An Iterated Greedy Heuristic for the 1/N Portfolio Tracking Problem

Oliver Strub and Norbert Trautmann

Department of Business Administration, University of Bern, Sch¨utzenmattstrasse 14, Bern, Switzerland

Keywords:

1/N Portfolio, Index Tracking, Portfolio Optimization, Iterated Greedy Heuristic.

Abstract:

The 1/N portfolio represents a simple strategy to invest money in the stock market. Investors who follow this

strategy invest an equal proportion of their investment budget in each stock from a given investment universe.

Empirical results indicate that this strategy leads to competitive results in terms of risk and return compared to

more sophisticated strategies. However, in practice, investing in all N stocks from a given investment universe

can cause substantial transaction costs if N is large or if the market is illiquid. The optimization problem

considered in this paper consists of optimally replicating the returns of the 1/N portfolio by selecting a small

subset of the N stocks, and determining the respective weight for each selected stock. For the ﬁrst time, we

apply the concept of iterated greedy heuristics to this novel portfolio-optimization problem. For analyzing

the performance of our heuristic approach, we also formulate the problem as a mixed-integer quadratic pro-

gram (MIQP). Our computational results indicate that, within a limited CPU time, our heuristic approach

outperforms the MIQP, in particular when the number of stocks N grows large.

1 INTRODUCTION

Stock-investment strategies aim at constructing port-

folios of stocks that maximize the expected return for

a given level of risk. Besides good risk-return charac-

teristics, a desirable property of an investment strat-

egy is low expenses for, e.g., transaction costs or in-

vestment research.

The 1/N portfolio represents an investment strat-

egy that delivers good risk-return characteristics at

low investment-research costs. An investor who fol-

lows the 1/N strategy invests an equal proportion of

the available budget in each of the N stocks from

a given investment universe. In an empirical anal-

ysis, (DeMiguel et al., 2009) showed that this sim-

ple strategy performed competitive in terms of risk

and return compared to other more sophisticated in-

vestment strategies like the mean-variance approach

(Markowitz, 1952) and extensions thereof. However,

applying the 1/N strategy can lead to substantial trans-

action costs if the investment universe consists of a

large number of stocks or if the market is illiquid.

The planning problem considered in this pa-

per consists of selecting a small subset of the N

stocks and determining each selected stock’s portfo-

lio weight such that the resulting tracking portfolio

minimizes the variance of the expected return differ-

ences (cf. (Roll, 1992)) between the tracking portfo-

lio and the 1/N portfolio at low transaction costs. The

following constraints must be fulﬁlled by the tracking

portfolio: The number of different stocks to be in-

cluded in the tracking portfolio is limited, the weight

of each selected stock must be within a speciﬁc range,

the whole budget must be invested, and short selling

is not allowed. We do not explicitly consider trans-

action costs, but implicitly limit the transaction costs

by limiting the number of stocks to be included in the

portfolio.

To the best of our knowledge, this planning prob-

lem has not been discussed in the literature. How-

ever, the 1/N portfolio can be interpreted as a stock-

market index with equal weights for each stock, and

therefore, so-called index-tracking methods known

from the literature (cf., e.g., (Beasley et al., 2003;

Canakgoz and Beasley, 2008; Guastaroba and Sper-

anza, 2012)) can be applied. However, in general,

a ﬁnancial index consists of stocks with different

weights, and general index-tracking methods do not

take these weights into account.

In this paper, we present a heuristic approach to

the planning problem stated above. To obtain a pure

combinatorial optimization problem, we only con-

sider tracking portfolios where each stock has the

same weight. We propose an iterated greedy heuris-

tic (cf., e.g., (Ruiz and St¨utzle, 2007)) that runs as

follows. First, a feasible portfolio is constructed by

applying a novel greedy insertion heuristic. Then, an

improvement phase runs until some termination cri-

424

Strub, O. and Trautmann, N.

An Iterated Greedy Heuristic for the 1/N Portfolio Tracking Problem.

