CONCEPTUAL FRAMEWORK FOR DEVELOPING AND VERIFICATION OF ATTRIBUTION MODELS. ARITHMETIC ATTRIBUTION MODELS

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1 CONCEPUAL FAMEOK FO DEVELOPING AND VEIFICAION OF AIBUION MODELS. AIHMEIC AIBUION MODELS Yui K. Shestopaloff, is Diecto of eseach & Deelopment at SegmentSoft Inc. He is a Docto of Sciences and has a Ph.D. in deeloping mathematical methods fo data intepetation. Yui deelops mathematical algoithms fo the financial industy and othe aeas and designs middle and lage-scale softwae applications and systems fo high-tech and financial companies, including pefomance measuement and tading systems fo majo financial institutions. He published about one hunded aticles in pee eiew scientific and pofessional jounals and 15 books. Abstact Aticle consides the aea of inestment attibution analysis. It intoduces mathematically gounded, objectie pinciples and eification citeia fo deeloping attibution models, which togethe ceate a conceptual attibution famewok. Existing models hae been scutinized using these new concepts; the known inadequacies of these models with egad to objectie epesentation of attibution paametes, such the inteaction effect, hae been confimed. New attibution models hae been deeloped based on intoduced pinciples, thooughly eseached, eified, and compaed with existing methods in tems of objectiity. he esults poed that the new attibution models poide moe objectie and meaningful esults than existing methods. Intoduction Attibution analysis, sometimes called as pefomance attibution, is a deeloped aea of financial mathematics widely used in pactical applications. Compehensie analysis of attibution methods fo equity potfolios, and some esults efeed to in this aticle, can be found in Shestopaloff, 008, 009a, 009b. In this aticle, we concentate on geneal pinciples how to deelop attibution models, intoducing a set of pinciples and alidation citeia. Based on these pinciples, togethe composing a conceptual deelopment famewok, we intoduce and alidate two new aithmetic attibution models. Besides, we conside two classic attibution models, fist pesented in Binson, 1985, called Binson-

2 Fachle model, and anothe intoduced in Binson, 1986, often efeed to as the BHB model. Pactical application of attibution models is consideed in Bacon, 008. Most of the intoduced pinciples and alidation citeia ae applicable to deeloping of one method. Howee, we also intoduce one specific citeion called atio alidation citeion that alidates one attibution method elatie to the othe. ith this egad, we analyze the geometic attibution model, pesented in Shestopaloff, 008, 009, and apply the afoementioned alidation citeion to compae this model and the newly deeloped aithmetic attibution model. In this pape, we conside mathematical fundamentals athe than paticula business pactices. he pesented attibution famewok is open to adjustment to paticula business equiements and pocesses in many diffeent ways. In this egad, we would like to efe the eade to aticle Hood, 005 and following esponses, which emphasize that the same fomula can be open to many diffeent intepetations and accodingly suppot diffeent business pocesses. By and lage, we pesent a compehensie and obust mathematical foundation on which many paticula methods and business pocesses can be built. ying to combine both tasks in one aticle would be athe impactical, gien the amount of pesented mateial. Howee, in the futue, we ae planning to addess moe paticula business pocesses, exploited by cetain financial institutions, based on this famewok. 1. Defining contibution e will stat fom contibution. he eason fo this is that some woks aise the question, if weights of secuities within the inestment potfolio should be defined elatie to thei beginning maket alue, o elatie to the ending maket alue. Suppose that potfolio consists of N goups of financial instuments it can be sepaate secuities, industy sectos, etc.. hen, accoding to a definition of ate of etun, we hae the following. p E p B B p p i N i N i N Bi 1 i Bi [ Bi 1 i Bi ] i N i 1 i 1 i 1 B p B p B i B i N i 1 p i 1 i i i 1

