Entropy Pooling - SEB [PDF]

Entropy Pooling SEB Investment Management House View Research Group

2015

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

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Editorial SEB Investment Management Sveavågen 8 SE-106 Stockholm Authors: Portfolio Manager, TAA: Peter Lorin Rasmussen Phone: +45 33 28 14 22 E-mail: [email protected] Portfolio Manager, Fixed Income & TAA: Tore Davidsen Phone: +45 33 28 14 25 E-mail: [email protected]

This paper describes the mathematical model which is used in SEB Investment Management to blend views on the markets with historical data. For example, views on future correlations, variances and/or the returns of specific asset classes. By using our proposed method, rather than manually setting 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 government bond yields have declined to what are historically very low levels. As a consequence hereof, it seems reasonable to assume that the future return 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 Government 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 technical considerations becomes accounted for by construction and without any demands on the user. An example hereof is views on the correlation structure, 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 implementation of general views. An example: Specifying the expected returns manually, 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.

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Introduction

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 expect 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 BlackLitterman (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 deliver 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 practice, 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 variables 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 expressed 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.

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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 en optimeringsmodel. gennem engennem optimeringsmodel. gennem en optimeringsmodel.

The first objective is a constraint for the optimization, which states that the output distribution must our required views. Iviord søger vi atmatch ender: fordeling, der:That Entropi I ord søger atI ord bestemme en søger vibestemme at fordeling, bestemme en fordeling, der: is, if we decide tiv Relativ Entropi Relativ Entropi that the expected return should beview 2%, then theview mean value 1. 1.Erofi an overensstemmelse med vores view med vores Er i asset overensstemmelse med vores Minimering mering Minimering 1. Er i overensstemmelse of thegennem resulting of distribution must bedata 2%. Thedatadata 2. som Er REM) så somfordelingen mulig på fordelingen af det historiske optimeringsmodel. 2. en Er(output så tæt mulig på af fordelingen det historiske 2. Ertæt så univariate tæt som mulig på af det historiske second objective is somewhat more complicated, as we need to define a gennem en optimeringsmodel. gennem en optimeringsmodel. measure the distance between two separate distributions. Forskellen, afstanden eller differencen mellem view og fordelingen af af IForskellen, ordwhich søger quantifies viafstanden atForskellen, bestemme en fordeling, der: tiv Entropi eller differencen mellem et view og et fordelingen af fordelingen afstanden eller differencen mellem et view og historiske data definerer Meucci (2010) gennem begrebet relativ entropi: In order to accomplish this we rely on the concept of relative entropy as 1. Erdata i overensstemmelse med vores view historiske definerer Meucci (2010) gennem begrebet relativ entropi: I ord søger vi at bestemme fordeling, der: historiske data (2010) gennem begrebet relativ entropi: I orddefinerer søger vi Meucci at bestemme en fordeling, der: en Relativ Entropi mering Relativ Entropi defined by2.Meucci Er så (2010): tæt som mulig på fordelingen af det historiske data 1. Er i overensstemmelse med vores view 1. Er i overensstemmelse med vores view Minimering

