AN INTRODUCTION TO THE BLACK &

LITTERMAN MODEL

THE BLACK & LITTERMAN MODEL We implement the Black & Litterman model because of some fundamental issues with Modern Portfolio Theory. Such as reliance on estimating anticipated future returns and the covariance of these returns from historical market data. This use of historical market data results in huge short and long positions in particular assets which is not viable in reality as there are many limitations on short sales. Alistair Spencer & Simon Long Finance 361 Modern Portfolio Theory

The Black & Litterman Model

Introduction Mega Bank Asset Management (MAM) has implemented the Black & Litterman (BL) model in order to gauge the most efficient weights to be used, in efforts to minimize the variation of returns for our client’s Global equity portfolio. The BL model enables us to incorporate our client’s opinion on US equity as well as avoiding the extreme allocations in short or long positions in particular assets that occur when using other portfolio optimization models such as the Markowitz optimization method which we will refer to as Modern Portfolio Theory (MPT).

Why Black & Litterman model We implement the BL model because of some fundamental issues with MPT. MPT relies on estimating anticipated future returns and the covariance of these returns from historical market data. This use of historical market data results in huge short and long positions in particular assets which is not viable in reality as there are many limitations on short sales Benninga, S (2013)i. The BL model is also preferred over MPT as MPT fails to incorporate the opinions of individual investors. This is not ideal for customizing a client’s portfolio to their particular needs and beliefs, for example our clients 0.5% alpha return expectation on the SPY index.

How does the BL model work The BL model can be broken into 2 parts:

Part 1: What does the market think?

To begin we assume that the investor chooses their portfolio from a given group of assets, as our client has done with indices SPY, EWJ, EZU, FXI and ILF. We then find the variances and covariances of these indices following these procedures.

1. Obtain the returns of our five chosen indices using available market data of prices over a given amount of time e.g. five years. Then find the variance of these historical returns.

2. Find the mean return of each of the indices and then demean each our historical returns from part 1) by taking the return of each less the mean of the total periods for each asset.

3. Calculate the variances and covariance between each of the assets using a variance co variance matrix (VCV). This is done my multiplying the demeaned returns matrix with the transpose of itself then dividing by the sum of the same calculation (the total variance and covariances). This provides the markets expectation surrounding the correlation of asset returns (variance and covariance) which is used for finding our final BL weights in part 2.

The portfolio chosen provides the efficient benchmark portfolio for the investor. This is the basis for the BL model; as it is very difficult to outperform a typical well diversified benchmark without additional information. This is opposite to the goal of MPT which tries to derive the efficient portfolio from a given set of assets. The implied asset returns from the well diversified benchmark is the starting point of our BL model.

Part 2: Incorporate investor opinion

Now that we have determined our efficient benchmark based on the choice of assets by the investor we can incorporate their opinion; which in this case is a 0.5% alpha on the US fund SPY. As each asset interacts with all other assets in the portfolio the investors opinion on the SPY fund impacts our expectation of the returns of all the other funds.

1. We multiply our VCV by the transpose of our benchmark weights derived from the percentage world GDP each index represents plus the risk free Tbill rate. When summed this gives us the expected return for our portfolio without any normalization factor i.e. what we expect the return on the efficient portfolio to be without additional information.

2. We can now incorporate MAM’s 4%/12 expectation of the global benchmark portfolio. We can use this expectation to normalize the returns. To achieve this we multiply our VCV by the transpose of our benchmark weights (the percentage world GDP each index represents) and multiply it by the expected portfolio return less the Tbill rate, divided by the non-normalized expected return found in step 1.

3. We then determine our tracking factors. Tracking factors provide sensitivities for the posterior estimates to various views Braga MD, Natale FP (2007)ii i.e. how much the analyst’s opinion delta on the SPY index is going to affect the other indices. We do this by dividing each VCV by the transposed variance of each index.

4. Using the analyst’s opinion delta of 0.5%/12 we can calculate the opinion adjusted returns for each of the indices. We take our normalized expected returns from step 2 and add the multiplied transpose of the tracking factors and analyst opinions. This effectively applies the analyst’s opinion to the normalized returns for each index.

5. Finally, we optimize the benchmark proportions by finding the weights of each assets in order to minimize variance. This involves multiplying two matrices 1) inverse of the VCV and 2) the opinion adjusted returns less the risk free rate (the Tbill rate). This effectively gives us the variance of each asset and the covariance of the asset with all other assets in the portfolio. We then divide each of these figures by the sum of all variances and covariance’s in the portfolio which yields the optimal weighting of each of the indices in order to minimize variance of the portfolio. The fundamental difference between the MPT and BL being that we have used the opinion adjusted returns found in step 4 instead of the mean return data from the historical returns. This provides an optimal portfolio of the assets SPY, EWJ, EZU, FXI and ILF.

By following steps 1-5 we achieve a set of weights that incorporates both MAM’s opinion of 4% expected return on the benchmark and our client’s opinion of 0.5% alpha on SPY’s expected return. Not only do they incorporate opinions the final weight’s ranging from 8.59% (ILF) to 43.79% (SPY) are achievable long positions without any need to short stock.

Why MPT fails: If we were to implement MPT we would calculate what the optimal weightings in each of the assets based on the VCV and the historical mean return data instead of the opinion adjusted returns. In our case this yielded a portfolio which included short positions in three of the indices and a 223.3% long position, which as previously discussed is difficult to implement in reality.

Shrinkage: an ineffective solution Instead of using BL some researchers have suggested implementation of a shrunk VCV matrix which is weighting (shrinkage) applied to the VCV to eliminate extreme portfolio weights. However, as expressed by Benninga, S (2013)iii negative historical returns when used to predict future expected returns will always produce a short in one of the given assets.

Are we confident in our opinion According to Theil (1971)iv, we could potentially add a degree of confidence to our analyst delta on the SPY index however this was not required as MAM was confident enough in the opinion for it to be irrelevant within our model. Additionally, we have made several assumptions that impact our confidence in the optimal BL portfolio. The main assumption is that the efficient market hypothesis holds, which means that the diversified benchmark is not outperformed without additional information i.e. all securities are fairly priced. Furthermore, the anticipated future returns are correlated with historical returns. This is important as historical demeaned returns are used in the generation of our VCV which is key in determining the BL optimal weights. Finally we have derived monthly returns simply by dividing the annual return by 12 where it is technically correct to use ((1+annual return)^12)-1.

Summary By implementing BL we have avoided the inconveniences that arrive from merely relying on market data to construct our optimal portfolio. We assume that the benchmark weights and the current Tbill rate are the more relevant predictors of anticipated expected returns. This enables us to incorporate analyst’s or investor opinions to derive optimal minimum variance portfolios Benninga, S (2013)v.

By Alistair Spencer & Simon Long

i Benninga, S (2013). Financial Modelling Third Edition, Chapter 13.2.1, The MIT Press Cambridge, Massachusetts London, England. ii Braga MD, Natale FP (2007). “TEV Sensitivity to Views in BlackLitterman Model”, 20th Australasian Finance & Banking Conference 2007 Paper, September, Available online at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1009635. iii Benninga, S (2013). Financial Modelling Third Edition, Chapter 13.2.3, The MIT Press Cambridge, Massachusetts London, England. iv Theil, Henri. 1971. Principles of Econometrics. New York: Wiley and Sons. v Benninga, S (2013). Financial Modelling Third Edition, Chapter 13.7, The MIT Press Cambridge, Massachusetts London, England.