Potfolio Pefoance Attibution Alia Biglova, Svetloa Rachev 2 Abstact In this pape, we povide futhe insight into the pefoance attibution by developent of statistical odels based on iniiing ETL pefoance isk with additional constaints on Asset Allocation (AA), Selection Effect (SE), and Total Expected Value Added by the potfolio anages (S). We analye daily etuns of 30 stocks taded on the Gean Stock Exchange and included in the DAX30-index. The benchak potfolio is the equally weighted potfolio of DAX30-stocks. The potfolio optiiation is based on iniiing the downside oveent of the DAX-potfolio fo the benchak subject to constaints on AA, SE and S. We investigate also the distibutional popeties of AA, SE and S sequences by testing the Gaussian distibution hypothesis vesus stable Paetian hypothesis. Finally, we popose an eical copaison aong suggested potfolio choice odels copaing the final wealth, expected total ealied etun of the optial potfolio, and pefoance atios fo obtained sequences of excess etuns. Reseache, Institute of Econoetics, Statistics and Matheatical Finance, School of Econoics and Business Engineeing, Univesity of Kalsuhe, Kollegiu a Schloss, Bau II, 20.2, R20, Postfach 6980, D-7628, Kalsuhe, Geany E-ail: glova@statistik.uni-kalsuhe.de. Alia Biglova s eseach wee suppoted by the Deutschen Foschungsgeeinschaft, and the Deutsche Akadeische Austausch Dienst. 2 Chai-Pofesso, Chai of Econoetics, Statistics and Matheatical Finance, School of Econoics and Business Engineeing, Univesity of Kalsuhe, Postfach 6980, 7628 Kalsuhe, Geany and Depatent of Statistics and Aed Pobality, Univesity of Califonia, Santa Babaa, CA 9306-30, USA E-ail: achev@statistik.uni-kalsuhe.de Svetloa Rachev gatefully acknowledges eseach suppot by gants fo Division of Matheatical, Life and Physical Sciences, College of Lettes and Science, Univesity of Califonia, Santa Babaa, the Deutschen Foschungsgeeinschaft, and the Deutsche Akadeische Austausch Dienst.
Key Wods: Pefoance attibution, isk easues, tacking eo, potfolio optiiation.. Intoduction We stat with the definition of pefoance attibution. David Spaulding in his book [2003] wote that attibution is the act of deteining the contibutos o causes of a esult o effect ; we efe to the extensive efeence list in this book fo coete eview of pefoance attibution fo financial potfolios. Ti Lod [997] stated that the pupose of pefoance attibution is to easue total etun pefoance and to exain that pefoance in tes of investent stategy and changes in aket conditions. Attibution odels ae designed to identify the elevant factos that ipact pefoance and to assess the contibution of each facto to the final esult. In this pape we evaluate the pefoance of the potfolio elative to the benchak apying the attibution technique. Ou ain goal is to deteine the souce of the potfolio s excess etun, defined as the diffeence between the potfolio s etun and the benchak s etun. To evaluate the effects, causing the excess etun, and pefoance-attibution effects, we foulate seveal optiiation pobles based on iniiing Expected Tail Lost (ETL) (see Rachev S., Otobelli S.et al [2007] fo a suvey on isk easues) with constaints on Asset Allocation (AA) and Selection Effect (SE). The ain otivation of the pape is two-fold:. Having a potfolio of assets o, fund of hedge funds fo the potfolio anage o, fund anage, would like (a) to iniie tacking eo ove a given bench-ak potfolio o bench-ak index, and at the sae tie; (b) guaanteeing excess ean etuns at a given benchak level, and (c) keeng potfolio attibutions of its assets (funds) in desiable bounds. 2. We pefo statistical analysis on the optial potfolios obtained in the back testing in ode to bette undestand ipotant statistical featues of the optial pefoance attibution constaint potfolios. Having an optial potfolio of type (a,b,c), o a potfolio- o fund-anage will have uch oe assuances that this optial potfolio will peseve the bounds on the pefoance attibution until the next e-allocation. In the classical pefoance attibution liteatue, the optiiation coponent is issing. As advocated in Betand Ph. [2005], such a dawback could lead to non-desiable discepancies between optial (tacking) potfolios and potfolios with given attibution 2
