# (1812-4906) A new quadratic deviation of fuzzy random variable and its application to portfolio optimization

Document Type: Research Paper

Authors

1 School of Economics and Management, Tianjin Polytechnic University, Tianjin 300387, China

2 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA

3 College of Mathematics & Information Science, Hebei University, Baoding 071002, Hebei, China

Abstract

The aim of this paper is to propose a convex risk measure in the framework of fuzzy random theory and verify its
advantage over the conventional variance approach. For this purpose, this paper defi nes the quadratic deviation (QD)
of fuzzy random variable as the mathematical expectation of QDs of fuzzy variables. As a result, the new risk criterion
essentially describes the variation of a fuzzy random variable around its expected value. For triangular and trapezoidal
fuzzy random variables as well as their linear combinations, we establish the analytical expressions of their QDs,
and obtain the desirable convexity about the analytical expressions with respect to critical parameters. To explore
the practical value of the proposed QD, we apply it to a portfolio selection problem to quantify the investment risk,
and develop three mean-QD models to fi nd the optimal allocation of the fund in different risky securities. Due to the
convexity of our QD, the original three mean-QD models can be turned into their equivalent convex parametric quadratic
programming problems, which can be solved by conventional optimization methods. The computational results clearly
demonstrate that our new QD signi cantly reduces the computational complexity that cannot be avoided when variance
is used as a risk criterion. Finally, the numerical comparison between the proposed mean-QD model and mean-variance
model is conducted to show the consistency between the optimal results in both techniques. Meanwhile, the comparison
between the proposed QD, variance, spread, and second moment is made to summarize the similarities and differences
between them, distinguish these four risk criteria and determine their respective application scopes in decision systems.

Keywords