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Diversification Returns and Asset Contributions

David G. Booth andEugene F. FamaFor a portfolio with a con-stant percentage invested ineach asset, the compoundreturn is the sum of thecontributions of the indi-vidual assets in theportfo-lio. Theportfolio com-pound return is greaterthan the weighted averageof the compound returnson the assets in the portfo-lio. The incremental returnis due to diversification.Thecontribution of eachasset exceeds its compoundreturn by the amount itadds to the portfolio diver-sification return.Thecompound return onan asset is approximatelythe asset'saverage returnminus one-half the asset'svariance. Aportfolio's aver-age return is the weightedaverage of each asset's av-erage return, but a portfo-lio's variance is theweighted average of eachasset'scovariance. Itfol-lows that the return contri-bution of an asset can bebetterapproximated by sub-tracting one-half the covari-ance than one-half the vari-ance.

The compound return on a port-folio is an important measure ofperformance because it providesthe connection between begin-

ning and ending portfolio values.Knowing the beginning portfoliovalue, the compound return andthe number of investment peri-ods, we can easily compute theending portfolio value.The compound return on an assetis, however, a misleading mea-sure of the contribution of theasset to the compound return ona portfolio. We will show that,with a constant percentage in-vested in each asset, the portfoliocompound return is greater thanthe weighted average of the com-pound returns on the assets in theportfolio. This means that thecontribution of each asset to theportfolio compound return isgreater than the asset's com-pound return. The difference isan incremental return due to di-versification.This incremental re-turn is the subject of this article.Table I presents a simple exam-ple using returnson the S&P500,Treasury bonds and a 50-50 port-folio of the two assets for the 50years 1941-90. The annualizedcompound returnfor the S&P500is 11.31%,and the annualized re-turn for Treasurybonds is 4.36%.The average of the two com-pound returns is 7.84%. But aportfolio that maintains a 50%weight in the S&P 500 and a 50%weight in Treasury bonds has ahigher compound return-8.11%. Diversification adds 28 ba-sis points a year to the portfolioreturn.Table I also shows the "returncontribution" of each asset. Thereturn contribution-the centralconcept of this article-is our es-timate of the contribution of eachasset to a portfolio's compoundreturn. For the 50-50 portfolio,the return contribution of theS&P500 is 11.79%,and the returncontribution of Treasurybonds is

4.44%. The return contributionsare higher than the compoundreturns (11.31% and 4.36%) be-cause of the gains from diversifi-cation. Portfolio diversificationadds 48 basis points to the S&P500 return and 8 basis points tothe Treasury bond return. Thebenefit of diversification for theportfolio (28 basis points) is theaverage of the benefits for theindividual assets.The increment of the return con-tributionof an asset over its com-pound return depends on howmuch the asset's risk is reducedthrough diversification. The in-crement is higher for the S&P500because more of its risk is diver-sified away. Conversely, from aportfolio perspective, judging as-sets in terms of their compoundreturns penalizes most those as-sets that benefit most from diver-sification.The Compound ReturnThe appendix shows that thecompound return (continuouslycompounded) on asset j is wellapproximated as:Eq. 1C,= ln[1 + E(R1)]

Sj22[1 + E(Rj)f

whereE(R.) = the expected(average) re-turn on asset j,

S2 = the variance(standard devi-ation squared)of the simplereturns on jand

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ln[l + E(Rf)] = the averagereturn on j, ex-pressed as acontinuouslycompoundedreturn.

The compound return on portfo-lio p can be expressed in terms ofthe mean and variance of itssimple returns:Eq. 2CP= ln[l + E(Rp)]

5p22[1 + E(Rp)]

Equations (1) and (2) say thatcompound returns are lower, thehigher the variance of returns.For a portfolio, variance is anappropriate measure of risk. Butthe riskof an asset in a diversifiedportfolio is less than the varianceof the asset's returns. We showbelow that the risk reduction dueto diversification is what causesan asset's compound return tounderstatethe contribution of theasset to the compound return onthe portfolio.