3 whee p is the potfolio ate of etun; i is the goup s ate of etun; i is the goup s weight within the whole potfolio; i - index denoting the goup s numbe; index p elates to potfolio; B denotes the beginning maket alue; E is the ending maket alue. Since fomula 1 uses simple ate of etun, in a stict mathematical sense, it is alid in the absence of cash flows. Howee, if cash tansactions ae small compaed to the beginning maket alue, it also poduces easonable appoximate esults. his fomula poes that when composing the total ate of etun of the potfolio o benchmak fom the ates of etun of included goups of secuities, we should use weights based on the beginning maket alues. his consideation is alid fo all tansfomations in case of aithmetic attibution models, including contibution to elatie etun. he last one is defined as D p, whee index b coesponds to a benchmak. In case of geometic attibution, the situation is diffeent. he ates of etun ae defined using 1, while the contibution to elatie etun D the paamete defining the diffeence between the benchmak s and potfolio s ates of etun on the goup leel, and excess etun sometimes this tem is used to denote the diffeence in ates of etun on the total leel ae defined by the ending maket alues. Accoding to the definition of the geometic contibution to elatie etun, we hae. b 1 D 1 1 Values of and denote the ates of etun of the same goup of secuities within the potfolio and benchmak accodingly. he intoduced elationship is not an obious one. e can ewite it as follows. e B E B e B E B e E e D B B E B B E B E E 1 B 3

4 whee e and E ae the ending maket alues of the potfolio s and benchmak s goups; B is the beginning maket alue that is chosen to be the same fo the potfolio and benchmak we can do this based on the definition of ate of etun. Since we use a simple ate of etun, as in 1, fomula 3 is alid in the absence of cash tansactions. So, in case of geometic attibution, the contibution to elatie etun is completely defined by the ending maket alues.. Pesent aithmetic attibution models he following notations will be in use. Capital lettes will efe to a benchmak, while the lowe-case lettes will efe to a potfolio. ate of etun will be accodingly and, weights and, diffeences between the ates of etun and weights ae defined as follows: d,. d BHB aithmetic attibution model he diffeence between the contibution by some goup of stocks into the potfolio ate of etun, and contibution of the same goup into the benchmak s ate of etun, is defined as follows. D d d d d d d 4 whee notations fo simplicity elate to one goup of stocks, so thee ae no indexes in this fomula. he fist tem on the second line in expession is a known industy selection o timing, the second tem is the stock selection. e also hae the thid tem in the BHB model, which is called inteaction. In fact, this tem epesents the model s noise facto and, gien the esults pesented in this aticle below, has no business meaning. hee wee a few attempts made to gie a business intepetation to the inteaction effect, but all of them ae diffeent and not eally conincing. hey seemed to hae appeaed due to ou human ability 4

5 to intepet whatee we encounte. In Spaulding, 003, the autho acknowledges: You may notice that I do not comment on the othe effect, since thee s no intuitie way to anticipate what the esult will be, whee othe is used fo denoting an inteaction effect. he existence of inteaction imposes limitations to the usage of two othe attibution paametes. Ideally, as long as we facto the diffeence in ates of etun between the benchmak and potfolio into the stock selection and industy selection, we would like fo these components to constitute the whole diffeence. he pesence of inteaction jeopadizes the factoing and makes the attibution paametes less aluable o maybe wothless. A numeical example in able demonstates this consideation, showing how the alue of inteaction can be compaable to the industy selection and stock selection alues. able 1. Sample potfolio with fou goups of stocks. Goup No otal able. BHB model. Attibution paametes fo the input data fom able 1. Goup No. Industy Sel Stock Sel. Inteaction otal Anothe impotant featue of the BHB model is the use of benchmak s paametes as pimay alues, compaed to potfolio s paametes. Legally, potfolio and benchmak ae 5

6 6 equal entities. So, thee ae no any easons to gie any pefeence to a benchmak. Equal entities should be teated equally to pesee the balance. Howee, this is not the case fo the BHB model. Let us explain this comment in moe detail. If we take a look at definition of stock selection and industy selection in 4, we can see that thee paametes fom fie belong to a benchmak, while only two paametes belong to a potfolio. his appeaance asymmety, in fact, leads to functional asymmety, giing highe weight to benchmak s paametes. Numeical esults will show this late. his potentially leads to stonge influence of benchmak paametes to the esults of attibution analysis. Binson-Fachle s attibution model he fomula that is usually attibuted to Binson-Fachle s attibution model diffes fom the BHB model in definition of industy selection, which is defined as follows in ou notations. B I 5 whee B denotes the total benchmak s ate of etun. Stock selection is the same, that is S 6 e can wite the contibution to elatie etun fo a single goup in the following fom. B B B B d d D ν 7 whee aiable B denotes the oeall benchmak etun.