Minimering

2. Erpåsåfordelingen tæt som mulig påhistoriske fordelingen af det historiske data 2. Er så tæt som mulig af det data ~~ ~ eller ~ differencen ~ ~mellem et view og fordelingen af Forskellen, afstanden ~ ~ ~ ) − lnx− flnx ( Xf x )( dXx) dx lnln) fln ε f x , f x data =ε fεfxx,(ffXxx ,)= ffx x((XX fx x( (X fln = f x))(−(2010) X fX x Meucci x ()Xd) historiske definerer gennem begrebet relativ entropi: Forskellen, afstanden eller differencen et view Forskellen, afstanden eller differencen mellem et view ogmellem fordelingen af og fordelingen af gennem en optimeringsmodel. historiske data definerer Meucci (2010) gennem begrebet relativ entropi: historiske data~definerer Meucci ~ ~ ̃(2010) gennem begrebet relativ entropi: f angiver her fordelingen af data, og f angiver en generisk fordeling, Here 𝑓𝑓 denotes the distribution of the historical data, 𝑓𝑓 denotes the ̃ xthe x generisk f x 𝑥𝑥denotes angiver her af fordelingen data, f xhistorical en fordeling, 𝑓𝑓𝑥𝑥 denotes the distribution of og the data, 𝑓𝑓denotes the ffordelingen angiver her afangiver data, og f x 𝑥𝑥angiver enthe generisk fordeling, HereHere distribution of the historical data, 𝑥𝑥 denotes I ord søger ~ vixat bestemme en fordeling, der: elativ Entropi generic distribution which is in alignment with our views, and 𝜖𝜖 defines the ~ ~ der opfylder de givne views. I takt med at vi ønsker at finde den fordeling generic distribution which is in alignment with our views, and 𝜖𝜖 defines the generic is fin (de alignment views, and der givne views. vi)Idour ønsker at at finde dendefines fordeling der givne views. takt med vi ønsker at finde den fordeling ( Xopfylder ) ln ) − lnmed (atXvores ε distribution fopfylder , f x1. =~de fiwhich XI takt f x with x view Er med xoverensstemmelse xtwo ~~ ~ Minimering therelative between the distributions. ~ xentropy ~ ~ ~ ~ ~ ~ ~ relative entropy between the two distributions. relative entropy between the two distributions. f Er ”mest ligner” den oprindelige fhistoriske finder vi fløse x , så x ,) f− f x , som2.”mest den f , finder vi f ved at fsom ,tæt som ”mest ligner” den oprindelige , finder vi fXx )ved atf x løse som på fordelingen af det data ( ) ( X ) dx ε f , f = f X ln −atlnløse ( ) ( ( ) εmulig foprindelige , f = f X ln f X ln f X dfxxx(ved x x x ligner” x x x x x x x x x ~ nedenstående optimeringsproblem: nedenstående optimeringsproblem: nedenstående optimeringsproblem: with our to the distribution As As stated, wewe seek to find the distribution which inalignment alignment with fstated, her fordelingen afdistribution data, og 𝑓𝑓f̃𝑥𝑥𝑓𝑓̃which angiver en generisk fordeling, whichisisisin in alignment with our Asstated, weseek seek tofind find the x angiver Forskellen, afstanden eller differencenx𝑥𝑥mellem et view ̃og fordelingen af ~ ~ and which is as “close” as possible to 𝑓𝑓 . Hence we find 𝑓𝑓 by solving: ̃ ourviews views and which is as “close” as possible to . Hence we find by f angiver her fordelingen af data, og f x angiver en generisk fordeling, 𝑥𝑥 𝑥𝑥 f angiver her fordelingen af data, og f angiver en generisk fordeling, views and which is as “close” as possible to 𝑓𝑓 . Hence we find 𝑓𝑓 by solving: der opfylder givne views. I takt med at vi ønsker at finde den fordeling ~dedata x 𝑥𝑥 𝑥𝑥 x x ~ historiske ~arg definerer Meucci (2010) gennem begrebet relativ entropi: { ( ) } f = min ε f , f ~ ~ { ( ) } x x f = arg min ε f , f { ( ) } f = arg min ε f , f solving:f x , som ”mest x ligner” x x de givne opfylder de givne views. I taktatmed vi ønsker at finde den fordeling der oprindelige opfylder views. viløse ønsker findeatden fordeling f der , finder viI takt f x med ved atat f ∈V f ∈den x f ∈V Ṽ ~ 𝑓𝑓𝑥𝑥𝑓𝑓~̃𝑥𝑥==min 𝜖𝜖(𝑓𝑓, 𝑓𝑓𝑥𝑥𝑓𝑓𝑥𝑥)x)~ ~ min 𝜖𝜖(𝑓𝑓, 𝑓𝑓∈𝑉𝑉 f x , som ligner” fden oprindelige f x , finder f x , som den ”mest oprindelige at løsevi f x ved at løse nedenstående optimeringsproblem: 𝑓𝑓∈𝑉𝑉”mest ligner” x , finder vi f x ved V~ Ver et Hvor generisk generisk af nedenstående bibetingelser (bemærk, heriheri er er optimeringsproblem: Hvorε V~ sæt af bibetingelser at der heriaterder et~ generisk sæt af) d(bemærk, bibetingelser (bemærk, at der nedenstående (the ) −sæt f x , er f x et=Hvor f x relative X )errelative ln fentropy. lnWith foptimeringsproblem: X xregards ~ x ( X entropy. x (With ~ That is, we minimize the regards to the set of conto the set of That is, we minimize ~ to theatset fom, of at f x ~ That wemin minimize entropy. With regards betingelse indeholdt den relative gængse fordelingsmæssige {indeholdt f x =is,arg ε (gængse f , fthe indeholdt den fordelingsmæssige betingelse om, gængse fordelingsmæssige betingelse fx x )} den x om, at straints: V contains both our views (an example was given in terms of the constraints: 𝑉𝑉 contains both our views (an example was given in terms of ~ ~ f V ∈ constraints: 𝑉𝑉integrerer containstilboth our views (an example was given in terms of {Det (Det )}også at ,vores 1til ogf1xkun antager også i V i, V f, xf Det =værdier). arg min εi V fer,,at fer {positive =positive arg min ε ( ~f that integrerer