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Portfolio Theory and Quantitative Investment / Robust Portfolio Construction

Black–Litterman

Hand a mean–variance optimiser a vector of historical average returns and it will tell you to hold 1,144% of your portfolio in US bonds and short 105% of international bonds. That is not a caricature: it is Idzorek's eight-asset example, and it is why most asset allocators do not use raw Markowitz optimisation. Expected returns are the input the optimiser is most sensitive to and the input that is hardest to estimate. The Black–Litterman model fixes the problem at its source. It starts from the expected returns that would make the market portfolio optimal (obtained by running the optimiser backwards), treats them as a Bayesian prior, and moves away from them only where the investor has a view and only as far as the investor's confidence justifies. The output is a return vector that produces diversified, intuitive portfolios, and a portfolio that differs from the market only in the assets the views are about. This note derives the implied returns, states the Black–Litterman formula, shows how views and their uncertainty are encoded, and works Idzorek's example end to end.
Learning objectives. By the end of this note you should be able to:
  • derive the implied equilibrium returns Π=λΣwmkt\Pi = \lambda\Sigma w_{\text{mkt}} by reverse optimisation and show that they are CAPM expected returns;
  • state the Black–Litterman formula and explain each of τ\tau, PP, QQ and Ω\Omega;
  • encode absolute and relative views, including market-capitalisation weighting within a relative view;
  • predict the direction of the tilt a view causes, including the counter-intuitive case of a view smaller than the equilibrium spread;
  • show that the optimal portfolio tilts away from the market only along the view portfolios, and compute the implied confidence of a view.

In this lesson

  1. Why expected returns need a prior
  2. Reverse optimisation: the implied equilibrium returns
  3. The Black–Litterman formula
  4. Building the inputs
  5. What the model does to the portfolio
  6. Code: reproducing Idzorek's example

Before you start

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