A combined stochastic programming and optimal control approach
to personal finance and pensions
Agnieszka Karolina Konicza∗
David Pisingera
[email protected]
[email protected]
Kourosh Marjani Rasmussena Mogens Steffensenb
[email protected]
[email protected]
April 4, 2013
a
DTU Management Engineering, Management Science, Technical University of Denmark,
Produktionstorvet 426, 2800 Kgs. Lyngby, Denmark
b
Department of Mathematical Sciences, University of Copenhagen,
Universitetsparken 5, 2100 Copenhagen, Denmark
Abstract
The paper presents a model that combines a dynamic programming (stochastic optimal control)
approach and a multi-stage stochastic linear programming approach (SLP), integrated into one SLP
formulation. Stochastic optimal control produces an optimal policy that is easy to understand and
implement. However, explicit solution may not exist, especially when we want to deal with constraints,
such as the limits on the portfolio composition, the limits on the insured sum, an inclusion of transaction
costs or taxes on capital gains, which are important issues regularly mentioned in the scientific literature.
Two applications are considered: (A) optimal investment, consumption and insured sum for an individual maximizing the expected utility of consumption and bequest, and (B) optimal investment for a
pension saver who wishes to maximize the expected utility of retirement benefits. Numerical results show
that among the considered practical constraints, the presence of taxes affects the optimal controls the
most. Furthermore, the individual’s preferences, such as impatience level and risk aversion, have even a
higher impact on the controlled processes than the taxes on capital gains.
Keywords: dynamic programming, multi-period stochastic linear programming, power utility, personal
finance, retirement.
JEL Classification: C61, D14, D91, G11, G22
1
Introduction
The purpose of this paper is to define a framework for solving various multi-period financial optimization
problems. The framework is based on a model which combines two optimization technologies: dynamic programming (stochastic optimal control), which is a classical approach for continuous time financial problems,
and stochastic programming, which is broadly applied in operations research and allows us to focus on the
practicalities of the problem. These two methods complement each other, especially in the areas where one
or the other does not perform well on its own, see e.g. Mulvey (2004). Stochastic optimal control produces an
optimal policy that is easy to understand and implement. However, explicit solution may not exist, especially,
when we want to deal with constraints, such as limits on portfolio composition, limits on the insured sum,
∗ Corresponding author at: DTU Management Engineering, Management Science, Technical University of Denmark, Produktionstorvet 426, 2800 Kgs. Lyngby, Denmark. Tel.: +45 4525 3109; Fax: +45 4525 3435; email: [email protected]
1
an inclusion of transaction costs or taxes on capital gains, which are important issues regularly mentioned in
the scientific literature. Stochastic programming (SLP) is a general purpose decision model with an objective function that can take a variety of forms. It is based on the scenario trees that represent the range of
possible outcomes for the uncertainties. Rather than finding a generic optimal policy, the optimal solution is
computed numerically at each node in the scenario tree, for the specified decision variables. SLP can easily
address realistic considerations and constraints, as long as they have an algebraic form. The main drawback
of this optimization method is that the problem size grows quickly as a function of the number of periods
and scenarios. In particular, taking into consideration the entire lifetime of an individual can be challenging
in terms of computational feasibility.
The problem of optimal consumption and investment has been widely investigated. In terms of dynamic
programming approaches, probably the most influential papers have been written by Merton (1969, 1971)
who defined the original problem of optimizing utility of consumption and terminal wealth over a fixed time
horizon for an investor. This work has inspired many researchers who later either expanded the original model
or investigated different objective functions, by introducing for instance the lifetime uncertainty, insured sum
and the labor income, Richard (1975), stochastic interest rate, Munk and Sørensen (2004), salary uncertainty,
Cairns et al (2006), multi-person household, Bruhn and Steffensen (2011), borrowing constraints, Byung and
Yong (2011), or constant linear taxation, Bruhn (2012). Applications within defined contribution pension
scheme with focus on the investment strategy either during the accumulation or post retirement phase (decumulation) together with the optimal time of annuitization, have been considered by Milevsky and Young
(2007) and Gerrard et al (2004, 2012).
The stochastic programming approach is more common within portfolio management and asset-liability
management than within the areas concerning the individual investors. See, for instance, Mulvey et al (2006,
2007), who argue that multi-period investment models combined with Monte Carlo simulation can address
significant issues for long-term investors, and Ferstl and Weissensteiner (2010), who present a stochastic
linear programming model for optimal asset allocation in a situation where a financial company wishes to
minimize the conditional value at risk. Nevertheless, applying stochastic programming or other numerical
approaches to find the optimal decisions from the individual investors’ point of view can also be found
in the literature. See, e.g. Horneff et al (2008), considering an individual characterized by Epstein-Zin
preferences, Cai and Ge (2012), investigating the asset allocation for an investor with a loss aversion objective,
a predetermined objective and a greedy objective, Consigli et al (2012), comparing investment opportunities
offered by traditional pension products and unit-linked contracts with variable life annuities, and Blake et al
(2013), deriving the optimal investment for a loss averse pension saver with an interim and final target.
Finally, Geyer et al (2009a) argue that two main optimization technologies, namely, dynamic programming and stochastic linear programming can be combined and both solutions can be integrated into one SLP
formulation. The authors show that it is possible to accurately replicate the closed-form solution obtained
by Richard (1975) by using a multi-stage stochastic programming approach. However, they do not include
the insured sum making the model less applicable. To the best of our knowledge, this is the only paper that
combines both methodologies.
We apply the idea of combining stochastic optimal control with multi-period stochastic programming presented in Geyer et al (2009a) and analyse two optimization problems:
(A) Optimal investment, consumption and insured sum for an individual maximizing the utility of consumption and a terminal utility of leaving a positive amount of money upon death over an uncertain
lifetime, Richard (1975). This problem has been considered in Geyer et al (2009a), but in contrast to
their work we have formulated the SLP model including an optimal control for the insured sum.
(B) A pension saver wishes to maximize the utility of the future retirement benefits by controlling the
optimal investment both before and after retirement and optimal benefits only after retirement. This
problem has not to our knowledge been considered in the literature, and we find it particularly relevant
for a pension product design in a defined contribution plan. The optimal controls help the pension fund
design a payout profile for a retiree based on her impatience level and risk preferences.
Except for working with different optimization problems, our paper differs from Geyer et al (2009a) in several
aspects, among which the main ones are:
• We derive the explicit formula for the optimal value function and optimal controls for problem (B) using
2
Hamilton-Jacobi-Bellman techniques. This formulation has not to our knowledge been considered in
the scientific literature.
• We focus on the optimal controls during the individual’s entire lifetime. Not only the decisions at the
beginning of the contract are of our interest, but particularly their development with time.
• We use a different method for the linearization of the objective function based on a non-linear optimization problem. We provide the evidence that the method is more suitable for our analysis and allows for
controlling the linearization error better.
• We follow a standard actuarial procedure and model the lifetime uncertainty in a continuous time
setting with parameters calibrated to the Danish mortality rates.
We find that:
• The results from the model combining two optimization methodologies are consistent with the optimal
controls that can be calculated explicitly using stochastic dynamic approach. The model defines a solid
framework for various life long optimization decisions, while allowing for incorporating a number of
practical constraints.
• Among limits on portfolio composition, limits on the insured sum, transaction costs, and taxes on
capital gains, only the latter has a significant impact on the optimal controls.
• The individual’s preferences, such as risk aversion and impatience level, have a higher impact on the
optimal controls, and especially, on the size and development of the benefits, than limits on the portfolio
composition, limits on the insured sum, transaction costs or taxes on capital gains.
• The presence of transaction costs in contrast to other considered constraints suggests a non-constant
investment in risky assets in the situation where there is no income (such as the period after retirement).
All other constraints are consistent with the classical result, which claims the the optimal proportion
in risky assets should be constant.
The paper is organized as follows. Section 2 describes problems (A) and (B) in more detail together with
their optimal solutions obtained via dynamic programming. Section 3 explains how to build a bridge between
dynamic programming and stochastic linear programming by incorporating both optimization methods in one
SLP formulation. Section 4 focuses on defining the SLP model, i.e. the objective function and the constraints,
the linearization of the objective and scenario tree generation. In Section 5 we present the numerical results
of the optimal controls obtained via two different optimization methods, and the impact on the controls of
the modifications considered during the first years of the contract. Finally, in Section 6 we summarize the
paper and suggest the future work. The paper also includes four appendices consisting of: derivation of the
optimal value function and controls for problem (B), App. A, the mutual fund theorem, App. B, derivation
of the expected values of the savings, App. C, and the comparison of two linearization methods, App. D.
2
Model description
The section develops the general model settings relevant for problems (A) and (B).
• We denote by Xt the amount of savings that belong to an individual interpreted as general personal
savings (problem (A)), and the level of savings on the pension account (problem (B)).
• The savings are invested in N underlying assets: one risk-free (bonds) and N − 1 risky assets (an
equity fund). The economy is represented by a standard Brownian motion W , which is defined on the
measurable space (Ω, F), where F is the natural filtration of W . The asset prices Sti are modelled by
Black-Scholes dynamics:
dSti = αi Sti dt + σi Sti dWti ,
S0i
=
(1)
si0 ,
3
where αi and σi are constants. The risk-free asset is defined by α3 = r and σ3 = 0. Merton (1971)
shows that without loss of generality we can assume that all risky assets are included in one mutual
fund, the prices of which follow the dynamics
dSt = αSt dt + σSt dWt ,
where α and σ are defined in App. B.
• The payments connected to the savings account are formalized by the continuous processes:
– a payment process lt that, depending on the problem, is interpreted either as the labor income that
is contributed to the account or the percentage of the labor income that is paid to the retirement
savings (premiums),
– a payment process ct determining the consumption of the savings for the private purposes or the
benefits that the pension fund pays to the retiree.
• The individual has an uncertain lifetime characterized by a finite state Markov chain Z, defined on a
measurable space (Ω, F). The jump intensities of the process Z, µ and µ∗ , denote the mortality rates
under the objective probability measure P and under the pricing probability measure P∗ , respectively.
P and P∗ are equivalent probability measures on (Ω, F) and the jump intensities are defined in Sec. 5.
• The individual is risk averse and has a utility function u characterized by a constant relative risk aversion
1 − γ, constant elasticity of intertemporal substitution 1/(1 − γ) (EIS) and time dependent weights wt
reflecting the importance of present consumption in contrast to future consumption characterized by
the impatience factor ρ:
u(t, c) = γ1 wt1−γ cγ ,
where
wt1−γ = e−ρt .
Parameter γ is defined for (−∞, 1) \ {0}, whereas for γ = 0 we have the case of the logarithmic utility.
Below, we define the model setup specific for the particular problems.
2.1
Problem (A) - optimal investment, consumption and insured sum
We keep the original settings defined in Richard (1975) and recall only the most important assumptions and
results that are crucial for this paper.
Savings dynamics. The individual has a bequest motive and upon death her heirs receive an amount on
the savings account plus the insured sum It . We assume that life insurance policies are tradable financial
contracts and the individual has access to a life insurance market. She buys a life insurance with a insured
sum It , and for that coverage she pays a premium at rate µ∗t It , where µ∗t is the natural premium intensity
decided by the life insurance company, also called pricing mortality. In particular, we allow for negative It ,
which means that the person sells the amount It to the pension fund and purchases life annuities.
The wealth dynamics while the person is alive develop as follows:
dXt = r + πt (α − r) Xt dt + πt σXt dWt + lt dt − ct dt − µ∗t It dt,
(2A)
X0 = x0 ,
where r is the return on the risk-free asset, α the return on the risky mutual fund, σ the volatility of the
mutual fund, π is the proportion invested in the mutual fund, and 1 − π in the risk-free asset.