tilreturn 1 og kun antager værdier). også xvores at vores integrerer og kun antager værdier). x )} the on asset) and specify 𝑓𝑓̃𝑥𝑥𝑓𝑓̃er must be expected return on an asset) specify that must be actual distrif x angiver her fordelingen af data, og fpositive angiver en generisk fordeling, theexpected expected return onananand asset) and beananactual actual ~ ~ V an f ∈V specify x that 𝑥𝑥f ∈must ~ views er indeholdt. Eksempelvis at middelværdien af f er 2 distribution; integrate one and attain positive x f x er eller views indeholdt. Eksempelvis atEksempelvis middelværdien af f x der er 2 fordeling eller indeholdt. atonly. middelværdien afer 2 eller bution; integrate to one and only positive values. distribution; integrate toer one andattain positive values only. Hvor er etviews generisk sæt afattain bibetingelser (bemærk, at derVeropfylder deto givne views. I takt med atvalues vi ønsker at finde denheri ~ standardafvigelsen er 3. ~ ~ standardafvigelsen erligner” 3. Hvor standardafvigelsen er er 3. et generisk Hvor generisk af bibetingelser (bemærk, at der heri er sæterof afet bibetingelser (bemærk, at der heri er indeholdt gængse fordelingsmæssige om, atenf x løse , som ”mest denVoprindelige f betingelse , V finder virelative f x ved atsæt In order tof xgive aden more intuitive illustration of the xthe InInorder to give a more intuitive illustration ofofconcept ofofrelative ~ ~ order to give a more intuitive illustrationindeholdt theconcept concept relative den gængse fordelingsmæssige om, at f x indeholdt den gængse betingelse om, atbetingelse fx V , Org at vores integrerer tilFor 11og kun antager positive værdier). DetThe erfordelingsmæssige også i named tropy, Figure 1 depicts two separate distributions. The one named dist nedenstående optimeringsproblem: grafisk at illustrere den relative entropi viser vi i Figur 1 to fordelinger. entropy, Figure depicts two separate distributions. one Org For grafisk at illustrere den entropi viser vientropi i Figur to entropy, Figure 1 depicts two separate distributions. The 1one named For grafisk atrelative illustrere den relative viser vi i FigurOrg 1 to fordelinger. ~ fordelinger. candist beviews thought of asene the distribution of the historical data, and the one integrerer tilpositive 1 og kun antager positive værdier). er også i V , at vores vores integrerer til 1 og kun antager værdier). Det erdata, også i V , atDet er indeholdt. Eksempelvis at middelværdien af f er 2 eller Den (org dist) kan betragtes som fordelingen af det historiske can be thought of as the distribution of the historical data, and the one x Dencan ene dist) kan sombetragtes fordelingen af fordelingen detdata, historiske data, dist be(org thought asbetragtes the distribution of the historical andafthe one Den of ene (org dist) kan som det historiske data, ~ ~ ~ named View be thought of aaer distribution which inalignment alignment with {anden )}ofas f xView =can arg min εthought fdet , f(view) den anden (view) som det view vi har på markedet. named View can be thought of asas isisismarkedet. in with views er indeholdt. Eksempelvis ataf middelværdien views indeholdt. Eksempelvis at middelværdien f x er 2 eller af f x er 2 eller standardafvigelsen er( 3. xview den anden (view) visom har på markedet. named can besom adistribution distribution which in alignment with den det view viwhich har på f ∈V ourour view; notnot necessarily the one that is “closest” the historical historical data. necessarily toto data. standardafvigelsen er 3. ourview; view;not necessarilythe theone onethat thatisis“closest” “closest” to3.the thehistorical data. standardafvigelsen er Figur 1: Illustration af to fordelinger For grafisk at illustrere den relative entropi viser vi i Figur 1 to fordelinger. Figur 1: Illustration afIllustration to fordelinger Figur 1: af to fordelinger V Hvor er et generisk sæt af bibetingelser at derdata, heri er Figure 1: Illustration of Two Distributions Den ene (org dist) kan betragtes som fordelingen af(bemærk, detathistoriske For grafisk illustrere relative entropi viser vi i Figur 1 to fordelinger. Figure0.41: Illustration of Two 0.4 ForDistributions grafisk at illustrere den relative entropiden viser vi 1 to fordelinger. ~ i Figur 0.4 Org dist indeholdt den gængse fordelingsmæssige betingelse at f Org dist Org dist om, den anden (view) som det view vi har på markedet. x Den ene (org dist) kan betragtes som fordelingen af det historiske data, 0.4 Den ene (org dist) kan betragtes som fordelingen af det historiske data, View 0.35 0.4 View View 0.35 0.35 Org dist Org dist , atview voresvi har på markedet. integrerer til 1 og kun antager positive værdier). Det erviogså i Vmarkedet. dendet anden det den anden (view) som view(view) harsom på View 0.35 View 0.35 0.3 ~ 0.3 0.3af to fordelinger Figur 1: Illustration views er indeholdt. Eksempelvis at middelværdien af f x er 2 eller 0.3 0.3 0.25 0.25 0.25 Figur Illustration af to fordelinger Figur 1: Illustration af to1:fordelinger 0.4 standardafvigelsen er 3. Org dist 0.25

