Equity Performance Attribution Methodology

Transcription

1 Equity erformance Attribution Methodoloy Morninstar Methodoloy aper December 5, Morninstar, Inc. All rihts reserved. The information in this document is the property of Morninstar, Inc. eproduction or transcription by any means, in whole or in part, without the prior written consent of Morninstar, Inc., is prohibited.

2 Contents Introduction Overview Effects versus Components eview of the Classic Approach rinson, Hood, and eebower eview of Attribution Components rinson and Fachler Top-Down versus ottom-up Approach Arithmetic versus Geometric Attribution Example Note asic Mathematical Expressions Formulas Explanation of Formulas Special Situation I: Groups without Holdins Special Situation II: Short ositions Top-Down Approach for Sinle eriod Overview Arithmetic Method Geometric Method Example ottom-up Approach for Sinle eriod Overview Arithmetic Method Geometric Method Multiple-eriod Analysis Overview Multi-eriod Geometric Method Multi-eriod Arithmetic Method Appendix A: eturn Gap Overview Top-Down Approach, Arithmetic Method Top-Down Approach, Geometric Method Multi-eriod Geometric eturn Gaps in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 2

3 Introduction Overview erformance attribution analysis consists of comparin a portfolio s performance to that of a benchmark and decomposin the excess return into pieces to explain the impact of various portfolio manaement decisions. This excess return is the Active eturn. For a portfolio denominated in the investor s home-currency, the investment manaer s active return is decomposed into Weihtin Effect, Selection Effect and eturn Gap. Weihtin effect refers to the portion of an investment manaer s value-add attributable to the manaer's decision on how much to allocate to each market sector, in other words, a manaer s decision to overweiht and underweiht certain sectors compared to the benchmark. Selection effect represents the portion of performance attributable to the manaer s stock pickin skill. eturn ap is the portion of the return that cannot be explained by the holdins composition at the beinnin of the analysis period, and this ap is usually caused by intra-period portfolio transactions, security corporate actions, etc. Attribution analysis focuses primarily on the explainable part of the active return the weihtin and selection effects. eturn Gap is discussed in Appendix A of this document. This document first reviews the classic attribution approach of rinson, Hood, and eebower (H), the principles upon which today's performance attribution methodoloies are founded. The H model in its oriinal form is not adopted, and two approaches, top-down and bottomup, are presented instead. In addition, each of these two approaches can be implemented usin the arithmetic or the eometric method. These four combinations and their uses are described in details in the subsequent sections, followed by how these attribution results can be accumulated in a multi-period analysis. Althouh multiple alternatives are presented in this document, the recommended method of Morninstar is the top-down eometric method. The top-down approach presents a uniform framework for comparin multiple investment manaers, and the eometric method has the merit of theoretical and mathematical soundness. This document focuses on equity attribution performed in the portfolio's base currency, and topics such as fixed income, currency, and transactions costs attribution analyses are outside of the scope of this document. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 3

4 Introduction (continued) Effects versus Components When performin attribution analysis, it is important to distinuish between Effects and Components. An Effect measures the impact of a particular investment decision. An Effect can be broken down into several Components that provide insiht on each piece of an overall decision, but each piece in isolation cannot represent the investment manaer's decision. For example, an investment manaer may make an active decision on sector weihtin by overweihtin certain sectors and underweihtin other sectors. Since overweihtin certain sectors necessitates underweihtin others and vice versa, the decision is on the entire set of sector weihts. To better understand the sector weihtin effect, one may examine contributions of individual sectors. These contributions are simply Components that provide additional insiht. However, each of these contributions cannot be used in isolation to measure the impact of a decision, as it is not meaninful to say that an investment manaer made a particular decision to time exposure to the Service sector, for example. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 4

5 Introduction (continued) eview of the Classic Approach rinson, Hood, and eebower Today's approaches to performance attribution are founded on the principles presented in an article 1 written by rinson, Hood, and eebower (H) and published in Therefore, it is important to review the H model even thouh the model in its oriinal form is not adopted. The study is based on the concept that a portfolio's return consists of the combination of roup (e.. asset class) weihts and returns, and decision makin is observed when weihts or returns of the portfolio vary from those of the benchmark. Thus, notional portfolios can be built by combinin active or passive roup weihts and returns to illustrate the value-add from each decision. The study deconstructs the value-added return of the portfolio into three parts: tactical asset allocation, stock selection, and interaction. The formulas for these terms are defined below: Tactical Asset Allocation = II - I = ( w w ) Stock Selection = III - I = w ( ) Interaction = IV - III - II + I = ( w w ) ( ) Total Value Added = IV - I = w w These formulas are based on four notional portfolios. These notional portfolios are constructed by combinin different weihts and returns, and they are illustrated in the chart below: 1 rinson, Gary., L. andolph Hood, and Gilbert L. eebower, "Determinants of ortfolio erformance," Financial Analysts Journal, July-Auust 1986, pp in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 5

