Full text of DOI:10.1007/s001860000070

Math Meth Oper Res ꢀ2000) 52:213±220
2000
999
Pension funds insurance individuation
Charles S. Tapiero, Assa Birati
ESSEC, Avenue Bernard Hirsch, B.P. 105, 95021 Cergy-Pontoise Cedex, France
ꢀe-mail: poquet@edu.essec.fr) and Bar Ilan University, Israel
Manuscript received: January 2000/Final version received: March 2000
Abstract. This paper considers an individual's problem and e¨ects on savings
for retirement. We show using a deterministicmodel, a linear utility function
and assuming an individual can bene®t from a tax break in savings, that under
reasonable circumstances, there is only one switch from not saving to starting
to save. Further, we draw some implications for the e¨ects of income in-
equalities expressed by individuals tax brackets on inequality after retirement.
Due to the model's simplicity, further research is required which will consider
the uncertain e¨ects of income and the future on current savings decisions.
Key words: Insurance, Savings, Control, Pensions
1 Introduction
Pension savers tend to underestimate the ®nancial requirements to support
future needs. This may lead to future economic distress as well as to a growing
social condition where governments signal their inability to meet future needs.
There are further darker clouds pending over the future of pension funds and
the need to overhaul past practices. This has a particularly important e¨ect on
persons that do not have su½cient wealth to support their future needs and
rely mostly on withdrawals made by employers and payments made to social
security. This state of a¨airs has led the search to new approaches in saving
rates and saving modes. Explicitly, the traditional pension saving approach
consisting in investment in ®xed income obligations, generally issued and
backed by the government is now amended to provide greater investment
¯exibility. This ¯exibility, it is believed, may be much more responsive to
market forces on the one hand and better tailored to individual savers needs
on the other.
Individuation of pension savings is applied in a number of countries in
several forms. In France for example, some pension funds allocate ``points to

214
C. S. Tapiero, A. Birati
savers'' according to their contribution, the value of these points however is
recalculated each year based on the aggregate performance of the fund. In the
US, TIAA and CREF retirement funds allow savers to select part of their re-
tirement income in terms of several investment vehicles, each exhibiting dif-
ferent risk pro®les. Such decisions may be binding over certain periods of
time. The current trend in the US is to allow greater ¯exibility in funds in-
vestments by diverting some pre-speci®ed proportion of the fund to the stock
market. In Israel, as well as in many other countries, the insu½ciency of ®xed
income retirement plans guaranteed by the government are increasingly com-
pensated by the creation of complementary pension savings that bene®t of tax
advantages ꢀsuch as tax de¨eral schemes and outright tax deductions for parts
of the investment). Although these schemes are increasingly popular and
varied, they share a common trend of ``disintermediation'' of pension savings,
reducing both the responsibility and the social role of governments in the
guarantee of pensions and increased ``individuation'' of pension schemes ±
much more tuned to individual possibilities and needs. These trends have a
major impact on investment policies of pension funds, on their portfolio and
of course express a far more liberal view of social policy.
There are many reasons for governments' disassociation of social and re-
tirement pension funds and their active support to pension individuation.
Among the many possible reasons, early retirement, a structural growth in
unemployment, increased life expectancy, the in¯ow of more women in the
labor market, a trend towards lower interest rates and so on combine to in-
crease the cost of national pension funds insurance and their dwindling eco-
nomicopportunities. The rcedo for smaller government, privatization and
disengagement by governments, the growth of global ®nancial markets are
also reducing the needs for de®cit ®nancing through guaranteed payments to
pension funds which have been in the past an important source of current
budget de®cit ®nancing. This paper provides a systematic assessment of some
of these issues by considering a simple deterministic dynamic model. The re-
sults obtained provide the conditions for savings. Due to the model's simplic-
ity, further research is required which will consider the uncertain e¨ects of
income and the future on current savings decisions.