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MedMed udgangspunkt i dei to afbildet i Figur 1, kan vi vise denden Med udgangspunkt i de to fordelinger, afbildet i Figur 1, kan vii Figur vise den udgangspunkt de fordelinger, to fordelinger, afbildet 1, kan vi vise Org dist 0.1 0.1 View 0.05 0.35 relative entropi. Denne afbildet i Figur 2. Det er is som skrevet mål,mål, The relative entropy distance between these distributions shown inin dette relative entropi. Denne er afbildet ierFigur 2. two Det eridistributions som skrevet mål, relative entropi. Denne er afbildet Figur 2. Det er som skrevet dette The relative entropy distance between these two isdette shown 0.05 vi søger at minimere. Bemærk, at vi har vist den kumulative relative Figure 2. Note that Figure 2 depicts the cumulative relative entropy, but in 0.05 at vi søger at -4minimere. Bemærk, vi har 6 vist at den kumulative 0.3 Figure 2.0-6 Note that Figure the cumulative entropy, but in vi søger at02 depicts minimere. Bemærk, virelative har vist denrelative kumulative relative -2 2 4 8 entropi. Det, vi i praksis søger at minimere, er således yderpunktet til til reality we are only trying to minimize the point to the right; marked with a entropi. Det,only vientropi. i trying praksis at minimere, er således yderpunktet tilyderpunktet 0 Det, vi i 0-6praksis søger at the minimere, er4-2 således reality we 0.25 are tosøger minimize the point to right; marked with a58 2 Page 4 6 -4 -2 0-6 2-4 60 red circle. The lower limit for the relative entropy is zero which can be shown Med udgangspunkt i defor tothe fordelinger, afbildetisizero Figurwhich 1, kancan vi be viseshown den red circle. The lower limit relative entropy 0.2

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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 400

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In practice it is generally not possible to solve the analytic problem described 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 represent 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 probability vector which specifies the probability of observing each row; the structure 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. Naturally 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 reconstructed a histogram! A histogram which can be thought of as a discrete representation of the distribution.