6 Introduction (continued) Where: w = The benchmark's weiht for roup w = The portfolio's weiht for roup = The benchmark's return for roup = The portfolio's return for roup The tactical asset allocation effect, also known as the weihtin effect, is the difference in returns between the notional portfolios II and I. Notional portfolio II represents a hypothetical tactical asset allocator that focuses on how much to allocate to each roup (e.. asset class) but purchases index products for lack of opinions on which stocks would perform better than others. Notional portfolio I is the benchmark which, by definition, has passive roup weihts and returns. These two notional portfolios share the same passive roup returns but have different weihts; thus, the concept intuitively defines the weihtin effect as the result of active weihtin decision and passive stock selection decision. The stock selection effect, also known as the selection effect, is the difference in returns between the notional portfolios III and I. Notional portfolio III represents a hypothetical security picker that focuses on pickin the riht securities within each roup but mimics how much money the benchmark allocates to each roup because the person is anostic on which roups would perform better. As described above, notional portfolio I is the benchmark which has passive roup weihts and returns. These two notional portfolios share the same passive roup weihts but have different roup returns; thus, the concept intuitively defines the selection effect as the result of passive weihtin decision and active stock selection decision. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 6

7 Introduction (continued) While the weihtin and the selection effects are intuitive, the interaction portion is not easily understood. The interaction term, as its name suests, is the interaction between the weihtin and the section effects, and it does not represent an explicit decision of the investment manaer. Due to its apparent lack of meanin, Morninstar believes that it is a better practice to incorporate it into either the weihtin or the selection effect, whichever of the two that represents the secondary decision of the investment manaer. The concept of primary versus secondary decision is discussed in more details in the next section of this document. The Morninstar methodoloy for equity performance attribution is founded on the principles of the H study, but the H model in its oriinal form is not adopted. First, the H model is an asset class level model and does not break down attribution Effects into roup level Components. The next section presents the rinson and Fachler model, this method addresses roup level Components. Furthermore, much has evolved in the field of performance attribution since the H study. Methodoloies are needed to incorporate the interaction term into the other two effects, accommodate for multiple hierarchical weihtin decisions, perform multiperiod analysis, etc. These topics are addressed in the subsequent sections of this document. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 7

8 Introduction (continued) eview of Attribution Components rinson and Fachler The H model presented in the previous section shows how attribution Effects are calculated. As discussed in the Effects versus Components section of this document, an Effect can be broken down into several Components. Today's approaches to Component level attribution are based on concepts presented in a study 2 by rinson and Fachler (F) in In this article, the impact of weihtin decision for a particular roup is defined as ( w w ) ( ). The ( w w ) portion of this formula is the same as the equation for the tactical asset allocation effect in the H study. It is the difference between the portfolio's weiht in this particular roup and the benchmark's weiht in the same roup, representin the investment manaer's weihtin decision. In the H model, the ( w w ) portion is multiplied by the benchmark's total return. This basic principle is preserved in the F model as the latter also uses the benchmark return. However, in order to ain insiht into each roup's value-add, the term is transformed into the return differential between the roup in question and the total return. Thus, this term intuitively illustrates that a roup is ood if it outperforms the total. This formula is not in conflict with the H model because their results match at the portfolio level, in other words, the sum of F results from all roups equals the H tactical asset allocation effect. With the two multiplicative terms of the formula combined, the F formula illustrates that it is ood to overweiht a roup that has outperformed and underweiht a roup that has underperformed. This is because overweiht produces a positive number in the first term of the formula, and outperformance yields a positive number in the second term, leadin to a positive attribution result. Similarly, a neative weihtin differential of an underweiht combined with a neative return differential of an underperformance produces a positive attribution result. Furthermore, it is bad to overweiht a roup that has underperformed and underweiht a roup that has outperformed because these combinations produce neative results. This concept is illustrated in the chart below: 2 rinson, Gary., and Nimrod Fachler, "Measurin Non-US Equity ortfolio erformance," Journal of ortfolio Manaement, Sprin 1985, pp in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 8

9 Introduction (continued) Top-Down versus ottom-up Approach There are several approaches to performance attribution, and we focus on two of them topdown and bottom-up. The choice of the approach depends on the investment decision process of the portfolio bein analyzed. The top-down approach to portfolio attribution is most appropriately used when analyzin an investment manaer with a top-down investment process that focuses on one or multiple weihtin allocation decisions prior to security selection. In this decision makin process, the Weihtin Effect is primary, and the Selection Effect is secondary. As discussed in the H section above, the interaction term of the H approach is incorporated in the effect of the secondary decision which is the Selection Effect in this case. The bottom-up approach is most relevant in analyzin an investment manaer with a bottom-up process that emphasizes security selection. In this decision makin process, the Selection Effect is primary, and the Weihtin Effect is secondary. Unlike the top-down approach which can measure the effects of multiple weihtin allocation decisions, there is only weihtin effect in the bottom-up approach. As discussed in the H section above, the interaction term of the H approach is included in the effect of the secondary decision which is the Weihtin Effect in this case. oth the top-down and the bottom-up approaches involve hierarchical decision. For example, in the case of a top-down analysis, an investment manaer may first decide on reional weihtin, followed by sector weihtin and market capitalization weihtin, before makin security selections. The analysis is hierarchical because weihtin at each decision level is anchored upon the weihtin of the prior decision. Similarly, in a bottom-up analysis, an investment manaer first decides on security selection before makin a weihtin decision such as sector weihtin. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 9

