seb investment management house view research group · page 5 relativ entropi mathematically, the...

2015

Entropy PoolingSEB Investment ManagementHouse View Research Group

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Black-Litterman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Relative Entropy Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5A Practical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8An Example: What Happens to my Portfolio if . . . . . . . . . . . . . . . . . . . .12Example: Markowitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13Non-linear Views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16Litteratur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

Editorial

SEB Investment ManagementSveavågen 8SE-106 Stockholm

Authors:

Portfolio Manager, TAA: Peter Lorin RasmussenPhone: +45 33 28 14 22E-mail: peter .lorin .rasmussen@seb .dk

Portfolio Manager, Fixed Income & TAA: Tore Davidsen Phone: +45 33 28 14 25 E-mail: tore .davidsen@seb .dk

This document produced by SEB contains general marketing information about its investment products . Although the content is based on sources jud-ged to be reliable, SEB will not be liable for any omissions or inaccuracies, or for any loss whatsoever which arises from reliance on it . If investment research is referred to, you should if possible read the full report and the disclosures contained within it . Information relating to taxes may become outdated and may not fit your individual circumstances . Investment products produce a return linked to risk . Their value may fall as well as rise, and historic returns are no guarantee of future returns; in some ca-ses, losses can exceed the initial amount invested . Where either funds or you invest in securities denominated in a foreign currency, changes in exchange rates can impact the return . You alone are responsible for your investment decisions and you should al-ways obtain detailed information before taking them . For more information, please see the relevant simplified prospectus for the funds, and the relevant information brochure for funds and for structured products . If necessary you should seek advice tailored to your individual circumstances from your SEB advisor .Skandinaviska Enskilda Banken AB (publ) is incorporated in Sweden as a Li-mited Liability Company . It is regulated by Finansinspektionen, and by the local financial regulators in each of the jurisdictions in which it has branches or subsidiaries . Skandinaviska Enskilda Banken AB, Sveavågen 8, SE-106 Stockholm

Disclaimer

Introduction

Editorial

This paper describes the mathematical model which is used in SEB Invest-ment Management to blend views on the markets with historical data . For example, views on future correlations, variances and/or the returns of spe-cific asset classes . By using our proposed method, rather than manually set-ting parameter values, we ensure that statistical interdependencies become accounted for in an elegant and mathematically correct fashion .

Let us start by giving an example: During the past three decades govern-ment bond yields have declined to what are historically very low levels . As a consequence hereof, it seems reasonable to assume that the future re-turn potential of all coupon bearing assets should be lower than what they have been in the past . A need therefore arises to input low expected return estimates on e .g ., Government Bonds, Investment Grade Bonds, and High Yield Bonds in one’s risk and portfolio optimization models . Furthermore, it is more than likely, in our view at least, that the low yields will have an effect on future correlations as well . Where safe haven assets such as US Govern-ment Bonds have “benefited” from past selloffs in equities, the general view of SEB Investment Management is now, that a severe rise in yields will have a very negative effect on equities . In other words, Government Bonds can, and should, not be treated as the same type of asset as it has been .

In terms of practicality the model also helps in so far that a range of techni-cal considerations becomes accounted for by construction and without any demands on the user . An example hereof is views on the correlation struc-ture, which has to obey certain statistical properties (the covariance matrix being positive semi-definite for example) . Using our proposed model such considerations does not need to be accounted for explicitly, as they follow automaticly from the model . Furthermore the model eases the implemen-tation of general views . An example: Specifying the expected returns manu-ally, which is the approach most often followed in practice, is easy given that one only has to change as many parameters as there are assets . However, changing correlations is more difficult/tedious as the number of parameters equals the squared number of assets . So if one wishes to change the general correlation between equities and coupon bearing assets, one has to change a lot of parameters . However using our proposed method, it can simply be implemented by introducing a view on the correlation between for example Equities and Government Bonds .

To overcome the problems and gain the advantages mentioned above, we propose to incorporate views by using a field of mathematics which is called Bayesian statistics .

