An Efficient Adaptive Real Coded Genetic Algorithm to Solve the Portfolio Choice Problem Under Cumulative Prospect Theory

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.

Appendix

Proof of Lemma 1: According to the definitions of the value functions and weighing functions, they all are differentiable. We can obtain

$$\begin{aligned}&\int \limits _{0}^{+\infty }v^+(t)dw^+(1-F_D(t)) \nonumber \\&= v^+(t)w^+(1-F_D(t))|_0^{+\infty } \nonumber \\&- \int \limits _{0}^{+\infty }w^+(1-F_D(t))dv^+(t) \end{aligned}$$

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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:

$$\begin{aligned} w^{+'}(F_D(t))f_D(t)=O\left( |t|^{-2-\varepsilon }\right) \nonumber \\ w^{-'}(F_D(t))f_D(t)=O\left( |t|^{-2-\varepsilon }\right) \end{aligned}$$

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$$\begin{aligned} w^{+'}(1-F_D(t))f_D(t)=O\left( |t|^{-2-\varepsilon }\right) \nonumber \\ w^{-'}(1-F_D(t))f_D(t)=O\left( |t|^{-2-\varepsilon }\right) \end{aligned}$$

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where $\varepsilon >0$.

Furthermore, from L’Hopital’s rule, we know that

$$\begin{aligned} \lim _{t\rightarrow \infty }v^+(t)w^+(1-F_D(t)) \nonumber \\ =\lim _{t\rightarrow \infty }\frac{w^+(1-F_D(t))}{\frac{1}{v^+(t)}}=0 \end{aligned}$$

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We can obtain that

$$\begin{aligned} \int \limits _{0}^{+\infty }v^+(t)dw^+(1-F_D(t)) \nonumber \\ =- \int \limits _{0}^{+\infty }w^+(1-F_D(t))dv^+(t) \end{aligned}$$

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Similarly, we have

$$\begin{aligned} \int \limits _{-\infty }^{0}v^-(t)dw^+(F_D(t)) \nonumber \\ =- \int \limits _{-\infty }^{0}w^-(F_D(t))dv^-(t) \end{aligned}$$

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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 $m=1$.