10 Introduction (Continued) Arithmetic versus Geometric Attribution Since attribution effects are the results of the portfolio's relative weihtin and performance to those of the benchmark, the return comparison can be performed usin arithmetic or eometric method. The arithmetic method refers to simple subtractions of return terms in formulas and is very intuitive; however, it works best in a sinle-period analysis, and additional "smoothin" is required to apply it in a multi-period settin. efer to the Multiple eriod Analysis section of this document for details. The eometric method takes a eometric difference by translatin returns into "return relatives" (that is, one plus the return), performin a division of the two return relatives, and subtractin one from the result. It is more complicated than the arithmetic method, but it has the benefit of bein theoretically sound for both sinle-period and multiperiod analyses when applied to Effect statistics. Example Since the top-down approach is more complex, as it may involve a hierarchy of weihtin decisions, it is more helpful to provide an example that illustrates this process throuhout the document. Let us assume a simple example where the investment process consists of decision makin in the followin order: Decision Level Decision Choice 1 eional weihtin Asia versus Europe 2 Sector weihtin Service versus Non-Service 3 Market capitalization weihtin Lare Cap versus Small Cap 4 Security selection Note All formulas assume that the individual constituents and the results are expressed in decimal format. For example, the number 0.15 represents fifteen percent. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 10

11 asic Mathematical Expressions Formulas Since attribution formulas use many mathematical expressions in common, these mathematical expressions and their formulas are defined in this section and are used throuhout the document. [1] benchmark weiht of stock if = M wø w = w < h if M h Ω w Ø [2] portfolio weiht of stock if = M wø w = w < h if M h Ω w Ø return on stock if = M [3] = wh h h Ω w if < M return on stock if = M [4] = wh h h Ω w if < M Where: w = The benchmark's weiht for roup w = The portfolio's weiht for roup = The benchmark's return for roup = The portfolio's return for roup = The vector that denotes the roup = The number of elements in the vector, representin the hierarchy level of the roup M = The level that represents the security level, that is, the last roupin hierarchy Ω = All of the sub-roups within the roup that are one hierarchy level below Ø = The total level, which is the equity portion of the portfolio or benchmark in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 11

12 asic Mathematical Expressions (continued) Explanation of Formulas A roup represents a basket of securities classified by the end-user, e.. economic sector, market cap, /E, reion, country, etc. Group is the most eneric term that represents a roup of securities or a sinle security. The symbol represents the vector that denotes the roup. In our example, Europe's Service sector's Small Cap is denoted as (2,1,2) since it is the second reion's first sector's second market cap bucket. This particular market cap's fifth stock is denoted as (2,1,2,5). When = Ø, it represents the null set that denotes the total level such as the total equity portfolio or the total equity benchmark. The symbol is the number of elements in the vector, representin the decision level of the roup in the hierarchy. In our example, = 2 stands for the sector level because it is the second level of decision. Note that = 0 represents the total level such as the total equity benchmark or the total equity portfolio. M denotes the level that represents the security level, that is, the last decision in the hierarchy. In our example, M = 4 because security level is the fourth level of decision. The Ω expression represents all of the sub-roups within the roup that are one hierarchy level below. Think of a family tree and let each decision level be a eneration of relatives, the Ω symbol represents all of the children of the same parent. In our example, Asia is denoted by = (1). When usin formula [1] to calculatin the benchmark weiht of Asia, the Ω (1) symbol represents all of the sub-roups within Asia. They are Service and Non-Service sectors, which are denoted by = (1,1 ) and = (1,2 ), correspondinly. The second part of formulas [1] and [2] simply states that the weiht of Asia is the sum of the weihts of Asian Service and Asian Non-Service sectors. Similarly, the second part of formulas [3] and [4] means that the return of Asia is the weihted sum of the returns of Asian Service and Asian Non-Service. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 12

13 asic Mathematical Expressions (continued) Special Situation I: Groups without Holdins If neither the portfolio nor the benchmark has holdins in a particular roup, this roup should be inored in order to provide a meaninful attribution analysis. If the portfolio does not have holdins in a particular roup but the benchmark does, the roup's portfolio weiht is zero, and the roup's portfolio return is assumed to be the same as the roup's benchmark return. This rule applies reardless of whether the roup represents lon or short positions. For example, if sector weihtin decision is bein evaluated, and the portfolio does not have holdins in the Asian Service sector while the benchmark does, the portfolio's return in the Asian Service sector is assumed to be the same as the benchmark's return in the Asian Service sector. The active return is attributable entirely to the sector weihtin effect and not the subsequent decisions such as market cap weihtin and security selection in a topdown model. Similarly, in a bottom-up model, the active return is attributable entirely to the sector weihtin effect and not to security selection effect. This makes intuitive sense as the decision to differ from benchmark's weiht is a weihtin effect. If the portfolio has holdins in a particular roup but the benchmark does not have holdins in the same roup, the roup's benchmark return is assumed to be the same as the roup's portfolio return. The only exception to this rule is the short position situation described below. Special Situation II: Short ositions When the portfolio or the benchmark has short positions, attribution analysis must be performed on the short positions separately from the lon positions. In other words, short positions and lon positions are in separate roups, and the number of roups is potentially double that of an analysis where only lon positions are present. To ensure that the separation is clear, lon and short positions must be separated at the first level of the decision hierarchy. For example, when the first level of decision hierarchy is reional allocation, and the reional classifications are Asia and Europe, a portfolio containin short positions should have four reional classifications: Asia Lon, Europe Lon, Asia Short, and Europe Short. For levels of the decision hierarchy other than the security level ( < M ), when the benchmark does not have holdins in a particular short position roup, this roup's benchmark's return is assumed to be the same as the benchmark's return of the same roup's lon position counterpart in order to allocate Effects correctly. For example, if the benchmark does not have short position holdins in the Asian Service sector, the return of this sector is assumed to be the same as the benchmark's return in the lon positions of the Asian Service sector. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 13