The Risk of an Asset in aPortfolioAsset j's beta relative to the port-folio is calculated by regressingits returns against the portfolio'sreturns. It can be expressed as:

Eq. 3= cov(Rj, Rp)SP 2Sp

Here cov(R.,Rp) s the covarianceof asset j's returns with the port-folio's returns and sp2 is the vari-ance of the portfolio's returns.RearrangingEquation (3), we canexpress the risk of asset j in termsof its contribution to portfolio p'srisk, as measured by its variance:Eq. 4

cov (Rj,Rp)= bjpspThe covariance measure of eachasset in the portfolio [cov(R,,Rp)],weighted by the asset's represen-tationin the portfolio, will sum tothe portfolio's variance,sP2.

The Return ContributionOne estimate of the contributionof asset j to the compound returnon portfolio p is:Eq. 5Di = In[1 + E(R1)]

2[1 + E(Rj)fComparingEquations (1) and (5)shows that the compound returnon asset j and its contribution tothe compound return on portfo-lio p have the same first term-ln[l + E(Rf)].The difference be-

Glossary*.Return Contribution:The contribution of an asset toa portfolio's compound re-turn.* Variance:The second moment of a dis-tribution around its mean; theaverage of the squared devia-tions from the mean.*'Asset-Class Investing:Investing to capture the re-turns of an asset class withoutregard to stock selection. Anindex fund is one example.*.Diversification Returns:For a portfolio, the differencebetween its compound returnand the weighted average ofeach asset's compound return.For an asset in a portfolio, thedifference between the returncontribution and the com-pound return.tween the compound return andthe return contribution is thusdue to the difference betweenb,pSp2and s.2. In a nutshell, thereturn contribution of asset j isgreater than its compound returnbecause the contributionof asset jto the variance of the return onthe portfolio, b. sp2, is less thanthe variance of thereturn on j, s;.This risk reduction, a result ofdiversification,enhances the con-tribution of asset j to the com-pound return on the portfolio.The weighted average of the re-turn contributions estimatedfrom Equation (5) is not exactlythe portfolio compound return,because the weighted average ofln[1 + E(P)] is not exactly ln[l +E(Rp)J.We can arrange for theweighted average of the returncontributions to equal the portfo-lio compound return given byEquation (2) if we change Equa-tion (5) to:Eq. 6

=E(RJ)n[1 + E(Rp)]E(Rp)

Table I Compound Returns, 1941-199020-YearS&P500 TreasuryIndex Bonds Portfolio

Portfolio Weights (%) 50.00 50.00 100.00AverageMonthly Returns (%) 1.03 0.39 0.71Standard Deviations (%) 4.14 2.39 2.62

Annualized Continuously Compounded Returns (%)Compound Return 8.11Compound Return of Each Asset 11.31 4.36 7.84and the AverageAsset Compound ReturnReturnDue to Diversification 0.48 0.08 0.28ReturnContribution 11.79 4.44 8.11

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b)jpsp2[1 + E(Rp)f

The difference between estimat-ing the return contribution usingEquation (5) and using Equation(6) is minuscule for most periods.We will use Equation (6) becauseof its additive property. Theweighted average of the first termon the right-hand side is exactlyln[l + E(Rp)Iand the weightedaverage of the second term issp2/2[1+ E(Rp)]2.Thus:Eq. 7Dp= ln[l + E(Rp)]

S2p -C2[1 + E(Rp)]2 PExamplesWe estimate below the returncontributions of seven assetclasses for different time periodsand portfolios.' We estimated be-

tas from quarterly data and aver-age returns and standard devia-tions from monthly data. Thecontinuously compounded re-turns and return contributionsare annualized from the monthlydata.

ResultsTable II and Table III show re-sults for domestic balanced port-folios for 1941-90. Table II usesdeciles 6 through 10 (D6-10) forsmall-cap stocks and Table IIIuses deciles 9 through 10 (D9-10). The results for the S&P 500,Treasurybonds and Treasurybillsare almost identical in the twotables. The annualized returncontribution for the S&P 500 inboth is 11.72%, an increase of0.41% over the index's com-pound return. The incrementalreturn for Treasury bonds is0.16%.The incremental returns due todiversification are greater forsmall-cap stocks than for the

other assets. The incremental re-turns (the difference between thecontribution of small stocks to thecompound return on the portfo-lio and the compound return onsmall stocks) are 0.99% for D6-10stocks and 1.36% for D9-10stocks. Measuring the contribu-tions of these assets to portfolioreturns, rather than the com-pound returns on the assets, sub-stantially increases the premiumof small-cap stocks over the S&P500. The difference between thereturn contributions of D9-10,15.42%,and the S&P500, 11.72%,is 3.70%. The difference betweenthe return contributions of D6-10, 13.96%, and the S&P 500 is2.24%. The corresponding differ-ences between the compound re-turns are only 2.75%and 1.65%.Why are the contributions ofsmall stocks to the compoundreturns on portfolios so muchlarger than the compound re-turns on small stocks? Smallstocks have high returnvariances,

