In last week’s commentary, we outline a number of problems faced by tactical asset allocators in actually implementing their views.

This week, we explore popular methods for translating a combination of strategic views and tactical views into a single, comprehensive set of views that can be used as the foundation of portfolio construction.

We explore Black-Litterman, which can be used to implement views on returns as well as the more recently introduced Entropy Pooling methodology of Meucci, which allows for more flexible views.

For practitioners looking to implement tactical views into a number of portfolios in a coherent manner, the creation of posterior capital market assumptions via these methods may be an attractive process.

Note: Last week’s commentary was fairly qualitative – and hopefully applicable for practitioners and non-practitioners alike. This week’s is going to be a bit wonkier and is primarily aimed at those looking to express tactical views in an asset allocation framework. We’ll try to keep the equations to a minimum, but if the question, “how do I create a posterior joint return distribution from a prior and a rank view of expected asset class returns?” has never crossed your mind, this might be a good week to skip.

In last week’s commentary, we touched upon some of the important details that can make the actual implementation and management of tactical asset allocation a difficult proposition.[1] Specifically, we noted that:

Establishing consistent measures across assets is hard (e.g. “what is fair value for a bond index and how does it compare to equities?”);

There often are fewer bets being made, so position sizing is critical;

Cross-asset dynamics create changing risk profiles for bets placed.

Tactical decisions often explicitly forego diversification, increasing the hurdle rate.

We’ll even add a fifth, sixth, and seventh:

Many attractive style premia (e.g. momentum, value, carry, and trend) trades require leverage or shorting. Many other tactical views (e.g. change in yield curve curvature or change in credit spreads) can require leverage and shorting to neutralize latent risk factors and allocate risk properly.

Combining (potentially conflicting) tactical views is not always straight forward.

Incorporating tactical views into a preexisting policy portfolio – which may include long-term strategic views or constraints – is not obvious.

This week, we want to address how points #2-7 can be addressed with a single comprehensive framework.[2]

What is Tactical Asset Allocation?

As we hinted in last week’s commentary, we’re currently smack dab in the middle of writing a book on systematic tactical asset allocation.

When we sat down to write, we thought we’d start at an obvious beginning: defining “what is tactical asset allocation?”

Or, at least, that was the plan.

As soon as we sat down to write, we got a case of serious writer’s block. Which, candidly, gave us deep pause. After all, if we struggled to even write down a succinct definition for what tactical asset allocation is, how in the world are we qualified to write a book about it?

Fortunately, we were eventually able to put digital ink to digital paper. While our editor would not let us get away with a two sentence chapter, our thesis can be more or less boiled down to:

Strategic asset allocation is the policy you would choose if you thought risk premia were constant; tactical asset allocation is the changes you would make if you believe risk premia are time-varying.[3]

We bring this up because it provides us a mental framework for thinking about how to address problems #2 – 7.

Specifically, given prior market views (e.g. expected returns and covariances) that serve as the foundation to our strategic asset allocation, can our tactical views be used to create a posterior view that can then serve as the basis of our portfolio construction process?

Enter Black-Litterman

Fortunately, we’re not the first to consider this question. We missed that boat by about 27 years or so.

In 1990, Fischer Black and Robert Litterman developed the Black-Litterman model while working at Goldman Sachs. The model provides asset allocators with a framework to embed opinions and views about asset class returns into a prior set of return assumptions to arrive at a bespoke asset allocation.

Part of what makes the Black-Litterman model unique is that it does not ask the allocator to necessarily come up with a prior set of expected returns. Rather, it relies on equilibrium returns – or the “market clearing returns” – that serve as a neutral starting point. To find these returns, a reverse optimization method is utilized.

Here, R is our set of equilibrium returns, c is a risk aversion coefficient, S is the covariance matrix of assets, and w is the market-capitalization weights of those assets.

The notion is that in the absence of explicit views, investors should hold the market-capitalization weighted portfolio (or the “market portfolio”). Hence, the return views implied by the market-capitalization weights should be our starting point.

Going about actually calculating the global market portfolio weights is no small feat. Plenty of ink has been spilled on the topic.[4] For the sake of brevity, we’re going to conveniently ignore this step and just assume we have a starting set of expected returns.