bounds. While in this pape we conside only elative optiiation (based on tacking downside eo), siila esults will be put fowad fo absolute optiiations in anothe fothcoing wok. We now biefly descibe ou optiiation poble: we solve the optiiation pobles daily, in the peiod fo 07.0.2003 to 02.03.2007 (total 632 days), aking foecast fo the next day, based on obsevations of pio yea (250 woking days), and obseved the ealied excess etun ove the benchak potfolio. We have chosen as a benchak an equally weighted potfolio of 30 shaes, included in DAX30-index. We assue that potfolio anage can outpefo the benchak aking diffeent allocation decisions acoss industy classes (AA-effect,) o cking diffeent secuities than ae in the benchak (called SE). To odel the AA-effect, we divide the assets into industy classes accoding to thei tading volues. Recall that AA effect consists in adjusting the weights of the potfolio in ode to outpefo the benchak, see fo exae Betand Ph. [2005] and the efeences thee in. The potfolio anage ight oveweight the secto elative to the benchak if she is bullish and undeweight the secto if she is beaish. Selection effect consists in the cking diffeent secuities that, by the anage s onion, outpefo those in the benchak. She can also ck the sae secuities as in the benchak, but buy oe o less of the than ae in the benchak. Spaulding D. [2003] foulated one of the Laws of Pefoance Attibution: The su of attibution effects usts equal the excess etuns. We need to account 00% of the excess etun. Consequently, when we ve calculated all the effects, thei su ust equal ou excess etun. It eans that 00% of the excess etun should be exained by analyed attibution effects. In ou setting, the AA and SE will define constaint sets in optiiation pobles of type tacking eo iniiation. We will addess the following questions: Ae ou optiiation odels successful at deteining the weights of sectos (o assets classes) to help anage outpefo the benchak? Ae ou optiiation odels successful at selecting stocks within each secto to help anage outpefo the benchak? To evaluate ou optial potfolio selection odels, we pefo the eical analysis of final wealth and expected total ealied etun of the obtained optial potfolios with espect to the benchak. We copae the esults ove diffeent suggested potfolio odels to deteine the ost pofitable odel fo potfolio anages which ae best at outpefoing the chosen benchak. 3
Futheoe, we investigate distibutional popeties of obtained (as esults of ou optiiations) values of AA, SE, and S, defined as the diffeence between the total expected etun of the potfolio and the total expected etun of the benchak. In ou study of those distibutional popeties we ephasie the skewness and kutosis we obseved in the values fo AA, SE, and S showing a non-gaussian (so called stable Paetian distibutional behavio.) Recall that the excess kutosis, found in Mandelbot's [963] and Faa's [963, 965] investigations on the eical distibution of financial assets, led the to eject the noal assuption (geneally used to justify the ean vaiance appoach) and to popose the stable Paetian distibution as a statistical odel fo asset etuns. The behavio, geneally stationay ove tie of etuns, and the Cental Liit Theoe and Cental Pe-liit Theoe fo noalied sus of i.i.d. ando vaiables theoetically justify the stable Paetian appoach poposed by Mandelbot and Faa. Thei conjectue was suppoted by nueous eical investigations in the subsequent yeas (see Biglova A. et al [2004a, b]; Rachev S. et al [2003]]) and the efeences theein. In ou wok we will povide additional eical evidence testing noal and stable Paetian hypotheses fo AA-, SE- and S-values. The eainde of the pape is oganied as follows. Section 2 povides a bief desciption of ou data and ethodology. Section 3 povides a desciption of the optiiation pobles. Section 4 povides a nueical analysis of suggested optiiation odels. Section 5 concludes the pape. 2. Data and Methodology 2. Desciption of the Data Ou sae copises of 30 stocks taded on the Gean Stock Exchange and included in the DAX30-index. We analye the daily etuns of these stocks fo the peiod between 07.0.2003 and 02.03.2007. Daily etuns wee calculated as ( t) log( S( t) / S( t )), whee S(t) is the stock daily closing at t (the stocks ae adjusted fo dividends). Eveyday, we solve the optiiation poble using the obsevations fo the pio 250 woking days and ake a foecast fo the next day. We analye two potfolios: the benchak potfolio and the potfolio of DAX30-stocks we want to optiie. Ou benchak potfolio is equally weighted potfolio of 30 shaes, included in DAX30-index. We assue that potfolio anage would like to outpefo the benchak aking diffeent allocation decisions acoss industy classes. Fo that, we divide shaes of DAX30 into 5 industy classes accoding to thei tading volue: 4