DOI: 10.5220/0005827704240431

In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 424-431

ISBN: 978-989-758-171-7

terion is met; this improvement phase consists of a

deconstruction and a construction sub-phase. Dur-

ing the deconstruction sub-phase, some randomly se-

lected stocks are deleted from the current solution. In

the construction sub-phase, stocks are added to the

portfolio by applying the greedy insertion heuristic

that has also been used to construct the initial solu-

tion. New solutions are accepted or discarded based

on an acceptance criterion that can also be satisﬁed for

worse solutions with a small probability. Because our

paper is the ﬁrst that considers the problem of track-

ing the 1/N portfolio, we also formulate the problem

as a mixed-integer quadratic program (MIQP) to an-

alyze the performance of our heuristic approach. We

used a standard commercial solver for the solution of

the MIQP. For a set of 23 test instances obtained from

real-world stock-marketdata, it turned out that, in par-

ticular for the larger instances, our iterated greedy

heuristic (IGH) devises better solutions within less

CPU time.

The paper is organized as follows. In Section 2,

we state the decision problem. In Section 3, we pro-

vide a short overview on the literature on index track-

ing and on iterated greedy heuristics. In Section 4,

we present our iterated greedy heuristic approach. In

Section 5, we report on our computational results. In

Section 6, we give some concluding remarks and an

outlook on future research.

2 PLANNING PROBLEM

The planning problem considered in this paper con-

sists of constructing a portfolio composed of a small

subset of the N stocks from a given investment uni-

verse to reproduce the returns of the 1/N portfolio.

More speciﬁcally, we want to construct a portfolio

that has the lowest possible variance of the relative

returns (VRR), where the relative returns correspond

to the differences between the returns of the tracking

and the 1/N portfolio. A lower VRR means more sta-

ble relative returns over time and therefore a smaller

risk of having large return differences in single peri-

ods. (Roll, 1992) formulates the problem of minimiz-

ing the VRR between a tracking portfolio and some

market index using the following quadratic objective

function of the decision variables x

representing the

portfolio weights of each stock i in the tracking port-

folio, the weights w

of each stock i in the index, and

the covariances σ

between the returns of stock i and

j (see Table 1 for the nomenclature):

Min.

∑

i=1

∑

j=1

− w

)(x

− w

) (1)

Table 1: Nomenclature for the MIQP.

Parameters

N Number of available stocks

k Maximum cardinality of the portfolio

> 0 Maximum weight of stock i if included

in the tracking portfolio

> 0 Minimum weight of stock i if included

in the tracking portfolio

Covariance between returns

of stock i and j

Weight of stock i in the index (=

)

Decision variables

Weight of stock i in the portfolio



= 1, if x

> 0

= 0, otherwise

To construct a portfolio that tracks the 1/N portfo-

lio, we replace the index weights w

. The follow-

ing constraints are considered: The maximum num-

ber of stocks to include in the tracking portfolio must

not exceed a prescribed value of k, and each stock in-

cluded in the portfolio is required to have a weight

within the predeﬁned range [ε

,δ

], i = 1,... ,N. This

range should include

such that we are able to con-

struct a portfolio of k stocks, each with a weight of

. Assigning a weight of

to each included stock

leads to good tracking portfolios according to (Fiter-

man and Timkovsky, 2001), who argue that a stock

included in the portfolio should have a relative weight

that is proportional to its relative weight in the origi-

nal index, i.e., to

. Finally, we must invest the whole

budget, and short selling is not possible. We obtain

the following mixed-integer quadratic program:

MIQP











Min.

∑

i=1

∑

j=1

−

)(x

−

) (2)

s.t.

∑

i=1

= 1 (3)

∑

i=1

≤ k (4)

≤ x

≤ δ

(i = 1,... ,N) (5)

≥ 0,z

∈ {0,1} (i = 1, ...,N) (6)

The objective function (2) correspondsto the VRR

between tracking and 1/N portfolio. Constraint (3)

ensures that the whole budget is invested, and con-

straint (4) limits the total number of stocks to include

in the tracking portfolio to k. The binary decision

variables z

are used to formulate the cardinality con-

straint (4). Constraints (5) both guarantee that the bi-

nary variables z

are equal to one if and only if the

portfolio weight x

is greater than zero, and that the

An Iterated Greedy Heuristic for the 1/N Portfolio Tracking Problem

425

portfolio weights x

are within the range [ε

,δ

] if and

only if z

= 1. Finally, constraints (6) specify the do-

mains of the decision variables z

and x

3 RELATED LITERATURE

To the best of our knowledge, there is no speciﬁc liter-

ature on the planning problem described in Section 2.

