Multiple-Period Attribution: Residuals and Compounding

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1 Multple-Perod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens S. Karnosky, Ph.D. and Bran D. Snger, CFA wrote Global Asset Management and Performance Attrbuton, a monograph that offers a performance attrbuton system for solatng the effects of market allocaton, currency management, and securty selecton on global portfolos. In response to publc nqures about ths method, the authors composed the followng artcle, whch addresses several ssues, and descrbes the mplementaton of the methodology over multple perods. Bran D. Snger, CFA s head of Investment Rsk Management at Brnson Partners, n Chcago, where he s responsble for the development and management of rsk managed and absolute return portfolos. As a member of the Asset Allocaton and Currency Commttee, Mr. Snger partcpates n the development and mplementaton of global nvestment strateges. Mr. Snger frequently contrbutes artcles on rsk management, global asset management and performance attrbuton to nvestment publcatons. He holds a BA from Northwestern Unversty and an MBA from the Unversty of Chcago. He was the recpent of a 1991 Graham and Dodd Scroll. In 1994, he co-authored the Karnosky-Snger monograph wth Dens S. Karnosky, Ph.D. He s also a member of ths publcaton s Advsory Board. Mguel Gonzalo s Assocate Drector and Research Assstant n the Asset Allocaton/Currency area at Brnson, where hs dutes nclude performance evaluaton and nvestment analytcs. He also asssts n the development and management of the frm s performance attrbuton and performance analytcs databases. Pror to jonng the frm, Mr. Gonzalo consulted for an nternatonal nformaton management company. He s a Level III Chartered Fnancal Analysts canddate. He s a graduate of the Internatonal Foundaton of Employee Benefts Plans Program and holds a BA from Notre Dame Unversty. Marc Lederman s a Performance Analyst for Brnson, responsble for performance evaluaton and nvestment analytcs and asssts n the frm s performance attrbuton and performance analytcs databases. Pror to jonng Brnson, Mr. Lederman was head of the development of quanttatve equty selecton processes at an nvestment research and management company. He s a Level I Chartered Fnancal Analysts canddate. The Karnosky-Snger monograph 1 ntroduces an analytcal framework desgned to overcome the defcences n sngle-country attrbuton models. Its usefulness n attrbutng a global portfolo manager s returns to market, securty, and currency management has been proven through mplementaton and analyss of performance. Although the monograph s theory s soundly based, n practce t sometmes fals to fully attrbute all of a portfolo s total added value to the decson-makng areas. Ths falure to fully explan the sources of added value s not due to a flaw n the theory, but rather s a result of at least four nterrelated factors: 1) cross-products, 2) the smplfyng assumptons made when collectng performance data and defntons of decsonmakng areas, 3) ntramonth changes n portfolo strategy, and 4) the mpact of compoundng the returns across tme. RESIDUAL EFFECTS These factors n concert create a resdual, whch s the dfference between the portfolo s actual added value and the sum of the added values from market, securty and currency selecton. Over tme, hopefully the re- The Journal of Performance Measurement

2 sdual returns from ntramonth strategy changes wll add value, whle all other sources of resduals wash out to leave a clear pcture of how the portfolo has performed. Through many years of mplementaton, t has been found that lnkng monthly data n a specfc manner produces analyses for longer perods wth small resduals and allows for an accurate assessment of portfolo performance over any selected tme perod. Ths paper revews a suggested method for lnkng sngle perod analyss nto a multple-perod analyss; ths s non-trval because the total added value compounds through tme. MARKET SELECTION MARKET SELECTION = Actve Market Weght {[ ] [( ) RP] } = w w r c Passve Market Return Weght Passve Market Premum Index Market Return Premum Equaltes Over any tme perod: Total Added Value = Portfolo Return - Benchmark Return And: Total Added Value = Market Selecton + Currency Selecton + Securty Selecton + resdual CONSIDER THE SOURCE where, w r c RP = weght of country assets = return from the assets of country, n localcurrency terms = return from country Eurodeposts = aggregate passve benchmark local-currency return premum It s helpful to consder the sources of sngle-perod resduals wthn the framework before dscussng multpleperod analyss. In practce, trade-offs must be made between the amount of data gathered to analyze portfolo total returns and the ncreased level of accuracy ganed from performng ths operaton. Further, the framework has cross-products, or small slvers of addedvalue that fall between the explaned returns. One example of such a cross-product s exchange rate gans, or losses, on the same perod s local market gans, or losses. 2 (Note the market and currency cross-products are accounted for wthn the framework, and do not end up n the resdual.) DEFINE PERIODS Also, the ablty to explan added values wthn sngle perods s dependent on the defnton of a perod. Each sngle perod analyss wll tend to become more accurate as the perods are defned as shorter lengths of tme (such as weekly versus monthly perods). Ths property can be observed wthn the market selecton added value term by studyng the sngle perod equaton. * Note: Plan lower-case letters ndcate portfolo weghts and returns; letters wth a bar over them ndcate passve benchmark weghts and returns. Statc Applyng statc, begnnng of the month weghts for both the portfolo and benchmark generally works well for computng added values. The effectveness depends on the relatve weghts durng the month not devatng sgnfcantly from the begnnng of the month snapshot, as the actve weghts of a portfolo depend drectly on the portfolo manager s decsons. For example, wthn a month a manager can decde to make a strategc shft n market weghts by reducng holdngs n one market and ncreasng them n another. Usng begnnng of the month weghts, the changes from the shft would not be reflected n the attrbuton s weghts untl the start of the followng month. So the portfolo s market selecton added value would be calculated based on the orgnal strategy, meanng any gans or losses from the strategy change would end up n the resdual untl the start of the next month The Journal of Performance Measurement

