Portfolio optimization model with transaction costs. (Q1862932)

The authors consider a class of portfolio selection problems with transaction costs with the \(L^\infty\) risk function; for the latter, see \textit{X. Cai}, \textit{K.-L. Teo}, \textit{X. Yang} and \textit{X. Y. Zhou} [Manage. Sci. 46, 957--972 (2000)]. An investor has total fund \(M\) which is to invested into possible assets \(S_1,\dots, S_n\). Let \(x_j\geq 0\) be the allocation from the total fund \(M\) for investment into \(S_j\). The transaction cost function of \(S_j\) is given by \[ c_j(x_j)= \begin{cases} 0,\quad &x_j= 0,\\ p_j u_j,\quad & 0< x_j< u_j,\\ p_j x_j,\quad & u_j\leq x_j\end{cases} \] for given constants \(u_j> 0\) and \(p_j>0\). Let \(R_j\) be the (random) rate of return of \(S_j\), and let \(r_j\) and \(q_j\) denote the expected return rate of \(S_j\) and, respectively, the expected absolute deviation of \(R_j\) from its mean. The portfolios \((x_1,\dots, x_n)\) under consideration are assumed to sastisfy \[ \sum^n_{j=1} (x_j+ c_j(x_j))= M. \] Thus the expected return of a portfolio \(x= (x_1,\dots, x_n)\) is given by \[ R(x)= \sum^n_{j=1} (r_j x_j- c_j(x_j)). \] The \(L^\infty\) risk function is defined as \(Q(x)= \max_{1\leq j\leq n} q_j x_j\). This leads to the following nonlinear optimization problem \[ (\text{BP})\;\begin{cases} \min(Q(x),-R(x)),\\ \text{s.t. }x_j\geq 0,\;\sum^n_{j=1} (x_j+ c_j(x_j))= M.\end{cases} \] The program (BP) is converted into a parametric optimization problem \((\text{PP}(\theta))\) \((0<\theta<\infty)\) with a single criterion: \[ (\text{PP}(\theta))\;\begin{cases} \min(\theta Q(x)- R(x)),\\ \text{s.t. }x_j\geq 0,\;\sum^n_{j=1} (x_j+ c_j(x_j))= M.\end{cases} \] Any solution of \((\text{PP}(\theta))\) is a solution of (BP). The authors obtain a characterization of solutions of \((\text{PP}(\theta))\) without transaction costs (i.e. \(c_j\equiv 0\), \(1\leq j\leq n\)) which plays a key role in characterizing the solutions of \((\text{PP}(\theta))\) in the presence of transaction costs. Finally, an efficient algorithm for solving \((\text{PP}(\theta))\) for sufficiently large \(M\) is given.