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Solving Dynamic Stochastic Control Problems in Finance Using Tabu Search with Variable Scaling


Abstract and Figures

Numerous multistage planning problems in finance involve nonlinear and nonconvex decision controls. One of the simplest is the fixed-mix investment strategy. At each stage during the planning horizon, an investor rebalances her/his portfolio in order to achieve a target mix of asset proportions. The decision variables represent the target percentages for the asset categories. We show that a combination of tabu search and variable scaling generates global optimal solutions for real world test cases, despite the presence of nonconvexities. Computational results demonstrate that the approach can be applied in a practical fashion to investment problems with over 20 stages (20 years), 100 scenarios, and 8 asset categories. The method readily extends to more complex investment strategies with varying forms of nonconvexities.
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Theory & Applications
Ibrahim H.Osman
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Solving Dynamic Stochastic Control Problems in Finance Using
Tabu Search with Variable Scaling
Fred Glover
School otBusiness
University of Colorado at Boulder
Boulder, CO 80309
John M. Mulvey·
Department of Civil Engineering and Operations Research
Princeton University
Princeton, NJ 08544
Kjetil Hoyland
Department of Managerial Economics and Operations Research
University of Trondheim
N-7034 Trondheim, Norway
Numerous multistage planning problems in finance involve nonlinear
and nonconvex decision controls. One of the simplest is the fixed-mix
• Research supported in part byNSF grant CCR-9102660 and Air Force grant
AFOSR-91-0359.We acknowledge the assistance of Michael T.Tapia in preparing
this paper for publication.
investment strategy. At each stage during the planning horizon, an in·
vestor rebalances herlhis portfolio in order to achieve a target mix of
asset proportions.The decision variables represent the target percent-
ages for the asset ~ategories. We show that a combination of Tabu
Search and Variable Scaling generates global optimal solutions for real
world test cases, despite the presence of nonconvexities. Computa-
tional results demonstrate that the approach can be applied in a practi.
cal fashion to investment problems with over 20 stages (20 years). 100
scenarios. and 8 asset categories.The method readily extends to more
complex investment strategies with varying forms of nonconvexities.
Dynamic stochastic control, financial modeling, tabu
search. variable scaling.
1. Introduction
Efforts to explain financial markets often employ stochastic differen-
tial equations to model randomness over time. For example, interest
rate modeling has been an active research area for the past decade.reo
suIting in numerous approaches for modeling the yield curve. See
Brennan and Schwartz (1982), Hull (1993), Ingersoll (1987) and Jar-
row (1995) for details.These equations form the basis for various
analytical studies such as estimating the "fair" market value of securi-
ties that display behavior conditional on the future path of interest
rates. Often,for tractability (Hull 1993), these teclmiques discretize
theplanning horizon into a fixed number of time steps
and the random variables into a finite number of outcomes.
Our goal is to employ a similar discretization of time and random-
ness. But instead of estimating fair market value, we are interested in
analyzing alternative investment strategies over extended planning pe-
riods -- 10 or even 20 years. The basic decision problem is called asset
allocation. The universe of investments is divided into a relatively
small number of generic asset categories (US stocks, bonds, cash, in-
ternational stocks) for the purposes of managing portfolio risks. There
is much evidence that the most critical investment decision involves
selecting the proportion of assets placed into these categories, espe-
cially for investors who are well diversified and for large institutions
such as insurance companies and pension plans.
In our model, we assume that future economic conditions are pre-
sented in the form of a modest number of scenarios.A scpnario is de-
scribed as a complete path of all economic factors and accompanying
returns for the assets over the planning horizon. Asset allocation deci-
sion are made at the end of each of the
time periods. They cannot be
changed during the period, but they can and do respond to the chang-
ing state of the world (including the condition of our portfolio) at the
beginning of the next period. To keep the problem in a form that can
be understood by fmancial planners and investment managers, we have
assumed a decision rule that is intuitive and commonly used -- the
fixed-mix rule. Under this rule, the portfolio has a predetermined mix
at the beginning of each period -- for instance, 60% stock, 30% bonds,
and 10% cash. The portfolio must be rebalanced by selling and buying
assets until the proper proportions are attained. This strategy has been
used successfully by Towers Perrin (Mulvey 1995) and other long
term investment managers. See Perold and Sharp (1988) for a detailed
description of the fixed-mix and other dynamic strategies for asset al-
In this paper, we show that Tabu Search can be combined with Vari-
able Scaling to solve important cases of the asset allocation problem.