14 Top-Down Approach for Sinle eriod Overview As discussed in the Introduction, the top-down approach to portfolio attribution is most appropriately used when analyzin an investment manaer with a top-down investment process that focuses on one or multiple weihtin allocation decisions prior to security selection. These decisions are hierarchical. In our example, the investment manaer first decides on reional weihtin, followed by sector weihtin, and market cap weihtin, before makin security selections. In this decision makin process, the Weihtin Effect is primary, and the Selection Effect is secondary. This section addresses the top-down approach in a sinle-period attribution analysis. The sinle-period methodoloy serves as a foundation for the multi-period attribution, and the latter is discussed in the last section of this document. Attribution can be performed usin the arithmetic or the eometric method. These methods and their merits are discussed in the Introduction section of this document. This section focuses on the presentation and the explanation of the formulas. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 14

15 Top-Down Approach for Sinle eriod (continued) Arithmetic Method w [5] CA = ( w w ) ( ) w [6] EA, n h Ω = h Ω CA EA h h, n if n = if n > [7] AAØ = Ø Ø = EA M n= 1 Ø, n Where: CA = Component that is attributable to roup, calculated based on arithmetic method EA, = Effect that is attributable to roup at decision level n, based on arithmetic method n AA Ø = The portfolio's active return, based on equity holdins, calculated based on arithmetic method = The roup where roup belons to in the prior roupin hierarchy level Ø = The portfolio's total equity return, calculated based on equity holdins Ø = The benchmark's total equity return, calculated based on equity holdins The arithmetic method refers to simple subtractions and additions. For example, simple subtractions are used when comparin returns of the portfolio and the benchmark, as shown in the second term of formula [5]. Furthermore, Active eturn in formula [7] is the simple addition of the total effects at various decision levels. These characteristics distinuish the arithmetic method from its eometric counterpart. The arithmetic method also serves as the foundation for the eometric method presented in the next section of this document. In the Component calculation in equation [5], there are some terms that are similar to the basic H and F models and many that are not. The H model is at the portfolio level while formula [5], as its name indicates, is at Components level. In other words, formula [5] calculates how Asia and Europe, as Components, each contributes towards the total reional weihtin effect. Thus, it is more appropriate to compare it to the F model. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 15

16 Top-Down Approach for Sinle eriod (continued) To fully understand the Component calculation, let us first focus on the first multiplicative term of equation [5]. The F model is founded on the concept of the weihtin effect bein the difference between portfolio and benchmark weihts, and the first term of the Component formula is essentially that difference. The dissimilarity between the F model and the Component formula stems from the latter bein modified for a hierarchical decision makin structure. For example, an investment manaer may first decide on reional weihtin, followed by sector weihtin and market capitalization weihtin before makin security selection. The analysis is hierarchical because weihtin at each decision level is anchored upon the weihtin of the prior decision. For example, let the portfolio's weiht in Asia be 60% and the benchmark's weiht in the same reion be 30%, representin a double weiht. Further assume that there are two sectors in Asia, and the benchmark has half the weiht in each sector, thus each sector has a 15% benchmark weiht. Since the portfolio has 60% in Asia, if it were to mimic the benchmark and place half its weiht in each of the two sectors, each sector would have a 30% portfolio weiht and looks overweihted even thouh it mimics the benchmark's allocation. Therefore, one must not compare the portfolio weiht of the Asia reion's Service sector directly with the weiht of the same sector in the benchmark. The fair comparison is to create an anchorin system like formula [5] where the benchmark's weiht in the Asia reion's Service sector is scaled to the proportion between the portfolio's weiht in Asia and that of the benchmark. In this example, the benchmark's weiht in the Asian Service sector must be multiplied by 2 before it can be compared to the portfolio's weiht in the same sector because 2 is the result of 0.6 divided by 0.3. is the roup where roup belons to in the prior decision level of the hierarchy. Followin the analoy of a family tree, represents the parent of. For example, the term for the Asian Service sector represents the Asia reion, as the Asian Service sector is part of the Asia reion, and reion is the decision level prior to sector. For simplicity, let us call this the "parent roup" to roup. When = Ø, when the parent roup is the total level, w = w 1. Ø Ø = Shiftin focus to the second term of equation [5], this term is similar to the F model. In order to adopt a hierarchical structure, the second term is transformed into the return differential between the roup in question and its parent roup. Thus, this term intuitively illustrates that a roup is ood if it outperforms the combined performance of all siblins, and vice versa. For example, if Europe's Service sector has a benchmark return of 8.40% while Europe has a benchmark return of 3.53%, the differential is 4.87%, a positive number demonstratin that this reion's Service sector has outperformed other sectors in the reion. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 16

