Global. Portfolio Analysis. Beating Benchmarks. A Stockpicker s Reality: Part II. Global. November 30, 1999

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1 Global Global Portfolio Aalysis November 3, 999 Aalysts Steve Strogi (New York) Melaie Petsch melaie.petsch@gs.com (New York) Greg Shareow greg.shareow@gs.com (New York) Beatig Bechmarks A Stockpicker s Reality: Part II The iability of active maagers to cosistetly outperform capitalizatio-weighted bechmarks ca be explaied by a mismatch betwee those bechmarks ad the uderlyig ature of active maagemet. We show that this mismatch caot be effectively addressed either through macro level risk cotrols or through improved stock selectio. However, we develop a ew approach to risk maagemet that emphasizes diversificatio at the idividual stock level ad offers sigificat icreases i risk-retur efficiecy ad portfolio maager cosistecy; it is also sigificatly easier to icorporate it ito a bottoms-up ivestmet process. Further, we show how pla sposors ca further improve the value of active maagemet through combiatios of ew, more portfolio-maager-friedly active maager bechmarks ad completio idices that move the overall allocatios back to their origial capitalizatioweighted bechmarks. Goldma Sachs Ivestmet Research Importat disclosures appear o the back cover. Electroic Documet Available via Ivestmet Research o GS Fiacial Workbech SM

2 Goldma Sachs Global Portfolio Aalysis Global Table of Cotets Makig Skill Cout The Nature of Skill 4 Mappig Skill Levels to Implied Returs 6 Gettig a Realistic Picture of the Stockpicker s Edge 8 A Guide to the Skill Scatter Charts ad Tables 9 Maagig Ucertaity: Turig a Thi Edge ito Cosistecy 3 Solvig the Risk Maagemet Problem 6 Large-Cap First The Russell Uiverse 6 Dealig with Stock-Specific Bechmark Risk 9 Performace Over Time 3 Real World Evidece 5 Mid- to Small-Cap, The Other Real World 7 Portfolio Maager Patters, Skill Sets ad Value-Ehacig Risk Maagemet 3 Risk Cotrol ad Log-Ru Portfolio Maager Performace 3 Appedix A: The Mathematics of Diversificatio 33 Derivatio of the Diversificatio Results 34 Effective 35 The /(+) Rule 36 Trackig Error 38 Appedix B: The Data ad Strategies 39 The Strategies 39 Portfolio Costructio 4 Adjustig for Size ad Sector 4 Appedix C: Key Results for a Hybrid or Growth at a Reasoable Price Strategy 4 Portfolio Costructio 4 Results 4 Maagig Ucertaity 44 Solvig the Risk Maagemet Problem 44 Appedix D: Volatility ad the Square Root of N 48 Appedix E: S&P 5 Idex Cocetratio 49 Appedix F: Maagig the Stock-Specific Risk i Large-Cap Bechmarks 5 Appedix G: How Cotrollig for Size Decreases Risk-Retur Efficiecy56 Appedix H: Rescaled Time Series Graphs 58 Appedix I: Data ad Methodology for Lipper Aalysis 59 Goldma Sachs Ivestmet Research

3 Global Makig Skill Cout Beatig bechmarks usig fudametal bottoms-up stock aalysis has at its core two parts: stock selectio (the orderig of stocks from best to worst usig fudametal aalysis) ad portfolio costructio (the traslatio of that orderig ito a actual portfolio). This paper focuses o portfolio costructio. I particular, we focus o how a portfolio maager skilled at stock selectio ca exploit that skill to beat a target bechmark by as much as possible ad as cosistetly as possible. Recet uderperformace by U.S. large-cap portfolio maagers has geerated a itese focus o valuatio methods for large-cap growth compaies ad o the geeral questio of macro-level risk maagemet for equity portfolios. We show that these efforts are likely to sigificatly reduce future returs without oticeably improvig the quality of risk or cosistecy of outperformace as these efforts are based o a misuderstadig of the true ature of the recet risk maagemet failure ad o a ureasoable otio of what eve the most skilled portfolio maager might be able to do i assessig the retur potetial of idividual compaies. I particular, we fid that for reasoable levels of portfolio maager skill (i.e., skill levels cosistet with the level of log-ru outperformace most portfolio maagers would be willig to claim), it is simply impossible to improve stock valuatio methods eough to solve the recet large-cap problem by better stock selectio. Further, our results demostrate that the key problems that have preveted large-cap portfolio maagers from geeratig cosistet outperformace over the last decade were either due to a macro level failure to properly cotrol size (or ay other macro risk factor) or due to a failure of fudametally drive stock pickig strategies (either growth or value) to discrimiate betwee high- ad low-performig stocks. Rather, almost the etirety of portfolio maager icosistecy ca be explaied by a failure to Stock selectio is the focus of our Jauary 4, 999 paper Style, Size ad Skill, Part I of A Stockpicker s Reality. Goldma Sachs Global Portfolio Aalysis properly take accout of a massive cocetratio of stock-specific risk i a small umber of ames at the top ed of the capitalizatio spectrum. While the differece betwee the macro cotrol of size ad cotrollig exposure to a small umber of stocks at the top of the capitalizatio spectrum might seem a rather academic distictio, the operatioal implicatios are eormous ad the resultig impact o portfolio maager performace dramatic. I particular, we show that cotrollig size risk at the macro level reduces returs almost directly i proportio to the amout of bechmark trackig error elimiated (i.e., risk ad retur fall i equal amouts as retur per uit of risk barely improves). I cotrast, compesatig for the cocetratio of stock-specific risk at the top ed of the capitalizatio spectrum through passive idividual stock positios reduces trackig error at double the rate that it reduces returs, substatially improvig the quality of risk ad the cosistecy of outperformace. For example, we fid that for a moderately skilled portfolio maager, simply market weightig the top 5 stocks will, o a pre-trasactios cost basis, double their Sharpe ratio from.58 to.3 for value ad from.6 to.3 for growth, icrease the percetage of quarters that such portfolio maagers outperform their bechmark from 58.3% to 73.8% for value ad from 6.3% to 74.3% for growth, radically smooth the time series of outperformace ad dramatically reduce both the size ad the frequecy of extreme uderperformace (icludig trasactio costs would oly make the improvemet more dramatic o a relative basis). The geeral presumptio that idices with large umbers of stocks diversify away most idiosycratic stock risk ad that the remaiig idex volatility is mostly macro i character oly holds if the weight o each stock is below a certai threshold. Whe stocks are added with weights above that threshold, more stock-specific risk is added tha diversified away, creatig idices with sigificat stock-specific volatility focused i those high-weight stocks. The mathematical details of this argumet are covered i Appedix A. Goldma Sachs Ivestmet Research