These decisions are difficult due to nonconvexities in the objective
function.Also, the inclusion of real world considerations, such as
taxes and other transaction costs (Mulvey 1993), may cause difficulties
for continuous optimization algorithms. In contrast, Tabu Search takes
advantage of the discretization of the solution space.
2. The Fixed-Mix Dynamic Control Problem
In this section we provide the mathematical description of the fixed·
mix control problem, a dynamic stochastic control model.Other real·
location rules could be employed, such as the life cycle concept; the1lC'
lead to similar nonconvex optimization problems.
To state the model, we define the following sets:
a set of discrete time steps in which the portfolio will rebalanced:
{1,2,... ,T};
a set of asset classes:
a set of scenarios, each of which describes the full economic
situation for each asset category
in each time period t:
{1,2,...,S} .
Define the following decision variables:
amount of money (in dollars) invested in asset category
time period
for scenario
wealth (in dollars) at the beginning oftime period
fraction of the wealth invested in asset category
across time), 0 ~ Ai ~
and LAi
(not al-
Define the following parameters:
WI:initial wealth in the beginning of time period 1 (in dollars);
percentage return for asset
in time period
under scenario
average return in time period
undcr scenarios.
(x/I .
"'"' J
L.. X"I
probability that sccnariosoccurs.
I.: percentage transaction cost forassetclass
to simplify the
presentation,symmetric transaction costs are
cost of selling equals costof buying);the implemented
model canalso handle.nonsymmetric transaction costs.
The fixed-mix control rule
thata fixed percentageofthe
wealih is invested in each asset category.Wealth at the beginning of
the see·and period willbe:
1=1 i-I
The second tenl1 of (2) takes
transaction cost into account assum-
inglinear transaction costs. To compute these values,tbeaveragere-
is calculated for each
period ( and for eacbs~nario s. A
portion ofthe asset classes withreturns
thantheaverage return
is sold, while the asset classes with belowaveragereturns.are bought.
Theeqmnion for rebalancing at the beginning-ofperiod
vlal1 S,an -
-'"l--' ,
L}/l-i/~«(I 'Vsand
{2, 3,..T}. (4)
Typically the model includes linear constraints on the asset mix
which we for now assume constant over time:
These constraints can take any linear form, for instance upper and
lower bounds on the proportions.A typical example will be for inves-
tors to limit the international exposure to some value (say 30%).
The objective function depends on the investor's risk attitude.
multiperiod extension of the Markowitz's mean-variance model
applied for this analysis.Two terms are needed -- the average total
wealth, Mean( wr), and the variance of the total wealth, Var( WT),
across the scenarios at the end of the planning horizon (at the end of
time period T). Other objective functions can also be used. For ex·
ample, we can maximize the von Neumann-Morgenstern expected
utility of the wealth at time
(Keeney and Raiffa 1993).Alterna-
tively, we can maximize a discounted utility function (Ziemba and
Vickson 1975). Other objective function forms, such as multiattributc:
utility functions, can also be used to model multiple investor goab
(Keeney and Raiffa 1993).
Our financial planning model is thus
Mean(wT)-(I -/3)Var(wr)
LPsw: ,
= ~
L[w: - Mean(wr)]2 ,
equations (1) - (5), and
T)measures the expected profit from the investment strategy,
while Var( wT)measures the risk of the strategy. A tradeoff exists
between the expected profit and the risk of an investment strategy.
solving the problem while allowing
to vary between 0 to 1 we obtain
the multiperiod efficient frontier. The quadratic equality constraint (3)
causes the ftxed-mix model to be nonconvex.
3. Specializations to Tabu Search
An advantage in solving the ftxed-mix control problem with Tabu
Search is that each application of Variable Scaling effectively discre-
tizes the solution space.We thus define the solution space as a dis-
crete set of possible investment proportions in each asset category.Let
us define the following vectors:
A(z):proportion of the total investment in asset category
lower bound on the investment proportion in asset category
u(i):upper bound on the investment proportion in asset category
Restricting the investment proportions in each asset category to be an
integer percentage share of the total investment means that we allow
A(i) to take the following percentage values:
2, ...,U(I) - 1, u(i) ,
where 0
100 (not allowing short sale) and
are integer.
