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META-HEURISTICS:

Theory & Applications

Ibrahim H.Osman

and

JamesP.

Kelly

••••

,~

Kiuwer

Acaaemi~

Publishers

BostonlUlndon!bordr~ht

Distributors for North America:

Kluwer Academic Publishers

101 Philip Drive

Assinippi Park

Norwell, Massachusetts 02061 USA

Distributors for all other countries:

Kluwer Academic Publishers Group

Distribution Centre

Post Office Box 322

3300 AH Dordrecht, THE NETHERLANDS

A C.I.P.Catalogue record for this book is available from the Library of

Congress.

All rights reserved.No part of this publication may be reproduced, stored in

a retrieval system or transmitted in any form or by any means, mechanical,

photo-eopying, recording, or otherwise, without the prior written permission of

the publisher, Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park,

Norwell, Massachusetts 02061

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

E-mail: fred.glover@colorado.edu

John M. Mulvey·

Department of Civil Engineering and Operations Research

Princeton University

Princeton, NJ 08544

E-mail: mulvey@macbeth.princeton.edu

Kjetil Hoyland

Department of Managerial Economics and Operations Research

University of Trondheim

N-7034 Trondheim, Norway

E-mail: kth@iok.unit.no

Abstract

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.

Key

Words:

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

t

E

{1•...•T}

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

T

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-

location.

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:

t

=

{1,2,... ,T};

a set of asset classes:

i

=

{1,2,...

,1};

a set of scenarios, each of which describes the full economic

situation for each asset category

i,

in each time period t:

s

=

{1,2,...,S} .

Define the following decision variables:

x:,1:

amount of money (in dollars) invested in asset category

i,

in

time period

t,

for scenario

s;

w;:

wealth (in dollars) at the beginning oftime period

t

under

scenarios;

Ai:

fraction of the wealth invested in asset category

i

(constant

/

across time), 0 ~ Ai ~

1,

i

=

1,2,...

,1

and LAi

=

1

(not al-

i-I

Define the following parameters:

WI:initial wealth in the beginning of time period 1 (in dollars);

rl~:

percentage return for asset

i,

in time period

t,

under scenario

r;':

average return in time period

t

undcr scenarios.

I

L

(x/I .

r/,)

'~I

I

"'"' •J

L.. X"I

1-1

S

p.,:

probability that sccnariosoccurs.

L

p"

=

1;

.'=1

I.: percentage transaction cost forassetclass

i--

to simplify the

presentation,symmetric transaction costs are

.<tssumed

(Le.

cost of selling equals costof buying);the implemented

model canalso handle.nonsymmetric transaction costs.

The fixed-mix control rule

ensures

thata fixed percentageofthe

wealih is invested in each asset category.Wealth at the beginning of

the see·and period willbe:

J

f

w~

=

L(t+ri~I)Xrl-

L:}1J

-rt~<ltl'

1=1 i-I

The second tenl1 of (2) takes

the

transaction cost into account assum-

inglinear transaction costs. To compute these values,tbeaveragere-

turn,

r"f.

is calculated for each

time

period ( and for eacbs~nario s. A

portion ofthe asset classes withreturns

higher

thantheaverage return

is sold, while the asset classes with belowaveragereturns.are bought.

Theeqmnion for rebalancing at the beginning-ofperiod

1

is

.1.1

w·

d.d·

f.-

{?...

T}

-\:i)=fi'rAj

vlal1 S,an -

-'"l--' ,

1

I

11':•.1

=

L<l+r/t)x!,/-

L}/l-i/~«(I 'Vsand

(=

{2, 3,..T}. (4)

I_I1-1

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.

A

multiperiod extension of the Markowitz's mean-variance model

i~

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

T

(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

Max

Z

=

/3

Mean(wT)-(I -/3)Var(wr)

s

s.t.Mean(wr)

=

LPsw: ,

s=1

s

Var(wr)

= ~

L[w: - Mean(wr)]2 ,

s=1

equations (1) - (5), and

O:S

13

:s

1.

Mean(

w

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.

By

solving the problem while allowing

f3

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

i;

l(i):

lower bound on the investment proportion in asset category

i;

u(i):upper bound on the investment proportion in asset category

i.

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:

1(1),

l(z)

+

1,

1(1)

+

2, ...,U(I) - 1, u(i) ,

where 0

<

l(i)

<

u{l)

<

100 (not allowing short sale) and

u(i)

and

l(i)

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:

A(Up)

=

A(up)

+

delta, and

A(down)

=

A

(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. 1b.is 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

fixed

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

=

A_start.

Calculate objective value of this solution, f(A _start), and set

best_objective

=

f(A_start).

