The mean-variance frontier (cone) with a risk-free asset is generally tangent to the frontier (hyperbola) formed from the risky assets only. The portfolio of risky assets at the tangency point is called the tangency portfolio. It belongs to both frontiers.

(a) Within the mean-variance optimization framework, solve analytically for the tan- gent portfolio, assuming that the risk-free rate is given by r. Show that it can be expressed as a ratio of mean-variance ratios.

(b) What would be an alternative way of deriving the tangent portfolio?