4 Goldma Sachs Global Portfolio Aalysis More broadly, we fid that the stadard top-dow macro risk approach fails bottoms-up portfolio maagers i three ways:. It completely misses the eed to offset the stockspecific risk embedded i large-capitalizatio bechmarks.. It misdirects stock selectio toward large-cap ames where active maagemet is, i geeral, less effective. 3. It causes portfolio maagers to cocetrate stock selectio risk ito too small a umber of positios to allow for cosistecy of outperformace. Our results idicate that for risk maagemet to actually aid portfolio maager performace, it is ecessary to refocus risk cotrol away from quatifyig the macro risk characteristics of the portfolio at a poit i time ad toward uderstadig ad cotrollig the quality (ot quatity) of risk that the portfolio maager takes over time. I particular, we fid that the key risk cotrol issue for bottoms-up fudametally drive portfolio maagers is beig able to distiguish betwee () habitual cocetratios of risk that arise either out of peculiarities of the bechmark or prejudices i the portfolio maager s ivestmet process that create cosistet risk positios that do ot reflect curret market coditios or the portfolio maager s skill, 3 ad () those risk positios that arise aturally out of the portfolio maager s assessmet of fudametals ad vary with market coditios. Global Habitual risk positios create a high level of bechmark risk without ay real expectatio of retur. Elimiatig such positios, either through passive offsets or costraits o portfolio costructio, ca sigificatly improve a portfolio maager s ability to cosistetly outperform bechmarks. I cotrast, we fid that the types of risk positios that arise aturally out of fudametal aalysis by skilled portfolio maagers are ot oly justified o a risk-retur basis, but are aturally diversified. The atural diversificatio suggests that the best way to reduce risk without sacrificig returs uecessarily is simply to icrease the umber of stock positios (both at a poit-i-time ad over time) i order to allow the high level of ucertaity that characterizes idividual stock positios to average out as much as possible ad, thus, allow the portfolio maager s skill at stock selectio to domiate rather tha the radomess of idividual stock returs. Macro level risk cotrols ca ad ofte do sigificatly iterfere with portfolio maagers takig advatage of this atural diversificatio ad actually reduce the riskretur efficiecy of portfolio maager performace. Cosequetly, we argue that, beyod idetifyig ad elimiatig chroic/habitual risk positios through passive offsets, 4 risk cotrols should be limited to () helpig exploit the atural diversificatio of the portfolio maager s stock choices ad () helpig the portfolio maager modestly emphasize takig risk i categories of stocks i which they ted to be more effective ad de-emphasize takig risk i areas i which they ted to be less effective. 5 3 Examples of habitual cocetratios of risk iclude () beig perpetually uderweight a cocetratio of stockspecific risk i the high idex-weight stocks discussed above or () a perpetual uderweight i the tech sector that reflects a portfolio maager s discomfort with ew techology rather tha their evaluatio of the actual compaies. 4 Such offsets ca be accomplished either through the use of derivatives or through appropriately costructed passive portfolios either at the portfolio maager or pla sposor level. 5 I particular, sector cotrols ca be useful i helpig defie categories of stocks i which the mappig betwee fudametals ad returs is more cosistet. Sector cotrols ca also provide a ability to compesate for the positive correlatios betwee stock picks that are created by the use of commo drivers of forecasted earigs withi sectors. Goldma Sachs Ivestmet Research

5 Global I summary, we argue that the risk maagemet process should be split ito three distict processes which, i order of decliig importace, are:. Compesatig for udue cocetratio of stockspecific risk i (large-cap) bechmarks.. Broadly diversifyig stock-specific risk chose by portfolio maagers i order to allow the high level of ucertaity i the idividual stock returs to average out. 3. Modestly cocetratig stock-specific risk i areas i which the portfolio maager has demostrated greater ability to idetify higherreturig stocks. Goldma Sachs Global Portfolio Aalysis The paper proceeds i three steps. First, we defie the ature of maager skill i a way that allows us to quatify the level of skill eeded to produce differig levels of log-ru outperformace. We the aalyze differet risk cotrol approaches, both i terms of performace ad i terms of portfolio maager cosistecy, i order to uderstad how differet risk maagemet approaches impact both returs ad cosistecy. We the eter a broader discussio of how bottoms-up maagers should approach risk cotrol, both coceptually ad as a matter of practice. I particular, we look at what is ecessary to tailor risk cotrol to improve overall performace rather tha simply reduce risk. Goldma Sachs Ivestmet Research 3