Tading _ volue( shae( i)) Volue( T ) * Stock _ value( T )), whee i is shae s nube, T- the tie peiod, Volue(T) is the aveage volue ove the entie peiod T, and Stock value (T) is the aveage stock value ove the entie peiod T. Evey class contains 6 shaes (6 shaes with the sallest tading volue efe to the fist class, 6 shaes with the lagest tading volue efe to the 5th class). The stuctue of ou benchak potfolio is pesented in Table 3 (see Section 4.2). 2.2 Calculation of the potfolios paaetes Daily, we calculate the following paaetes sepaately fo each potfolio, based on obsevations of pio yea (250 woking days): Potfolio of DAX-stocks Benchak potfolio is the ando daily etun of asset l in the DAX-index l In the tie seies setting, (t) is the etun of the asset l at tie peiod (day) t, l l,..., n; t,..., T, wee n is a nube of assets equal to 30, T250.We will use the sae notation fo obseved histoical values of l (t). Rl - ean etun (expected value of l ), R E ) - the weight of asset l in the potfolio of DAX30-stocks. l ( l bl - the weight of asset l in the benchak potfolio. p... p p2 pn b b b... 2 bn n l l n l fo the DAX30-potfolio is the total ando etun n bl l In ou case, the benchak potfolio is equally weighted, / 30. n l bl l bl is the total ando etun fo the benchak potfolio 5
w l U i is the weight of asset class in the potfolio of DAX30-stocks U i w bl l U i is the weight of asset class in the benchak potfolio U i w p w w... w p p2 p, whee is a nube of classes (in ou case is equal to 5) w b w w... w b b2 b Fo ou equally weighed benchak potfolio, w / 5. w l w l ( i) w w l U i is the weight of asset l in its asset class of the potfolio of DAX30-stocks l U i w ( i) l U i is the ando etun fo asset class i of the potfolio of DAX-stocks. n l l i w, ( i) bl bl wbl bl w l U i is the weight if asset l in its asset class of the benchak potfolio U i Fo ou equally weighed benchak ( i) / 30 potfolio, w bl / 5 6 ( i) w l U i is the ando etun fo asset class i in the benchak potfolio. n bl l bl l l i w is the total DAX30-potfolio (daily) etun R E ) ( is the expected etun of asset class i of the potfolio of DAX-stocks is the total benchak-potfolio (daily) etun R E ) ( is the expected etun of asset class i of the benchak 6
R E( ) n l R l i is the total expected (daily) etun of the potfolio of DAX-stocks ove all classes w R R E( ) n l bl R l i is the total expected (daily) etun of the benchak potfolio w R Afte these paaetes ae calculated, we calculate S, the total expected value added by the potfolio anages: S R R ( w R w R ) ( ) R. i l The ai of potfolio attibution is to beak down total value added into its ain souces, naely: asset allocation (AA), secuity selection (SE), and inteaction (I). n bl l Asset Allocation (AA): The contibution of the asset class i to the total value added easued by: AA i ( w w )( R R (b) ) The total asset allocation effect is easued by: AA i ( w w )( R R ) Selection Effect (SE): The contibution of the total out pefoance of the choice of secuity within each asset class is given by: SE i w ( R R The total selection effect is given by: ) SE i w ( R R ) 7
Inteaction: Betand Ph. [2005] stated that the su of the asset allocation and selection effects is not equal to the total outpefoance of asset class i,. To ensue equality, it is necessay to add a te efeed to as inteaction that is defined by: I w w )( R R i S i ( It can be intepeted as the pat of the excess etun jointly exained by the asset allocation and selection effects. It can be defined as an extension of the effect of secuity selection: it is the secuity selection effect on the ove- o unde-weighted pat of asset class i. ) S ( w R w R ) ( AA + SE + I ). i 3. Desciption of Optiiation Pobles Ou potfolio optiiation odels ae based on iniiing a downside (tail) isk easue, called Expected Tail Loss (ETL), also known as Total Value-at-Risk (TVaR), Expected Shotfall, Conditional Value-at-Risk (CVaR ), and defined as δ ETLδ ( X ) δ VaRq ( X ) dq, 0 i Whee ( X ) F ( δ ) inf{ x / P( X x) δ} VaR X is the Value-at-Risk (VaR) of the δ ando etun X. If we assue a continuous distibution fo the pobality law of X, then ETL ( X ) E( X / X VaR ( X )) and thus, ETL can be intepeted as the aveage δ loss beyond VaR, see Rachev S., Otobelli S. at al. [2007]. δ i i i 3. Optiiation Tacking Eo Poble with Constaints on AA, SE, S, and individual asset weights Ou goal is to find an optial potfolio iniiing the tacking eo easued by ETL ( ). We shall exaine vaious optiiation pobles choosing diffeent δ δ 0.0, 0.05, 025 and 0.50, subject to constaints on the AA, SE and S in contast to the standad tacking eo given by the standad deviation STD( ), by using ETL ( δ ).Thus, we do not penalie fo positive deviations of ou potfolio fo the benchak; we only iniie the downside oveent of the optial DAX30-8
