Section 24.3

Portfolio Variance

Minimum-variance and riskless combinations of assets, Sharpe-ratio maximisation.

Toolkit

For a portfolio $P=xA+yB$ (fixed or with one free weight), variance is a quadratic form

\operatorname{Var}(P)=x^2\sigma_A^2+y^2\sigma_B^2+2xy\operatorname{Cov}(A,B),

so minimising over a single free parameter reduces to calculus/completing the square.

Minimum-variance hedge ratio (holding one unit of $A$ , $k$ units of $B$ ):

k^*=-\frac{\operatorname{Cov}(A,B)}{\operatorname{Var}(B)}=-\rho\frac{\sigma_A}{\sigma_B}, \qquad \operatorname{Var}_{\min}=\sigma_A^2(1-\rho^2).

A riskless combination exists iff $\rho=\pm1$ .

Sharpe ratio of a portfolio with weights $w$ (summing to 1), excess returns $\mu$ , covariance $\Sigma$ :

S(w)=\frac{w^\top\mu}{\sqrt{w^\top\Sigma w}}.

Maximised by the tangency portfolio with weights proportional to $\Sigma^{-1}\mu$ , rescaled so $\mathbf{1}^\top w=1$ ; for two assets this reduces to a one-variable optimisation in the relative weight.

Problems

24.7 Minimum-Variance Hedge Ratio+3 variations Easy Free
24.8 Tangency Portfolio of Two Assets Medium