6 Goldma Sachs Global Portfolio Aalysis The Nature of Skill For our purposes, portfolio maager skill is defied as the ability to rak stocks based o future fudametals. I Part I of this series, Style, Size ad Skill, we looked at how the market pays for future fudametals. I particular, we showed that a portfolio maager who raks stocks based o future fudametals usig either a growth methodology (i which stocks are raked from best to worst based o forward earigs growth) or a value methodology (i which stocks are raked from best to worst based o a future ormalized P/E ratio) ca separate stocks ito higher- ad lower-performig groups with a high degree of cosistecy. (More details o the growth ad value measures we use, as well as the data, ca be foud i Appedix B.) A limitatio of the prior aalysis was that, to clearly defie what the market was pricig ito the market, we allowed the portfolio maager a ureasoable level of foresight ito future earigs, allowig the maager to perfectly predict earigs. I the curret cotext, we eed to be able to allow the portfolio maager a fixed, but limited, ability to rak stocks based o isights ito forward earigs behavior. We ca the use these rakigs to see how differet risk maagemet approaches will allow a portfolio maager with a give level of stock-selectio skill to create portfolios capable of outperformig a give bechmark. The way we model skill is to allow the portfolio maager to rak stocks relative to the true (i.e., perfect foresight) fudametal rakigs for their style of ivestig with differig degrees of statistical accuracy. This allows us to hold the stock-selectio skill level costat ad ivestigate how differet risk-cotrol approaches work with differet ivestmet styles ad portfolio costructio approaches through large-scale simulatios. I the mai body of the paper, we focus o pure growth ad value styles. I Appedix C, we repeat some of the key results for a hybrid valuatio or growth at a reasoable price style. To create the true rakigs for pure value ivestig, every caledar quarter, stocks are raked based o P/E ratios (which are based o the average realized earigs for the ext four quarters) from the least to most expesive, uder the expectatio that less Global expesive stocks will outperform more expesive stocks. For pure growth ivestig, true rakigs are based o forward earigs growth the ext four quarters for the S&P 5 ad Russell (largecap) simulatios, the ext two quarters for Russell (small-cap) simulatios. 6 The we create simulated rakigs i which the portfolio maager is able to approximate the true rakig more or less closely, based o their skill level. The specifics of the statistical modelig are quite simple. I the zero-skill case, the portfolio maager s stock ratigs follow a uiform radom distributio from to (thik of this as the percetile rak of the stock). Thus, i the absece of stockpickig skill, each stock is equally likely to have ay ratig from to, regardless of fudametals. To create skill, we tilt the distributio such that stocks with better fudametals are more likely to receive higher ratigs ad less likely to receive low ratigs. We do this simply by tiltig the uiform distributio based o the true rakig of the stock ad the skill of the maager. Figure shows the resultig valuatio ratig distributios for the best, media ad worst stocks, for a zero-skill, moderateskill ad max-skill portfolio maager. For zero skill, the top stock (true ratig value of. measured o a scale from. to.) has roughly a 5% probability of receivig a ratig betwee.95 ad. ad a % probability of gettig a ratig betwee.8 ad. (i.e., a top quitile ratig). Similarly, i the zero-skill case, the top stock also has a % probability of gettig a ratig betwee. ad. (i.e., a bottom quitile ratig). With max stock-selectio skill, the top stock has a 9.75% chace of gettig a ratig betwee.95 ad. ad a 36% chace of a top quitile ratig, while oly a 4% chace of gettig a bottom quitile ratig. 6 I Style, Size ad Skill, we showed that the optimal horizo for earigs isight for growth strategies is shorter for smaller-cap stocks. These horizos (four quarters forward for large-cap, two quarters forward for small-cap) were chose because, of oe to four quarters forward, they provide the best performace for the growth strategies for these particular data samples. 4 Goldma Sachs Ivestmet Research

7 Global Aother way of thikig about this measure of stock-selectio skill, which gives some additioal ituitio about the level of skill implied by these tilts, is to ask how accurate the rakig is relative to the true (that is, perfect foresight) rakig. Oe way of doig this is to look at the quitile accuracy. That is, if the portfolio maager raks stocks from to 5 where is the best quitile ad 5 is the worst quitile of stocks, how likely is the portfolio maager to rak stocks i the correct quitile? Table shows the map from the tilt of the skill distributio to the percetage of the stocks the portfolio maager raks i the correct quitile bucket. No skill (% of maximum tilt) gets it right % of the time. Moderate skill (54% of maximum Goldma Sachs Global Portfolio Aalysis tilt) gets it right 3% of the time, high skill (8% of maximum tilt) 5% of the time ad max skill (% of maximum tilt) 6.4% of the time. At first glace, the implied level of accuracy appears quite low (eve whe the distributio is tilted as far as possible), but as we see i the ext sectio, whe we traslate these skill levels ito implied log-ru excess returs that would be geerated by a portfolio maager with these levels of stock-selectio skill, the rage of implied returs covers the full rage of what might reasoably be expected from portfolio maagers ad eve reaches levels well beyod what eve the most skilled portfolio maager could be expected to deliver. Figure : Skill No Skill Moderate Skill Max Skill Pael : Worst Stock Probability Desity Ratig Variable for "Worst" Stock, No Skill Probability Desity Ratig Variable for "Worst" Stock, Moderate Skill Probability Desity Ratig Variable for "Worst" Stock, Max Skill Pael : Media Stock Probability Desity Ratig Variable for "Media" Stock, No Skill Probability Desity Ratig Variable for "Media" Stk, Moderate Skill Probability Desity Ratig Variable for "Media" Stock, Max Skill Pael 3: Best Stock Probability Desity Ratig Variable for "Best" Stock, No Skill Probability Desity Ratig Variable for "Best" Stock, Moderate Skill Probability Desity Ratig Variable for "Best" Stock, Max Skill Goldma Sachs Ivestmet Research 5