Table II Portfolio with Small-CapStocks, 1941-1990US. 20-Year One-MonthSmall-Cap Treasury TreasuryS&P500 D6-10 Bonds Bills Portfolio

Portfolio Weights (%) 50.00 10.00 20.00 20.00 100.00Average Monthly Returns (%) 1.03 1.24 0.39 0.36 0.79Standard Deviations (%) 4.14 5.51 2.39 0.28 2.70

Annualized Continuously Compounded Returns (%)Compound Return 9.02Compound Return of Each Asset 11.31 12.96 4.36 4.30 8.68

and the Average Asset Compound ReturnReturn Due to Diversification 0.41 0.99 0.16 -0.00 0.33Return Contribution 11.72 13.96 4.52 4.29 9.02

Table III Portfolio with Very Small-CapStocks, 1941-1990U.S.Small-Cap 20-Year One-MonthS&P500 D9-10 Treasury Bonds TreasuryBills Portfolio

Portfolio Weights (%) 50.00 10.00 20.00 20.00 100.00AverageMonthly Returns (%) 1.03 1.37 0.39 0.36 0.80StandardDeviations (%) 4.14 6.20 2.39 0.28 2.72Annualized Continuously Compounded Returns (%)Compound Return 9.17Compound Returnof EachAsset 11.31 14.06 4.36 4.30 8.79and the AverageAsset Compound ReturnReturn Due to Diversification 0.41 1.36 0.16 -0.00 0.37ReturnContribution 11.72 15.42 4.52 4.29 9.17

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which lower their compound re-turns (see Equation (1)). Becausesmall stocks are not highly corre-lated with other assets, however,their riskin a diversified portfolio(the covariance of their returnswith portfolio returns) is muchless than their return variance.This diversification benefit sub-stantially increases the contribu-tion of small stocks to the com-pound returns on portfolios. (Thestory is similar for internationalstocks, large and small.)Tables II, IV and V show resultsfor the same domestic portfolioover different time periods. Theresults for the last 20 years (TableV) are similar to the results forthe last 50 years (Table II). Theresults for the Great Depression(Table IV) are quite different.During the Depression, it was im-portant to be diversified. The re-turn contribution for small-capstocks in this period is greaterthan the return contribution forthe S&P 500, but the compoundreturn for small-cap stocks is

lower than the compound returnfor the S&P 500.Tables VI and VII show results fortwo internationally diversifiedportfolios. With two additionalequity asset classes, the negativebias of the asset compound re-turn is more apparent. Table VIshows that the compound returnunderstatesthe annualized returncontributions for stocks by thefollowing percentages:S&P500 0.56U.S. Small-CapStocks 1.31(D6-10)International Large-Cap 1.30StocksInternational Small-Cap 1.23StocksBecause the S&P 500 benefits theleast from diversification amongthe four asset classes and it has alarge weight in the portfolio, itsreturn increment due to diversifi-cation is the lowest of the fourasset classes. The U.S. small-cappremium over the S&P is thus

understated by 0.75%(1.31-0.56).The international large-cap pre-mium over the S&P is under-stated by 0.74%.Table VII shows the results for aportfolio that equally weights allfour equity asset classes. Itsmonthly standard deviation isonly slightly greater than that ofthe portfolio in Table VI-2.99%versus 2.94%. Even though theS&P 500 standard deviation ismuch lower than the standarddeviations of the other equity as-set classes, the portfolio standarddeviation does not increase sig-nificantly because the S&P 500commitment is reduced. The im-provement in diversificationabout offsets the increase in aver-age standard deviation.Finally, the difference betweenthe contribution of Treasurybonds to the compound returnson portfolios and the compoundreturnon the bonds has widenedin the last 20 years (see Tables I toVII). The reason is that the vari-

