Abstract
Cumulative prospect theory (CPT) has become one of the most popular approaches for evaluating the behavior of decision makers under conditions of uncertainty. Substantial experimental evidence suggests that human behavior may significantly deviate from the traditional expected utility maximization framework when faced with uncertainty. The problem of portfolio selection should be revised when the investor’s preference is for CPT instead of expected utility theory. However, because of the complexity of the CPT function, little research has investigated the portfolio choice problem based on CPT. In this paper, we present an operational model for portfolio selection under CPT, and propose a real-coded genetic algorithm (RCGA) to solve the problem of portfolio choice. To overcome the limitations of RCGA and improve its performance, we introduce an adaptive method and propose a new selection operator. Computational results show that the proposed method is a rapid, effective, and stable genetic algorithm.
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Notes
We use the definition of He and Zhou (2011).
The objective function (12), which involves numerical integration, is solved using Matlab’s “integral” and “normcdf”functions.
The data are derived by Hu and Kercheval (2010) and the four stocks are Disney, Pfizer, Altria, Intel.
All results in these tables are global optimal solutions given by the exhaustive method to three decimal places.
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Appendix
Appendix
Proof of Lemma 1: According to the definitions of the value functions and weighing functions, they all are differentiable. We can obtain
using integration by parts.
He and Zhou (2011) showed that, if the return on a portfolio follows a normal distribution and |t| and \(0<F_D(t)<1\) are sufficiently large, then (5) and (6) satisfy:
where \(\varepsilon >0\).
Furthermore, from L’Hopital’s rule, we know that
We can obtain that
Similarly, we have
Proof of Proposition 1: Hogg and Craig (1995) have shown that a linear transformation of multivariate normal random vectors has a multivariate normal distribution, namely, suppose that \({\varvec{R}}\) has the distribution \(N_n({\varvec{\mu }},{\varSigma })\) and \(D={\varvec{AR}}+b\), where \({\varvec{A}}\) is an \(m\times n\) matrix and \(b\in {\varvec{R}}^m \). Then, D has the distribution \(N_m({\varvec{A}}{\varvec{\mu }},{\varvec{A}}{\varSigma }{\varvec{A}}')\). Proposition 1 can be proved when \(m=1\).
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Gong, C., Xu, C. & Wang, J. An Efficient Adaptive Real Coded Genetic Algorithm to Solve the Portfolio Choice Problem Under Cumulative Prospect Theory. Comput Econ 52, 227–252 (2018). https://doi.org/10.1007/s10614-017-9669-5
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DOI: https://doi.org/10.1007/s10614-017-9669-5