Question

For Example, construct a trade-off curve between the chosen portfolio’s expected return and variance. This is often called the efficient frontier.

Solution

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We have two objectives, maximize expected return and minimize variance. These objectives are given by the formulas

maxz1=0.14x1+0.11x2+0.1x3minz2=0.2x12+0.08x22+0.18x32+0.1x1x2+0.04x1x3+0.06x2x3\begin{align*} \max z_1&=0.14x_1+0.11x_2+0.1x_3\\ \min z_2&=0.2x_1^2+0.08x_2^2+0.18x_3^2+0.1x_1x_2+0.04x_1x_3+0.06x_2x_3 \end{align*}

We have the restrains:

x1+x2+x3=1000x1,x2,x30\begin{align*} x_1+x_2+x_3&= 1000\\ x_1,x_2,x_3&\geq 0 \end{align*}

If we want to maximize expected return we obtain the solution

(z1,z2)=(140,20000)(z_1,z_2)=(140 , 20000)

If we want to minimize variance we obtain the solution

(z1,z2)=(111.85,63988.8)(z_1,z_2)=\left(111.85,63988.8\right)

Adding a restraint z1>=110+5iz_1>=110+5i for i{1,2,,6}i\in\{1,2,\dots,6\} we obtain the following values for the trade-off curve:

z1z2111.8563989115656671207523812593286130119810135155000140200000\begin{array}{|c|c|}\hline z_1&z_2\\\hline 111.85&63989\\\hline 115&65667\\\hline 120&75238\\\hline 125&93286\\\hline 130&119810\\\hline 135&155000\\\hline 140&200000\\\hline \end{array}

Bellow we show a graph of this trade-off curve.

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