Lecture / Ponencia xx
Solving the Portfolio Optimization Problem under realistic
constraints
R. Kizys1, A.A. Juan2, B. Sawik3 and A. Linares2
1
University of Portsmouth, Department of Economics and Finance, UK,
renatas.kizys@port.ac.uk
2 IN3 – Open University of Catalonia, Department of Computer Science, Spain,
{ajuanp, alinaresz}@uoc.edu
3 AGH University of Science & Technology, Faculty of Management, Department of
Applied Computer Science, Poland, b_sawik@cal.berkeley.edu
Abstract
A hybrid algorithm for realistic portfolio optimization is introduced in this
paper. Our matheuristic algorithm combines quadratic programming with a
metaheuristic framework in order to solve complex variants of the meanvariance portfolio optimization problem. Thus, it is able to quickly generate
near-optimal asset-allocation solutions –in terms of minimum risk for a given
expected return rate– under realistic conditions. These realistic conditions
include the well-known cardinality, pre-assignment, and quantity
constraints. The algorithm proceeds in two steps. First, a feasible initial
solution is constructed by allocating portfolio weights according to the
individual return rate. Secondly, an iterated local search process gradually
improves the initial solution throughout an iterative combination of a
perturbation stage and a local search stage, which includes quadratic
programming optimization. According to the experimental results obtained,
our approach is competitive when compared against existing state-of-theart approaches, both in terms of the quality of the best solution generated
as well as in terms of the computational times required to obtain it.
1.
Introduction
Mean-variance optimization has become the workhorse model for portfolio
selection since Markowitz (1952). An important assumption underlying this
model is that investors are concerned about both the expected returns from
their investment and the risk from that investment, where risk is defined as
the variance of future returns. By optimally allocating appropriate weights
to imperfectly correlated risky assets, investors can reduce the variance of
future portfolio returns and thus diversify their investment. An optimal
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allocation of assets produces the so-called unconstrained efficient frontier
(UEF), which consists of a 2D plot representation of portfolio configurations
with a minimum degree of risk (horizontal axis) for each level of expected
return (vertical axis) (Figure 1).
Figure 1: 2D plot representing the UEF.
Since the mean-variance portfolio formulation, an enormous amount of
papers have been published extending or modifying the basic model in three
directions (Sawik 2013). The first path goes to simplification of the amount
or the type of input data. The second path focuses on the introduction of an
alternative measure of risk. The third path involves incorporation of the
additional criteria and constraints. Traditional optimization methods include,
among others, linear programming (Mansini et al. 2014) and also quadratic
programming (Sawik 2012). Unfortunately, the complexity of the meanvariance model for portfolio optimization increases dramatically upon
adding additional constraints that are needed to provide a realistic
representation of the investor’s portfolio choice, which reduces the
efficiency of exact methods in other than small-size instances.
Metaheuristic approaches have been proposed in the literature as a way to
surpass these limitations (Maringer 2005). This paper focuses on the
single-period version of the so-called constrained mean-variance portfolio
optimization problem. We will assume that all assets are risky assets
(stocks). However, the solving approach introduced here could be also
used for similar optimization problems regarding commodities, futures,
options and swaps. The objective function is the portfolio variance, which
is considered to be the measure of risk on investment in the portfolio. It is
obtained as a weighted average of return co-variances between each pair
of assets included in the portfolio. The decision problem involves
minimizing the portfolio variance while ensuring a return level not inferior to
the one required by the investor. Portfolio weights add up to one and are
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constrained to take on non-negative values only. Real-world portfolio
selection problems may involve additional constraints. The adoption of an
initial solution that is based on a well-chosen criterion makes our solver
more flexible when solving tight instances. Also, the rate of return on an
individual asset is used as a main criterion to construct the initial solution.
It is worth noting that this criterion provides the best possible solution in
terms of guarantying feasibility of the requested portfolio return –i.e., if this
initial solution is not feasible, then the problem has no feasible solution. By
contrast, in most of the existing approaches, an initial solution is randomly
drawn and, hence, the feasibility of these initial solutions cannot be
guaranteed. Finally, our solver features several efficient sub-procedures
that seek to avoid getting trapped into a local optimum.
For a
comprehensive overview of the formulations of the portfolio selection
problem, see Di Tollo and Rolli (2008).
2.
Formal description of the problem
A set of n assets is given by the market, A  a1 , a2 ,..., an  , where:


i 1, 2,..., n , ai has a known expected return, ri  0 .
i, j 1, 2,..., n , the pair  ai , a j  has a known expected risk index,
 ij   ji  R .