Tabu Search operates under the assumption that a neighborhood can:
be constructed to identify adjacent solutions. The fixed-mix problem
is constrained such that the sum of all the investment proportions must
equal 100%.We define a neighbor solution by choosing two vari-
ables, A(Up)and 'A,(down)and assigning them new values:
delta, and
(down) - delta,
such that A(Up)
u(up) and A(down)
l(down), where up is the index
of the variable increasing its value, down is the index of the variable
decreasing its value,and delta> 0 is the stepsize for the neighborhood.
For a given discrete solution space, the following neighborhood
search algorithm (Glover and Laguna 1993) is an attempt to solve the
fixed-mix control problem. approach gives the global opti"~
solution (in the discrete solution space) if the problem is com'c.
However, if the problem is nonconvex, as is the case with the
mix control problem, the neighborhood search algorithm will tennlMu
at the first (discrete) local optimal solution. The area of the SolUlh)f\
space that has been searched will be limited using this approach.
Step 1:Initialization.
Select a feasible starting solution, A_start.
Set current_solution
Calculate objective value of this solution, f(A _start), and set
Step 2: Stopping criterion.
Calculate the objective value of the neighbors of current 30/ution.
Uno neighbors have a better objective value than best_objective, stop.
Otherwise,go to Step 3.
Step 3:Update.
Select a neighbor, A_next, with the best objective value, f(AJleXt) (or se-
lect any neighbor that gives an improving objective value).
Update: current_solution
A_next, best_objective
Go to Step 2.
Due to nonconvexities, we need a more intelligent method for find,
ing a global solution.Tabu Search extracts useful information from
previous moves to direct the search to productive areas of the soluti(m
space. It uses specialized memory .structures to characterize move.
with certain attributes as tabu,and assigns other moves values Of
penalties in order to guide the search through the solution space with·
out becoming trapped at local optima.
We define the total neighborhood of the current solution,
N(current_solution), as all possible ways of increasing the amount
vested in one asset category and decreasing the amount invested in
another asset category, while satisfying the constraints. This neigh·
borhood is modified to N(H, current_solution), where Hrepresents the
history of the moves in the search process. Therefore, the search his·
tory determines which solutions might be reached from the current
point.In short term Tabu Search strategies, N(H, current_solution) is
typically a subset of N(current _solution).In intermediate and longer
term strategies N(H, current_solution) may contain solutions not in
N(current_solution). See Glover (1995) for further details.
3.1 The Memory Structures
Tabu Search makes use of adaptive (recency based and frequency
based) memory structures to guide the search.The recency and the
frequency memory structures each consist of two parts. The first part
of the recency based memory keeps track of the most recent iteration at
which each variable received each of the values assigned to it in the
past (tabu_time(i, value)). The second part of the recency based mem-
ory keeps track of the most recent iteration at which each variable
changed its value (tabu_last_time(i)).
The first part of the frequency based memory keeps track of the
number of times each variable has been assigned its specific values
(tabu_count(i, value)). The second part measures the number of itera-
tions the variables have resided at the assigned values (tabuJes(i,
value)). Let duration(z) measure the number of iterations since the
variable received its current value.
3.2 Move Attributes
To make use of these memory structures, Tabu Search uses the concept
of move attributes.Generally a move ",_next(up)
delta and
'" next(down)
",(down) - delta has four attributes:
(1) ",(up):value of increasing variable, prior to increase;
(2) ",_next(up): value of increasing variable, after increase;
(3) f..{down):value of decreasing variable, prior to decrease;
(4) ",_next(down):value of decreasing variable, after decrease.
If a move attribute is tabu,this attribute is said to be tabu-active. A
move is classified tabu as a function of the tabu status of its attrihul~"
In the next section we show what COmbinations of tabu-active mo\'
attributes classifies a move tabu.
3.3 Tabu Rules Based on Recency Memory Structure
An .important parameter of this process is tabu_tenure, that
number of iterations in which a move attribute is tabu-active. For
approach, tabu_tenure will have two values, tJrom and t_to,
(.from is the number of iterations an assignment
~alue is tabu.
active, to discourage moving
"from." its present value (now_va/uC'
and t_to is the number of iterations an assig~ent
value is tahu-
active,to discourage moving
"to" a specific value (next_vall44ft
(Glover and Laguna 1993).
