#### Question

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

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#### Step 1

1 of 3

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

\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:

\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

$(z_1,z_2)=(140 , 20000)$

If we want to minimize variance we obtain the solution

$(z_1,z_2)=\left(111.85,63988.8\right)$

Adding a restraint $z_1>=110+5i$ for $i\in\{1,2,\dots,6\}$ we obtain the following values for the trade-off curve:

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

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

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