|
Have you given
up on mean-variance optimization because the resulting asset allocations are unintuitive
and anything but diversified? Or perhaps you feel the need to use mean-variance
optimization coupled with tight optimization constraints? The inclusion of the
sophisticated Black-Litterman asset allocation model into AllocationADVISOR will
help you realize the benefits of mean-variance optimization.
Portfolios created
using the Black-Litterman approach and mean-variance optimization are well diversified
and intuitively reflect the investor’s own forecasts about future market
performance.
Harry Markowitz’s
mean-variance optimization is widely regarded as the holy grail of asset allocation.
Markowitz seminal work demonstrated how to form efficient or optimum portfolios
based on three inputs – returns, standard deviation, and correlations. His
work resulted in a Nobel Prize. Unfortunately, Markowitz never told us how to
derive the inputs, especially the estimated expected returns.
Unlike a number
of other Nobel Prize winning ideas, mean-variance optimization has not enjoyed
a high level of practitioner acceptance. This is because the Markowitz algorithm
is very powerful, perhaps too powerful for its own good. The algorithm is very
sensitive to the return forecasts, which have traditionally been created using
historical returns. If two asset classes are similar, but one has a slightly higher
forecasted return, the optimizer allocates everything to the asset with the higher
forecasted return and nothing to the other asset. Because of this input sensitivity,
mean-variance optimizers can lead to highly concentrated asset allocations that
contradict the common sense notion of diversification. The input sensitivity also
makes it difficult for investors to incorporate their own forecasts into a historical
model.
Many investors
have attempted to overcome these shortcomings in mean-variance optimization by
using tight constraints on the optimization. These artificial limits interfere
with the optimization in a way that renders the output sub-optimal. The tighter
the constraints, the further you move toward dictating the allocations to the
optimizer instead of optimizing the allocations. The Black-Litterman model, on
the other hand, creates better return forecasts so that it is not necessary to
constrain the optimization in order to create diversified portfolios. This allows
investors to harness the power of mean-variance optimization in a practical and
intuitive way.
In the first section
below we take a closer look at the poorly diversified asset allocations that typically
result from using purely historical inputs. Next we look at the solution, the
Black-Litterman model, and the well-diversified asset allocations that result
from using returns from the Black-Litterman model. In the second section below
we demonstrate why the Black-Litterman approach to incorporating investors’
forecasts of future market performance is superior to an ad-hoc approach.
Diversified
Portfolios
Investors have
traditionally used historical returns, standard deviations and correlations as
the inputs for mean-variance optimization. Figure 1 shows an efficient frontier
created using 9 years of historical data (the longest period available for this
set of indices) for the forecasts.
Figure 1: Efficient
Frontier Based on Black-Litterman Returns
While this appears
to be a perfectly usable efficient frontier, when you examine the allocations
of the portfolios along the frontier we see that they are poorly diversified.
Figure 2 compares five the asset allocations of five mixes corresponding to five
approximate standard deviation levels, 5%, 7.5%, 10%, 12.5%, and 15%, respectively.
Figure 2: Asset
Allocation Mixes Based on Historical Returns
 |
| Historical Returns lead to concentrated portfolios |
Of the eight available
asset classes, none of the “optimal” asset allocation mixes contain
more than two asset classes. Historical returns lead to poorly-diversified
asset allocations!
Recall that the
goal of mean-variance optimization is to capture the benefit of diversification
and to find asset allocations that maximize expected return for a given level
of risk. Fortunately, there is a solution, albeit a solution that emerged almost
50 years after the creation of mean-variance optimization. AllocationADVISOR 6.0
has a better tool for estimating expected returns – the Black-Litterman
model. The Black-Litterman approach tackles the weakest point of mean-variance
optimization—its sensitivity to the return forecasts. Black-Litterman uses
the historical standard deviations and correlations, values which tend to be stable
and make good forecasts, but develops better estimates of expected returns.
The foundation
of the Black-Litterman model is the Implied Returns. To calculate the Implied
Returns we define the market as the set of assets available to the investor. The
Implied Returns are calculated by assuming that the market is in equilibrium (supply
for the assets equals demand) and using reverse optimization to back out the returns
that would bring about this equilibrium. These calculations require three pieces
of information for the markets—the risk premium, covariance and market capitalizations.
If the investor
does not wish to add their own views to the forecasts, the Implied Returns are
the Black-Litterman forecasts and are used to create the efficient frontier. Figure
3 shows an efficient frontier created using Implied Returns.
Figure 3: Efficient
Frontier Based on Black-Litterman Returns
The superior diversification
of the asset allocation mixes that result from the Black-Litterman Implied Returns
is demonstrated in Figure 4.
Figure 4: Asset
Allocation Mixes Based on Black-Litterman Returns
 |
| Black-Litterman Implied Returns lead to diversified
portfolios |
Of the eight asset classes, all of the "optimal" asset allocation mixes contain at least six of the eight asset
classes.