Black-Litterman The somewhat naïve intuition behind Bayesian statistics is that a view on one parameter affects all other parameters as well . For example, if you ex-pect the volatility of Equities to rise, you are probably also expecting the volatility of High Yield Bonds to rise . Bayesian statistics help quantify this relationship .

The most famous application of Bayesian statistics in finance is the Black-Litterman (1990) model . This model allows for the implementation of linear views on normally distributed variables . That is, a way to incorporate views on level and spread returns; and only those! For example: Equities will deli-ver an expected return of 10% and/or equities will deliver a return which is 5%-point higher than that of bonds .

The main shortfalls of the Black-Litterman model are:

• It is only possible to implement linear views

• It is only valid for Gaussian variables

As should be apparent to all who has worked with asset allocation in prac-tice, these restrictions are very limiting . More often than not, we wish to incorporate views on correlations, and trying to find normally distributed variables in finance is often difficult . Hence, a need arises for a more general framework than the one proposed by Black-Litterman .

In SEB Investment Management we propose to expand the Black-Litterman model by using the concept of Relative Entropy Minimization (REM); the definition of which will be given in the following . The advantages of REM, compared to Black-Litterman, are:

• The possibility of incorporating views on all (defined) moments . For example views on standard deviations and/or the correlation structure

• The possibility of incorporating views on non-normally distributed va-riables

The input for REM is the same as for the traditional Black-Litterman model: A set of defined views and a certainty hereof . The certainty can be expres-sed as a parameter with values between 0% and 100% . The output of REM, as in the Black-Litterman model, is a new distribution, which can be used to re-estimate all the moments needed for a risk or portfolio optimization model .

Relativ Entropi Minimering

Mathematically, the objective of REM is to find a distribution which:

1 . Is in perfect alignment with our views

2 . Is as close as possible to the distribution of the historical data

The first objective is a constraint for the optimization, which states that the output distribution must match our required views . That is, if we decide that the expected return of an asset should be 2%, then the mean value of the resulting (output of REM) univariate distribution must be 2% . The second objective is somewhat more complicated, as we need to define a measure which quantifies the distance between two separate distributions . In order to accomplish this we rely on the concept of relative entropy as defined by Meucci (2010):

Here

gennem en optimeringsmodel.


I ord søger vi at bestemme en fordeling, der: 1. Er i overensstemmelse med vores view 2. Er så tæt som mulig på fordelingen af det historiske data

Forskellen, afstanden eller differencen mellem et view og fordelingen af historiske data definerer Meucci (2010) gennem begrebet relativ entropi:

( ) ( ) ( ) ( )[ ]dxXfXfXfff xxxxx −= ln~ln~,~ε

xf angiver her fordelingen af data, og xf~ angiver en generisk fordeling,

der opfylder de givne views. I takt med at vi ønsker at finde den fordeling

xf~ , som ”mest ligner” den oprindelige , finder vi xf xf~ ved at løse

nedenstående optimeringsproblem:

( ){ }xVf

x fff ,minarg~ ε∈

=

Hvor er et generisk sæt af bibetingelser (bemærk, at der heri er

indeholdt den gængse fordelingsmæssige betingelse om, at

V

xf~

integrerer til 1 og kun antager positive værdier). Det er også i V , at vores

views er indeholdt. Eksempelvis at middelværdien af xf~ er 2 eller

standardafvigelsen er 3. For grafisk at illustrere den relative entropi viser vi i Figur 1 to fordelinger. Den ene (org dist) kan betragtes som fordelingen af det historiske data, den anden (view) som det view vi har på markedet. Figur 1: Illustration af to fordelinger

-6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Den

sity

Org distView

Med udgangspunkt i de to fordelinger, afbildet i Figur 1, kan vi vise den relative entropi. Denne er afbildet i Figur 2. Det er som skrevet dette mål, vi søger at minimere. Bemærk, at vi har vist den kumulative relative entropi. Det, vi i praksis søger at minimere, er således yderpunktet til