Optimization problem. Since the individual has a bequest motive, she obtains the utility from the amount
that she will leave upon death to her heirs:
U (t, x) = γ1 vt1−γ xγ ,
where v is the weight for this utility. It sounds reasonable to define v in terms of the weights for the utility
of consumption w, hence v denotes the weight put on her heir’s consumption relative to her own.
vt1−γ = λ−γ wt1−γ ,
4
where λ is a constant.
The individual can control consumption, investment and insured sum, in order to maximize the utility of
consumption and bequest. Given the wealth dynamics, X, the problem is mathematically formulated as:
#
"Z e
T
R
A
− ts µτ dτ
u(s, c) + µs U (s, Xs + Is ) ds ,
(3A)
V (t, x) =
sup
Et,x
e
t
π,c,I∈Q[t,Te)
V A (Te, x) = 0,
where Et,x is the conditional expectation under P, given that the person is alive at time t and holds wealth
Xt = x, and Q[t, Te) is the set of control processes for the time [t, Te) which are admissible at time t. Te
is aR fixed time point at which the investor is dead with certainty. Both utilities are multiplied by factor
s
e− t µτ dτ denoting the probability that the individual survives until time s > t, given she has survived until
time t. Furthermore, the utility of bequest is multiplied by the mortality intensity rate µs , which represents
the probability that the person dies within a short period after time s. The utility functions u and U are
assumed to be strictly concave in c and X + I, respectively.
The optimal value function for this problem is given by
V A (t, x) =
γ
1
fA (t)1−γ x + gA (t) ,
γ
(4A)
where
Z
Te
fA (t) =
e
1 R
− 1−γ ts µτ −γ(µ∗
τ +ϕ) dτ
t
Z
Te
gA (t) =
e−
Rs
t
(r+µ∗
τ )dτ
"
ws +
µs
(µ∗s )γ
#
1/(1−γ)
vs ds,
ϕ=r+
(α−r)2
2σ 2 (1−γ) ,
ls ds,
(5A)
(6A)
t
and the optimal investment, consumption and insured sum are of the form:
πt∗ =
2.2
α−r
σ 2 (1−γ)
Xt +gA (t)
,
Xt
c∗t =
wt
fA (t)
It∗ =
Xt + gA (t) ,
µt
µ∗
t
1/(1−γ)
vt
fA (t)
(Xt + gA (t)) − Xt .
(7A)
Problem (B) - optimal investment with optimal life annuities
The individual is considered specifically as a pension saver who wishes to maximize the expected utility
function of her future retirement benefits. She purchases a pension contract at time t0 with an assumption
of retiring at time T and receiving the benefits until time Te. For Te < ∞, the pension saver is interested in
temporary life annuities, whereas for Te → ∞ she prefers life annuities. Furthermore, we assume that the
benefits are directly linked to the market and no guarantees are provided. One can think of this model as a
special case of problem (A), where the individual is a pension saver with no bequest motive. In case of her
death, the pension fund inherits all her savings. Moreover, she obtains no utility from consumption before
retirement. We interpret the consumption process specifically as a pension benefit process, and process lt as
pension contributions (premiums).
Savings dynamics. We split the problem into a before retirement (accumulation) phase and an after
retirement (decumulation) phase. In practise the pension saver is not allowed to withdraw the pension
savings for the consumption reasons or receive the benefits before retirement. Thus, ct = 0 for t ∈ [t0 , T ).
It is also reasonable to assume that the payment process lt denoting the premiums paid to the pension fund
is set to 0 after retirement, i.e. lt = 0 for t ∈ [T, Te). Since the person has no bequest motive, she has an
additional income of µ∗t Xt , which is the amount that the pension fund pays to be her only inheritor. With
these assumptions the definition of savings dynamics are of the form
(r + πt (α − r) + µ∗t ) Xt dt + πt σXt dWt + lt dt, t ∈ [t0 , T ),
dXt =
(2B)
(r + πt (α − r) + µ∗t ) Xt dt + πt σXt dWt − ct dt, t ∈ [T, Te),
Xt = x0 .
5
Optimization problem. The pension saver wishes to maximize the utility of pension benefits, or in other
words, the utility of consumption after retirement. She obtains no utility from consumption before retirement,
i.e. u(t, c) = 0 for t ∈ [t0 , T ), hence:
"Z e
#
T
V B (t, x) =
sup
e−
Et,x
π∈Q[t,Te), c∈Q[max(t,T ),Te)
Rs
t
µτ dτ
u(s, c)ds ,
(3B)
max(t,T )
V B (Te, x) = 0.
where, as before, Et,x is the conditional expectation under P, given that the person is alive at time t and
holds savings Xt = x, and Q[t, Te) is the set of control processes for time [t, Te) which are admissible at time t.
The control processes in this problem are the proportion in the risky fund πt and the benefits ct . Note that
even though the individual obtains no utility from consumption before retirement, the investment process
πt is controlled both before and after retirement. Also, as in problem (A), we take into
consideration the
R
− ts µτ dτ
. At time Te the
uncertain lifetime of the individual by multiplying the utility function by the factor e
investor is assumed to be dead with certainty. The utility function u is assumed to be strictly concave in c.
To our knowledge this problem has not been considered in the literature, therefore we derive the optimal
value function and the optimal controls. We refer the reader to Appendix A for details. The optimal value
function is of the form:
γ
(4B)
V B (t, x) = γ1 fB (t)1−γ x + gB (t) ,
where

1 RT
R Te − 1 RTs
(µ −γ(µ∗
−

τ +ϕ))dτ
1−γ
fB (t) = e 1−γ t τ
T e
1 Rs
∗
R
e
µ −γ(µτ +ϕ) dτ
T −

fB (t) = t e 1−γ t τ
ws ds,
(
µτ −γ(µ∗
τ +ϕ) dτ
ws ds,
t ∈ [t0 , T ),
(5B)
t ∈ [T, Te),
Rs
RT
∗
gB (t) = t e− t (r+µτ )dτ ls ds, t ∈ [t0 , T ),
gB (t) = 0,
t ∈ [T, Te),
(6B)
and the optimal controls are given by
πt∗ =
α−r Xt +gB (t)
,
σ 2 (1−γ)
Xt
t ∈ [t, Te),
c∗t =
w(t)
fB (t) Xt ,
t ∈ [T, Te).
(7B)
Since the utility function u is concave in c, the retirement benefits must be non-negative. Furthermore, we
must have that XTe = 0, which ensures that all the savings that belong to the pension saver have been paid
out during the distribution phase.
In both problems the functions gA (t) and gB (t) are of the same form. The first function have been defined by Richard (1975) as human capital and represents the expected present value of future earnings, or,
in other words, what an individual’s labor force is worth in the financial market. Function gB (t) represents
the present value of the future premiums.
3
Combined SLP and optimal control
One of the main contributions of the paper is constructing a model which combines two optimization methodologies: dynamic programming and stochastic programming. We are motivated by the fact that the explicit
solutions only exist for simple models, hence often cannot be applied in practise. As soon as we add some
realistic constraints, such as limits on portfolio composition, limits on the insured sum, transaction costs or
capital gains taxes, explicit solutions do not exist. The SLP approach fills this gap and allows for incorporating a number of constraints in an easy way. However, the main drawback of SLP is limited ability to
handle many time periods under sufficient uncertainty. The scenario tree grows exponentially with each time
period, and solving the problem becomes soon computationally infeasible. Therefore, we split the individual’s
lifetime into two periods. We apply the SLP approach to obtain the optimal decisions for the first years,
t ∈ [t0 , TSLP ), whereas the decisions for the remaining lifetime of the individual, t ∈ [TSLP , Te), are solved
6
Figure 1: The first years decisions are solved by SLP model, whereas the decision over the remaining lifetime of the individual
are solved using HJB techniques.
explicitly. Fig. 1 presents the concept. The mathematical formulation for problem (A), defined in eq. (3A)
can be rewritten using its recursive properties in the following way:
Z TSLP
RT
R
− t SLP µτ dτ A
A
− ts µτ dτ
V (TSLP , XTSLP ) .
u(s, c) + µs U (s, Xs + Is ) ds + e
e
V (t, x) = sup Et,x
π,c,I∈Q[t,TSLP )
t
(8A)
The above definition is convenient because it separates the problem into two periods, which allows for applying
different optimization methodologies in each period. The crucial part of the model is to insert the optimal
value function derived explicitly into the objective function of the SLP model, thus we can ensure that the SLP
formulation incorporates the decisions for the entire lifetime and not only for the first years. We start with
the latest period, t ∈ [TSLP , Te), and calculate the optimal value function at time TSLP , i.e. V A (TSLP , XTSLP )
according to formula (4A). Then, insert this function into eq. (8A) and solve the optimization problem using
multi-period stochastic linear model. In this way we construct one SLP formulation that covers the entire
lifetime of the individual.
We proceed analogously with problem (B). The mathematical formulation for the optimization problem
defined in eq. (3B) can be rewritten as
V B (t, x) =
Z
sup
TSLP
Et,x
π∈Q[t,TSLP ), c∈Q[max(t,T ),TSLP )
e−
Rs
t
µτ dτ
u(s, c)ds + e−
R TSLP
t
µτ dτ
V B (TSLP , XTSLP ) ,
max(t,T )
(8B)
where V B (TSLP , XTSLP ) is given in eq. (4B).
From here, we can focus on modelling the first years decisions via SLP.
4
SLP formulation
Stochastic programming is a method for the study of optimal decision making under uncertainty, Birge and
Louveaux (1997). It can be compared with dynamic programming, though it allows for more flexibility when
defining an objective function and emphasizes constraints accompanying the problem.
A multi-period stochastic linear program with recourse combines anticipative and adaptive models in
a common mathematical framework. For example, the investor specifies the composition of a portfolio by
taking into account both possible future movements of asset returns (anticipation) and that the portfolio will
be rebalanced (taking recourse decisions) as prices change (adaptation), Zenios (2008).
To start with, let us formulate a two-stage version of the problem with recourse. We need two vectors for
decision variables to distinguish between the anticipative and adaptive policy:
• y0 - a vector of first-stage decisions, which are made before the random variables are observed; the
decisions does not depend on the future observations but anticipate possible future realizations of the
random vector,
• ỹ1 = y1 (y0 , ω) - a vector of second-stage decisions which are made after the random variables ω have
been observed. They are constrained by decisions y0 and depend on the realizations of the random
vector ω.
7
The objective is to maximize/minimize a certain function f , whose value depends on the decisions y0 and
y1 (y0 , ω) taken at stages t0 and t1 , accordingly. We keep the notation from Geyer et al (2009a) and formulate
the problem as follows:
Maximize
f (y0 ) + E[f (y1 (y0 , ω))],
A0 y0 ≤ b0 ,
s.t.
A1 (ω)y1 (y0 , ω) ≤ b1 (ω),
where A(ω) and b(ω) are the model parameters, which can also be dependent on the random variable ω.
Multi-period stochastic programs have a similar structure. Observations are made at TSLP different stages,
hence the decision vectors taken at stage t, ỹt = yt (ω1 , . . . , ωt , y0 , . . . , yt−1 ), depend on the decisions taken
in all previous stages and also on the history of the random variables (ω1 , . . . , ωt ). In this way, the problem
has the following structure
Maximize
f (y0 ) +
TX
SLP
E[ft (ỹt )],
t=1
At ỹt ≤ bt ,
s.t.
t = t0 , . . . , TSLP ,
and the information constraints are given as
decide y0
→
observe ω1 , A(ω1 ), b(ω1 )
→
→
observe ωT , A(ωTSLP ), b(ωTSLP )
...