However, since we can interpret the 1/N portfolio as

a special index with equal weights for each stock, the

returns of the 1/N portfolio could be replicated by ap-

plying index-tracking methods. Index-tracking meth-

ods are applied to replicate the returns of large market

indices with a small set of stocks to save transaction

costs. In Subsection 3.1, we give an overview on the

literature on index tracking. In Subsection 3.2, we

summarize the literature on iterated greedy heuristics.

3.1 Index Tracking

Index-tracking methods are applied to solve the

index-tracking problem, which consists of construct-

ing a portfolio that best-possibly replicates the index

returns by investing in a small subset of some set of

stocks. With the exception of (Roll, 1992), most ap-

proaches to the index-tracking problem assume that

the true index weights are unknown. For example,

(Beasley et al., 2003) develop an evolutionary heuris-

tic to the index-tracking problem. They minimize

some function of the differences between the portfo-

lio and index returns and consider various practical

portfolio constraints such as a maximum number of

stocks to include in the portfolio, minimum and max-

imum weights for each included stock, and a budget

for transaction costs. (Canakgoz and Beasley, 2008)

develop a regression-based approach for the index-

tracking problem. Their objective is to construct a

portfolio that has an intercept of zero and a slope of

one when regressing the portfolio returns on the in-

dex returns. They consider similar practical portfolio

constraints as (Beasley et al., 2003). (Guastaroba and

Speranza, 2012) present a mixed-integer linear pro-

gram as well as a MIP-based heuristic called Kernel

Search for the index-tracking problem. They consider

similar practical portfolio constraints as (Canakgoz

and Beasley, 2008), but additionally consider ﬁxed

transaction costs. By assuming the true index weights

to be unknown, these approaches have the advantage

that they can be applied to track any index by select-

ing a subset of stocks from any set of stocks, even if

the index is not composed of this set of stocks.

3.2 Iterated Greedy Heuristics

Iterated greedy heuristics start by constructing an ini-

tial solution. This solution is then improved dur-

ing an improvement phase that consists of the two

sub-phases deconstruction and construction (Ruiz and

St¨utzle, 2007). During the deconstruction sub-phase,

some elements of the current candidate solution are

removed. In the construction sub-phase, a new can-

didate solution is constructed by adding elements in

a greedy way back to the deconstructed solution.

The new candidate solution is then either accepted

or discarded based on a speciﬁc acceptance crite-

rion. To escape local minima, the acceptance crite-

rion is designed such that also worse solutions are ac-

cepted with some probability. By repeating the decon-

struction and construction sub-phases, iterated greedy

heuristics overcome the drawbacks of a simple greedy

heuristic such as those mentioned in (Gutin et al.,

2002).

Iterated greedy heuristics are simple to understand

and easy to implement in practice, yet very effective

in providing high quality solutions for various prob-

lems. For example, (Jacobs and Brusco, 1995) apply

IGH to the set covering problem. (Ruiz and St¨utzle,

2007) and (Ruiz and St¨utzle, 2008) demonstrate the

effectiveness of iterated greedy heuristics for vari-

ants of the ﬂowshop problem. IGH is also applied

to the unrelated parallel machine scheduling problem

(Fanjul-Peyro and Ruiz, 2010). However, to the best

of our knowledge, iterated greedy heuristics have not

been applied to portfolio-optimization problems.

4 ITERATED GREEDY

HEURISTIC

In this section, we present our iterated greedy heuris-

tic, which is based on an IGH developed for the

permutation ﬂowshop scheduling problem (Ruiz and

St¨utzle, 2007). In Subsection 4.1, we give an

overviewon the design of our IGH. In Subsection 4.2,

we present the greedy insertion heuristic that is used

as a subroutine in our IGH.