7 So, we can see that in the Binson-Fachle s model the industy selection B is epesented by the second tem on the last line, fom the total fou that compose the contibution to elatie etun. Othe tems ae familia stock selection, inteaction, and a new summand B, which beside the inteaction epesents anothe model s noise facto. Simila to the inteaction tem, we can ty making some business sense of it, but this would not eally be conincing and doubtfully hae any pactical alue. Neithe this tem, no the inteaction tem, ae accepted by the community as legitimate attibution paametes. Unlike the inteaction, the alue of this tem is always zeo on the total potfolio leel. able 3 pesents attibution paametes fo the same data fom able 1. able 3. Attibution paametes computed using the Binson-Fachle model fo the data fom able 1. Goup No. Industy Sel Stock Sel. Inteaction otal ith egad to appeaance asymmety, this model includes fie benchmak paametes fom total seen, and two potfolio s paametes. 3. New scientific pinciples and alidation citeia fo deeloping attibution models e cannot ealuate how objectie ae the attibution paametes found by BHB o Binson- Fachle s methods in the aboe numeical examples. e just hae to take the esults fo ganted. At the same time, the lage alue of inteaction indicates that the attibution paametes may be inalid. So, we need some tools and citeia to ealuate objectiity of attibution models. 7

8 Peiously, we noticed the appeaance asymmety of fomulas defining attibution paametes. On the othe hand, benchmak and potfolio legally ae equally impotant entities. his assumes the necessity of appeaance symmety of attibution fomulas. he fomulas must be such, that it should not be diffeent if we compae benchmak to the potfolio, o potfolio to the benchmak. he esults can diffe in algebaic signs, but not in absolute alues. Besides, we should also be able to compae two benchmaks, o two potfolios. Appeaance and functional symmeties So, attibution methods hae to pesee the appeaance symmety, which assumes at least the same numbe of paametes coesponding to potfolio and benchmak. Secondly, fomulas must hae functional symmety as well. It means that if we substitute the benchmak s data by potfolio s data, and ice esa, then the esults can diffe in algebaic signs, but they hae to be the same in absolute alues. e will call this opeation and alidation pinciple as a data swapping symmety. Independence of attibution paametes hen we facto contibution to elatie etun into the stock selection and industy selection, we would like to obtain independent paametes. he analogy can be two-dimensional plane. he location of each point on this plane is defined by two independent alues, which ae coodinates of this point on the abscissa and odinate. ith this egad, we cannot guaantee that the stock selection and industy selection defined by the BHB and Binson-Fachle s models ae independent. In fact, fom the mathematical pespectie, which we discuss below, they ae not. Algeba has well deeloped fomal appaatus to define such independent alues fo mathematical expessions composed of combination of aiables. he cental idea is the intoduction of canonical foms of such mathematical expessions. Sometimes the canonical fom can be found by simple algebaic eaangements of the initial expessions, and we will use this appoach as one of the methods. he othe appoach is to use matix algeba and find 8

9 the eigenalues of the appopiate matix whose elements ae composed fom the tems of oiginal mathematical expession. e will use this method too. Note that both appoaches ae independent mathematical methods. So, if we manage to obtain the same esult by both methods, this will be a stong poof of the esult s alidity. Besides, if we manage to facto the contibution to elatie etun by these methods, it will necessaily guaantee the independence and uniqueness of such obtained paametes, and the absence of any othe associated alues which we consideed as noise factos in the peious models. hatee alue of contibution to elatie etun we will hae, it will be always factoed into two, and only two, independent alues without any inteaction o othe noise tems. If we take a look at fomulas 4 and 7, we can see that they do not epesent the canonical foms of contibution to elatie etun because of the pesence of noise factos. So, the existence of a canonical fom of contibution to elatie etun should be intoduced as the next pinciple in deeloping of attibution methods, which guaantees the independence and uniqueness of attibution paametes. Howee, this is not always possible to poe because of mathematical complexity of chaacteistic equation, when its solution cannot be found. In such a situation, the heuistic appoach can help, while the est of pinciples and alidation citeia of the deelopment famewok help to alidate such a heuistically discoeed method. A symmetical data set Anothe appoach to alidate attibution models is to use a symmetical data set. If we compose such potfolio and benchmak that the weights and ates of etun ae symmetical, then attibution paametes hae to be symmetical as well. he diffeence between the discussed ealie pinciple of data swapping symmety and a symmetical data set test is that in the last case we hae to do only one set of computations, while data swapping alidation equies two sets. Of couse, the data swapping alidation can be applied to a symmetical data set, although this is a paticula case of a moe geneal pinciple of data swapping symmety. 9