17 Top-Down Approach for Sinle eriod (continued) Formula [5] illustrates the same intuitive concepts in the F article. It is ood to overweiht a roup that has outperformed and underweiht a roup that has underperformed. It is bad to overweiht a roup that has underperformed and underweiht a roup that has outperformed. Formula [6] shows that the Effect of a parent roup is the sum of the Components of all of its children if the children are components of this decision. For example, when analyzin the sector weihtin effect, the sectors are components of the decision, so Asia's sector weihtin effect is the sum of the sector weihtin components of Asian Service and Asian Non-Service. The Effect of a randparent roup is the sum of the Effects of all of its children if the children's descendents are components of this decision. For example, when analyzin the selection effect, the securities are components of this decision. Thus, the Asian Service sector's selection effect is the sum of selection effects of Asian Service Lare Cap and Asian Service Small Cap, and these two are in term sums of selection components of the underlyin constituent stocks. The formulas for components and effects are universal to all roupin levels. When n < M, the result of the formula is referred to as a weihtin effect. When n = M, the result of the formula is referred to as a selection effect. For example, EA (1,2), 4 is the selection (fourth decision) effect of the first reion's second sector. Active eturn in formula [7] is the value-add of equity securities, and it is the difference between the return of the equity portion of the portfolio and that of the equity portion of the benchmark. Expressed in attribution terms, the Active eturn is the simple addition of the total effects at all decision levels. In other words, it is the sum of the total effects of all four decisions made in the portfolio: reional weihtin, sector weihtin, market cap weihtin, and security selection. This Active eturn represents the value-add of the equity portion of the portfolio, and it is calculated based equity holdins as of the beinnin of the analysis period. efer to the Appendix section of this document for the value-add of the total portfolio and return aps that account for the difference between actual and calculated returns. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 17

18 Top-Down Approach for Sinle eriod (continued) Geometric Method [8] [9] EA HL Ø = H ( L 1) Ø,L + CA = H if if L = 0 L > 0 [10], n = h Ω h Ω h h, n if n = if n > M Ø [11] AG Ø = 1 = (1 + Ø, ) 1 n Ø n= 1 Where: HL = eturn of the hybrid portfolio at level L = Component that is attributable to roup, calculated based on eometric method, = Effect that is attributable to roup at decision level n, based on eometric method n AG Ø = The portfolio's active return, based on equity holdins, calculated based eometric method Equation [5] in the last section presents the hierarchical anchorin system used in the Component formula. The denominator of formula [9] in this section demonstrates another method of hierarchical anchorin, and it is facilitated by the use of the "hybrid" portfolio defined in formula [8]. The hybrid portfolio may look unfamiliar when presented in its concise presentation in formula [8], but it is based on the already familiar hierarchical anchorin system in equation [5]. ecall that in equation [5] the benchmark's weiht in the Asia reion's Service sector is scaled to the proportion between the portfolio's weiht in Asia and that of the benchmark. The hybrid portfolio is similar to the benchmark portfolio in that the benchmark weiht in each sector is combined with the benchmark return in the sector, but the scaled benchmark weihts are used instead of the raw benchmark weihts. Mathematically the concise form in formula [8] yields the same result as combinin the scaled benchmark weihts with benchmark returns. The concise form in formula [8] has the benefit of re-usin numbers that are already calculated in the arithmetic method. Formula [8] shows that at the total level, no anchorin is required, and the hybrid portfolio is the same as the benchmark portfolio. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 18

19 Top-Down Approach for Sinle eriod (continued) At levels other than the total level, the hybrid portfolio's return is the sum of the arithmetic total effect of this decision level and the hybrid return of the prior decision level. For example, the sector-level hybrid portfolio is the sum of the arithmetic total sector effect and the return of the reional hybrid portfolio. The Effect calculation in formula [10] needs no further explanation as it is similar to its arithmetic counterpart in formula [6]. The Active eturn in formula [11] is also similar to its arithmetic counterpart in formula [7], but eometric operations are used instead of arithmetic operations. The Active eturn is the eometric difference between the returns of the equity portion of the portfolio and the equity portion of the benchmark. The Active eturn can also be computed by eometrically linkin the total effects from all decision levels. Example Followin the example described earlier, below is a top-down investment process that consists of decision makin in the followin order: reional weihtin, sector weihtin, market capitalization weihtin, and security selection. Hierarchical Structure Illustration Total Ø, 1 Asia (1) eion Wt Sector Wt Market Cap Wt Sec Selection Active et Ø, 2 Ø, 3 Ø, 4 AG (1), 2 (1), 3 (1), 4 (1,1), 3 (1,1), 4 (1,1,1), 4 (1,1,2), 4 (1,2), 3 (1,2), 4 (1,2,1), 4 (1,2,2), 4 (2), 2 (2), 3 (2), 4 (2,1), 3 (2,1), 4 (2,1,1), 4 (2,1,2), 4 (2,2), 3 (2,2), 4 (2,2,1), 4 (2,2,2), 4 Service (1,1) Lare Cap (1,1,1 ) Small Cap (1,1,2 ) Non-Service (1,2) Lare Cap (1,2,1) Small Cap (1,2,2 ) Europe (2) Service (2,1) Lare Cap (2,1,1) Small Cap (2,1,2 ) Non-Service (2,2) Lare Cap (2,2,1) Small Cap (2,2,2) in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 19