4.1 Overview

For our iterated greedy heuristic to track the 1/ N port-

folio, we made two simpliﬁcations: (1) We always

include as many stocks as possible (= k) in the track-

ing portfolios, and (2) each stock in the portfolio has

the same weight (=

). These simpliﬁcations allow

us to save the CPU time required for determining the

portfolio weights of each stock, e.g., by solving a

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

426

quadratic program. We assume that for all instances

≤

≤ δ

, i = 1, ..., N, i.e., a portfolio weight of

for every stock is feasible (cf. Section 2).

Our iterated greedy heuristic proceeds as follows

(cf. Algorithm 1): We start with an empty portfolio π,

where all stocks have a portfolio weight of zero. We

then greedily add k stocks with a weight of

to π by

applying the greedy insertion heuristic discussed in

Subsection 4.2 such that the objective function value

by adding one new stock declines the most. The best

known solution is stored in π

and will be outputted

at the end of Algorithm 1.

Algorithm 1: Iterated greedy heuristic (IGH).

π :=

for i := 1 to k do

Add best stock to portfolio π with weight

applying Algorithm 2

:= π;

while termination criterion not satisﬁed do

{Improvement phase}

′

:= π

p := ⌊random∗ d⌋ + 1

for i := 1 to p do {Deconstruction sub-

phase}

remove a randomly selected stock from

portfolio π

′

for i := 1 to p do {Construction sub-phase}

Add best stock to portfolio π

′

with

weight

by applying Algorithm 2

if VRR(π

′

) < VRR(π) then

π := π

′

;

if VRR(π) < VRR(π

) then

:= π;

else

if random ≤ e

−

VRR(π

′

)−VRR(π)

temp

then

π := π

′

;

return π

;

After having found an initial solution, Algorithm1

enters the improvement phase and tries to improve

until some termination criterion is met, e.g., un-

til the limit on total CPU time is reached. From the

current solution, p random stocks are removed dur-

ing the deconstruction sub-phase, where p is calcu-

lated as ⌊random ∗ d⌋ + 1. The parameter d denotes

the maximum number of stocks to remove. Because

random is a uniformly distributed random number

in the interval [0,1), p is a uniform random integer

from the set {1,...,d}. During the construction sub-

phase, p stocks are added back to the portfolio with

a weight of

. These p stocks are selected greedily

by applying our greedy insertion heuristic (cf. Sub-

section 4.2). The new portfolio π

′

is then compared

to the old portfolio π before removing and adding

back p stocks. If the solution π

′

is better than π, the

new portfolio is immediately accepted as new solu-

tion. Algorithm 1 then also checks if the new port-

folio improves the best known solution π

. If this

is the case, π

is updated. With some probability,

the solution π

′

is accepted as new solution even if

it is worse than π. Due to the greedy nature of our

heuristic, Algorithm 1 could be trapped in a local min-

imum if only better solutions were accepted (John-

son et al., 1989). Therefore, worse solutions are ac-

cepted with a small probability to enable Algorithm 1

to escape such local minima. The probability to ac-

cept worse solutions depends on the parameter temp,

which can be interpreted as the temperature used in

simulated annealing heuristics (Johnson et al., 1989;

Johnson et al., 1991). To compare the objective func-

tion values (VRR(π),VRR(π

′

),VRR(π

)) of the solu-

tions, Algorithm 1 does not actually evaluate the ob-

jective function, but uses the information about the

possible improvement in the objective function value

for adding a given stock. This information is gener-

ated by Algorithm 2, which is presented in the follow-

ing subsection.

4.2 Greedy Insertion Heuristic

In this subsection, we explain the greedy insertion

heuristic that is used both to add back stocks to the

portfolio during the construction sub-phase and to

construct an initial solution at the beginning of the

IGH. A naive approach to ﬁnd the best stock to add

to a current portfolio would proceed as follows: For

each stock that has not been selected yet, a portfo-

lio is constructed by adding the stock to the current

portfolio with a weight of

. Then, the objective

function value is computed by using (2) for each of

these portfolios. The stock that leads to the portfo-

lio with the lowest objective function value is then

selected. To calculate the objective function value

for a given portfolio, N

times two subtractions and

three multiplications need to be performed, i.e., in to-

tal 5N

ﬂoating point operations (ﬂops). If the cur-

rent portfolio includes m stocks, this approach needs

(N − m) = 5N

− 5N

m ﬂops to select the best

stock.