“Customizing”
the Implied Returns with Views
As demonstrated above,
the Implied Returns are excellent forecasts for use with mean-variance optimization.
Investors, however, often have their own opinions about how the market is going
to behave in the future. These investors often want to adjust the Implied Returns
so that so that the forecasts better reflect their opinions, or “views,”
on future performance.
How should an investor’s
views be incorporated into the return forecasts? Why not just directly edit the
Implied Returns so that they reflect the views? As we demonstrate below, trying
to edit the Implied Returns directly leads once again to non-diversified portfolios.
The Black-Litterman model, on the other hand, gives you an intuitive way to incorporate
views without losing the advantage of diversification which comes from using the
Implied Returns.
For our examples
we will use two sample views:
View 1: US Large Cap Value will out perform US Large Cap Growth by 1%
View 2: US Small Cap will have a return of 11.5%
These Views can
be better understood by comparing them to the Implied Returns. The Implied Returns
forecast that US Large Cap Value will under perform US Large Cap Growth. View
1, then, is a bullish relative view on US Large Cap Value relative to US Large
Cap Growth. View 2 reflects the investor’s bullish view on US Small Cap.
Why? Because the View return of 11.5% is greater than the Implied Return for US
Small Cap (the weighted average of US Small Cap Growth and US Small Cap Value)
which is 10.92%.
Let’s look
at one possible ad-hoc approach to implementing these views as an example.
For View 1, we could increase the return of US Large Cap Value and simultaneously
decrease the return of US Large Cap Growth until the difference is 1%. And for
View 2, we could increase the return of the two components of US Small Cap so
that the weighted average return equals 11.5%. Figure 5 shows the
resulting efficient frontier and mixes.
Figure 5: Efficient
Frontier and Mixes (Ad-hoc Approach)
 |
| Ad-hoc approaches to incorporating views lead to concentrated portfolios |
We can see from
the allocations in Figure 5 that even though we started with the Black-Litterman
Implied Returns, which we have seen lead to well-diversified asset allocations,
the ad-hoc approach to incorporating the views leads to relatively concentrated
asset allocations.
The Black-Litterman
model offers an alternative to this kind of ad-hoc adjustment of the return forecasts.
The Black-Litterman model uses a sophisticated mixed estimation technique to incorporate
views into return forecasts which continues to derive returns as a relatively
balanced function of risk. The result is diversified portfolios whose allocations
reflect the views of the investor.
Figure 6 shows
the mixes created by using the Black-Litterman approach to incorporate our two
sample views.
Figure 6: Asset
Allocation Mixes Based on Black-Litterman Approach
 |
| Black-Litterman approach leads to well-diversified portfolios that reflect views |
Notice in Figure
6 that at each of the five standard deviation levels we have achieved improved
diversification using the Black-Litterman model relative to the ad-hoc approach.
The other advantage
to using the Black-Litterman approach to include views in the optimization is
that the resulting portfolios are affected in an intuitive way. This means that
if you are bullish on Small Cap, the mixes will increase the allocation to Small
Cap by a reasonable amount. While this may sound like an obvious (and necessary)
result, those of you who have attempted to adjust return forecasts manually will
recognize that it is not necessarily easy to achieve.
Figure 7 compares
the asset allocation mixes at each of the five standard deviation levels using
the Black-Litterman case without views (just using the Implied Returns) and the
Black-Litterman case with views.
Figure 7: Asset
Allocation Mix Comparison
The upper panel
of Figure 7 shows the allocations using the Black-Litterman approach without views
while the lower panel shows the allocations using the Black-Litterman approach
with our two sample views. Remember that the sample views were bullish on US Large
Cap Value and Small Cap. Note that the allocations reflect these views in an intuitive
way, with increased allocations to these two assets.
What’s
Next
Hopefully by now
you are anxious to start using the AllocationADVISOR’s Black-Litterman Forecast
Model. If you haven’t already done so, you will need to download AllocationADVISOR
6.0. Also, the AllocationADVISOR workbook file that was used to create all of the examples
in this newsletter is available for download:
Black Litterman Asset Allocations (.exe file)
Black Litterman Asset Allocations (.zip file)
In the Spring of 2005,
we will be hosting a number of AllocationADVISOR 6.0 WebEx online training sessions.
Our online training schedule is available here.
For a non-technical
discussion of how the Black-Litterman model calculates forecasts, see How
does the Black-Litterman Model Calculate Return Forecasts?
For those of you
who would like to know more about the mathematics of the Black-Litterman model
and our implementation of it see Black-Litterman Forecast Methodology in the AllocationADVISOR
manual. This section of the manual explains reverse optimization, the process
of creating Market Cap Assets, and the Zephyr Asset Palettes. The AllocationADVISOR
manual is located in your Style folder as well as the Start Menu StyleADVISOR
Program Group.
For the mathematically
inclined, a copy of “A Step-By-Step Guide to the Black-Litterman Model:
Incorporating User-Specified Confidence Levels” is available upon request
(support@styleadvisor.com).
Back to the Newsletters page.
|