10 able 4 pesents an example of a symmetical data set, in the peious notations, while able 5 and able 6 show attibution paametes calculated fo this data set using the BHB and Binson-Fachle s models. able 4. Sample symmetical data set Goup No otal able 5. Attibution paametes fo the symmetical data set fom able 4, calculated using the BHB model. Goup No. Industy Sel Stock Sel. Inteaction otal able 6. Attibution paametes fo the symmetical data set computed by Binson-Fachle s model. Goup No. Industy Sel Stock Sel. Inteaction otal

11 e can see that both models poduce asymmetical attibution paametes. So, they do not sustain the alidation by a symmetical data set. atio alidation citeion his alidation citeion has been discoeed afte we deeloped seeal new attibution models based on intoduced conceptual famewok. So, to some extent, the discoey of this alidation citeion can see as a poof of alidity of the whole conceptual famewok and these new methods. he idea behind this citeion is the following. Suppose we ealuate attibution paametes by two methods and obtain two sets of attibution paametes. Let it be the industy selection and stock selection obtained by some aithmetic and geometic attibution models. hen, if the methods ae coect, it is easonable to expect that the atio of attibution paametes industy selection alue diided by the stock selection should be the same fo diffeent methods. his consideation has to be alid both on the total leel and on the goup leel. In a symbolic fom, it can be witten as follows. I S G G G A I I j I j ; 8 G A S S S A A j j whee indexes G and A elate to geometic and aithmetic attibution models accodingly; index denotes the total alues; index j denotes the j-th goup of financial instuments. So, we can intoduce the atio alidation citeion as follows: he atio of the appopiate attibution paametes fo the same input data should be identical acoss diffeent attibution methods designed to pocess this type of data within the same compaison context. e hae to stess that context of compaed methods should be the same. Fo instance, we can compae the aithmetic attibution method, consideing the whole alue of contibution to elatie etun, to the geometic attibution method that also factos into simila attibution paametes the whole alue of the geometic contibution to elatie etun. 11

12 Howee, if anothe method adjusts the ate of etun fo some theshold alue, then the context of methods becomes diffeent, and the atio alidation citeion cannot be applied. 4. Deeloping a new aithmetic attibution model Now, as we intoduced the geneal pinciples and alidation citeion of the new conceptual famewok, we will deelop new attibution methods to illustate the application of declaed pinciples. At fist, we find the canonical fom of a contibution to elatie etun fo the aithmetic attibution model. e can ewite 4 as follows. D 9 his fomula is simila to what we hae seen in the BHB model. he diffeence is that we use the weight fom a potfolio, not fom a benchmak. Altenatiely, we can egoup tems in 4 as follows. D 10 Now, we can expess alue D as a half sum of two expessions using 9 and 10. he esult will be as follows. D 11 Fomula 11 pesents the mathematical canonical fom of a contibution to elatie etun. It includes only two attibution paametes, which ae independent as functions of aguments and, which we conside as independent aiables fo the puposes of attibution analysis. eiewe of the fist esion of this aticle suggested anothe wit deiation of 11. So, thee ae seeal ways to obtain the same esult. he fist tem in 11 1

13 should be intepeted as a new industy selection, while the second tem is a new stock selection. Fo cetainty, we can call these paametes as a symmetical industy selection and a symmetical stock selection. he eason is that these paametes satisfy to symmety equiements intoduced ealie. he model itself will be called accodingly as a symmetical aithmetic attibution model SAA. he attibution paametes ae defined as follows. I s S s 1 13 whee 1 defines a symmetical industy selection, 13 defines a symmetical stock selection. he question that can be asked is as follows. hy these paametes ae moe objectie than the ones poduced by the BHB o Binson-Fachle s attibution models? he answe contains conceptual, mathematical, business, and just common sense consideations. Fom the business pespectie, in the new model, we do not hae inteaction effect o any othe noise factos, nee. So, unlike the case of the pesent models, whatee input data ae, the contibution to elatie etun will be always decomposed into the sum of two independent meaningful paametes without any othe noise factos. Fom stictly mathematical pespectie, fomula 11 epesents a canonical fom of the oiginal mathematical expession 4 with egad to attibution paametes. his means that thus defined paametes ae independent and unique, accoding to the popeties of canonical foms fo this type of mathematical expessions. On the opposite side, attibution paametes defined by the BHB o Binson-Fachle s models ae neithe independent, no unique we alone intoduced thee diffeent foms 4, 9, and 10 fo the BHB model. Let us ealuate the coespondence of this model to the conceptual famewok. Appeaance and data swapping symmety 13

