= Solution
Let $\mu$ be the vector of <expected returns> and $\Sigma$ the positive definite <covariance matrix> of risky returns. For risky <investment portfolio> weights $w$ satisfying $\mathbf1^Tw=1$, the <portfolio opportunity set> consists of
$$
(\sigma,m)=\bigl(\sqrt{w^T\Sigma w},\mu^Tw\bigr).
$$
<Short selling> permits negative weights. For a fixed <expected return>, the least possible <variance> is attained on the left boundary of this set. Its upper branch is the <mean-variance efficient frontier>: every other attainable point has either a larger <standard deviation> for the same <expected return> or a smaller <expected return> for the same risk. The lower boundary is therefore inefficient.
For example, writing $A=\mathbf1^T\Sigma^{-1}\mathbf1$, $B=\mathbf1^T\Sigma^{-1}\mu$, $C=\mu^T\Sigma^{-1}\mu$ and $D=AC-B^2$, minimization of $w^T\Sigma w$ subject to its two linear constraints gives
$$
w=\Sigma^{-1}\left(\frac{C-Bm}{D}\mathbf1+\frac{Am-B}{D}\mu\right),\qquad
\sigma_{\min}^2(m)=\frac{Am^2-2Bm+C}{D}=\frac1A+\frac{A}{D}\left(m-\frac BA\right)^2.
$$
To obtain these weights, use <Lagrange multipliers>: the stationary equation is $\Sigma w=\lambda\mathbf1+\nu\mu$, and the budget and mean constraints determine $\lambda,\nu$. Positive definiteness makes this the unique constrained minimum. Thus the boundary is a hyperbola in risk/mean coordinates.
Adding the risk-free asset allows borrowing and lending at $r$. In the usual positive-premium case the <market portfolio> is the risky tangency portfolio, and combinations of it with the risk-free asset form the upper <capital market line>
$$
\boxed{m=r+\theta\sigma,\qquad \theta=\frac{m_M-r}{\sigma_M}.}
$$
The <Sharpe ratio> measures excess <expected return> per unit of <standard deviation>; its maximum is the <market price of risk> $\theta$. Points between the risk-free point and the tangency point lend some wealth; points beyond it borrow to leverage the <market portfolio>. The first panel illustrates the actual data here. The second anticipates the change in part (g), where the normalization of the tangency direction reverses its orientation.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-35-frontiers.png]
{title=Risky opportunity sets and efficient risk-free portfolio combinations at risk-free returns of 3 and 25 percent}
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