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

=

f(A_next)

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

ih'

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)

=

",(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

i9

IIw

number of iterations in which a move attribute is tabu-active. For

0'-

approach, tabu_tenure will have two values, tJrom and t_to,

when.•

(.from is the number of iterations an assignment

Ai

=

~alue is tabu.

active, to discourage moving

Ai

"from." its present value (now_va/uC'

I.

and t_to is the number of iterations an assig~ent

Ai

=

value is tahu-

active,to discourage moving

AI

"to" a specific value (next_vall44ft

(Glover and Laguna 1993).

An attribute

(i,

value) is declared tabu-active when

tabu_time(i, value)"*0, tabu_time(i, value) 2:current_iteration

_I,

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 _

I.

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

(from

to a constant value between 1 and 3 and

t_to

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

==

2...{;

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;

ii)

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

S

denote

the sequence of solutions generated by the search process up to the

present point and define two subsets,

Sl

and

S2,

which contain high

quality and low quality solutions (in terms of objective function val-

ues), respectively. A high residence frequency in the subsequence

Sl

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,

S3,

containing

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

pn".·

dures sometimes used in optimization) to generate subsets of

gll< ••

1

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"

11M'

variables to receive only a relatively smalll}umber of discrete valu(",

However, we allow the scaling to be variable, permitting the seal

11111

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 T.hu

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

glW!'

rise to a restricted neighborhood.A local neighborhood search is

dON

over a given restricted neighborhood until no improvement is posslb&f

When local optimality is reachedin that neighborhood (for one

fW'

ticular stepsize), we switch to another restricted neighborhood vi•••

new stepsize.We choose a set of stepsizes in decreasing order.

(h.

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.

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

=

stepsize

j.

Construct an initial solution, A_start.Let currentjolution

=

A_start.

Set current_iteration

=

O.

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:

Algorithm:COMPLETE ALGORITIIM

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

_u

_counter

==

O.

3.2:Set div_it_counler

==

div_it_counter

+

1.

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

=

100

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

Ps

=

1/S

=

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-

I

Number of stepsizes for the Variable Scaling approach (NS): 3; Variable Scaling

stepsizes

stepsize_J

=

5%,

stepsize_2

=

3%,

:1tepsize_3

=

1%; stepsize for the Tabu

Search part of the algorithm

(stepsize _div):

10%; maximum number of diversifying

steps

(max_div

_it): 25; tabu tenures:

t..from=

2,

t_to

=

4.

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).

All

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

13

>

0.94 in the model where transaction costs and tax is included, it

i5

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

a.ll

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

P

>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%.

•....

--

•.••-u!.._--------

..-_._

....

~----

•.'..I.'y··

,

/'

-8";1'(

I.

!r~-'"

•.... ,f't

JI~·t...···

'I

i

!

Figure

1:

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.

f

,

I

I

l

j·~!ic;~",,xl.

oOI,..,"':n71~1

.JII.('~[~

I

.",,".I

alu::.Ua::.::.

.~flid"""'i'l}t.\o'b

.11~U

&""'1''»

I

.e...;;,

&;.~_.....:

»

11 11

~

11

.~

11 11 11

»

11

!l

3

~

d

:;

3

:~

3~

3

::

3

btQ~

~

::

0

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

8

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

OUT

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

UnJ.

versity (1994).

M.J. Brennan and E.S. Schwartz,An equilibrium model of bond pric-

ing and a test of market efficiency", Journal of Financial and

Quantitative Analysis, 17 (1982) 75.

F.Glover, Tabu search: fundamenta1~ and usage, Working paper, Uni·

versity of Colorado, Boulder (1995).

F.Glover and M. Laguna,Tabu seaIch, in:Heuristic Techniques for

Combinatorial Problems, ed. C. Ree:ves(1992).

J.C. Hull, Options. Futures and other Derivative Securities; (prentice

Hall, 1993).

J.E. Ingersoll, Jr. Theory of Financial Decision Making, (Rowman

&

Littlefield, 1987).

R.Jarrow, Pricing interest rate options, in:Finance, eds: R.Jarrow, V.

Madsimovic, and W.T. Ziemba,(North Holland,Amsterdam, 1995).

R.Keeney and H. Raiffa,Decisions with Multiple Objectives:Prefer-

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

Interfaces 1995).

A.F. Perold and W.F. Sharpe, Dynamic strategies for asset allocation,

Financial Analysts Journal, (1988) 16.

W.T. Ziemba and R.G. Vickson (eds.), Stochastic Optimization Mod-

els in Finance, (Academic Press, New York, 1975).