The idea behind Black-Litterman is to then use a Bayesian approach to combine our subjective views with these prior equilibrium views to create a posterior set of capital market assumptions.

Specifically, Black-Litterman gives us the flexibility to define:

Absolute asset class return views (e.g. “I expect U.S. equities to return 4%”)

Relative asset class return views (e.g. “I expect international equities to outperform U.S. equities by 2%”)

The confidence in our views

Implementing Black-Litterman

We implement the Black-Litterman approach by constructing a number of special matrices.

P: Our “pick matrix.” Each row tells us which asset classes we are expressing a view on. We can think of each row as a portfolio.

Q: Our “view vector.” Each row tells us what our return view is for the corresponding row in the pick matrix.

O: Our “error matrix.” A diagonal matrix that represents the uncertainty in each of our views.

Given these matrices, our posterior set of expected returns is:

If you don’t know matrix math, this might be a bit daunting.

At the highest level, our results will be a weighted average of our prior expected returns (R) and our views (Q). How do compute the weights? Let’s walk through it.

t is a scalar. Generally, small. We’ll come back to this in a moment.

S is the prior covariance matrix. Now, the covariance matrix represents the scale of our return distribution: i.e. how far away from the expectation that we believe our realized returns could fall. What we need, however, is some measure of uncertainty of our actual expected returns. g. If our extracted equilibrium expected returns for stocks is 5%, how certain are we it isn’t actually supposed to be 4.9% or 5.1%? This is where t comes back. We use a small t (generally between 0.01 and 0.05) to scale S to create our uncertainty estimate around the expected return. (tS)^{-1}, therefore, is our certainty, or confidence, in our prior equilibrium returns.

If O is the uncertainty in our view on that portfolio, O^{-1} can be thought of as our certainty, or confidence, in each view. Each row of P is the portfolio corresponding to our view. P’O^{-1}P, therefore, can be thought of as the transformation that turns view uncertainty into asset class return certainty.

Using our prior intuition of (tS)^{-1}, (tS)^{-1}R can be thought of as certainty-scaled prior expected returns.

Q represents our views (a vector of returns). O^{-1}Q, therefore, can be thought of as certainty-scaled P’O^{-1}Q takes each certainty-scaled view and translates it into cumulative asset-class views, scaled for the certainty of each view.

With this interpretation, the second term – (tS)^{-1}R + P’O^{-1}Q – is a weighted average of our prior expected returns and our views. The problem is that we need the sum of the weights to be equal to 1. To achieve this, we need to normalize.

That’s where the first term comes in. (tS)^{-1} + P’O^{-1}P is the sum of our weights. Multiplying the second term by ((tS)^{-1} + P’O^{-1}P)^{-1} is effectively like dividing by the sum of weights, which normalizes our values.

Similar math has been derived for the posterior covariance matrix as well, but for the sake of brevity, we’re going to skip it. A Step- by-Step Guide to Black-Litterman by Thomas Idzorek is an excellent resource for those looking for a deeper dive.

Black-Litterman as a Solution to Tactical Asset Allocation Problems

So how does Black-Litterman help us address problems #2-7 with tactical asset allocation?

Let’s consider a very simple example. Let’s assume we want to build a long-only bond portfolio blending short-, intermediate-, and long-term bonds.

For convenience, we’re going to make a number of assumptions:

Constant durations of 2, 5, and 10 for each of the bond portfolios.

Use current yield-to-worst of SHY, IEI, and IEF ETFs as forward expected returns. Use prior 60 months of returns to construct the covariance matrix.

This gives us a prior expected return of:

E[R]

SHY

1.38%

IEI

1.85%

IEF

2.26%

And a prior covariance matrix,

SHY

IEI

IEF

SHY

0.00005

0.000177

0.000297

IEI

0.000177

0.000799

0.001448

IEF

0.000297

0.001448

0.002795

In this example, we want to express a view that the curvature of the yield curve is going to change. We define the curvature as:

Increasing curvature implies the 5-year rate will go up and/or the 2-year and 10-year rates will go down. Decreasing curvature implies the opposite.

