Lesson 8 of 14 · All lessons in this module
  1. Portfolio expected return and varianceVideo lesson
  2. Portfolio returns over time: drift and rebalancing
  3. The covariance matrix and many-asset portfolios
  4. Correlation and diversification
  5. Scenario-based portfolio risk and the many-asset limit
  6. Integrated cases: three-asset risk from volatilities, correlations and scenarios
  7. Minimum-variance and efficient frontiersVideo lesson
  8. Finding the minimum-variance portfolio and optimising weights
  9. Risk aversion and utilityVideo lesson
  10. Indifference curves and the capital allocation lineVideo lesson
  11. Theory assumptions, risk-aversion measures and the Sharpe ratio of the CAL
  12. Optimal risky portfolio and the optimal investor portfolioVideo lesson
  13. The risk-free asset, the market portfolio and the CMLVideo lesson
  14. Beta, the security market line and the CAPM

This lesson is part of the 2027 curriculum. You are following the 2026 curriculum, where it is not taught in this form. Switch if you are sitting the exam under the 2027 curriculum.

Lesson 8 of 14 · 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 σP2\sigma_P^2 subject to weights summing to 1, a target return kk, and often no short selling (wi≥0w_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: wA=(σB2−σAB)/(σA2+σB2−2σAB)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: wA=(rP−rB)/(rA−rB)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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