P

denotes the distribution of the historical data,










( ){ }xVf


=



V

xf~




-6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Den

sity

Org distView


P

denotes the generic distribution which is in alignment with our views, and

3

Here 𝑓𝑓𝑥𝑥 denotes the distribution of the historical data, 𝑓𝑓𝑥𝑥 denotes the generic distribution which is in alignment with our views, and 𝜖𝜖 defines the relative entropy between the two distributions. As stated, we seek to find the distribution 𝑓𝑓𝑥𝑥which is in alignment with our views and which is as “close” as possible to 𝑓𝑓𝑥𝑥. Hence we find 𝑓𝑓𝑥𝑥 by solving:

𝑓𝑓𝑥𝑥 = min𝑓𝑓∈𝑉𝑉

𝜖𝜖(𝑓𝑓,𝑓𝑓𝑥𝑥)

That is, we minimize the relative entropy. With regards to the set of constraints: 𝑉𝑉 contains both our views (an example was given in terms of the expected return on an asset) and specify that 𝑓𝑓𝑥𝑥 must be an actual distribution; integrate to one and attain positive values only. In order to give a more intuitive illustration of the concept of relative entropy, Figure 1 depicts two separate distributions. The one named Org dist can be thought of as the distribution of the historical data, and the one named View can be thought of as a distribution which is in alignment with our view; not necessarily the one that is “closest” to the historical data. Figure 1: Illustration of Two Distributions

The relative entropy distance between these two distributions is shown in Figure 2. Note that Figure 2 depicts the cumulative relative entropy, but in reality we are only trying to minimize the point to the right; marked with a red circle. The lower limit for the relative entropy is zero which can be shown by simple algebra. Figure 2: Cumulative Relative Entropy of the Distributions

-6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Den

sity

Org distView

defines the relative entropy between the two distributions .

As stated, we seek to find the distribution










( ){ }xVf


=



V

xf~




-6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Den

sity

Org distView


P

which is in alignment with our views and which is as “close” as possible to










( ){ }xVf


=



V

xf~




-6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Den

sity

Org distView


P

. Hence we find










( ){ }xVf


=



V

xf~




-6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Den

sity

Org distView


P

by solving:

That is, we minimize the relative entropy . With regards to the set of con-straints: V contains both our views (an example was given in terms of the expected return on an asset) and specify that










( ){ }xVf


=



V

xf~




-6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Den

sity

Org distView


P

must be an actual distri-bution; integrate to one and only attain positive values .

In order to give a more intuitive illustration of the concept of relative en-tropy, Figure 1 depicts two separate distributions . The one named Org dist can be thought of as the distribution of the historical data, and the one named View can be thought of as a distribution which is in alignment with our view; not necessarily the one that is “closest” to the historical data .










( ){ }xVf


=



V

xf~




-6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Den

sity

Org distView


P

3

Here 𝑓𝑓𝑥𝑥 denotes the distribution of the historical data, 𝑓𝑓𝑥𝑥 denotes the generic distribution which is in alignment with our views, and 𝜖𝜖 defines the relative entropy between the two distributions. As stated, we seek to find the distribution 𝑓𝑓𝑥𝑥which is in alignment with our views and which is as “close” as possible to 𝑓𝑓𝑥𝑥. Hence we find 𝑓𝑓𝑥𝑥 by solving:

𝑓𝑓𝑥𝑥 = min𝑓𝑓∈𝑉𝑉

𝜖𝜖(𝑓𝑓,𝑓𝑓𝑥𝑥)

That is, we minimize the relative entropy. With regards to the set of constraints: 𝑉𝑉 contains both our views (an example was given in terms of the expected return on an asset) and specify that 𝑓𝑓𝑥𝑥 must be an actual distribution; integrate to one and attain positive values only. In order to give a more intuitive illustration of the concept of relative entropy, Figure 1 depicts two separate distributions. The one named Org dist can be thought of as the distribution of the historical data, and the one named View can be thought of as a distribution which is in alignment with our view; not necessarily the one that is “closest” to the historical data. Figure 1: Illustration of Two Distributions

The relative entropy distance between these two distributions is shown in Figure 2. Note that Figure 2 depicts the cumulative relative entropy, but in reality we are only trying to minimize the point to the right; marked with a red circle. The lower limit for the relative entropy is zero which can be shown by simple algebra. Figure 2: Cumulative Relative Entropy of the Distributions

