The Black Littermanmodel was developed in 1990 at Goldman Sachs by Fischer Black and RobertLitterman and published in 1992. It is a sophisticated portfolio constructionmethod that overcomes the problem of unintuitive, highly-concentratedportfolios, input-sensitivity, and estimation error maximization, in applyingmodern portfolio theory in practice.
The model usesa Bayesian approach to combine the subjective views of an investor regarding theexpected returns of one or more assets with the market equilibrium vector ofexpected returns to form a new, mixed estimate of expected returns. The resultingnew vector of returns leads to intuitive portfolios with sensible portfolioweights. The building of the required inputs is complex, so in this work aretrying to explain it so everybody can use this model. Let’s see now how to useBlack Litterman model in practice.
Model provides a closed-form solution. This means that it can be applied by anyone, without necessarily understanding the mathematics or economics behind it. As every other novelt,y the Black-Litterman Model was studied in-depth and modi ed in multiple ways to t the speci c needs of investors and overcome some limitations. Excel-based software application for estimating portfolio returns and performing asset allocation using the Black-Litterman model. These free excel spreadsheets are related to business finance, including valuations, cash flow models, forecasting, and fundamental analysis. Note: The site is not responsible for any errors in the models.
The inputsof the Black Litterman Model
The Black Littermanmodel is a mathematical model for portfolio allocation, which creates stable,mean-variance efficient portfolios, based on an investor’s unique insights. Theprincipal formula of Black Litterman model is the formula for the new CombinedReturn Vector, which is found by the following expression:
E[R] = [(τΣ)-1 + P’Ω-1P]-1[(τΣ)-1Π + P’Ω-1Q]
Where:
- E[R] is the new (posterior) CombinedReturn 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 theassets involved in the views (K x N matrix or 1 x N row vector in the specialcase of 1 view).
- Ω is a diagonal covariance matrix of error terms from the expressedviews representing the uncertainty in each view (K x K matrix).
- is the ImpliedEquilibrium Return Vector (N x 1 column vector).
- Q is the View Vector (K x 1 columnvector).
The errorterm (ω) that form thediagonal elements of the covariance matrix of the error term (Ω) is found bythe following expression:
ɯk = (pkΣpk’)τ
Where:
- pk k is a single 1 x Nrow 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
Where:
- λ is the risk aversion coefficient.
- wmkt is the marketcapitalization weight (N x 1 column vector) of the assets.
In returnthe risk aversion coefficient is found by the following expression:
λ = (E[Rm] − Rf) / σ2
Where:
- E[Rm] is the expectedmarket (or benchmark) total return.
- Rf is the risk-free rate.
- σ2 is the variance of the market (orbenchmark) excess returns.
Thevariance of Black Litterman model is found by the following expression:
Σp = Σ + [(τΣ)-1 + (P’Ω-1P)]-1
The ExpectedReturn Vector and the variance can now be used as inputs in the mean-variancemodel. So the new optimal “combined”weights are found by the followingexpression:
wp = (λΣ)-1E[R]
The modeldoes not require that investors specify views on all assets, so in this casethe 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 aremany software that can be used to implement the Black Litterman model inpracites, but for many of them it is necessary to pay a lot of money. The implementationof the model can be done using Excel but it is not simple to do.
The Black-Litterman (BL) model is a model in finance proposed by Fischer Black and Robert Litterman. The model was developed in 1990 when both were working at Goldman Sachs. The model offers a simple way for managers to include ‘views’. In fact, the model is an extension of the mean-variance portfolio optimization approach of Markowitz. The resulting portfolio allocation will deviate from the market-capitalization weights.
On this page, we discuss the Black-Litterman model assumptions, the Black-Litterman model formulas, and finally we explain how to extract market-implied returns from the market-capitalization weights. It is important to note that, to implement the model, matrix algebra is required. Thus, it is generally easier to implement the model using Python or MATLAB than by using Excel. Still, we include a Black-Litterman in Excel spreadsheet at the bottom of the page.
Black-Litterman assumptions
The Black-Litterman model starts from market-implied expected returns as the natural starting point. Next, these market-implied expected returns are adjusted to reflect the views of the fund manager. The BL model makes two important assumptions:
- Asset returns are normally distributed
- The covariance matrix of asset returns is known
Black-Litterman formulas
Black Litterman Model Excel Example
Let’s start with the formula for market-implied expected returns (pi)
Where Sigma is the covariance matrix, omega is the risk-aversion parameter, and w is the vector with the market capitalization of the assets in the portfolio.
Next, let’s calculate the expected returns under the BL model. This is done by changing the market-implied returns using the views of the fund manager. First, we need to define a number of variables:
- P: matrix with the views. Each row is a view and each entry of the row represents the portfolios’ weights of each asset
- Omega: a diagonal covariance matrix with the uncertainty within each view
- Q: the expected return matrix of the portfolios from the views described in matrix P
Now we have all the ingredients to calculate expected returns
Black-Litterman versus Treynor-Black
It is important not to confuse the Black-Litterman model with the Treynor-Black model. For details on the latter model, see Treynor-Black model. This model is used to perform security analysis, whereas the Black-Litterman model is used for asset allocation.
Black Litterman Model Excel Free
Summary
We discussed the basics behind the BL, a model that can be used to shrink expected returns such that they are consistent with market-implied returns. Despite the elegance of the model, it is not used that often.