14 Fomula 11 includes thee benchmak paametes and thee potfolio paametes. So, the appeaance symmety is fulfilled, at least with egad to the numbe of paametes. e can see that if we swap the potfolio data and benchmak data, then the fomula, in fact, emains the same. e just compae the benchmak to a potfolio in the same way as we compaed the potfolio to a benchmak. So, the attibution paametes will be the same in absolute alues, but diffeent in algebaic signs. e can do the appopiate algebaic tansfomations substituting the benchmak s paametes by potfolio s paametes and ice esa, but fo 11 this is an obious consideation. his confims the pesence of the data swapping symmety equied by the conceptual famewok. able 7 pesents these new attibution paametes fo the data fom able 1. If we swap the benchmak s and potfolio s data, the absolute alue of attibution paametes will emain the same, but the algebaic signs will change to opposite. So, both the analytical analysis and numeical example confim the model s data swapping symmety. able 7. Attibution paametes found using the SAA model. Data set fom able 1. Goup No. Industy Selec. Stock Selec. Inteaction otal Application of 11 to a symmetical data set fom able 4 poduces the attibution paametes pesented in able 8. able 8. Attibution paametes fo the symmetical data set calculated using the symmetical aithmetic attibution model SAA. Goup No. Industy Sel Stock Sel. Inteaction

15 otal e can see that the attibution paametes ae symmetical, as it is equied by the intoduced conceptual famewok. So, the model passed the test by a symmetical data set as well. 5. Finding a canonical fom of aithmetic contibution to elatie etun using eigenalues It is well known fom linea algeba that linea foms can be tansfomed into diagonal ones using thei eigenectos. If we intepet this fact in a moe pactical fom, it means that we can ceate a space of new independent infomatie alues fom the set of alues gien in a non-diagonal fom. e can pesent the industy selection, stock selection and inteaction fo the aithmetic contibution to elatie etun fom 4 in the following matix fom we use the ealie notations. p p p p 14 hat we did hee is the following. he half of inteaction alue p is added to the industy and stock selections located on the main matix diagonal. he matix tace will be equal to contibution to elatie etun, while the second diagonal has been filled with hales of inteaction alue. hus constucted matix will always poduce zeo inteaction alue assuming we can find its eigenalues. Adding elements on the second diagonal pusues the idea to edistibute the inteaction between eigenalues. Soling chaacteistic polynomial of this matix, we find the following eigenalues. [ 4 ] 1 I S 1 λ 1 p 15 [ 4 ] 1 I S 1 λ p 16 15

16 whee I, S. Note that disciminant in these fomulas is always non-negatie alue, so that these fomulas ae alid fo any combination of ates of etun and weights. he eade can think of eigenalues 15 and 16 as a mathematically coect way to edistibute the alues of non-diagonal tems togethe composing inteaction to the diagonal elements epesenting the industy selection and stock selection accodingly, added with a half of inteaction alue. e can also check compatibility with the BHB model scutinizing eigenalues. Fo example, if d 0, then the fist eigenalue poduces paamete simila to industy selection, while the second eigenalue poduces the stock selection, when d 0. hat is λ 1 17 λ 18 Note that, in fact, we obtained exactly the same attibution paametes as in 1, 13, although this time we did this using independent appoach. Values 17, 18 ae guaanteed to be independent and unique, because they epesent a mathematical canonical fom of the contibution to elatie etun. 6. A new efeential aithmetic attibution model Peiously, we showed that the Binson-Fachle s attibution model does not satisfy to data swapping symmety pinciple, and does not suppot the symmety of attibution paametes fo the symmetical input data set. Hee, we intoduce a new attibution model implementing the idea of some efeence alue, as the Binson-Fachle s model does efeing to the total benchmak s ate of etun. Howee, we will not define this alue fo now. Let us denote it as some theshold alue. e will use the following substitutions:,, that is the ates of etun adjusted fo the efeence alue. As a paticula case, alue can 16

