Portfolio selection problems in practice:
a comparison between linear and quadratic
optimization models
Francesco Cesarone� , Andrea Scozzari‡ , Fabio Tardella§
� Università
degli Studi Roma Tre - Dipartimento di Economia
fcesarone@uniroma3.it
‡
UNISU - Università Telematica delle Scienze Umane “Niccolò Cusano”
Facoltà di Economia
andrea.scozzari@unisu.it
§ Università
di Roma “La Sapienza”
Dipartimento di Metodi e Modelli per l’Economia, il Territorio e la Finanza
fabio.tardella@uniroma1.it
(July 2010)
ABSTRACT
Several portfolio selection models take into account practical limitations on the number of assets to
include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz
(LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional
Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and
cardinality constraints.
We propose a completely new approach for solving the LAM model, based on reformulation as a
Standard Quadratic Program and on some recent theoretical results. With this approach we obtain
optimal solutions both for some well-known financial data sets used by several other authors, and for
some unsolved large size portfolio problems. We also test our method on five new data sets involving
real-world capital market indices from major stock markets. Our computational experience shows that,
rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the
linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer
linear programming (MILP) problems.
Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of
the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained
models and with that of the official capital market indices.
KEYWORDS
Mixed Integer Linear and Quadratic Programming; Ex-post Performance; Portfolio Management; Conditional Value-at-Risk; Mean-Variance; Mean Absolute Deviation.
1
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
1
2
Introduction
The classical Mean-Variance (MV) portfolio selection model of Markowitz [40, 41, 42] has
been widely recognized as one of the cornerstones of modern portfolio theory. However,
its success has inevitably drawn many criticisms and proposals of alternative or more
refined models (see, e.g., [19, 28, 32, 43, 44, 46, 47] and references therein).
Among the many refinements that have been proposed to make the Markowitz model
more realistic, we analyze in this paper the one that limits the number of assets to be held
in an efficient portfolio (cardinality constraint), and also the one that prescribes lower and
upper bounds on the fraction of the capital invested in each asset (quantity constraints).
These requirements come from real-world practice, where the administration of a portfolio
made up of a large number of assets, possibly with very small holdings for some of them, is
clearly not desirable because of transactions costs, minimum lot sizes, complexity of management, or policy of the asset management companies. We call Limited Asset Markowitz
(LAM) model the Markowitz model with the above restrictions. Because of its practical
relevance, this model (often called cardinality constrained Markowitz model ), and some
variations thereof, have been fairly intensively studied in the last decade, especially from
the computational viewpoint [4, 9, 10, 14, 16, 17, 18, 20, 21, 24, 26, 35, 39, 45, 50, 52, 55].
In these studies it appears that the computational complexity for the solution of the
LAM model is much greater than the one required by the classical Markowitz model or
by several other of its refinements. Indeed, the standard Markowitz model is routinely
solved for markets with thousands of assets. This practical difference in computational
complexity is also theoretically justified by the fact that the classical Markowitz model is
a convex quadratic programming problem that has a polynomial worst-case complexity
bound, while the LAM model is usually modeled by adding binary variables, thus becoming a mixed integer quadratic programming (MIQP) problem, that falls into the class of
considerably more difficult NP-hard problems (see, e.g., [10, 52]).
We remark that some attempts have been made to construct simpler portfolio selection
models, for instance, by linearizing the quadratic objective function. These approaches
involve either the approximation or the decomposition of the covariance matrix (see, e.g.,
[44]). Some researchers have also introduced alternative risk measures for portfolio planning. In many cases these measures are linear, leading to a corresponding simplification
from the computational viewpoint. Konno [30] introduced the mean absolute deviation
(MAD) model. This return-risk model together with the return-standard deviation model
are among the most important portfolio models, that use dispersion measures. Speranza
[53] considered the downside mean semi-deviation, i.e., the mean absolute value of negative deviations. She also proved that it is always equal to half of the mean absolute
deviation from the mean. Roy [49] laid the basis for the development of downside risk
measures. The objective in portfolio selection models with downside risk measures is the
maximization of the probability that the portfolio return is above a certain minimal acceptable level. Markowitz [41] proposed the semi-variance as an alternative risk measure,
but he suggested that the use of variance is computationally more tractable and reveals
the same information. In the 1970s, many papers (see, e.g., [6, 22]) provided a natural generalization of semi-variance with the lower partial moment risk measure, whose
justification proceeds from the observation that an investor’s true risk is the downside
risk. Young [59] developed a minimax approach for the portfolio management, measuring
risk as the minimum return (maximum loss) that the portfolio would have achieved over
all of the past observation periods. In 1994 JP Morgan proposed what is now probably
the most famous downside risk measure: Value-at-Risk (VaR, see [46]). This is a very
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
3
important risk management tool in the financial industry, but with some drawbacks: the
VaR optimization problem is not convex, and VaR is not sub-additive, i.e., it does not
express the benefits of diversification. Sub-additivity is one of a set of desirable properties
that define a coherent risk measure (see the results described by Artzner et al. [5] for
downside risk measures, and by Rockafellar et al. [48] for dispersion measures). One of
the most important coherent risk measures is Conditional Value-at-Risk (CVaR, see [47]).
Some researchers call it Expected Shortfall [2]. Enhanced CVaR measures has been suggested by Mansini et al. [36]. Acerbi [1] proposed a spectral risk measure which involves
a weighted average of the quantiles of the loss. CVaR is a special case of spectral risk
measure. Although much work has been done on risk measures and mean-risk models,
the question of which risk measure is most appropriate is still the subject of much debate.
In this paper we compare the LAM model with the CVaR and MAD portfolio models
under additional constraints that limit the number of assets to be held in a portfolio and
prescribe lower and upper bounds on the fraction of the capital invested in each asset.
We call them Limited Asset CVaR (LACVaR) model and Limited Asset MAD (LAMAD)
model, respectively. These two models are usually formulated as Mixed Integer Linear
Programming models (MILP) and fall into the class of NP-hard problems too.
The LAM model is solved here with a completely new approach that is based on a
reformulation as a Standard Quadratic Programming problem and exploits recent theoretical results for Quadratic Programming by Tardella [57, 58] and by Scozzari and Tardella
[51]. Our method is able to solve to optimality the well-known five benchmark problems
described in [14] and publicly available in Beasley’s OR Library [7], whose optimal solution has been reported only very recently by Di Gaspero et al. [18]. In addition to
these five problems, we report solutions of much larger and unsolved real-world problems,
one with around 500 assets and two with more than 2000 assets also taken from the ORLibrary [13]. We provide some computational results comparing our solution method with
the exact MIQP solver implemented in CPLEX 11.0. We also test our method on five
new data sets involving real-world capital market indices from major stock markets. Our
experimental analysis shows that the practical computational complexity for most exact
algorithms for the LAM model seems to be related not only to the number of variables
but also to the number of assets with positive weight in the solution of the unconstrained
Markowitz model.
An important issue highlighted in this study is that, rather unexpectedly, it easier to
solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and
LAMAD models with CPLEX, one of the best commercial codes for MILP problems.
Finally, on the new data sets we have compared, using out-of-sample analysis, the
performance of the portfolios obtained by the Limited Asset models with the performance
provided by the unconstrained models and with that of the official capital market indices.
We made our data sets and the solutions that we found publicly available for use by
other researchers in this field.
The paper is organized as follows: Section 2 introduces the three Limited Asset models
considered in this study along with a review of the methods used to solve them. In Sections
3 and 4 we present our new approach to solve the LAM model based on a reduction to
a Standard Quadratic Programming problem. Section 5 reports some computational
results showing that the quadratic LAM model can be solved more efficiently than the
linear LACVaR and LAMAD models. In Section 6 we present a comparison among the
ex-post performances of the portfolios obtained by the models.
