CHAPTER5

RiskandRatesofReturnStand-aloneriskPortfolioriskRisk&return:CAPM/SMLInvestmentreturnsTherateofreturnonaninvestmentcanbecalculatedasfollows:

(Amountreceived–Amountinvested)

Return=

________________________

AmountinvestedForexample,if$1,000isinvestedand$1,100isreturnedafteroneyear,therateofreturnforthisinvestmentis: ($1,100-$1,000)/$1,000=10%.Whatisinvestmentrisk?TwotypesofinvestmentriskStand-aloneriskPortfolioriskInvestmentriskisrelatedtotheprobabilityofearningalowornegativeactualreturn.Thegreaterthechanceoflowerthanexpectedornegativereturns,theriskiertheinvestment.ProbabilitydistributionsAlistingofallpossibleoutcomes,andtheprobabilityofeachoccurrence.Canbeshowngraphically.ExpectedRateofReturnRateofReturn(%)100150-70FirmXFirmYInvestmentalternativesEconomyProb.T-BillHTCollUSRMPRecession0.18.0%-22.0%28.0%10.0%-13.0%Belowavg0.28.0%-2.0%14.7%-10.0%1.0%Average0.48.0%20.0%0.0%7.0%15.0%Aboveavg0.28.0%35.0%-10.0%45.0%29.0%Boom0.18.0%50.0%-20.0%30.0%43.0%Return:CalculatingtheexpectedreturnforeachalternativeSummaryofexpectedreturnsforallalternatives

Expreturn HT 17.4% Market 15.0% USR 13.8% T-bill 8.0% Coll. 1.7%HThasthehighestexpectedreturn,andappearstobethebestinvestmentalternative,butisitreally?Havewefailedtoaccountforrisk?WhyistheT-billreturnindependentoftheeconomy?DoT-billspromiseacompletelyrisk-freereturn?T-billswillreturnthepromised8%,regardlessoftheeconomy.No,T-billsdonotprovidearisk-freereturn,astheyarestillexposedtoinflation.Although,verylittleunexpectedinflationislikelytooccuroversuchashortperiodoftime.T-billsarealsoriskyintermsofreinvestmentraterisk.T-billsarerisk-freeinthedefaultsenseoftheword.HowdothereturnsofHTandColl.behaveinrelationtothemarket?HT–Moveswiththeeconomy,andhasapositivecorrelation.Thisistypical.Coll.–Iscountercyclicalwiththeeconomy,andhasanegativecorrelation.Thisisunusual.Risk:CalculatingthestandarddeviationforeachalternativeStandarddeviationcalculationComparingstandarddeviationsUSRProb.T-billHT0813.817.4 RateofReturn(%)CommentsonstandarddeviationasameasureofriskStandarddeviation(σi)measurestotal,orstand-alone,risk.Thelargerσiis,thelowertheprobabilitythatactualreturnswillbeclosertoexpectedreturns.Largerσiisassociatedwithawiderprobabilitydistributionofreturns.Difficulttocomparestandarddeviations,becausereturnhasnotbeenaccountedfor.ComparingriskandreturnSecurityExpectedreturnRisk,σT-bills8.0%0.0%HT17.4%20.0%Coll*1.7%13.4%USR*13.8%18.8%Market15.0%15.3%*Seemoutofplace.CoefficientofVariation(CV)Astandardizedmeasureofdispersionabouttheexpectedvalue,thatshowstheriskperunitofreturn.Riskrankings,

bycoefficientofvariation

CV T-bill 0.000 HT 1.149 Coll. 7.882 USR 1.362 Market 1.020Collectionshasthehighestdegreeofriskperunitofreturn.HT,despitehavingthehigheststandarddeviationofreturns,hasarelativelyaverageCV.IllustratingtheCVasameasureofrelativeriskσA=σB,butAisriskierbecauseofalargerprobabilityoflosses.Inotherwords,thesameamountofrisk(asmeasuredbyσ)forlessreturns.0ABRateofReturn(%)Prob.SelectedRealizedReturns,

1926–2001

AverageStandard

Return

DeviationSmall-companystocks 17.3% 33.2%Large-companystocks 12.7 20.2 L-Tcorporatebonds 6.1 8.6L-Tgovernmentbonds 5.7 9.4U.S.Treasurybills 3.9 3.2 Source:BasedonStocks,Bonds,Bills,andInflation:(ValuationEdition)2002Yearbook(Chicago:IbbotsonAssociates,2002),28.InvestorattitudetowardsriskRiskaversion–assumesinvestorsdislikeriskandrequirehigherratesofreturntoencouragethemtoholdriskiersecurities.Riskpremium–thedifferencebetweenthereturnonariskyassetandlessriskyasset,whichservesascompensationforinvestorstoholdriskiersecurities.ProbabilitydistributionsAlistingofallpossibleoutcomes,andtheprobabilityofeachoccurrence.Canbeshowngraphically.ExpectedRateofReturnRateofReturn(%)100150-70FirmXFirmYPortfolioconstruction:

RiskandreturnAssumeatwo-stockportfolioiscreatedwith$50,000investedinbothHTandCollections.Expectedreturnofaportfolioisaweightedaverageofeachofthecomponentassetsoftheportfolio.Standarddeviationisalittlemoretrickyandrequiresthatanewprobabilitydistributionfortheportfolioreturnsbedevised.CalculatingportfolioexpectedreturnAnalternativemethodfordeterminingportfolioexpectedreturnEconomyProb.HTCollPort.Recession0.1-22.0%28.0%3.0%Belowavg0.2-2.0%14.7%6.4%Average0.420.0%0.0%10.0%Aboveavg0.235.0%-10.0%12.5%Boom0.150.0%-20.0%15.0%CalculatingportfoliostandarddeviationandCVCommentsonportfolioriskmeasuresσp=3.3%ismuchlowerthantheσiofeitherstock(σHT=20.0%;σColl.=13.4%).σp=3.3%islowerthantheweightedaverageofHTandColl.’sσ(16.7%).\Portfolioprovidesaveragereturnofcomponentstocks,butlowerthanaveragerisk.Why?Negativecorrelationbetweenstocks.GeneralcommentsaboutriskMoststocksarepositivelycorrelatedwiththemarket(ρk,m

0.65).σ

35%foranaveragestock.Combiningstocksinaportfoliogenerallylowersrisk.

Returnsdistributionfortwoperfectlynegativelycorrelatedstocks(ρ=-1.0)-101515252525150-10StockW0StockM-100PortfolioWMReturnsdistributionfortwoperfectlypositivelycorrelatedstocks(ρ=1.0)StockM01525-10StockM’01525-10PortfolioMM’01525-10Creatingaportfolio:

Beginningwithonestockandaddingrandomlyselectedstockstoportfolioσpdecreasesasstocksadded,becausetheywouldnotbeperfectlycorrelatedwiththeexistingportfolio.Expectedreturnoftheportfoliowouldremainre