8 Goldma Sachs Global Portfolio Aalysis Mappig Skill Levels to Implied Returs To get a solid idea of what these skill levels mea i terms of actual stock selectio, we ru statistical simulatios i which a simulated portfolio maager of a give skill level creates, rakigs, which are the traslated ito log-short portfolios. The portfolios are log the top % of the stocks (by the simulated rakigs) ad short the bottom %. 7 Returs are the calculated for, log-short portfolios. We the treat half the average of these returs as a reasoable estimate of the potetial excess returs relative to market that a log-oly portfolio maager of this skill level should be able to attai over time. 8 The use of log-short portfolios i this sectio of the paper allows us to cocetrate o the impact of skill o stock selectio (i.e., the rakig of stocks) ad how that relates to returs i a way that is largely idepedet of bechmark choice. This turs out to be importat as it allows us i later sectios to clearly separate which risk cotrol problems relate to the portfolio maager s skills (ad prejudices) ad which relate to the particular bechmark they are attemptig to beat. This, i tur, acts as a guide to which results are likely to hold for all portfolio maagers ad which eed to be adjusted to reflect the particular skills/process of a specific portfolio maager. The average returs ad implied log-ru excess returs geerated by value ad growth strategies at the various skill levels are reported i Table (alog with two rak correlatios betwee the skilled raks ad the perfect foresight raks). As would be expected, a portfolio maager with o skill geerates Table : Stockpickig Edge Associated with Skill Tilt Percet of Maximum Percet Skill i Correct Skill Tilt (%) Quitile (%) Name No Skill Moderate High 6.4 Max ---- Perfect Global o excess returs. What might be more surprisig is how little of a edge i stock selectio is eeded to geerate extraordiarily high excess returs. Over this period, the max skill level, which oly has a 6.4% chace of rakig a stock i the correct quitile, produces.9% aualized returs for a market-eutral log-short growth maager ad 9.9% for a market-eutral value maager, implyig logru log-oly outperformace of 5.4% ad 4.9%, respectively. Such retur umbers would suggest that the reasoable rage for portfolio maager skill would be betwee 3% ad 5% or betwee 3% ad 5% better tha radom selectio, which we refer to as moderate ad high skill respectively. I most cases, we also look at higher skill levels to demostrate that our results hold eve at extraordiary skill levels that are cosistet with returs well beyod historical precedet. 7 Uless otherwise idicated, all of the log-oly strategies are log the top % of stocks ad the log-short strategies are log the top % ad short the bottom %. The exceptios are graphs that deal with the impact of the umber of stocks i the portfolio. 8 Trasactio costs, of course, would reduce these returs, but such costs are too depedet o positio size to be dealt with i a geeral maer. As a first approximatio, reduce reported returs by 5 basis poits to approximate realized returs for roughly a $- billio portfolio. For a more complete aalysis of how trasactios costs would impact these results, see Appedix A of Style, Size ad Skill. 6 Goldma Sachs Ivestmet Research

9 Global Goldma Sachs Global Portfolio Aalysis Table : Implied Log-Ru Excess Returs ad Rak Correlatios for Skill Levels (Estimated Russell Sample, Q987-Q998) Average Implied Log-Ru Log-Short Rts (%) Excess Rts (%) Percet Quitile Skill i Correct Growth Value Growth Value Bucket Rak Name Bucket (%) Strategy Strategy Strategy Strategy Correlatio Correlatio No Skill Moderate High Max Perfect Goldma Sachs Ivestmet Research 7

10 Goldma Sachs Global Portfolio Aalysis Gettig a Realistic Picture of the Stockpicker s Edge The thiess of the stock-pickig edge described above hits that oe of the core risk maagemet problems facig portfolio maagers is that they have a high probability of beig wrog o idividual stocks. Such radomess ca oly be tured ito cosistet short- or medium-term performace by takig a large umber of positios ad allowig the radomess to average out. This radomess turs out to be deeply fudametal to the whole portfolio maagemet process. Stock returs are very, very disperse ad mostly radom with respect to ay particular otio of fudametals, eve for portfolio maagers with very, very high levels of skill based o early perfect foresight of fudametals, let aloe those with more reasoable levels of skill. Global To provide a baselie represetatio of the uderlyig radomess of stock returs, Figure shows a represetative scatter plot from a particular quarter of data of the relatioship of idividual stock returs to their rak for a portfolio maager with o skill, plus the regressio lie characterizig the relatioship betwee rakigs ad returs ad the limits of the 5% ucertaity bad aroud the regressio lie. Because this type of graphic ad the associated table are cetral to this paper, the cotets are explaied more fully i the sidebar o the ext page. Not that surprisigly, for the o-skill case, the regressio lie is flat ad the idividual equities ofte fall quite far from the lie. Based o, simulatios of the rakig process at this skill level, we calculate that 5% of the stocks will have quarterly returs withi ±7.8% of portfolio maager s expectatios. 9 This is a very wide bad, highlightig the large extet of the uderlyig radomess i stock returs. Figure : No Skill Relatioship of Rakig Criterio to Returs (Estimated Russell Sample, Q987-Q998) Quarterly Retur Oe Quarter Forward (%) % Ucertaity Bad Stock Selectio Rakig Criterio No Stock Selectio Skill (% of Stocks i Correct Bucket) High-Mid Spread. % High-Low Spread. % 5% Ucertaity Bad +/- 7.8 % Average Slope. % Stadard Deviatio 4.4 % Average R-Squared. % Aualized Log-Short Retur. % Aualized Implied Log-Ru. % Excess Log-Oly Retur 9 We also iclude a estimated stadard deviatio which is estimated by calculatig a 95.4% cofidece iterval ad the dividig the width of that iterval by 4. This provides a more robust measure of the stadard deviatio tha the more 9 covetioal calculatio as it reduces the impact of outliers. 8 Goldma Sachs Ivestmet Research

11 Global Goldma Sachs Global Portfolio Aalysis A Guide to the Skill Scatter Charts ad Tables The scatter plots depict oe simulatio of each style ad skill level from the third quarter of 997. Each dot represets a idividual stock s oe-quarter forward retur i our estimated Russell sample. At the various levels of skill, the simulated portfolio maager raks stocks from to, where is the worst ad is the best. The graphs show the relatioship of this rakig variable ad the subsequet retur o the stocks, alog with a regressio lie to summarize that relatioship. The statistics i the accompayig tables refer to the average of, simulatios for each quarter from the first quarter of 987 to the first quarter of 998. The figure above illustrates how some of the key skill statistics are displayed i the scatter charts; the table below describes the statistics we report. High-Mid Spread The differece betwee the expected returs of the top-raked stock ad the media-raked stock. High-Low Spread The differece betwee the expected returs of the top-raked stock ad the bottom-raked stock. 5% Ucertaity Bad 5% of the realized returs of the stocks falls withi these bouds of the retur predicted by the valuatio model. That is, 5% of the realized returs are betwee the predicted retur plus this percetage ad the predicted retur mius this percetage. Average Slope Average of the slope from the, simulated regressios of the forward retur o the rakig criterio. Stadard Deviatio Stadard deviatio of the realized returs aroud the expected retur lie. Average R-Squared Average of the R-squared from the, simulated regressios of the forward retur o the rakig criterio. Aualized Log-Short Retur Aualized Implied Log-Ru Excess Log-Oly Retur Retur Oe Quarter Forward (%) Stock Selectio Model Predictio Idividual Stock Stock Selectio Rakig Criterio 5% of stocks fall withi this rage of retur forecast errors Aualized average of the excess returs for, simulated portfolios log the top % of the stocks by the rakig criteria ad short the bottom %. That is, go log the fastest growig ad short the slowest growig or go log the least expesive ad short the most expesive. Aualized average excess retur oe might expect over the log ru from a maager with this style ad skill level. This umber is half of the historical log-short retur show above. Goldma Sachs Ivestmet Research 9