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
2
2.1
4
Limited Asset Models
The Limited Asset Markowitz Model
The classical Mean-Variance (MV) portfolio optimization model introduced by Markowitz
aims at determining the fractions xi of a given capital to be invested in each asset i
belonging to a predetermined set or market so as to minimize the risk of the return of the
whole portfolio, identified with its variance, while restricting the expected return of the
portfolio to attain a specified value.
More precisely, we assume that n assets are available, and we denote by µi the expected
return of asset i, and by σij the covariance of returns of asset i and asset j. We also denote
by ρ the required level of return for the portfolio. The classical MV model is:
Min
n
n �
�
σij xi xj
i=1 j=1
st
n
�
i=1
n
�
(1)
µi x i = ρ
xi = 1
i=1
xi ≥ 0
i = 1, . . . , n
This is a convex quadratic programming problem which can be solved by a number of
efficient algorithms with a moderate computational effort even for large instances. We
denote by φ(ρ) the optimal value of (1) as a function of ρ. Let ρmin denote the value of
�
n
i=1 µi xi at an optimal solution of the problem obtained by deleting the first constraint in
(1), and let ρmax = max{µ1 , . . . , µn }. Then the graph of φ(ρ) on the interval [ρmin , ρmax ]
coincides with the set of all non-dominated (or efficient) portfolios (efficient frontier ),
and is usually approximated by solving (1) for several (equally spaced) values of ρ in
[ρmin , ρmax ].
Proposition 1 The convexity of (1) implies that, for ρ ≥ ρmin , the function φ(ρ) is
increasing and convex.
Proof. Let ρ0 , ρ1 ∈ [ρmin , ρmax ] and let x0 , x1 be two corresponding solutions of (1).
Then xλ = (1 − λ)x0 + λx1 is a feasible solution to (1) for ρλ = (1 − λ)ρ0 + λρ1 , for any
λ ∈ [0, 1] (due to the linearity of expected return). The convexity of φ(ρ) follows from
the convexity of the variance σ 2 (x) = x� Σx:
φ(ρλ ) ≤ σ 2 (xλ ) = σ 2 ((1 − λ)x0 + λx1 ) ≤ (1 − λ)σ 2 (x0 ) + λσ 2 (x1 ) = (1 − λ)φ(ρ0 ) + λφ(ρ1 )
To prove isotonicity of φ, take any ρ0 < ρ1 ∈ [ρmin , ρmax ]. Then for some λ ∈ (0, 1) we
have ρ0 = λρmin + (1 − λ)ρ1 , so that
φ(ρ0 ) ≤ λφ(ρmin ) + (1 − λ)φ(ρ1 ) ≤ λφ(ρ1 ) + (1 − λ)φ(ρ1 ) = φ(ρ1 ),
where the last inequality follows from the fact that φ(ρmin ) ≤ φ(ρ) for all ρ ∈ [ρmin , ρmax ],
by definition of ρmin .
�
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
5
From the above proposition we immediately derive the following result:
Corollary 2�The solution of (1) does not change if we replace the expected return constraint with ni=1 µi xi ≥ ρ.
�
� �
Note that for every solution x̄ of problem (1) the point ( ni=1 µi x̄i , ni=1 nj=1 σij x̄i x̄j )
is an efficient point of the multi-objective problem of minimizing risk and maximizing
return. Conversely, every such efficient point can be obtained in this manner for some ρ.
We now add to the MV model the realistic constraint that no more than K assets
should be held in the portfolio (a cardinality constraint), and furthermore that the quantity xi of each asset that is included in the portfolio should be limited within a given
interval [�i , ui ] (a quantity constraint or buy-in threshold ). Thus we obtain the following
Limited Asset Markowitz model:
Min
n �
n
�
σij xi xj
i=1 j=1
st
n
�
i=1
n
�
µi x i = ρ
(2)
xi = 1
i=1
xi = 0 or �i ≤ xi ≤ ui , i = 1, . . . , n
|supp(x)| ≤ K,
where supp(x) = {i : xi > 0}.
Problem (2) is no longer a convex optimization problem because of the non-convexity
of its feasible region. As a consequence the optimal value function φK (ρ) of (2) need
not be increasing nor convex. Furthermore, φK (ρ) does not any longer coincide with
the �
optimal value function φ�K (ρ) of problem (2) where the first constraint is replaced
by ni=1 µi xi ≥ ρ. Indeed, φ�K (ρ) = φK (ρ) if and only if the point (ρ, φK (ρ)) is on the
efficient frontier.
On the other hand, if φ�K (ρ) �= φK (ρ), then there exist no points (u, v) on the efficient
frontier with u = ρ. We deem important to point out that the non-convexity of problem
(2) (due to non-convexity of the feasible region) makes it incorrect to use a convex combination approach to find the efficient frontier for the multi-objective problem of minimizing
risk and maximizing return. More precisely, for any λ ∈ [0, 1] and any solution x̄ of the
problem
Min λ
n �
n
�
σij xi xj − (1 − λ)
i=1 j=1
n
�
µi x i
i=1
st
n
�
xi = 1
(3)
i=1
xi = 0 or �i ≤ xi ≤ ui ,
i = 1, . . . , n
|supp(x)| ≤ K,
�n �n
�n
the point ( i=1 µi x̄i , i=1 j=1 σij x̄i x̄j ) is an efficient point. However, as observed by
Jobst et al. [26], not all points on the efficient frontier can be obtained in this way.
This difficulty seems to have been overlooked by some authors that used the convex
combination approach to build the efficient frontier [20]. We remark that a necessary and
6
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
DAX 100 (85 Stocks) port2 in OR−Library
−3
x 10
9
9
8
8
7
7
6
5
x 10
6
5
φ′κ(ρ), Κ=2
φ′κ(ρ), Κ=3
φ′κ(ρ), Κ=4
φ′κ(ρ), Κ=5
φ(ρ)
4
3
2
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
Std deviation
φκ(ρ), Κ=2
φκ(ρ), Κ=3
φκ(ρ), Κ=4
φκ(ρ), Κ=5
φ(ρ)
4
3
2
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
Std deviation
Figure 1: Graphs of φ(ρ) and φ�K (ρ)
Figure 2: Graphs of φ(ρ) and φK (ρ)
DAX 100 (85 Stocks) port2 in OR−Library
−3
10
DAX 100 (85 Stocks) port2 in OR−Library
−3
10
RETURN
RETURN
10
x 10
9
8
RETURN
7
6
5
Κ=2
Κ=3
Κ=4
Κ=5
φ(ρ)
4
3
2
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
Std deviation
Figure 3: Efficient frontiers
sufficient condition for obtaining all the points on the efficient frontier by solving problem
(3) for all λ ∈ [0, 1] is that the value function φK (ρ) is convex.
Figs. 1, 2 and 3 illustrate the graphs of φ(ρ) and φ�K (ρ), of φ(ρ) and φK (ρ), and the
efficient frontiers for some values of K, in an instance based on real-world data. The lower
(�i ) and upper (ui ) bounds are set to 0.01 and 1, respectively. Note that these figures
are based on the exact optimal solutions to the following problem (4) obtained with the
algorithm described in Section 4. Fig. 3 can be compared with Fig. 9 in [14] that is based
on approximate solutions to (4) found with heuristic algorithms.
