Uncertain mean-risk index portfolio selection considering inflation: Chaos adaptive genetic algorithm

Abstract

This paper discusses a mean-risk index model and the solution algorithm for portfolio selection considering inflation under the uncertain environment. Firstly, we propose an uncertain mean-risk index model considering inflation which is one of the most general multiplicative background risks. To get the optimal solution of the proposed model, we provide a chaos adaptive genetic algorithm (CAGA), which is an improvement of the adaptive genetic algorithm (AGA). Through numerical experiments, the performances of the proposed algorithm are tested. Comparison with other genetic algorithms shows the better performance of the proposed algorithm. Finally, a numerical example is given to demonstrate the application of the proposed model.

Appendix

Definition 2

(Liu [2]) Let $\text{ L }$ be a $\sigma$-algebra over a nonempty set $\Gamma$. Each element $\Lambda \in \text{ L }$ is called an event. A set function $\text{ M } \{\Lambda \}: \text{ L }\rightarrow [0,1]$ is called an uncertain measure if it satisfies the following axioms:

• (i) (Normality) $\text{ M } \{\Gamma \}=1$.

• (ii) (Duality) $\text{ M } \{\Lambda \}+\text{ M } \{\Lambda ^c\}=1.$

• (iii) (Subadditivity) For every countable sequence of events$\{\Lambda _i\}$,

  $$\begin{aligned} \text{ M }\displaystyle \left\{ \bigcup \limits _{i=1}^{\infty }\Lambda _i\right\} \le \sum \limits _{i=1}^{\infty } \text{ M }\{\Lambda _i\}. \end{aligned}$$

Definition 3

(Liu [2]) An uncertain variable $\xi$ is a function from an uncertainty space $(\Gamma , \text{ L }, \text{ M})$ to the set of real numbers such that for any Borel set of B of real numbers, the set

$$\begin{aligned} \{\xi \in B\}=\{\gamma \in \Gamma |\xi (\gamma )\in B\} \end{aligned}$$

is an event.

Definition 4

(Liu [18]) An uncertain variable $\xi$ is called a linear uncertain variable if it has a linear uncertainty distribution function

$$\begin{aligned} \Phi \left( t \right) = \left\{ {\begin{array}{ll} 0, &{} t < e - {\sqrt{3} }\sigma \\ \frac{t-e+{\sqrt{3} }\sigma }{2{\sqrt{3} }\sigma }, &{} e -{\sqrt{3} } \sigma \leqslant t \leqslant e + {\sqrt{3} }\sigma \\ 1, &{} t > e + {\sqrt{3} }\sigma \\ \end{array} } \right. \end{aligned}$$

we denote it by $\xi \sim L(e -{\sqrt{3} }\sigma ,e+{\sqrt{3} }\sigma ),$ where e and $\sigma$ are real numbers. The inverse uncertain function of the linear uncertainty distribution function $\Phi (t)$ is as follows:

$$\begin{aligned} \Phi ^{-1}(\alpha )=e+{\sqrt{3} }( {2\alpha - 1})\sigma ,~0< \alpha <1. \end{aligned}$$

Definition 5

(Liu [18]) An uncertain variable $\xi$ is called a normal uncertain variable if it has a normal uncertainty distribution function

$$\begin{aligned} \Phi \left( t \right) = \left[ {1 + \exp \left( {\frac{\pi \left( {e - t} \right) }{\sqrt{3} \sigma }} \right) } \right] ^{ - 1},~t \in R~,~\sigma > 0. \end{aligned}$$

we denote it by $\xi \sim N(e,\sigma )$, and an inverse function of the normal uncertainty distribution function $\Phi (t)$ is as follows:

$$\begin{aligned} \Phi ^{-1}\left( \alpha \right) = e + \frac{\sqrt{3} }{\pi }\sigma \ln \left( {\frac{\alpha }{1 - \alpha }} \right) ~,~0< \alpha < 1. \end{aligned}$$

Theorem 2

(Liu [18]) Let $\xi$ be an uncertain variable with a regular uncertainty distribution $\Phi$. If its expected value exists, then

$$\begin{aligned} E\left[ \xi \right] = \int _0^1 {\Phi ^{ - 1}\left( \alpha \right) d\alpha } \end{aligned}$$

When the uncertain variables are represented by uncertainty distributions, the operational law is given by [18] as follows:

Theorem 3

(Liu [18]) Let $\xi _1, \xi _2, \cdots ,\xi _n, \xi _{n+1}, \cdots , \xi _{n+m}$ be independent uncertain variables with uncertainty distributions $\Phi _1, \Phi _2,\cdots , \Phi _n, \Phi _{n+1}, \cdots , \Phi _{n+m},$ respectively. Let $f(t_1,t_2,\cdots ,t_n)$ be strictly increasing with respect to $t_1,t_2,\cdots ,t_n,$ and $g(t_1$, $t_2$, $\cdots$, $t_n$, $t_{n+1}$, $\cdots ,t_{n+m})$ be strictly increasing with respect to $t_1,t_2,\cdots ,t_n$ and strictly decreasing with respect to $t_{n+1}$, $t_{n+2}$, $\cdots$, $t_{n+m}.$ Then

$$\begin{aligned} \xi =f(\xi _1,\xi _2,\cdots ,\xi _n) \end{aligned}$$

is an uncertain variable with inverse uncertainty distribution function

$$\begin{aligned} \Psi ^{-1}(\alpha )=f(\Phi ^{-1}_1(\alpha ), \Phi ^{-1}_2(\alpha ), \cdots , \Phi ^{-1}_n(\alpha )), \quad 0< \alpha <1, \end{aligned}$$

and

$$\begin{aligned} \eta =g(\xi _1,\xi _2,\cdots ,\xi _n,\xi _{n+1},\cdots ,\xi _{n+m}) \end{aligned}$$

is an uncertain variable with inverse uncertainty distribution function

$$\begin{aligned} \Psi ^{-1}(\alpha )=g(\Phi ^{-1}_1(\alpha ), \cdots , \Phi ^{-1}_n(\alpha ),\Phi ^{-1}_{n+1}(1-\alpha ),\cdots , \Phi ^{-1}_{n+m}(1-\alpha )), \quad 0< \alpha <1, \end{aligned}$$

if $\Phi _1^{-1}(\alpha ), \Phi _2^{-1}(\alpha ), \cdots , \Phi _n^{-1}(\alpha ),\Phi _{n+1}^{-1}(\alpha ),\cdots ,\Phi _{n+m}^{-1}(\alpha )$ are unique for each $\alpha \in (0,1).$

Theorem 4

(Huang [10]) Let $\xi$ be an uncertain portfolio return with regular uncertainty distribution $\Phi$, and $r_0$ is the base reference return rate, and $\Phi ^{ - 1}$ denote an inverse function of the $\Phi$. Then the risk index of the portfolio can be calculated via:

$$\begin{aligned} RI\left( \xi \right) = \int _0^\beta {\left( {r_0 - \Phi ^{ \text{- }1}\left( \alpha \right) } \right) } d\alpha , \end{aligned}$$

where $\beta$ is defined by $\Phi ^{ - 1}\left( \beta \right) = r_0.$