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Citation: Gerrard, R. J. G., Haberman, S. and Vigna, E. (2004). Optimal investment
choices post-retirement in a defined contribution pension scheme. Insurance: Mathematics
and Economics, 35(2 SPEC), doi: 10.1016/j.insmatheco.2004.06.002
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Permanent repository link: https://openaccess.city.ac.uk/id/eprint/11900/
Link to published version: http://dx.doi.org/10.1016/j.insmatheco.2004.06.002
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City Research Online
Optimal Investment Choices Post Retirement in a Defined
Contribution Pension Scheme
Russell Gerrard
∗
Steven Haberman
†
Elena Vigna
‡
June 17, 2004
Abstract
In defined contribution pension schemes, the financial risk is b orne by the member. Financial
risk occurs both during the accumulation phase (investment risk) and at retirement, when the
annuity is bought (annuity risk). The annuity risk faced by the member can be reduced through
the “income drawdown option”: the retiree is allowed to choose when to convert the final capital
into pension within a certain period of time after retirement. In some countries, there is a
limiting age when annuitization becomes compulsory (in UK this age is 75). In the interim, the
member can withdraw periodic amounts of money to provide for daily life, within certain limits
imposed by the scheme’s rules (or by law).
In this paper, we investigate the income drawdown option and define a stochastic optimal
control problem, looking for optimal investment strategies to be adopted after retirement, when
allowing for periodic fixed withdrawals from the fund. The risk attitude of the member is also
considered, by changing a parameter in the disutility function chosen. We find that there is a
natural target level of the fund, interpretable as a safety level, which can never be exceeded when
optimal control is used.
Numerical examples are presented in order to analyse various indices — relevant to the pen-
sioner — when the optimal investment allocation is adopted. These indices include, for example,
the risk of outliving the assets before annuitization occurs (risk of ruin), the average time of
ruin, the probability of reaching a certain pension target (that is greater than or equal to the
pension that the member could buy immediately on retirement), the final outcome that can be
reached (distribution of annuity that can be bought at limit age), and how the risk attitude of
the member affects the key performance measures mentioned above.
Keywords: income drawdown option, stochastic optimal control, decumulation phase, imme-
diate annuitization.
JEL classification: C61, G11, J26.
We gratefully acknowledge the interesting observations made by the participants of the 7
th
IME
Congress, Lyon, and of the 13
th
AFIR Colloquium, Maastricht. A special thanks goes to Andrew
Cairns, Bjarne Højgaard and an anonymous referee, for their valuable and useful comments.
This research was funded by a research grant provided by the Institute and Faculty of Actuaries
(UK).
∗
Faculty of Actuarial Science and Statistics, Cass Business School, City University, London. Email:
r.j.gerrard@city.ac.uk
†
Faculty of Actuarial Science and Statistics, Cass Business School, City University, London. Email:
s.hab erman@city.ac.uk
‡
Dipartimento di Statistica e Matematica Applicata, University of Turin. Email: elena.vigna@econ.unito.it
1 Introduction
The income drawdown option in defined contribution (DC) pension schemes allows the member who
retires not to convert the accumulated capital into an annuity immediately at retirement but to
defer the purchase of the annuity until a certain point of time after retirement. During this perio d,
the member can withdraw periodically a certain amount of money from the fund within prescribed
limits. The period of time can also be limited: usually freedom is given for a fixed number of years
after retirement and at a certain age the annuity must be bought.
In the UK, where the option was introduced in 1995, the periodic income drawn is bounded between
35% and 100% of the amount that the member would have received if she bought a level annuity
at retirement. At age 75, the annuity must be bought with the remaining fund.
Comparing the drawdown option with the purchase of an annuity at retirement, we observe two
important points: the member is given complete investment freedom (instead of locking the fund
into bond-based assets, as is usual with annuities) and a bequest desire can be satisfied should the
member die before buying the annuity (because, in the case of death, the fund remains as part of
the individual’s estate).
On the other hand, the drawdown option does not provide any hedge against longevity risk and
financial risk: the retiree faces both the risk of outliving her own assets and the risk of buying, after
the deferment period, a lower annuity than the one which was possible to buy at retirement.