As observed by several authors [10, 14, 26], problem (2) can be reformulated as a
Mixed Integer Quadratic Program (MIQP) with the addition of n binary variables:
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
Min
n �
n
�
7
σij xi xj
i=1 j=1
st
n
�
i=1
n
�
i=1
n
�
µi x i = ρ
xi = 1
(4)
yi ≤ K
i=1
�i yi ≤ xi ≤ ui yi i = 1, . . . , n
xi ≥ 0
i = 1, . . . , n
yi ∈ {0, 1}
i = 1, . . . , n
A number of exact approaches have been proposed to solve problem (4). Bienstock [10]
proposes a branch-and-cut algorithm and reports good computational results for some
real-life problems (not available for comparison). However, his method seems to become
extremely slow for small values of K. Bertsimas and Shioda [9] extend the algorithm of
Bienstock [10] presenting a tailored procedure, based on Lemke’s pivoting algorithm [34],
that takes advantage of the special structure of the problem. They present computational
results only on randomly generated data for fairly large values of K. A branch-and-bound
algorithm for mixed integer nonlinear programs, including portfolio selection problems,
is presented in [12]. Li et al. [35] propose a convergent Lagrangian method as an exact
solution scheme for a problem slightly more general than (4) and they describe some computational results for problems with at most 30 assets. Another Lagrangian relaxation
method is proposed in [52] with application to some undisclosed real-life problems with up
to 500 assets. Lee and Mitchell [33] develop an interior-point algorithm within a parallel
branch-and-bound framework for solving nonlinear mixed-integer programming problems.
Preliminary computational results on three randomly generated quadratic portfolio models are reported. Frangioni and Gentile [23] use a method based on perspective cuts to
solve randomly generated LAM problems (4) without cardinality constraints involving
up to 400 assets. Furthermore, some commercial or free optimization softwares provide
tools to solve general MIQPs, and thus (4), although only for problems with few hundreds
variables at most.
Since exact methods are able to solve only a fraction of practically useful LAM models,
a variety of heuristic procedures have also been proposed for solving (4). Local search
techniques are discussed in [50], while Chang et al. [14] present three heuristics based
upon genetic algorithms, tabu search, and simulated annealing. In [26] two heuristic solution approaches are proposed for problems subject to buy-in threshold, cardinality and
roundlot constraints. A hybrid local search algorithm combining principles of simulated
annealing and of evolutionary strategies is used in [39] to solve problem (4) in the absence
of quantity constraints. Other evolutionary algorithms, combined with local search techniques in order to improve the quality of the solutions, are described in [55, 56]. Fieldsend
et al. [21] introduce a parallel solution method by extending techniques developed in the
multi-objective evolutionary optimization domain. Finally, Di Gaspero et al. [18] present
a heuristic solver, based on a hybrid technique that combines a local search metaheuristic
with a quadratic programming procedure. Experimental results seem to show that their
approach is very promising for medium size problems. They also consider pre-assignment
constraints, which specify a subset of asset that has to be included in the chosen portfolio.
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
8
In Section 4 we will show that such constraints actually make the problem easier for the
new approach that we propose in this paper.
We should mention that, while several authors experiment their algorithms on undisclosed or randomly generated data, a selection of the cited papers [4, 17, 18, 26, 45, 50,
55, 56] report results obtained on the five real-world data sets introduced in [14] that have
been made available by Beasley in his OR-Library [7].
2.2
2.2.1
The Limited Asset CVaR and MAD Models
CVaR
The Limited Asset CVaR (LACVaR) model is similar to the previous one in that it also
consists in a risk-return model with realistic constraints, represented by cardinality and
quantity constraints, but with the risk measured by CVaR. The model can be written as
follows:
min CV aR(x, �)
st
n
�
µ i xi = ρ
i=1
n
�
(5)
xi = 1
i=1
xi = 0 or �i ≤ xi ≤ ui , i = 1, . . . , n
|supp(x)| ≤ K,
Fig. 4 shows the optimal solutions of the LACVaR model in the risk-return plane for
several data sets and for some values of K. As described in [3], problem (5) can be
reformulated as a Mixed Integer Linear Program (MILP) with the addition of n binary
variables:
�
min
ζ + 1� T1 Tj=1 dj
st
�n
i=1 −rij xi − dj − ζ ≤ 0 j = 1, . . . , T
n
�
µi x i = ρ
i=1
n
�
i=1
n
�
xi = 1
(6)
yi ≤ K
i=1
�i yi ≤ xi ≤ ui yi
xi ≥ 0
yi ∈ {0, 1}
dj ≥ 0
ζ∈R
i = 1, . . . , n
i = 1, . . . , n
i = 1, . . . , n
j = 1, . . . , T
This is a MILP problem with n + T + 1 continuous variables, n binary variables and
T + n + 3 constraints. In Fig. 4, the bold line represents the frontier in the unconstrained
case, while the dashed and the thin lines are the frontiers for some values of K. The lower
(�i ) and upper (ui ) bounds are set to 0.01 and 1, respectively.
9
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
HS31
−3
11
x 10
Dax85
−3
10
10
x 10
9
9
8
8
Return
Return
7
7
6
6
5
5
4
4
Uncon.
K=2
K=3
K=5
K=10
3
2
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Uncon.
K=2
K=3
K=5
K=10
3
2
0.02
0.12
0.04
0.06
CVaR
FTSE89
−3
x 10
9
3.5
8
3
7
2.5
6
5
0.12
0.14
x 10
2
1.5
4
1
Uncon.
K=2
K=3
K=5
K=10
3
2
0.02
0.1
Nikkei225
−3
4
Return
Return
10
0.08
CVaR
0.025
0.03
0.035
0.04
0.045
CVaR
0.05
0.055
0.06
0.065
0.07
Uncon.
K=2
K=3
K=5
K=10
0.5
0
0.03
0.035
0.04
0.045
0.05
0.055
0.06
0.065
0.07
0.075
CVaR
Figure 4: Frontiers of the Limited Asset CVaR model
In order to compare the computational time required to solve the LAM and the LACVaR
models with respect to the same data sets, we have considered in both models the target
return levels ρ used to construct the frontiers for the LAM model. Thus ρ varies in
the interval [ρmin , ρmax ] for several (equally spaced) values, where ρmin is the value of
�
n
i=1 µi xi at an optimal solution of the problem obtained by deleting the constraint on
net portfolio mean return in (1), and ρmax = max{µ1 , . . . , µn }. The optimal solutions and
the corresponding frontiers are illustrated in Fig. 4.
2.2.2
MAD
The MAD model requires the solution of a simple linear programming problem. However,
when we add cardinality and quantity constraints, it also becomes a Mixed Integer Linear
Programming (MILP) for which it is harder to find an optimal solution. This model,
called Limited Asset MAD (LAMAD) model, can be formulated as follows [3]:
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
min E[|
n
�
10
(rij − µi )xi |]
i=1
st
n
�
µ i xi = ρ
i=1
n
�
(7)
xi = 1
i=1
xi = 0 or �i ≤ xi ≤ ui , i = 1, . . . , n
|supp(x)| ≤ K,
If we use n additional binary variables we can re-write the LAMAD model as the following
MILP:
min
1
T
T
�
dj
j=1
st
dj ≥
n
�
(rij − µi )xi
i=1
n
�
−dj ≤
n
�
i=1
n
�
i=1
n
�
(rij − µi )xi j = 1, . . . , T
i=1
µ i xi = ρ
(8)
xi = 1
yi ≤ K
i=1
�i yi ≤ xi ≤ ui yi
xi ≥ 0
yi ∈ {0, 1}
dj ≥ 0
i = 1, . . . , n
i = 1, . . . , n
i = 1, . . . , n
j = 1, . . . , T
This problem has n+T continuous variables, n binary variables and n+2T +3 constraints.