2A dynamic model of individual savings and retirement
Assume a saver whose tax bracket is known and given by t > 0. Let Sꢀt† be
the current account state of the saver at time t ꢀthis may stand for the number
of point accumulated by the saver, the outstanding amount of money saved
and accumulated over time etc.). Funds are accumulated by individual savers
in three ways: First, through an obligatory payment ꢀusually, paid for by the
employer and determined as a function of the saver's salary), aꢀw; t† where
wꢀt† is the individual salary at time t. Second, through a voluntary contribu-
tion vꢀt†, measured in dollars and contributed by the individual saver. This
contribution, has usually some tax advantages usually regulated by govern-
ments that seek to provide an incentive for retirement saving. For our pur-
pose, we state that the amount contributed at a given time is paid con-
tinuously and is proportional to the salary, or
0 U nꢀt† U nÃ
ꢀ1†

Pension funds insurance individuation
215
where 0 < nà U 1 is a parameter expressing the maximal amount a saver can
devote to a pension fund at a given time. Third, investment yield of the fund
generates an income that is reinvested. For our purpose, we shall assume that
the investment yield is known and given by a rate of return r. As a result, we
can represent the saving's process by the following di¨erential equation:
dSꢀt†
ˆ aꢀw; t† ‡ yꢀt†nꢀt†wꢀt† ‡ rSꢀt†; Sꢀ0† ˆ S0
ꢀ2†
dt
where yꢀt† is a parameter expressing the transaction cost associated to an in-
dividual contribution. Now assume that at time T, a retirement time, the saver
stops working-contributing, and collects a pension over his remaining life
time, which is a function of the accumulated fund at time T, SꢀT†. In partic-
ular, if 1 À Fꢀz† is the probability that the saver is still alive at time z ꢀafter
retirement), then the current value at retirement of the pension is given by:
ꢀy
aꢀSꢀT††‰1 À Fꢀz†ŠeÀrꢀzÀT† dz
ꢀ3†
T
„
y
which we write for simplicity by paꢀSꢀT††, p ˆ
‰1 À Fꢀz†ŠeÀrꢀzÀT† dz where
r is a riskless discount rate expressing the time vTalue of money for the saver.
The rate thus expresses the valuation of future consumption relative to the
present one. As a result, we can formulate an individual saver's objective by
the following:
ꢀ
T
Max VꢀT† ˆ eÀrT paꢀSꢀT†† ‡
Qꢀt†eÀrt dt
ꢀ4†
vꢀt† A _
0
where Qꢀt† denotes the utility of consumption of the individual saver. Let cꢀt†
be consumption, then if all disposable income is consumed, we have wꢀt† ˆ
cꢀt† ‡ ꢀ1 À t†vꢀt†wꢀt† and therefore,
Qꢀt† ˆ uꢀcꢀt†† ˆ uꢀwꢀt†ꢀ1 À ꢀ1 À t†vꢀt†††
ꢀ5†
where uꢀ:† is a utility function while ꢀ1 À t†vꢀt†wꢀt† is the individual's volun-
tary contribution net of taxes. The wealthier the individual the larger the tax-
ation rate. A solution of the optimal control problem de®ned by ꢀ1)±ꢀ4) pro-
vides initial results for discussing the individual's saving problem. We shall
summarize these results by the following proposition proved below. Further,
since there is no uncertainty, we consider a linear utility function.
Proposition 1: Assume a linear utility of consumption and for simplicity, assume
®xed wage rates, ®xed tax brackets.Then:
ꢀa) If r > r, it is not optimal to save.
ꢀb) If r > r, there is only one single saving switching time de®ned at time T Ã
and savings occur in bT Ã; Tc
Ã
Ã
ꢀ1 À t†
eÀrT À eÀrT ˆ keÀrT À eÀrT
;
k ˆ
> 1 ‡ eÀrT À eÀrT
;
yap
ꢀ6†
dT Ã
dT Ã
dT
dT Ã
> 0;
< 0 if r=r < T=T Ã and vice versa;
> 0
dr
dr

216
C. S. Tapiero, A. Birati
ꢀ1 À t†
ꢀc) If; k ˆ
< 1 ‡ eÀrT À eÀrT ; it is optimal to save all of the time.
yap
Proof: See Appendix 1.