In this paper, we consider the income drawdown option and investigate, by means of stochastic
optimal control techniques, what should be the optimal investment allocation of the fund after
retirement until the purchase of the annuity, given that the pensioner wishes to achieve a certain
target when she buys the annuity. We assume here that the pensioner has no bequest motive
and that the only reason for choosing the drawdown plan is the hop e of being able to buy a better
annuity in the future than the one which she could buy at retirement. It seems therefore a reasonable
suggestion that the pensioner would have a certain “income target” in mind and attempt to pursue
it when investing in the financial market. We deal with the bequest motive in a parallel paper
(Gerrard, Haberman, Højgaard and Vigna, 2004). In this work, we do not consider the impact that
some additional pre-existing annuity (either a public pension or a private one) would have on the
decision whether to annuitize or take the income drawdown option. Also, the choice of the target
does not take into account the possibility of future unexpected inflation.
The remainder of the paper is organized as follows. In section 2, we introduce the model. In section
3, we define the stochastic optimal control problem. In section 4, we solve the problem in the finite
time horizon with two different forms for the target. In section 5, we solve the problem in the
infinite time horizon. In section 6, we introduce the simulation part of the work, report the results
and carry out a sensitivity analysis. In section 7, we draw conclusions and present some ideas for
further research.
A number of authors have dealt with the problem of managing the financial resources of a pensioner
after retirement, which arises from the fact that whole life annuities are felt by policyholders to
be “poor value for money” (Orszag, 2000) and have investigated the other alternatives available
to a retiree. Khorasanee (1996) compares the purchase of an index-linked annuity with two alter-
natives: level annuity and income withdrawal option. Milevsky (1998) proposes a strategy for the
post-retirement period where the pensioner’s consumption exactly matches what a level annuity
purchased at retirement would pay and the pensioner invests the remaining part. Milevsky calcu-
lates the ruin time in a deterministic scenario and estimates the probability of being able to buy
(after a certain number of years) at least a larger annuity than the one that can be purchased at
Version: June 17, 2004 Page 2
retirement in the case when market returns are stochastic. Kapur and Orszag (1999) consider in-
vestment decisions in the decumulation phase of a DC plan by means of stochastic optimal control,
choosing between equities and annuities, and find that complete annuitization eventually occurs.
Mitchell, Poterba, Warshawsky and Brown (1999), in an attempt to answer the question why people
do not buy annuities, compare the expected present value of payments from an annuity (calculated
with a proper term structure of interest rates) with the amount of premium charged, and compare
the expected utility from the annuity payments with the expected utility from an optimal consump-
tion path, if there were no annuities in the market. Milevsky and Robinson (2000) consider the
adoption of a drawdown option assuming a fixed amount withdrawn every year and investment of
the remaining fund in one risky asset. They calculate exactly the eventual probability of ruin and
then approximate the probability that ruin occurs before the random time of death, comparing their
approximations with the frequency of ruin found via Monte Carlo simulations. Findlater (1999) and
Wadsworth, Findlater and Boardman (2001) propose a product for the post- retirement period (the
second paper being a more detailed exposition of the product) in which pensioners have flexibility
in the way that they invest the fund and withdraw money, as in the drawdown option, with the
difference that mortality credits are given to the survivors and there is no bequest at death. Blake,
Cairns and Dowd (2003) use expected utility to compare immediate annuitization at retirement
with two kinds of drawdown plan (one approach has survival bonuses but no bequest — as in the
proposal by Wadsworth et al. — and the other is more traditional, with a bequest in the case
of death and no mortality credits); they also investigate the optimal annuitization age. Albrecht
and Maurer (2002) consider the probability that the p ensioner outlives her own assets when taking
the income drawdown option, withdrawing exactly the amount of money that an immediate level
annuity bought at retirement would provide. Charupat and Milevsky (2002) find the optimal mix
(constant over time) between a fixed immediate annuity and a variable immediate annuity, with
different mortality assumptions, via the maximization of expected utility, and then compare it with
the optimal mix found in the accumulation phase of a DC scheme. Lunnon (2002) lists nine alter-
natives to immediate and complete annuitization at retirement: three kinds of annuity, three kinds
of income drawdown and three kinds of combinations of the two; he proposes criteria for the design
of a post- retirement product and analyzes all of the choices with reference to these criteria.
