Asset Allocation: Black – Litterman

Introduction – Black-Litterman Model

  • Based on Black and Litterman’s 1992 paper entitled “Asset Allocation: Combining Investor Views with Market Equilibrium”.
  • The model incorporates an investor’s views on the constituent securities as well as his confidence around them.
  • Main objective is to generate an efficient frontier which marks the optimal assets combinations that create the highest returns with the lowest risk.
  • Model does not account for skewness and kurtosis : tail risk is present.
  • It provides diversified portfolios as opposed to the typical Mean-Variance Optimization (“MVO”) framework.


  • Create a market equilibrium implied excess return vector π :

π = λ Σ w_mkt

where λ is the risk aversion coefficient, Σ is the covariance matrix and w is the market cap weight of each constituent

  • Estimate the views adjusted return vector:

π  = [(τΣ)^(-1)+P′w^(-1) P]^(-1) ∗ [(τΣ)^(-1) π+P′w^(-1) Q]

w = τ P Σ P′

where τ is the uncertainty scalar, Σ is the covariance matrix, P is the link matrix, w is the market cap weight, Q is the views vector

  • If no personal views are included we set P to zero and the views adjusted returns equal the implied returns vector.
  • The link matrix is used to show if a constituent over or underperforms another constituent.
  • The uncertainty scalar τ is based on “The Canonical Reference Model, Walters, 2013” and is equal to:  1 / number of observations used in the covariance matrix.

Portfolio Optimization

  • Combination of securities that achieve the specified rate of return while yielding the lowest amount of risk possible.
  • The two critical points in the optimization is the minimum variance portfolio and the maximum return portfolio.
  • Also it is determined than no portfolio consumes more than 35% and that all constituents are present.

Efficient Frontier graph


Asset allocation graph


Expected returns graph


Overall, there three graphs provide useful insight into the process of selecting the weights of the portfolio constituents.