12 Goldma Sachs Global Portfolio Aalysis To show how stock-selectio skill impacts this radomess, Figures 3 ad 4 show equivalet scatter plots ad relatioship characteristics for growth ad value portfolio maagers, respectively, for three skill levels: Moderate skill, 3% edge: stocks are placed i the right quitile 3% of the time more tha pure radom chace, 3% vs. % of the time, cosistet with 3.% aualized log-ru log-oly outperformace for growth (before trasactios costs) ad.8% for value, High skill, 5% edge cosistet with 4.5% outperformace for growth ad 4.% for value Perfect skill, 8% edge rakig of stocks reflect actual future earigs with perfect foresight cosistet with 6.5% outperformace for growth ad 5.3% for value. The core observatio is that, while icreases i portfolio maager skill geerate higher ad higher returs, as evideced by the slope of the regressio lie ad the expected excess retur of the top-raked stock, higher levels of skill do ot oticeably reduce the uderlyig level of ucertaity at the idividual stock level with respect to the lik betwee valuatio ad returs. Eve whe we allow for perfect foresight of future fudametals ad extraordiary rates of implied excess returs (6.5% per year for growth ad 5.3% for value), the scatter diagrams show o reductio i ucertaity visible to the aked eye ad, without the regressio lies ad related statistical aalysis, it would be impossible to assess whether the valuatio method was i fact addig value. At the idividual stock level, this high level of ucertaity domiates the risk-retur problem. For a highly skilled growth maager, the top-rated stock Global would be expected to outperform the media stock by oly.5% i ay give quarter (this is the hi-mid spread i the tables). The ucertaity aroud that.5% of outperformace is eormous. The 5% ucertaity bad is ±7.8%, meaig that 5% of the time that top stock would outperform the media stock by more tha 9.3% ad 5% of the time the top stock would uderperform the media stock by more tha 6.3%. (To ease compariso to the log-ru retur statistics, the hi-mid spread aualizes to 5.8% outperformace with a ucertaity bad of ±3.%.) Eve for a growth maager with perfect foresight, the 5% ucertaity bad at the idividual stock level is still ±7.7% or ±3.8% o a aualized basis, eve though such a portfolio maager would be expected to produce log-ru returs a full 6.5 percetage poits above ad beyod their bechmark. The situatio is eve worse if we shift our focus from the top stock to what could be called key idex drivers (stocks with high idex weights) with middlig ratigs. Such stocks are clearly capable of very strog ad very weak performace, but it is simply impossible to forecast those returs with ay accuracy through eve the most isightful fudametal aalysis. Further, these results are true o matter how skillful the portfolio maager or the type of valuatio methods employed. The reaso for this is that the level of dispersio of stock returs is so high that ay method of valuatio that meaigfully reduces the ucertaity of returs at the idividual stock level would geerate such high returs as to defy historical reality ad commo sese. Put more simply, eve though skilled stock selectio is capable of geeratig sigificat returs at the portfolio level, it is simply impractical to get idividual stocks or eve small groups of stocks right. See Appedix C to see this work repeated for a hybrid valuatio method. Goldma Sachs Ivestmet Research

13 Global Goldma Sachs Global Portfolio Aalysis Figure 3: Growth Relatioship of Rakig Criterio to Returs (Estimated Russell Sample, Q987-Q998) Pael : Portfolio Maager with Moderate Skill Quarterly Retur Oe Quarter Forward (%) % Ucertaity Bad Stock Selectio Rakig Criterio Moderate Stock Selectio Skill (3% of Stocks i Correct Bucket) High-Mid Spread. % High-Low Spread.9 % 5% Ucertaity Bad +/- 7.8 % Average Slope.9 % Stadard Deviatio 4.4 % Average R-Squared.6 % Aualized Log-Short Retur 6. % Aualized Implied Log-Ru 3. % Excess Log-Oly Retur Pael : Portfolio Maager with High Skill Quarterly Retur Oe Quarter Forward (%) % Ucertaity Bad Stock Selectio Rakig Criterio High Stock Selectio Skill (5% of Stocks i Correct Bucket) High-Mid Spread.5 % High-Low Spread.9 % 5% Ucertaity Bad +/- 7.8 % Average Slope.9 % Stadard Deviatio 4.4 % Average R-Squared.46 % Aualized Log-Short Retur 9. % Aualized Implied Log-Ru 4.5 % Excess Log-Oly Retur Pael 3: Portfolio Maager with Perfect Skill Quarterly Retur Oe Quarter Forward (%) % Ucertaity Bad Stock Selectio Rakig Criterio Perfect Stock Selectio Skill (% of Stocks i Correct Bucket) High-Mid Spread 5. % High-Low Spread. % 5% Ucertaity Bad +/- 7.7 % Average Slope. % Stadard Deviatio 4. % Average R-Squared 4.7 % Aualized Log-Short Retur 33. % Aualized Implied Log-Ru 6.5 % Excess Log-Oly Retur Goldma Sachs Ivestmet Research

