Modeling Black-Litterman; Part 2 - Incorporating Manager Views

 

"The 'radical' of one century is the 'conservative' of the next."

-Mark Twain

In this series, I'm going to explore some of the advances in portfolio management, construction, and modeling since the advent of Harry Markowitz's Nobel Prize winning Modern Portfolio Theory (MPT) in 1952.

MPT's mean-variance optimization approach shaped theoretical asset allocation models for decades after its introduction.  However, the theory failed to become an accepted industry practice, so we'll explore why that is and what advances have developed in recent years to address the shortcomings of the original model.

The Black-Litterman Formula

The Black-Litterman formula incorporates two distinct inputs; the first is the Implied Equilibrium Return Vector we constructed in Part 1, the second is a series of vectors and matrices that incorporate a manager's views/forecasts of the market.  The product of the formula is an updated Combined Expected Excess Return vector.

Here's the Black-Litterman Forumula:

where:

One Asset Absolute Example

The formula certainly looks daunting and perhaps the best way to begin to understand it is to work through a simple - one asset class - example to begin to see how the variables are working together.

For this example, we'll use the energy sector ETF (XLE) from Part 1.

Let's say that, due to the high energy prices of late, we expect XLE to produce excess monthly returns of 2% instead of its equilibrium implied return of 0.82%.  Additionally, we have a high level of confidence about this forecast, so we'll assign a low uncertainty parameter of 0.25%.  Lastly, for this forecast, we'll set the risk scalar equal to 1 and also assign a value of 1 to the link matrix.

XLE has:

• Pi-Value:         0.82%
• Variance:         0.636%
• Q-Value:          2%
• Omega:            0.25%
• Tau:                  1
• P-Value:           1

The produces the following:

(click on image to enlarge)

The formula returns a new monthly expected excess return of 1.67%.  By combining the implied equilibrium excess return and our view of sector performance, the formula calculates an average of the two forecasts weighted by the uncertainty variable, Omega.

In looking at the second half of the formula, we can see this weighted average process...

• The 'Pi' and 'Q' variables represent the two expected excess return inputs

• The risk-scaled variance, the 'P' link matrix, and Omega matrix are measures of confidence that serve as the weights for the weighted average

The first term of the formula serves to balance out the weights in the second term to make sure they sum to 1.

Portfolio Level Example with One Manager View

Now, let's look at what happens when a portfolio manager wants to express a relative performance view in the portfolio construction process.

In the last example, we expressed an absolute view in stating that we expected the XLE monthly excess returns to be 2%... absolute views can be used for instances of specific catalysts or when an asset is expected to meet a target valuation.  

However, relative views would be more useful for portfolios whose performance is measured against a benchmark.  Portfolio managers of benchmarked portfolios will express their views by overweighting favored assets and underweighting out of favor assets that are part of their benchmark.

Building on our previous XLE example, let's say that we are managing a benchmarked portfolio and we expect the inflationary impacts to benefit some asset classes and hurt others.  We might expect it to help sectors that enjoy price elasticity like energy companies, real estate, and commodities and hurt fixed income and cyclical assets.  

Let's assume that the sectors benefited will outperform the sectors hurt by a total of 5% (Q-value) of expected excess return a month and, again, we're quite confident in our forecast so we'll assign a low Omega value of 0.125%.

To communicate our forecasts to the model, we now have to incorporate the link matrix; but when only using one forecast/scenario, it will be a link vector.

(click on image to enlarge)

When constructing a link matrix, each row represents a separate manager view or forecast.  The row must sum to zero, all the 'positive' views must add to +1, and all the 'negative' views must add to -1.  It's a zero-sum game.

You'll notice that we haven't distributed our weights equally in the vector.  For the positive views, we expect this bout of inflation to most benefit energy, so we allocated +0.6 to XLE, followed by real estate with +0.3, and lastly commodities with +0.1.  

For the negative views, we are projecting the largest impacts will be on cyclical assets, small caps (IWM) and high yield debt (HYG) while we expect investment grade fixed income assets to be hurt less.  You'll notice that HYG gets a bit of a double whammy here as it's both a cyclical and fixed income asset class.

After feeding all of our variables through the formula, we get an updated vector of combined expected excess returns:

As expected, we see increases in expected excess return for all of our 'positive' view asset classes.  However, there were some odd changes to the combined 'negative' view returns.... most notably, the expected return for small caps increased from 10.5% to 11.33%.  

Why did this happen?

This sector had our second largest negative weight allocation in the link vector.  Honestly, I'm not sure just yet but I suspect it's related to the weighted average calculation in the formula.

When we use the combined returns in a mean-variance optimization to create a 10% vol portfolio, here are the updated asset weights versus the implied equilibrium excess returns:

(click on image to enlarge)

You'll notice, the allocations for all three of our 'positive' view asset classes increased (DBC, VNQ, and XLE).  Among the 'negative' view assets (IWM, HYG, LQD, TSY, and MBB), only the allocation to MBS increased but that's for the sake of risk reduction as it's a diversifying asset class.

A Zero-Sum Game

The last point to highlight is the key difference between having an absolute view and a relative view with regard to the levels of expected portfolio return.

If we were to include the absolute view of monthly excess return for XLE of 1.67% (which is 22% annual excess return), in a mean-variance optimization at the 10% risk profile, the level of expected excess return increases from 5.98% to 8.70%... almost a 50% increase.

However, when we incorporate relative views into Black-Litterman, overall levels are mostly unchanged... as we said before, relative views are a zero-sum game.

Here's the change in the efficient frontier from incorporating our relative view of inflation (original implied equilibrium excess returns is the blue line and updated relative views is the orange line)...

Under the relative views, the 10% risk profile portfolio's expected excess return only increases from 5.98% to 6.38% as the level of returns are mostly unchanged.

*References:     Microsoft Word - BL Draft with Graphs.doc (duke.edu)

                    Phil Davies - YouTube