Soft Computing
https://doi.org/10.1007/s00500-018-3281-z
METHODOLOGIES AND APPLICATION
Multi-period mean–semivariance portfolio optimization based on
uncertain measure
Wei Chen
1
· Dandan Li
1
· Shan Lu
1
· Weiyi Liu
2
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract
In this paper, we discuss a multi-period portfolio selection problem when security returns are given by experts’ estimations.
By considering the security returns as uncertain variables, we propose a multi-period mean–semivariance portfolio opti-
mization model with real-world constraints, in which transaction costs, cardinality and bounding constraints are considered.
Furthermore, we provide an equivalent deterministic form of mean–semivariance model under the assumption that the secu-
rity returns are zigzag uncertain variables. After that, a modified imperialist competitive algorithm is developed to solve the
corresponding optimization problem. Finally, a numerical example is given to illustrate the effectiveness of the proposed
model and the corresponding algorithm.
Keywords Multi-period portfolio optimization · Uncertain variable · Semivariance · Cardinality constraint · Imperialist
competitive algorithm
1 Introduction
Portfolio selection discusses the problem of how to allocate
a certain amount of investor’s wealth among different assets
and from a satisfying portfolio. The mean–variance (M–V)
model formulated by Markowitz (1952) lays the basis for
single-period portfolio selection. It combines probability the-
ory with optimization techniques to model the investment
behavior with some uncertainties. Quantifying investment
return as the mean of returns, and investment risk as the
variance from the mean, Markowitz formulated his mod-
els mathematically in two ways: minimizing variance for
Communicated by V. Loia.
B
Wei Chen
chenwei@cueb.edu.cn
Dandan Li
ldd@cueb.edu.cn
Shan Lu
lushan@cueb.edu.cn
Weiyi Liu
liuweiyi@cueb.edu.cn
1
School of Information, Capital University of Economics and
Business, Beijing, China
2
School of Finance, Capital University of Economics and
Business, Beijing, China
a given expected value, or maximizing expected value for
a given variance. Since then, variance is widely used as a
risk measure, and the mean–variance models are well investi-
gated such as Yoshimoto (1996), Best and Hlouskova (2000),
Liu et al. (2003), Corazza and Favaretto (2007), etc. How-
ever, as pointed out by Grootveld and Hallerbach (1999), the
distinguished drawback is that variance treats high returns
as equally undesirable as low returns because high returns
will also contribute to the extreme of variance. In particular,
when probability distributions of security returns are asym-
metric, variance becomes a deficient measure of investment
risk because it may have a potential danger to sacrifice too
much expected return in eliminating both low and high return
extremes. To overcome this disadvantage, semivariance was
proposed to replace variance in portfolio selection. Semivari-
ance only measures the variability of returns below the mean
and gauges no variability of returns above the mean and thus
better matches investors’ intuition of risk than the variance.
Afterward, several researchers have employed the semivari-
ance as risk measure to study portfolio selection problems
in random environment, such as Markowitz (1959, 1993),
Hogan and Warren (1974), Choobineh and Branting (1986),
Ballestero (2005).
Note that all the models mentioned above are single-period
portfolio selection models, which provide an one-off deci-
sion at the beginning of the investment period and suggest to
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