However, by applying Algorithm 2, the best stock

can be found in a more efﬁcient way, i.e., in 2N

−

2Nm ﬂops. To see this, assume that we want to calcu-

late the possible improvement in the objective func-

tion value if we increase the weight of some stock j

′

that has not been selected yet from 0 to

. The only

summands in the objective function affected by this

An Iterated Greedy Heuristic for the 1/N Portfolio Tracking Problem

427

change are:

′

−

)

∑

i∈{1,...,N}\ j

′

[σ

′

−

)(x

′

−

)]+

∑

i∈{1,...,N}\ j

′

[σ

′

−

)(x

−

)]

≡ σ

′

−

)

∑

i∈{1,...,N}\ j

′

[σ

′

−

)(x

′

−

)]

(7)

Because the covariance matrix is symmetric, i.e.,

= σ

, i, j ∈ {1,...,N}, the two sums in (7) are

equal. If we add stock j

′

to the portfolio, there are

three different cases how the summands in (7) are af-

fected.

1. Consider some stock i 6= j

′

that has not been se-

lected yet. The contribution to (7) of the two

stocks i and j

′

before adding stock j

′

to the port-

folio is 2σ

′

(0 −

)(0 −

). After changing the

weight of stock j

′

from 0 to

, this contribu-

tion changes to 2σ

′

(0−

)(

−

). So, the im-

provement can be calculated as the difference be-

tween these two contributions and corresponds to

2σ

′

. In Algorithm 2, we use w

as fac-

tor to weight the covariances that belong to this

ﬁrst case to compute the possible improvementfor

stock j

′

2. Consider some other stock i 6= j

′

that has al-

ready been selected and thus, has a weight of

in the portfolio. Therefore, the contribution

of the stocks i and j

′

to the objective function

value before changing the weight of stock j

′

2σ

′

(

−

)(0−

), and becomes 2σ

′

(

−

)

after changing the weight of stock j

′

. So, the dif-

ference is 2σ

′

(

−

). In Algorithm 2, we use

= w

−

to weight the covariances for this

second case.

3. The variance of the returns of stock j

′

contributes

as follows to the objective function value before

changing its weight: σ

′

(0−

)

. After adding

stock j

′

to the portfolio, the contribution trans-

forms to σ

′

(

−

)

. So, the difference is

′

(

−

). For this third case, we use the

weight w

= w

−

in Algorithm 2.

In total, to compute the possible improvement for

increasing the weight of stock j

′

from 0 to

, we can

use Algorithm 2 that multiplies the covariances be-

tween the returns of stock j

′

and stocks i ∈ {1,... ,N}

with the respective weights w

, w

, or w

. The

weighted covariances are then added up to calculate

the total improvement. So, in total, this takes N mul-

tiplications and N additions (ignoring the ﬂops to cal-

culate the weights w

). Therefore, to select the

best stock from the N − m stocks that have not been

selected yet, i.e., the stock that leads to the largest de-

cline ω

∗

in the objective function value, Algorithm 2

takes 2N

− 2Nm ﬂops.

Algorithm 2: Greedy insertion heuristic.

′

:= current portfolio (input);

;

:= w

−

;

:= w

−

;

∗

:= −∞

for j := 1 to N do

if stock j has a weight of 0 in π

′

then

ω := 0

for i := 1 to N do

if i = j then

ω := ω+ w

;

else

if stock i has a weight of 0 in π

′

then

ω := ω+ w

;

else

ω := ω+ w

;

if ω > ω

∗

then

∗

:= ω

∗

:= j

return j

∗

and ω

∗

;

5 COMPUTATIONAL

EXPERIMENT

We tested our iterated greedy heuristic on a set of

test instances generated from real-world stock-market

data, and compared the best results obtained by the

heuristic to those obtained by the MIQP. In Subsec-

tion 5.1, we present the test instances. In Subsec-

tion 5.2, we explain the test setting. In Subsection 5.3,

we report on the computational results.

5.1 Test Instances

In total, we used 23 test instances for our computa-

tional experiment (cf. Table 2). We generated the ﬁrst

eight of these instances from index-tracking bench-

mark instances that can be downloaded from the OR-

Library (Beasley, 1990). We constructed 15 further

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

428

Table 2: Test instances.

