The canonical form of a shifted hyperbola in the \((\sigma, \mu)\) plane is \(\sigma^{2} / c_{\sigma}^{2}-\) \(\left(\mu-\mu_{0}ight)^{2} / c_{\mu}^{2}=1\).

(a) Derive the expression for the two asymptotes of the shifted hyperbola ( \(\mu\) intercept, slopes).

(b) Formula 3.9 shows that the weights for any portfolio on the MVF are a linear function of \(\mu, w^{*}(\mu)=\mathbf{a} \mu+\mathbf{b}\). Using this, show that the variance can be expressed as \(\sigma^{2}(\mu)=A \mu^{2}+B \mu+C\) for all points on the MVF, and express the parameters \(A, B, C\) in terms of \(\mathbf{a}, \mathbf{b}, \mathbf{c}\). Hint: For any \(\mu\), the MVF's variance is

\[\begin{aligned}\sigma^{2}(\mu) & =(\mathbf{a} \mu+\mathbf{b})^{T} \mathbf{C}(\mathbf{a} \mu+\mathbf{b}) \\ & =A \mu^{2}+B \mu+C\end{aligned}\]

(c) Complete the square above to show that MVF can be written as \(\sigma^{2} / c_{\sigma}^{2}-\left(\mu-\mu_{0}ight)^{2} / c_{\mu}^{2}=1\) and, hence, is a hyperbola, and express \(c_{\sigma}, c_{\mu}, \mu_{0}\) in terms of \(A, B, C\).

(d) For the two risky assets case, \(R_{1} \sim\left(\mu_{1}, \sigma_{1}ight)\) and \(R_{2} \sim\left(\mu_{2}, \sigma_{2}ight)\) with correlation \(ho, \mu=w \mu_{1}+(1-w) \mu_{2}\) and, hence, \(w\) is a linear function of \(\mu\)
\[w=\frac{1}{\mu_{1}-\mu_{2}} \mu-\frac{\mu_{2}}{\mu_{1}-\mu_{2}}\]

and the variance can be expressed as \(\sigma^{2}=A \mu^{2}+B \mu+C\). Express \(A, B, C\) and the asymptotes in terms of \(\mu_{1}, \mu_{2}, \sigma_{1}, \sigma_{2}, ho\).