Portfolio selection problem under uncertainty, taking account of risks and regrets

In this paper new solution for portfolio selection problem under uncertainty is formalized. It is based on three concepts: guaranteed result, regret function, Pareto optimal solution.
This paper deals with a portfolio selection problem under uncertainty. Let us consider $n$ securities with return rate $\xi_i$ , $i \in {1, . . . , n}$ and denote by $x_i$ the proportion of total amount of funds invested in the i-th security. A portfolio selection problem consists in finding the investment rate vector $x = (x_1, . . . , x_n)$, where $x_i ≥ 0$, $\sum\limits_{n} , i ∈ {1, . . . , n}$. Vector $x \in R^n$ maximizes the total portfolio return $\xi_P = \sum\limits_{i=1}^n x_i\xi_i$. The traditional Markowitz models of portfolio selection treat the return rates as a random variables vector following a probability distribution with vector $\xi = (\xi_1, . . . , \xi_n)$ and a covariance matrix $V = || \sigma_{ij} ||$, where \begin{equation*}
\sigma_{ij} =
\sigma_i^2,&\text{$i = j$}\\
cov(\xi_i, \xi_j), &\text{ $i \ne j$}

In particular, the well known model proposed by Markowitz, consists in minimizing the portfolio variance $\sigma_P^2 = \sum\limits_{i=1}^n x_i x_j \sigma_{ij} = x^T V x.$
In this paper, the total portfolio return equals $\xi_P = \sum\limits_{i=1}^n x_iy_i$ , where uncertainty $y_i \in [a_i, b_i]$, parameters $a_i > 0$, variables $x_i ≥ 0$, $\sum\limits_{i} x_i = 1$, $i \in {1, . . . , n}$ . We have two-criteria problem under uncertainty $P_1 = \langle X, Y, {f_1(x, y), f_2(x, y)}\rangle$, where the set
\begin{equation*} X = \begin{Bmatrix}
x - (x_1,..., x_n), | x_i \geq 0, \sum\limits_{i=1}^n x_i = 1
\end{Bmatrix} , \end{equation*}
the set $Y = \{y = (y_1, . . . , y_n)|y_i \in [a_i, b_i] \}$, first criteria $f_1(x, y) = \sum\limits_{i} x_iy_i$ second criteria $f_2(x, y) = f_1(x, y) −\underset{x∈X}{max} f_1(x, y)$. Then we consider two-criteria problem $P_2 = \langle X, {g_1(x), g_2(x)}\rangle$, here $g_1(x) = −R_V (x), g_2(x) = −R_S(x)$ and

$R_V (x) = \underset{x}{max}\: \underset{x}{min} f_1(x, y) − \underset{x}{min}\: f_1(x, y)$ is risk function,
$R_S(x) = \underset{y}{max} \: Φ(x, y) − \underset{x}{min} \: \underset{y}{max}\: Φ(x, y)$ - regret function.

Further we look for vector $x_P ∈ R^n$, which minimizes the function
\begin{equation*}G(x) = R^2_V(x) + R^2_S(x), x ∈ X.\end{equation*}
Solution $x_P ∈ X$ is Pareto optimal for given problem $P_2$. In this paper, we construct the algorithm finding optimal portfolio selection.

Key words: portfolio selection, uncertainty, risk, regret, two-criteria problem, Pareto
optimal solution.