An attribute
value) is declared tabu-active when
tabu_time(i, value)"*0, tabu_time(i, value) 2:current_iteration
where t
t_to if value
next_value. ('Ibis classification will discour.
age a variable from returning to a value it recently has moved away
from.) Also, an attribute i(implicitly, an attribute (i, value) for value ••
now_value) is declared tabu-active When
tabu_Iast_time(i)"*0, tabu_Iast_time(i) >= current_iteration _
where t
tJrom. (This classification 'mIl discourage moving a vari.
able from its present value if it recently received this value.)
Both dynamic and static rules can be applied to set tabu_tenure. For
our purposes, we have used simple static rules, such as setting
to a constant value between 1 and 3 and
to a constant value be-
tween 4 and 6.Our choice of values for t_to and tJrom is based on
preliminary experimentation. We choose t_to
as a rule of
thumb in deciding the size of this parameter (in our test cases n =8).
In the implementation we introduce one other kind of tabu-active
classification: an attribute (i, value) is "strongly from tabu-active" if
tabu_last_time(i, value) >= current_iteration - 1.
For this study, the basic rule for defining a move tabu is if one of the
following is true:
i) either attribute (1) or attribute (3) is strongly tabu-active;
two or more of the remaining attributes are tabu-active.
3.4 Tabu Rules Based on Frequency Memory Structure
The frequency memory structure is usually applied by assigning pen-
alties and inducements to the choice of particular moves. Let
the sequence of solutions generated by the search process up to the
present point and define two subsets,
which contain high
quality and low quality solutions (in terms of objective function val-
ues), respectively. A high residence frequency in the subsequence
indicates that an attribute is frequently a participant in high quality
solutions, while a high residence frequency in the subsequence S2 in-
dicates that an attribute is frequently a participant in low quality solu-
tions. These relationships guide the search in the direction of high
quality solutions, by increasing or decreasing the incentive to choose
particular moves based on the quality of past solutions that contain at-
tributes provided by these moves.TIus constitutes an instance of a
Tabu Search intensification process. Define another set,
both high and low quality solutions. Assigning a penalty to high fre-
quency attributes in this set pushes the search into new regions,creat-
ing a diversification process. A high JraDsition frequency might indi-
cate that an attribute is a "crack filler". which is an attribute that alter-
nates in and out of the solution to "perform a fine tuning function"
(Glover and Laguna 1993).
4. Solution Strategy
Tabu Search methods often do not "turn on" their memory or reslr .
tions until after reaching a flrst local optimum. In our solution stratq'\
we take advantage of an efficient method to fmd the local optirn\lfll
namely a Variable Scaling approaGh.Our Variable Scaling proccd\lf.
is an instance of a candidate list ml~thod, which is the name given
III ;1
class of strategies used in Tabu Search (by extension of related
dures sometimes used in optimization) to generate subsets of
gll< ••
moves without the effort of examining all moves possible.In the prr \
ent setting, where the number of values available to each variahk .'
effectively infmite, we elect a candidate list strategy that "scales"
variables to receive only a relatively smalll}umber of discrete valu(",
However, we allow the scaling to be variable, permitting the seal
interval to change each time a 10Galoptimum is reached in the n'
stricted part of the neighborhood defined by the current scaling.Wh"u
no change of scaling (from the options specifled) discloses an imprm->
ing move --so that the current solution is truly a local optimum reI.·
tive to the scalings considered -- we apply a recency based
Search approach to drive the solution away from its current locatiu«.
and then again seek a local optimum by our Variable Scaling approach
The Variable Scaling approach is outlined in section 4.1, while ,,,-
complete strategy in outlined in section 4.2.
4.1 Variable Scaling Approach
We defme a set of stepsizes (or scaling intervals), each of which
rise to a restricted neighborhood.A local neighborhood search is
over a given restricted neighborhood until no improvement is posslb&f
When local optimality is reachedin that neighborhood (for one
ticular stepsize), we switch to another restricted neighborhood vi•••
new stepsize.We choose a set of stepsizes in decreasing order.
our purposes, typically the biggest stepsize is 5% and the smaU•••
stepsize is 1%.) The smallest stepsize determines the accuracy of
identifying a solution as a local optimum.