Number of stocks

Nr. Index N k

1 Hang Seng 31 10

2 DAX100 85 10

3 FTSE100 89 10

4 S&P100 98 10

5 Nikkei225 225 10

6 S&P500 457 40

7 Russell2000 1,319 70

8 Russell3000 2,152 90

9 SMI 20 10

10 Hang Seng 49 10

11 EUROSTOXX50 50 10

12 FTSE100 96 10

13 S&P100 99 10

14 NASDAQ100 101 10

15 DAX100 102 10

16 SPI 198 10

17 Nikkei225 220 10

18 S&PASX300 254 10

19 S&P500 489 40

20 FTSE All Share 567 40

21 STOXXEURO600 575 40

22 S&P1200 1,179 70

23 Nasdaq Composite 2,140 90

test instances from real-world stock-market indices in

the same way as (Beasley, 1990). Each of the 23 in-

stances represents a speciﬁc investment universecom-

posed of N stocks, and consists of the weekly prices

of all the stocks that constitute the given market index

and do not have any missing prices. The 1/N portfo-

lio is then composed of each of these N stocks with a

weight of

for each stock, and can be seen as an ar-

tiﬁcial index that is equally weighted and rebalanced

every week. The objective is to replicate the returns

of this artiﬁcial index.

5.2 Test Setting

For our computational experiments, we deﬁned the

speciﬁc values for the different parameters from Ta-

ble 1 as follows. The number of stocks in the in-

vestment universe (N) depends on the speciﬁc test

instance and is depicted in Table 2. This table also

shows the maximum number of stocks to include in

a tracking portfolio (k) for each instance. For the

ﬁrst eight instances, we used the same values for k

as (Guastaroba and Speranza, 2012), whose objective

was to track the original market index. For the re-

maining 15 instances, the corresponding k of the ﬁrst

eight instances with a similar size N was used. As

lower and upper limits on the portfolio weights for

each stock included in the tracking portfolio, we used

a minimum weight of 0.5% (ε

= 0.005, i = 1,... ,N)

and a maximum weight of 20% (δ

= 0.2, i = 1,...,N)

for each instance. These values allowed us to con-

struct tracking portfolios composed of k stocks with

a weight of

for all instances. The covariances be-

tween the returns of the stocks σ

(i, j = 1,... ,N)

were computed based on the historical stock returns

over a time window of 104 weeks. Since the result-

ing MIQP is only convex if the matrix of covariances

is positive semi-deﬁnite, we use the covariance esti-

mator developed by (Ledoit and Wolf, 2004) that al-

ways yields a positive semi-deﬁnite covariance ma-

trix (even if a test instance contains more stocks than

weekly prices). The last parameter in Table 1 is the

weight of each stock in the artiﬁcial index we want to

replicate (w

, i = 1, ..., N). Because our target index

corresponds to the 1/N portfolio, we used

for this

parameter.

Furthermore, we had to deﬁne the parameters d

and temp that are relevant for the iterated greedy

heuristic only. For the maximum number of stocks

d to remove and add during the deconstruction and

the construction sub-phase, we tested the four values

2,3,4,5 for all instances. For the largest instances

with N > 500 and k ≥ 40, we also tested the values

d = 10 and d = 20. For the parameter temp, which

deﬁnes the acceptance criterion for new solutions, we

used a constant value of 0.5 for all instances. This

means that we tested four and six different variants

of the iterated greedy heuristic for the smaller and the

larger test instances, respectively.

For the MIQP and the IGH, we limited the CPU

time to 300 and 100 seconds per instance, respec-

tively. For IGH, the 100 seconds CPU time limit were

used as only termination criterion, which means that

the deconstruction and construction sub-phases were

executed repeatedly until this time limit was reached.

We used the Gurobi solver 6.0 to solve the MIQP. The

iterated greedy heuristic was implemented in C (com-

piler: gcc 4.9.3). All computations were performed

on a standard PC with an Intel i7 CPU with 3.4 GHz

and 4 GB RAM.