→
decide ỹ1
→
→
observe ω2 , A(ω2 ), b(ω2 )
decide ỹTSLP .
The probability distribution of the random variable ωi , which in our case reflects the distribution of the
returns on risky assets, is described in terms of a discrete number of outcomes (scenarios). A scenario tree
consists of nodes nν(t) . Each of them is assigned to a certain period t and state ν(t), whereas the links
between the nodes indicate possible transition between states as time evolves. Node n0 is the root node and
it has no predecessor, whereas every other node has a unique predecessor from the previous set of states.
Each sequence {nν(t0 ) , nν(t1 ) , . . . , nν(TSLP ) }, where nν(tj ) is a predecessor of nν(tj+1 ) for j = 0, . . . , TSLP − 1
defines a scenario with the associated probability P rnν(tj ) ≥ 0. The total number of scenarios in the tree
is specified by the product of the number of outcomes in each stage t. An example showing a three-stage
problem is illustrated on Fig. 2.
Figure 2: A 3-period scenario tree with a constant branching factor bf = 3.
4.1
Objective and constraints
The stochastic programming model determines the optimal decisions at each node of the event tree, given
the information available at that point. An important assumption is that randomness in asset returns Rni ν(t)
is exogenous and not affected by decisions.
The following decision variables and parameters are used in the model formulation:
Indices:
t, stage (point in time), t = t0 , . . . , TSLP ,
ν(t), state in stage t, ν(t) = 1, . . . , Kt ,
8
nν(t) , node corresponding to stage t and state ν(t),
i, available asset, i = 1, . . . , N .
Parameters:
Kt , number of nodes (states) in period t,
N , number of available assets,
lt , labor income/premiums paid to the savings account at time t,
x0 , initial value of the savings,
µt , mortality rate for a y+t-year old individual at time t (probability to die between time t and t + 1),
µ∗t , pricing mortality rate for a y+t-year old individual at time t (probability to die between time t and t + 1
used by the pension fund for pricing life insurance / life annuities),
Rni ν(t) , return on asset i at node nν(t) , obtained via scenario generation, see Sec. 4.4,
P rnν(t) , probability of being in node nν(t) obtained via scenario generation, see Sec. 4.4.
Decisions variables:
Peni ν(t) ≥ 0, amount of asset i purchased at node nν(t) ,
Seni ν(t) ≥ 0, amount of asset i sold at node nν(t) ,
eni , total holdings of asset i at node nν(t) ,
X
ν(t)
en− , total holdings in all assets at node nν(t) including the accumulated capital gains before any payments
X
ν(t)
i
en
en− = x0 1{t=t } + PN 1 + Rni
1
.
X
or trades are made, i.e. X
0
i=1
ν(t−1) {t>t0 }
ν(t)
ν(t)
e
Cnν(t) ≥ 0, consumption/benefit at node nν(t) ,
Ien , insured sum at node nν(t) (problem (A) only).
ν(t)
Expression
1{(·)=t} denotes an indicator function equal to 1 if (·) = t and 0 otherwise.
The entire SLP formulation that replicates the assumptions made in dynamic programming model setup,
consists of the objective function and three constraints: the budget constraint, the asset inventory constraint
and the constraint on positivity of certain variables. With only these three constraints we can replicate the
continuous time models presented in Sec. 2. Afterwards, in Sec. 4.2, we modify these equations in order to
investigate the impact of various factors, such as limits on portfolio composition, limits on the insured sum,
transaction costs, and taxes on capital gains.
4.1.1
Problem (A)
The problem of optimizing the expected utility function of consumption and a terminal utility of leaving a
positive amount of money upon death, where the investment, consumption and insured sum are the controlled
processes, can be modeled with the set of the following equations.
The objective function is obtained by substituting the integrals with the sums and the expectation operator E with its discrete definition in the eq. (8A).
The budget constraint (10A), or alternatively, the cash flow balance constraint specifies that the amount
invested in the purchase of new securities plus consumption must be equal to the amount gained from the
sale of the securities plus labor income that is paid to the savings account plus any initial savings x0 that
the person has at the beginning of the contract. Moreover, we add the term µ∗t Ienν (t) to the left hand side of
the equation, which is the price for the life insurance that the investor pays at each period. The inclusion of
the insured sum variable Ienν(t) in the budget constraint and in the objective function for problem (A) is an
important part of the model that distinguishes our work from Geyer et al (2009a).
Constraint (11A) calculates the value of the savings on the pension account. It is equal to the accumulated
capital gains/losses on the assets held in portfolio from the previous period plus any amount purchased in
the current period minus sold securities.
The continuous time versions of the problems assume the utility function u to be strictly concave in c,
and U to be strictly concave in Xt + It . Furthermore, for γ < 0, u(0) = U (0) = −∞, therefore we define the
positivity constraints (12A). For γ ∈ (0, 1) we have u(0) = 0 and U (0) = 0, so the positivity constraints are
substituted with non-negativity constraints.
9
This leads to the model:
max
TSLP
X−1
X
s=t0
nν(s)
X
+
P rnν(s) · e
P rnν(TSLP ) · e−
−
Rs
t0
µτ dτ
R TSLP
t0
h
µτ dτ
i
en ) + µs U s, X
e − + Ien
u(s, C
nν(s)
ν(s)
ν(s)
en−
· V A TSLP , X
ν(T
,
SLP )
(9A)
nν(TSLP )
N
X
en (t) + µ∗t Ien (t) = x0 1{t=t } +
Peni ν (t) + C
ν
ν
0
i=1
N
X
Seni ν (t) + lt ,
(10A)
i=1
for t = t0 , . . . , TSLP − 1, ν(t) = 1, . . . , Kt ,
i
eni
en
X
= 1 + Rni ν(t) X
1
+ Peni ν(t) − Seni ν(t) ,
ν(t)
ν(t−1) {t>t0 }
(11A)
for t = t0 , . . . , TSLP − 1, ν(t) = 1, . . . , Kt , i = 1, . . . , N, nν(t−1) is the ancestor of nν(t) .
e − + Ien
> 0,
X
nν(t)
ν(t)
en
C
> 0,
ν(t)
4.1.2
for t = t0 , . . . , TSLP − 1, ν(t) = 1, . . . , Kt .
(12A)
Problem (B)
The problem of maximizing the expected utility of retirement benefits, where the investment process is controlled both before and after retirement, whereas the benefits are controlled only after retirement, can be
modelled using SLP formulation as follows.
The objective function (9B) is a discrete version of eq. (8B) where the integrals are substituted with the
sums and the expectation operator E with its discrete definition.
The budget constraint (10B) specifies that the amount invested in the purchase of new securities plus
consumption must be equal to the amount gained from the sale of the securities plus premium that is paid
to the savings account plus any savings x0 that the person has at the beginning of the contract. We also add
en (t) to the right hand side of the balance equation, which is the price that the pension fund
the term µ∗t X
ν
pays the pension saver to be her only heir, and the pension benefit to the side of the outgoing payments for
t ≥ T.
Constraint (11B) calculates the value of the savings on the pension account. This constraint is identical
to the asset inventory balance for problem (A).
Similarly as in problem (A), the utility function u is assumed to be strictly concave in c, and for γ < 0,
we have u(0) = U (0) = −∞. Therefore, we assume that the benefits are positive, (12B), or, for γ ∈ (0, 1),
non-negative.
This leads to the following SLP formulation:
max
TSLP
X−1
X
P rnν(s) · e
−
Rs
t0
µτ dτ
en )
u(s, C
ν(s)
s=max(t0 ,T ) nν(s)
X
+
P rnν(TSLP ) · e−
R TSLP
t0
µτ dτ
en−
· V B TSLP , X
ν(T
SLP )
,
(9B)
nν(TSLP )
N
X
en (t) 1{t≥T } = x0 1{t=t } +
Peni ν (t) + C
ν
0
i=1
N
X
Seni ν (t) + lt + µ∗t
i=1
N
X
eni ,
X
ν(t)
(10B)
i=1
for t = t0 , . . . , TSLP − 1 ν(t) = 1, . . . , Kt .
i
eni
en
X
= 1 + Rni ν(t) X
1
+ Peni ν(t) − Seni ν(t) ,
ν(t)
ν(t−1) {t>t0 }
for
(11B)
t = t0 , . . . , TSLP − 1, ν(t) = 1, . . . , Kt , i = 1, . . . , N, nν(t−1) is the ancestor of nν(t) ,
en (t) > 0,
C
ν
for
t = max(t0 , T ), . . . , TSLP − 1, ν(t) = 1, . . . , Kt :
10
(12B)
4.2
Additional constraints
The reason for modelling the first years decisions with SLP is that this optimization method can easily capture
the practical constraints such as limits on portfolio composition, limits on the insured sum, transaction costs,
and taxes on capital gains. Below, we present how to modify the original constraints or add other constraints
to the SLP formulation presented in Sec. 4.1.
Limits on portfolio composition. A feature that is interesting from the point of view of a private investor,
a pension saver and a pension fund, is the limit on the portfolio composition. For instance, no pension fund
allows for borrowing or short selling of the assets. Alternatively, if the investor has certain preferences about
the minimum and maximum percentage of her wealth invested in a certain asset, we would like to include it in
the optimization model. The limits on the portfolio composition can be incorporated in the SLP formulation
by adding the following constraints for t = t0 , . . . , TSLP , ν(t) = 1, . . . , Kt and i = 1, . . . , N :
eni
≥ di
X
ν(t)
N
X
eni ,
X
ν(t)
eni
X
≤ ui
ν(t)
i=1
N
X
eni ,
X
ν(t)
(16)
i=1
where di and ui are the lower and upper limits for the holding of asset i. In particular, di and ui can be
extended to be time dependent, which would be suitable for a person with specific preferences only for a
certain year.
Limits on the insured sum. Richard (1975)’s model does not assume any constraint on the size or the
sign of the insured sum It . Including such a constraint in the dynamic programming approach is definitely
not trivial to solve and to our knowledge the explicit solution for this case has not been derived. The SLP
model allows for adding the limits on the insured sum in a straightforward way:
Ienν(t) ≥ dins ,
Ienν(t) ≤ uins .
(17)
As shown later in Sec. 5.1, being able to control the sign of the insured sum is especially important from the
practical point of view. A negative insured sum means that each period the pension fund pays the amount
µ∗t Ienν(t) in order to receive a part of the individual’s savings equal to Ienν(t) upon her death. This situation
describes purchasing life annuities. In reality the individual is not allowed to purchase life annuities before
she retires, so being able to set dins = 0 for t < T is an important extension.
Transaction costs. Similarly, the presence of transaction costs q i makes the considered problems more
realistic. Investigating how the transaction costs affect the optimal controls is of practical importance. We
add the costs as a percentage of the traded amount by modifying the budget equations (10A) and (10B). We
subtract the costs from the amount of the assets sold in a given period, and add the costs to the amount
purchased. The modified budget equations are defined as follows:
N
X
en (t) + µ∗ Ien (t) = x0 1{t=t } +
Peni ν (t) (1 + q i ) + C
t
ν
ν
0
i=1
N
X
N
X
Seni ν (t) (1 − q i ) + lt ,
(10A’)
i=1
en (t) 1{t≥T } = x0 1{t=t } +
Peni ν (t) (1 + q i ) + C
ν
0
i=1
N
X
Seni ν (t) (1 − q i ) + lt + µ∗t
i=1
N
X
eni ,
X
ν(t)
(10B’)
i=1
for t = t0 , . . . , TSLP − 1 and ν(t) = 1, . . . , Kt .