Page 7

A Practical Implementation

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J 𝑝𝑝 historiske data definerer 𝜖𝜖(𝑝𝑝 �, �, 𝑝𝑝)𝑝𝑝)= �)�) − (𝑝𝑝)⌉ 𝜖𝜖(𝑝𝑝 =� ��⌈ln(𝑝𝑝 𝑝𝑝�⌈ln(𝑝𝑝 −lnln (𝑝𝑝)⌉Meucci (2010) gennem begrebet relat

~ ε ( ~p , p ) =  p[ln ( ~ p ) − ln ( p )] 𝑗𝑗=1

J

 ~p[ln( ~p ) − ln( p )]

j =1

𝑗𝑗=1

j =1

J

(

[

)

]

~ vectorofofthe ~thehistorical ~ distribution, Where 𝑝𝑝�x𝑝𝑝� ~ ε ( ~p , p ) =  ~p[ln ( ~p ) − ln ( p )] Where Where𝑝𝑝 𝑝𝑝denotes denotesthe theprobability probability ) lnsandsynlighedsvektor, ) − ln f x (and ) dand ε fvector f x ( Xhistorical f x ( Xdistribution, Xand p denotes the probability vector the historical distribution, x , fden x =of Hvor p angiver oprindelige  j =1 denotes the probability vector of our view. Again, we try to minimize this og p angiver e

denotes the probability vector of our view. Again, we try to minimize this denotes the probability ouranalytical view. we tweaked try to minimize giver den oprindelige sandsynlighedsvektor, og ~ p expression angiver en sandsynlighedsvektor, derAgain, overholder vores view. byby tweaking 𝑝𝑝�;𝑝𝑝vector asas wewe inofin the solution 𝑓𝑓̃.𝑓𝑓̃. ~ this expression tweaking �; the analytical solution tweaked ~ edsvektor, derHvor overholder voresden view. her fordelingen data, og f x .angiver en generis expression by tweaking as wef x inangiver solutionaftweaked p angiver oprindelige sandsynlighedsvektor, og p ; angiver enthe analytical I det følgende illustrerer vi metoden gennem to atænkte eksempler. Tænkt Eksempler a asimple approach, that der deassume givne views. I takt med at vi ønsker at finde de sandsynlighedsvektor, der overholderTovores view. Togive give simpleexample exampleofofthe theopfylder approach, assume thatwe wehave have asingle single To give a simple example of the approach, assume that we have a single vade illustrerer vi metoden gennem to tænkte eksempler. ~ ~ variable; a avector. Based hereon variable;therefore, therefore,our ourpanel panel merely vector. Based hereonwewecan fisx ,ismerely som ”mest ligner” den oprindelige fcan x , finder vi f x v riable; therefore our panel is merely a vector. Based hereon we can estimate ( ) Betragt først en normalfordelt, stokastisk variabel . F X ~ N 0 , 1 estimate historical mean as:as: tothe tænkte eksempler. sempler I det følgende illustrerer vi metoden gennem estimate the historical mean nedenstående optimeringsproblem: the historical mean as: st en normalfordelt, stokastisk variabel X ~ N (0,1) . Fra denne fordeling trækker vi 10.000 tilfældige observationer, o den deling trækker vi 10.000 observationer, 𝑝𝑝〉𝑝𝑝〉sit histogram i Figur 4. )~.𝐸𝐸(𝑋𝑋) Betragt først entilfældige normalfordelt, stokastisk og variabel Xpræsenterer Fra=gennem ~ N (0,1𝐸𝐸(𝑋𝑋) 〈𝑋𝑋, =〈𝑋𝑋, f x = arg min{ε ( f , f x )} den gennem sit histogram i Figur 4. denne fordeling trækker vi 10.000 tilfældige observationer, og f ∈V Figur 4:(vector Histogram af en standard normalfordelt variabel Where X denotes the panel in this specific case). Now say that we Where 𝑋𝑋 𝑋𝑋 denotes the panel (vector inin case). Now wewe Where denotes the panel (vector thisspecific specific case). Nowsay saythat that præsenterer den gennem sit histogram i Figur 4. togram af en standard normalfordelt variabel have a view that the asset’s expected return should be 2. Then we find a new have expected return should wewefind havea aview viewthat thatthe theasset’s asset’s expected shouldbesæt be2.2.Then finda a (bemærk, at 0.09 V erreturn Hvor et generisk afThen bibetingelser probability vector of which the inner product with the panel equals 2. new probability vector of which the inner product with the panel equals 2. new probability vector of Figur 4: Histogram af en standard normalfordelt variabel 0.08which the inner product with the panel equals 2. indeholdt den gængse fordelingsmæssige betingelse om 0.09

