Incorporating market expectations and forecasts into asset allocation used to be more of an art than an analytical process in the 80s. In this post we will review Fisher Black’s elegant and very practical solution to portfolio construction, going through a sector allocation example.

## The Black Litterman Model

The BL (Black-Litterman) model can be considered one of the greatest contributions in portfolio optimisation, after Markovitz’s MPT(modern portfolio theory), due to its practicality and wide adoption by the industry. The BL model is based on MPT and is using bayesian theory to incorporate views on future outlook (subjective forecasts) to the portfolio optimization process. Let’s unpack Bl mathematical structure below.

### 1. Model Structure

BLM needs 3 main inputs, the CAPM derived weights (as proposed by MPT), the vector of subjective/expected returns and the confidence level of those. The final weights are given by using a mean-variance optimiser on the posterior mean and covariance. Let’s dive into details.

### 2.** ****Implied Excess Equilibrium Returns**

**Implied Excess Equilibrium Returns**

The starting point of BL model is the returns as derived by the CAPM market portfolio.

*r _{f }the risk* free rate.

*r _{m} The excess return of the market portfolio.*

**** a regression coefficient

**** The residual, or asset specific (idiosyncratic) excess return

Then a reverse optimization method is used to calculate the vector of implied excess equilibrium returns (prior) **Π**.

*Π the Implied Excess Equilibrium Return Vector*

*λ the risk aversion coefficient*

*Σ the covariance matrix of excess returns*

*w _{mkt} the market capitalization weight of the assets*

### 3. The Black-Litterman Formula

*E[R ] the new (posterior) Combined Return Vector*

*Σ the covariance matrix of excess returns*

*P matrix that identifies the assets involved in the views *

*τ scalar*

*E[R ] the new (posterior) Combined Return Vector*

*Π the Implied Equilibrium Return Vector*

*Ω diagonal covariance matrix of error terms from the expressed views(uncertainty in each view)Q the View Vector*

The covariance matrix of the error term ( Ω ) has the following form:

The assumption here is that the variance in the view errors will be proportional to prior variance and thus can be estimated only from the prior.

**More analytically the steps of the procedure are as follows:**

- For each view (k), calculate the New Combined Return Vector E[R] using the Black-Litterman formula under 100% certainty.
- Calculate the weight vector based on 100% confidence w
_{k}. - Get the weight tilts by multiplying the confidence vector with the pair-wise difference from the market capitalization weights.
- Estimate the target weights adding the tilts.
- Finally we solve for w
_{k}that optimize for uncertainty. - Repeat process to derive the K x K diagonal Ω matrix.

## Sector Allocation under Black-Litterman model

In this example we will compare the allocations suggested by **MV** (Mean-Variance Optimization) and **BL** before **COVID** shocked the markets.

First we select 10 sector ETFS as our investment universe. The historical data used for the optimization are from 2010-01-01 to 2021-02-01.

**Investment Universe**

- XLC : Communication Services Select Sector SPDR
- XLP : Consumer Staples Select Sector SPDR
- XLY : Consumer Discretionary Select Sector SPDR
- XLE : Energy Select Sector SPDR
- XLF : Financial Select Sector SPDR
- XLV : Health Care Select Sector SPDR
- XLI : Industrial Select Sector SPDR
- XLRE : Real Estate Select Sector SPDR
- XLK : Technology Select Sector SPDR
- XLU : Utilities Select Sector SPDR

**Views****on** **sector outlook **

Now it’s time to introduce our own views for the US economy. We will base our sector outlook for the US market on the Business Cycle Framework according to Fidelity Research. The graph below shows the current economy state as it was presented by Fidelity for the Q2 of 2021.

Based on the suggested sector overweight-underweight recommendations, we will be adjusting our expected sector returns by adding or subtracting a standard deviation from the historical mean. For example, **++** suggest expected sector return equal to historical return plus 2 standard deviations.

According to the report US economy is the Rebound state (Early cycle) and there are 6 sectors suggested as overweight. In that case we set the maximum asset weight to 20% to allow the model allocate weight to all overweighted sectors. Thus providing an allocation closer to our expectations.

Moreover, we construct our confidence vector using random values between 0.3 and 0.9. This is because we do not have any insight regarding to the conviction of the sector weights provided by Fidelity. Below we see how the historical returns compare with our views, the CPM implied returns and the BL posterior returns (with and without taking the confidence input into account).

### Analyse the results

The BL allocation overweights the **real estate** and **communication services** sector. BL allocates weight to all sectors. In contrast, the mean-variance optimization overweights the **consumer staples** and **health care** sector.

**BL Expected annual return: 5.1% v**s** MV Expected annual return: 9.7%**

**BL Sharpe Ratio: 0.30 **vs** MV Sharpe Ratio: 0.52**

**BL Annual volatility: 10.3% **vs** MV Annual volatility: 14.8%**

As we see the attractiveness of the **BL** model stems from its convenience as an analytical tool rather than effective asset allocation. **In the end all comes down to the accuracy of the expectations and assumptions used in the optimization.**

## References

**[1]** Idzorek, Thomas. (2004). A step-by-step guide to the Black-Litterman model: Incorporating user-specified confidence levels**[2]** Walters, Jay and Walters, Jay. (2014). The Black-Litterman Model in Detail**[3]** https://institutional.fidelity.com/app/literature/item/9901406.html