14 Goldma Sachs Global Portfolio Aalysis Global Figure 4: Value Relatioship of Rakig Criterio to Returs (Estimated Russell Sample, Q987-Q998) Pael : Portfolio Maager with Moderate Skill Quarterly Retur Oe Quarter Forward (%) % Ucertaity Bad Stock Selectio Rakig Criterio Moderate Stock Selectio Skill (3% of Stocks i Correct Bucket) High-Mid Spread.9 % High-Low Spread.7 % 5% Ucertaity Bad +/- 7.8 % Average Slope.7 % Stadard Deviatio 4.4 % Average R-Squared.7 % Aualized Log-Short Retur 5.6 % Aualized Implied Log-Ru.8 % Excess Log-Oly Retur Pael : Portfolio Maager with High Skill Quarterly Retur Oe Quarter Forward (%) % Ucertaity Bad Stock Selectio Rakig Criterio High Stock Selectio Skill (5% of Stocks i Correct Bucket) High-Mid Spread.3 % High-Low Spread.6 % 5% Ucertaity Bad +/- 7.8 % Average Slope.6 % Stadard Deviatio 4.4 % Average R-Squared.47 % Aualized Log-Short Retur 8.3 % Aualized Implied Log-Ru 4. % Excess Log-Oly Retur Pael 3: Portfolio Maager with Perfect Skill Quarterly Retur Oe Quarter Forward (%) % Ucertaity Bad Stock Selectio Rakig Criterio Perfect Stock Selectio Skill (% of Stocks i Correct Bucket) High-Mid Spread 4.6 % High-Low Spread 9.3 % 5% Ucertaity Bad +/- 7.7 % Average Slope 9.3 % Stadard Deviatio 4. % Average R-Squared 4.83 % Aualized Log-Short Retur 3.6 % Aualized Implied Log-Ru 5.3 % Excess Log-Oly Retur Goldma Sachs Ivestmet Research

15 Global Maagig Ucertaity: Turig a Thi Edge ito Cosistecy Coquerig this type of radomess is easy i theory ad ot that difficult i practice. The solutio is diversificatio. While idividual stocks are subject to a high degree of ucertaity, as we icrease the umber of stocks, the radomess of the portfolio falls roughly as a fuctio of the iverse of the square root of ( ). See Figure 5, where is the umber of stocks i a equally weighted portfolio (more accurately, where is the umber of statistically idepedet risk positios). (Appedix A shows i detail how to calculate the level of diversificatio i o-equally weighted portfolios.) Thus, the stadard deviatio ad 5% ucertaity bads fall to half their origial sizes if the portfolio has 4 stocks i it ad to oe-teth their origial sizes if the portfolio has stocks. I practice, the effectiveess of diversificatio is strogly impacted by the correlatios betwee idividual stock positios high correlatios imply a smaller reductio i ucertaity, while egative correlatios would imply eve larger reductios i ucertaity. I fact, as will become clear, the core risk cotrol issue for fudametally drive portfolio maagers is elimiatig highly correlated risk positios so that diversificatio works ad the uderlyig radomess i returs ca be averaged out. A simple way to see how effective diversificatio is at reducig the ucertaity of stock selectio is to look at the impact of chagig the umber of equally weighted stocks i the portfolio o returs, trackig errors ad Sharpe ratios. Returs fall as stocks with lower expected returs are added to the portfolio, but volatility (that is, trackig error) falls; thus, the riskretur efficiecy as measured by the Sharpe ratio ca rise as log as the resultig reductio i ucertaity more tha offsets the reductio i returs. Figure 6 shows the aualized average returs, the volatilities (as measured by the stadard deviatios) ad the Sharpe ratios for log-short portfolios with If the log-short portfolios were held as a overlay to a bechmark portfolio, these stadard deviatios would be the trackig errors. Goldma Sachs Global Portfolio Aalysis Figure 5: Equal-Weight Portfolio Volatility Falls with the Iverse of the Square Root of Percetage of Origial Idiosycratic Risk (%) Number of Stocks i Portfolio differet umbers of stocks. We iclude a lie of the iverse of the square root of ( ) 3 i the volatility graph to bechmark how well diversificatio is workig i each case. These graphs show that diversificatio works quite well without ay risk cotrol at all. The close correspodece betwee the square root of baselie ad the average stadard deviatio of the simulated portfolios, which correspods to the trackig error of a log-short overlay portfolio, shows that the stock picks that arise from fudametal aalysis are relatively ucorrelated across stocks. This is very good ews i that it strogly suggests that, as we go forward to look at risk maagemet approaches, it will ot be ecessary to distort or eve guide the stock selectio process i ay strog way as fudametals ad the uderlyig diversity of stocks will create all the diversificatio that is eeded to geerate much more cosistet portfolio maager performace. The Sharpe ratio graph shows that, iitially, addig stocks to the portfolio causes a rapid icrease i the risk-retur efficiecy of the portfolio, but as the Due to covergece issues for portfolios of small umbers of stocks, graphs with results from such portfolios represet, simulatios rather tha,. 3 Appedix D provides a explaatio for why we use the square root of i this cotext. Goldma Sachs Ivestmet Research 3

16 Goldma Sachs Global Portfolio Aalysis Global Figure 6: Impact of Icreasig Number of Positios for the Average of Value ad Growth, Moderate Skill, Log-Short Portfolios (Estimated Russell Sample, Q987-Q998,, Simulatios) Pael : Aualized Returs Average Excess Returs (%) Log-Short 5 5 Number of Stocks i Portfolio Figure 7: Impact of Icreasig Number of Positios for the Average of Value ad Growth, Moderate Skill, Log-Oly Portfolios (Estimated Russell Sample, Q987-Q998,, Simulatios) Pael : Aualized Returs Average Excess Returs (%) Log-Oly. 5 5 Number of Stocks i Portfolio Pael : Volatility Stadard Deviatio (%) Log-Short 5 /Sqrt() 5 5 Number of Stocks i Portfolio Pael : Trackig Errors Trackig Error (%) Log-Oly 5 /Sqrt() 5 5 Number of Stocks i Portfolio Pael 3: Sharpe Ratios 3.5 Sqrt() Scaled to Log-Short Sharpe Ratio 3. Pael 3: Sharpe Ratios 3. Sqrt() Scaled to Log-Oly Sharpe Ratio.5 Sharpe Ratio Log-Short Sharpe Ratio..5. Log-Oly Number of Stocks i Portfolio. 5 5 Number of Stocks i Portfolio 4 Goldma Sachs Ivestmet Research