-6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Den

sity

Org distView

Figur 1: Illustration of Two Distributions

-6 -4 -2 0 2 4 6 80

0 .05

0 .1

0 .15

0 .2

0 .25

0 .3

0 .35

0 .4

Den

sity

Org distView

The relative entropy distance between these two distributions is shown in Figure 2 . Note that Figure 2 depicts the cumulative relative entropy, but in reality we are only trying to minimize the point to the right; marked with a red circle . The lower limit for the relative entropy is zero which can be shown by simple algebra .

Figur 2: Cummulative Relative Entropy of the Distributions

-6 -4 -2 0 2 4 6 8-100

0

100

200

300

400

Rela

tive

entr

opi

A Practical Implementation

In practice it is generally not possible to solve the analytic problem descri-bed above, as we do not necessarily have a functional form of the output distribution . To solve this, we turn to the discrete solution where we repre-sent the historical as well as the “target” distribution by their histograms . To be specific, by the representation of the histogram in terms of a panel and a corresponding probability vector .

To illustrate our line of thought, we start by considering the case where we have estimated a distribution of some historical data . From this distribution we draw a large set of independent observations and assign to it a probabi-lity vector which specifies the probability of observing each row; the struc-ture being illustrated in Figure 3 .

Figur 3: Illustration of a Panel Structure

Here J denotes the number of random draws, and N denotes the dimension of our distribution; which can be thought of as the number of assets . Na-turally all the elements in the probability vector must be non-negative and must sum to one .

Now, to illustrate that this representation is equivalent to a histogram try to imagine dividing the panel into a multidimensional grid . For example, we split the return of one asset into those lower than -2%, those between -2% and -1% and so forth . With this grid we then summarize the corresponding probability mass, based on our probability vector, and thereby we have re-constructed a histogram! A histogram which can be thought of as a discrete representation of the distribution .

Figur 4: A Histogram Can be Expressed by a Panel with a Corresponding Pro-bability Vector and Vice Versa

To tie the knot, we now express the relative entropy between the two distri-butions in terms of the probability vector:

Where p denotes the probability vector of the historical distribution, and

( ) ( ) ([ ]=

−=J

jppppp

1ln~ln~,~ε )

Hvor p angiver den oprindelige sandsynlighedsvektor, og p~ angiver en sandsynlighedsvektor, der overholder vores view.

Tænkt Eksempler I det følgende illustrerer vi metoden gennem to tænkte eksempler. Betragt først en normalfordelt, stokastisk variabel . Fra denne fordeling trækker vi 10.000 tilfældige observationer, og præsenterer den gennem sit histogram i Figur 4.

( 1,0~ NX )

Figur 4: Histogram af en standard normalfordelt variabel

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Den

sitiy

Antag at vi nu ønsker at pålægge et view om, at standardafvigelsen på denne fordeling er 2 i stedet for 1. Det første vi betragter er den tilhørende sandsynlighedsvektor for både dette view og vores oprindelige fordeling. For det ”oprindelige” data er sandsynligheds-vektoren uniformt fordelt, men ved at pålægge vores view på standardafvigelsen får vi en ny sandsynlighedsvektor, hvor ikke alle elementer er ens. Nogle rækker i panelet bliver således tildelt en tungere vægt end andre. Figur 5 afbilder de to sandsynlighedsvektorer. Figur 5: Sandsynlighedsvektorer: Oprindeligt data og med pålagt view

S

denotes the probability vector of our view . Again, we try to minimize this

expression by tweaking

( ) ( ) ([ ]=

−=J

jppppp

1ln~ln~,~ε )



( 1,0~ NX )


-5 -4 -3 -2 -1 0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Den

sitiy


S

; as we in the analytical solution tweaked










( ){ }xVf


=



V

xf~




-6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Den

sity

Org distView


P

.

To give a simple example of the approach, assume that we have a single va-riable; therefore our panel is merely a vector . Based hereon we can estimate the historical mean as:

Where X denotes the panel (vector in this specific case) . Now say that we have a view that the asset’s expected return should be 2 . Then we find a new probability vector of which the inner product with the panel equals 2 .