Expected return constraint: a user-provided value, R  0 , represents
the minimum expected return from the investment.
A portfolio is a vector X   x1 , x2 ,..., xn  such that each xi represents
the fraction of the total budget invested in asset ai , i.e., 0  xi  1 and
n
x
i 1
i
1.

i 1, 2,..., n , zi  1
otherwise.

Cardinality constraint: the number of assets in the portfolio,
if
xi  0 (i.e., ai in portfolio) and zi  0
n
z
i 1
i
, is
bounded by user-defined values, kmin and kmax .

Quantity constraints: for each asset ai , its associated quantity in the

portfolio, xi , is bounded by user-defined values,  i and  i .
Pre-assignment constraints: the user can pre-select certain assets to
be included in the portfolio. i 1, 2,..., n , pi  1 if ai is preassigned in portfolio (i.e., xi  0 ) and pi  0 otherwise
Then, the problem consist in selecting the optimal combination of fractions
of each asset, so that the overall variance (risk) is minimized while satisfying
all the aforementioned constraints. More formally:
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n
n
min f ( X )   ij xi x j
i 1 j 1
n
subject to:
rx
i 1
i i
n
x
i 1
i
R
(1)
1
(2)
n
kmin   zi  kmax
(3)
0  xi  1 , i 1, 2,..., n
(4)
 i zi  xi  i zi , i 1, 2,..., n
(5)
xi  zi  pi , i 1, 2,..., n
(6)
zi 0,1 , i 1, 2,..., n
(7)
i 1
3.
Solving Approach
The proposed matheuristic makes use of an Iterated Local Search (ILS)
framework (Lourenço et al. 2003). ILS is a conceptually simple yet powerful
metaheuristic that has proven to be very efficient in solving complex
combinatorial optimization problems. The underlying idea behind ILS is to
narrow the search for candidate local optimal solutions returned by some
embedded algorithm, typically a local search heuristic. At some stages of
the ILS, a quadratic programming optimizer is called as a local search in
order to improve the investment levels assigned to each of the assets in the
current portfolio. Figure 2 provides a high-level description of our approach
(the low-level details are not included here due to space limitations).
01 Procedure Matheuristic(inputs)
02 baseSol <- generateInitialSolution % generate a feasible solution
03 bestSol <- baseSol
04 while {elapsedTime <= maxTime} do
% modify portfolio configuration without losing feasibility
05
newSol <- perturbate(baseSol)
% improve assets levels via quadratic programming
06
newSol <- localSearch(newSol)
07
if {risk(newSol) < risk(baseSol)} then
08
baseSol <- newSol
09
if {risk(newSol) < risk(bestSol)} then
10
bestSol <- newSol
11
end if
12
end if
13 % (optional) include an acceptance criterion
14 end while
15 return bestSol
16 end procedure
Figure 2: Overview of our matheuristic approach.
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4.
Computational experiments
The algorithm has been implemented as a Java application. A standard
personal computer, Intel Core i5 CPU at 3.2 GHz and 4 GB RAM with Linux
Ubuntu, was used to perform all tests. In this research, we experiment with
sets of stock market data already used in previous studies by MoralEscudero et al. (2006) and Di Gaspero et al. (2011). The data set comprises
constituents of five stock market indices, Hang Seng (Hong Kong), DAX 100
(Germany), FTSE 100 (United Kingdom), S&P 100 (United States) and
NIKKEI 225 (Japan). Figure 3 shows a comparative among different
approaches, where gap refers to the gap with respect to the unconstrained
problem –lowerbound for the optimal solution of the constrained problem.
Notice that our algorithm is able to outperform the other approaches in 4 out
of the 5 sets.
Figure 3: Visual comparison among different approaches.
5.
Conclusion
We have proposed hybrid algorithm, combining a metaheuristic framework
with quadratic programming, to efficiently deal with the constrained meanvariance optimization problem. The specific constraints we use in our tests
include the cardinality and quantity constraints. The ensuing complexity of
the problem rules out closed-form solutions and conventional optimization
methods. Specifically, the cardinality and quantity constraints render the
problem computationally expensive to solve real-world instances.
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Therefore, the use of metaheuristics –that handle such problems more
flexibly and efficiently– is needed. According to the computational
experiments, our solver outperforms –in terms of average percentage loss–
some of the most recent approaches in the literature.
Acknowledgements
This work has been partially supported by the Spanish Ministry of Economy
and Competitiveness (grant TRA2013-48180-C3-3-P) and FEDER.
Likewise we want to acknowledge the support received by the Department
of Universities, Research & Information Society of the Catalan Government
(Grant 2014-CTP-00001) and by the NCN grant (6459/B/T02/2011/ 40).
References
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