Taxes on capital gains. Worth consideration are also taxes on capital gains τ i . Generating a scenario
tree for the SLP model allows us to subtract taxes only from the positive capital gains, i.e.
(
Rni ν(t) (1 − τ i ), Rni ν(t) > 0,
i
(18)
net Rnν(t) =
Rni ν(t) ,
Rni ν(t) ≤ 0.
This requires changing the asset inventory balance constraints (11A) and (11B) as follows:
i
eni
en
X
= 1 + net Rni
X
1{t>t } + Peni − Seni ,
ν(t)
ν(t)
ν(t−1)
0
ν(t)
ν(t)
11
(11A’,11B’)
for t = t0 , . . . , TSLP − 1, ν(t) = 1, . . . , Kt and i = 1, . . . , N . This is also convenient in contrast to the
stochastic optimal control approach, where the asset prices are modelled by the Black Scholes model, and
one can only adjust the expected returns and volatility of the assets by introducing αi,net = αi (1 − τ i ) and
σi,net = σi (1 − τ i ). The taxes are then subtracted from both positive (gains) and negative (losses) returns.
The latter can be interpreted as a possibility to deduct the taxes from the negative capital income.
Other modifications. There are plenty of economic factors whose impact on the optimal control would be
interesting to investigate but are beyond the scope of this paper. Probably the most relevant from a practical
perspective is modelling the asset returns using a different model than Black Scholes. Geyer et al (2009b)
extended their previous work by applying the first-order unrestricted vector autoregressive process (VAR)
to model asset returns. They find that there is a substantial difference in asset allocation which reflects the
impact of time-varying investment opportunities, which, not surprisingly, shows the sensitivity of the model
to the assumptions in the scenario generation.
Note that all the constraints are linear, though the objective function is not. We prefer the linear solvers
due to their robustness and speed, therefore in the next section we describe how to linearize the objective
function.
4.3
Linearization
To be able to use linear programming solvers for solving problems (A) and (B), the objective function in each
problem needs to be linearized. Alternatively, the problems could be solved using the nonlinear optimization
techniques, however the linear solvers are preferable.
One of the differences between our work and Geyer et al (2009a) lies in the method of linearization. Our
approach differs in two ways:
• the position of breakpoints,
• the formula for the linear approximation.
In this section we describe our method of linearization, whereas in Appendix D we provide the evidence that
this method is more suitable for our problem.
Breakpoints. The linearization of the objective function requires choosing the number and the position
of breakpoints. This choice can affect the standard errors of the SLP optimization results. The parameter
m denoting the number of breakpoints is chosen freely, as well as the minimum and maximum breakpoint.
To obtain the positions of the remaining breakpoints we solve a non-linear optimization problem, where the
objective is to maximize the area underneath the power function:


X
X

(19)
max
(bpm − bpj ) h(bpj ) − h(bpj−1 ) + 0.5
(bpj − bpj−1 ) h(bpj ) − h(bpj−1 )  ,
bpj ,j=1,...,m
j6=m
∀j
h(bpj ) =
j6=1
γ
1
γ bpj .
(20)
The code for the optimization problem is accessible in the online GAMS library, Rasmussen (2011). The
solution to this problem is locally optimal which is sufficient for our purposes. In a situation, when the
solution consists of the repetitive breakpoints (∃i6=j bpj = bpi ), we use a heuristic method and rerun the
model for different minimum and maximum breakpoints, bp1 and bpm , in order to find m different points.
The choice of bp1 and bpm is crucial (next to the choice of the number of breakpoints m) and reflects
how well we can reproduce the closed-form solutions for the optimal controls. We estimate bp1 and bpm
using the properties of geometric Brownian motion, which allow for deriving the explicit formulae for the
expectation of the savings process with incorporated optimal controls, E[Xt∗ ], see App. C, hence also the
expected values of E[c∗t ], E[πt∗ ] and E[It∗ ]. These values can guide us in specifying the possible range of the
outcomes of the SLP variables, i.e. the length of the intervals [bp1 , bpm ].
12
Exact utility vs. linearized utility of consumption
4
−3
x 10
exact utility
linearized utility
−4
−5
−6
−7
−8
−9
n0
n30
n60
n90
n120
n150
n180
nodes
n210
n240
n270
n300
n330
Figure 3: Exact utility versus linearized utility of consumption for the nodes associated with the first five periods in problem (A).
When the cross and circular markers cover each other, the linearization is accurate. Minimum breakpoint, bpc1 = 0.7E[c∗T
]=
SLP
] = 0.042, number of breakpoints, m = 40.
0.015, maximum breakpoint, bpcm = 2E[c∗T
SLP
Approximating function. Since the utility functions and the optimal value functions are concave, they
can be approximated as closely as desired by m−1 linear segments between the breakpoints bpj (j = 1, . . . , m),
Hillier et al (2010). The argument x is defined in terms of non-negative decision variables zj associated with
each segment
x=ε+
m
X
zj ,
(21)
j=1
where ε ≥ 0 is the lower limit of the argument of the function chosen as small as possible for γ ∈ (−∞, 0)
and ε = 0 for γ ∈ (0, 1), subject to constraints:
0 ≤ z1 ≤ bp1 − ε,
0 ≤ zj ≤ bpj − bpj−1 ,
j = 2, . . . , m − 1,
0 ≤ zm ,
with a special restriction
zj+1 = 0
whenever zj < bpj − bpj−1 .
(22)
For any concave function h(x), its slope decreases as zj is increased, and the algorithm for maximizing h(x)
automatically gives the highest priority to using z1 , the next highest priority to using z2 , and so on. Thus,
the special restriction is implicitly included in the model and all the constraints can be modelled without the
use of indicator variables. Using the above, any concave function h(x) is approximated by:
h(x) ≈ h(ε) +
m
X
h(bp
j )−h(bpj−1 )
zj .
bpj −bpj−1
(23)
j=1
As explained in App. D the results of the linerization are sensitive to the choice of the minimum and maximum
breakpoints, the length of the interval, and the number of breakpoints. The approximations are best for short
intervals and for small values of x such as x ∈ [0.01, 1]. Therefore, to ensure a good approximation, we rescale
all input data regarding payments, e.g. initial savings, x0 , and labor income/premiums, lt . Fig. 3 shows an
example of the accurate approximation of the utility of consumption for each node associated with the first
five years of the contract (341 nodes).
4.4
Scenario tree
The uncertainty associated with the market returns is modeled by an N -dimensional random process. The
multivariate return process evolves in discrete time, and the underlying probability distributions are approximated by discrete distributions in terms of a scenario tree.
13
Our goal is to reproduce the model assumptions that lead to the explicit solutions in the stochastic optimal
control approach. The prices for the securities follow the Black Scholes model defined in eq. (1), therefore,
we aim for constructing
a multi-period scenario tree with a discrete representation of a normal distribution
N αi − σi 2 /2, σi .
There are plenty of scenario generation methods for stochastic programming, whose evaluation has been
described in Kaut and Wallace (2005). To find a discrete representation of a normal distribution, while
keeping the minimum number of nodes, we choose to follow the approach presented in Høyland and Wallace
(2001), i.e. we match the first four moments plus the correlations of the simulated processes. This algorithm
has been improved in Høyland et al (2003), where a scenario tree is obtained by a number of transformations.
According to the authors, the new algorithm is efficient for large trees, and does not require solving a
nonlinear optimization problem, which can be time consuming. For our work, however, we choose the older
algorithm, i.e. Høyland and Wallace (2001). In our experiments it performs better for small trees, and allows
to define the probabilities of each branch as variables (in contrast to Høyland et al (2003) which requires
the pre-specified probabilities). The approach is improved by a priori adding the non-arbitrage constraints
suggested by Klaasen (2002).
5
Numerical results
In this section we first present realistic examples illustrating problems (A) and (B) in the original model setup
(continuous time) and the SLP solutions obtained by the SLP model presented in Sec. 4.1. Afterwards, we
add or modify the particular constraints as described in Sec. 4.2 and investigate their impact on the optimal
controls during the first years of the contract.
Parameters. If not specified in the captions of the tables and figures, the following parameters have been
chosen for testing the models:
• Market: the number of assets, N = 3; two risky and one risk-free asset, the expected returns are,
respectively, α1 = 0.05, α2 = 0.07, r = α3 = 0.02, the volatility of the assets, σ1 = 0.2, σ2 = 0.25,
σ3 = 0, and the correlation between the risky assets is corr = 0.5; all parameters are adjusted for
inflation.
• Utility function: risk aversion, 1 − γ = 4, corresponding to the optimal proportion in risky asset after
retirement πt∗ equal to 25%, the impatience factor for the utility weights, ρ = 0.04, intuitively chosen
such that ρ ≥ r, and the weight on the utility of bequest relatively to the utility of consumption, λ = 5,
implying that the optimal amount which the inheritors receive upon death is approximately equal to
five years of the person’s consumption while she is alive, Xt∗ + It∗ ≈ 3.5c∗t (λ is only relevant for problem
(A)).
• Contract: age at the beginning of the contract, age0 = 45, retirement age, ageT = 65, the final age at
which the person is assumed to be dead with certainty, Te = 110, type of retirement benefits (problem
(B)): life annuity.
• Lifetime uncertainty: the mortality intensity model is of the form µt = µ∗t = θ + 10β+δ(y+t)−10 , where
y is the age of the person at time t0 , and θ, β and δ are constants2 . The model has been calibrated to
the mortality rates among Danish women in 2010 over age 40, Finanstilsynet (2010), where θ = 0.0,
β = 4.59364, δ = 0.05032.
• Scenario tree: number of SLP stages, TSLP = 6, branching factor in a single tree (number of nodes),
bf = 4, number of trees notree = 50, which implies in total 45 × 50 = 51.200 scenarios.
• Linearization: number of breakpoints, m = 40, minimum and maximum breakpoints for the consumption, bpc1 = 0.7E[c∗TSLP ], bpcm = 2E[c∗TSLP ], minimum and maximum breakpoints for the savings plus the
x+g
= 2E[XT∗SLP + gTSLP ],
present value of labor income/premiums, bp1x+g = 0.5E[XT∗SLP + gTSLP ], bpm
x+ins
minimum and maximum breakpoints for the savings plus insured sum, bp1
= 0.8E[XT∗SLP +
∗
x+ins
∗
∗
ITSLP ], bpm
= 1.5E[XTSLP + ITSLP ],
2 In principle the mortality intensity model does not assume that an individual is dead at time T
e with probability 1. However,
for Te = 110, this error is negligible.
14
• Payments:
– Problem (A). Labor income lt = 27.000 EUR, corresponding to the average Danish disposable
income (after taxes) for a 45-year old individual, Danmarks Statistik (2010), average savings of a
45-year old individual, x0 = 60.000 EUR,
– Problem (B). Before retirement: premiums, lt = 4.000 EUR, corresponding to 15% of the salary.
Assuming that the person has been contributing to the pension account for around 15 years,
accumulated with some capital gains, gives the value of the initial savings of x0 = 75.000 EUR;
After retirement: premiums, lt = 0, the value of the savings is estimated from the expected savings
B
for a 70 year old person (see before retirement case), i.e. x0 = E[X25
] = 225.000 EUR.
