zephyr concepts - black-litterman

8/11/2019 Zephyr Concepts - Black-Litterman

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22 Investments&Wealth MONITOR

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raditionally, the standard proce-

dure to come up with the required fore-

casts for the expected return, standard

deviation, and correlation coeffi cients

for the chosen asset classes has been torely on historical data. Running MVO

with these historical estimates often

results in unintuitive and poorly diversi-

fied portfolios. Moreover, if an investor

makes small changes to the historical

return forecasts based on her own

assessment of future market behav-

ior, then the resulting changes in the

optimal portfolio allocations are often

unintuitive and erratic.

The Black-Litterman Model

Te BLM, introduced by Fisher Black

and Robert Litterman in 1992, is a

mathematical model for obtaining

stable and consistent return forecasts

for a set of asset classes. When these

return forecasts are used as input to the

MVO calculation, the resulting portfo-

lios tend to be much more intuitive and

less erratic than the ones that are based

on historical return estimates.

Te BLM does not address the issue

of finding estimates for the standard

deviations and correlation coeffi cients.

As far as those are concerned, historical

values have turned out to be reasonably

stable. Moreover, the MVO calculation

is much less sensitive to these inputs

than it is to the return estimates.

Te Black-Litterman approach

consists of two separate parts. Te first

part addresses the problem of MVO

portfolios being unintuitive and poorly

diversified. Te second part addresses

the fact that MVO portfolios are erratic;

that is, they are overly sensitive to small

changes in the return forecasts.

BLM Part 1: Achieving Well-

Diversified Portfolios

Te fact that classical MVO portfolios,

based on historical return forecasts,

tend to be unintuitive and poorly di-

versified should not come as a surprise.

Te mathematics of MVO is purely

focused on maximizing the expected re-

turn for each level of risk. It simply does

not know what we human investors

consider to be intuitive and well-diver-sified. o achieve an output that meets

our idea of an intuitive portfolio, we

must find a way of making intuitive-

ness of a portfolio part of the input

of the MVO algorithm. Tat is exactly

what the first part of the BLM does.

Te first step is to identify the consensus

portfolio, that is, the default portfolio that

an investor should buy in the absence of

any special information, inclination,

opinion, speculation, or the like. Tis is

how the investor communicates to theBLMand thus, indirectly, to the MVO

calculationwhat is to be viewed as an

intuitive, well-balanced portfolio.

By far the most common choice for

the consensus portfolio is the market

portfolio; that is, the portfolio where the

weights of the asset classes are propor-

tional to their respective total market

capitalizations. Terefore, it may be

FIGURE 2: MVO WITH BLM RETURN FORECASTS

FIGURE 3: MVO WITH SMALL-VALUE FORECAST LOWERED

Cas : Black-Litterman Return vs. Risk (Standard Deviation)

0 1 15 20 2 3

3

4

5

6

7

8

9

10

11

Risk (Standard Deviation

Return

Asse AllocationUS Bond -Aggrega e

US Large p Growh

Large p lu

ll r th

US Small ap Value

Cas : Historical Forec sts Return vs. Ri k (Stan ar Deviation)

6 7 9 10 11 12 13 14 1 16

10

11

12

13

14

Risk (Standard Devi tion

Return

Asset Allocation

- t

Large ap ro h

L al

US Sm ll Cap rowth

mall ap al e

2009 Investment Management Consultants Association. Reprint with permission only.


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23January/February 2009

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worth pointing out that the mathemat-

ics of the BLM depends in no way on

how the investor came up with the con-

sensus portfolio. From a purely math-

ematical point of view, any choice is as

good as the next one. It is true, however,

that the market portfolio remains the

consensus portfolio of choice for most

investors. Tat is the reason why asset

allocation programs such as Zephyrs

AllocationADVISOR come with regu-

larly updated market capitalizations for

the most common asset classes, and

they use the market portfolio as the

default consensus portfolio.

