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Lesson 7 of 13 · 14 min

Finding the minimum-variance portfolio and optimising weights

Portfolio optimisation picks weights that minimise variance for a target return (or maximise return or the Sharpe ratio); for two assets the minimum-variance weights have a closed-form answer.

In short

Optimisation: minimise $\sigma_P^2$ subject to weights summing to 1, a target return $k$ , and often no short selling ( $w_i \ge 0$ ).
With no short selling, any target return must lie between the lowest and highest asset expected returns.
Two-asset global minimum-variance weight: $w_A = (\sigma_B^2 - \sigma_{AB})/(\sigma_A^2 + \sigma_B^2 - 2\sigma_{AB})$ . Expected returns play no role.
For a target return on the two-asset frontier: $w_A = (r_P - r_B)/(r_A - r_B)$ ; portfolios with targets above the minimum-variance return are efficient.
Three ways to optimise: return-constrained (min risk), risk-constrained (max return), Sharpe-ratio maximisation (the tangency portfolio).

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