Similarly as Geyer et al (2009a), we have chosen to consider the small-scale optimization problems rather than
the large-scale problems. On the first place, we are interested in replicating the optimal controls obtained
from the continuous time models. The small-scale models are sufficient to evaluate whether a similar solution
can be achieved by running the SLP model. With only three assets: one risk-free and two risky assets, the
first four moments of a normal distribution can be approximated with 4 nodes. 6 periods with 4 nodes each
give in total 45 = 1.024 scenarios. Additionally, we rerun each program notree times for different scenario
trees, and present the results in terms of means and standard errors from sampling. We have tested the results
for the various number of trees, and choose notree = 50, since for ntree > 50 the results remain unchanged.
Each problem has been implemented using Matlab 7.12.0 (R2011a) for calculation of the explicit solutions,
GAMS 23.7.3 with CONOPT 3 (3.15A) solver for scenario generation and finding the position of breakpoints,
and GAMS with CPLEX 12.3.0.0 solver for the SLP formulation and solution of the stochastic programming
models. The running time of each model for one scenario tree takes approximately 20 seconds on a Dell
computer with an Intel Core i5-2520M 2.50 GHz processor and 4 GB RAM.
5.1
Numerical results for Problem (A)
We start with analyzing the optimal controls obtained explicitly by solving Richard (1975)’s model and
the SLP solutions replicating this model (no additional constraints). Fig. 4 (top left and right) shows the
development in the expected values of the savings account, Xt∗ , optimal consumption, c∗t , and insured sum, It∗ ,
in 1000 EUR (left figure), as well as the optimal proportion in both risky assets, πt∗ , and optimal proportion
in the first risky asset πt∗1 (right figure) during the entire lifetime of the individual. The markers, which
represent the optimal decisions obtained from the SLP model, closely follow the continuous lines representing
the explicit solutions. Table 1 shows the exact numbers plotted on the figure for the first five years of the
contract. Since we compare a continuous time model with a combined continuous and discrete time model,
we expect the discrepancies. The explicit solutions are calculated based on the assumption that the labor
income, consumption, insured sum, as well as capital gains occur simultaneously. In the SLP solutions, we
first account for the payments, and then make the investment decisions based on the value of the savings
after the payments. Therefore, the investment control obtained in the SLP formulation should be compared
with the average continuous time investment over the two adjacent time periods. Furthermore, with each
time period the discretization error accumulates, which can be seen on the development of the values of
savings, Xt∗ , hence the discrepancies in all optimal controls are expected to occur. The standard deviations
calculated from testing different scenario trees are negligibly small, which means that increasing the number
of scenarios will not necessarily improve the precision of the results.
Richard (1975)’s model implies that the initial proportion in risky assets for our choice of parameters is
given by πt∗0 = 181%, where the distribution between the risky assets is specified by the mutual fund theorem
as πt∗1
= 0.33πt∗0 and πt∗2
= 0.67πt∗0 , see Appendix B. The optimal solution favours the asset with a higher
0
0
return but also a higher volatility, therefore πt∗1 < πt∗2 , ∀t . These proportions decrease smoothly over time
as the individual manages to increase her savings due to labor income and capital gains. After retirement
πt∗ remains constant, which is a classical result known in the literature. The SLP solution closely follow the
explicit solutions, though it consistently remains a few percentages below the original solutions. In particular,
the SLP solution slightly favors the first risky asset - with a higher return and higher volatility - and the
ratio πt∗1 /πt∗ = 36%. The optimal consumption rate increases at a very slow pace over the entire lifetime.
The increase is caused by the choice of impatience factor ρ and risk tolerance γ in relation to the choice of
market parameters αi , σi , and mortality rates, as explain later in the analysis of Figure 6. An interesting
fact is that for our parameters choice the insured sum It∗ is positive only during the first year of the contract.
Afterwards, It∗ is negative, meaning that it is optimal to purchase the life annuities which would increase
15
Expected value of the savings, optimal consumption and insured sum
Optimal investment in risky assets
X*t − expl.
X*t − SLP
−I*t − expl.
−I*t − SLP
c*t − expl.
c*t − SLP
350
300
250
200
150
π*1
− SLP
t
2
π*t − SLP
π*1
− expl.
t
1.5
π*t − expl.
1
100
0.5
50
0
45
65 − retirement
0
45
85
55
Optimal investment in risky assets, π*t
1.8
Optimal investment in the first risky asset, π*1
t
1.8
π*t : orig. constraints
1.6
1.2
π*t : qi=0.5%
1.4
π*t : τi=20%
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
45
46
47
48
49
π*t : orig. constraints
1.6
π*t : πt ≤ 1 and It ≥ 0
1.4
65 − retirement
0
50
π*t : πt ≤ 1 and It ≥ 0
π*t : qi=0.5%
π*t : τi=20%
45
46
47
48
49
50
Optimal insured sum, I*t
10
0
−10
−20
−30
I*t : original constraints
−40
I*t : qi=0.05%
−50
I*t : πt≤ 1 and It ≥ 0
I*t : τi=20%
45
46
47
48
49
Figure 4: Problem (A). Explicit solution and SLP solution for the expected value of the savings, optimal consumption and insured
sum (left, top, in 1000 EUR). Optimal investment in both risky assets and in the first risky asset (right, top, πt∗ = πt∗1 + πt∗2 ).
The SLP solution for the expected optimal investment in both risky assets (left, middle) and in the first risky asset (right,
middle) given different constraints. The SLP solution for the expected optimal insured sum given different constraints (bottom,
in 1000 EUR).
the individual’s savings rather than to purchase a life insurance from the pension company. This result is
also consistent with Richard (1975) conclusions. Recall that the amount that is paid out to the inheritors
upon the person’s death is not equal to the insured sum, but to the value of the savings plus the insured
sum. Therefore It∗ can be negative as long as Xt∗ + It∗ > 0. The size and the sign of insured sum depends
on parameter λ. Choosing λ = 1 means that the individual puts an equal weight on her consumption and
the bequest motive, therefore the optimal legacy amount is approximately equal to the optimal consumption
while the person is alive, i.e. Xt∗ + It∗ ≈ c∗t . Fig. 4 and Table 1 show the case for λ = 5, where the optimal
death sum corresponds to approximately five years of optimal consumption while the person is alive, i.e.
Xt∗ + It∗ ≈ 3.5c∗t .
The example takes as input some realistic parameters, however, not all the optimal controls are actually
realistic. First, the insured sum is negative for t > t0 . In practise the life annuities can first be offered to
a person when she retires, and not 20 years before retirement. Secondly, πt∗ > 1 for t ≤ t3 . It may happen
that the individual does not have an opportunity to borrow money and short the risk-free asset. We have
16
control
model
expl.
orig. constraints
πt ≤ 1 and It ≥ 0
Xt∗
q i = 0.5%
τ i = 20%
expl.
orig. constraints
πt ≤ 1 and It ≥ 0
πt∗
q i = 0.5%
τ i = 20%
expl.
orig. constraints
πt ≤ 1 and It ≥ 0
πt∗1
q i = 0.5%
τ i = 20%
expl.
orig. constraints
π t ≤ 1 and It ≥ 0
c∗t
q i = 0.5%
τ i = 20%
expl.
orig. constraints
πt ≤ 1 and It ≥ 0
It∗
q i = 0.5%
τ i = 20%
t0
60.0
60.0
(0.0)
60.0
(0.0)
60.0
(0.0)
60.0
(0.0)
t1
72.3
72.9
(0.1)
70.9
(0.0)
71.3
(0.0)
68.5
(0.2)
t2
84.9
85.9
(0.2)
82.6
(0.1)
83.9
(0.1)
77.4
(0.4)
t3
97.6
99.0
(0.3)
94.9
(0.1)
97.0
(0.2)
86.5
(0.5)
TSLP − 1
110.5
112.3
(0.4)
107.6
(0.1)
110.3
(0.2)
95.8
(0.7)
1.81
1.79
(0.02)
1.00
(0.00)
1.44
(0.02)
0.81
(0.09)
1.51
1.49
(0.02)
1.00
(0.00)
1.39
(0.01)
0.70
(0.08)
1.30
1.26
(0.01)
1.00
(0.00)
1.25
(0.01)
0.61
(0.07)
1.12
1.10
(0.01)
0.96
(0.01)
1.12
(0.01)
0.54
(0.07)
0.99
0.96
(0.02)
0.89
(0.02)
1.03
(0.02)
0.48
(0.06)
0.60
0.64
(0.02)
0.14
(0.03)
0.46
(0.02)
0.06
(0.06)
0.50
0.53
(0.02)
0.21
(0.01)
0.44
(0.01)
0.06
(0.05)
0.43
0.46
(0.02)
0.26
(0.02)
0.42
(0.01)
0.05
(0.05)
0.37
0.40
(0.02)
0.28
(0.02)
0.39
(0.01)
0.04
(0.05)
0.33
0.35
(0.02)
0.28
(0.03)
0.37
(0.01)
0.04
(0.05)
20.8
20.7
(0.0)
20.7
(0.0)
20.7
(0.0)
20.7
(0.0)
20.8
20.8
(0.1)
20.6
(0.1)
20.6
(0.1)
20.4
(0.1)
20.8
20.9
(0.1)
20.7
(0.0)
20.7
(0.0)
20.3
(0.1)
20.8
20.9
(0.1)
20.7
(0.0)
20.7
(0.0)
20.3
(0.1)
20.8
21.0
(0.1)
20.7
(0.0)
20.7
(0.0)
20.1
(0.1)
9.5
9.8
(0.4)
9.3
(0.0)
9.0
(0.2)
8.8
(0.3)
-2.8
-3.2
(0.2)
4.8
(0.3)
-1.9
(0.2)
-0.2
(0.3)
-15.2
-16.0
(0.3)
3.1
(0.2)
-14.5
(0.2)
-9.4
(0.4)
-27.9
-29.1
(0.4)
1.7
(0.1)
-27.4
(0.2)
-18.8
(0.6)
-40.8
-42.3
(0.5)
0.9
(0.1)
-40.6
(0.2)
-28.4
(0.7)
Table 1: Problem (A). Explicit solution and SLP solution with various modifications for the expected value of the savings,
optimal consumption and insured sum (in 1000 EUR) and optimal investment in both risky assets and in the first risky asset
(πt∗ = πt∗1 + πt∗2 ). The numbers are presented in terms of means and standard errors (in parenthesis).
mentioned that one of the drawbacks of dynamic programming is that the explicit solutions exist only if we
consider simple models, such as the one above. As soon as we add realistic constraints, such as, not allowing
for purchasing a life annuity before retirement, or not allowing for being short in any asset, the explicit
solutions cannot be derived. Being able to add realistic constraints is the main reason to include the SLP
17
part, and hence to combine the stochastic optimal control approach with stochastic programming approach
into one formulation. We have shown that the closed-form solutions can be replicated well. The numbers are
not the exact match, but the patterns and the level of the optimal controls are consistent with the continuous
time models. Therefore, we find the model reliable and use it as a tool to analyze the optimal controls under
the effects of various modifications during the first years of the contract.
A borrowing constraint and an annuity purchase constraint. Figure 4 (middle left and right, and
bottom plot) and Table 1 show the optimal solutions that simultaneously account for two additional constraints: (i) the individual does not have a possibility to borrow money in order to invest more than she
has in other assets (this constraint is defined in eq. (16) where ui = 100%), and (ii) the individual is not
allowed to purchase a life annuity before retirement, It > 0 for t < T , as defined in eq. (17). As expected,
the constraints directly affect the optimal investment and insured sum. The optimal investment πt∗ is now
equal to its upper limit 100% and decrease faster than in the original model setup. The constraint naturally
also implies a change in the distribution between the risky assets. Before, the ratio of the first risky asset
to the total risky investment was given by πt∗1 /πt∗ ≈ 36%. With the new constraints the ratio starts with
14% at t0 and with time converges to the original optimal ratio, which implies that πt∗1 increases until the
optimal ratio between the risky assets is reached. The optimal insured sum is positive and converges to its
lower limit 0. Finally, the value of the savings, and as follows, the optimal consumption, is lower, due to
more conservative investment and positive costs for the insured sum µ∗t It .