17 17 be calculated as an aithmetic aeage of the potfolio s and benchmak s total ates of etun. e will define this new efeential aithmetic attibution model AA as follows. I s 19 S s 0 whee s I and s S ae new industy selection and stock selection adjusted fo a theshold alue. hus, the defined attibution model does not hae noise factos. he total contibution to elatie etun consists of only industy selection and stock selection. It can be poed as follows. s s D I S 1 1 his fomula discoes that the notion of contibution to elatie etun D has to be applied to the secto s ate of etun that is adjusted fo the efeence alue. Ealie, we used the contibution to elatie etun defined as D. Suppose, we would like to keep this definition, which is easonable assumption, because this way we pesee the consistency of the oeall appoach and acquie one moe connection with the poen method. hen, the equality D D has to be alid, that is

18 Solutions of this equation ae as follows. One is ν fo all 0. he second solution is 0. It is alid fo any alues of the sectos weights. he fist solution does not hae business sense to us, gien the natue of the poblem. he second solution is an inteesting deelopment. In fact, this paticula case is a symmetical aithmetic attibution model SAA that we intoduced peiously. Such a conesion of one model into anothe on the boundaies of paametes domain is one moe poof of both models alidity, because we deeloped them independently based only on the geneal pinciple of symmety. he definition of contibution to elatie etun fo the adjusted ates of etun can be intoduced as follows. Compaison of ates of etun elatie to some theshold alue means that we should weight the ates of etun elatie to this theshold. Othewise, thee is no way to weight the theshold alue popotionally to the secto s weight. his confims the alidity of intoducing contibution to elatie etun as a paamete that is adjusted fo the theshold alue, using the oiginal weights of goups. he attibution model 19, 0 pesees the oiginal idea to compute attibution paametes elatie to some efeence alue. his model is moe objectie in tems of factoing the contibution to elatie etun into two paametes. his is because it confims to symmety pinciples, and it does not hae noise factos such as inteaction and othe tems. Let us find the efeence alue. It tuns out that the goup leel attibution paametes satisfy to symmety equiement only if the efeence alue is such that it does not change when we swapped the data. Othewise, the attibution paametes fo the oiginal and swapped data ae asymmetical. Fo example, this condition fulfills when the efeence alue is popotional to aithmetic aeage of the potfolio s and benchmak s etuns. If we denote the total benchmak s ate of etun as B and the total potfolio s ate of etun as P, then the efeence alue in the AA model can be defined as follows. B P C 3 18

19 whee C is a eal numbe. If we denote the efeence alue as a function of B and P, that is B, P, then any alue of this function can be used as a efeence alue, poided the function satisfies the following condition. B, P P, B 4 In the numeical examples below, we use the efeence alue of B P. able 9 epesents attibution paametes fo the symmetical data set, when the efeential aithmetic model is used. esults demonstate that these attibution paametes ae symmetical. his is an expected esult gien the popeties of the AA model that should poduce symmetical attibution paametes fo the symmetical input data. able 9. Attibution paametes fo the symmetical data set when the AA model is applied. Goup No. Industy Sel Stock Sel. Inteaction otal he next test should be the appeaance symmety and the data swapping symmety. e can see that the appeaance symmety is peseed fo 19 and 0 when we define the theshold alue as in 3 o, in a moe geneal case, as in 4. Data swapping symmety can be poed analytically ewiting 19 and 0 fo the swapped potfolio s and benchmak s data. ables 10 and 11 pesent numeical examples confiming the popety of data swapping symmety fo this model, when we use the data set fom able 1. e can see that the attibution paametes ae the same in absolute alues, but hae the opposite algebaic signs. 19

20 able 10. Attibution paametes fo the symmetical efeential aithmetic attibution model. Data set fom able 1. Goup No. Industy Sel Stock Sel. Inteaction otal able 11. Attibution paametes fo the same swapped data. Goup No. Industy Sel Stock Sel. Inteaction otal he following consideations should be taken into account when defining the alue of theshold. e discussed befoe that the absolute alues of attibution paametes cannot depend on the diection of compaison. he eason is that these paametes objectiely pesent the diffeence between two entities of the same type. hen we intoduced the efeence alue, we changed the compaed alues. his change of compaed alues has to be balanced in such a way that the data swapping does not change the efeence alue, and the appeaance symmety has to be alid too. e also should conside the equiement that the attibution model mathematically has to be pesented by a canonical fom. Fomulas 19, 0 can be consideed as a canonical fom of the contibution to elatie etun adjusted fo the theshold alue with egad to independent aiables and, poided thei coefficients do not depend on these aiables. hen we choose the theshold alues accoding to 3 o 4, this condition holds tue. So, 19, 0 epesent a mathematical canonical fom. 0

