M3: Black-Litterman Model

0. Background

The return of any asset or asset class can be separated into three parts: 

– Risk-Free Return 

– Return Correlated with Benchmark 

– Return Not Correlated with Benchmark 

Returns that are correlated with the benchmark result in beta risk (systematic risk, benchmark risk, non-diversifiable risk, or market risk) • Beta risk is the type of risk that is rewarded with a premium

1.  The Black-Litterman model uses a Bayesian approach in which the investors' subjective views regarding the expected returns of one or more assets are combined with the market equilibrium vector (the prior distribution) of expected returns. The result is a new mixed estimate of expected returns (the posterior distribution), which can be described as a weighted average of the investor’s views and the market equilibrium.

2. Sharpe index for the market portfolio:

3. Beta  = covariance

4. CAPM is based on the concept that there is a linear relationship between risk (as measured by standard

deviation of returns) and return. Further, it requires returns to be normally distributed. This model is of

the form

5. 

6. While it is true that PSO requires a lot of simulations, it benefits from quasi random number simulations as these provide a more uniform distribution of simulated values that in return improve the accuracy of numerical integration procedures in higher dimension problems, which are the norm in PSO.

7. 

8. Excess returns result from reverse optimization Π=d.Σ.w