The Black-Litterman model was published in the Financial Analyst’s journal in 1992. It provides an estimate of future expected returns by combining equilibrium returns with an investor’s views. The resulting new vector of returns leads to potentially well diversified, intuitive, and stable portfolios. Implied returns are computed by using the following equation (reverse optimization):

with risk aversion coefficient (δ), asset covariance matrix (∑), and equal asset weights (w). Since the true expected returns (µ) are unknown, it is assumed that the equilibrium returns are reasonable estimate of the true expected returns:


ϵΠ ~ N (0,τ∑).

The matrix τ∑ corresponds to the confidence in how well the implied returns can be estimated.

K views in the model can be expressed as


P is a K x N matrix, each row of which provides a combination of assets for which we have a view.

q is a K-dimensional vector of the views about expected returns.

εq is a K-dimensional vector of errors, ϵ ~ N(0,Ω).

is a K x K matrix expressing the confidence in the views.

Equations (2) and (3) can be stacked in the form


withI denoting the N x N identity matrix. Calculating the Generalised Least Squares estimator for µ results in the following Black-Litterman combined expected returns:

AlternativeSoft’s software platform uses the Black-Litterman combined returns as the input to compute optimal portfolios on the efficient frontier which will have: (i) low volatility (Mean-Variance Optimization), or (ii) small average extreme loss (Mean-Conditional VaR optimization), or (iii) low volatility, high skewness, and low kurtosis (mean-Modified VaR optimization). As an example of diversification achieved by using AlternativeSoft’s Black-Litterman based optimization, two figures are provided below. Figure 1 exhibits the portfolio weights on the efficient frontier obtained from classical mean-variance optimization. Portfolio weights computed by using BL model based mean-variance optimization are shown in Figure 2. Figure 2 displays that BL model based optimization produces a portfolio with more hedge funds in the portfolio leading to highly diversified optimized portfolio.

Figure 1: Distribution of weights based on classical MVO

Figure 2: Distribution of weights based on BL MVO

Read the full paper here.

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