Transaction costs. Secondly, we investigate the effect of transaction costs q i . These have been defined
as a constant percentage of a traded amount and are the same for purchases and sales and for all assets,
see eq. (10A’) As a result, the amount invested in risky assets is lower during the first few years than in
the original problem, but after a few years stays above the original πt∗ . We conclude that introducing the
transaction costs implies that the development of the risky investment over time is smoother, i.e. it decreases
as in all other cases but the difference between the maximum and minimum value is smaller. We explain it
by the fact that rebalancing portfolio is expensive. A smoother optimal investment implies the lower traded
amount, hence lower costs. Other optimal controls, i.e. consumption and insured sum, are not much affected
by the transaction costs. As well as the total savings, the optimal values are slightly lower (lower absolute
value for the insured sum). Fig. 4 (middle, left and right) shows the results for q i = 0.5%, but we have also
investigated higher costs, q i = 1%, and the patterns for optimal controls remain similar.
Taxes on capital gains. Finally, we are interested in the impact of capital gains taxes on the optimal
decisions. We have integrated a linear taxation on the positive returns on asset i, τ i , as defined in eq. (18).
In comparison with other constraints, the taxes on capital gains are the most influential for the optimal
decisions. Only τ i = 20% causes already a loss of 17% of the expected savings after 5 years relatively to
the original problem. Less savings implies less money to consume and to be paid for the life annuity. Both
the consumption and the absolute value of the insured sum decrease. But the biggest impact can be seen
on the investment controls. The optimal percentage in stocks is only πt∗0 = 81% comparing to the original
πt∗0 = 179%, while the proportion in the first risky asset is only πt∗1
= 6%. Subtracting taxes only from
0
the positive returns causes that the scenario tree is asymmetric and favours the risk-free assets (the positive
outcomes are lower, whereas the negative outcomes are as severe as they were before). A similar conclusion
has been drawn by Geyer et al (2009a). The SLP solutions are also more volatile for this case than for other
considered constraints. Especially, the investment controls varies significantly for different scenario trees.
5.2
Numerical results for problem (B)
This optimization problem is relevant for a pension product design in a defined contribution pension scheme.
In practice the individual cannot control the size of the pension benefits. This decision belongs to the
pension fund. However, the pension fund can take into consideration the pension saver’s risk preferences γ,
the market parameters hidden under the variable ϕ, the impatience factor ρ and both regular and pricing
mortality intensities µt and µ∗t and design the benefits (payout profile) such that the individual’s preferences
are included.
t
Xt , see eq. (7B). Inserting the definitions of
The optimal pension benefits are of the form c∗t = fBw(t)
18
functions wt and fB (t) leads to
ct =
Xt
āy+t ,
t ∈ [T, Te)
(24)
where āy+t is a traditional single-life annuity that pays 1 EUR per year continuously to an individual who is
y + t years old, i.e.
Z Te R
− ts r̄+µ̄τ dτ
āy+t =
e
ds,
t
where
r̄ =
1
1−γ ρ
−
γ
1−γ ϕ,
ϕ=r+
(α−r)2
2σ 2 (1−γ) ,
µ̄τ =
1
1−γ µτ
γ
∗
1−γ µτ .
−
Parameters r̄ and µ̄t are the utility adjusted interest rate and mortality rate, respectively.
Figures 5 and 6 explain the concept of optimal annuity design and show the expected optimal life annuities
c∗t for the individuals with different risk aversion 1 − γ and impatience factor ρ. Fig. 5 shows the initial
value of the benefit at time of retirement, c∗T , and its sensitivity on γ and ρ. Given the value of the savings
account XT = 265.000 EUR and no premiums after retirement, the size of the life annuities for t = T varies
between 12.800 EUR and 19.500 EUR. The lowest initial benefit is for patient persons who prefer more risky
investment, i.e. those characterized by f.ex. ρ = 0.0 and γ = 0, whereas the highest benefits will be received
still by those who prefer more risky investment but are impatient, i.e. ρ = 0.04 and γ = 0. The tendency is
similar for the risk averse retirees - the impatient ones will receive the highest annuities in the beginning of
the retirement.
19
18
17
16
15
14
13
0.04
0.02
0
0
−0.5
−1
−1.5
−2
−2.5
−3
γ
ρ
Figure 5: Sensitivity of the optimal life annuities at the beginning of retirement, c∗T , to risk aversion 1 − γ and impatience factor
ρ. The numbers are in 1000 EUR. xT = 265. Other parameters as specified in Sec. 5.
Fig. 6 shows the development of the size of the benefits during retirement (the payout profile) for different
values of γ and ρ. We immediately notice that the optimal payout profile is different for individuals with
different patience levels and risk preferences. The annuity payments can be constant, decreasing or increasing.
To be precise, it is the value of the impatience factor ρ relatively to the average return on investment adjusted
for the risk aversion level 1−γ that drives the wealth process, which controls the development of the payments.
Impatient persons (f.ex. with ρ = 0.08) wish to receive the highest benefits in the first years of the retirement,
whereas those who are indifferent between the benefits received just after retirement or later in the future
(ρ = 0.0) receive lower benefits in the beginning. The latter allows for larger capital gains during retirement,
thus, those who are patient and survive until age 95, will on average receive between 2.5-5 times higher benefit
at age 95 (depending on their risk aversion) than upon retirement. Parameter γ affects mostly the size of the
benefits. The more risky investment gives on average a higher return, hence higher benefits. Furthermore,
risk averse persons (f.ex. γ = −3) will receive each year smoother benefits (the difference between c∗65 and
c∗95 is smaller) than for a person with a low risk aversion, such as γ = 0.
19
model
t0
t1
t2
t3
TSLP − 1
Xt∗
expl.
orig. constraints
πt∗
expl.
orig. constraints
πt∗1
expl.
orig. constraints
75.0
75.0
(0.0)
0.45
0.44
(0.01)
0.15
0.16
(0.01)
82.0
82.2
(0.0)
0.42
0.42
(0.01)
0.14
0.15
(0.01)
89.1
89.6
(0.0)
0.40
0.40
(0.01)
0.13
0.15
(0.01)
96.5
97.2
(0.1)
0.38
0.39
(0.01)
0.13
0.14
(0.01)
104.1
105.1
(0.1)
0.37
0.36
(0.01)
0.12
0.13
(0.01)
225.0
225.0
(0.0)
225.0
(0.0)
225.0
(0.0)
225.0
(0.0)
216.9
216.7
(0.0)
216.7
(0.0)
215.7
(0.1)
214.3
(0.1)
208.7
208.5
(0.1)
208.6
(0.2)
207.4
(0.1)
203.8
(0.1)
200.6
200.2
(0.2)
200.3
(0.2)
198.9
(0.1)
193.4
(0.2)
192.5
191.8
(0.2)
192.0
(0.2)
190.4
(0.1)
183.0
(0.3)
0.25
0.25
(0.00)
0.29
(0.00)
0.26
(0.00)
0.11
(0.01)
0.25
0.25
(0.00)
0.29
(0.00)
0.25
(0.00)
0.11
(0.01)
0.25
0.25
(0.00)
0.29
(0.00)
0.25
(0.00)
0.11
(0.01)
0.25
0.25
(0.01)
0.29
(0.00)
0.25
(0.00)
0.11
(0.01)
0.25
0.25
(0.01)
0.29
(0.00)
0.24
(0.00)
0.11
(0.01)
0.08
0.09
(0.00)
0.15
(0.00)
0.09
(0.00)
0.01
(0.01)
0.08
0.09
(0.00)
0.15
(0.00)
0.09
(0.01)
0.01
(0.01)
0.08
0.09
(0.00)
0.15
(0.00)
0.09
(0.00)
0.01
(0.01)
0.08
0.09
(0.01)
0.15
(0.00)
0.09
(0.00)
0.01
(0.01)
0.08
0.09
(0.01)
0.15
(0.00)
0.08
(0.00)
0.01
(0.01)
17.8
17.8
(0.0)
17.8
(0.0)
17.7
(0.1)
17.3
(0.0)
17.9
17.7
(0.1)
17.7
(0.1)
17.6
(0.1)
17.1
(0.1)
17.9
17.7
(0.0)
17.7
(0.0)
17.6
(0.0)
17.1
(0.0)
17.9
17.7
(0.0)
17.7
(0.0)
17.6
(0.0)
17.0
(0.0)
17.9
17.7
(0.0)
17.8
(0.0)
17.6
(0.0)
16.9
(0.0)
control
before retirement
after retirement
expl.
orig. constraints
π ∗1 ≥ 15%
Xt∗
q i = 0.5%
τ i = 20%
expl.
orig. constraints
π ∗1 ≥ 15%
πt∗
q i = 0.5%
τ i = 20%
expl.
orig. constraints
π ∗1 ≥ 15%
πt∗1
q i = 0.5%
τ i = 20%
expl.
orig. constraints
π ∗1 ≥ 15%
c∗t
q i = 0.5%
τ i = 20%
Table 2: Problem (B) before and after retirement. Explicit solution and SLP solution with various modifications for the expected
value of the savings and optimal benefits (in 1000 EUR) and optimal investment in both risky assets and in the first risky asset
(πt∗ = πt∗1 + πt∗2 ). The numbers are presented in terms of means and standard errors (in parenthesis).
20
50
ρ=0.00, γ=−3
ρ=0.00, γ=−1
ρ=0.00, γ=0
40
30
20
10
65 − retirement
70
75
80
85
90
30
ρ=0.04, γ=−3
ρ=0.04, γ=−1
ρ=0.04, γ=0
20
10
65 − retirement
95
70
75
80
85
90
30
95
ρ=0.08, γ=−3
ρ=0.08, γ=−1
ρ=0.08, γ=0
20
10
65 − retirement
70
75
80
85
90
95
Figure 6: Optimal benefits c∗t for the first 30 years of retirement. Sensitivity of the benefits to various choices of the impatience
factors ρ = {0.00, 0.04, 0.08} and risk preferences, γ = {−3, −1, 0}. The numbers are in 1000 EUR. xT = 265. Other parameters
are as specified in Sec. 5.
Basic constraints. Analogously as in Sec. 5.1 we present the results for problem (B) for the time before
retirement, t < T , and after retirement, t > T , on Fig. 7 and Table 2. Fig. 7 shows the development in the
expected values of the savings account, Xt∗ , and optimal life annuities, c∗t (in 1000 EUR), before retirement
(top, left) and after retirement (middle, left). The figures on the right show the optimal proportion in the
first risky asset πt∗1 during the lifetime of the individual for the case before retirement (top, right) and after
retirement (middle, right). Given that at age 45 the individual has x0 = 75.000 EUR on her savings account
and she continues to pay the premiums of lt = 4.000 EUR every year, which are invested in a conservative
way, i.e. γ = −3, she should expect that her savings at retirement will reach 265.000 EUR, which will define
the pension benefits at almost c∗t = 18.000 EUR at age 65. The size and development of the benefits is
optimal given she puts more weight on the benefits just after retirement than on those in the future, and
given she is risk averse (the percentage in risky assets at retirement is only about 25%), i.e. ρ = 0.04 and
γ = −3. The investment before retirement is more conservative than in problem (A). The process lt denoting
the pension contributions is much lower than in the previous case, where lt was defined as the labor income.
