The risk of a portfolio consisting of two risky assets with risks \(\sigma_{1}\) and \(\sigma_{2}\) and correlation \(ho\) in proportions \(w_{1}\) and \(w_{2}\) is

\[ \sigma_{\mathrm{P}}=\sqrt{w_{1}^{2} \sigma_{1}^{2}+w_{2}^{2} \sigma_{2}^{2}+2 w_{1} w_{2} ho \sigma_{1} \sigma_{2}} \]

Let us require that \(w_{1}+w_{2}=1\) and \(w_{1}, w_{2} \geq 0\), that is, no short selling.

(a) Show that for \(ho<1\), the value of \(w_{1}\) that minimizes the risk is

\[ w_{1}=\frac{\sigma_{2}^{2}-ho \sigma_{1} \sigma_{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}-2 ho \sigma_{1} \sigma_{2}} \]

What is the minimum if \(ho=1\) ?

(b) Suppose there is a risk-free asset with fixed return \(\mu_{\mathrm{F}}\).

Determine the tangency portfolio (that is, the weights of the two risky assets).

Determine the expected return \(\mu_{\mathrm{T}}\) and the risk \(\sigma_{\mathrm{T}}\) of the tangency portfolio.

(c) Consider the case of two risky assets and a risk-free asset as above. Now, assume \(ho=1\).

What are the efficient frontier and the capital market line? Consider various scenarios with respect to \(\mu_{1}, \mu_{2}, \sigma_{1}\), and \(\sigma_{2}\).