Uncertain portfolio adjusting model using semiabsolute deviation

Abstract

Since the financial markets are complex, sometimes the future security returns are represented mainly based on experts’ judgments. This paper discusses a portfolio adjusting problem with risky assets in which security returns are given subject to experts’ estimations. Here, we propose uncertain mean-semiabsolute deviation adjusting models for portfolio optimization problem in the trade-off between risk and return on investment. Various uncertainty distributions of the security returns based on experts’ evaluations are used to convert the proposed models into equivalent deterministic forms. Finally, numerical examples with synthetic uncertain returns are illustrated to demonstrate the effectiveness of the proposed models and the influence of transaction cost in portfolio selection.

Appendix

Proof of Theorem 1

Note that $\sum _{i=1}^n\xi _ix_i$ is also an uncertain variable. It immediately follows from Lemmas 1 and 3 of Sect. 2 that the theorem holds.

Proof of Theorem 2

Assume that there exists $k\in \{1,2,\ldots ,n\}$ such that $\hat{x}_k^+>0$ and $\hat{x}_k^->0$. Without loss of generality, it is assumed that $\hat{x}_k^+ > \hat{x}_k^-$. The optimal holding quantity of security $i$ after adjusting is $\hat{x}_k=x_k^0+\hat{x}_k^+-\hat{x}_k^-$. We set $\tilde{x}_k^+=\hat{x}_k^+-\hat{x}_k^-$ and $\tilde{x}_k^-=0$. It is evident that $\tilde{x}_k^+\cdot \tilde{x}_k^-=0$, $\tilde{x}_k^+,\tilde{x}_k^-\ge 0$ and $\tilde{x}_k=x_k^0+\tilde{x}_k^+-\tilde{x}_k^-=\hat{x}_k$ which implies that $(\hat{x}_1^+,\ldots ,\hat{x}_{k-1}^+,\tilde{x}_k^+,\hat{x}_{k+1}^+,\ldots ,\hat{x}_n^+, \hat{x}_1^-,\ldots ,\hat{x}_{k-1}^-,\tilde{x}_k^-,\hat{x}_{k+1}^-,\ldots ,\hat{x}_n^-)$ is a feasible solution of Model (8). Note that

$$\begin{aligned}&r(\hat{x}_1,\ldots ,\hat{x}_{k-1},\tilde{x}_k,\hat{x}_{k+1},\ldots ,\hat{x}_n)\\&\qquad -r(\hat{x}_1,\ldots ,\hat{x}_{k-1},\hat{x}_k,\hat{x}_{k+1},\ldots ,\hat{x}_n)\\&\quad =(b_k+s_k)\hat{x}_k^->0 \end{aligned}$$

which means that $E[r(\hat{x}_1,\ldots ,\hat{x}_{k-1},\tilde{x}_k,\hat{x}_{k+1},\ldots ,\hat{x}_n)] >E[r(\hat{x}_1,\ldots ,\hat{x}_{k-1},\hat{x}_k,\hat{x}_{k+1},\ldots ,\hat{x}_n)]$. In addition, since $\tilde{x}_k=\hat{x}_k$, the return on the portfolio $(\hat{x}_1,\ldots ,\hat{x}_{k-1},\tilde{x}_k,\hat{x}_{k+1},\ldots ,\hat{x}_n)$ has the same semiabsolute deviation as that on the portfolio $(\hat{x}_1,\ldots ,\hat{x}_{k-1},\hat{x}_k,\hat{x}_{k+1},\ldots ,\hat{x}_n)$. Therefore, it is in contradiction to that $(\hat{x}_1^+,\ldots ,\hat{x}_n^+,\hat{x}_1^-,\ldots ,\hat{x}_n^-)$ is Pareto optimal. The theorem is completed.

Proof of Theorem 3

It follows from the operational law of uncertain variables that the portfolio return $\sum _{i=1}^n\xi _ix_i =(\sum _{i=1}^nx_ic_i,\sum _{i=1}^nx_id_i)$ is also a linear uncertain variable with expected value $\sum _{i=1}^nx_i(d_i+c_i)/2$. Further, we have

$$\begin{aligned}&E\left[ \sum _{i=1}^n\xi _ix_i\right] -\sum _{i=1}^n(b_ix_i^++s_ix_i^-)\\&\qquad =\frac{1}{2}\left( \sum _{i=1}^nx_i(d_i+c_i)-2\sum _{i=1}^n(b_ix_i^++s_ix_i^-)\right) . \end{aligned}$$

Therefore, the second objective is equivalent to maximize the term in parentheses on the right-hand side of the above equation. In addition, it follows from Liu and Qin (2012) that

$$\begin{aligned} Sa\left[ \sum _{i=1}^n\xi _ix_i\right] =\frac{1}{8}\sum _{i=1}^nx_i(d_i-c_i). \end{aligned}$$

Note that since $x_i\ge 0$ and $d_i>c_i$ for $i=1,2,\ldots ,n$, we have $\sum _{i=1}^nx_i(d_i-c_i)\ge 0$ which implies that the first objective is equivalent to minimize it. The theorem is proved.

