bkmsolch249e corrected 8910.docx
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bkmsolch249ecorrected8910
CHAPTER24:
PORTFOLIOPERFORMANCEEVALUATION
PROBLEMSETS
1.Asestablishedinthefollowingresultfromthetext,theSharperatiodependsonbothalphafortheportfolio（
）andthecorrelationbetweentheportfolioandthemarketindex（ρ）:
Specifically,thisresultdemonstratesthatalowercorrelationwiththemarketindexreducestheSharperatio.Hence,ifalphaisnotsufficientlylarge,theportfolioisinferiortotheindex.Anotherwaytothinkaboutthisconclusionistonotethat,evenforaportfoliowithapositivealpha,ifitsdiversifiableriskissufficientlylarge,therebyreducingthecorrelationwiththemarketindex,thiscanresultinalowerSharperatio.
2.TheIRR（i.e.,thedollar-weightedreturn）cannotberankedrelativetoeitherthegeometricaveragereturn（i.e.,thetime-weightedreturn）orthearithmeticaveragereturn.Undersomeconditions,theIRRisgreaterthaneachoftheothertwoaverages,andsimilarly,underotherconditions,theIRRcanalsobelessthaneachoftheotheraverages.Anumberofscenarioscanbedevelopedtoillustratethisconclusion.Forexample,considerascenariowheretherateofreturneachperiodconsistentlyincreasesoverseveraltimeperiods.Iftheamountinvestedalsoincreaseseachperiod,andthenalloftheproceedsarewithdrawnattheendofseveralperiods,theIRRisgreaterthaneitherthegeometricorthearithmeticaveragebecausemoremoneyisinvestedatthehigherratesthanatthelowerrates.Ontheotherhand,ifwithdrawalsgraduallyreducetheamountinvestedastherateofreturnincreases,thentheIRRislessthaneachoftheotheraverages.（Similarscenariosareillustratedwithnumericalexamplesinthetext,wheretheIRRisshowntobelessthanthegeometricaverage,andinConceptCheck1,wheretheIRRisgreaterthanthegeometricaverage.）
3.Itisnotnecessarilywisetoshiftresourcestotimingattheexpenseofsecurityselection.Thereisalsotremendouspotentialvalueinsecurityanalysis.Thedecisionastowhethertoshiftresourceshastobemadeonthebasisofthemacro,comparedtothemicro,forecastingabilityoftheportfoliomanagementteam.
4.a.Arithmeticaverage:
;
b.Dispersion:
σABC=7.07%;σXYZ=13.91%
StockXYZhasgreaterdispersion.
（Note:
Weused5degreesoffreedomincalculatingstandarddeviations.）
c.Geometricaverage:
rABC=（1.20×1.12×1.14×1.03×1.01）1/5–1=0.0977=9.77%
rXYZ=（1.30×1.12×1.18×1.00×0.90）1/5–1=0.0911=9.11%
Despitethefactthatthetwostockshavethesamearithmeticaverage,thegeometricaverageforXYZislessthanthegeometricaverageforABC.ThereasonforthisresultisthefactthatthegreatervarianceofXYZdrivesthegeometricaveragefurtherbelowthearithmeticaverage.
d.Intermsof“forwardlooking”statistics,thearithmeticaverageisthebetterestimateofexpectedrateofreturn.Therefore,ifthedatareflecttheprobabilitiesoffuturereturns,10%istheexpectedrateofreturnforbothstocks.
5.a.Time-weightedaveragereturnsarebasedonyear-by-yearratesofreturn:
Year
Return=（capitalgains+dividend）/price
2007−2008
[（$120–$100）+$4]/$100=24.00%
2008−2009
[（$90–$120）+$4]/$120=–21.67%
2009−2010
[（$100–$90）+$4]/$90=15.56%
Arithmeticmean:
（24%–21.67%+15.56%）/3=5.96%
Geometricmean:
（1.24×0.7833×1.1556）1/3–1=0.0392=3.92%
b.
Date
Cash
Flow
Explanation
1/1/07
–$300
Purchaseofthreesharesat$100each
1/1/08
–$228
Purchaseoftwosharesat$120lessdividendincomeonthreesharesheld
1/1/09
$110
Dividendsonfivesharesplussaleofoneshareat$90
1/1/10
$416
Dividendsonfoursharesplussaleoffoursharesat$100each
416
110
Date:
1/1/071/1/081/1/091/1/10
228
300
Dollar-weightedreturn=Internalrateofreturn=–0.1607%
6.
Time
Cashflow
Holdingperiodreturn
0
3×（–$90）=–$270
1
$100
（100–90）/90=11.11%
2
$100
0%
3
$100
0%
a.Time-weightedgeometricaveragerateofreturn=
（1.1111×1.0×1.0）1/3–1=0.0357=3.57%
b.Time-weightedarithmeticaveragerateofreturn=（11.11%+0+0）/3=3.70%
Thearithmeticaverageisalwaysgreaterthanorequaltothegeometricaverage;thegreaterthedispersion,thegreaterthedifference.
c.Dollar-weightedaveragerateofreturn=IRR=5.46%
[Usingafinancialcalculator,enter:
n=3,PV=–270,FV=0,PMT=100.Thencomputetheinterestrate,orusetheCF0=−300,CF1=100,F1=3,thencomputeIRR].TheIRRexceedstheotheraveragesbecausetheinvestmentfundwasthelargestwhenthehighestreturnoccurred.
7.a.Thealphasforthetwoportfoliosare:
αA=12%–[5%+0.7×（13%–5%）]=1.4%
αB=16%–[5%+1.4×（13%–5%）]=–0.2%
Ideally,youwouldwanttotakealongpositioninPortfolioAandashortpositioninPortfolioB.
b.Ifyouwillholdonlyoneofthetwoportfolios,thentheSharpemeasureistheappropriatecriterion:
UsingtheSharpecriterion,PortfolioAisthepreferredportfolio.