5.3 Results

Table 3 presents the computational results for the 23

test instances sorted by the number of stocks N in the

instance. Columns three and four show the best ob-

jective function values and the best lower bounds on

the objective function value for the MIQP (scaled by

a factor of 10

). Columns ﬁve to ten report the results

obtained by the variants of our iterated greedy heuris-

tic. The ﬁgures in these columns represent the relative

An Iterated Greedy Heuristic for the 1/N Portfolio Tracking Problem

429

Table 3: Computational results.

Instance MIQP IGH with temp = 0.5

N Nr. Best obj. Best bound d = 2 d = 3 d = 4 d = 5 d = 10 d = 20

20 9 9.84 9.84 25.4% 25.4% 25.4% 25.4% − −

31 1

32.92 32.92 18.3% 16.1% 16.1% 18.3% − −

49 10 19.45 12.13 17.0% 17.0% 17.0% 17.0% − −

50 11

18.06 7.86 4.1% 4.1% 4.1% 4.1% − −

85 2 42.04 3.85 −2.1% −2.1% −2.1% − 2.1% − −

89 3 47.57 3.39 1.1% −1.0% −1.0% −1.0% − −

96 12

20.99 1.29 8.2% 5.5% 5.5% 5.5% − −

98 4 58.80 2.46 2.9% 0.5% 0.5% 0.5% − −

99 13

16.45 1.08 4.5% 4.5% 4.5% 4.5% − −

101 14 44.91 2.73 6.3% −1.2% −1.2% −1.2% − −

102 15

37.39 1.96 3.0% 0.9% 0.9% 0.9% − −

198 16 147.85 0.84 0.0% 0.0% 0.0% 0.0% − −

220 17

31.08 0.06 2.3% 0.2% 0.2% −0.3% − −

225 5 29.85 0.05 23.9% 20.6% 19.3% 20.0% − −

254 18

116.18 0.27 3.2% −2.8% −3.6% −3.6% − −

457 6 14.74 0.03 −9.7% − 9.3% −11.4% −12.2% − −

489 19

4.98 0.01 −11.8% −14.7% −13.9% −14.1% − −

567 20 10.94 0.01 −12.7% −14.8% −15.1% −14.5% −14.2% −12.3%

575 21

8.19 0.01 −7.5% −11.1% −9.2% −12.5% −6.7% −2.5%

1179 22 5.39 0.00 −30.5% −32.2% −31.7% −31.2% −29.7% −26.9%

1319 7

37.75 0.04 −25.2% −26.6% −26.4% −26.7% −26.6% −26.9%

2140 23 44.48 0.01 −22.3% −22.3% −22.3% −22.4% −22.3% −22.3%

2152 8

103.64 0.01 −84.4% −84.5% −84.4% −84.6% −84.4% −84.3%

differences between the best objective function values

obtained by the iterated greedy heuristics (VRR(π

)

from Algorithm 1) and the MIQP (referred to as ofv

MIQP) within the CPU time limits. These relative

differences are computed as

VRR(π

)−ofv MIQP

ofv MIQP

, and are

negative if IGH obtained better (smaller objective

function value) solutions than the MIQP within the

given CPU times.

For the smaller instances, the Gurobi solver ob-

tained better solutions than the iterated greedy heuris-

tics and could also prove optimality of the solutions

for the two smallest instances. However, with re-

spect to the average over all instances, and in par-

ticular for the larger problem instances, the iterated

greedy heuristic obtained portfolios with a consid-

erably lower objective function value. Over all 23

instances, the variants of the iterated greedy heuris-

tic obtained solutions whose objective function val-

ues were roughly 5% lower than those obtained by

the MIQP. For the larger instances with N > 500 the

average relative difference was approximately −35%.

Furthermore, Table 3 also indicates that the greedy

heuristics delivered good results independently of the

choice for the parameter d.

6 CONCLUSIONS

In this paper, we considered the problem of replicat-

ing the 1/N portfolio by investing only in a subset of

the N stocks from a given investment universe. To

ﬁnd solutions to this problem, we presented a mixed-

integer quadratic program (MIQP) and developed an

iterated greedy heuristic. In a computational experi-

ment, it turned out that the iterated greedy heuristic

was able to obtain better solutions for the larger in-

stances in shorter CPU time.