20 Top-Down Approach for Sinle eriod (continued) Attribution Weihts eturns Attribution ortfolio ench ortfolio ench eion Sector MktCap Selection Active eturn Total Asia Service Lare Cap Small Cap Non-Service Lare Cap Small Cap Europe Service Lare Cap Small Cap Non-Service Lare Cap Small Cap First Decision: eional Weihtin H,0 = Ø = ( w(1) (1) + w(2) (2) ) / wø = ( ) /1.00 = H,0 (1) = ( w(1) wø / wø w(1) ) ( (1) Ø ) /(1 + ) = ( / ) ( ) /( ) = H,0 (2) = ( w(2) wø / wø w(2) ) ( (2) Ø ) /(1 + ) = ( / ) ( ) /( ) = Total reional weihtin effect: = + = Ø,1 (1) (2) = in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 20

21 Top-Down Approach for Sinle eriod (continued) Second Decision: Sector Weihtin H,1 H,0 H,0 H,0 = EAØ,1 + = Ø,1 (1 + ) + = ( ) = H,1 (1,1) = ( w(1,1) w(1) / w(1) w(1,1) ) ( (1,1) (1) ) /(1 + ) = ( / ) ( ) /( ) = H,1 (1,2) = ( w(1,2) w(1) / w(1) w(1,2) ) ( (1,2) (1) ) /(1 + ) = ( / ) ( ) /( ) = ( 1),2 = (1,1) + (1,2) = = H,1 (2,1) = ( w(2,1) w(2) / w(2) w(2,1) ) ( (2,1) (2) ) /(1 + ) = ( / ) ( ) /( ) = H,1 (2,2) = ( w(2,2) w(2) / w(2) w(2,2) ) ( (2,2) (2) ) /(1 + ) = ( / ) ( ) /( ) = ( 2),2 = (2,1) + (2,2) = ( ) + ( ) = Total sector weihtin effect: = + = ( ) = Ø,2 (1),2 Third Decision: Market Capitalization Weihtin H,2 H,1 H,1 H,1 = EAØ,2 + = Ø,2 (1 + ) + = ( ) ( ) = H,2 (1,1,1) = ( w(1,1,1) w(1,1) / w(1,1) w(1,1,1) ) ( (1,1,1) (1,1) ) /(1 + ) = ( / ) ( ) /( ) = H,2 (1,1,2) = ( w(1,1,2) w(1,1) / w(1,1) w(1,1,2) ) ( (1,1,2) (1,1) ) /(1 + ) = ( / ) ( ) /( ) = ( 1,1),3 = (1,1,1) + (1,1,2) = ( ) + ( ) = H,2 (1,2,1) = ( w(1,2,1) w(1,2) / w(1,2) w(1,2,1) ) ( (1,2,1) (1,2) ) /(1 + ) = ( / ) ( ) /( ) = H,2 (1,2,2) = ( w(1,2,2) w(1,2) / w(1,2) w(1,2,2) ) ( (1,2,2) (1,2) ) /(1 + ) = ( / ) ( ) /( ) = ( 1,2),3 = (1,2,1) + (1,2,2) = ( ) + ( ) = = + = ( ) + ( ) = (2),2 ( 1),3 (1,1),3 (1,2),3 H,2 (2,1,1) = ( w(2,1,1) w(2,1) / w(2,1) w(2,1,1) ) ( (2,1,1) (2,1) ) /(1 + ) = ( / ) ( ) /( ) = H,2 (2,1,2) = ( w(2,1,2) w(2,1) / w(2,1) w(2,1,2) ) ( (2,1,2) (2,1) ) /(1 + ) = ( / ) ( ) /( ) = = + = ( ) + ( ) = ( 2,1),3 (2,1,1) (2,1,2) in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 21

22 Top-Down Approach for Sinle eriod (continued) Third Decision: Market Capitalization Weihtin (continued) H,2 (2,2,1) = ( w(2,2,1) w(2,2) / w(2,2) w(2,2,1) ) ( (2,2,1) (2,2) ) /(1 + ) = ( / ) ( ( )) /( ) = H,2 (2,2,2) = ( w(2,2,2) w(2,2) / w(2,2) w(2,2,2) ) ( (2,2,2) (2,2) ) /(1 + ) = ( / ) ( ( ))/( ) = 0 ( 2,2),3 = (2,2,1) + (2,2,2) = = ( 2),3 = (2,1),3 + (2,2),3 = ( ) = Total market capitalization weihtin effect: = + = ( ) = Ø,3 (1),3 (2),3 Fourth Decision: Security Selection H,3 H,2 H,2 H,2 = EAØ,3 + = Ø,3 (1 + ) + = ( ) ( ) = Instead of showin every stock in the portfolio and benchmark, let us show ust one example and assume that w 0. 15, w 0, and = ( 1,1,1,1) = ( 1,1,1,1) = ( 1,1,1,1) (1,1,1,1) = H,3 (1,1,1,1) = ( w(1,1,1,1) w(1,1,1) / w(1,1,1) w(1,1,1,1) ) ( (1,1,1,1) (1,1,1) ) /(1 + ) = ( / ) ( ( ) /( ) = Further, assume that the total security selection effect: Ø,4 = Active eturn AG Ø = ( Ø,1) (1 + Ø,2 ) (1 + Ø,3) (1 + Ø, 4) 1 = ( ) ( ) ( ) ( ) 1 = in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 22