0.07

0.08

0.06

Densitiy

integrerer til 1=og 𝐸𝐸(𝑋𝑋) ==〈𝑋𝑋, 𝑝𝑝�〉𝑝𝑝�〉 〈𝑋𝑋, 𝐸𝐸(𝑋𝑋) =2kun 2 antager positive værdier). Det er også i V ~ views er indeholdt. Eksempelvis at middelværdien af f x 0.05 0.07 If Ifone 2,2,the onerow rowininthe thepanel panelis isexactly exactly theproblem problem solvedbybyjust just standardafvigelsen ercould 3.couldbebesolved 0.04 0.06 If one row in the panel is exactly 2, the problem could be solved byHowever, just asassigning that a weight However, assigning thatrow row aJweightofof1; 1;with withallallother otherrows rowsa weight a weightofof0.0. ~ ~ 0.05 signing that weight of 1; (with all aalso weight ofto0. However, this this ofFor as arearealso trying thiswill willnot not solution REM trying tominimize minimize the vi i Figur 1 to ( ~pbe,abepthe )the ( pas)we ]atwerows εrow = solution p0.03[ln pof)REM −grafisk lnother illustrere den relative entropithe viser 0.02 will not be the solution of REM as we are also trying to minimize the entropy j =1historical 0.04 entropy distribution. Thus, all rows will be assigned a a af det histo entropytowards towardsthe thehistorical distribution. Thus, all rows will be assigned Den ene (org dist) kan betragtes som fordelingen 0.01 towards the historical distribution. Thus, all rows will be assigned a positive positive probability, but some rows more than others. 0.03 positive probability, but someden rows more(view) than som others. anden det view vi har på~markedet. probability, Hvor but some rows more than others. p angiver den oprindelige sandsynlighedsvektor, og 5p angiver en 0 0.02 -5 -4 -3 -2 -1 0 1 2 3 4 To illustrate the approach in practice, we start by focusing on two Examples To illustrate the approach inFigur practice, we startvores byaffocusing on twosynthetic synthetic sandsynlighedsvektor, der overholder view. 0.011 1: Illustration to fordelinger -3 -2 -1 0 2 3 Examples 4 5 examples: One univariate and one multivariate. Antag at vi nu ønsker at pålægge et view om, at standardafvigelsen p examples: One univariate and one multivariate. To2 illustrate the approach in practice, we start by focusing on two synthetic 0 Examples 0.4 -5 -3 -2 -1 0 1 3 4 5 denne fordeling er 2 i stedet for 1. Det første viOrg dist betragter er de nu ønsker at pålægge et-4 view om, at standardafvigelsen på I det følgende illustrerer vi metoden gennem to tænkte eksempler. Tænkt Eksempler examples: One univariate and one multivariate. View 0.35 tilhørendestandard sandsynlighedsvektor for både dette view og vores oprindelig eling er 2 i Antag stedet atforvi 1. første vi betragter er consider den a aunivariate 𝑋𝑋 𝑋𝑋 ~𝑁𝑁(0,1). First, consider univariate standard Gaussiandistribution: distribution: ~𝑁𝑁(0,1). nuDet ønsker at pålægge etFirst, view om, at standardafvigelsen på Gaussian fordeling. For det ”oprindelige” data er sandsynligheds-vektoren andsynlighedsvektor for både dette view og vores oprindelige 0.3 From this we draw 10,000 observations and construct a histogram; Figure First, consider standard distribution: Betragt først en normalfordelt, stokastisk variabel X ~Figure N (05.,15.) . Fra uniform weadraw 10,000 observations and construct a histogram; denne fordeling er 2 i stedet for 1.From Det this første viunivariate betragter er den Gaussian fordelt, men ved atviand pålægge vores på standardafvigelsen or det ”oprindelige” data er sandsynligheds-vektoren uniformt From this wedenne drawog 10,000 observations construct aview histogram; Figure 5. og får vi en n fordeling trækker 10.000 tilfældige observationer, 0.25 tilhørende sandsynlighedsvektor for både dette view vores oprindelige sandsynlighedsvektor, hvor ikke alle elementer er ens. Nogle rækker ved at pålægge vores view på standardafvigelsen får vi en ny præsenterer den gennem sit histogram i Figur 4. fordeling. For det ”oprindelige” data er sandsynligheds-vektoren uniformt 0.2 panelet således tildelt en tungere vægt end andre. Figur 5 afbild edsvektor, hvor ikke men alle ved elementer er ens. Nogle i fordelt, at pålægge vores viewrækker på standardafvigelsen får bliver vi en ny 0.15 de to sandsynlighedsvektorer. er således tildelt en tungere vægt end andre. Figur 5 afbilder Figur Histogram af en standard normalfordelt variabel 5 5 sandsynlighedsvektor, hvor ikke alle elementer er 4: ens. Nogle rækker i 0.1 ynlighedsvektorer. panelet bliver således tildelt en tungere vægt end 0.09 andre. Figur 5 afbilder Figur 5: Sandsynlighedsvektorer: Oprindeligt data og med påla 0.05 de to sandsynlighedsvektorer. 0.08 view ndsynlighedsvektorer: Oprindeligt data ogPage med 8 pålagt Density