As before, the lower (�i ) and upper (ui ) bounds are set to 0.01 and 1, respectively. The
frontiers have been computed for several (equally spaced) values of ρ in [ρmin , ρmax ], where
the interval is determined as described in the previous subsection. In Fig. 5 we report the
optimal solutions in the MAD-return plane, computed using CPLEX, for some instances
based on real-world data and for some values of K.
2.2.3
Literature on Mixed Integer LP portfolio models
The need of solving large portfolio problems with real-world constraints justifies a long
tradition in the literature of mixed integer LP portfolio models. Konno [29] tackles the
portfolio optimization model, using a piecewise linear risk function. Historical monthly
data of 50 stocks for 5 years are used. Konno and Yamazaki [32] compare the MAD model
ex-post performance with that obtained by the MV model and the Single Index Model
using 224 stocks with monthly data for 5 years. Linear programming based heuristics
are used by Speranza [54], considering the negative semiMAD model with cardinality
constraints, transaction costs and minimum transaction units. The minimax model has
been presented by Young [59], who also describes how this model can be adapted to
11
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
HS31
−3
11
x 10
DAX85
−3
10
10
x 10
9
9
8
8
Return
Return
7
7
6
6
5
5
4
4
Uncon.
K=2
K=3
K=5
K=10
3
2
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Uncon.
K=2
K=3
K=5
K=10
3
2
0.005
0.05
0.01
0.015
0.02
CVaR
FTSE89
−3
x 10
9
3.5
8
3
7
2.5
6
5
0.035
0.04
x 10
2
1.5
4
1
Uncon.
K=2
K=3
K=5
K=10
3
2
0.01
0.03
Nikkei 225
−3
4
Return
Return
10
0.025
CVaR
0.012
0.014
0.016
0.018
0.02
CVaR
0.022
0.024
0.026
0.028
0.03
Uncon.
K=2
K=3
K=5
K=10
0.5
0
0.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
0.028
0.03
0.032
CVaR
Figure 5: Frontiers of the Limited Asset MAD model
include linear transaction costs. A set of historical data of 7 stock indices is examined
using the minimax and mean-variance portfolio selection rules. A comparison between
these model is done, evaluating the optimal solution on 30 monthly data and testing the
performance on the following 30 months. Bertsimas et al. [8] implement a mixed-integer
programming model, using CPLEX as a solver, to construct a portfolio that is close
to a target portfolio and controls frictional costs with cardinality constraints. Mansini
and Speranza [37] show that finding a feasible solution to the portfolio selection problem
using the Mean Semi-absolute Deviation model with roundlots is NP-complete. Some
computational experiences are performed on data sets with at most 277 securities on a
time period of 2 years. Different mixed integer linear programming models are presented
by Kellerer et al. [27]. They compare the solutions of the semi-absolute deviation with
fixed costs and possibly minimum lots, obtained by heuristic procedures and by CPLEX.
The computational results are performed on the monthly rates of return of 244 securities
for a time interval of 3 years. Konno and Wijayanayake [31] suggest a branch and bound
algorithm for calculating a globally optimal solution of a portfolio problem, using MAD
as risk measure under concave transaction costs and minimal transaction unit constraints.
Some numerical tests are done with at most 60 monthly data of 200 stocks. Chiodi et
al. [15] present a mixed integer linear programming model to solve a portfolio selection
problem on mutual funds. Some heuristics are proposed, testing the problems on the
historical data consisting of 310 mutual funds over the time period of 36 monthly returns
for each fund. A portfolio selection problem with transaction costs and integer constraints
on the quantities selected for the securities is presented by Mansini and Speranza [38].
12
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
The number of stocks considered in the experiments are between 50 and 1000 on a time
period varying from 2 to 6 years (this means that 100 ≤ T ≤ 300).
While the MAD model with integer constraints has been widely analyzed in the literature, to the best of our knowledge few papers has been devoted to the CVaR model
with real-world constraints. A financial and computational comparison of MAD and
CVaR models with real features has been performed by Angelelli et al. [3]. Two different
mixed integer linear programming models with integer stock units, transaction costs and
a cardinality constraint are taken into account, analyzing their performance on real size
instances. They use 104 weekly rates of return for the in-sample analysis and they test
the performance of the optimal solutions obtained on the following 52 weeks.
3
Reduction to a Standard Quadratic Programming
Problem
We propose here a new method for solving (2) that avoids the explicit use of additional
binary variables. Our approach is based on the reduction of the LAM model (2) to a
Standard Quadratic Programming (StQP) problem, as defined by Bomze [11], and is able
to solve to optimality Beasley’s problems and problems of greater dimension.
A StQP is the problem of minimizing a (possibly indefinite) quadratic form over the
standard simplex ∆, that is
Min x� Qx
st
n
x ∈ ∆ = {x ∈ R :
n
�
(9)
xi = 1, xi ≥ 0,
i = 1, . . . , n}
i=1
Despite its formal simplicity, this problem is theoretically difficult to solve (NP-hard)
when Q is indefinite [11]. Indeed, its actual optimal solution for instances with more than
40 variables have not been reported in the literature until the recent paper by [51], where
instances with more than 1000 variables have been solved.
We also point out that there is no loss of generality in restricting to quadratic forms
instead of considering a general quadratic objective function. Indeed, over ∆ a quadratic
function f (x) = x� P x + 2q � x coincides with the quadratic form x� Qx, where Q = P +
eq � + qe� , and e denotes the all-ones vector.
Problem (1) can be easily transformed into a (convex) StQP problem by using a
quadratic penalty for the return constraint:
Min fM (x) =
n
n �
�
i=1 j=1
st
σij xi xj + M [
n
�
i=1
µi xi − ρ]2
(10)
x∈∆
Adding the cardinality constraint |supp(x)| ≤ K to (10) amounts to minimizing fM on
the faces of dimension not greater than K of the standard simplex ∆. If we further add
the condition that �i ≤ xi ≤ ui for i = 1, . . . , n, we obtain a StQP with cardinality and
upper and lower bound constraints which is equivalent to (4).
13
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
In the next section we describe how to solve a StQP with cardinality and upper and
lower bound constraints by adapting the algorithms developed by [51] for the unconstrained case.
4
Theoretical Results and Solution Method
We consider the cardinality constrained StQP problem:
min f (x) = x� Qx
st
x ∈ ∆ = {x ∈ Rn :
n
�
xi = 1, xi ≥ 0,
i = 1, . . . , n}
(11)
i=1
|supp(x)| ≤ K
In order to restrict the search for its global minimizers, we use the following QP
extension of the fundamental theorem of Linear Programming.
Theorem 3 [51, 57, 58]. A quadratic function f that is bounded below on a (pointed)
polyhedron P attains its minimum on P in the relative interior of a face of P where f is
strictly convex.
�
Let N = {1, . . . , n}. Every face of ∆ has the form ∆I = {x ∈ ∆ : i∈I xi = 1}, where
I ⊆ N is a subset of indices. Furthermore, the dimension dim(∆I ) of ∆I coincides with
the cardinality |I| of I. Let IK denote the family of all subsets of N with cardinality at
most K. Then the cardinality constrained StQP (11) can be reformulated as:
x∈
�min
I∈IK
∆I
f (x) = x� Qx
(12)
Hence we obtain the following straightforward consequence of Theorem 3:
Corollary 4 At least one global minimizer of (11) must be in the relative interior of a
face ∆I of ∆ where f is strictly convex and |I| ≤ K.
For I ⊆ N , let QI denote the submatrix of Q formed by those elements with row
and column indices in I. When QI is positive �
definite, the unique global minimizer
�
of the quadratic form x QI x on the hyperplane
i∈I xi = 1 is attained at the point
−1 −1
e.