The optimal investment control that we derive in problem (B) follows the classical result saying that the
optimal investment is proportional to the sum of the value of the savings and the present value of the pension
Xt +gB (t)
contributions relatively to the value of savings, recall eq. (7B), πt∗ = σ2α−r
. Therefore, πt∗ is much
(1−γ)
Xt
lower than in problem (A) starting with 45% at age 45 and decreasing afterwards. After retirement, the
optimal proportion in risky assets remains constant at 25%. The ratio of the investment in the first risky
asset to the total risky assets according to the mutual fund theorem remains the same, i.e. πt∗1 /πt∗ = 33%,
both before and after retirement.
Similarly as in problem (A) we conclude that the SLP solution closely follows the optimal controls derived
explicitly. The value of the savings obtained from the SLP formulation is slightly higher (before retirement)
or lower (after retirement) with each time period, whereas as in problem (A) the ratio between the risky
assets is a few percentage higher than for the continuous time model. As mentioned before, we expect the
discrepancies due to the discretization. The SLP model, in contrast to the continuous time model, assumes
that the decision on investment is made after all the payments occur, and not simultaneously.
In contrast to problem (A), we find that the model provides a realistic solution that can be applied by
a pension fund. Nevertheless, there is a number of constraints whose effect on the optimal control is worth
investigating. Incorporating the pension saver’s preferences, such as the limits on the portfolio composition,
or practical constraints, e.g. transaction costs and taxes on capital gains in a dynamic programming approach
is non trivial. Therefore, we have included the SLP part in our model, which adds flexibility to the model
21
Expected value of the savings and optimal benefits
Optimal investment in risky assets
X*t − expl.
X*t − SLP
c*t − expl.
300
250
π*1
− SLP
t
0.5
π*t − SLP
π*1
− expl.
t
0.4
π*t − expl.
200
0.3
150
0.2
100
0.1
50
0
45
65 − retirement
0
45
85
55
Expected value of the savings and optimal benefits
65 − retirement
Optimal investment in risky assets
X*t − expl.
X*t − SLP
c*t − expl.
c*t − SLP
200
150
π*1
− SLP
t
π*t − SLP
0.25
π*1
− expl.
t
π*t − expl.
0.2
0.15
100
0.1
50
0.05
0
70
75
0
70
80
75
Optimal investment in risky assets, π*t
80
Optimal investment in the first risky asset, π*1
t
π*t : orig. constraints
π*t : orig. constraints
0.3
π*t : π1t ≥ 15%
0.3
π*t : π1t ≥ 15%
0.25
π*t : qi=0.5%
0.25
π*t : qi=0.5%
π*t : τi=20%
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
45
46
47
48
49
π*t : τi=20%
0.2
50
0
45
46
47
48
49
50
Figure 7: Problem (B) before and after retirement. Explicit solution and SLP solution for the expected value of the savings and
optimal benefits before retirement (left, top, in 1000 EUR). Explicit solution and SLP solution for the expected value of the
savings and optimal benefits after retirement (left, middle, in 1000 EUR). Optimal investment in risky assets before retirement
(right, top) and after retirement (right, middle). The SLP solution for the expected optimal investment after retirement given
different constraints in both risky assets (bottom, left) and in the first risky asset (bottom, right). πt∗ = πt∗1 + πt∗2 .
22
and allows for investigating the desired factors.
Limits on the portfolio composition. Imagine a 70 year old pension saver who simply is not satisfied
with the way the pension fund invests her savings. In particular, she believes that the first risky asset is
more valuable and does not agree that after retirement only 9% of her savings are invested in this asset
(see Table 2, original constraints). She prefers a higher percentage in the first risky asset and defines her
new preferences, for instance, as minimum 15% should be invested in the first risky asset. Such a constraint
can easily be incorporated in the SLP model by adding eq. (16), where constant d1 is set to be equal 15%.
The optimal solution for this case is shown on Figure 7 (middle left and right) and Table 2. Both the total
investment in risky assets and the investment in each of the risky assets remain constant as before. Though,
both values are now higher. The pension saver’s preferences are taken into account and πt∗1 raises from 9%
in the original model setup to 15%, whereas the investment in both risky assets increases from 25% to 29%.
The fact the πt∗ increases as well can be explained by the fact that there exists an optimal ratio between the
risky assets which the model tries to remain as closely as possible.
The SLP formulation allows for adding various limits on the portfolio. The limits can be both from above
and below, and separately for each asset class. We have investigated various combinations of limits on a
particular asset, and without presenting the numbers, we came to the following conclusions. None of these
constraints have much effect on the overall value of the savings or the optimal benefits, but adding a limit
just to one asset class directly affects the distribution between all three asset classes. Imposing a higher
percentage on an asset class than the percentage given by the optimal control in the case without constraints,
results in setting this asset exactly to this limit. And, vice versa, limiting the amount of savings in a given
asset class from above when the original optimal investment suggest a higher amount, results in new optimal
investment equal to the limit. Furthermore, the limit set on one of the risky assets, triggers the change in
the same direction of the percentage of the second risky assets.
Transaction costs. In problem (A) we concluded that adding transaction costs, defined as a percentage
of a traded amount, implies a smoother development in the proportion invested in risky assets. In this
case, where the original optimal investment is constant, the development of πt∗ is not constant as well, but
actually decreases very slowly. The reason for that is that while the value of the savings decreases, trying
to rebalance the portfolio each year to a constant proportion is more expensive than rebalancing to the
decreasing proportion. As before, the changes in the optimal benefits are so small that it is hard to conclude
how the transaction costs affect the benefits, except for the obvious fact, that the savings are slightly lower,
which implies lower benefits.
Taxes on capital gains. Finally, to strengthen the conclusions we drew from investigating the effect
of adding taxes on the positive returns in problem (A), we repeat this investigation for problem (B). The
results are similar: the impact of the taxes on the optimal investment is significant. Without taxes the
optimal investment in both risky assets was given πt∗ = 25% and in the first risky asset πt∗1 = 9%. With the
presence of taxes, the optimal controls are down to πt∗ = 11% and πt∗1 = 1%. Again, we explain this fact
by the asymmetry in the scenario tree caused by subtracting taxes only from the positive capital gains. It is
interesting though that taxes affect the size of the investment but not its development, i.e. the proportions
remain constant as in the case of the original problem. Furthermore, the SLP solutions are not as volatile
as for problem (A). The standard deviations are low, which shows a stability for different scenario trees. In
terms of other controls, similarly as in problem (A) the taxes significantly affect the value of the savings,
implying lower and decreasing pension benefits.
6
Conclusions and future work
We have presented a model combining two optimization technologies: multi-period stochastic linear programming and dynamic programming, to obtain optimal decisions related to various problems within personal
finance and retirement. The optimization methods complement each other very well. SLP allows for building
flexible and practical models, which are difficult to solve with the stochastic optimal control approach. The
explicit formulas, however, can handle any number of periods and are useful for the validation of the results
obtained by the SLP.
23
The numerical results show that among considered limits on portfolio composition, limits on the insured
sum, transaction costs and taxes on capital gains, the presence of taxes affects the optimal controls the most,
and the changes in both the value of savings and the controlled processes are significant.
Except for the contribution regarding computational methods, the paper introduces an important aspect
regarding designing pension benefits in a defined contribution pension scheme. Problem (B) suggests how
the pension fund could decide on the initial size of the life annuities and their development during the
decumulation phase such that the individual’s risk and impatience preferences are captured. Interestingly,
the pension saver’s preferences have a higher impact on the size and development of the benefits than limits
on portfolio composition, transaction costs or taxes on capital gains.
The model can be improved in various ways. The most interesting would be the extension of the SLP
part to even more periods while still keeping the running time of the program within a couple of minutes.
Another suggestion is to introduce the bond market instead of the risk-free asset and/or the uncertainty
on the labor income/premiums. Finally, since the stochastic programming approach allows the objective
function to take a variety of forms, it could be of interest to consider a different utility function, for instance,
loss aversion framework, where an individual can control both the gains and losses in savings relative to a
pre-defined target, or a multi-objective function, taking into account other preferences than risk aversion and
impatience.
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Appendices
A
Derivation of the optimal controls for problem (B)
To our knowledge problem (B) has not been considered in the literature. We derive here the optimal value
function and the optimal controls for the problem of maximizing the utility of consumption, where the control
processes are specified over different time periods: investment is controlled both before and after retirement,
25
πt ∈ Q[t0 , Te), benefits are controlled only after retirement, ct ∈ Q[max(t0 , T ), Te). The problem is solved first
for the period after retirement, where we refer to Bruhn and Steffensen (2011), and then for the period before
retirement, which is one of the contributions of this paper. In the latter we insert the optimal value function
obtained in after retirement period as the boundary condition at time T .
After retirement period. The problem of optimizing expected utility of consumption after retirement in
the case where the person controls investment πt∗ , life insurance purchase It∗ and consumption c∗t , and has
no bequest motive has been solved in Bruhn and Steffensen (2011). The authors derive the optimal value
function
V (t, x) = γ1 ft1−γ xγ ,
(25)
and the explicit formulas for the optimal controls
πt∗ Xt =
where
Z
ft =
c∗t =
α−r
σ 2 (1−γ) Xt ,
Te
e
It∗ = 0,
wt
ft Xt ,
1 R
− 1−γ ts µτ −γ(µ∗
τ +ϕ) dτ
ws ds,
t
ϕ=r+
(α−r)2
2σ 2 (1−γ) .
(26)
Before retirement period. We apply Hamilton-Jacobi-Bellman techniques to solve the problem. We first
define the HJB equation for the considered period, based on the savings dynamics defined in eq. (2B):
n
o
2
(t,x)
(t,x)
∂V (t,x)
− µt V + sup lt ∂V∂x
+ r + πt (α − r) + µ∗t x ∂V∂x
+ 12 πt2 σ 2 x2 ∂ V∂x(t,x)
= 0,
2
∂t
V (T, x) =
π
1 1−γ γ
XT .
γ fT
The boundary condition is equal to the optimal value function at time T given by eq. (25).
To solve the problem we first guess the solution and verify that it is a correct guess:
γ
V (t, x) = γ1 ft1−γ x + gt .
We find the functions ft and gt by plugging in the derivatives to the HJB equation:
γ
γ−1 ∂gt
1−γ −γ ∂ft
+ ft1−γ x + gt
γ ft
∂t x + gt
∂t
γ−1
1−γ
γ
1 1−γ
= µt γ ft (x + gt ) − lt ft
x + gt
− (r + µ∗t )(x + gt )ft1−γ (x + gt )γ−1 + (r + µ∗t )gt ft1−γ (x + gt )γ−1 −
2
1−γ
1 (α−r)
(x
1−γ 2σ 2 ft
+ gt )γ ,
and obtain for t ∈ [t0 , T ):
Z T
Rs
∗
gt =
e− t (r+µτ )dτ ls ds,
(27)
t
ft = e
1 R
− 1−γ tT (µτ −γ(µ∗
τ +ϕ))dτ
fT = e
1 R
− 1−γ tT (µτ −γ(µ∗
τ +ϕ))dτ
Z
Te
e
1 R
− 1−γ Ts µτ −γ(µ∗
τ +ϕ) dτ
ws ds.
(28)
T
The optimal investment is then given by:
∂
∂π
B
:
π = − α−r
σ2 x
∂V (t,x)
∂x
∂ 2 V (t,x)
∂x2
=
1 α−r x+gt
1−γ σ 2
x .
Mutual fund
Theorem 1 (Merton (1971)). Given n assets with prices Si whose changes are log-normally distributed,
then (1) there exists a unique pair of ”mutual funds” constructed from linear combinations of these assets
such that, independent of preferences (i.e., the form of utility function), wealth distribution, probability of
distribution of lifetime individuals will be indifferent between choosing from a linear combination of these two
funds or a linear combination of the original n assets.