Densitiy



0.07

0 -6

Figur 5: Sandsynlighedsvektorer: Oprindeligt data og med pålagt

-4

-2

0

2

4

6

8

Figur 5: Histogram of a Standard Gaussian Variable 0.09 0.08 0.07

Densitiy

0.06 0.05 0.04 0.03 0.02 0.01 0 -5

0

5

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 above. As can be seen in Figure 6, the probability vector of the original distribution 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

Page 9

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 0.05 Org

0.04

Density

0.03 0.05 Org

0.045

0.02 0.04

0.035

Density

0.03

0.01 0.025

0.02 0.015

0

0.01 -6

-4

-2

0

2

4

6

0.005 0 -6

-4

-2

0

2

4

6

illustrateREM REMininaabivariate bivariate setting, setting, say system: ToTo illustrate saywe weobserve observethe thefollowing following system: 𝑋𝑋 1 1 0.5 � 1 � ~𝑁𝑁 �� � , � �� 𝑋𝑋2 0.5 0.5 2

Thatis,is,we wehave have two of which havehave positive mean mean values values and That twovariables, variables,both both of which positive a positive correlation. To To illustrate REM in ainnew way, Figure 8 depicts a a and a positive correlation. illustrate REM a new way, Figure 8 depicts scatter plot of 500 observations from this distribution. scatter plot of 500 observations from this distribution. Figure 8: Plot of Observations from a Multivariate Distribution 5 4 3

Aktiv 2

2 1 0 -1 -2 -3 -3

Page 10

-2

-1

0 Aktiv 1

1

2

3

Say we implement the view that the mean value of 𝑋𝑋1 equals -2. Figure 9 depicts the same observations as in Figure 8, but in this example we have

0.005 0 -6

-2

-4

0

4

2

6

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

Figur 8: Plot of Observations from a Multivariate Distribution 𝑋𝑋 1 1 0.5 �� � 1 � ~𝑁𝑁 �� � , � 𝑋𝑋2 0.5 0.5 2

6

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 4 plot of 500 observations from this distribution. scatter Figure 8: Plot of Observations from a Multivariate Distribution 2

Aktiv 2

5 4

03

Aktiv 2

2 1

-2

0 -1

-4-2 -3 -3 -3

-2 -2

-1 -1

0 Aktiv 1

0 Aktiv 1 1

1 2

2

3

3

implement viewthat thatthe themean meanvalue value of of 𝑋𝑋1 equals SaySay we we implement thetheview equals-2. -2.Figure Figure9 9 depicts the same observations as in Figure 8, but in this example we have depicts the same observations as in Figure 8, but now we have colored the colored the(rows observations (rowswhich in theare panel) whicha probability then are assigned a observations in the panel) assigned mass greprobability mass greater than 1/𝐽𝐽. ater than 1/J. Figure 9: Plot of Observations from a Multivariate Distribution

Figur 9: Plot of Observations from a Multivariate Distribution 6 P1/J 4

2

Aktiv 2

7

0

-2

-4 -3

-2

-1

0 Aktiv 1

1

2

3

Page 11

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

An Example: What Happens to my Portfolio, if...