Thus
the
quadratic
form
f
(x)
has a global minimizer on ∆ in the
x∗I = (e� Q−1
e)
Q
I
I
relative interior rint(∆I ) of a face ∆I where f is strictly convex only if x∗I ∈ rint(∆I ).
To every subset I ⊆ N we associate the (nonlinear) weight
w(I) = min{f (x) : x ∈ ∆I }.
Corollary 4 and simple matrix algebra imply that
x∈
�min
I∈IK
∆I
−1
x� Qx = min w(I) = min f (x∗I ) = min (e� Q−1
I e) ,
I∈CK
I∈CK
I∈CK
(13)
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
14
where CK is the subset of IK defined by
CK = {I ∈ IK : QI is positive definite and x∗I ∈ rint(∆I )}.
In view of (13), the cardinality constrained StQP could be solved by evaluating
−1
(e Q−1
for all elements I ∈ CK , but this is clearly not practical for large values of
I e)
n and K. However, another recent theoretical result can be used to restrict the search for
a global minimizer:
�
Theorem 5 [51]. If x∗ is a global minimizer of a quadratic function f on a polyhedron
P , then there exists a nested sequence of faces F 1 ⊂ F 2 ⊂ . . . ⊂ F k of P , with dimension
dim(F i ) = i, where f is strictly convex, has an interior global minimizer x̂F i , and x∗ =
x̂F k .
For j ≤ K, let C j = {I ∈ CK : |I| = j}. Then the above theorem guarantees that
for any I ∗ minimizing w(I) on CK there exists a sequence I1 ⊆ I2 ⊆ · · · ⊆ Ih = I ∗ , such
that Ij ∈ C j for all j = 1, . . . , h. Thus we can apply the following algorithm to solve the
cardinality constrained StQP (11) or, equivalently, to minimize w(I) on CK :
Increasing Set Algorithm
1 Set C0 = ∅, C1 = {{i}, i ∈ N }
2 MIN(1) = minI∈C1 w(I) = min1≤i≤n qii
3 for j = 1 to K
4
5
do construct C j+1 by increasing, if possible, all elements in C j
if C j+1 = ∅
6
then MIN(h) = MIN(j) for h = j + 1, . . . , K, return(MIN(K))
7
else MIN(j + 1) = min{MIN(j) , minI∈C j+1 w(I)
8
return (MIN(K))
Note that, by Theorem 5, at any iteration j, MIN(j) contains the minimum value of
w(I) among all sets in Cj . Furthermore, if C j+1 = ∅ in step 5, then, again by Theorem 5,
C h must be empty for all h ≥ j + 1. Hence the algorithm correctly stops with the global
minimizer in CK . In fact, at each iteration j the Increasing Set Algorithm provides in
MIN(j) the solution to the StQP problem with cardinality constraint |supp(x)| ≤ j.
We have proved that the Increasing Set Algorithm is exact. Unfortunately, it has
exponential complexity in the worst case, and may be too slow in practice for large size
problems. However, we obtain a very good heuristic if we bound at each iteration the
size of C j by keeping only a limited number of sets I with the best values of w(I). From
a theoretical viewpoint, we can achieve polynomial time complexity in this way, but
of course we lose the guarantee of optimality. In practice, however, we have observed
considerable reduction in the running time without losing optimality in all real-world
instances, described in Section 5, that have been solved with both our algorithm and with
CPLEX.
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
15
We should point out that in order to apply our algorithm to the (reformulated) LAM
model that also includes lower and upper bounds �i and ui on the variables xi > 0, we need
to further modify the basic Increasing Set Algorithm described above. Indeed, to solve a
�
��
StQP with cardinality and lower and upper bound constraints, we find the sets C j and C j ,
�
��
�
where C j = {I ∈ C j : � ≤ x∗I ≤ u} and C j = C j \ C j . We replace minI∈C j+1 w(I) in step 7 of
�
��
��
= K
the algorithm with minI∈C �j+1 w(I), and we memorize the list of all sets I in CK
j=1 C j .
�
�
At the end of the algorithm, we then replace MIN(K) with min MIN(K), minI∈CK�� w(I) .
��
This can be done efficiently by observing that, for all I ∈ CK
, w(I) can be computed by
solving a convex quadratic programming problem of dimension |I|, and that we only need
��
to solve such problems for those I ∈ CK
for which f (x∗I ) < MIN(K).
Furthermore, if we want to find the best portfolio among those that contain a given
subset J of assets (i.e., satisfy the pre-assignment constraints of Di Gaspero et al. [18]),
then we just need to modify the Increasing Set Algorithm so that it starts with the family
C|J| = {J}. Thus the pre-assignment constraints actually simplify the solution of the
problem with the Increasing Set Algorithm, which needs less iterations to terminate.
5
5.1
Data Sets and Computational Results
Data Sets
An important issue when evaluating computational results for a class of problems is
the availability of benchmark data sets, possibly with solutions, that can be used by
researchers to compare the efficiency of their algorithms, and the quality of the solutions
obtained in the case of heuristics. Unfortunately, in the case of the LAM model, such
benchmark data sets are currently only partially available.
The most popular publicly available data sets based on real-world data for the LAM
model seem to be the ones described by Chang et al. [14]. They include covariance
matrices and expected return vectors of sizes ranging from 31 to 225 built from weakly
price data from March 1992 to September 1997 for the Hang Seng, DAX, FTSE 100,
S&P 100, and Nikkei 225 capital market indices. The weakly price data are contained
in the files indtrack1, indtrack2,..., indtrack5 available from Beasley’s OR-Library [7] at
http://people.brunel.ac.uk/˜mastjjb/jeb/orlib/indtrackinfo.html, where one can also find
weakly price data for the S&P 500 (457 assets), Russell 2000 (1318 assets), and Russell
3000 (2151 assets) capital market indices in the files indtrack6, indtrack7, indtrack8, as
described in [13]. The return rates for these eight markets have been computed as logarithmic variations of the quotation prices (ln(Pt /Pt−1 )). The historical realizations consist
of 290 rates of return. It should be mentioned however that, for commercial reasons, the
data sets have been anonymized, in the sense that the names of the stocks associated to
the data are not disclosed. Thus we decided to construct, and to make available in the web
page http://w3.uniroma1.it/Tardella/datasets.html, five additional data sets that refer to
the EuroStoxx50 (Europe), FTSE 100 (UK), MIBTEL (Italy), S&P 500 and NASDAQ
(USA) capital market indices. These data sets contain the names of all the stocks included. For each stock we obtained 263 weakly price data, adjusted for dividends, from
Yahoo Finance for the period from March 2003 to March 2008. Stocks with more than
two consecutive missing values were disregarded. The missing values of the remaining
stocks were interpolated. We thus obtained data sets of 47 stocks for EuroStoxx50, 76
for FTSE 100, 221 for MIBTEL, of 476 for S&P 500, and 2191 for NASDAQ. We then
computed (logarithmic) weekly returns, expected returns, and covariance matrices based
16
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
OR-Library
Number of
Hang Seng
DAX 100
FTSE 100
S&P 100
Nikkei
S&P 500
Russell 2000
Russell 3000
EuroStoxx50
FTSE 100
MIBTEL
S&P 500
NASDAQ
K=5
K = 10
assets (N )
CPLEX
INCR. SET
CPLEX
INCR. SET
31
85
89
98
225
457
1318
2151
47
76
221
476
2191
8
1015
2978
197816
161
30
354
-
37
377
663
1223
375
5863
12172
48098
165
345
3603
7510
55500
6
136
986
85912
61
17
79
-
55
797
1750
4184
752
19710
14611
52643
341
717
19220
45491
63671
Table 1: Running times in seconds to solve the LAM model for 500 return values with K
assets
on the (in-sample) data for the period March 2003 - March 2007. The remaining data,
for the period April 2007 - March 2008, have been used as out-of-sample data to evaluate
the ex-post performance of the portfolios obtained with the models (see Section 6).
