

c

/‘(X(t)

+ u)P’+” du

t=O 0

I

= (H-L)Z,

x

ei* u’(1 -

q)[l

- (1 -q)‘] 4

r=O =(H_

(1-Q)

ij~,-~&il~,l_~~gq

1

ii_ _ 1 - (1 - q)Y’

x

<‘I [

1 -(l

_ Cc+‘(l -q)[l

-q)r

I

- (1 -q)“]

q(1 i-q/i)6

. (5.h)

x

ii,--[

1.

1 - (1 - q) c UC I-(1-q)u

(5.2a)

For the annually increasin, 0 salary scale (3.lh) vve have the corresponding expression

K. P. Sharp / Ear!), reriremrnr incet7ric~e.s

124

E_4*(x.

i)

= (U-

_

L)zi,

ccc

(l-4)

-CZi:q

(1 -

d[l - (1 - d’l 40 +4/G

. (5.5b)

Equations (5Sa) and (5.5b) simplify dramatically for real interest rate i = 0: EA(so,

0) = eC(m + l/2

- l/q).

EA*(y;.Oj=~C(nr+l-l/q).

(5.6a) (5.6b)

6. Practical example In a practical problem a school board sought advice on the lump sum early retirement incentive which should be offered to older teachers. The salary scale was approximated as linear with continuous increases, as illustrated in Figure 1, with parameters: Maximum salary rvlinimum salary

H = $40,688 per annum L = $22.734 per annum

Length of salary rise

m=ll

Rate of rise of salary scale

C=(H-L)jm=$1.632

Service at retirement

p =

that the induced retirements result in overall salary savings of $2.904 for each year by which retirement is earlier than it otherwise would be. If it were appropriate to assume a zero real rate of interest, a rate of termination q = 0.2, and retirement one year early then the lump sum equivalent of the long-term saving as given by (5.6a) is $10,608. Thus the assumed rate of termination has a major effect on the lump sum incentive which can be offered. 7. Summary From theoretical arguments. results have been presented which have practical value in deciding the appropriate amount of the lump sum early retirement incentive which should be offered. As a step towards the derivation of expressions of general applicability. the special cases of zero termination rates and of high termination rates were analysed. The results from the special cases will give guidance m the solution of the case of moderate terminations. Further work is under way to determine the extent to which a workforce can be regarded as a mix of zero termination and high termination rate employees.

years Acknowledgements 34 years

If it were appropriate to assume a zero rate of termination and zero real rate cf interest then the insertion of e = 1 into equation (3.3a) indicates

This work was performed while the author was in receipt of a gram from the Natural Sciences and Engineering Research Council of Canada. The author is grateful for helpful comments from both the editor. Stuart Klugman, and the referees.

Appendix A: IEffuse distribution for t wheu i > 0 Following from equation (2.5) we wish to make an alternative interpretation of the ERIP lump sum as being the expectation of A( t, i) where I has a diffuse distribution. A choice of diffuse distribution must be made. A natural choice is the limit of the exponential distributio n as the parameter p tends to zero. Noting first that for integer r and i > 0

(A.1)

we find for i > 0, using the notation

N(T) = Int(

N(T)-

r/p

j

for the integer part

1

exp( - pr) dt

= Iim lim p-0 r+r.

i-=0 N(T)-

i

c pexp(

-t

-grp)orp[~,P(f.

i)

exp(-pr)dt

r=O = lim

&P. i) lim r (1 - exp( -r-La)) l _ I/P

pm0 7-m

1

x ((l-exp(

-pNN(T)p)

_

1 - exp( --~fV(t)p)~"'~'~ +lJ

hhnE[A(t.

i)] =

-pp)op

P / I=0

A([.

exp(

--pp)uP

i) exp( -,uLt) dt

Ah 4 l-UP

where (A.2) corresponds

l-exp(

-pAr(T)p)u‘"T)p~ 1 -

1 -ev(-pp)

\

=

l-exp(

1 f

(A-2)

’

to (2.5). Hence for i > 0 we have A( P. if = lim A(i. 1 -UP I+ %

i).

(A.3)

And we have demonstrated the consistency of the interpretation of the discounted value of all future salary savings as being an expectation of the discounted value of future salary savings up to a random final A]^.___ _.._.. :-- ULILG. ULC”U“II‘I~

Appendix

B: Analysis

One method

of equation

(2.6)

of finding a value for A(%.

0) involves taking the limit of (2.5) as the force of interest rule to (2.5) and

i + 0 from above. We notice that from (2.3) and (2.4). A( p_ 0) = 0. We apply L’Hopital’s

integrate

by parts:

A( x), 0) = ,li~+ ,\;A(

=-

1

p

A(u,O) P I0

t. i) = ;li~+ (1 - P~)-‘A(

du-A.

where 2 is defined as the mean value of non-existence of !im , _ %A( t, O;, the double in the opposite order, except in the Cesaro double limit leads us to look for alternative

p_ i)

(B.1) A(t, 0) over one period of length p years. In view of the ‘nit at the start of fB.1) does not exist if the iimits are taken 11 (Cl) sense [see e.g.. Bartle (1964)]. The non-existence of the interpretations.

