Expected returns generated by the Black–Litterman model can be derived by using risk budgeting, which is a widely used mean–variance optimization (MVO) approach to active investing. The author presents a simple framework to show how the expected returns from the Black–Litterman model can be used to generate portfolio weights that accurately reflect the underlying investment views.

The Black–Litterman model for expected returns is well known in the investment management industry. But the way the model works lacks clarity because it uses Bayesian statistics, which clouds the practical working of the model for prospective users. The author shows how the expected returns generated by the Black–Litterman model can also be derived using a risk-budgeting approach.

The expected returns from the Black–Litterman model can be divided into two components that reflect passive and active investment views. Equilibrium expected returns reflect the passive view and are derived from the weights of a benchmark portfolio by using reverse optimization. This approach ensures that when these returns are used in an unconstrained MVO, the benchmark weights will be the output. Reverse optimization can also be used to obtain active expected returns, which produces a particular active portfolio when used in an unconstrained MVO.

By using risk-budgeting MVO with expected risks, correlations, and Bayesian-adjusted alphas of active strategies as inputs, an active portfolio can be generated. The reverse-optimized returns of the active portfolio will be equal to the alphas derived from the Black–Litterman model.

The author concludes that the risk-budgeting perspective provides an instinctive explanation of how the Black–Litterman model generates portfolios that are less sensitive to the expected return inputs and more accurately reflect underlying investment views.

Using the risk-budgeting framework, the author replicates a case with two active views that were used in a previous study.

The Black–Litterman model calculates two sets of expected returns that reflect passive and active viewpoints. The passive viewpoint is represented by equilibrium expected returns, which are derived directly from the benchmark portfolio for a universe of assets. The second set of expected returns reflects the active investment viewpoints. Views are expressed as expected returns for a portfolio of assets rather than expected returns for individual assets. The views are adjusted for subjective measures of uncertainty that reflect the degree of confidence in the active strategies. The Black–Litterman model converts the expected returns of the confidence-adjusted portfolio into asset-level expected returns consistent with the active views.

Next, the author blends the active and passive expected returns and uses them in an unconstrained MVO, which results in an optimal portfolio that reflects both the active and passive investment perspectives.

Risk budgeting, which is a practical approach to active investment management, uses MVO to maximize the active risk–return trade-off across strategies while targeting a level of active risk. The optimal weights are then used to calculate the risk exposures of each strategy to the overall portfolio.

To generate the results of the Black–Litterman model using a risk-budgeting approach, the author uses a covariance matrix of asset returns, active strategy alphas, and the P-matrix of holdings for each active strategy as inputs. He modifies the strategy alphas using Bayesian adjusting to account for the level of confidence and then solves for MVO optimal weights using an appropriate level of risk aversion. The author scales the strategy portfolios by the risk-budgeting weights and adds them to produce a vector of asset-level active holdings. Finally, he adds the active holdings to the weight of a benchmark and uses reverse optimization to calculate the Black–Litterman expected returns.

The risk-budgeting approach is widely used and not only simplifies the Black–Litterman model but also makes it more understandable. The author adds to the existing literature that explains the Black–Litterman model by using the risk-budgeting perspective. Such studies encourage greater understanding among prospective users of the Black–Litterman model for active investment management.