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QUESTION

Consider a portfolio of 4 risky assets $X^{(1)},\ X^{(2)}$ which
are held in proportion $\theta_1$ and $\theta_2=1-\theta_1$. Let
them be distributed normally with $\mu_1,\ \mu_2$ and variance
$\sigma_1^2,\ \sigma_2^2$ respectively.

\begin{description}

\item[(a)]
Show that the mean value of the portfolio is
$\mu=\theta_1\mu_1+\theta_2\mu_2$.

\item[(b)]
Show that if the prices of the risky assets are uncorrelated, then
the variance of the portfolio is given by
$\theta_1^2\sigma_1^2+\theta_2^2\sigma_2^2$.

\item[(c)]
Show that if the prices of the risky assets have some correlation
then the variance is given by
$\theta_1^2\sigma_1^2+\theta_2^2\sigma_2^2+2\theta_1\theta_2\rho_{12}$
where the correlation between the prices
$\rho_{12}=\left<X{(1)}X{(2)}\right>-\mu_1\mu_2$.

\item[(d)]
Given the following data evaluate the mean and variance of the
portfolio: $\mu_1-0.2,\ \sigma_1=0.75,\ \mu_2=0.16,\
\sigma_2=0.5,\ \rho_{12}=-0.60$.

\end{description}


ANSWER

$\begin{array}{ll}X^{(1)}:\theta_1&X^{(1)}\in
N(\mu_1,\sigma_1^2)\\ X^{(2)}:\theta_2=1-\theta_1&X^{(2)}\in
N(\mu_1,\sigma_2^2)\end{array}$

\begin{description}

\item[(a)]
$\left<\theta_1X^{(1)}+\theta_2X^{(2)}\right>$ is mean value of
portfolio$=\mu=\theta_1\left<X^{(1)}\right>+
\theta_2\left<X^{(2)}\right>=\theta_1\mu_1+\theta_2\mu_2=\mu$

\item[(b)]
Variance
\begin{eqnarray*}
\sigma^2&=&\left<[(\theta_1X^{(1)}+\theta_2X^{(2)})-\mu]^2\right>\\
&=&\left<(\theta_1(X^{(1)}-\mu_1)+\theta_2(X^{(2)}-\mu_2))^2\right>\\
&=&\left<\theta_1^2\underbrace{(X^{(1)}-\mu_1)^2}_{<>=\sigma_1^2}+\theta_2^2\underbrace{(X^{(2)}-\mu_2^2)}_{<>=\sigma^2_2}+2\theta_1\theta_2(X^{(1)}-\mu_1)\times(X^{(2)}-\mu_2)\right>\\
&=&\theta_1^2\sigma_1^2+\theta_2^2\sigma_2^2+2\theta_1\theta_2\underbrace{\left<(X^{(1)}-\mu_1)(X^{(2)}-\mu_2)\right>}_{\textrm{correlation
term}=0}\\ &=&\theta_1^2\sigma_1^2+\sigma_2^2\theta_2^2
\end{eqnarray*}

\item[(c)]
If correlation $\neq0$ must include

\begin{eqnarray*}
&=&2\theta_1\theta_2\left<(X^{(1)}-\mu_1)(X^{(2)}-\mu_2)\right>
\textrm{ term}\\
&=&2\theta_1\theta_2\left<x^{(1)}X^{(2)}-\mu_1X^{(2)}
-\mu_2X^{(1)}+\mu_1\mu_2\right>\\ &=&2\theta_1\theta_2\bigg[
\left<X^{(1)}X^{(2)}\right> -\mu_1
\underbrace{\left<X^{(2)}\right>}_{=\mu_2}
-\mu_2\underbrace{\left<X^{(1)}\right>}_{=\mu_1}+\mu_1\mu_2\bigg]\\
&=&2\theta_1\theta_2\underbrace{\left[\left<X^{(1)}X^{(2)}\right>
-\mu_1\mu_2\right]}_{=\rho_{12}}
\end{eqnarray*}

Hence

$$\sigma^2=\theta_1^2\sigma_1^2+\sigma_2^2\sigma_2^2+2\rho_{12}\theta_1\theta_2$$

\item[(d)]
$\left.\begin{array}{ll}\mu_1=0.2,&\sigma_1=0.75\\\mu_2=0.16,&\sigma_2
=0.5\end{array}\right\}\rho_{12}=0.60.$

$\mu=0.2\theta_1+0.16\theta_2=0.2\theta_1+0.16(1-\theta_1)=0.16+0.04\theta_1$

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\begin{eqnarray*}
\sigma^2&=&\theta_1(0.75)^2+(0.5)^2\theta_2^2-2\times0.6\times\theta_1\theta_2\\
&=&\theta_1^2(0.5625)+0.25(1-\theta_2)^2-1.2\theta_1(1-\theta_1)\\
&=&2.0125\theta_1^2-1,70\theta_1+0.25
\end{eqnarray*}

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Use combination $\theta_1$, $\theta_2$ from region $A$ to eliminate
$\sigma^2$, i.e. minimise variance and osciallations in portfolio values.

$\sigma^2=0$ when
$\theta_1=\frac{1.70+\sqrt{1.7^2-4\times2.0125\times0.25}}{2\times2.0125}=0.6551$
or $0.1896$.

Better to pick the 0.6551 value since then $\mu$ is higher

$\mu=0.16+0.04\times0.6551-0.1862$

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