26
Corollary 1 (Merton (1971)). If one of the assets is ”risk-free” [assume with the index i = N ], then the
proportions of each asset held by the mutual funds are
∀i=1,...,N −1
θi =
PN −1
[σij ]−1 (αi −r)
PN −1i=1
PN −1
.
−1 (αj −r)
i=1
j=1 [σij ]
Hence, without loss of generality we can work with only two assets, one risk-free and the other a mutual fund
of risky assets with a log-normally distributed price St defined by
dSt = αSt dt + σSt dWt ,
with
α=
N
−1
X
σ2 =
θi αi ,
i=1
C
N
−1 N
−1
X
X
θi θj σij ,
dW =
i=1 j=1
N
−1
X
θi σσi dWi .
i=1
Derivation of the expected values of the savings
The expected values of the savings account, optimal investment, optimal consumption and optimal insured
shown in the plots in Sec. 5 can be derived explicitly. Knowing these values is not only useful for analyzing
the numerical results, but also for estimating the minimum and maximum breakpoints when linearizing the
utility function, 4.3.
Denote Zt := Xt∗ + g(t), then dZt = dXt∗ +
∂g(t)
∂t
and
∂g(t)
∂t
= (r + µ∗t )g(t) − l(t).
Problem (A). The dynamics of process Z are given dZt = dXt∗ + ∂g(t)
∂t . Inserting the dynamics of process
X together with the optimal controls gives
1/(1−γ)
(α−r)2
∗
∗ µt
vt
wt
α−r
dZt = r + µt + σ2 (1−γ) − fA (t) − µt µ∗
fA (t) Zt dt + σ(1−γ) Zt dWt
t
1/(1−γ)
vt
α−r
t
= ψt − fAw(t)
− µ∗t µµ∗t
fA (t) Zt dt + σ(1−γ) Zt dWt ,
t
Z0 = x0 + gA (0),
2
with ψt = r + µ∗t + σ(α−r)
2 (1−γ) .
The solution to the equation is given by:
Z t 1/(1−γ)
s
Zt = Z0 exp
ψs − fAw(s)
− µ∗s µµ∗s
s
0
vs
fA (s)
−
(α−r)2
2σ 2 (1−γ)2
ds +
α−r
σ(1−γ) Wt
,
(29)
implying that Xt > −gA (t), ∀t . Process Z is a geometric Brownian motion, whose expected value is given
explicitly:
Z t 1/(1−γ)
∗ µs
ws
vs
E[Zt ] = Z0 exp
ψs − fA (s) − µs µ∗
fA (s) ds .
0
s
From the definition of Z we can calculate the expected value of the savings process:
E[Xt∗ ] = E[Zt ] − gA (t).
Problem (B). Analogically, we derive the expected value of savings for problem (B) before and after
retirement:

nR
o
t
 x0 + gB (0) exp
ψ
ds
− gB (t), t ∈ [t0 , T ),
s
0 nR o
E[Xt∗ ] =
t
s
 x0 exp
ψs − fBw(s)
ds ,
t ∈ [T, Te).
0
27
D
Comparison of linearization methods
In this section we compare the linearization method presented in our work with the method suggested in
Geyer et al (2009a). There are two main differences between our approaches: finding the position of breakpoints and defining the formula for the linear approximation.
First, we compare the position of breakpoints defined by the method described in Geyer et al (2009b) and in
our work defined by eq. (19)-(20). The first method defines the minimum and maximum breakpoints and uses
the curvature of the utility function to obtain optimal positions of the remaining breakpoints. We refer the
reader to Geyer et al (2009a) for details. Figures 8 and 9 show, respectively, the curvature of the considered
utility functions and the utility function with the most defined curvature (γ = −3) with the optimal positions
of the breakpoints for different minimum and maximum breakpoints, as well as for the different lengths of
the intervals. The plots to the left in Fig. 6 present the position of breakpoints obtained from the curvature
method, and the plots to the right, breakpoints given by the optimization method applied in our work. In
the comparison we have used m = 40 breakpoints.
γ=−3
γ=−1
γ=0
−0.2
−0.4
κ(x)
−0.6
−0.8
−1
−1.2
−1.4
1
2
3
4
5
x
6
7
8
9
10
Figure 8: Curvature of utility functions with γ ∈ {−3, −1, 0}.
We have indicated two main differences between the two methods. Firstly, the curvature of the utility
function does not inherit one of the important properties of the utility functions, rescalability, i.e. the
shape of the utility function remains the same over all intervals A × [bpmin , bpmax ], ∀A > 0, where A is a
constant. Therefore, the optimal position of the breakpoints in the curvature method is different for intervals
Aj × [bpmin , bpmax ] and Ak × [bpmin , bpmax ], where Aj 6= Ak . On the contrary, the position of the breakpoints
obtained from the optimization method is rescaled, and remains the same for the mentioned intervals, as
can be seen on Fig. 9a and 9b, where we investigate the positions of the breakpoints over different intervals,
x ∈ [1, 10] and x ∈ [0.1, 1] = 0.1[1, 10]. Secondly, the breakpoints in the curvature method are concentrated
in the interval, where the absolute value of the curvature is the highest, whereas the breakpoints in the
optimization method are more spread out, though with more points around the arguments for which the
absolute value of the curvature is the highest. This is shown on Fig. 9c, where the chosen interval is longer,
x ∈ [0.1, 10].
Before we can compare the approximations based on the positions of breakpoints we have to define the
formula for the approximation. Geyer et al (2009b) approximate the utility function by
h̃(x) ≈
h(bp2 )−h(bp1 )
z1
bp2 −bp1
+
m
X
h(bp
j )−h(bpj−1 )
zj
bpj −bpj−1
+
h(bpm )−h(bpm−1 )
zm+1 ,
bpm −bpm−1
(30)
j=2
where
x=
m+1
X
zj ,
0 ≤ z1 ≤ bp1 ,
0 ≤ zj ≤ bpj − bpj−1 ,
j = 2, . . . , m,
0 ≤ zm+1 .
j=1
Both the above formula and the one defined in eq. (23) approximate a concave function by a certain number
of segments between the breakpoints, and define argument x in terms of non-negative decision variables zj
associated with each segment. The reason we changed eq. (30) to eq. (23) is to be able to control the
28
0
0
b40
−0.05
−0.1
−0.1
−0.15
−0.15
u(x)
u(x)
−0.05
b40
−0.2
−0.2
−0.25
−0.25
−0.3
−0.3
b1
−0.35
1
b1
2
3
4
5
6
7
8
9
10
−0.35
1
2
3
4
5
x
6
7
8
9
10
x
(a) x ∈ [1, 10]
0
b39
−50
−100
−100
−150
−150
u(x)
u(x)
0
−50
−200
−200
−250
−250
−300
b40
−300
b1
−350
0.1
b1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−350
0.1
1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
x
(b) x ∈ [0.1, 1]
0
b40
−50
−100
−100
−150
−150
u(x)
u(x)
0
−50
−200
−200
−250
−250
−300
−300
b1
−350
0
b40
b1
1
2
3
4
5
x
6
7
8
9
10
−350
0
1
2
3
4
5
x
6
7
8
9
10
(c) x ∈ [0.1, 10]
Figure 9: Positions of the breakpoints for the power utility function u(x) =
(plots to the left) and the optimization method (plots to the right).
1 γ
x ,
γ
γ = −3, obtained via the curvature method
linearization error. Notice, that all the considered utility functions are negative, and u(0) = −∞, whereas
function h̃(x) always returns a positive number. Therefore, we added a small number ε in our formulation,
0 < ε < bpmin , to have a starting point for which the approximated function is negative.
Table 3 compares the values of the utility function with their approximations based on two methods for
finding the position for the breakpoints: the optimization method (columns (3) and (5)) and the curvature
method (columns (4) and (6)) calculated accordingly to formula (23) (columns (3) and (4)) and formula (30)
(columns (5) and (6)). As also shown on Fig. 9, we have investigated three different intervals x ∈ [1, 10],
x ∈ [0.1, 1] and x ∈ [0.1, 10], and for each interval we choose five values of x equally spaced over the interval.
We first analyze columns (3) and (4), which differ with the positions of the breakpoints. The numbers in
bold show the approximation closest to the value of the utility function. Both optimization methods have the
same minimum breakpoint, hence for x = bp1 we obtain the same result. Out of 12 samples, the curvature
method outperforms the optimization method only twice. The results over interval x ∈ [0.1, 1] are particularly
interesting, since for the small values of x, such as x = 0.325 and x = 0.550 the approximation based on the
breakpoints obtained via the curvature method is very imprecise, whereas for x = 0.775 and x = 1.0 it slightly
outperforms the optimization method. Fig. 9b (left) supports these results. We notice that all the remaining
breakpoints in the curvature method are placed in the interval (0.65, 1.0), therefore the approximation is
good for x ≥ 0.65 and imprecise for x ≤ 0.65. The running time of finding positions of breakpoints for both
considered methods is a couple of seconds.
29
As for the linearization error - compare columns (2) with (3) and (4) - we conclude that both methods
are sensitive to the choice of the minimum and maximum breakpoints, as well as to the length of the interval.
The approximations are the best and can be precise for short intervals, such as x ∈ [0.1, 1] and for the small
values of x. Therefore, to ensure a good approximation, we rescale all input data regarding payments, i.e.
initial savings, x0 , and labor income/premiums, lt .
It is hard to say much about the results in columns (5) and (6). As mentioned before, we chose formula (23)
over (30) for the ease of controlling the linearization error. We believe that these two formulae approximate
the utility function similarly well because there is only a small difference between them - the starting point.
We have tested both formulae, and when it comes to the optimization, the values of decision variables zj are
in both cases the same.
formula for approximation
h(x), eq. (23)
h̃(x), eq. (30)
opt.
curv.
opt.
curv.
x
u(x)
x ∈ [1, 10]
1.000
3.250
5.500
7.750
10.000
-0.333
-0.010
-0.002
-7.16100E-4
-3.33333E-4
-0.312
-0.162
-0.181
-0.123
-0.124
-0.312
-0.174
-0.312
-0.312
-0.250
0.941
1.265
1.272
1.274
1.274
0.972
1.276
1.286
1.295
1.305
x ∈ [0.1, 1]
0.100
0.325
0.550
0.775
1.000
-333.333
-9.710
-2.004
-0.716
-0.333
-333.312
-9.703
-2.090
-0.675
-0.436
-333.312
-198.354
-63.422
-0.726
-0.426
941.122
1264.735
1272.432
1273.739
1274.122
59.970
194.902
329.834
392.587
392.970
x ∈ [0.1, 10]
0.100
2.575
5.050
7.525
10.000
-333.333
-0.020
-0.003
-7.82275E-4
-3.33333E-4
-333.312
-0.026
-0.047
-0.060
0.062
-333.312
-0.172
-0.312
-0.250
-0.250
928.620
1261.933
1261.950
1261.952
1261.953
65.364
398.651
398.666
398.682
398.697
Table 3: Comparison of the value of the utility function with its approximations based on two methods for finding the position
for the breakpoints: the optimization method (column (3) and (5)) and the curvature method (column (4) and (6)) calculated
accordingly with formulae (23) (column (3) and (4)) and formula (30) (column (5) and (6)) for different intervals x ∈ [1, 10],
x ∈ [0.1, 1] and x ∈ [0.1, 10], and different values of x equally spaced over the interval. The numbers in bold shows the
approximation closest to the value of the utility function.
30