Lecture 03(Introduction to Risk and Diversificatio

INTRODUCTIONTORISKANDDIVERSIFICATIONImreKonyáriInvestmentsinCEELecture3OUTLINE
AttitudeTowardsRisk
MeasuringExpectedReturn
MeasuringRisk
MeasuringRiskofaTwo-stockPortfolio
MeasuringRiskofaPortfolioofThreeorMore
MeasuringRiskofaPortfolioofThreeorMoreSecurities
UniqueRiskandMarketRisk
BetaandUniqueRisk
MeasuringCountryRisk
SovereignRatings
CountryRiskScores
Market-basedMeasures2ATTITUDETOWARDSRISK/1
Objective:Understandhowinvestorsperceiverisk.
Considertwolotteries:
Lottery1:youreceive5HUFforsure;
Lottery2:youreceive10HUFwithprobability0.5and0HUFwithprobability0.5;3
Inlottery2youget10HUFwith50%chanceand0HUFwith50%chance.Yourexpectedpayoffisthesameasinlottery1:5HUF5HUF0.50.510HUF0HUF0.50.5HUF5HUF05.0HUF105.0=×+×ATTITUDETOWARDSRISK/2
Thechoicebetweenlotteriesisdeterminedbytheattitudetowardsrisk.
Moregenerally,consideranindividualwhochoosesbetweentwolotteries:Lottery1:givesYHUFforsure;4
Lottery1:givesYHUFforsure;
Lottery2:givesYHUFonaverage,i.e.insomestatesyoureceivemorethanYHUFwhileinotherslessthanYHUF.
Aninvestorissaidtobe:riskaverseifchooseslottery1riskneutralifindifferentrisklovingifchooseslottery2ATTITUDETOWARDSRISK/3
Investorsareusuallyriskaverse.
Therefore,evenifstockpaysonaveragethesamereturnsastreasurybillstheywouldprefertobuytreasurybills;
Asaresult,stockpricesadjustinsuchawayastopay5
Asaresult,stockpricesadjustinsuchawayastopaypremiumforholdingriskystocks.MEASURINGEXPECTEDRETURN/1
ConsiderarandomvariableZthattakesmpossiblevaluesZ1,Z2,…,Zmwithprobabilitiesp1,p2,…,pm.
Forexample,rollingadicehassixoutcomesZ1=1,Z2=2,Z3=3,Z4=4,Z5=5,andZ6=6,eachwithprobability1/6.623456probability1/6.
Theprobabilitiesrepresentthefrequencieswithwhichtheoutcomesoccur.
ExpectedvalueofrandomvariableE[Z]isgivenby:E[Z]=Z1×p1+Z2×p2+Z3×p3+…+Zm×pmMEASURINGEXPECTEDRETURN/2
Suppose,youhaveN(e.g.,N=10000000)observationsofrandomvariableZ:X1,X2,….XN,whereXiisoneofthenumbersZ1,Z2,…,Zm.
Thesampleaverageisdefinedas:7
WhenNbecomesverylarge,thesampleaverageconvergestoexpectedvalueE[Z].Therefore,forlargeN:NXXXXN+++=...21XZE≈][MEASURINGEXPECTEDRETURN/3
Inanexamplewithdicerollingwehavesixoutcomes(i.e.m=6):Z1=1,Z2=2,Z3=3,Z4=4,Z5=5,andZ6=6,eachwithprobability1/6.53616615614613612611.][=×+×+×+×+×+×=ZE8
SupposeyourolladiceN=1000000timesandobserveoutcomes:X1=1,X2=3,X3=6,X4=5,X5=3,X6=2,X7=5,X8=6,X9=4,X10=5,X11=6,X12=2,X13=4,X14=4,….
Ifyoucomputetheaverageofthatnumbersitshouldbecloseto3.5.
