Black-Litterman model

The idea in plain words

Plain mean-variance takes historical returns at face value and produces wild weights. Black and Litterman turned the problem around: begin with the returns that would make a sensible balanced portfolio optimal (the prior), state your views with a confidence, and blend the two the way a statistician combines evidence. No view, no tilt; a confident view, a strong tilt.

How it is calculated

Prior: our "market" portfolio is equal-risk (inverse volatility; a currency or index basket has no market capitalisation), and the implied returns are π = δΣw_mkt with δ = 2.5. Views: the expected annual returns you enter for some tickers, each with a confidence from 5% to 95%; without views, the historical means at 25% confidence. The blend (posterior) is then fed to mean-variance with risk aversion δ.

π = δ Σ w_mkt                     (implied returns)
views:  P μ = Q + ε,  ε ~ N(0, Ω)
μ_BL = [(τΣ)⁻¹ + PᵀΩ⁻¹P]⁻¹ [(τΣ)⁻¹π + PᵀΩ⁻¹Q]
τ = 0.05;  Ωₖₖ = (1 − cₖ) ÷ cₖ × (P τΣ Pᵀ)ₖₖ

How to use it

• Enter views only where you have one: "gold 12% a year, 60% sure". The other tickers stay neutral.
• Raise the confidence to move the weights further; at 50% the view and the prior weigh about equally.
• Use it as a sensible default when you do not trust history's returns: without views it leans only lightly on them.

Drawing conclusions

• Weights close to the equal-risk mix mean the views were weak or close to the prior.
• Compare the implied and posterior returns (in the study's notes) to see what your views changed.

Watch out for

• A confident view moves weights a lot: confidence is a strong lever, use it honestly.
• Views are about returns relative to the prior, not about prices; "gold will rise" needs a number.

A little history

Fischer Black and Robert Litterman developed it at Goldman Sachs in 1990 and published "Global Portfolio Optimization" in the Financial Analysts Journal in 1992. It became the standard way for asset managers to put their views into an optimiser.