For a convex problem, applying such a candidate list approach based
on Variable Scaling speeds up the search since we move in bigger
steps toward the optimal solution in the beginning of the search, and
only as we get "close to the top" (given we are maximizing) do we de-
crease the stepsize.For a nonconvex problem (like ours) with several
local optima,the approach can reduce the number of necessary itera-
tions to get to a local optimum, and it can also move the search away
from a local optimum. If one stepsize is "stuck" in a local solution, a
change in the stepsize can take us away from the local pptimum. All
of these ideas are incorporated in Algorithm: VARIABLE SCALING.
Step 0: Initialize.
Construct a stepsize list:stepsize_J, depsize_2, .., stepsize_NS, where NS
is the number of stepsizes.Choose stepsizes in decreasing order.
Set stepsize
Construct an initial solution, A_start.Let currentjolution
Set current_iteration
Let best_objective
f(A_start), the objective value of A_start.
Step 1:Calculate neighbors.
Step 2: Evaluate neighbors.
If objective value of at least 1 of the neighbors is better than best_objective:
Pick the neighbor with the best ob}~ve value, A_next-.
Update current_solution
A_next and best_objective =f(A_next);
Set current_iteration =current_iteration +1;
Go to Step I.
If none of the neighbors improves the solution:
If stepsize ,;,stepsize _NS, use next !:tepsize from list and go to Step I;
If stepsize
stepsize_NS, check if objective has improved for any of the
last NS stepsizes: If the objective has improved,set stepsize to
stepsize_l, and go to Step I.[fit has not improved,go to Step 3.
Step 3. STOP:Local optimal solution is o!:>tained.
4.2 The Complete Algorithm
The implemented algorithm is:
Step 0, Step 1and Step 2 from the Variable Scaling approach.
Step 3: Incorporate Recency Based Tabu Search (diversifying search).
3.1:Set stepsize
stepsize_div (stepsize used for the diversifying search).
Set div
3.2:Set div_it_counler
For all neighborhood moves do
check ifmove is tabu.
if tabu, go to next neighbor;
.if not tabu, calculate the objective value of the move.
3.3: Pick the (nontabu) neighbor with the best objective value.Denote this
neighbor A_best_div, and its objective value f(A_best_div).
If f(A_best_div)
best_objective set stepsize
stepsizej and go to
Step I in the Variable Scaling approach.
If f(A_best_div)
best_objective go to Step 3.4.
3.4:If div_it_counter
max_diY_it,update tabu status (as done in Steps 1-3
in memory updating in section 3.1) and go to Step 3:2.
If div_it_counter
max_itJounter, go to Step 4.
Step 4: STOP.The solution is equal or sufficiently close to the true (discrete)
global optimal solution.
The procedure imposes the Tabu Search approach in conjunction
with the candidate list strategy of Variable Scaling. This is just
one of many ways that Tabu Search can be coordinated with a candi-
date list strategy. (More typical is to subject such a strategy directly to
the guidance of Tabu Search memory, rather than invoking this guid-
ance only at particular stages and for particular neighborhood in-
stances.) Nevertheless, we have fo\md this approach convenient for
our present purposes. Specifically, in this alternating approach,we
switch from the Variable Sc~ing method (over its chosen set of scal-
ings) to the Tabu Search method (over another set of seatings) when-
ever Variable Scaling no longer improves the current solution.If Tabu
Search generates a solution better than the one obtained from the Vari-
able Scaling, we return to Variable Scaling, with the solution from
Tabu Search as an initial solution. If Tabu Search fails to generate an
improved solution, the algorithm will stop when the maximum number
of iterations is reached. In this procedure the choice of the stepsizes is
crucial for the algorithm's success. We typically choose small step-
sizes in the range of 0.5% to 5% for the Variable Scaling, while a big-
ger one, typically 5% to 15% is applied for the Tabu Search.