Proof of Theorem 4

It follows that the portfolio return

$$\begin{aligned} \sum _{i=1}^n\xi _ix_i =\left( \sum _{i=1}^nx_i(a_i-\alpha _i),\sum _{i=1}^nx_ia_i,\sum _{i=1}^nx_i(a_i+\beta _i)\right) \end{aligned}$$

is also a zigzag uncertain variable. According to the definition of expected value, we have $E[\sum _{i=1}^n\xi _ix_i]=\sum _{i=1}^nx_i(4a_i+\beta _i-\alpha _i)/4$ which implies that the second objective is equivalent to maximize $\sum _{i=1}^nx_i(4a_i+\beta _i-\alpha _i)-4\sum _{i=1}^n(b_ix_i^++s_ix_i^-)$. Further, by the definition of semiabsolute deviation of uncertain variable, it is obtained that

$$\begin{aligned} Sa\left[ \sum _{i=1}^n\xi _ix_i\right] \!=\!\frac{\left[ \sum _{i=1}^n2x_i(\alpha _i\!+\!\beta _i)\!+\!\left| \sum _{i=1}^nx_i(\alpha _i\!-\!\beta _i)\right| \right] ^2}{\sum _{i=1}^nx_i(\alpha _i\!+\!\beta _i)\!+\!\left| \sum _{i=1}^nx_i(\alpha _i\!-\!\beta _i)\right| }. \end{aligned}$$

Substituting the semiabsolute deviation of the portfolio return into the first objective, the theorem is proved.

Proof of Theorem 5

It follows from the operational law of normal uncertain variables that the portfolio return $\sum _{i=1}^n\xi _ix_i\sim \mathcal {N}(\sum _{i=1}^nx_ie_i,$ $\sum _{i=1}^nx_i\sigma _i)$ is also a normal uncertain variable. Further, it follows from the definitions of expected value and semiabsolute deviation of uncertain variables that

$$\begin{aligned}&E\left[ \sum _{i=1}^n\xi _ix_i\right] =\sum \limits _{i=1}^ne_ix_i\ge 0,\\&Sa\left[ \sum _{i=1}^n\xi _ix_i\right] =\frac{\sqrt{3}\ln 2}{\pi }\sum \limits _{i=1}^n\sigma _ix_i\ge 0 \end{aligned}$$

in which non-negativity holds due to non-negativity of $x_i,e_i$ and $\sigma _i$ for $i=1,2,\ldots ,n$. Substituting them into the two objective functions in Model (9), the theorem is proved.

Proof of Theorem 6

The second objective holds since $E[\xi _i]=\int _0^1\Phi _i^{-1}(\alpha )\mathrm{d}\alpha$ for $i=1,2,\ldots ,n$ by Lemma 1. According to the definition of semiabsolute deviation of uncertain variable, we have

$$\begin{aligned} Sa\left[ \sum _{i=1}^n\xi _ix_i\right]&= \int _0^{+\infty }\mathcal {M}\left\{ \min \left\{ \sum _{i=1}^n\xi _ix_i\right. \right. \\&\left. \left. -\sum _{i=1}^nx_i\int _0^1\Phi _i^{-1}(\alpha )\mathrm{d}\alpha ,0\right\} \ge r\right\} \mathrm{d}r\\&= \int _{-\infty }^{\sum _{i=1}^nx_i\int _0^1\Phi _i^{-1}(\alpha )\mathrm{d}\alpha }\mathcal {M}\left\{ \sum _{i=1}^n\xi _ix_i{\le } r\right\} \mathrm{d}r. \end{aligned}$$

Note that since $x_i\ge 0$ for $i=1,2,\ldots ,n$, it follows from the operational law (Liu 2010) that $\xi _1x_1+\xi _2x_2\cdots +\xi _nx_n$ has an inverse uncertainty distribution $\Psi _1^{-1}(\alpha ) = x_1\Phi _1^{-1}(\alpha )+x_2\Phi _2^{-1}(\alpha )\cdots +x_n\Phi _n^{-1}(\alpha )$. For any given r, the value of $\Psi (r)=M\{\xi _1x_1+\cdots +\xi _nx_n\le r\}$ is just the root of the equation $\Psi _1^{-1}(\alpha ) = r$, i.e., $x_1\Phi _1^{-1}(\alpha )+\cdots +x_n\Phi _n^{-1}(\alpha )=r$. Substituting it into the expression of $Sa[\xi _1x_1+\xi _2x_2\cdots +\xi _nx_n]$, the first objective function is obtained. The theorem is proved.