The Black Litterman model was developed in 1990 at Goldman Sachs by Fischer Black and Robert Litterman and published in 1992. It is a sophisticated portfolio construction method that overcomes the problem of unintuitive, highly-concentrated portfolios, input-sensitivity, and estimation error maximization, in applying modern portfolio theory in practice.
The model uses a Bayesian approach to combine the subjective views of an investor regarding the expected returns of one or more assets with the market equilibrium vector of expected returns to form a new, mixed estimate of expected returns. The resulting new vector of returns leads to intuitive portfolios with sensible portfolio weights. The building of the required inputs is complex, so in this work are trying to explain it so everybody can use this model. Let’s see now how to use Black Litterman model in practice.
The inputs of the Black Litterman Model
The Black Litterman model is a mathematical model for portfolio allocation, which creates stable, mean-variance efficient portfolios, based on an investor’s unique insights. The principal formula of Black Litterman model is the formula for the new Combined Return Vector, which is found by the following expression:
E[R] = [(τΣ)-1 + P’Ω-1P]-1[(τΣ)-1Π + P’Ω-1Q]
- E[R] is the new (posterior) Combined Return Vector (N x 1 column vector).
- τ is a scalar (from 0 to 1).
- Σ is the covariance matrix of excess returns (N x N matrix).
- P is a matrix that identifies the assets involved in the views (K x N matrix or 1 x N row vector in the special case of 1 view).
- Ω is a diagonal covariance matrix of error terms from the expressed views representing the uncertainty in each view (K x K matrix).
- is the Implied Equilibrium Return Vector (N x 1 column vector).
- Q is the View Vector (K x 1 column vector).
The error term (ω) that form the diagonal elements of the covariance matrix of the error term (Ω) is found by the following expression:
ɯk = (pkΣpk’)τ
- pk k is a single 1 x N row vector from Matrix P that corresponds to the kth view and Σ is the covariance matrix of excess returns.
In return, the Implied Equilibrium Return Vector is found by the following expression:
Π = λΣwmkt
- λ is the risk aversion coefficient.
- wmkt is the market capitalization weight (N x 1 column vector) of the assets.
In return the risk aversion coefficient is found by the following expression:
λ = (E[Rm] − Rf) / σ2
- E[Rm] is the expected market (or benchmark) total return.
- Rf is the risk-free rate.
- σ2 is the variance of the market (or benchmark) excess returns.
The variance of Black Litterman model is found by the following expression:
Σp = Σ + [(τΣ)-1 + (P’Ω-1P)]-1
The Expected Return Vector and the variance can now be used as inputs in the mean-variance model. So the new optimal “combined”weights are found by the following expression:
wp = (λΣ)-1E[R]
The model does not require that investors specify views on all assets, so in this case the formula of the Combined Return Vector is:
E[R] = Π
Now the expected returns of the Black Litterman model coincide with the implied equilibrium returns.
The implementation of Black Litterman model
The following diagram (Idzorek 2005), summarizes the procedure for implementing the Black Litterman model.
There are many software that can be used to implement the Black Litterman model in pracites, but for many of them it is necessary to pay a lot of money. The implementation of the model can be done using Excel but it is not simple to do.