If one row in the panel is exactly 2, the problem could be solved by just as-signing that row a weight of 1; with all other rows a weight of 0 . However, this will not be the solution of REM as we are also trying to minimize the entropy towards the historical distribution . Thus, all rows will be assigned a positive probability, but some rows more than others .

( ) ( ) ([ ]=

−=J

jppppp

1ln~ln~,~ε )



( 1,0~ NX )


-5 -4 -3 -2 -1 0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Den

sitiy


S

Examples To illustrate the approach in practice, we start by focusing on two synthetic examples: One univariate and one multivariate .

First, consider a univariate standard Gaussian distribution:

( ) ( ) ([ ]=

−=J

jppppp

1ln~ln~,~ε )



( 1,0~ NX )


-5 -4 -3 -2 -1 0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Den

sitiy


S

. From this we draw 10,000 observations and construct a histogram; Figure 5 .

5

probability mass, based on our probability vector, and thereby we have reconstructed a histogram! A histogram which can be thought of as a discrete representation of the distribution. Figure 4: A histogram can be expressed by a panel with a corresponding probability vector and vice versa

To tie the knot, we now express the relative entropy between the two distributions in terms of the probability vector:

𝜖𝜖(𝑝𝑝�,𝑝𝑝) = �𝑝𝑝�⌈ln(𝑝𝑝�) − ln (𝑝𝑝)⌉𝐽𝐽

𝑗𝑗=1

Where 𝑝𝑝 denotes the probability vector of the historical distribution, and 𝑝𝑝� denotes the probability vector of our view. Again, we try to minimize this expression by tweaking 𝑝𝑝�; as we in the analytical solution tweaked 𝑓𝑓. To give a simple example of the approach, assume that we have a single variable; therefore, our panel is merely a vector. Based hereon we can estimate the historical mean as:

𝐸𝐸(𝑋𝑋) = ⟨𝑋𝑋, 𝑝𝑝⟩ Where 𝑋𝑋 denotes the panel (vector in this specific case). Now say that we have a view that the asset’s expected return should be 2. Then we find a new probability vector of which the inner product with the panel equals 2.

𝐸𝐸(𝑋𝑋) = ⟨𝑋𝑋,𝑝𝑝�⟩ = 2 If one row in the panel is exactly 2, the problem could be solved by just assigning that row a weight of 1; with all other rows a weight of 0. However, this will not be the solution of REM as we are also trying to minimize the entropy towards the historical distribution. Thus, all rows will be assigned a positive probability, but some rows more than others.

Examples To illustrate the approach in practice, we start by focusing on two synthetic examples: One univariate and one multivariate. First, consider a univariate standard Gaussian distribution: 𝑋𝑋 ~𝑁𝑁(0,1). From this we draw 10,000 observations and construct a histogram; Figure 5.

Panel: JxN

Prob

abili

ty v

ecto

r: Jx

1

-4 -3 -2 -1 0 1 2 3 40

1

2

3

4

5

6

7

8x 10

-4

5

probability mass, based on our probability vector, and thereby we have reconstructed a histogram! A histogram which can be thought of as a discrete representation of the distribution. Figure 4: A histogram can be expressed by a panel with a corresponding probability vector and vice versa

To tie the knot, we now express the relative entropy between the two distributions in terms of the probability vector:

𝜖𝜖(𝑝𝑝�,𝑝𝑝) = �𝑝𝑝�⌈ln(𝑝𝑝�) − ln (𝑝𝑝)⌉𝐽𝐽

𝑗𝑗=1

Where 𝑝𝑝 denotes the probability vector of the historical distribution, and 𝑝𝑝� denotes the probability vector of our view. Again, we try to minimize this expression by tweaking 𝑝𝑝�; as we in the analytical solution tweaked 𝑓𝑓. To give a simple example of the approach, assume that we have a single variable; therefore, our panel is merely a vector. Based hereon we can estimate the historical mean as:

𝐸𝐸(𝑋𝑋) = ⟨𝑋𝑋, 𝑝𝑝⟩ Where 𝑋𝑋 denotes the panel (vector in this specific case). Now say that we have a view that the asset’s expected return should be 2. Then we find a new probability vector of which the inner product with the panel equals 2.