In future research, we will evaluate our approach

on a larger set of test instances, and compare the re-

sults of our approach with standard index-tracking ap-

proaches that ignore the actual index weights. Fur-

ther, we will analyze the impact of the minimum

and maximum weights of each stock included in the

tracking portfolio as well as the maximum number of

stocks to invest in. Also, a local search method should

be developed that can be applied to improve the so-

lutions in each iteration of the heuristic. Further-

more, our IGH approach could be extended by relax-

ing the two simpliﬁcations that exactly k stocks with

an equal weight have to be included in a tracking port-

folio; for this purpose, we should apply quadratic pro-

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

430

gramming to determine optimal portfolio weights for

each included stock. According to the result of (Balas

and Saltzman, 1991)that max-regret greedy heuristics

outperform simple greedy heuristics, it could also be

interesting to develop an iterated max-regret greedy

heuristic for the problem considered in this paper.

REFERENCES

Balas, E. and Saltzman, M. J. (1991). An algorithm for

the three-index assignment problem. Operations Re-

search, 39:150–161.

Beasley, J. E. (1990). OR-Library: Distributing test prob-

lems by electronic mail. The Journal of the Opera-

tional Research Society, 41:1069–1072.

Beasley, J. E., Meade, N., and Chang, T.-J. (2003). An

evolutionary heuristic for the index tracking problem.

European Journal of Operational Research, 148:621–

643.

Canakgoz, N. A. and Beasley, J. E. (2008). Mixed-integer

programming approaches for index tracking and en-

hanced indexation. European Journal of Operational

Research, 196:384–399.

DeMiguel, V., Garlappi, L., and Uppal, R. (2009). Optimal

versus naive diversiﬁcation: How inefﬁcient is the 1/N

portfolio strategy? The Review of Financial Studies,

22:1915–1953.

Fanjul-Peyro, L. and Ruiz, R. (2010). Iterated greedy

local search methods for unrelated parallel machine

scheduling. European Journal of Operational Re-

search, 207:55–69.

Fiterman, A. E. and Timkovsky, V. G. (2001). Basket

problems in margin calculation: Modelling and algo-

rithms. European Journal of Operational Research,

129:209–223.

Guastaroba, G. and Speranza, M. G. (2012). Kernel Search:

An application to the index tracking problem. Euro-

pean Journal of Operational Research, 217:54–68.

Gutin, G., Yeo, A., and Zverovich, A. (2002). Traveling

salesman should not be greedy: Domination analysis

of greedy-type heuristics for the TSP. Discrete Ap-

plied Mathematics, 117:81–86.

Jacobs, L. W. and Brusco, M. J. (1995). A local search

heuristic for large set-covering problems. Naval Re-

search Logistics Quarterly, 42:1129–1140.

Johnson, D. S., Aragon, C. R., McGeoch, L. A., and

Schevon, C. (1989). Optimization by simulated an-

nealing: An experimental evaluation; Part I, graph

partitioning. Operations Research, 37:865–892.

Johnson, D. S., Aragon, C. R., McGeoch, L. A., and

Schevon, C. (1991). Optimization by simulated an-

nealing: An experimental evaluation; Part II, graph

coloring and number partitioning. Operations Re-

search, 39:378–406.

Ledoit, O. and Wolf, M. (2004). A well-conditioned estima-

tor for large-dimensional covariance matrices. Journal

of Multivariate Analysis, 88:365–411.

Markowitz, H. (1952). Portfolio selection. The Journal of

Finance, 7:77–91.

Roll, R. (1992). A mean/variance analysis of tracking error.

The Journal of Portfolio Management, 18:13–22.

Ruiz, R. and St¨utzle, T. (2007). A simple an effective

iterated greedy algorithm for the permutation ﬂow-

shop scheduling problem. European Journal of Op-

erational Research, 177:2033–2049.

Ruiz, R. and St¨utzle, T. (2008). An Iterated Greedy heuris-

tic for the sequence dependent setup times ﬂowshop

problem with makespan and weighted tardiness ob-

jectives. European Journal of Operational Research,

187:1143–1159.

An Iterated Greedy Heuristic for the 1/N Portfolio Tracking Problem

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