17 Global umber of stocks exceeds or roughly % of the stock uiverse, the rate of reductio i ucertaity begis to fall off, causig the declie i expected returs to take a more sigificat toll o the riskretur efficiecy of the portfolio. Net, the data strogly supports the otio that risk-retur tradeoffs are improved by broadeig rather tha deepeig research ad stock-selectio criterio. Similarly, these results suggest that extedig holdig periods will (by reducig the umber of idividual stock choices) reduce rather tha improve the risk-retur tradeoff. These graphs also poit to a sigificat trade-off betwee log-ru returs ad short-term risk-retur efficiecy. We will address this i more detail later, but it is already clear that cosistecy will have a price ad that fidig ways to improve this trade-off will have sigificat log-ru beefits for portfolio maagers ad ivestors alike. Goldma Sachs Global Portfolio Aalysis The problem facig real world portfolio maagers, who caot short stocks ad thus caot egage i log-short strategies, becomes evidet if we redo this aalysis usig oly the log portio of the portfolio ad measure results agaist a Russell bechmark. 4 Figure 7 shows the retur, trackig error ad Sharpe ratio results ad the lie of the iverse of the square root of for log-oly portfolios. For log-oly portfolios, diversificatio fails after the first few stocks. I particular, the trackig error graph, which shows covergece to a higher level of odiversifiable risk tha show by the lie rather tha simply a slower covergece to a commo risk level, implies some commo risk positio i all stock positios relative to the bechmark that diversificatio i the active portfolio is idetifyig rather tha elimiatig. As we shall show, it is this commo risk positio ad ot stock selectio that has made it so difficult eve for skilled maagers to cosistetly outperform bechmarks. 4 As described i Appedix B, the data sample we use is a approximatio of the stocks i the Russell idex. Because the differece i the cap-weighted mea retur of our sample ad the actual Russell retur could bias the results, the excess returs we report are actually the excess above the cap-weighted mea of our Russell sample rather tha above the idex retur. Uless otherwise specified, i this report, Russell ad Russell refer to our estimated samples. Goldma Sachs Ivestmet Research 5

18 Goldma Sachs Global Portfolio Aalysis Solvig the Risk Maagemet Problem Large-Cap First The Russell Uiverse So what is the commo risk factor ad what ca portfolio maagers do to elimiate it? Oce the commo risk factor is elimiated, what other risk cotrol is eeded/desired? The commo risk factor turs out to be stocks with large idex weights. The simplest ad most effective risk correctio is to hold a passive positio 5 i those stocks (or a equivalet derivative) to offset that cocetratio of stockspecific risk. Beyod that, as was implied by the log-short results, little risk maagemet will tur out to be ecessary, although as we will discuss later, some additioal risk cotrols ca help portfolio maagers, but those cotrols eed to be carefully tailored to the specific portfolio maager ad deped importatly o that portfolio maager s specific skills, weakesses ad research methodologies. These coclusios might seem surprisigly simple give the broad failure of risk models to oticeably improve portfolio maager performace over the last decade, but as we show, macro approaches do ot correctly address the problems of a bottoms-up fudametally drive portfolio maager ad, oce the perspective is shifted to the idividual stock level, the problems become much simpler. Put differetly, we fid that the types of macro risk positios that arise aturally out of bottomsup aalysis are, i fact, justified o a risk-retur basis ad do ot eed to be cotrolled. The risks that prove to be both importat i size ad ujustified are those that arise out of mismatches betwee the portfolio maager s atural base portfolio ad the bechmark. Such mismatches geerate persistet risk positios that do ot reflect the portfolio maager s judgemet about ivestmet opportuities, ad, hece, are rarely justified o a 5 Because our portfolios are rebalaced every caledar quarter, the positios i the largest stocks we describe as passive are ot etirely passive. Chage is due to turover i the set of the largest stocks, which, for our purposes, is more of a issue i the Russell tha it is i the Russell or the S&P 5. Global risk-retur basis. Macro risk systems idiscrimiately work at reducig both types of risk ad are, i geeral, more effective at elimiatig the good risk drive by a portfolio maager s stock selectio tha they are at elimiatig the habitual risk patters that do ot offer reasoable expectatios of retur. To show this formally ad uderstad the key drivers of these coclusios ad the real world solutios to the risk maagemet problem, we eed to aalyze the match betwee fudametally drive stock selectio ad various risk cotrol approaches. We first look at these questios from the stadpoit of a orthodox maager followig a pure ivestmet style with complete research coverage across all sectors ad size groups i their ivestmet uiverse ad o idiosycratic biases i accuracy across categories or types of stocks. I the fial sectio, we re-examie the questio from the perspective of a more idiosycratic portfolio maager with research stregths ad weakesses, ivestmet prejudices, o-stadard valuatio methods ad correlated patters of forecast accuracy. I the curret cotext, we idetify three obvious potetial cocetratios of commo risk size, sector ad idividual stock positios. Some readers may woder at the otio that bechmarks ca cotai large cocetratios of idividual stock risk. I fact, oe of the key uderlyig assumptios i the way most portfolio maagers ad most macro risk models approach idices is that stock-specific risk i idices has bee diversified away. I the case of large-cap idices, this assumptio is patetly false. (It is more reasoable for mid- ad small-cap idices as we show later.) Equal-weighted idices quickly diversify away stock-specific risk followig the iverse of the square root of N ( N ) rule discussed earlier. For cap-weighted idices, the questio of diversificatio is much more subtle. Appedix A develops the mathematics i some detail, but the key poit is quite simple if the weight of the stock i a idex exceeds /(N+) where N is the umber 6 Goldma Sachs Ivestmet Research