To implement this trade with bonds, however, we want to neutralize duration exposure to limit our exposure to changes in yield curve level and slope. The portfolio we will use to implement our curvature views is the following:

We also need to note that bond returns have an inverse relationship with rate change. Thus, to implement an increasing curvature trade, we would want to short the 5-year bond and go long the 2- and 10-year bonds.

Let’s now assume we have a view that the curvature of the yield curve is going to increase by 50bps over the next year. We take no specific view as to how this curvature increase will unfold (i.e. the 5-year rate rising by 50bps, the 5-year rate rising by 25bps and each of the 2-year and 10-year rates falling by 25bps, etc.). This implies that the curvature bond portfolio return has an expected return of negative 5%.

Implementing this trade in the Black-Litterman framework, and assuming a 50% certainty of our trade, we end up with a posterior distribution of:

E[R]

SHY

1.34%

IEI

1.68%

IEF

1.97%

And a posterior ovariance matrix,

SHY

IEI

IEF

SHY

0.000049

0.000182

0.000304

IEI

0.000182

0.000819

0.001483

IEF

0.000304

0.001483

0.002864

We can see that while the expected return for SHY did not change much, the expected return for IEF dropped by 0.29%.

The use of this model, then, is that we can explicitly use views about trades we might not be able to make (due to leverage or shorting constraints) to alter our capital market assumptions, and then use our capital market assumptions to build our portfolio.

For global tactical style premia – like value, momentum, carry, and trend – we need to explicitly implement the trades. With Black-Litterman, we can implement them as views, create a posterior return distribution, and use that distribution to create a portfolio that still satisfies our policy constraints.

The Limitations of Black-Litterman

Black-Litterman is a hugely powerful tool. It does, however, have a number of limitations. Most glaringly,

Returns are assumed to be normally distributed.

Expressed views can only be on returns.

To highlight the latter limitation, consider a momentum portfolio that ranks asset classes based on prior returns. The expectation with such a strategy is that each asset class will outperform the asset class ranked below it. A rank view, however, is inexpressible in a Black-Litterman framework.

Enter Flexible Views with Entropy Pooling

While a massive step forward for those looking to incorporate a variety of views, the Black-Litterman approach remains limited.

In a paper titled Fully Flexible Views: Theory and Practice[5], Attilio Meucci introduced the idea of leveraging entropy pooling to incorporate almost any view a practitioner could imagine. Some examples include,

A prior that need not be normally distributed – or even be returns at all.

Non-linear functions and factors.

Views on the return distribution, expected returns, median returns, return ranks, volatilities, correlations, and even tail behavior.

Sounds great! How does it work?

The basic concept is to use the prior distribution to create a large number of simulations. By definition, each of these simulations occurs with equal probability.

The probability of each scenario is then adjusted such that all views are satisfied. As there may be a number of such solutions, the optimal solution is the one that minimizes the relative entropy between the new distribution and the prior distribution.

How is this helpful? Consider the rank problem we discussed in the last section. To implement this with Meucci’s entropy pooling, we merely need to adjust the probabilities until the following view is satisfied:

Again, our views need not be returns based. For example, we could say that we believe the volatility of asset A will be higher than asset B. We would then just adjust the probabilities of the simulations until that is the case.

Of course, the accuracy of our solution will depend on whether we have enough simulations to accurately capture the distribution. A naïve numerical implementation that seeks to optimize over the probabilities would be intractable. Fortunately, Meucci shows that the problem can be re-written such that the number of variables is equal to the number of views.[6]

A Simple Entropy Pooling Example

To see entropy-pooling in play, let’s consider a simple example. We’re going to use J.P. Morgan’s 2017 capital market assumptions as our inputs.

In this toy example, we’re going to have the following view: we expect high yield bonds to outperform US small-caps, US small-caps to outperform intermediate-term US Treasuries, intermediate-term US Treasuries will outperform REITs, and REITs will outperform gold. Exactly how much we expect them to outperform by is unknown. So, this is a rank view.

We will also assume that we are 100% confident in our view.

The prior, and resulting posterior expected returns are plotted below.