The examples given above are naturally very stylized. To give a more practical one, let us illustrate by using a portfolio consisting of the weights presented in Table 1. Table 1: Portfolio Weights (example) Asset Equities High Yield Bonds Investment Grade Bonds Government Bonds Cash Total

Weight 20% 20% 10% 45% 5% 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. Multiplying this set of probability vectors onto our panel will result in the sensitivity 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

-0.5

-1.5

-0.4

-1.7

-2.0

0.0

2.0

-0.8

-0.1

-1.0

-0.4

4.0

6.0

8.0

10.0

0.5

1.1

1.6

2.3

3.3

3.7

4.1

3.8

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 2.8

0.1

-2.4

-2.1

-1.6

-1.1

-0.5

0.0

0.6

1.1

1.7

2.2

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 expected 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%.

Page 12

As a last practical example, we illustrate the usefulness of REM by the Markowitz 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 purely historical returns and covariances; these are estimated on the basis of weekly observations from January 1999 to October 2012.

Figur 10: Markowitz - Historical Distribution 100 IG DK 5Y Gov EMD HY SP 500 DAX

80

Weight, %

60

40

20

0

1

1.2

1.4

1.6 1.8 2 Standarddeviation, %

2.2

2.4

2.6

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.

Page 13

Example: Markowitz

Figur 11: Historical Returns for the Relevant Asset Classes 12

10

Return, %

8

6

4

2

0

IG

DK 5Y Gov

EMD

HY

SP 500

DAX

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 towards 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.

Page 14

Figur 12: Markowitz Based on the Original Data and a View that American Equities will return 15% 100 IG DK 5Y Gov EMD HY SP 500 DAX

80

Weight, %

60

40

20

0

1

2

3 4 Standarddeviation, %

5

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 German equities and High Yield Bonds, which both rise compared to the historical returns.

Figur 13: The Implied Returns Given Our View 18 16 14

Return, %

12 10 8 6 4 2 0

IG

DK 5Y Gov

EMD

HY

SP 500

DAX

Page 15

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 normalized 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 computational advantage.

J

ε ( ~p , p ) =  ~p[ln ( ~p ) − ln ( p )]

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 Hvor p angiver den oprindelige sandsynlighedsvektor, og ~ p . angiver en this is a very large optimization problem and probability vector Naturally, therefore sandsynlighedsvektor, der overholder vores view. difficult to solve numerically. To overcome this problem, Meucci (2010) proposes to solve the dual problem instead which reduces the proI det følgende illustrerer vi metoden gennem tænkte eksempler. blemtoso that we are merely optimizing over the number of views. Put differently, if we have three views then we have tree variables for the optimiBetragt først en normalfordelt, stokastisk variabel X ~ N (0,1) . Fra zation. j =1

denne fordeling trækker vi 10.000 tilfældige observationer, og Although the approach of Meucci (2010) reduces the computational burpræsenterer den gennem sit histogram i Figur 4.

den, it invalidates the optionality of views on non-linear moments. We therefore suggest solving the primal problem only, using large scale optimizaFigur 4: Histogram af en standard normalfordelt variabel tion packs such as e.g., KNITRO or those found in the standard version of 0.09 GAMS. It is our experience that this results in solutions which are equal to 0.08 those of the dual problem, although they must be judged on a case by case 0.07 basis; as one is not guaranteed to find an optimal solution. 0.06

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

Conclusion Densitiy

pler

Non-Linear Views

0.05

0.04 0.03 0.02 0.01

Through a series of examples this paper illustrates how the theory can be 3 4 in practice, 5 used both for risk and portfolio optimization purposes. It is our belief that using Bayesian statistics rather than setting parameter values diAntag at vi nu ønsker at pålægge et view om, at standardafvigelsen på rectly, results in far more desirable results and saves us a lot of technical denne fordeling er 2 i stedet for 1. Det første vi betragter er den problems such as ensuring statistical properties of the covariance matrix tilhørende sandsynlighedsvektor for både dette view og vores oprindelige are fulfilled. fordeling. For det ”oprindelige” data er sandsynligheds-vektoren uniformt 0 -5

-4

-3

-2

-1

0

1

2

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

Page 16

Black, 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

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Litterature

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