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QUESTION

Reconsider the above question from the point of view of
diversification to reduce risk. Does this make sense when the
assets are

\begin{description}

\item[(a)]
perfectly positively correlated ($\rho_{12}=1$);

\item[(b)]
perfectly uncorrelated ($\rho_{12}=0$);

\item[(c)]
the shares are perfectly negatively correlated ($\rho_{12}=-1$).

\end{description}
In the case of (c), show that it is theoreticly possible to obtain
a risk-free portfolio.


ANSWER

Reduction of risk is equivalent to  minimising $\sigma^2$

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

\begin{description}

\item[(a)]
$\rho_{12}=1\Rightarrow$

\begin{eqnarray*}
\sigma^2&=&\theta_1^2\sigma_1^2+\theta_2^2\sigma^2_2+2\theta_1\theta_2\\
&>&\theta_1^2\sigma_1^2+\theta_2^2\sigma_2^2\\
&>&\sigma_1^2\textrm{ or }\sigma_2^2
\end{eqnarray*}

Thus it doesn't make sense to diversify into positively correlated
assets as the variance (uncertainty) increases. (Or if one goes
down, so does the other).

\item[(b)]
$\rho_{12}=0\Rightarrow$

\begin{eqnarray*}
\sigma^2&=&\theta_1^2\sigma_1^2+\theta_2^2\sigma_2^2+0\\
&=&\theta_1^2\sigma_1^2+(1-\theta_1)^2\sigma_2^2\\
&=&\theta_1^2(\sigma_1+\sigma_2)-2\sigma_2^2\theta_1+\sigma_2^2
\end{eqnarray*}

Which as a plot against $\theta$ like:

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There is a minimum at
$\theta_1=\frac{\sigma_2^2}{\sigma_1^2+\sigma_2^2}\Rightarrow\sigma^2$
valueless$<\sigma_2^2$ or $\sigma_1^2$ so it is better to
diversify $(=\frac{\sigma_1^2\sigma_2^2}{\sigma_1^2+\sigma_2^2})$
optional portfolio.

\item[(c)]
$\rho_{12}=-1\Rightarrow$

\begin{eqnarray*}
\sigma^2&=&\theta_1^2\sigma_1^2+\theta^2_6\sigma_2^2-2\theta_1\theta_2\\
&=&\theta_1^2\sigma_1^2+(1-\theta_1)^2\sigma_2^2-2\theta_1(1-\theta_1)\\
&=&\theta_1^2(\sigma_1^2+\sigma_2^2+2)-\theta_1(2\sigma_2^2+2)+\sigma_2^2
\end{eqnarray*}

\end{description}




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