𝐸𝐸(𝑋𝑋) = ⟨𝑋𝑋,𝑝𝑝�⟩ = 2 If one row in the panel is exactly 2, the problem could be solved by just assigning that row a weight of 1; with all other rows a weight of 0. However, this will not be the solution of REM as we are also trying to minimize the entropy towards the historical distribution. Thus, all rows will be assigned a positive probability, but some rows more than others.

Examples To illustrate the approach in practice, we start by focusing on two synthetic examples: One univariate and one multivariate. First, consider a univariate standard Gaussian distribution: 𝑋𝑋 ~𝑁𝑁(0,1). From this we draw 10,000 observations and construct a histogram; Figure 5.

Panel: JxN

Prob

abili

ty v

ecto

r: Jx

1

-4 -3 -2 -1 0 1 2 3 40

1

2

3

4

5

6

7

8x 10

-4

Figur 5: Histogram of a Standard Gaussian Variable

-5 0 50

0 .01

0 .02

0 .03

0 .04

0 .05

0 .06

0 .07

0 .08

0 .09

Den

sitiy

Let us say we have a view that the standard deviation of the variable should be 2 instead of 1 . First observe the two probability vectors as described abo-ve . As can be seen in Figure 6, the probability vector of the original distribu-tion is weighted equally, but the probability vector which corresponds to our view assigns higher weights to some rows .

Figur 6: Probability Vectors - Original and Posterior

With the new probability vector, and the original panel, we can reconstruct the histogram which is in line with our view; that the standard deviation is 2 . This new histogram, the output of REM, is shown in Figure 7 . Please notice that the tails of our distribution are now assigned more probability mass in order to fit our view .

Figur 7: Histogram of the Posterior Distribution

-6 -4 -2 0 2 4 60

0 .01

0 .02

0 .03

0 .04

0 .05D

ensi

ty

Org

To illustrate REM in a bivariate setting, say we observe the following system:

That is, we have two variables, both of which have positive mean values and a positive correlation . To illustrate REM in a new way, Figure 8 depicts a scatter plot of 500 observations from this distribution .

7


�𝑋𝑋1𝑋𝑋2�~𝑁𝑁�� 1

0.5� , � 1 0.50.5 2 ��

That is, we have two variables, both of which have positive mean values and a positive correlation. To illustrate REM in a new way, Figure 8 depicts a scatter plot of 500 observations from this distribution. Figure 8: Plot of Observations from a Multivariate Distribution

Say we implement the view that the mean value of 𝑋𝑋1equals -2. Figure 9 depicts the same observations as in Figure 8, but in this example we have colored the observations (rows in the panel) which then are assigned a probability mass greater than 1/𝐽𝐽. Figure 9: Plot of Observations from a Multivariate Distribution

-6 -4 -2 0 2 4 60

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Den

sity

Org

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

4

5

Aktiv

2

Aktiv 1

Figur 8: Plot of Observations from a Multivariate Distribution

-3 -2 -1 0 1 2 3-4

-2

0

2

4

6

Aktiv

2

Aktiv 1

Say we implement the view that the mean value of

7


�𝑋𝑋1𝑋𝑋2�~𝑁𝑁�� 1

0.5� , � 1 0.50.5 2 ��

That is, we have two variables, both of which have positive mean values and a positive correlation. To illustrate REM in a new way, Figure 8 depicts a scatter plot of 500 observations from this distribution. Figure 8: Plot of Observations from a Multivariate Distribution

Say we implement the view that the mean value of 𝑋𝑋1equals -2. Figure 9 depicts the same observations as in Figure 8, but in this example we have colored the observations (rows in the panel) which then are assigned a probability mass greater than 1/𝐽𝐽. Figure 9: Plot of Observations from a Multivariate Distribution

-6 -4 -2 0 2 4 60

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Den

sity

Org

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

4

5

Aktiv

2

Aktiv 1

equals -2 . Figure 9 depicts the same observations as in Figure 8, but now we have colored the observations (rows in the panel) which are assigned a probability mass gre-ater than 1/J .