5. Computational Results
The algorithm described in section 4.2 is applied to an investment
problem with I
8 asset categories, T
20 time periods and S
scenarios. The 8 different asset categories are cMh equivalents,
Treasury bonds, large capitalization US stocks, international bonds,
international stocks, real estate, government/corporate bond index, and
small capitalization US stocks. One hundred scenarios were generated
by the technique introduced in Mulvey (1995). Each scenario is given
equal probability
1%. Each scenario consists of returns for
each asset, in each time period. Hence the total number of returns
generated is equal to 16000.The initial wealth is set equal to unity.
In all the experiments, each point on the multiperiod efficient fron-
tier is obtained as explained in section 4.2. The entire efficient frontier
is obtained by solving the problem f()r 22 values of J3.Through all the
experiments, we have a set of basic test parameters 1
5.1 Comparison with Global Methud
Our solutions are compared with the solutions obtained from the
global method described in Androulakis et al. (1994). This determi-
nistic global optimization algorithm guarantees finite e-convergence to
the global optimum. In this case we assume no transaction cost or tax.
The solutions obtained by our approach· are very close to the solutions
obtained by the global method and are often slightly better.The opti-
Number of stepsizes for the Variable Scaling approach (NS): 3; Variable Scaling
1%; stepsize for the Tabu
Search part of the algorithm
(stepsize _div):
10%; maximum number of diversifying
_it): 25; tabu tenures:
mal mixes generated by the two methods are indeed practically identi
cal.These results are obtained despite the presence of nonconvexitics
5.2 Including Transaction Costs and Tax
We asswned percentage transaction costs for the asset categories}.
The tax rate is asswned to be 28%. Figure 1 shows the efficient fron-
tiers obtained by solving the model that takes tax and transaction cost
in to account (tax-model) and the one that does not (no-tax model).
expected, the efficient frontier obtained by the tax-m9del is tilted
down.More surprisingly, perhaps, the efficient frontiers cross in the
low end of the variance area (for low (3.'s). Transaction costs and taxel!
dampen the variance of the expected value and therefore a lower vari-
ance can be achieved for the tax-modeL From Figure 2 we see that for
0.94 in the model where transaction costs and tax is included, it
optimal to have all the funds in the asset category with the highest ex-
pected value. This is in contrast to the case without taxes or transac-
tion costs, where a more diversified optimal solution was found
soon as the variance was assigned a weight in the objective function.
There is an explanation for this phenomenon. When all of the fund.
are concentrated in one asset category, no buying or selling is neces-
sary to update the portfolio at the end of each time period. When the
funds are split between asset categories, however, trading must be
done at the end of each time period to reconstruct the portfolio to tho
predetermined weights of the fixed-mil{ rule. This implies cash out-
flows because of the transaction cosu: and taxes on assets sold for
profit.So for
>0.94 the gain from reduced variance by diversifying
is offset by a larger loss in the expected value.
2Cash equivalents:0%; Treasury bonds:0.25%; large capitalization stocks:0.25%;
international bonds:0.5%; international stocks: 1.0%; real estate: 6.0%;govern-
ment/corporate bond index: 1.0%; small capitalization stocks:1.0%.
•.... ,f't
Efficient frontiers for model with and without tax and
transaction costs. Notice how the efficient frontier for the tax-model is
tilted down (as one would expect). Also see how the efficient frontiers
crosses in the low variance end.
11 11
11 11 11
Figure 2: Optimal solutions for model with and without tax and trans-
action costs. Notice how the solutions obtained from the tax-model
are less diversified for high Ws (i.e. :inthe high variancelhigh expected
value end of the efficientfrontier), and more diversified for low I3's.
At the other end of the efficient frontier (low 13,i.e. low variancc),
we see that tax-model recommends solutions that are more diversified
than the solutions from the no-tax model.Tax-model recommends"
portfolios consisting of all 8 asset categories for 13:5 0.4, with nu
dominating asset category (except for 13
0). The no-tax model, how-
ever, recommends a portfolio concentrated in fewer asset categoricli
There is a simple explanation for this ;~esult.With many asset c1assclI.
the return in some of asset classes will be close to the average return.
Little trading is required in these asset categories to rebal~ce the port-
folio to the prefixed weights. More tIading is done in the asset cate-
gories with returns far from average, but these asset categories are
fraction of our total portfolio. This is not the case for a portfolio con·
centrated in fewer asset categories. Consider, for instance, the portfo-
lio recommended by the no-tax model for 13
0.2 -- approximately
80% in cash equivalents and Treasury bonds. Since it is unlikely that
both dominating asset categories have close to average returns, more
trading is needed to rebalance the portfolio. This portfolio is likely to
have more cash outflows from transal:tion costs and taxes than the
more diversified one -- the gain in reduced variance by concentrating
the portfolio in cash equivalents and Treasury bonds is offset by a loss
in expected value caused by the increase in cash outflows.