A drawback of Beasley’s data sets is the lack of optimal (or best known) solutions
to the LAM model based on them, although some statistics and some indicators that
measure the quality of the solutions obtained are presented in [14, 17, 18, 26, 45, 50]. We
fill this gap by providing the optimal (or best known) solutions to the LAM model both
for our data sets and for the ones contained in Beasley’s OR-Library.
5.2
Computational Results for the LAM Model
In this section we provide some computational results comparing our heuristic algorithm
with the exact MIQP solver in CPLEX 11.0. We point out that although optimality is not
guaranteed for our algorithm, we have observed that in all instances where CPLEX could
solve the problem, the solutions found by the two algorithms coincided up to numerical
precision. Hence we need not report the accuracy of the solutions found by our algorithm.
Our algorithm is coded in MATLAB 7.4 and executed on a workstation with Intel
Core2 Duo CPU (T7500, 2.2 GHz, 4Gb RAM) under Windows Vista. CPLEX 11.0 is
also called from MATLAB with the TOMLAB/CPLEX toolbox [25].
For each data set, we computed ρmin and ρmax as described in Section 2.1 by solving the
classical (unconstrained) Markowitz model. We then repeatedly solved the LAM model
(4) for 500 equally spaced returns between ρmin and ρmax thus obtaining 500 values of the
function φK (ρ). A simple post-processing of these values allowed us to compute φ�K (ρ),
and to determine the points on the Efficient Frontier of the LAM model (4), also called
LAMEF. The graphs obtained for some data sets are shown in Fig. 6.
As in [14, 17, 18, 26, 39, 45, 50], we report results for problems with cardinality
constraints K = 5 and K = 10, lower bound �i = 0.01, and upper bound ui = 1 for all
17
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
S&P 100 (98 Stocks) port4 in OR−Library
−3
10
x 10
Russell 3000 (2151 Stocks) indtrack8 in OR−Library
0.014
9
0.012
8
0.01
RETURN
RETURN
7
6
5
0.008
0.006
4
φ′κ(ρ), Κ=2
φ′κ(ρ), Κ=3
φ′κ(ρ), Κ=5
φ′κ(ρ), Κ=10
φ(ρ)
3
2
1
0.01
0.015
0.02
0.025
0.03
0.035
Std deviation
0.04
0.045
0.05
0.055
φ′κ(ρ), Κ=2
φ′κ(ρ), Κ=3
φ′κ(ρ), Κ=5
φ′κ(ρ), Κ=10
φ(ρ)
0.004
0.002
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Std deviation
Figure 6: Examples of Efficient Frontiers of the LAM model
i = 1, . . . , n. The choice of K = 10 as the largest cardinality constraint is also justified by
the observation that for several data sets the optimal portfolio in the classical Markowitz
model does not include more than 10 stocks for more than half of the ρ values (see also
Fig. 7). Furthermore, we observed that the number of stocks with positive weight in the
optimal portfolio for the classical Markowitz model might be an important indicator of
the practical computational complexity for most exact algorithms for the LAM model.
This is certainly the case both for CPLEX and for our Increasing Set algorithm, as clearly
results by comparing in Table 1 the computation time for S&P 100, (98 assets) with the
one for Nikkei (225 assets). The computation for S&P 100 takes much longer because
it has many more assets in the optimal portfolio for the classical Markowitz model, as
shown in Fig. 7. In fact, for Nikkei the number of assets in the optimal portfolio is less
than or equal to 10 for about half of the target return values, so that the cardinality
constraint is not active. The maximum number of assets in the Markowitz portfolios is
15. For the S&P100 data set the maximum number of assets in a portfolio is 34 and only
for about 33% of the target return values the Markowitz portfolio contains less than 10
assets. Indeed, when the cardinality constraint is not active, the unconstrained and the
Limited Asset Markowitz Efficient Frontier (LAMEF) coincide, and both CPLEX and the
Increasing Set Algorithm have no difficulties to solve the LAM model. Hence, the hardness
of computation of the LAMEF seems to be related not only to the number of variables
but also to the number of assets satisfying xi > 0 in the solution of the unconstrained
Markowitz model.
We should make some remarks concerning the running times presented in Table 1.
First, the Increasing Set algorithm is currently a prototype algorithm coded in MATLAB
tailored for the LAM model, while the solver in CPLEX is a highly optimized general
purpose MIQP solver. Furthermore, the times reported to solve the LAM model for a
given K with the Increasing Set algorithm should actually be read as the times required
to solve the model for all K � ≤ K, as observed in Section 4. Thus such times are clearly
increasing with K, but they refer to solving a family of problems. On the other hand, the
running times of CPLEX seem to almost always decrease with K. Hence, CPLEX might
be used as a complementary tool with respect to the Increasing Set algorithm. However,
it should be noted that CPLEX is currently unable to solve the largest problems in our
data sets.
Some authors [17, 18, 45, 50] have measured the quality of the results obtained by
their heuristic algorithms by computing an Average Percentage Loss (APL) comparing
18
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
S&P (98 stocks) Port4 in OR−Library
50
50
50
40
30
20
10
3
4
5
6
7
Return
8
10
−3
x 10
S&P (457 stocks) indtrack6 in OR−Library
30
20
10
2
3
4
5
6
Return
7
8
30
20
10
2
4
6
Return
8
10
12
100
50
0
2
4
−3
x 10
6
8
Return
10
12
14
−3
x 10
20
10
1
2
Return
3
4
−3
x 10
NASDAQ (2196 stocks)
Number of stocks with xi > 0
Number of stocks with xi > 0
40
30
150
150
50
40
0
0
9
−3
x 10
Russell 3000 (2151 stocks) indtrack8 in OR−Library
60
0
40
0
9
Number of stocks with xi > 0
60
0
Number of stocks with xi > 0
Nikkei (225 stocks) Port5 in OR−Library
60
Number of stocks with xi > 0
Number of stocks with xi > 0
Hang Seng (31 stocks) Port1 in OR−Library
60
100
50
0
0.005
0.01
0.015
Return
0.02
0.025
Figure 7: Number of assets in the unconstrained Mean-Variance optimal portfolio
the risk obtained by the algorithms for the LAM model with a given required return ρ
to the optimal risk for the same return in the classical (unconstrained) Markowitz model.
Since the definitions of APL considered by these authors are slightly different, we compare
our results separately. More precisely, with our notation, the APL considered by MoralEscudero et al. [45] is defined as
APL1 =
100
�
φK (ρj ) − φ(ρj )
j=1
φ(ρj )
,
(14)
where φK (ρj ) is the optimal value function of problem (2), the returns ρj , with j =
1, . . . , 100, are equally distributed in the interval [ρmin , ρmax ], K = 10, �i = 0.01 and
ui = 1 for all i. While, the APL considered in [17, 18] and in [50] is obtained as
100
�
φ�K (ρj ) − φ(ρj )
,
APL2 =
φ(ρ
)
j
j=1
(15)
where φ�K (ρj ) is the optimal value function of�problem (2), where the equality constraint
on the target expected return is replaced by ni=1 µi xi ≥ ρ.
