Portfolio optimization

Consider a portfolio problem with $n$ investments held over a period of time. Let $x_j$ denote the amount of investment $j$ throughout the period with a unit total budget, i.e.,

$$\sum_{j=1}^n x_j=1,$$

and let $p_j$ denote the price change of investment $j$ over the period. Then, the overall expected return is given by $r=p^Tx.$ Furthermore, let $\sigma_{jj'}$ denote the covariance between the returns for investment $j$ and $j'.$ The variance of the return is

$$\sigma =\sum_{jj'}x_j\sigma_{jj'}x_{j'},$$

referred to the risk. Here we consider that the $\sigma_{jj'}$ are fixed and computed from historical data. A similar application can also be found in [15].

One approach is to minimize the risk over the return with a lower bound $r_{min}$,

$$\begin{array}{rcl} \mbox{min}&\sum_{jj'}x_j\sigma_{jj'}x_{j'}... ...\geq r_{min},&\\ &\sum_j x_j=1,& x_j\geq 0, j=1:n. \end{array}$$ (3-7)

Another approach is to maximize the return over the risk with an upper bound $\sigma_{max},$

$$\begin{array}{rcl} \mbox{max}&\sum_j p_jx_j&\\ \mbox{s.t.}&\... ...\sigma_{max},&\\ &\sum_j x_j=1,& x_j\geq 0, j=1:n. \end{array}$$ (3-8)

The above models are based on historical data.

Example 4. Two test cases from (3-7) and (3-8) are cast as in formula (1-1) with the real data from optrisk.mod in [16]. Another case which combines (3-7) and (3-8),

$$\begin{array}{rcl} \mbox{min}&\sum_{jj'}x_j\sigma_{jj'}x_{j'}... ...j&\\ \mbox{s.t.}&\sum_j x_j=1,& x_j\geq 0, j=1:n, \end{array}$$

is also considered.