potfolio fo the benchak (see Rachev S., Otobelli S. et al [2007]). Optiiation Poble 3..: Miniu ETL-Tacking Eo with Constaints on asset weights, AA and SE: such that in ETL δ ( (i) > 0, wee is the weight of individual asset l in the potfolio of DAX30-stocks, (ii) (iii) a AA c SE i i n p l ( w w )( R R ) b ; w ( R R ) d. ; The constants a,b,c,d can be pe-specified to eet paticula needs of the potfolio anage. In ou case they can take atay values. Optiiation Poble 3..2: Miniu ETL-Tacking Eo with Constaints on asset weights, AA, SE and S: in ETL δ ( ) ) such that (i),(ii),(iii) hold and (iv) S R R s, whee s>0 the excess total (benchak) expected value added we want to achieve with iniu ETL-tacking eo. 3.2 Optiiation Tacking Eo Poble with constaints AA, SE, S, and asset classes weights Optiiation Poble 3.2.: Miniu ETL-Tacking Eo with Constaints on classes weights, AA, and SE: in ETL w δ ( ) 9
such that (I) w > 0, wee w is the weight of asset class i in the potfolio of DAX-stocks, w l, and (ii) and (iii) ae hold. In this optiiation poble, afte the optial potfolio of DAX30-stocks is found, we deteine the weights of assets within the classes in coespondence with the stuctue of the benchak potfolio. As ou benchak potfolio is equally weighted potfolio, we ipose that the classes in the optial potfolio of DAX-stocks ae also equally weighted potfolios. Each class contains 6 shaes. It eans that the weight of asset in the class will be found accoding to this stategy as w / 6, wee w is the optial weight of the class i in the potfolio of DAX30-stocks. Optiiation Poble 3.2.2: Miniu ETL-Tacking eo with constaints on classes weights, AA, SE and S: in ETL w ( ) δ, such that (I), (ii), (iii) and (iv) hold. The weights of assets ae found as descibed in Optiiation Poble 3.2.. 4. Eical Analysis of Potfolio Pefoance. Suppose an investo has an initial wealth of W0 on Septebe 20, 2004. Evey day she solves the optiiation poble descibed above using daily obseved etuns fo the pio yea. Once she deteines the optial potfolio of DAX30-stocks, at tie t, based on the histoical etun values until t (including), that is, (t), l,,n the potfolio wealth at tie (t+) geneated by the potfolio allocation at tie t is evaluated accoding to W ( t + ) W ( t)( + ( t + )), whee the potfolio s etun (t+) at tie t+ is given by ( t + ) p( t) ( t + ) +... + pn ( t) n ( t + ). 0
The cuulative potfolio etun CR (t+) at tie t+, geneated by the potfolio allocation ade at tie t, is defined iteatively by CR( t + ) CR( t) + ( t + ). Values of the final wealth and cuulative etun fo the benchak potfolios wee calculated in the sae way. 4. Results Suay. In Table (Panels A, B, C, D), we fist pesent esults obtained by the fou diffeent optiiations with diffeent constaints on AA, SE, and S. Ou esults show that Optiiation Poble 3..2, based on ETL 0.05 -tacking eo-iniiation, and Optiiation Poble 3.., based on ETL 0.05 -tacking eo-iniiation with constaints 0.03 AA 0.3, 0.03 SE 0.3, 0, povide the lagest ealied wealth and total ealied etun at the end of the peiod T. These stategies wee efeed to the ost pofitable stategies. Futheoe, Optiiation Poble 3..2 povides consistently ost pofitable stategies ove a vaiety of constaints sets, while Optiiation 3.. is best only in the long shot stategy whee the Optiiation Poble 3..2 is aginally second. Oveall, the best pefoing potfolio, egadless whethe we can conside long-only o long-shot stategy, is given Optiiation Poble 3..2. [Inset Table hee] As the investigation of the stategies based on copaison of ealied wealth and total ealied etun doesn t take isk into account, we futhe analye the sequences of ealied excess etuns, obtained ove the ost pofitable optiiation pobles, conside thei tail-isk pofile and select the ones with best isk-etun pefoance. Ou next goal is to deteine a odel which achieves the Best Tacking Eo Potfolio with Pefoance Attibution Constaints (we call shotly this potfolio BTEP) taking tail-isk (pobality fo lage losses) into account. Fo that, we conside the sequences of ealied excess etuns: s( t) ( t) ( t), t,,t