Table IV Domestic Portfolio, 1926-1940US. Small-Cap 20-Year One-MonthS&P500 D6-10 TreasuryBonds TreasuryBills Portfolio

Portfolio Weights (%) 50.00 10.00 20.00 20.00 100.00Average Monthly Returns (%) 0.78 0.99 0.42 0.11 0.59Standard Deviations (%) 9.56 13.62 1.45 0.14 6.09

Annualized Continuously Compounded Returns (%)Compound Return 4.94Compound Return of Each Asset 3.96 1.73 4.87 1.31 3.39and the Average Asset Compound ReturnReturn Due to Diversification 2.09 4.79 0.10 0.03 1.55Return Contribution 6.05 6.53 4.96 1.34 4.94

Table V Domestic Portfolio, 1971-1990U.S. Small-Cap 20-Year One-Month

S&P 500 D6-10 Treasury Bonds Treasury Bills PortfolioPortfolio Weights (%) 50.00 10.00 20.00 20.00 100.00Average Monthly Returns (%) 0.99 1.11 0.75 0.62 0.88Standard Deviations (%) 4.65 6.10 3.30 0.22 3.13

Annualized Continuously Compounded Returns (%)Compound Return 9.95Compound Return of Each Asset 10.57 10.97 8.36 7.39 9.53and the Average Asset Compound ReturnReturn Due to Diversification 0.48 1.19 0.28 -0.00 0.42Return Contribution 11.06 12.16 8.64 7.39 9.95

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ance of long-bond returns hasincreased. In terms of variability,long bonds are now more likestocks. Thus it is now more im-portant that long bonds be heldin diversified portfolios that in-clude stocks.Diversification Returnsvs. InvestmentUncertaintyThus far we have shown that di-versification improves portfoliocompound returns. We will nowshow that the incremental returnsmay be lost by engaging in activemanagement.In order to measure diversifica-tion returns, we engaged in as-set-class investing, investing inasset classes in fixed proportions.This runs counter to active man-agement. The essence of activemanagement is to tryto "beat themarket"by shifting asset weights.Active management can either in-crease or decrease the portfoliocompound return, depending on

both skill and luck.As we will see,the luck component can be largerthan the skill component.To see how shifting weights intro-duces uncertainty, suppose werandomly invest in the S&P 500and Treasurybonds on a monthlybasis. One-half of the months weinvest 65% in the S&P 500 and35% in Treasurybonds. The otherhalf of the months we invest 35%in the S&P 500 and 65% n Trea-sury bonds.The returns from this strategywilldepend on the luck of the draw.To show how uncertain the out-comes can be, we simulated theapproach 1000 times for the fiveyears 1986-90. It is as though wehad 1000 managers engaged in amarket timing contest. The sum-mary of the results are displayedin Table VIII.We calculated the five-yearcom-pound return for each of the sim-ulations, then ranked from high-est to lowest. As TableVIIIshows,10% of the annualized returns

were 14.33% or higher, while10% were 10.52% or lower.The average asset mix is 50%S&P500 and 50% Treasury bonds.However, the average standarddeviation of returns is somewhathigher for the timing portfoliosthan it is for the 50/50 portfolio,because of the shifting of assetweights. The average standardde-viation for the random timingportfolio is equal to that of aconstant-mix portfolio invested53% in the S&P 500 and 47% inTreasury bonds. This 53/47 port-folio serves as a suitable bench-mark for comparison with therandom timing portfolios. Thebenchmark outperforms the ran-domly generated portfolios. Its12.53% compound return is 14basis points higher than the aver-age compound returnfor the ran-domly generated portfolios, andhas a 52-basis-pointannual diver-sification return.Thus shifting weights lowers ex-pected compound return. But,perhaps more importantly, shift-

TableVI 10% International Portfolio, 1971-1990US. Small Intl. Intl. 20-Year One-Month

S&P 500 D6-10 Large Small T-Bonds T-Bills PortfolioPortfolio Weights (%) 45.00 5.00 5.00 5.00 20.00 20.00 100.00Average Monthly Returns (%) 0.99 1.11 1.63 2.02 0.75 0.62 0.96StandardDeviations (%) 4.65 6.10 5.86 5.50 3.30 0.22 2.94

