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Definition: Black-Litterman Model (BLM)
The Black-Litterman model is a financial model that is used to determine the optimal allocation of assets in a portfolio. Developed in 1990 by Fischer Black and Robert Litterman from Goldman Sachs, the model seeks to overcome the shortcomings of the modern portfolio theory such as extreme portfolio allocations and highly concentrated portfolios based on historical data. The model uses the Capital Asset Pricing model (CAPM) to provide an intuitive starting point or prior to estimate the portfolio returns by following a Beyesian approach. It also provides a means to integrate investors views of expected market behaviour with the intuitive prior.
The CAPM model measures the investors’s expected returns for the systematic risk only. The Black-Litterman model converts the equilibrium CAPM market returns using reverse optimization to generate a stable distribution of the expected portfolio returns. These implied returns are not the same as historical returns. The model then incorporates the investor’s view of the market behaviour which determines the optimal portfolio weights and the expected return.
The following equation gives the expected returns derived from the Black-Litterman Model:
E[R] = [(τΣ)-1 + PTΩP]-1 [(τΣ)-1 π + PTΩQ]
E[R] is the posterior combined return, τ is a scalar;
Σ is the covariance matrix of excess returns;
P is the matrix identifying the assets associated with investor views;
Ω is a diagonal covariance matrix with error terms from the investor views thereby, accounting for uncertainty;
Π is the implied equilibrium returns; and
Q is the expected return of the portfolios described by the views presented by matrix P.
The model uses the implied returns as the starting point and makes use of reverse optimization using:
Π = λΣωmkt
λ is the risk-aversion coefficient and ωmkt is the market capitalization weight of the assets.
In the absence of any views about the market behaviour, investors should hold the market portfolio. The advantage of the BLM model is that it allows investors to incorporate their individual views and their confidence levels thus, optimize their portfolio to get a return that differs from the implied returns. It leads to more stable and well-diversified portfolios. However, the model does not give the best possible portfolio. It gives the most optimized portfolio given the investor’s views.