23 ottom-up Approach for Sinle eriod Overview As discussed in the Introduction, the bottom-up approach to portfolio attribution is most appropriately used when analyzin an investment manaer with a bottom-up investment process that focuses on security selection. The weihtin effect is secondary to the decision makin process. Unlike the top-down process that may involve a series of weihtin decisions, there is only one weihtin effect in the bottom-up process. This section addresses the bottom-up approach in a sinle-period attribution analysis. Multiperiod attribution is discussed in the last section of this document. Arithmetic Method w [12] CA = ( w w ) ( ) w [13] EA, n h Ω = h Ω CA EA h h, n if n = + 1 if n > + 1 [14] AA = = EAØ,1 + EAØ, 2 Ø Ø Ø Where: CA = Component that is attributable to roup, calculated based on arithmetic method EA, = Effect that is attributable to roup at decision level n, based on arithmetic method n AA Ø = The equity portfolio's active return, based on equity holdins, calculated based on arithmetic method = The roup where roup belons to in the prior roupin hierarchy level n = Decision level, where n = 1 is the weihtin decision, and = 2 Ø = The total level, which is the total equity n is the security selection decision in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 23

24 ottom-up Approach for Sinle eriod (continued) These formulas are similar to their counterparts in the Top-Down Approach section of this document. The Component formula in equation [12] demonstrates a hierarchical anchorin structure that is similar to that of its top-down counterpart in equation [5]. In the case of formula [12], it is the portfolio weiht of a roup that is scaled to the proportion between the benchmark's weiht and the portfolio's weiht in the parent roup. Once scaled, the portfolio weiht can be fairly compared to the benchmark weiht. In other words, when evaluatin the stock selection component of a particular stock, one should not compare the portfolio's weiht in the stock directly with the benchmark's weiht in the same stock. One must scale the portfolio's weiht in this stock by the proportion between the benchmark's weiht in the sector and the portfolio's weiht in the sector, assumin that the investment manaer roups stocks by sector. To accompany this anchorin system, it is the portfolio's return in the security that is compared to the benchmark's return in the sector in the second term of formula [12]. Similarly, when evaluatin a sector, it is the portfolio's return in the sector that is compared to the benchmark's total return, and this is consistent with incorporatin the Interaction term of the H model into the Weihtin Effect in a bottom-up approach. The Effect formula in equation [13] is intentionally written to be the same as its top-down counterpart in equation [6], a concept that is already familiar. In order to achieve this, n = 1 is set to denote the weihtin decision and n = 2 the security selection decision, even thouh security selection is the primary decision. This order is more intuitive as it matches the roupin hierarchy structure where = 1 represents the sector and = 2 the security. Formula [13] shows that the Effect of a parent roup is the sum of the Components of all of its children if the children are components of this decision. For example, when analyzin the selection effect, the securities are components of the decision, so the Service sector's selection effect is the sum of the selection components of all stocks in the sector. The Effect of a randparent roup is the sum of the Effects of all of its children. For example, the total equity portfolio's selection effect is the sum of selection effects of Service and Non-Service sectors, and these two are in term sums of selection components of the underlyin constituent stocks. Active eturn in formula [14] is the same as its top-down counterpart in equation [7], but only two decisions are involved: weihtin and selection. Active eturn is the value-add of the portfolio above the benchmark, and it is the difference between the return of the equity portion of the portfolio and that of the equity portion of the benchmark. Expressed in attribution terms, the Active eturn is the simple addition of the total weihtin effect and the total selection effect. This Active eturn represents the value-add of the equity portion of the portfolio. efer to the Appendix section of this document for the value-add of the total portfolio. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 24

25 ottom-up Approach for Sinle eriod (continued) Geometric Method CA EAØ,2 + [15] = CA Ø [16], n h Ω = h Ω h h, n Ø if if if = 1 if = 2 n = + 1 n > + 1 Ø [17] AG Ø = 1 = (1 + Ø,1) (1 + Ø,2 ) 1 Ø Where: = Component that is attributable to roup, calculated based on eometric method, = Effect that is attributable to roup at decision level n, based on eometric method n AG Ø = The equity portfolio's active return, based on equity holdins, calculated based eometric method The eometric component formula in equation [15] is similar to its top-down counterpart in formulas [8] and [9]. While the top-down approach allows multiple decisions and is better presented with two formulas, the bottom-up approach only requires one formula as there are only two decisions. Similar to equation [9], the component formula in equation [15] shows the use of the "hybrid" portfolio in the denominator to facilitate hierarchical anchorin. This anchorin system is similar to the one used in equation [12] for the arithmetic component calculation. ecall that in equation [12] the portfolio's weiht in a stock is scaled to the proportion between the benchmark's weiht in the sector and that of the portfolio. The hybrid portfolio is similar to the actual portfolio in that the portfolio's weiht in each stock is combined with the portfolio return in each stock, but the scaled portfolio weihts are used instead of the raw portfolio weihts. Mathematically the concise form in the denominator of equation [15] yields the same result as combinin the scaled portfolio weihts with portfolio returns, and the concise form has the benefit of re-usin numbers that are already calculated in the arithmetic method. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 25