Figur 9: Plot of Observations from a Multivariate Distribution

-3 -2 -1 0 1 2 3-4

-2

0

2

4

6

Aktiv

2

Aktiv 1

P<1/JP>1/J

The examples given above are naturally very stylized . To give a more prac-tical one, let us illustrate by using a portfolio consisting of the weights pre-sented in Table 1 .

Table 1: Portfolio Weights (example)

Asset WeightEquities 20%High Yield Bonds 20%Investment Grade Bonds 10%Government Bonds 45%Cash 5%Total 100%

Based on these portfolio weights we want to quantify the sensitivity of the portfolio with regard to a range of events in the German 10Y Government Bond rate and MSCI World .

To do this, we first estimate the distribution of assets as well as the two risk series . We then estimate a set of probability vectors which are in alignment with a grid of observations in the government yield and MSCI World . Mul-tiplying this set of probability vectors onto our panel will result in the sensi-tivity analysis presented in Table 2 .

Tabel 2: Portfolio Sensitivity Towards Simultaneous Movements in Equities and Interest Rates

GE GOVT 10Y \MSCI W

-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0

-0 .5 -1 .5 -0 .8 -0 .1 0 .5 1 .1 1 .6 2 .3 3 .3 3 .7 4 .1 3 .8

-0 .4 -1 .7 -1 .0 -0 .4 0 .2 0 .7 1 .3 1 .9 2 .5 3 .3 3 .9 4 .8

-0 .3 -1 .9 -1 .2 -0 .6 -0 .1 0 .5 1 .0 1 .6 2 .1 2 .7 3 .4 4 .2

-0 .2 -2 .1 -1 .5 -0 .9 -0 .3 0 .2 0 .8 1 .3 1 .9 2 .4 3 .0 3 .6

-0 .1 -2 .2 -1 .7 -1 .1 -0 .6 -0 .0 0 .5 1 .1 1 .6 2 .2 2 .7 3 .3

0 .0 -2 .3 -1 .9 -1 .3 -0 .8 -0 .3 0 .3 0 .8 1 .4 1 .9 2 .5 3 .0

0 .1 -2 .4 -2 .1 -1 .6 -1 .1 -0 .5 0 .0 0 .6 1 .1 1 .7 2 .2 2 .8

0 .2 -2 .7 -2 .4 -1 .9 -1 .3 -0 .8 -0 .2 0 .3 0 .9 1 .4 2 .0 2 .5

0 .3 -3 .0 -2 .8 -2 .2 -1 .6 -1 .0 -0 .5 0 .1 0 .6 1 .2 1 .7 2 .2

0 .4 -3 .5 -3 .4 -2 .7 -2 .0 -1 .4 -0 .8 -0 .2 0 .3 0 .9 1 .4 1 .9

0 .5 -3 .9 -3 .8 -3 .3 -2 .6 -1 .8 -1 .2 -0 .6 -0 .0 0 .5 1 .0 1 .5

The table should be read as follows: The first row shows monthly changes in MSCI World and the first column shows monthly absolute changes in the German 10Y Government Bond yield . Each cell in the table depicts the ex-pected portfolio return, given defined movements in the yields and equities . An example: If equities rise with 4% and yields drop by 0 .1%-points then the portfolio would be expected to deliver a monthly return of 1 .6% .

An Example: What Happens to my

Portfolio, if...

As would be expected, the observations which are more in line with our view get assigned a higher weight .

As a last practical example, we illustrate the usefulness of REM by the Mar-kowitz model . Here, we want to estimate the efficient frontier for a universe consisting of

• Government Bonds

• High Yield Bonds (HY)

• Investment Grade Bonds (IG)

• Emerging Markets Bonds (EMD)

• US Equities (S&P 500)

• German Equities (DAX)

Figure 10 depicts the weights across the efficient frontier based on pure-ly historical returns and covariances; these are estimated on the basis of weekly observations from January 1999 to October 2012 .

