**Introduction – Black-Litterman Model**

- Based on Black and Litterman’s 1992 paper entitled “Asset Allocation: Combining Investor Views with Market Equilibrium”.
- The model incorporates an investor’s views on the constituent securities as well as his confidence around them.
- Main objective is to generate an efficient frontier which marks the optimal assets combinations that create the highest returns with the lowest risk.
- Model does not account for skewness and kurtosis : tail risk is present.
- It provides diversified portfolios as opposed to the typical Mean-Variance Optimization (“MVO”) framework.

**Methodology**

- Create a market equilibrium
__implied excess return__vector**π**:

**π **= λ **Σ w_mkt**

where λ is the risk aversion coefficient, Σ is the covariance matrix and w is the market cap weight of each constituent

- Estimate the
__views adjusted return__vector:

**π **= [(τΣ)^(-1)+P′w^(-1) P]^(-1) ∗ [(τΣ)^(-1) π+P′w^(-1) Q]

w = τ P Σ P′

where τ is the uncertainty scalar, Σ is the covariance matrix, P is the link matrix, w is the market cap weight, Q is the views vector

- If no personal views are included we set P to zero and the views adjusted returns equal the implied returns vector.

- The link matrix is used to show if a constituent over or underperforms another constituent.

- The uncertainty scalar τ is based on “The Canonical Reference Model, Walters, 2013” and is equal to: 1 / number of observations used in the covariance matrix.

**Portfolio Optimization**

- Combination of securities that achieve the specified rate of return while yielding the lowest amount of risk possible.

- The two critical points in the optimization is the minimum variance portfolio and the maximum return portfolio.
- Also it is determined than no portfolio consumes more than 35% and that all constituents are present.

**Efficient Frontier graph**

**Asset allocation graph**

**Expected returns graph**

Overall, there three graphs provide useful insight into the process of selecting the weights of the portfolio constituents.