The proposition sets out the conditions for retirement savings. Of course, if
tax brackets T Ã ꢀdeductions) are su½ciently large, life after retirement p and
the proportion of savings per period at retirement a are large, then it will pay
to save as long as possible and vice versa. Interestingly, the switching time is
independent of the amount saved at retirement time because of the risk neu-
trality ꢀlinear) assumption made here. Under uncertainty and risk aversion,
this will not be the case. We consider next the optimal retirement time T,
summarized by Proposition 2. The proof of this proposition is based of course
on the ®rst proposition. Further, we consider only the case k > 1 which is the
more interesting one.
Proposition 2: The optimal retirement time is found by the following:
ꢀa) If r < r and k U 1 then T à ˆ 0 as stated in the proposition and the
optimal retirement time is given by reaching the wealth state at time T given by:
‰ꢀꢀ1 À t†nà À apynà À 1Šw ‡ apaꢀw†
SꢀT† ˆ
ꢀ7†
ꢀr À r†ap
ꢀb) If r U r and k > 1 then T Ã < T and the optimal retirement time is given
by the solution of &equation &8)):
‰w ‡ apaꢀw† ‡ ꢀr À r†apSꢀT†Š
a: CꢀT† ˆ
w‰ꢀ1 À t† ‡ apyŠvÃ
Ã
CꢀT†kꢀr=r† À CꢀT†eÀꢀrÀr†T ˆ erT ꢀ1 À eÀꢀrÀr†T
†
Ã
Ã
b: eÀrT À eÀrT ˆ keÀrT À eÀrT
Proof:
Let the case r < r, and k > 1
ꢀ
ꢀ
T Ã
T
VꢀT† ˆ Sꢀ0†eÀrꢀ0†
‡
AeÀrt dt ‡
BeÀrt dt
0
T Ã
A ˆ ‰w À w‰ꢀ1 À t† À apyŠvà ‡ apaꢀw† ‡ ꢀr À r†apSŠ;
B ˆ ‰w ‡ apaꢀw† ‡ ꢀr À r†apSŠ
and therefore, the ®rst order necessary condition for the optimal retirement
dVꢀT†
time is found by
ˆ 0 leading to a solution of:
dT
dVꢀT†
dT Ã
dT
dT
dT Ã
dT
Ã
Ã
ˆ AeÀrT
‡ BeÀrT
dT
À BeÀrT
ˆ 0 or
dT
Ã
BꢀT Æ À AꢀT Æ ˆ w‰ꢀ1 À t† ‡ apyŠvÃ
BꢀT† ˆ ‰w ‡ apaꢀw† ‡ ꢀr À r†apSꢀT†Š
†
BꢀT†eÀrꢀTÀT
dT Ã
ˆ
with
BꢀT Æ À AꢀT Æ
dT

Pension funds insurance individuation
217
From proposition 1, we have however:
dT Ã
dT
eꢀrÀr†T À 1 > 0
Ã
ˆ
> 0 ! if kꢀr=r†eꢀrÀr†T > 1
Ã
Ã
kꢀr=r†eÀrT À eÀrT
and therefore,
Ã
†
BꢀT†eÀrꢀTÀT
eꢀrÀr†T À 1
ˆ
or
Ã
Ã
BꢀT Æ À AꢀT Æ kꢀr=r†eÀrT À eÀrT
‰w ‡ apaꢀw† ‡ ꢀr À r†apSꢀT†Š
w‰ꢀ1 À t† ‡ apyŠvÃ
erT À erT
CꢀT† ˆ
ˆ
Ã
kꢀr=r† À eÀꢀrÀr†T
Further manipulations lead to:
Ã
CꢀT†kꢀr=r† À CꢀT†eÀꢀrÀr†T ˆ erT ꢀ1 À eÀꢀrÀr†T
as stated in the proposition. Now set k < 1 and
†
ꢀ
VꢀT† ˆ Sꢀ0†eÀrꢀ0†
‡
T ꢀB À A†eÀrt dt
0
A ˆ ‰ꢀꢀ1 À t† À apy†nÃwŠ; B ˆ ‰w ‡ apaꢀw† ‡ ꢀr À r†apSŠ
dVꢀT†
and therefore,
ˆ 0 ! A ˆ B with
dT
‰ꢀꢀ1 À t†nà À apynà À 1Šw ‡ apaꢀw†
ꢀr À r†ap
SꢀT† ˆ
Q.E.D
The results obtained in the two propositions above can be studied further. We
consider ®rst the e¨ects of the tax Brackets. Note that from the optimality
condition we have that saving for retirement will increase when tax brackets
are increased. In other words, the richer a person, the more he will save for
retirement. In this sense, individuation of retirement will increase inequalities.