Figur 10: Markowitz - Historical Distribution

1 1 .2 1 .4 1 .6 1 .8 2 2 .2 2 .4 2 .60

20

40

60

80

100

Standarddeviation, %

Wei

ght,

%

IGDK 5Y GovEMDHYSP 500DAX

The portfolio with the highest expected return consists purely of EMD which is an artifact of the data, since EMD has the best performance for the period; see Figure 11 .

Example: Markowitz

Figur 11: Historical Returns for the Relevant Asset Classes

IG DK 5Y Gov EMD HY SP 500 DAX0

2

4

6

8

10

12

Retu

rn, %

It comes as no surprise that the equity returns for the chosen period are very influenced by the development from 2001 through 2008 . Furthermore, the overall drop in yields over the period elevates the returns of the coupon bearing assets .

Let us assume we want to impose the view that US equities, going forward, will return 15% . If we implement this view with 100% certainty, the result is the efficient front weights shown in Figure 12 . What should be noted is that the maximum return portfolio now consists of German equities . This is a consequence of the fact that German equities have a positive beta to-wards US equities . Hence, when we impose our view that US equities will return 15% we are implicitly saying that German equities will deliver an even higher return . This serves to illustrate that having a view on one asset class implies views on the other asset classes as well .

Figur 12: Markowitz Based on the Original Data and a View that American Equities will return 15%

1 2 3 4 50

20

40

60

80

100

Standarddeviation, %

Wei

ght,

%

IGDK 5Y GovEMDHYSP 500DAX

Figure 13 illustrates the implied returns given the view on US equities . As can be visualized the view on US’ equities has ramifications for both Ger-man equities and High Yield Bonds, which both rise compared to the histo-rical returns .

Figur 13: The Implied Returns Given Our View

IG DK 5Y Gov EMD HY SP 500 DAX0

2

4

6

8

10

12

14

16

18

Retu

rn, %

On a technical note, the solution as proposed by Meucci (2010) does not allow for views on non-linear moments: For example views on the norma-lized skewness and/or kurtosis . This shortcoming is due to Meucci (2010) focusing on the dual problem and not the primal in order to gain a compu-tational advantage .

To illustrate the difference between the primal and dual problem, note that in the primal problem we are solving over J variables; each element in the probability vector

( ) ( ) ([ ]=

−=J

jppppp

1ln~ln~,~ε )



( 1,0~ NX )


-5 -4 -3 -2 -1 0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Den

sitiy


S

. Naturally, this is a very large optimization problem and therefore difficult to solve numerically . To overcome this problem, Meucci (2010) proposes to solve the dual problem instead which reduces the pro-blem so that we are merely optimizing over the number of views . Put dif-ferently, if we have three views then we have tree variables for the optimi-zation .

Although the approach of Meucci (2010) reduces the computational bur-den, it invalidates the optionality of views on non-linear moments . We the-refore suggest solving the primal problem only, using large scale optimiza-tion packs such as e .g ., KNITRO or those found in the standard version of GAMS . It is our experience that this results in solutions which are equal to those of the dual problem, although they must be judged on a case by case basis; as one is not guaranteed to find an optimal solution .

This paper presents a method to incorporate views on historical data by Bayesian statistics . The method extends the approach taken by Black-Lit-terman (1990), as it allows for views to be incorporated on all well-defined moments .

Through a series of examples this paper illustrates how the theory can be used in practice, both for risk and portfolio optimization purposes . It is our belief that using Bayesian statistics rather than setting parameter values di-rectly, results in far more desirable results and saves us a lot of technical problems such as ensuring statistical properties of the covariance matrix are fulfilled .

Non-Linear Views

Conclusion

LitteratureBlack, Fisher and Litterman, Robert (1990), Global Portfolio Optimization, Financial Analyst Journal, September 1992, pp . 28-43

Meucci, Attilio (2010), Fully Flexible Views: Theory and Practise, Version 4 . December 2010, SSRN

seb investment management house view research group · page 5 relativ entropi mathematically, the...

Documents