Annualized Continuously Compounded Returns (%)Compound Return 10.95Compound Return of Each Asset 10.57 10.97 17.46 22.21 8.36 7.39 10.44and the Average Asset Compound ReturnReturn Due to Diversification 0.56 1.31 1.30 1.23 0.31 -0.00 0.50Return Contribution 11.13 12.28 18.76 23.45 8.67 7.39 10.95

Table VII 30% International Portfolio, 1971-1990US. Small Intl. Intl. 20-Year One-Month

S&P 500 D6-10 Large Small T-Bonds T-Bills PortfolioPortfolio Weights (%) 15.00 15.00 15.00 15.00 20.00 20.00 100.00Average Monthly Returns (%) 0.99 1.11 1.63 2.02 0.75 0.62 1.14Standard Deviations (%) 4.64 6.10 5.86 5.50 3.29 0.22 2.99

Annualized Continuously Compounded Returns (%)Compound Return 13.04Compound Return of Each Asset 10.57 10.97 17.46 22.21 8.36 7.39 12.33and the Average Asset Compound ReturnReturn Due to Diversification 0.66 1.35 1.18 1.08 0.36 -0.01 0.71Return Contribution 11.23 12.32 18.64 23.30 8.72 7.39 13.04

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ing weights also introduces un-certainty. The median return isthe most likely return, but achiev-ing the median return is highlyuncertain. About 10%of the ran-dom portfolios have returns 200basis points a year higher than thebenchmark return. About 10%ofthe portfolios have returns 200basis points a year lower than thebenchmark return. Most investorswould take 12.53% for certainrather than invest in a strategythat has uncertain returns and anequal chance of a 14.33% returnor a 10.52% return.The uncertainty of achievingbenchmark returns extends toreal portfolios. Table IX displaysthe distribution of total plan re-turns for institutional funds in theSEICorporationdatabase. The in-stitutional portfolios behave likethe randomly generated portfo-lios. About 10%of the funds havereturns 200 basis points a yearabove the median. About 10% ofthe funds have returns 200 basispoints a year below the median.The 400-basis-point spread of re-turns between the 10th and 90th

percentiles measures the uncer-tainty introduced by active man-agement. The implication is clear:Investors should prefer the cer-tain return from a portfolio thatmaintains fixed asset weights to astrategy whose outcome is uncer-tain and has an equal chance ofunderperforming or outperform-ing the benchmark by 200 basispoints per year. Stateddifferently,investors need a large premiumto be willing to incur the addi-tional uncertainty of active man-agement.Recentstudies indicate that inves-tors cannot expect such a premi-um.2 The premium return fromactive management is negativerather than positive. Every year,about half the funds outperformtheir benchmarks. Those in theupper half in one year have onlyabout a 50% chance of repeatingin any other year.Activemanagement introduces somuch uncertainty that we cannotdocument a premium return overbenchmark returns. By contrast,fully diversified portfolios reli-ably increase portfolio com-pound returns through the diver-sification process and eliminate"benchmark risk."

ConclusionThe portfolio compound return isuseful for describing portfolio re-sults. But the compound returnofan asset in a portfolio understatesthe contribution of the asset tothe portfolio compound returnbecause it ignores the benefit ofdiversification.The "return contribution" de-fined by Equation (6) is an im-proved measure of the contribu-tion of an asset to the portfoliocompound return. The portfoliocompound return is the averageof the asset return contributions.The increments of return contri-bution over compound returnsare especially large for small-capstocks and international stocks,primarily because diversification

benefits these assets (reducestheir risks) the most. For the S&P500, the difference is smaller, butimportant. The increased vari-ance of long-bond returns overthe last 20 years has widened thespread between their returncon-tributions and their compoundreturns.By shifting asset weights, activemanagement introduces uncer-tainty about portfolio compoundreturns. Diversification re-turns are assured only for thoseportfolios that maintain relativelyfixed asset weights.

