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Portfolio Theory and Quantitative Investment / Portfolio Optimisation and Quant Portfolio Construction

Mean-Variance Optimisation

Mean–variance optimisation is the engine room of quantitative portfolio construction: every weight in this course is ultimately the solution of a quadratic program. This note builds the framework and solves, in closed form, the four "simpler versions" of L11 plus the full fully-invested problem — each derivation is short, each solution has the same Σ−1\Sigma^{-1}-shape, and each teaches a different lesson about how return, risk, correlation and constraints interact. The second half re-poses everything in the active (benchmark-relative) setting that institutional managers actually live in, and closes with the CAPM as the equilibrium consequence of everyone optimising.
Learning objectives. By the end of this note you should be able to:
  • set up the MV framework and the canonical quadratic program;
  • derive the unconstrained solution w∗=1λΣ−1μw^* = \frac1\lambda\Sigma^{-1}\mu and interpret each ingredient;
  • execute the Lagrange-multiplier method for the minimum-variance and characteristic portfolios;
  • reduce the with-cash problem to the unconstrained one via excess returns;
  • state the fully-invested solution and decompose it into anchor + tilt;
  • formulate active management (w=b+aw = b + a), derive the optimal active weights, and discuss portable alpha and the long-only caveat;
  • state the CAPM and its role as the 1-factor baseline.

In this lesson

  1. 2.1 The mean–variance framework
  2. 2.2 The four "simpler versions" — know all derivations
  3. 2.3 Active portfolio management
  4. 2.4 CAPM and beta

The full lesson, with worked examples and exercise solutions, is for members.

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