3 More frequent snapshots of the portfolo would correct for ths problem by reducng the length of tme untl changes n the actve weghts enter nto the attrbuton calculatons (attrbuton weghts would accurately model the actual weghts as they change wth tme). Yet, the margnal mprovement n accuracy that would come from usng less dated weghts (especally on a regular bass) would nvolve a very large ncrease n the data mantaned, technologcal resources requred, and tme spent analyzng the attrbuton. Keepng monthly perods, whle beng somewhat flexble n selectng the date for each perod s actve weghts, s more easly mplemented, f a slghtly less rgorous soluton. For example, f the above descrbed shft n assets occurred on the thrd day of a gven month, t would probably be more accurate to use actve weghts from the fourth day, rather than the begnnng of the month. Compoundng When movng from sngle perod returns analyses to analyses over more than a sngle perod, t s necessary to consder the mpact of compoundng. Portfolos and ther benchmarks compound ndependently through tme; so added values, or dfferences n returns, are also approxmately compounded through tme. Compoundng wthn attrbutons s necessary for smple U.S. balanced portfolos as well as global portfolos wth hedgng. Table A (see page 25) s an example of a multpleperod attrbuton analyss for a three naton global portfolo. Each market added value area can be vewed as a collecton of weghted rsk premums. So, by addng or subtractng specfc weghted rsk premum returns t s possble to construct any of the market decson makng areas, as descrbed by the followng equatons: Securty selecton: { w [( r c ) ( r c )]} = w r c Market cross-product: { [( ) ( )]} w w r c r c = - w [ r c] + w [ r c] w r c w r c - w r c - The frst step toward a multple-perod attrbuton of market added-values s calculatng the weghted rsk premum returns for each market n every perod. Then, the weghted rsk premum returns represented n Table B (see page 26) are compounded over the perods wthn the multple-perod attrbuton. Next, the resultng compounded rsk premums are added or subtracted as necessary to form the multple perod market explaned added values as descrbed n Table C (see page 26). The procedure to calculate multple-perod added values for currency selecton, hedge selecton and currency cross-product s completely analogous to ther market counterparts. 3 Fnally, t s helpful to restate the equalty that gves resduals. The total added value over any selected perod equals the dfference n compounded returns between a portfolo and ts benchmark. Ths equalty determnes the multple-perod resdual once the total multple-perod explaned added value has been calculated. Market selecton: { [( ) ]} w w r c RP = [ ] - w [ RP ] + w [ RP ] w r c w r c - CONCLUSION The Karnosky-Snger attrbuton framework s structured to provde nformaton at all levels of decson makng. Lnkng the sngle-perod analyss nto a multple-perod attrbuton study s a powerful tool for The Journal of Performance Measurement

4 The Journal of Performance Measurement -25- PASSIVE BENCHMARK Passve market weght ( w ) TABLE A EXAMPLE OF A MULTIPLE-PERIOD ATTRIBUTION Passve market return ( r ) Passve hedge weght ( h ) Eurodepost return ( c ) Exchange rate return ( e ) Index average rsk premum (RP) = w ( r c ) U.S % 9.25% N/A 5.25% N/A 2.95% Perod 1 Germany 30.00% 7.00% N/A 5.50% -8.00% 2.95% Japan 30.00% 8.50% N/A 5.50% % 2.95% U.S % 8.50% N/A 4.75% N/A 0.60% Perod 2 Germany 30.00% 6.00% N/A 4.00% 5.00% 0.60% Japan 30.00% -2.00% N/A 3.00% 17.00% 0.60% U.S % -5.25% N/A 4.75% N/A -3.85% Perod 3 Germany 30.00% 3.75% N/A 4.25% 6.00% -3.85% Japan 30.00% 4.50% N/A 3.50% 2.00% -3.85% ACTIVE PORTFOLIO Actve market weght ( w ) Actve market return ( r ) Actve hedge weght ( h ) Eurodepost return ( c ) Exchange rate return ( e ) U.S % 10.00% 15.00% 5.25% N/A Perod 1 Germany 38.50% 6.00% 0.00% 5.50% -8.00% Japan 25.25% 8.75% % 5.50% % U.S % 7.50% 34.00% 4.75% N/A Perod 2 Germany 34.25% 6.75% -6.00% 4.00% 5.00% Japan 33.75% -4.00% % 3.00% 17.00% U.S % -1.75% 14.00% 4.75% N/A Perod 3 Germany 42.00% 5.25% 0.00% 4.25% 6.00% Japan 28.25% 4.00% % 3.50% 2.00%