Append'xThe continuously compoundedreturn on an asset is In[1 + ,where R. s the simple return. Theexpected value of the compoundreturn of an asset, E[ln(l + R.)],can be derived from the Taylorseries expansion of the naturallogarithm about the mean returnof the asset:Eq. AlCi= In[1 + E(R1)]

M2 M3+2[1 + E(R)]2 3[ + E(RjM4

_ 4 +4[1 + E(Rj)]where Mk = the kth moment ofthe asset return about its mean,or:Eq. A2Mk= E[(R- E(R1))k].Equation (1) is justEquation (A1)with the higher momentsdropped. For the assets and port-folios in Tables I through VII,these higher-order terms are al-ways minuscule, except duringthe Great Depression. Table AIdisplays the accuracy of Equation(1) for the portfolios displayed inTables I through VII.

TableVIII Random TimingStrategy, AnnualizedCompound Returns(%),1986-199010th Percentile 14.3350th Percentile 12.3990th Percentile 10.52lOth-90th 3.81Benchmark Portfolio 12.53S&P 500 13.14Treasury Bills 10.74

Table IX Total Plan SponsorAnnualized Com-pound Returns (%),1986-1990*10th Percentile 12.3850th Percentile 10.4790th Percentile 8.35lOth-90th 4.03Source: SEICorporation.

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TableAI Annualized Portfolio Compound Returns Obtained by Linking Monthly Returns Comparedwith Compound Returns Derived from Equation (1)Actual Derived Difference

Portfolio Compound Compound (Formulaby Time Return Return Error)

Table Period % % %I 1941-90 8.11 8.11 0.00II 1941-90 9.02 9.02 -0.00III 1941-90 9.17 9.17 -0.00LV 1926-40 4.97 4.94 0.03V 1971-90 9.95 9.95 -0.00VI 1971-90 10.95 10.95 -0.00VII 1971-90 13.04 13.04 -0.00

Footnotes1. S&P500 and long-term Treasurybond data from R. G.Ibbotson andR. A. Sinquefield, Stocks, Bonds, Billsand Inflation (Chicago:Ibbotson As-sociates). D9-10 is the DimensionalFund AdvisorsInc. (DFA)US. 9-10Small Company Portfolio, net of allfees, from 1982 on and the CRSP9-10 index (courtesy of the Centerfor Researchin SecuritiesPrices, Uni-versity of Chicago)for 1926-81. TheD6-10 is the DFA US. 6-10 SmallCompany Portfolio, net of adminis-trative ees, from June 1986 on andthe CRSP6-10 index for 1926through May 1986. The interna-tional small-cap portfolio uses thefollowing data. For the UK: theHoare GovettSmaller Co. Index(courtesyLondon Business School)for 1956 throughMarch 1986 andthe DFA United Kingdom Small Com-pany Portfolio, net of allfees, forApril 1986 on. ForJapan: thesmaller half of thefirst section of theTokyoStockExchange (courtesy ofNomura SecuritiesInvestment TrustManagement Co., Ltd.,Tokyo) or1970 throughMarch 1986 and theDFAJapanese Small Company Port-folio, net of allfees, for April 1986on. The international large-capport-folio uses thefollowing data. For theUK: the Financial TimesAll SharesIndex for 1956 on. For Japan: Thelarger half of thefirst section of theTokyo StockExchange (courtesyNo-mura Securities) or 1970 throughJune 1986 and theJapan NationalIndex (courtesy of Morgan StanleyCapital International) for July 1986on. Data on large-cap continentalstocks (excluding the UK) and onAsia and Australia (excluding Japan)from Morgan Stanley. Theinterna-tional large and small-cap portfolioswere weighted as follows. For 1970-June 1988& 0% Japan, 50% UK

ForJuly 1988-September 1989:50%Japan, 30% Continent, 20% UK ForOctober 1989-March 1990: 40%Japan, 30% Continent, 20% UK,10% Asia-Australia.For April 1990on: 40% Japan, 35% Continent,15% UK, 10% Asia-Australia.Re-turnsfor various portfolios over vari-ous years are available from the au-thors.2. See G.P. Brinson, L. R. Hood andG.L.Beebower, "DeterminantsofPortfolio Performance, FinancialAnalysts Journal, July/August 1986,and G. P. Brinson, B. D. Singer andG.L.Beebower, "Determinants ofPortfolio Performance II:An Up-date," Financial Analysts Journal,May/June 1991.

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