Since we could compute the exact values of φK (ρj ) and φ�K (ρj ), the Average Percentage
Loss that we obtain is the best possible with respect to the return values used (we call it
Exact APL). The results that we obtain show that, in spite of the theoretical difference,
the values of AP L1 and AP L2 are the same or almost the same for all the data sets that
we consider. Table 2 shows a comparison between the results of the exact AP L1 computed
by the Increasing Set Algorithm with the APL reported in [45]. Similarly, we compare
in Table 3 the results of the exact AP L2 , with the values computed by Di Gaspero et al.
[18] and by Schaerf [50]. We remark that the values that we obtain are slightly better
than those reported by the other authors. This seems to contrast with the optimality of
the results claimed by Di Gaspero et al. [18]. However, the small differences in the results
might also be explained by the choice of the points on the frontier, or by the numerical
19
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
Data set
# of Assets Exact AP L1
Hang Seng
DAX 100
FTSE 100
S&P 100
Nikkei
31
85
89
98
225
0.00312
2.50749
1.90225
4.64937
0.19978
[45]
0.00321
2.53180
1.92150
4.69507
0.20198
Table 2: Comparison of Average Percentage Loss AP L1
Data set
Hang Seng
DAX 100
FTSE 100
S&P 100
Nikkei
# of Assets Exact AP L2
31
85
89
98
225
0.00312
2.50742
1.90203
4.64937
0.19978
[18]
[50]
0.00321 0.00409
2.53139 2.53617
1.92146 1.92597
4.69371 4.69816
0.20199 0.20258
Table 3: Comparison of Average Percentage Loss AP L2
precision of the algorithms. In order to make the comparison with our results easier,
we have made available in the web page http://w3.uniroma1.it/Tardella/APL.html the
100 return values and the covariance matrices obtained from Beasley’s OR Library that
we used in our computations of the APL, together with the optimal solutions φK (ρ) and
φ�K (ρ) that we found for problem (2) with equality or inequality constraint on the expected
return.
5.3
Computational Results for the LACVaR and LAMAD Models
In this section we report the running times required by the commercial solver CPLEX
to find exact or approximate (within a specified tolerance) solutions to the LACVaR and
LAMAD models for the instances considered in Section 5.2 for the LAM model with the
same settings:
• the lower bound �i and upper bound ui are set to 0.01 and 1, respectively;
• the maximum number K of securities in the portfolio (cardinality constraint) is
fixed at 5 and 10;
• for each data set we vary ρ in the interval [ρmin , ρmax ] defined in Section 2.1 for the
LAM model.
We have solved the LACVaR (6) and LAMAD (8) models for 500 equally spaced
returns between ρmin and ρmax , obtaining 500 points for each frontier. The graphs for
some data sets are shown in Figs. 4 (LACVaR) and 5 (LAMAD).
In Tables 4 and 5, for each data set we report the number of assets, variables, and
constraints, and the corresponding running times. We point out that the number of
20
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
Number of
LACVaR
OR-Library
assets (N)
Hang Seng
DAX 100
FTSE 100
S&P 100
Nikkei
S&P 500
Russell 2000
Russell 3000
EuroStoxx50
FTSE 100
MIBTEL
S&P 500
NASDAQ
31
85
89
98
225
457
1318
2151
47
76
221
476
2191
K=5
variables constraints
353
461
469
487
613
1205
2927
4593
305
363
653
1163
4593
324
378
382
391
454
750
1611
2444
260
289
434
689
2404
Uncon
15
22
22
24
33
97
577
648
14
17
26
42
470
OPT APPR
20
670
9097
934
77
337
13773
-
19
530
6440
394
62
238
9393
-
K = 10
OPT
APPR
19
155
3753
198
30
65
21666
-
18
45
216
7904
101
21
38
1061
-
Table 4: Running times in seconds to solve the LACVaR model for 500 return values with
K assets
Number of
LAMAD
OR-Library
assets (N)
Hang Seng
DAX 100
FTSE 100
S&P 100
Nikkei
S&P 500
Russell 2000
Russell 3000
EuroStoxx50
FTSE 100
MIBTEL
S&P 500
NASDAQ
31
85
89
98
225
457
1318
2151
47
76
221
476
2191
K=5
variables constraints
352
460
468
486
740
1204
2926
4592
304
362
652
1162
4592
614
668
672
681
744
1040
1901
2734
470
499
644
899
2614
Uncon
22
56
64
67
66
235
1148
2090
24
32
71
130
1148
OPT APPR
136
4956
1951
341
11299
-
90
4856
16255
6057
661
4823
124
1337
9809
2103
-
K = 10
OPT
APPR
95
13173
814
112
14980
-
68
488
288
723
553
1407
79
178
1536
900
-
Table 5: Running times in seconds to solve the LAMAD model for 500 return values with
K assets
variables and constraints in the LACVaR and LAMAD models depends not only on the
number of assets n but also on the length of the in-sample period T chosen (T = 290 for
the OR-Library data sets and T = 210 for our data sets). Thus, the number of variables
tends to become fairly large making the mixed integer problem difficult to solve. In Tables
4 and 5 we present the running times of CPLEX for finding both optimal (denoted by
OPT) and approximate (denoted by APPR) solutions. The optimal values are obtained
using default tolerances (10−6 ), while the approximate values are computed relaxing to
10−4 the absolute tolerance on the gap found by CPLEX. We also report the running
times for finding the optimal solutions to the unconstrained models (Uncon).
In Table 6 we summarize the running times of the Increasing Set (IS) algorithm for the
LAM model and of CPLEX, both with small tolerance (OPT) and with larger tolerance
(APPR), for all models. The values are missing when the code has not been able to solve
the problem within 2 days. From the table it appears that, apart from the simplest problems, the running time of the Increasing Set algorithm is comparable to that of CPLEX
with large tolerance (APPR). However, CPLEX is not able to solve, even approximatively,
the largest problems. Furthermore, it should be recalled that the Increasing Set algorithm
N of
OR-Library
assets
Hang Seng
DAX 100
FTSE 100
S&P 100
Nikkei
S&P 500
Russell 2000
Russell 3000
EuroStoxx50
FTSE 100
MIBTEL
S&P 500
NASDAQ
31
85
89
98
225
457
1318
2151
47
76
221
476
2191
K=5
CPLEX
LAMAD
K = 10
K=5
IS
CPLEX
IS
OPT
8
37
1015
377
2978
663
197816 1223
161
375
- 5863
- 12172
- 48098
30
165
354
345
- 3603
- 7510
- 55500
6
136
986
85912
61
17
79
-
55
797
1750
4184
752
19710
14611
52643
341
717
19220
45491
63671
136
4956
1951
341
11299
-
APPR
LACVaR
K = 10
K=5
OPT
APPR
OPT
90
95
4856 13173
16255
6057
661
814
4823
124
112
1337 14980
9809
2103
-
68
488
288
723
553
1407
79
178
1536
900
-
20
670
9097
934
77
337
13773
-
APPR
K = 10
OPT
APPR
19
19
530
155
6440 3753
394
198
62
30
238
65
9393 21666
-
18
45
216
7904
101
21
38
1061
-
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
LAM
Table 6: Running times in seconds to solve the Limited Asset models for 500 return values with K assets
21
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
22
actually finds a solution for all K � ≤ K, while CPLEX solves the problem for a single
K. Thus from the computational viewpoint it turns out that quite surprisingly in most
cases the quadratic LAM model can be solved more efficiently with our algorithm than
the linear LAMAD and LACVaR models with current state-of-the-art solvers. Therefore,
unless more efficient ad-hoc algorithms are developed for the mixed integer linear models,
the LAM model should be preferred for large size problems.