whee ( ) p ( t) is the DAX30-potfolio etun at tie t geneated by the potfolio allocation at tie (t-), obtained by the optial stategies obtained in solving the coesponding optiiation pobles 3.., 3..2, 3.2., and 3.2.2; ( ) b ( t) is the equally weighted benchak potfolio etun at tie t. The sequence of excess etuns contains total T 632 obsevations stating fo the 25st day of the peiod exained as the fist-day optiiation poble is solved based on fist 250 obsevations of the fist yea. We stat ou isk-analysis of the optial potfolios by coputing the ost coonly accepted isk-ewad easue, the Shape Ratio (Shape W. [994]) (see () below), using the sequences of ealied excess etuns s(t) fo t,,t. Howeve, in ode to include in the isk-etun analysis, the obseved non-noality distibution of the ealied excess etuns s(t), we also calculate the STAR Ratio (STARR) and R-Ratio (see (2) and (3) below) as altenatives to the Shape atio eacing the standad deviation in the Shape atio with the tail-isk easued by ETL. We analye and copae STARR Ratio (0.05), and R-Ratio (0.05, 0.05) using the 5% of the excess highest and lowest etuns. The choice of those quantiles is based on the pefoance-evidence we have collected in ou pevious papes on potfolio optiiation. See fo exae Biglova A. at al [2004a,b]. We now give a suay of the thee pefoance atios:. The Shape Ratio (see Shape W. [994]) is the atio between the expected excess etun and its standad deviation of the ealied excess etuns s with stable distibution deteined by the sae T s(t), t,,t : E( s) ρ ( s ) () STD( s) whee E(s) and STD(s) is the ean and sae standad deviation s. Fo this Ratio it is assued that the second oent of the excess etun exists, thus the stable distibution we use fo odeling the pobality distibution of s is, in fact, Gaussian. (We give the definition and discuss the basic popeties of stable distibutions fo odeling asset etuns in the next section.) 2
2. STARR (0.05) (see Rachev S. et al [2007a]) is the atio between the expected excess etun and its Expected Tail Loss: E( s) ρ ( s) (2) ETL ( s) whee ETL δ (s) is defined in Section 3. δ 3. R-Ratio (0.05, 0.05) The R-Ratio is the atio between the Expected Tail Retun ETR(s) ETL(-s) at a given confidence level and the ETL of the excess etun at anothe confidence level. ETLγ ( s) ρ( s) (3) ETL ( s) γ 2 We analye the R-Ratio fo paaetes γ γ 2 0.05. Fo (2) and (3) to exist (to be well defined) we only need that the index of stality of s is geate than, which is the ean of s exists. All eical analysis on the distibution of asset etuns show that, without estiction, one can assue that that the ean of asset etuns is finite (see Rachev S.[2007a] and the efeences thee in). Table 2 epots values of stable distibution paaetes, pefoance atios: Shape Ratio, STARR-Ratio (0.05), and R-Ratio (0.05, 0.05) fo ealied excess etuns ove analyed stategies. Results, pesented in Table 2, show that Shape Ratio is not suitable to be aed as the coefficients of stable fit confi that the ealied excess etuns ae non-gaussian, heavy-tailed and skewed, hence STARR and R-Ratio ae oe eliable. This table shows that Optiiation Poble 3..2 based on ETL (0.05) with long-shot constaints on AA and SE: 0.03 AA 0.3, 0.03 SE 0.3, S 0, 0 povides the best values of STARR equal to 0.0622 and R-Ratio equal to.883, theefoe it povides the best potfolio when we take into account the tail-isk of the ealied excess etuns. We call this optial potfolio the Best Tacking Eo Potfolio with Pefoance Attibution Constaints (BTEP) and we shall analye it now in oe detail. [Inset Table2 hee] 3
4.2 Analysis of the BTEP. Having solved the optiiation poble daily, a total of 632 ties, in the peiod fo 07.0.2003 to 02.03.2007, we pesent ean statistics of the obtained daily optial potfolios of DAX30-stocks in Table 3. [Inset Table 3 hee] Table 3 shows that fo ost cases, the ain pats of optial potfolios of DAX30-stocks wee the shaes of the second and the thid classes (39% and 26%). It eans that in ost cases the potfolio of DAX-stocks consisting of shaes with aveage tading volues outpefos the potfolio of benchak. Figue pesents gaphs of the ealied final wealth of the potfolio of DAX-stocks and benchak- potfolio. This exhit shows that the potfolio of DAX-stocks sae paths doinate the benchak sae paths and they yield the axiu wealth of 2. at the end of the peiod exained, the axiu wealth of the benchak potfolio is equal to.47 at the end of the peiod exained. Figue 2 pesents sae paths of cuulative etuns fo the potfolio of DAX-stocks and the benchak potfolio. The ots also show that the potfolios of DAX-stocks always pefos bette than the benchak potfolio and it yields the axiu total ealied annualied etun equal to 77.53% at the end of the peiod exained (and thus the annualied value is 30.67%, the total ealied etun of the benchak potfolio is equal to 4.