26 ottom-up Approach for Sinle eriod (continued) The effect formula in equation [16] is the same to its arithmetic counterpart in formula [13] and its top-down counterpart in formula [10]. Formula [16] shows that the Effect of a parent roup is the sum of the Components of all of its children if the children are components of this decision. The Effect of a randparent roup is the sum of the Effects of all of its children if the children's descendents are components of this decision. The Active eturn calculation in formula [17] is essentially the same as its top-down counterpart in formula [11], demonstratin that Active eturn is achieved by takin the eometric difference between the portfolio's equity return and the benchmark's equity return or by eometrically linkin the total weihtin and selection effects. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 26

27 Multiple-eriod Analysis Overview The previous sections of this document demonstrate how Effects and Components are calculated for each sinle holdin period. A holdin period is the time between reported portfolio holdins. For example, if portfolio holdins are available on 1/31/2008, 3/31/2008 and 6/30/2008, the period between 2/1/2008 and 3/31/2008 represents the first holdin period, and the period between 4/1/2008 and 6/30/2008 is the second holdin period. When applyin the formulas in the previous sections, weihts are taken from the beinnin of the period, and returns are based on the entire holdin period. For example, when analyzin the first holdin period, weihts are based on 1/31/2008, and returns are from 2/1/2008 to 3/31/2008. It is often desirable to perform an analysis that spans over several portfolio holdins dates, for example, from 2/1/2008 to 6/30/2008. Althouh one miht think of treatin this as a sinle period, that is, takin the weihts as of 1/31/2008 and applyin them to returns from 2/1/2008 to 6/30/2008, valuable information could be lost. ortfolio constituents and their weihts miht have chaned between 1/31/2008 and 3/31/2008 due to buys, sells, adds, trims, corporate actions, etc. For the most meaninful analysis, portfolio holdins should be updated frequently, especially for hiher turnover portfolios. Frequent portfolio holdin updates create multiple sinle periods, and the next sections demonstrates how these sinle period attribution results can be accumulated into an overall multi-period outcome. Multi-period attribution effects consist of accumulatin sinle period results. Similar to a sinle period analysis, results can be calculated usin the arithmetic method or the eometric method. Use the multi-period arithmetic method to accumulate sinle-period arithmetic attribution results, and use the multi-period eometric method to link sinle-period eometric results. These multi-period methods apply to attribution results from both the top-down and the bottom-up approaches. Multi-eriod Geometric Method The eometric method is the method recommended by Morninstar. The eometric method has the merit of bein theoretically and mathematically sound. As stated in the Introduction of this document, it is important to distinuish between Effects and Components when performin an attribution analysis. An Effect measures the impact of a particular investment decision. An Effect can be broken down into several Components (e.. individual sectors such as Service) that provide insiht on each piece of an overall decision, but each piece in isolation cannot represent the impact of decision makin. Therefore, theoretically, multi-period linkin is only applicable to an Effect and not a Component. From a mathematical viewpoint, accumulatin Components over time either by addin or compoundin, and addin them back toether either by simple summation or eometric linkin, does not equal the Active eturn. in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 27

28 Multiple-eriod Analysis Use the followin formulas to link sinle-period eometric attribution Effects into multi-period results. T,, n, t ) 1 t= 1 [18] n, T, Cum = ( y, n, T, Ann (, n, T, Cum m [19] = ) 1 T M Ø,T,Cum [20] AGØ, T, Cum = 1 = (1 + AG t = Ø, ) 1 Ø,T,Cum t= 1 y Ø, T, Ann ( Ø, T, Cum m [21] AG = AG ) 1 n= 1 Ø, n, T, Cum Where:, n, T, Cum = Cumulative effect for roup decision level n, calculated based on eometric method, cumulative from sinle holdin periods 1 to T, n, T, Ann = Annualized effect for roup decision level n, calculated based on eometric method, over the time period from 1 to T AG Ø, T, Cum = Cumulative active return of the portfolio, calculated based on eometric method, cumulative from sinle holdin periods 1 to T AG Ø, T, Ann = Annualized active return of the portfolio, calculated based on eometric method, over the time period from 1 to T, n, t = Effect that is attributable to roup at decision level n, calculated based on eometric method, for sinle period t y = The number of periods in a year, for example, it is 12 when data are in monthly frequency m = The total number of periods, for example, it is 40 when the entire time period spans over 40 months Ø,T,Cum = The portfolio's return for the total level (total equity portfolio), cumulative from sinle holdin period from 1 to T Ø,T,Cum = The benchmark's return for the total level (total equity portfolio), cumulative from sinle holdin period from 1 to T AG Ø, = Active return of the portfolio, calculated based on eometric method, for sinle period t t M = The level that represents the security level, that is, the last roupin hierarchy in whole or part, without the prior written consent of Morninstar, Inc., is prohibited. 28