19 Global Goldma Sachs Global Portfolio Aalysis Table 3: Effect of Macro Risk Cotrol Methods (Estimated Russell Sample, Q987-Q998) Pael : Moderate Skill (3% Edge) Value Growth Mea Trackig Sharpe Percet Mea Trackig Sharpe Percet Returs (%) Error (%) Ratio Positive (%) Returs (%) Error (%) Ratio Positive (%) Log-Oly Uadjusted Cotrol for Size Cotrol for Sector Cotrol for Sector ad Size Log-Short Pael : High Skill (5% Edge) Value Growth Mea Trackig Sharpe Percet Mea Trackig Sharpe Percet Returs (%) Error (%) Ratio Positive (%) Returs (%) Error (%) Ratio Positive (%) Log-Oly Uadjusted Cotrol for Size Cotrol for Sector Cotrol for Sector ad Size Log-Short of stocks i the idex, the that stock adds more stock-specific risk tha it diversifies away. 6 Lookig at the S&P 5 7 like this would suggest that somewhere betwee the 5 ad 75 largest stocks are addig sigificat stock-specific risk. Such cocetratio of stock-specific risk ca act as a commo risk positio agaist all of the portfolio maager s idividual stock positios. I Table 3, we show the relative effectiveess of differet macro risk cotrol approaches for our value ad growth maagers. For portfolio maagers with moderate ad high skill levels, we simulate, portfolios based o simulated rakigs for differet risk maagemet approaches focused o cotrollig size, sector ad cocetratios of idividual stock risk. We the report the aualized averages for the returs, trackig errors ad Sharpe ratios for each approach. 6 Formally, Appedix A defies a effective N that takes accout of the rage of idex weights applied across the capitalizatio spectrum. 7 Appedix E provides some statistics o the cocetratio of market capitalizatio i the largest stocks i the S&P 5. The macro risk cotrols are imposed by stratified risk samplig methods ofte used i the costructio of pollig data. This meas that stock pickig is oly allowed withi groups of cotrolled categories. Thus, for the size risk cotrol results, the stocks uiverse is divided ito decile rages (smallest %, ext larger %,..., largest %) ad the best stocks i each decile are chose accordig to the fudametal rakig criterio. The stocks chose i each size decile are equally weighted. The, each decile portfolio is give a portfolio weight equal to that decile s share of the idex. For example, if we have a sample of, stocks ad we wat to costruct a 5-stock log-oly portfolio, we start by dividig the, stock uiverse ito size deciles of stocks each. The, we pick the best 5 stocks from each decile. Withi each decile, the 5 stocks are equal-weighted. The, the decile portfolios are combied by weightig the portfolios by the share of the idex market capitalizatio i that decile. As of July 3, 999, that meat the largest decile i a estimated Russell sample was weighted at roughly 6%. Similarly, for sector cotrols, we break the data ito the Compustat ecoomic sectors, the best stocks Goldma Sachs Ivestmet Research 7

20 Goldma Sachs Global Portfolio Aalysis are chose withi each sector, ad the sector is the weighted by the capitalizatio of the sector i the idex. For the joit size ad sector cotrols, the stocks are broke up ito size/sector groupigs, the best stocks withi each size/sector group are chose, ad the the portfolio is assembled from these subportfolios by cap-weightig. Usig this type of risk cotrol allows us to look at how well stock selectio is workig, both withi the categories ad how well it is workig at geeratig cross-category risk positios. That is, we ca examie whether it is better to pick stocks withi sectors or it is better to allow sector overweights that arise aturally out of bottoms-up aalysis. A first pass at iterpretig Table 3 would suggest some moderate gai from macro risk cotrols, Global especially cotrollig for size. Trackig error is reduced dramatically, but the Sharpe ratio oly improves modestly as returs also fall dramatically. Give the large drop off i returs that arise from reduced risk takig ad the modest improvemet i the quality of risk, it is little woder that portfolio maagers view risk cotrol with more tha modest suspicio that it is doig more harm tha good over the log haul. The size bets would be expected to average out over time, but reduced risk takig would still impact the portfolio maager s cumulative returs exactly i proportio to the quarterly reductios i returs. Sector cotrols appear to have little value i risk cotrol as they have little impact o returs or Sharpe ratios. 8 Goldma Sachs Ivestmet Research

21 Global Dealig with Stock-Specific Bechmark Risk However, because these results igore the potetial impact of cocetratios of stockspecific risk, they are actually highly misleadig. As macro size risk ad stock-specific risk are focused i the same large-cap stocks, it is easy to mistake oe for the other. However, the operatioal methods of offsettig the two risks are completely differet ad the resultig impact o quality of risk is equally differet. Cotrollig cocetratios of stock-specific risk is quite simple. The portfolio maager ca simply market-weight the largest stocks. The dowside of this approach is that every dollar used to offset these cocetratios is o loger available for geeratig outperformace through active maagemet, so the Goldma Sachs Global Portfolio Aalysis loss i log-ru outperformace is equal to the percetage of fuds used to offset stock-specific risk. (I Appedix F, we examie strategies aimed at reducig the ecessary fuds.) The impact of such a risk cotrol approach o Sharpe ratios is dramatic, especially i compariso with the modest impact of the size-based risk cotrols. I Table 4, we show the results of addig a market-weightig of the largest stocks (from to ) to a otherwise u-risk-cotrolled log-oly portfolio. For compariso, we also iclude the uadjusted log-short returs, which ca be thought of as a measure of the total ucostraied portfolio maager s potetial for extractig value from their ability to rak stocks. After offsettig the stock-specific risk of the top 5 stocks, the log-oly portfolio maager has doubled their Sharpe ratio ad recaptured approximately Table 4: Effect of Offsettig Stock-Specific Risk (Estimated Russell Sample, Q987-Q998) Pael : Moderate Skill (3% Edge) Value Growth Number of Largest Mea Trackig Sharpe Percet Mea Trackig Sharpe Percet Stocks Idex-Weighted Returs (%) Error (%) Ratio Positive (%) Returs (%) Error (%) Ratio Positive (%) Log-Oly Portolios Log-Short Portfolios Pael : High Skill (5% Edge) Value Growth Number of Largest Mea Trackig Sharpe Percet Mea Trackig Sharpe Percet Stocks Idex-Weighted Returs (%) Error (%) Ratio Positive (%) Returs (%) Error (%) Ratio Positive (%) Log-Oly Portolios Log-Short Portfolios Goldma Sachs Ivestmet Research 9