Thisisbecausethefrequencyofeachoutcomeis1/6.666666MEASURINGRISK/1
Variancemeasureshowfaronaveragetherealizationsofrandomvariablesdeviatefromtheirmeans:2222211)]([...)]([)]([)var(ZEZpZEZpZEZpZmm−++−+−=distancebetweenZandE[Z]9
Inourexampleofrollingadice:
Standarddeviationisdefinedas:Z2andE[Z]916726536535534533532531222222.).().().().().().(]var[=−+−+−+−+−+−=Z)var()(ZZ=σMEASURINGRISK/2
Suppose,youhaveN(e.g.,N=10000000)observationsofrandomvariableZ:X1,X2,….XN
Samplevarianceiscomputedas:10
ForlargeNitapproximatesthevariance.122221−−++−+−≈N)XX(...)XX()XX()Zvar(NMEASURINGRISK/3
Varianceisagooddescriptionofrisk.Suppose,youcaninvest$1000inoneoftheprojectswithpayoffs:$2000$20000.50.5$4000$00.50.5$6000–$20000.50.5$8000–$40000.50.511
Alltheseprojectshavethesameexpectedpayoff$2000.However,thestandarddeviationisdifferent.Amongalltheavailableprojectswiththesameexpectedpayoffariskaverseinvestorwillchoosetheonewithlowervariance.$2000$0–$2000–$4000Project1E[Z]=$2000σ[Z]=0Project2E[Z]=$2000σ[Z]=$2000Project3E[Z]=$2000σ[Z]=$4000Project4E[Z]=$2000σ[Z]=$6000MEASURINGRISKOFATWO-STOCKPORTFOLIO/1
Suppose,aninvestorcaninvest$500inproject1and$500inproject2.Hence,thepayoffswillbeasfollows:$3000boom$0boomboom$150012
Notethatbycombiningprojectsweeliminatedriskscompletely.
Conclusion:Accountingforthejointbehaviourofstocks/projectcashflowsallowstoreducerisk!$0slump$3000slumpProject1Project2$1500slump0.5×Project1+0.5×Project2MEASURINGRISKOFATWO-STOCKPORTFOLIO/2
Thejointbehaviouroftworandompayoffsiscapturedbycovariancewhichisdefinedasfollows:IfcovarianceisnegativerandompayoffsZandWmove[]])[(])[(),cov(WEWZEZEWZ−×−=13
IfcovarianceisnegativerandompayoffsZandWmovein“opposite”direction,i.e.WtendstobelowwhenZishigh.
Covariancecanbeestimatedfromthedataasfollows:whereX1,…,XNarerealizationsofZ,Y1,…,YNrealizationsofW.111−−−++−−≈NYYXXYYXXWZNN))((...))((),cov(MEASURINGRISKOFATWO-STOCKPORTFOLIO/3
NotethatifZandWmoveinthesamedirectionthencov(Z,W)0.
Thisisbecauseinthiscasetendtobepositive.
ifZandWmoveintheoppositedirectionscov(Z,W)0.))((YYXXii−−14
ifZandWmoveintheoppositedirectionscov(Z,W)0.
Thisisbecauseinthiscasetendtobenegative.))((YYXXii−−)()(),cov(),(WZWZWZσσ=corrMEASURINGRISKOFATWO-STOCKPORTFOLIO/4
Correlationisacovariancescaledbystandarddeviationsofrandomvariables:
Thescalingisdoneforconvenience,because:1≤≤),(VZ-corr11≤≤),(VZ-corr1Z1=$3Z2=$0boomslumpboomslumpZWW1=$0W2=$61−=),(WZcorrZ1=$3Z2=$0boomslumpW1=$6boomslumpZW1=),(WZcorr15
Howcorrelationofvariousassetscontributedtothefinancialcrisis?VIDEO:=imbT9PRt8-oW2=£0MEASURINGRISKOFATWO-STOCKPORTFOLIO/5
Suppose,youhaveinitialwealt