% at the end of the peiod exained (with annualied value of 6.26%). [Inset Figues,2 hee] We now focus on the statistical analysis of the tie seies of AA, SE and S-values in the optial DAX30- potfolio and the benchak potfolio. We view the obsevations of Asset Allocation (AA), Selection Effect (SE), and Total expected value, added by potfolio anages (S), calculated in solving the optiiation pobles 632 ties, as thee saes of sie 632 each, and we would like to study the 4
distibutional popeties of the AA, SE, and S, and in paticula ean-values, dispesion, skewness and kutosis. The fist obsevation we ade concens the non-noality of the distibution of the saes fo AA, SE, and S. We obseve that by testing the hypotheses about noal (Gaussian) vesus stable (non-gausian, Paetian) distibutions fo the AA, SE and S values. Let us fist ecall soe basic facts on stable distibutions. The α -stable distibutions descibe a geneal class of distibution functions which include leptokutic and asyetic distibutions. A ando vaiable X is stable distibuted if thee exist a sequence of i.i.d. ando vaiables { Yi} i N, a sequence of positive eal values { d i} i N and a sequence of eal values { a } such that, as n + : whee d i i N d n Y i n i + a n d X, points out the convegence in distibution. The chaacteistic function which identifies a stable distibution is given by: α α exp γ u ( iβ sgn( u) tan( πα 2) ) + iµ u Φ X ( u) E(exp( iux )) 2 exp γ u + iβ sgn( u)log( u) + iµ u π { } if α if α Thus, an α -stable distibution is identified by fou paaetes: the index of stality α (0,2] which is a coefficient of kutosis, the skewness paaete β [, ], µ R + and γ R, which ae espectively, the location and the dispesion paaete. If X is a ando vaiable whose distibution is α -stable, we use the following notation to undeline the paaete dependence (see Saoodnitsky G., Taqqu M. [994]): S γ, β µ. X ( ) α, When α 2 and β 0 the α -stable distibution has a Gaussian density. Theα -stable distibutions with α < 2 ae leptokutotic and pesent fat tails. While a positive skewness paaete ( β > 0 ) identifies distibutions whose tails ae oe extended towads ight, the negative skewness paaete ( β < 0) tycally chaacteies distibutions whose tails ae extended towads the negative values of the distibution. If α < 2, then X is called stable (non-gaussian, o Paetian ) ando vaiable. We estiate the stable distibution paaetes of the sequences AA, SE and S by axiiing the likelihood function (see McCulloch J. [998], Stoyanov S. and Racheva- Iotova B. [2004a,b]). It is possible to obtain optial appoxiations of the stable 5
paaetes with STABLE poga, developed and descibed in Stoyanov S., Racheva B. [2004a,b]). We copute the ain paaetes of the stable law: the index of stalityα, skewness paaete β, which will chaacteie the heavy-tailedness and asyety of the obsevations distibutions espectively. We also copute µ and σ in the Gaussian fit. The noality tests eoyed ae based on the Kologoov distance (KD) and coputed accoding to KS sup F x R S ( x) Fˆ ( x) F S ( ) ( x) whee x is the eical sae distibution and Fˆ is the standad noal cuulative distibution function evaluated at x fo the Gaussian o stable fit, espectively. Ou esults show that we can eject the noality using the standad Kologoov- Sinov test fo obsevations of AA, SE and S values at the exteely high confidence level of 99%. In contast, the stable-paetian hypothesis is not ejected fo these sequences at the sae confidence level. Figue 3 pesents the gaphs of distibution densities of AA, SE, and S sequences. Figue 4 pesents the histogas of AA, SE, and S values with espect to noal distibuted values. [Inset Figues 3,4 hee] Gaph show that the analyed obsevations exhit heavie tails than that the noal. The fit of stable non-gaussian distibution is now aed to the obsevations and the paaetes of stable distibution ae obtained. Table 4 pesents obtained paaetes and K-S statistics fo the noal and stable non-gaussian cases. The ean values of annualied AA, SE and S ae also pesented in Table 4. [Inset Table 3 hee] Table 4 shows that the K-S distances in the stable case ae 0 ties salle than the K-S distances in the Gaussian case fo the analyed sequences. So showing clealy that the stable fit outpefos the Gaussian one. 5. Conclusions In this study, we futhe develop pefoance attibution ethods intoducing new 6