5 Table B Weghed Rsk Premums [ ] w [ r c ] w [ RP ] w r c w RP w [ r c ] w [ r c] Perod % 1.60% 1.07% 1.18% 1.90% 1.72% U.S. Perod % 1.50% 0.19% 0.24% 1.10% 0.88% Perod % -4.00% -1.15% -1.54% -2.60% -1.93% Compounded -0.39% -1.00% 0.10% -0.14% 0.34% 0.63% Perods 1-3 Perod % 0.45% 1.14% 0.89% 0.15% 0.19% Germany Perod % 0.60% 0.21% 0.18% 0.83% 0.94% Perod % -0.15% -1.62% -1.16% 0.30% 0.42% Compounded 1.05% 0.90% -0.30% -0.10% 1.28% 1.56% Perods 1-3 Perod % 0.90% 0.74% 0.89% 0.98% 0.82% Japan Perod % -1.50% 0.20% 0.18% -2.10% -2.36% Perod % 0.30% -1.09% -1.16% 0.15% 0.14% Compounded -0.66% -0.32% -0.15% -0.10% -1.00% -1.42% Perods 1-3 nvestment management by provdng a means for crtcal revew of the decson process over long perods of tme. As the ssues nvolved wth multple-perod attrbuton analyss are not partcular to any one framework, the explanaton s drawn broadly. Stll, the explanaton lays out n detal an approach for handlng the complextes of multple-perod attrbuton analyss for the beneft of analysts seekng to mplement the Karnosky-Snger framework. ENDNOTES Table C Multple-Perod Added Values Market selecton Securty selecton Market crossproduct Total market added value U.S 0.37% 1.34% -0.32% 1.39% Germany 0.35% 0.38% 0.13% 0.85% Japan -0.30% -0.68% -0.08% -1.06% Total 0.42% 1.04% -0.27% 1.19% Note: Due to roundng sums and totals may devate from expected amounts. construct a performance attrbuton system that solates the effects of market allocaton, currency management, and deep theoretcal ssues of asset prcng or optmal nvestment strateges but, rather, on the ssue of developng useful measures of the market and currency components of global nvestment ssues. 1 Ths monograph develops an analytcal framework for evaluatng global asset markets and uses that framework to Ths work reflects the ongong efforts wth Brnson Partners to address practcal ssues n the management of glo- The Journal of Performance Measurement

6 bal portfolos. The analytcal framework and the performance attrbuton system are ntegral parts of our nvestment process, and we beleve that open dscusson of these tools wll enhance general understandng of global nvestment ssues. The analyss provdes a consstent framework for all who are nvolved n the evaluaton of nvestment opportuntes, Performance, and rsks. Ths monograph reflects the dscussons and research of many Brnson Partners nvestment managers and analysts, to all of whom we owe a great debt. The presentaton benefted partcularly from the thoughts and arguments of Gary Brnson, Rchard Carr, Khaled Salama, Raymond Chan and Norman Cummng. Ray Chan was also ndspensable n solvng the large number of techncal ssues nvolved n the performance attrbuton program. Robert Clarke was nstrumental n the early development of the attrbuton program. We also thank the Research Foundaton of the Insttute of Chartered Fnancal Analysts, and AIMR, for ther support and encouragement n preparng ths monograph. 2 For example, a foregn asset wth a 10% local market return and a 5% exchange rate return would have a total return (measured n base currency) of 15.5 percent. TOTAL RETURN = (1+10%)x(1+5%)-1 = 15.5% = LOCAL RETURN + FX RETURN + CROSS-PRODUCT The local return and exchange rate returns are 10% and 5% respectvely. The cross-product s 0.5%, the exchange rate gan on the addtonal market exposure obtaned through the 10% local return. +, C + and [ w + h] ), compounded and assembled nto the decson-makng areas. 3 The Eurodepost returns n base currency ( [ c e] and [ c + e] ) are weghted by the currency weghts ( [ w h] The Journal of Performance Measurement

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