We can see that our rank views were respected in the posterior. That said, since the optimizer seeks a posterior that is as “close” as possible to the prior, we find that the expected returns of intermediate-term US Treasuries, REITs, and gold are all equal at 3%.

Nevertheless, we can see how our views altered the structure of other expected returns. For example, our view on US small-caps significantly altered the expected returns of other equity exposures. Furthermore, for high yield to outperform US small-caps, asset class expectations were lowered across the board.

Conclusion

Tactical views in multi-asset portfolios can be difficult to implement for a variety of reasons. In this commentary, we show how methods like Black-Litterman and Entropy Pooling can be utilized by asset allocators to express a variety of views and incorporate these views in a cohesive manner.

Once the views have been translated back into capital market assumptions, these assumptions can be leveraged to construct a variety of portfolios based upon policy constraints. In this manner, the same tactical views can be embedded consistently across a variety of portfolios while still acknowledging the unique objectives of each portfolio constructed.

[2] For clarity, we’re using “addressed” here in the loose sense of the word. As in, “this is one potential solution to the problem.” As is frequently the case, the solution comes with its own set of assumptions and embedded problems. As always, there is no holy grail.

[3] By risk premia, we mean things like the Equity Risk Premium, the Bond Risk Premium (i.e. the Term Premium), the Credit Risk Premium, the Liquidity Risk Premium, et cetera. Active Premia – like relative value – confuse this notion a bit, so we’re going to conveniently ignore them for this discussion.

## Combining Tactical Views with Black-Litterman and Entropy Pooling

By Corey Hoffstein

On July 17, 2017

In Portfolio Construction, Weekly Commentary

This post is available as a PDF download here.SummaryNote: Last week’s commentary was fairly qualitative – and hopefully applicable for practitioners and non-practitioners alike. This week’s is going to be a bit wonkier and is primarily aimed at those looking to express tactical views in an asset allocation framework. We’ll try to keep the equations to a minimum, but if the question, “how do I create a posterior joint return distribution from a prior and a rank view of expected asset class returns?” has never crossed your mind, this might be a good week to skip.In last week’s commentary, we touched upon some of the important details that can make the actual implementation and management of tactical asset allocation a difficult proposition.[1] Specifically, we noted that:

We’ll even add a fifth, sixth, and seventh:

This week, we want to address how points #2-7 can be addressed with a single comprehensive framework.[2]

What is Tactical Asset Allocation?As we hinted in last week’s commentary, we’re currently smack dab in the middle of writing a book on systematic tactical asset allocation.

When we sat down to write, we thought we’d start at an obvious beginning: defining “what is tactical asset allocation?”

Or, at least, that was the plan.

As soon as we sat down to write, we got a case of serious writer’s block. Which, candidly, gave us deep pause. After all, if we struggled to even write down a succinct definition for what tactical asset allocation

is,how in the world are we qualified to write a book about it?Fortunately, we were eventually able to put digital ink to digital paper. While our editor would not let us get away with a two sentence chapter, our thesis can be more or less boiled down to:

We bring this up because it provides us a mental framework for thinking about how to address problems #2 – 7.

Specifically, given prior market views (e.g. expected returns and covariances) that serve as the foundation to our strategic asset allocation, can our tactical views be used to create a

posteriorview that can then serve as the basis of our portfolio construction process?Enter Black-LittermanFortunately, we’re not the first to consider this question. We missed that boat by about 27 years or so.

In 1990, Fischer Black and Robert Litterman developed the Black-Litterman model while working at Goldman Sachs. The model provides asset allocators with a framework to embed opinions and views about asset class returns into a prior set of return assumptions to arrive at a bespoke asset allocation.

Part of what makes the Black-Litterman model unique is that it does not ask the allocator to necessarily come up with a prior set of expected returns. Rather, it relies on equilibrium returns – or the “market clearing returns” – that serve as a neutral starting point. To find these returns, a reverse optimization method is utilized.

Here, R is our set of equilibrium returns, c is a risk aversion coefficient, S is the covariance matrix of assets, and w is the market-capitalization weights of those assets.

The notion is that in the absence of explicit views, investors should hold the market-capitalization weighted portfolio (or the “market portfolio”). Hence, the return views implied by the market-capitalization weights should be our starting point.