By the same token we can study the e¨ects of an extended life of savers by the
comparative static of parameter a. Other e¨ects can be studied as well, as this
will be the case in subsequent research. For example, so far we have assumed
a linear utility function of consumption. When we have a nonlinear utility of
consumption, we can expect greater savings when the saver is risk averse.
However this will depend on the discount rate used. Further, uncertainty re-
garding premium at retirement will also a¨ect the savings rate such that for a
risk averse person there may be an incentive to start saving earlier.
3 Conclusion
This paper has considered a pension saving insurance problem using an inter-
temporal framework. We have assumed a deterministicframework in order

218
C. S. Tapiero, A. Birati
to obtain tractable and yet revealing results regarding the propensity to save
for retirement. Introducing uncertainties in wage rates, funds rates of return,
retirement time and life, provide avenues for further study although these un-
certainties are likely to have few substantial e¨ects on the problem's conclu-
sion. The essential conclusions we have obtained include the condition for a
single switch. This means that generally, a saver will decide at some time prior
to retirement, to start saving. When such a decision is taken, this decision will
be maintained consistently till the retirement time. Because of the linear ob-
jective used, saving rates were found to be of the bang-bang type. Of course,
for a nonlinear utility of consumption, this result is likely to be amended. This
would be the case, if the utility of consumption were to be some convex
function in consumption, yielding thereby interior solutions.
The e¨ects of taxes were also considered important, re¯ecting savers wage
income. The richer a saver, the higher the tax rate and therefore, there are
advantages for saving to high wage earners. In other words, saving for retire-
ment will tend to maintain over time income inequality since a high wage earners
will save and later on collect greater bene®ts. Of course, in some countries,
such as in France, tax is deferred to the retirement time. At this time however,
chances are that the individual's tax rate will be smaller and thereby providing
another indirect bene®t to savers. If the tax rate at retirement equal t0, our
results are then still valid, if we replace the parameter a by aꢀ1 À t0† which
measures the income received at retirement net of takes.