2 The model
In our model, we consider the position of an individual who chooses the drawdown option at
retirement, i.e. withdraws a certain income until she achieves the age at which the purchase of
the annuity is compulsory.
The fund is invested in two assets, a riskless asset, with constant instantaneous rate of return, r,
and a risky asset, whose price follows a geometric Brownian motion with drift λ and diffusion σ.
The pensioner withdraws an amount b
0
in the unit time. Therefore, the stochastic differential
equation that describes the growth of the fund is the following (see, for instance, Merton, 1969):
dX(t) = [X(t)(y(t)(λ − r) + r) − b
0
]dt + X(t)y(t)σdW (t)
X(0) = x
0
(1)
where
X
(
t
) is the fund at time
t
(with
X
(0) being the fund at retirement),
y
(
t
) is the proportion
of the fund invested in the risky asset and W (t) is the standard Brownian motion.
We introduce the following quadratic loss (or disutility) function:
L(t, X(t)) = (F (t) − X(t))
2
(2)
Version: June 17, 2004 Page 3
The function of time F (t) is a target that the individual wishes to achieve: deviations from this
target are penalized so that a “cost”, measured by the loss function, is paid when the fund is
different from the target.
The use of a quadratic loss function is not new in the context of pension funds. Some examples
are Boulier, Trussant and Florens (1995), Boulier, Michel and Wisnia (1996), and Cairns (2000).
From a theoretical point of view, the quadratic loss function also penalizes any deviations above
the target, and this can be considered as a drawback to the model. However, the choice of trying
to achieve a target and no more than this has the effect of a natural limitation on the overall risk
of the portfolio: once the target is reached, there is no reason for further exposure to risk and
therefore the surplus becomes undesirable. The idea that people act by following subjective targets
is accepted in the decision theory literature. For example, Kahneman and Tversky (1979) support
the use of targets in the cost function, and, more recently, Bordley and Li Calzi (2000) investigate
and support the target-based approach in decision making under uncertainty. Another example of
the use of the targets in an insurance context is provided by Browne (1995), who derives optimal
investment policies by minimizing the probability that the wealth hits a certain bottom level (ruin)
before hitting a certain upper level (target).
In addition, as will be shown later, with a proper and not unreasonable choice of the target, the
fund never exceeds the target. Hence, the choice of a quadratic loss function can be considered
appropriate and has the advantage of leading to closed-form solutions.
The terminal cost at the terminal time T , if the fund is different from the target is:
ε(F (T ) − X(T ))
2
= K(T, X(T ))
K(·) has the same form of the loss function L(·), multiplied by a constant ε. The weighting factor
ε can be useful if the cost experienced at final time T is to be considered more important than the
running costs experienced before then.
Adopting the same approach as in Haberman and Vigna (2002), it can be shown that the target
F (t) is also a parameter that measures the risk attitude of the individual at time t: the higher its
value, the lower the risk aversion of the individual. In fact, for a loss function L(z), with z being
the loss, the coefficient of absolute risk aversion is
L
00
(z)
L
0
(z)
, which in this case is:
L
00
(z(t))
L
0
(z(t))
=
1
z(t)
=
1
F (t) − X(t)
This relation between the target and the risk attitude is also intuitive: the less risk averse the
individual is, the higher the target that she will pursue (and vice versa).
The targets are time dep endent because, as time passes, the individual becomes older and her
future life expectancy decreases: hence, the value of the annuity that would be purchased at the
interruption of the income drawdown option decreases, ceteris paribus. We also observe that, as
time passes, the fund on the one hand decreases due to the periodic income drawn, and on the other
hand changes in value (and hopefully increases) due to the investment return from the two assets
in which it is invested.
In a later section, we will produce results for different specification of the targets.
We now define the op en set U ⊂ R
+
× R, where the couples (t, X(t)) are allowed to range:
U = (0, T ) × (−∞, +∞), (3)
where T is the time when purchase of the annuity becomes compulsory.
Version: June 17, 2004 Page 4