Going about actually calculating the global market portfolio weights is no small feat. Plenty of ink has been spilled on the topic.[4] For the sake of brevity, we’re going to conveniently ignore this step and just assume we have a starting set of expected returns.

The idea behind Black-Litterman is to then use a Bayesian approach to combine our subjective views with these prior equilibrium views to create a posterior set of capital market assumptions.

Specifically, Black-Litterman gives us the flexibility to define:

Implementing Black-LittermanWe implement the Black-Litterman approach by constructing a number of special matrices.

Given these matrices, our posterior set of expected returns is:

If you don’t know matrix math, this might be a bit daunting.

At the highest level, our results will be a weighted average of our prior expected returns (R) and our views (Q). How do compute the weights? Let’s walk through it.

^{-1}, therefore, is ourcertainty,or confidence, in our prior equilibrium returns.^{-1}can be thought of as our certainty, or confidence, in each view.Each row of P is the portfolio corresponding to our view. P’O

^{-1}P, therefore, can be thought of as the transformation that turns view uncertainty into asset class return certainty.^{-1}, (tS)^{-1}R can be thought of as certainty-scaled prior expected returns.^{-1}Q, therefore, can be thought of ascertainty-scaledP’O^{-1}Q takes each certainty-scaled view and translates it into cumulative asset-class views, scaled for the certainty of each view.With this interpretation, the second term – (tS)

^{-1}R + P’O^{-1}Q – is a weighted average of our prior expected returns and our views. The problem is that we need the sum of the weights to be equal to 1. To achieve this, we need to normalize.That’s where the first term comes in. (tS)

^{-1}+ P’O^{-1}P is the sum of our weights. Multiplying the second term by ((tS)^{-1}+ P’O^{-1}P)^{-1}is effectively like dividing by the sum of weights, which normalizes our values.Similar math has been derived for the posterior covariance matrix as well, but for the sake of brevity, we’re going to skip it.

A Step- by-Step Guide to Black-Littermanby Thomas Idzorek is an excellent resource for those looking for a deeper dive.Black-Litterman as a Solution to Tactical Asset Allocation ProblemsSo how does Black-Litterman help us address problems #2-7 with tactical asset allocation?

Let’s consider a very simple example. Let’s assume we want to build a long-only bond portfolio blending short-, intermediate-, and long-term bonds.

For convenience, we’re going to make a number of assumptions:

This gives us a prior expected return of:

And a prior covariance matrix,

In this example, we want to express a view that the

curvatureof the yield curve is going to change. We define the curvature as:Increasing curvature implies the 5-year rate will go up and/or the 2-year and 10-year rates will go down. Decreasing curvature implies the opposite.

To implement this trade with bonds, however, we want to neutralize duration exposure to limit our exposure to changes in yield curve level and slope. The portfolio we will use to implement our curvature views is the following:

We also need to note that bond returns have an inverse relationship with rate change. Thus, to implement an increasing curvature trade, we would want to

shortthe 5-year bond and go long the 2- and 10-year bonds.Let’s now assume we have a view that the curvature of the yield curve is going to increase by 50bps over the next year. We take no specific view as to how this curvature increase will unfold (i.e. the 5-year rate rising by 50bps, the 5-year rate rising by 25bps and each of the 2-year and 10-year rates falling by 25bps, etc.). This implies that the curvature bond portfolio return has an expected return of

negative5%.Implementing this trade in the Black-Litterman framework, and assuming a 50% certainty of our trade, we end up with a posterior distribution of:

And a posterior ovariance matrix,

We can see that while the expected return for SHY did not change much, the expected return for IEF dropped by 0.29%.

The use of this model, then, is that we can explicitly use views about trades we might not be able to make (due to leverage or shorting constraints) to alter our capital market assumptions, and then use our capital market assumptions to build our portfolio.

For global tactical style premia – like value, momentum, carry, and trend – we need to explicitly implement the trades. With Black-Litterman, we can implement them as views, create a posterior return distribution, and use that distribution to create a portfolio that still satisfies our policy constraints.