optiiation odels based on ETL-isk easue. We deteine the ost pofitable odel fo potfolio optiiation, which best outpefoed the benchak potfolio. In addition, we analye the distibutional popeties of Asset Allocation (AA), Selection Effect (SE) and Total Expected Value, Added by potfolio anages (S), and stongly eject fo those sequences the noality assuption in favo of the stable Paetian Hypothesis. In the futue, we expect to confi the obtained esults on a lage dataset and futhe develop suggested odels. Refeences. Biglova A., Otobelli S., Rachev S. and Stoyanov S. (2004a). Copaison aong diffeent appoaches fo isk estiation in potfolio theoy, Jounal of Potfolio Manageent, New Yok, Vol. 3, pp. 03-2. 2. Biglova A., Otobelli S., Rachev S., Stoyanov S. (2004b) potfolio selection and Risk anageent: A copaison between the stable paetian appoach and the Gaussian one, S.Rachev (edt.)handbook of Coputational and Nueical Methods in Finance, Bikhause, Boston, pp.97-252, 3. Betand Ph. (2005) A note on potfolio pefoance attibution: taking into account, Jounal of Asset Manageent, Vol. 5, pp. 428-437. 4. Faa E. (963). Mandelbot and the Stable Paetian Hypothesis, Jounal of Business, Vol. 36, pp. 394-49. 5. Faa, E. (965). The Behavio of Stock Maket Pices, Jounal of Business, Vol. 38, pp. 34-05. 6. Lod T. (997) The Attibution of Potfolio and Index Retuns in Fixed Incoe, The Jounal of Pefoance Measueent. Vol.2. 7. Mandelbot B. (963). The Vaiation in Cetain Speculative Pices, Jounal of Business, Vol. 36, pp. 394-49. 8. McCulloch J. (998). Linea egession with stable distubances, A Pactical guide to heavy Tailed Data, R. Adle at al. (edt) Bikhause, Boston. 9. Rachev S., Tokat Y., and Schwat E., (2003). The Stable non-gaussian Asset Allocation: A copaison with the Classical Gaussian Appoach, Jounal of Econoic Dynaics and Contol, Vol. 27, pp. 937-969. 0. Rachev S., Matin D., Racheva-Iotova B. and Stoyanov S.(2007a), Stable ETL optial potfolios and extee isk anageent', fothcoing in Decisions in Banking and Finance, Spinge/Physika. Rachev S., Otobelli S., Stoyanov S., Faboi F., Biglova A. (2007b) Desiable Popeties of an Ideal Risk Measue in Potfolio Theoy, fothcoing in 7
Intenational Jounal of Theoetical and Aed Finance 2. Saoodnitsky G. and Taqqu M. (994). Stable Non-Gaussian Rando Pocesses, Stochastic odels with Infinite Vaiance. Chapan and Hall, New Yok. 3. Shape W. (994). The Shape Ratio, Jounal of Potfolio Manageent, pp. 45-58. 4. Spaulding D. (2003). Investent pefoance attibution: a guide to what it is, how to calculate it, and how to use it. New Yok, NY: McGaw-Hill. 5. Stoyanov S., Racheva-Lotova B. (2004a). Univaiate stable laws in the field of finance-appoxiations of density and distibution functions, Jounal of Concete and Acable Matheatics, Vol. 2/, pp. 38-57. 6. Stoyanov S., Racheva B. (2004b). Univaiate stable laws in the field of financepaaete estiation, Jounal of Concete and Acable Matheatics, Vol. 2/4, pp. 24-49. 8
Table Suay statistics ove diffeent analyed optiiation odels ETL(0.0) ETL(0.05) ETL(0.25) ETL(0.5) Annualied Annualied Annualied Annualied Potfolios Realied Total Realied Total Realied Total Realied Total wealth Realied wealth Realied wealth Realied wealth Realied Retun (%) Retun (%) Retun (%) Retun (%) Benchak.47 6.26%.47 6.26%.47 6.26%.47 6.26% Panel A: long only constaints on AA and SE Optiiation Poble 3.. ( 0 AA,0 SE, 0 ) Potfolio.9 26.68% 2.04 29.33%.82 24.60%.88 26.0% Optiiation Poble 3..2 ( 0 AA,0 SE, S 0, 0 ) Potfolio Potfolio Potfolio.94 27.4% 2.09 30.23%.83 24.88%.92 26.89% Optiiation Poble 3.2. ( 0 AA,0 SE, w 0 ) NO feasible solution Optiiation Poble 3.2.2 ( 0 AA,0 SE, S 0, w 0 ) NO feasible solution Panel B: long-shot constaints on AA and SE Optiiation Poble 3.. ( 0.03 0.3, 0.03 SE 0.3, 0 AA ) potfolio potfolio potfolio potfolio.88 26.03% 2.3 3.03%.85 25.38%.89 26.09% Optiiation Poble 3..2 ( 0.03 0.3, 0.03 SE 0.3, S 0, 0 AA ).88 26.03% 2. 30.67%.82 24.60%.9 26.62% Optiiation Poble 3.2. ( 0.03 0.3, 0.03 SE 0.3, w 0 AA ) 0.82-5.56%.9 8.70%.20 8.89% 0.92 0.08% Optiiation Poble 3.2.2 ( 0.03 0.3, 0.03 SE 0.3, S 0, w 0 AA ) NO feasible solution 9