21 7. atio alidation citeia Fo this pupose, we compae attibution paametes fo the symmetic aithmetic attibution model and symmetic geometical attibution model SGA deeloped in [4], and also descibed in [5]. e use the same data fom able 1 fo both models. Note that these models hae the same attibution context. Namely, both models facto the whole contibution to elatie etun into two attibution paametes. e calculated the atio of stock selection to industy selection on the total and goup leels fo these models. he esults ae pesented in able 1. e can see that they ae identical. So, both models confim to the atio alidation citeia, that alidates both models. able 1. Application of atio alidation citeion to SAA and SGA attibution models. Goup No. atio of Attib. paametes fo SAA atio of Attib. paametes atio fo SGA otal Conclusion he intoduced conceptual famewok fo deeloping attibution models poed to be a aluable and pactical instument, which includes a set of pinciples and alidation tools to which the deeloped attibution models should confim. e analyzed two existing attibution models and discoeed that they do not satisfy some equiements fom the pespectie of this new deelopment famewok. e intoduced two new attibution models that see as the illustation how the new concepts can be applied in pactice, completing eey step fom the models inception and though all alidation and testing phases. Futhe, we compaed two diffeent models in ode to demonstate the application of a atio alidation citeion. 1

22 his is a special alidation citeion. Unlike the peious alidation tools, this one is used fo alidation acoss diffeent models. Lack of space did not allow pesenting moe petinent issues, such as, fo instance, elationship of intoduced concepts and methods with absolute pofit and loss, which could be an inteesting deelopment too. Howee, since the only basis of poposed famewok is ates of etun, such elationships can be exposed faily staightfowadly once we know some absolute monetay alue, such as the beginning maket alue. Anothe inteesting aspect is linking of attibution paametes, of which some esults can be found in Shestopaloff, 009a. e think that oeall the esults conincingly demonstate the obustness and objectiity of the poposed conceptual famewok fo deeloping attibution models, and can be successfully applied to deelopment of othe attibution models. Although we esticted ouseles to analysis of equity potfolios, the same pinciples ae tansfeable to the attibution analysis of fixed income secuities. he summay of intoduced pinciples and alidation tools is pesented below in a tabula fom below. 1. A coect compaison method has a popety of symmety, which assumes that the absolute alue of the esult of compaison should not depend on the diection of compaison, whethe the fist entity is compaed to the second, o ice esa, although algebaic signs of the esults of compaison can be opposite. In attibution analysis, this pinciple leads to a data swapping symmety when swapping of the benchmak s and potfolio s data has to poduce the same absolute alues of attibution paametes, while the algebaic signs ae opposite.. he data swapping symmety esults in appeaance symmety of mathematical fomulas defining attibution paametes. 3. Existence and uniqueness of a canonical fom of the contibution to elatie etun. 4. Attibution paametes calculated fo the symmetical data set hae to be symmetical too. 5. atio alidation citeion assumes that the atios of attibution paametes on the

23 total and goup leels hae to be the same fo diffeent attibution methods deeloped within the same context. EFEENCES 1. Binson G.P., Hood.L., Beebowe G.L. Deteminants of Potfolio Pefomance, Financial Analysis Jounal, July-August 1986, p Binson, G. P., Fachle N., Measuing Non-U.S. Equity Potfolio Pefomance, Jounal of Potfolio Management, Sping 1985, p Bacon, C. Pactical Potfolio Pefomance Measuement and Attibution, iley, 008, 40 p. 4. Hood L.. "Deteminants of Potfolio Pefomance: 0 Yeas Late", Financial Analysts Jounal, 005, Vol. 61, No. 5, pp Shestopaloff Yu. K. Geometic Attibution Model and a Symmety Pinciple, Jounal of Pefomance Measuement, 008, No. 4, pp Shestopaloff Yu. K. Science of Inexact Mathematics. Inestment Pefomance Measuement. Motgages and Annuities. Computing Algoithms. Attibution. isk Valuation, AKVY PESS, 009a, 59 p. 7. Shestopaloff Yu. K. A Model fo a global Inestment Attibution Analysis, he Jounal of Pefomance Measuement, 009b, Vol. 13, No. 3, pp Spaulding, Daid, Inestment Pefomance Attibution, 003, McGaw-Hill, 54 p. 3

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