As expected, the solution time increases considerably when transac-
tion costs and taxes are included to the model.The average solution
time per f3-problem for the model with basic test parameters (see be-
ginning of this section) is 46.7 CPU sel:onds Gust over 17 CPU min-
utes to obtain the entire efficient frontier) on a Silicon Graphics Iris
workstation.The results are highly attractive, particularly given that
we are developing a long term projecting system.
5.3 The Effect of Including Tabu Sean:h Restrictions.
To see the effect of the Tabu Search restrictions we consider the tax
problem and compare the solutions of our approach (COMPLETE
ALGORITHM) with the solutions of a pure Variable Scaling approach
(the candidate list strategy used in
method). The largest stepsize in
the pure Variable Scaling approach is set equal to the stepsize of the
Tabu Search part of our algorithm (stepsize_div
10%).The other
stepsizes are also set equal for both approaches (5%, 3% and 1%).
Hence, the sole difference between the two approaches is the recency
memory restrictions on the 10% search applied in our algorithm. For ~
0.5 the solutions are identical. Th~:pure Variable Scaling approach
"gets stuck" in a local solution for ~
0.4, in contrast to the approach
that includes Tabu Search memory guidance.
6. Conclusions
Tabu Search is an efficient method ft>robtaining the efficient frontier
for the fixed-mix investment problem.The computational results
show that the solutions obtained by the Tabu Search are very close to
(often slightly better than) the e-tolenmce global optimal solutions ob-
tained by the method of Androulakis et al. (1994) for the case with no
taxes or transaction costs. In an expanded model, which addresses
transaction costs and taxes for greateI realism,our approach continues
to obtain optimal solutions efficiently.The expanded model is beyond
the capability of the global optimization approach, and its complicat-
ing features in general pose significant difficulties to global optimiza-
tion solvers based on currently standard designs.
Some areas for future research are: (1) develop and test related dy-
namic stochastic control strategies (with nonconvexities);(2) design an
approximation scheme for updating iilformation between iterations in
order to improve computational efficiencies; and (3) incorporate addi-
tional strategic elements of Tabu Search. Since the discretization of
the solution space depends upon an investor's circwnstances, research
in this area is critical for successful use of this methodology.
7. References
J.P.Androulakis et al.Solving stochastic control problems in financ:.
via global optimization,Working paper, SOR-94-01, Princeton
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M.J. Brennan and E.S. Schwartz,An equilibrium model of bond pric-
ing and a test of market efficiency", Journal of Financial and
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F.Glover, Tabu search: fundamenta1~ and usage, Working paper, Uni·
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J.C. Hull, Options. Futures and other Derivative Securities; (prentice
Hall, 1993).
J.E. Ingersoll, Jr. Theory of Financial Decision Making, (Rowman
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Madsimovic, and W.T. Ziemba,(North Holland,Amsterdam, 1995).
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ences and Value Tradeoffs, (John Wiley, New York, 1976; reprinted
by Cambridge University Press, 1993).
J.M. Mulvey,Incorporating transaction costs in models for asset allo-
cation, in:Financial Optimization, ed: S.Zenios, (Cambridge Uni-
versity Press, 1993).
J.M.Mulvey, Generating scenarios for the Towers Perrin investment
system, Working paper, SOR 95-04, Princeton University (to appear
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A.F. Perold and W.F. Sharpe, Dynamic strategies for asset allocation,
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... Imposing proportional plus fixed combines the effect of the two costs, as increasing them will reduce portfolio diversification. So, considering all typologies, global transaction costs tend to reduce portfolio diversification, but this assertion must be taken cum grano salis as investor behavior depends on subjective factors too: In Glover et al. (1995) it is explicitly stated that if the investor is risk-averse the portfolio held is more diversified with taxes and transaction costs, whilst diversification is not requested by investors with low risk-aversion if taxes and transaction costs are included in the model. It is clear however that only Di Tollo and Roli: Metaheuristics for the Portfolio Selection Problem IJOR Vol. 5, No. 1, 13-35 (2008) 21 proportional costs are suitable to be included in the continuous model, as the remainder is sensitive to the invested amount. ...
