12 The Black–Litterman model
This chapter covers
- Understanding the Black–Litterman model for asset allocation
- Implementing the Black–Litterman model in Python
- Examples of the model in action, including one with cryptocurrency as an asset class
In chapter 10, we learned how to use convex optimization to build portfolios. The key inputs to these optimization problems are the expected returns of the assets and the covariance matrix of asset returns. We saw that portfolios obtained via optimization can be very sensitive to the expected returns inputs.
The Black–Litterman model, developed by Fischer Black and Robert Litterman while working at Goldman Sachs in the early 1990s, aims to accomplish two goals:
- Address the sensitivity of portfolio optimization to its inputs
- Allow the user the flexibility to express opinions about the future returns of the assets in the portfolio.
In this chapter, we’ll cover both the theory and implementation of the Black–Litterman model.
12.1 Equilibrium returns
chapter 3 showed multiple ways of estimating expected returns. Of course, each method results in different values. Then, in chapter 10, we saw that portfolio optimization can be highly sensitive to expected returns—portfolios built using different methods of estimating expected returns can be wildly different. We then showed that we can use constraints to keep the optimized portfolio “close” to a reference portfolio.