ETL(0.0) ETL(0.05) ETL(0.25) ETL(0.5) Annualied Annualied Annualied Annualied Potfolios Realied Total Realied Total Realied Total Realied Total wealth Realied wealth Realied wealth Realied wealth Realied Retun (%) Retun (%) Retun (%) Retun (%) Benchak.47 6.26%.47 6.26%.47 6.26%.47 6.26% Panel C: no constaints on AA, long only constaints on SE Optiiation Poble 3.. ( AA,0 SE, 0 ) Potfolio.88 26.03% 2.0 30.53%.82 24.63%.89 26.6% Optiiation Poble 3..2 ( AA,0 SE, S 0, 0 ) Potfolio Potfolio Potfolio.88 26.03% 2.2 30.74%.83 24.87%.90 26.38% Optiiation Poble 3.2. ( AA,0 SE, w 0 ) NO feasible solution Optiiation Poble 3.2.2 ( AA,0 SE, S 0, w 0 ) NO feasible solution Panel D: no constaints on SE, long only constaints on AA Optiiation Poble 3.. ( 0, SE, 0 AA ) potfolio potfolio potfolio potfolio.92 26.87% 2.06 29.66%.8 24.40%.89 26.08% Optiiation Poble 3..2 ( 0 AA, SE, S 0, 0 ).94 27.4% 2.09 30.23%.83 24.92%.93 26.9% Optiiation Poble 3.2. ( 0 AA, SE, w 0 ).23 8.62%.28 0.28%.33.90%.36 2.52% Optiiation Poble 3.2.2 ( 0, SE, S 0, w 0 AA ) NO feasible solution 20
Table epots values of ealied wealth and annualied total ealied etun obtained ove diffeent atheatical odels with diffeent estictions on AA, SE, and S. The sae includes a total of 30 stocks taded on the Gean Stock Exchange duing the peiod of Octobe 2003 and Mach 2007. Table 2 Suay statistics of excess ealied etuns ove the ost pofitable optiiation pobles alpha beta siga u Shape Ratio STARR Ratio (0.05) R-Ratio(0.05,0.05) Optiiation Poble 3..2 based on ETL(0.05) long only constaints on AA and SE: 0 AA,0 SE, S 0, 0.5557 0.3066 0.003 7.02e-004 0.0879 0.0397.4005 Optiiation Poble 3..2 based on ETL(0.05) long-shot constaints on AA and SE: 0.03 AA 0.3, 0.03 SE 0.3, S 0, 0.557 0.2927 0.003 7.059e-004 0.0907 0.0622.883 Optiiation Poble 3..2 based on ETL(0.05) no constaints on AA, long only constaints on SE: AA,0 SE, S 0, 0.557 0.2927 0.003 7.096e-004 0.09 0.0324.3483 Optiiation Poble 3.. based on ETL (0.05) no constaints on SE, long only constaints on AA: 0.03 AA 0.3, 0.03 SE 0.3, 0.5325 0.320 0.003 7.92e-004 0.0928 0.0278.937 Table 2 epots values of stable distibution s paaetes fo the sequences of ealied excess etuns and values of pefoance Ratios fo those sequences. 2
Table 3 Mean statistics of the Best Tacking Eo Potfolio with Pefoance Attibution Constaints (BTEP) Optiiation Poble BTEP : Constaints: 0.03 0.3, 0.03 SE 0.3, S 0, 0 ETL 0. AA, 05 ean of ean of the Classes Shaes Tading volue (Euo, Millions) the asset weights in the optial asset class weights in the optial ean of the asset weight in the class potfolio potfolio hx 24.55 0.024 0.2044 fe 27.39 0.666 0.59 st class henkel 30.37 0.066 0.59 0.047 alt 33.05 0.066 0.59 lin 39.67 0.066 0.59 tui-n 4.26 0.066 0.59 lha 49. 0.066 0.0432 an 49.84 0.066 0.0432 2nd class eo 5.43 0.066 0.0432 0.3856 con 65.73 0.234 0.3202 db 67. 0.788 0.4638 tka 69.6 0.0332 0.0863 sch 69.84 0.764 0.6792 dpw 75.9 0.066 0.064 3d class cbk 93.43 0.066 0.064 0.2598 ifx 93.87 0.066 0.064 bw 96.64 0.066 0.064 baye 35.37 0.066 0.064 vow 5.45 0.066 0.2 we-a 58.8 0.045 0.303 4th class bas 62.37 0.066 0.2 0.498 ads 83.6 0.0380 0.2537 uv2 95.43 0.066 0.2 sap 227.3 0.066 0.2 5th class dcx 238.95 0.066 0.000 0.666 22
eoa 262.08 0.066 0.666 dbk 305.46 0.066 0.666 sieens 330.33 0.066 0.666 alv 333.44 0.066 0.666 dte 347.28 0.066 0.666 Table 3 epots shaes, divided into 5 classes accoding to thei tading volues, values of tading volues of appopiate shaes, pesented in illions of Euos, eans of the asset weights, asset class weights and asset weights in the class ove 632 optial potfolios of DAX-stocks, obtained solving the Optiiation Poble 3..2 daily, a total 632 ties, and based on iniiing of ETL0.05 with constaints 0.03 AA 0.3, 0.03 SE 0.3, S 0, 0 duing the peiod of Octobe 2003 and Mach 2007. 23
Figue Realied Wealth of the Optiied Potfolio (BTEP) and the Benchak Potfolio 24
Figue 2 Total Realied Retun of the Optiied Potfolio (BTEP) and the Benchak Potfolio 25
Figue 3 Quantile-quantile (QQ) ots of the AA, SE and S quantiles and coesponding the noal(gaussian) quantiles in the BTEP 26
Figue 4 Histoga of AA, SE and S values and the noal density fit Table 4 Estiated paaetes alpha, beta of AA, SE, and S sequences and K-S distances unde the noal and the stable distibution in the BTEP alphas betas K-S distances(noal case) K-S distances(stable case) Mean annualied values (%) AA.4599 0.5932 0.7594 0.0664.36% SE.3624-0.49 0.7563 0.0537 2.99% S.3670-0.4307 0.7705 0.0569 67.35% 27