The Limitations of Black-LittermanBlack-Litterman is a hugely powerful tool. It does, however, have a number of limitations. Most glaringly,

To highlight the latter limitation, consider a momentum portfolio that ranks asset classes based on prior returns. The expectation with such a strategy is that each asset class will outperform the asset class ranked below it. A rank view, however, is inexpressible in a Black-Litterman framework.

Enter Flexible Views with Entropy PoolingWhile a massive step forward for those looking to incorporate a variety of views, the Black-Litterman approach remains limited.

In a paper titled

Fully Flexible Views: Theory and Practice, Attilio Meucci introduced the idea of leveraging entropy pooling to incorporate almost any view a practitioner could imagine. Some examples include,[5]Sounds great! How does it work?

The basic concept is to use the prior distribution to create a large number of simulations. By definition, each of these simulations occurs with equal probability.

The

probability of each scenariois then adjusted such that all views are satisfied. As there may be a number of such solutions, the optimal solution is the one that minimizes the relative entropy between the new distribution and the prior distribution.How is this helpful? Consider the rank problem we discussed in the last section. To implement this with Meucci’s entropy pooling, we merely need to adjust the probabilities until the following view is satisfied:

Again, our views need not be returns based. For example, we could say that we believe the volatility of asset A will be higher than asset B. We would then just adjust the probabilities of the simulations until that is the case.

Of course, the accuracy of our solution will depend on whether we have enough simulations to accurately capture the distribution. A naïve numerical implementation that seeks to optimize over the probabilities would be intractable. Fortunately, Meucci shows that the problem can be re-written such that the number of variables is equal to the number of views.[6]

A Simple Entropy Pooling ExampleTo see entropy-pooling in play, let’s consider a simple example. We’re going to use J.P. Morgan’s 2017 capital market assumptions as our inputs.

In this toy example, we’re going to have the following view: we expect high yield bonds to outperform US small-caps, US small-caps to outperform intermediate-term US Treasuries, intermediate-term US Treasuries will outperform REITs, and REITs will outperform gold. Exactly how much we expect them to outperform by is unknown. So, this is a

rank view.We will also assume that we are 100% confident in our view.

The prior, and resulting posterior expected returns are plotted below.

We can see that our rank views were respected in the posterior. That said, since the optimizer seeks a posterior that is as “close” as possible to the prior, we find that the expected returns of intermediate-term US Treasuries, REITs, and gold are all equal at 3%.

Nevertheless, we can see how our views altered the structure of other expected returns. For example, our view on US small-caps significantly altered the expected returns of other equity exposures. Furthermore, for high yield to outperform US small-caps, asset class expectations were lowered across the board.

ConclusionTactical views in multi-asset portfolios can be difficult to implement for a variety of reasons. In this commentary, we show how methods like Black-Litterman and Entropy Pooling can be utilized by asset allocators to express a variety of views and incorporate these views in a cohesive manner.

Once the views have been translated back into capital market assumptions, these assumptions can be leveraged to construct a variety of portfolios based upon policy constraints. In this manner, the same tactical views can be embedded consistently across a variety of portfolios while still acknowledging the unique objectives of each portfolio constructed.

[1] https://blog.thinknewfound.com/2017/07/four-important-details-tactical-asset-allocation/

[2] For clarity, we’re using “addressed” here in the loose sense of the word. As in, “this is one potential solution to the problem.” As is frequently the case, the solution comes with its own set of assumptions and embedded problems. As always, there is no holy grail.

[3] By risk premia, we mean things like the Equity Risk Premium, the Bond Risk Premium (i.e. the Term Premium), the Credit Risk Premium, the Liquidity Risk Premium, et cetera. Active Premia – like relative value – confuse this notion a bit, so we’re going to conveniently ignore them for this discussion.

[4] For example, see: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2352932

[5] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1213325

[6] Those looking to implement can find Meucci’s MatLab code (https://www.mathworks.com/matlabcentral/fileexchange/21307-fully-flexible-views-and-stress-testing) and public R code (https://r-forge.r-project.org/scm/viewvc.php/pkg/Meucci/R/EntropyProg.R?view=markup&root=returnanalytics) available. We have a Python version we can likely open-source if there is enough interest.