Using the exponential A(co, 0) =

distribution

jl;E[A(?,

O)]

for the final accounting

= liia”,mA(t,

ZXlim E ;tIPA(r.O) BdQ,,a 0

day t, as is done in Appendix

0)~ exp( -pr)

exp(-y(t+-:-p))

A, we have

dt

dt

= limp(l-exp(-pp)~~‘~~A(t,0)exp(-pr)dt=~~p~(~.O)di~~, 0 P-‘O

(B-2)

where z is defined as the mean value of A( t, 0) over one period of length p years. The fact that the double limit in (B.l) can be evaluated only for one of the two possible orders for the limit makes the case i = 0 mathematically interesting. For practical purposes, the problem is of little relevance since an actuary is unlikely 10 have a great preference for a real rate of interest i = 0 as opposed to a value of, say, i = 0.0000001. The convergence to the limit Ais well-behaved. and use of an extremely low interest rate will give a result close to x so for practical purposes Acan be taken as the required result at a zero interest rate.

Appendix

C: Derivation

of equation

(3.3a)

The derivation of equation (2.5) involved the summing of a geometric series which cannot be SI,mmed if the real force of interest i = 0. Therefore it is incorrect to derive equaticn (3.3a) by substituting i = 0 in:o equation (3.2a), which was derived from equation (2.5). We will derive equation (3.3a) from equation (2.6) for the case of a salary scale given by (3.la):

(3.3a)

Appendix

D: Derivation

of equation

(5.3)

Introducrion

Note that the salary difference between the two scenarios at a time f where t is non-integer. S( Y( r )) - S( X( t )), does not depend on which of salary scales (3.la) and (3.1 b) we use. Distribution

This mteger. history The

of X(t) conditional

on Xfi,,) = xc,

result is a generalisation of (4.1) where now we allow the initial value of X( t,,) to be any positive Note that one termination ‘between i, and P is sufficient for the process lo ‘forget’ totaiiy its up to the time of the termination including its value x,) at lo. conditional probability distribution for integer t. where t - t,, is positive. of X( t ). the length of

I;. P.

service under

scenario

(i) just

after

time

Sharp / /:aril t. is given

0 I x 5 t - t,, - 1.

I\ (1 -q)’

the conditional

expectation

E[X(t)IX(t,,)=x,]

=

of X(r)

“:.

is given

for integer

(I-4)[1-(I -4Yl

(D-1)

X = i - f,, + X,,.

t 2 I,, by

+x

(1

_4)‘-‘,,

0

4 Ecaluution of EV(x,,

127

:n<~l’nlll’l’\

by

/Y(l-d.

Prob(X(t)=xIX(t,)=s,,)=‘. Then

r“I,,-e~,lctlr

F,,, i)

in this derivation we assume that the linear rise in salary continues indefinitely, that is IH = x. This assumption should be viewed in conjunction with the assumption that q is large so only a small proportion of employees would reach the plateau of the salary scale. We use (D.?) for the conditional expectation of x(r) and of y(r) and we use the sa!ary scales (3.la) or (3.lb) and the definition (4.4) to give EV( Xg. lit. i)=CE

I~[S(Y(I))-S(.~(~))]~‘-“‘~~IX(I,,)=-~,,. [ - 11,

=q

=Cc

T’(E[S(Y(t)) ‘(1

/y4 11+i

=A,]

/“‘{E[S(Y(l)jlY(i,,)=~~,] f = ,,, ,

=“gC In order for

I Y(h)

2 (_vo-sJ(l f = r,,

-q)f+V’G*=

to
Y(t,,)=.ti,]I 1

- E[ S( X( t )) 1 X( I,,) = _q,] > L.’ ‘,, dr

-E[S(X(t))jX(r,,)=s,,]~~‘-“~“,du ( _v,,- x0 ) ci S(i+4) . be summable.

(5.3 is true if and

if (D.3

’

or specifically, i>max(-q. In particular,

q-2). the result

(D-4) holds

for all i 2 0.

References Anderson. A.W. (1985). Pension Marhemarics for Acruaries. Society of Actuaries. Chicago. IL. Baltz. R.B. (1984). An incentive early retirement model for college and university faculty. Journal of Risk and Insurance 51. no. i. 477-497. Bartie. R.G. (1964). Tile Elements of Reul Ana!rsis. Wiley. New York. Burkhauser. R.V. (1979). The pension acceptance decision of older workers. Jourtlal of Human Resoarces 14. 63-75. Clark. R.L. and A. McDermed (1982). Inflation. pension beneiits and retirement. Journal of Risk and Insurance 49. no. 1. 19-33. Hansen. W.L. and KC. Holden (1985). Critica! iinkages in higher education: Age composition and labor costs. insura,rce: Marhematics and Economics 4. 55-64.

Ibbotson.

(1982). Srocks. Bonds. Ihe Faurure. Financial Analysts Research Foundation. Charlottesville. VA. Keilison. S.G. (1970). 7Ae Tizco~v qflnterest. Irvin. Homewood. IL. Meyer. M. and H. Fox (1971). Eur!\, reriremerlr programs. Conference Board Report no. 532. New York. Parnes. H.S. (1983). Health. pension polic> and retirement. Aging und Work 6. no. 2. 93-104. Wolfe. J.R. (i983). Perceived longevity and early retirement. Bil/s

R.G. and R.A. Sinquefield

und

Inflorion:

The

The Recrer. of Economics

Pas:

and

and Srarrsrics 65. 544-551.

Wolfe. J.R. ( i685). A model of declining health and retirement. Journui of Polrr~col Ecouon~y 93. no. 6. 1259-1267.