6
Evaluation of Ex-post Performance
Out-of-sample experiments allow evaluation of the effectiveness of portfolio models for
actual risk management purposes. The computed portfolios are built by solving the
unconstrained models and the Limited Asset ones (K = 5, 10; �i = 0.01, ui = 1) on a
given sample interval [March 2003, March 2007] for a fixed value of the target return ρ.
After that, we simulate the holding of such portfolios for the time interval [April 2007,
March 2008], and we evaluate their ex-post performance using out-of-sample data for
the EuroStoxx 50, FTSE 100, MIBTEL, S&P500 and NASDAQ capital markets. Such
performances are compared to that of the official capital market index in the same period.
Some results are illustrated in Figs. 8 and 9. In Fig. 8 we show the results obtained for
low risk strategies (ρ = ρmin ). These results seem to indicate that, although the portfolios
selected by the Limited Asset models contain at most 5 or 10 securities, they have a better
performance than the ones provided by the unconstrained models.
We also notice that the performance of the LAM and LAMAD portfolios seems to be
at least as good as that of the LACVaR portfolio. Moreover, all the portfolios found by
the models generally present a better performance than the market index in longer-term
investment horizons.
In Fig. 9 we present similar experiments for higher risk strategies (ρ = ρmin +
1/2(ρmax − ρmin )). For this target return level, it seems that the assets limitation for
the linear models is less important for the portfolio composition. Indeed, the performance
of the unconstrained MAD and CVaR portfolios are similar to the constrained ones, while
the optimal portfolios for the LAM model with high values of ρ still provide portfolios with
different performance. This is because the Markowitz model leads to a more diversified
portfolio than the MAD and CVaR models [36].
7
Conclusions and Further Research
In this paper we have presented an efficient algorithm for a Mean-Variance portfolio
selection model with constraints, coming from real-world practice, that are difficult to
handle computationally. With this algorithm we can solve to optimality not only some
well-known benchmark problems, but also larger problems with more than 2000 variables.
Our algorithm is based on a completely new approach that starts from a pair of assets
and tries to add one asset at a time in an optimal manner by exploiting some recent
theoretical results on Quadratic Programming.
We have also analyzed the CVaR and MAD models with cardinality constraints and
buy-in thresholds. These are mixed integer linear programming (MILP) models, that have
been solved using CPLEX, a state-of-the-art commercial solver. Although one expects a
MILP model to be more tractable than a MIQP one, the computational results have shown
that CPLEX requires more time to solve the Limited Asset CVaR and MAD models than
the time needed by our algorithm to solve the Limited Asset Markowitz one, particularly
23
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
MV: FTSE (76 stocks) ρ=ρmin
MV: SP (476 stocks) ρ=ρmin
115
115
FTSE_100 Index
Portfolio K=5
Portfolio K=10
Portfolio Markowitz
110
110
105
Portfolio Value
Portfolio Value
105
100
95
100
95
90
90
85
85
80
SP_500 Index
Portfolio K=5
Portfolio K=10
Portfolio Markowitz
18Apr07
07Jun07
27Jul07
15Sep07
04Nov07
24Dec07
80
12Feb08
18Apr07
MAD: FTSE (76 stocks) ρ=ρ
24Dec07
12Feb08
SP_500 Index
Portfolio K=5
Portfolio K=10
Portfolio MAD
110
105
Portfolio Value
Portfolio Value
04Nov07
min
105
100
95
100
95
90
90
85
85
18Apr07
07Jun07
27Jul07
15Sep07
04Nov07
24Dec07
80
12Feb08
18Apr07
CVaR: FTSE (76 stocks) ρ=ρmin
07Jun07
27Jul07
15Sep07
04Nov07
24Dec07
12Feb08
CVaR: SP (476 stocks) ρ=ρmin
115
115
FTSE_100 Index
Portfolio K=5
Portfolio K=10
Portfolio CVaR
110
SP_500 Index
Portfolio K=5
Portfolio K=10
Portfolio CVaR
110
105
Portfolio Value
105
Portfolio Value
15Sep07
115
FTSE_100 Index
Portfolio K=5
Portfolio K=10
Portfolio MAD
110
100
95
100
95
90
90
85
85
80
27Jul07
MAD: SP (476 stocks) ρ=ρ
min
115
80
07Jun07
18Apr07
07Jun07
27Jul07
15Sep07
04Nov07
24Dec07
12Feb08
80
18Apr07
07Jun07
27Jul07
15Sep07
04Nov07
24Dec07
12Feb08
Figure 8: Evaluation of ex-post performances with ρ = ρmin for the Markowitz (top),
MAD (middle) and CVaR (bottom) models on the FTSE100 (left) and SP500 (right)
data sets
24
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
MV: FTSE (76 stocks) ρ=ρ
+1/2(ρ
min
−ρ
max
)
MV: SP (476 stocks) ρ=ρ
min
FTSE_100 Index
Portfolio K=5
Portfolio K=10
Portfolio Markowitz
125
125
Portfolio Value
Portfolio Value
120
115
110
105
100
110
105
100
95
90
90
85
85
18Apr07
07Jun07
27Jul07
15Sep07
04Nov07
MAD: FTSE (76 stocks) ρ=ρ
24Dec07
+1/2(ρ
min
80
12Feb08
−ρ
max
)
07Jun07
27Jul07
15Sep07
04Nov07
24Dec07
+1/2(ρ
min
12Feb08
−ρ
max
)
min
135
FTSE_100 Index
Portfolio K=5
Portfolio K=10
Portfolio MAD
130
125
SP_500 Index
Portfolio K=5
Portfolio K=10
Portfolio MAD
130
125
120
Portfolio Value
120
115
110
105
100
115
110
105
100
95
95
90
90
85
85
18Apr07
07Jun07
27Jul07
15Sep07
04Nov07
24Dec07
80
12Feb08
CVaR: FTSE (76 stocks) ρ=ρmin+1/2(ρmax−ρmin)
18Apr07
07Jun07
27Jul07
15Sep07
04Nov07
24Dec07
12Feb08
CVaR: SP (476 stocks) ρ=ρmin+1/2(ρmax−ρmin)
135
135
FTSE_100 Index
Portfolio K=5
Portfolio K=10
Portfolio CVaR
130
125
SP_500 Index
Portfolio K=5
Portfolio K=10
Portfolio CVaR
130
125
120
Portfolio Value
120
Portfolio Value
18Apr07
MAD: SP (476 stocks) ρ=ρ
min
135
Portfolio Value
115
95
115
110
105
100
115
110
105
100
95
95
90
90
85
85
80
)
min
SP_500 Index
Portfolio K=5
Portfolio K=10
Portfolio Markowitz
130
120
80
−ρ
max
135
130
80
+1/2(ρ
min
135
18Apr07
07Jun07
27Jul07
15Sep07
04Nov07
24Dec07
12Feb08
80
18Apr07
07Jun07
27Jul07
15Sep07
04Nov07
24Dec07
12Feb08
Figure 9: Evaluation of ex-post performances with ρ = ρmin + 1/2(ρmax − ρmin ) for the
Markowitz (top), MAD (middle) and CVaR (bottom) models on the FTSE100 (left) and
SP500 (right) data sets
Cesarone Scozzari Tardella - Portfolio Selection Problems in Practice
25
for the more difficult problems.
Finally, a comparison of ex-post performances of the Limited Asset models, of the
unconstrained models and of the market indices seems to indicate that a strong limitation
on the number of assets to hold in the optimal portfolio could generally provide more
robust and convenient portfolios.
We plan to improve the computational efficiency of the Increasing Set Algorithm also
by exploiting the possibility of parallel computing. Furthermore, we intend to consider
quadratic models with additional complex constraints and possibly with different objectives (e.g., index tracking).
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