8.
a.
StockA
StockB
（i）
Alpha=regressionintercept
1.0%
2.0%
（ii）
Informationratio=
0.0971
0.1047
（iii）
*Sharpemeasure=
0.4907
0.3373
（iv）
**Treynormeasure=
8.833
10.500
*TocomputetheSharpemeasure,notethatforeachstock,（rP–rf）canbecomputedfromtheright-handsideoftheregressionequation,usingtheassumedparametersrM=14%andrf=6%.Thestandarddeviationofeachstock’sreturnsisgivenintheproblem.
**ThebetatousefortheTreynormeasureistheslopecoefficientoftheregressionequationpresentedintheproblem.
b.（i）Ifthisistheonlyriskyassetheldbytheinvestor,thenSharpe’smeasureistheappropriatemeasure.SincetheSharpemeasureishigherforStockA,thenAisthebestchoice.
（ii）Ifthestockismixedwiththemarketindexfund,thenthecontributiontotheoverallSharpemeasureisdeterminedbytheappraisalratio;therefore,StockBispreferred.
（iii）Ifthestockisoneofmanystocks,thenTreynor’smeasureistheappropriatemeasure,andStockBispreferred.
9.Weneedtodistinguishbetweenmarkettimingandsecurityselectionabilities.Theinterceptofthescatterdiagramisameasureofstockselectionability.Ifthemanagertendstohaveapositiveexcessreturnevenwhenthemarket’sperformanceismerely“neutral”（i.e.,haszeroexcessreturn）,thenweconcludethatthemanagerhasonaveragemadegoodstockpicks.Stockselectionmustbethesourceofthepositiveexcessreturns.
Timingabilityisindicatedbythecurvatureoftheplottedline.Linesthatbecomesteeperasyoumovetotherightalongthehorizontalaxisshowgoodtimingability.Thesteeperslopeshowsthatthemanagermaintainedhigherportfoliosensitivitytomarketswings（i.e.,ahigherbeta）inperiodswhenthemarketperformedwell.Thisabilitytochoosemoremarket-sensitivesecuritiesinanticipationofmarketupturnsistheessenceofgoodtiming.Incontrast,adecliningslopeasyoumovetotherightmeansthattheportfoliowasmoresensitivetothemarketwhenthemarketdidpoorlyandlesssensitivewhenthemarketdidwell.Thisindicatespoortiming.
Wecanthereforeclassifyperformanceforthefourmanagersasfollows:
SelectionAbility
TimingAbility
A.
Bad
Good
B.
Good
Good
C.
Good
Bad
D.
Bad
Bad
10.a.Bogey:
（0.60×2.5%）+（0.30×1.2%）+（0.10×0.5%）=1.91%
Actual:
（0.70×2.0%）+（0.20×1.0%）+（0.10×0.5%）=1.65%
Underperformance:
0.26%
b.SecuritySelection:
（1）
（2）
（3）=
（1）×
（2）
Market
Differentialreturn
withinmarket
（Manager–index）
Manager's
portfolioweight
Contributionto
performance
Equity
–0.5%
0.70
−0.35%
Bonds
–0.2%
0.20
–0.04%
Cash
0.0%
0.10
0.00%
Contributionofsecurityselection:
−0.39%
c.AssetAllocation:
（1）
（2）
（3）=
（1）×
（2）
Market
Excessweight
（Manager–benchmark）
Index
Return
Contributionto
performance
Equity
0.10%
2.5%
0.25%
Bonds
–0.10%
1.2%
–0.12%
Cash
0.00%
0.5%
0.00%
Contributionofassetallocation:
0.13%
Summary:
Securityselection–0.39%
Assetallocation0.13%
Excessperformance–0.26%
11.a.Manager:
（0.30×20%）+（0.10×15%）+（0.40×10%）+（0.20×5%）=12.50%
Bogey:
（0.15×12%）+（0.30×15%）+（0.45×14%）+（0.10×12%）=13.80%
Addedvalue:
–1.30%
b.Addedvaluefromcountryallocation:
（1）
（2）
（3）=
（1）×
（2）
Country
Excessweight
（Manager–benchmark）
IndexReturn
minusbogey
Contributionto
performance
U.K.
0.15
−1.8%
−0.27%
Japan
–0.20
1.2%
–0.24%
U.S.
−0.05
0.2%
−0.01%
Germany
0.10
−1.8%
−0.18%
Contributionofcountryallocation:
−0.70%
c.Addedvaluefromstockselection:
（1）
（2）
（3）=
（1）×
（2）
Country
Differentialreturn
withincountry
（Manager–Index）
Manager’s
countryweight
Contributionto
performance
U.K.
0.08
0.30%
2.4%
Japan
0.00
0.10%
0.0%
U.S.
−0.04
0.40%
−1.6%
Germany
−0.07
0.20%
−1.4%
Contributionofstockselection:
−0.6%
Summary:
Countryallocation–0.70%
Stockselection−0.60%
Excessperformance–1.30%
12.Support:
Amanagercouldbeabetterperformerinonetypeofcircumstancethaninanother.Forexample,amanagerwhodoesnotiming,butsimplymaintainsahighbeta,willdobetterinupmarketsandworseindownmarkets.Therefore,weshouldobserveperformanceoveranentirecycle.Also,totheextentthatobservingamanageroveranentirecycleincreasesthenumberofobservations,itwouldimprovethereliabilityofthemeasurement.
Contradict:
Ifweadequatelycontrolforexposuretothemarket（i.e.,adjustforbeta）,thenmarketperformanceshouldnotaffecttherelativeperformanceofindividualmanagers.It
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