Te BLM also requires the investor

to make a forecast for the risk-free rate

and for the risk premium of the consen-

sus portfolio. It is important to know

that the chosen risk premium will not

ultimately affect the portfolio weights. It

is merely a way to ensure that the BLMreturn forecasts have plausible values.

Te risk-free rate, on the other hand, is

an essential input to the model.

Based on these inputs, the BLM will

calculate a set of return forecasts called

the implied returns. Te implied returns

have the following property: When

used as inputs to the MVO calculation,

they will make the consensus portfolio

appear at one particular point on the

effi cient frontier. More precisely, the

consensus portfolio will appear exactly

at the point on the effi cient frontier

where the Sharpe ratio is maximal.

In summary, we can describe the

first part of the BLM as follows: Te

model allows the investor to describe

what she considers to be the consensus

portfolio. Te model then produces

a set of implied returns, which, when

used as input to the MVO calcula-

tion, force the effi cient frontier to be

anchored to the consensus portfo-

lio. Rather than producing arbitrary,

unpredictable portfolios, MVO now

will produce the consensus portfolio at

the point of the maximum Sharpe ratio,

and it will smoothly deviate from the

consensus portfolio as one moves away

from the point of the maximum Sharpe

ratio on the effi cient frontier.o see the difference between

classical MVO with historical return

forecasts and MVO with BLM return

forecasts, compare figures 1 and 2. Te

historical forecasts result in a concen-

trated portfolio that contains no growth

stocks at all. Te BLM return forecasts

result in a well-diversified portfolio that

takes its cues from the market portfolio.

BLM Part 2: Making Return

Forecasts Consistent

Recall that the second complaint about

classical MVO is that small changes in

the return forecasts often result in large,erratic changes to the resulting port-

folios on the effi cient frontier. Again,

this phenomenon should not come as

a surprise. Suppose you are running

an MVO with a standard set of asset

classes that include small-value stocks,

as in figure 1.

Looking at the return forecast for

your small-value asset class, you decide

that you expect it to return less in the

future; therefore, you slightly lower the

return forecast. Now if you run your

MVO again with just that one small

modification, you will find that your

portfolio allocations have changed in a

dramatic and unintuitive manner. For

example, figure 3 shows what happens if

you start with the MVO of figure 1 and

then lower the 14.75-percent historical

forecast for small-value stocks slightly to

13 percent. Te result is that small-value

stocks completely disappear from the

portfolio, leaving us with only bonds and

large-value stocks. Why did this happen?

Recall that in addition to the return fore-

casts, you also gave the MVO calculation

a set of correlation coeffi cients, which

describe the dependencies between the

returns of the asset classes. Terefore,

when you modified the return estimate

of small-value stocks, these correlations

implied corresponding changes in the

other assets classes return estimates. But

you did not make those corresponding

changes. You have thus created an incon-

sistent set of inputs. Te result is a bad

case of GIGO (garbage in, garbage out).

Given inconsistent input, MVO producesmeaningless, random portfolios.

Te second part of the BLM fixes

this problem in a way that is, although

mathematically very complex, easy to

understand on a high level. When the

investor makes changes to the return

forecasts that were obtained in the first

part of the BLM, the BLM does not take

these changes at face value. Instead, it

FIGURE 4: MVO WITH BLM VIEW RAISING SMALL-VALUE FORECAST

Case: Bl ck-Litterman Return vs. Risk (Stan ard Devi ti n

5 10 15 2 25 0

3

4

5

6

7

8

9

1

11

1

Risk (Standard Deviation

Return

L h

US Large p V lu

m ll ap rowth

mall al

As t Allocations



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makes the modified return forecasts

consistent in the sense that it takes into

account how the investors modifica-

tions affect other asset classes based

on the correlations between the asset

classes. When these new, consistent

return forecasts are used as input to the

MVO calculation, the resulting change

in the portfolio allocations no longer

is erratic and unintuitive. o see this,

compare figures 2 and 4. Figure 4 shows

the effect of raising the BLM return

forecasts for small-value stocks from

7.63 percent to 9 percent. Te allocation

to small-value stocks increases, leaving

the weights of the other asset classes

relatively untouched.