... Iterative improvement is usually incorporated into a more complex strategy, rather than being used as a stand alone local search. For instance, in Glover et al. (1995) iterative improvement is the local search component of a variable neighborhood search technique. As another example, we mention Armañanzas and Lozano (2005) who use a greedy search to refine solutions found by an ant algorithm. ...
... This simple mechanism, enhanced with the exploitation of the search history for intensifying and diversifying the search, makes TS one of the best performing local search strategies. The application of TS to the PSP has its milestones in the works by Rolland (1997) and Glover et al. (1995). These works refer to different formulations of the problem and moreover Glover tackles a multi-period formulation. ...
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The Portfolio selection problem is a relevant problem arising in finance and economics. Some practical formulations of the problem include various kinds of nonlinear constraints and objectives and can be efficiently solved by approximate algorithms. Among the most effective approximate algorithms, are metaheuristic methods that have been proved to be very successful in many applications. This paper presents an overview of the literature on the application of metaheuristics to the portfolio selection problem, trying to provide a general descriptive scheme.
... mean deviations below and above the portfolio rate of return) resulted in a model equivalent to the MAD model. In Speranza [19] Tabu search (TS) was developed by Glover [22] and was applied by Glover, Mulvey and Hoyland [23] to a portfolio optimisation problem involving dynamic rebalancing to maintain constant asset proportions, using a scenario approach to model forecast asset returns. ...
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... Di Tollo and Roli (2008) provided an overview of the literature on the application of meta-heuristics to the portfolio selection problem. These methods consist of simulated annealing (SA) (Crama & Schyns, 2003; Armañanzas & Lozano, 2005), threshold accepting (TA) (Dueck & Winker, 1992; Gilli et al., 2006), tabu search (TS) (Glover et al. 1995; Rolland, 1997) genetic algorithm (GA) (Arnone, 1993; Yang, 2006) and ant colony optimization (ACO) (Armañanzas & Lozano, 2005; Maringer, 2001). Chang et al. (2000) proposed GA, SA and TS for cardinality constrained problem and the proposed model of this paper will be compared with the results of this paper. ...
... More recently, a simplified instance of the 3-2-3 algorithm has been embedded in such a strategy in Gardeux et al. (2010) with highly promising results. A related approach for the fixed-mix problem in finance is employed in Glover, Mulvey and Hoyland (1996) which simultaneously changes the values of two variables at a time. In this latter case the variables are first scaled and an increase is one variable is matched by a decrease in another, thus effectively inducing a unidimensional change within a transformed coordinate system. ...
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This note describes three line search algorithms for multi-modal function optimization that incorporate a combination of binary search and direct search, called the 3-2-3, the Stratified Split and Nested Interval Algorithms. The methods are designed to be used as subroutines of a more general global optimization procedure over a bounded search space, as a way to perform effective line searches without the limiting assumption of unimodality.
... TS have been widely applied to different optimization problems such as nonlinear covering problems, shop scheduling, quadratic assignment and traveling salesman problems. For portfolio optimization problem, Glover, Mulvey and Hoyland [18] applied TS to the problem involving rebalancing a portfolio to maintain a fixed proportion in each asset category. Another financial application is Consiglio and Zenios [19] who implemented TS to optimize a callable bond design. ...
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... The manner in which we assign tabu penalties --and thus define the fiJnction t(x) --to each potential move relies on exploitation of short-term search history; the methodology is based on the technique developed in Glover, Mulvey, and Hoyland [1996]. The assignment is based on a comparison of the move anributes of each potential move and those of the iterates comprising the recent search history. ...
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The use of formal mathematical models and optimization in finance has become common practice in the 1980s and 1990s. This book clearly presents the exciting symbiosis between the fields of finance and management science/operations research. Prominent researchers present the state of the art in financial optimization, while analysts from industry discuss the latest business techniques practised by financial firms in New York, London and Tokyo. The book covers a wide range of topics: portfolio management of equities and fixed income investments, the pricing of complex insurance, mortgage and other asset-backed products, and models for risk-management and diversification.
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