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Appendix 1: Proof of Proposition 1
The problem ꢀ1)±ꢀ4) has an objective we can write as follows ꢀwhere the time
index is dropped for convenience and rꢀ0† ˆ 0):

Pension funds insurance individuation
219
ꢁ
ꢂ
ꢀ
ꢀ
ꢀ
T
dS
Qꢀt†‡ap ÀrapS eÀrt dt
VꢀT† ˆ Sꢀ0†eÀrꢀ0†
ˆ Sꢀ0†eÀrꢀ0†
‡
‡
‡
dt
0
T
‰wꢀ1Àꢀ1Àt†v†‡ap‰aꢀw†‡yvw‡rSŠÀrapSŠeÀrt dt
‰wꢀ1Àꢀ1Àt†v‡apyv†‡apaꢀw†ÀꢀrÀr†apSŠeÀrt dt
0
T
ˆ Sꢀ0†eÀrꢀ0†
0
The Hamiltonian is given by:
HꢀS; v; z; t† ˆ ‰apaꢀw† ‡ wꢀ1 À ꢀ1 À t†n ‡ apyv† À ꢀr À r†apSŠeÀrt
‡ z‰aꢀw† ‡ yvw ‡ rSŠ
where zꢀt† is a Lagrange multiplier given by:
dzꢀt†
qH
qS
ˆ À
ˆ ꢀr À r†apeÀrt À rz; zꢀT† ˆ 0
dt
The solution of this di¨erential equation, an ordinary linear di¨erential
equation, is:
zꢀt† ˆ zꢀ0† ‡ apeÀrtb1 À eÀꢀrÀr†tc and therefore;
zꢀ0† ˆ ÀapeÀrT b1 À eÀꢀrÀr†T
c
Using the ®nal boundary,
zꢀt† ˆ apeÀrtbeÀꢀrÀr†t À eÀrT‡rt À 1 ‡ eÀrꢀTÀt†
c
The optimal control is thus:
ꢃ
nà if fꢀt† V 0
v ˆ
where
0
if fꢀt† < 0
fꢀt† ˆ ‰Àꢀ1 À t† ‡ apyŠeÀrt ‡ z‰yŠ
and therefore,
fꢀt†ert
ꢀ1 À t†
ˆ Àk ‡ ert‰eÀrt À eÀrT ‡ eÀrT Š; k ˆ
and
yap
yap
ꢃ
nà if ‰eÀrt À eÀrT ‡ eÀrT Š V keÀrt
v ˆ
0
if ‰eÀrt À eÀrT ‡ eÀrT Š < keÀrt
The condition for not saving initially is:

220
C. S. Tapiero, A. Birati
b1 À eÀrT ‡ eÀrT c < k
and for T large this is equivalent to: 1 < k and therefore, the smaller the tax
bracket the more we are likely not to save and vice versa, the larger the tax
bracket the more likely we are to save. In other words, the richer a person the
greater the propensity to save. Of course, if k < 1, we start by saving if:
‰1 À kŠ V eÀrT ꢀeÀꢀrÀr†T À 1†
which is always the case. Say that initially we do not save, then a switch occurs
if the switching function is increasing. That is if:
d
ꢀeÀrT À eÀrT À keÀrt ‡ eÀrt† > 0 or
dt
rkeÀrt À reÀrt > 0 ! ꢀrk=r† > eÀꢀrÀr†t
:
Thus if r > r we hÃave a single switch that occurs at the following time:
Ã
eÀrT À eÀrT ˆ keÀrT À eÀrT . If r < r then if initially we do not save, we
never switch thereafter to saving. In other words the necessary condition for
saving is r > r or the rate of return is greater than the discount rate. In this
case, by implicit di¨erentiation we have:
ÀrT Ã
ÀrꢀTÀT Ã
†
dT à ÀreÀrT ‡ kre
1 À ꢀ1=k†e
ˆ
!
!
> 0
Ã
Ã
Ã
1 À ꢀr=kr†eÀꢀrÀr†T
dr
kreÀrT À reÀrT
Ã
Ã
dT Ã
reÀrT À reÀrT
eÀrT‡rT À 1
ˆ
Ã
Ã
Ã
ꢀkr=r†eÀꢀrÀr†T À 1
dr
kreÀrT À reÀrT
Ã
Ã
eÀrT‡rT > 1 ! r=r > T=T Ã; ꢀkr=r†eꢀrÀr†T > 1
dT Ã
dT
eꢀrÀr†T À 1 > 0
Ã
ˆ
> 0 ! if kꢀr=r†eꢀrÀr†T > 1
Ã
Ã
kꢀr=r†eÀrT eÀrT À eÀrT eÀrT
and the earlier we start saving, the earlier we retire and vice versa, the later we
start saving the later we retire as indicated by the proposition. Further, the
larger the discount rate the greater the propensity not to save while the larger
the rate of return, the larger the propensity to save.