Let us now take a closer look at how

this second part of the BLM presents

itself to the investor. After completing

the first part of the BLM, where the con-

sensus portfolio is entered, the investor

is presented with the implied returns.

At this point, it is possible to state views

on how the investor thinks the implied

returns ought to be modified.

In theory, a view can be any linear

relationship between the returns. In

practice, only two special kinds of views

are relevant:

An absolute view is a view of the

form, I believe that asset class A will

have a return of x percent.

A relative view is a view of the form,

I believe that asset class A will out-perform asset class B by x percent.

In addition, the investor may attach

a level of confidence to each view; that

is, a view can be held with more or less

confidence. Te more confidence the

investor has in a view, the more the view

will affect the outcome of the calculation.

When the investor has entered the

views, the BLM will modify the implied

returns according to the views. In

doing so, it will take into account how

the returns of the different asset classes

affect each other via the correlations.

As mentioned before, this is a math-

ematically very complex task. In fact,

it may seem like an impossible task:

When an investor states more than one

view, it is quite likely that these views

will be contradictory to some extent.

Suppose, for example, that the investor

states two views:

1. I believe that asset class A will

return x percent.

2. I believe that asset class A will out-

perform asset class B by y percent.

When the BLM processes the first

view, it will set the return forecast of

asset class A to x percent, and it will

modify the return forecasts of all other

asset classes according to their correla-

tions with asset class A. In particular, it

will arrive at a certain return forecast

for asset class B. In all likelihood, the

difference between the return forecasts

for asset classes A and B will not equal

y percent, as stipulated by the second

view. Terefore, to process the second

view, the BLM has to modify the return

forecasts of asset class A or asset class

B again, thus violating the result of

processing the first view.

So what the BLM typically does is

a balancing act between several views

that contradict each other. In otherwords, the BLM does not really make

the return forecasts consistent; rather,

it makes them statistically consistent in

the sense that it finds the best compro-

mise between the views. One might

expect that finding a compromise in

a situation like that would involve an

element of discretion. In other words,

one might expect that there would be

different schools of thought on how

to define that compromise. Somewhat

surprisingly, it turns out that this is

not the case. Te situation at hand is

an instance of a very solid and well-founded mathematical theory known

as generalized least squares estimation,

which is backed by more than 200 years

of mathematical research. Least squares

estimation has countless applications

in navigation, surveying, oil drilling,

geodesy, and target finding, and many

other areas. Its validity and appropri-

ateness are not a matter of conten-

tion. It thus is fair to say that the BLM

provides us with scientifically sound

return-forecasts.

Summary

Te Black-Litterman model is a front

end for classical mean-variance opti-

mization. Te purpose of the BLM is

twofold:

By anchoring the MVO portfolios to

a consensus portfolio that the inves-

tor provides, the BLM causes the

MVO calculation to produce intui-

tive, well-diversified portfolios.

When an investor states views

regarding return estimates, the BLM

processes these views to make them

statistically consistent. As a result,

the portfolios produced by the MVO

will reflect the investors view in a

plausible manner rather than jump-

ing erratically as they would without

the BLMs preprocessing.

Te BLM model achieves its

purpose using sound mathematical

principles whose value has been proven

in a wide variety of fields in science and

engineering.

Thomas Becker, PhD, is a mathema-

tic ian and scienti f i c software engi-

neer at Zephyr Associates, Inc. in

Zep hy r Co ve , NV. He ear ne d a Ph D

in mathematics from the University

of Heidelberg, Germany. Contact him

at [email protected].

. . . the BLM does not really make the re-turn forecasts cons istent; rather, it makes th em

statistically consistent in the sense that it findsthe best compromise between the views.


zephyr concepts - black-litterman

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