任务-1讲义-v12016cfa一级基础数量组合

【梦轩考 专业提供CFAFRM全 ),'▁ঃ׀੟੟tativeԥ㩮䙹୐३ݾਃ୐୐TopicTopicWeightingsinCFALevelSessionStudySessionEthics&ProfessionalStudySession2-StudySession4-StudySession7-FinancialReportingandStudySessionCorporate7StudySessionPortfolioManagementandWealth7StudySession13-EquityStudySession15- StudySession5StudySessionAlternative4tativeTimeValueR5TheTimeValueofR6DiscountedCashFlowProbability&R7StatisticalConceptsandMarketR8ProbabilityR9CommonProbabilityInferentialR10SamplingandR11HypothesisTechnicalR12Technical【梦轩考 专业提供CFAFRM全 R5TimeValueofTimeValueofRequiredinterestrateonasecurityⲴ㓴AnnuitiesⲴ䇑㇇FV,PV,requiredR5TimeValueofRequiredrateofreturnaffectedbythesupplyanddemandoffundsinthethereturnthatinvestorsandsaversrequiretogetthemtowillinglylendtheirfunds;usuallyforparticularDiscountratetheinterestrateweusetodiscountpaymentstobemadeinusuallyusedinterchangeablywiththeinterestOpportunitycostalsounderstoodasaformofinterestrate.Itisthevaluethatinvestorsforgobychoosingaparticularcourseofaction.R5TimeValueofposerequiredrateofNominalrisk-freerate=realrisk-freerate+expectedinflationRequiredinterestrateona=nominalrisk-freerate+defaultriskpremium+liquidityriskpremium+maturityriskpremium㘳ሏᯩ⌅Realrisk-freerateਃnominalrisk-freerateⲴޣ【梦轩考 专业提供CFAFRM全 R5TimeValueofEARrmEAR(1+periodic 1 1 m䛓Ѹྲᯩᱟsemi,m=2;ྲᯩᱟquarterly,ྲᯩᱟ䘎㔝༽ᯩˈޜᔿᯩਈѪEAR=eannualint-ሏᯩ⌅㇇——㇇EARˈᡆᱟ㇇ᯩ⅑ᇊᙗ˄EARਃ䇑᚟⅑᚟ᴹޣThegreaterthecompoundingthegreatertheEARwillbeincomparisontothestatedthegreaterthedifferencebetweenEARandthestatedR5R5Example:TimeValueofAmoneymanagerhas$1,000,000toinvestforoneyear.Shehasidentifiedtwoalternativeone-year sofdeposit(CD)shownbelow:CompoundingAnnualinterestWhichCDhasthehighesteffectiveannualrate(EAR)andhowmuchinterestwillitearn?HighestInterestR5TimeValueofFuturevalue(FV):Amounttowhichinvestmentgrowsafteroneormorecompoundingperiods.Presentvalue(PV):CurrentvalueofsomefuturecashIfinterestsarecompoundedmtimesperyear,andinvest1Ifinterestsarecompoundedmtimesperyear,andinvestnFV=PV˄1+r/m˅mnWhere:misthecompoundingristhenominal/quotedannualinterestWhenwecalculatethefuturevalueofcontinuouslycompounding,formula FV=PVlim(1+ 【梦轩考 专业提供CFAFRM全程+讲R5TimeValueofWhat’s---isastreamofequalcashflowsthatoccursatequalintervalsoveragiven᚟ᇩN=numberofI/Y=interestrateperPV=presentPMT=amountofeachperiodicFV=futureሏᯩ⌅˖㇇——NI/Y,PMTFV,PVѝԫ᚟ᇊഋњˈ≲ਖཆаAnAnexampleofordinaryannuities˄ঃԈᒤ˅Example1:What’stheFVofanordinaryannuitythatpays150peryearattheendofeachofthenext15years,giventhediscountrateisSolutions:enterrelevantdataforN15,I/Y6,PMT-150,PV0,CPT→FVNotice:ifweweregiventhatFV=3491.4,N=15,I/Y=6,PMT=-150,wealsocouldcalculatePV. Ă Ă R5TimeValueofR5TimeValueofAboutanannuitydue˄ݸԈᒤDefinition:anannuitywheretheannuitypaymentsoccuratthebeginningofeachcompoundingperiod.Measure1:putthecalculatorintheBGNmodeandinputrelevantdata.Measure2:treatasanordinaryannuityandsimplymultipletheresultingPVby(1+I/Y)【梦轩考 专业提供CFAFRMConstructanamortizationscheduletoshowtheinterestandprincipalcomponentsoftheendofyearpaymentsfora10%,5year,$10,000loan.Theamountoftheloanpayments:N=5;I/Y=10;PV=$10,000;FV=0;CPT:【梦轩考 专业提供CFAFRMConstructanamortizationscheduletoshowtheinterestandprincipalcomponentsoftheendofyearpaymentsfora10%,5year,$10,000loan.Theamountoftheloanpayments:N=5;I/Y=10;PV=$10,000;FV=0;CPT:balancetozero.R5Example:TimeValueofAmortization12345R5Example:TimeValueofSupposeyoumustmakefiveannual$1,000payments,thefirstonestartingatthebeginningofYear4(endofYear3).Toaccumulatethemoneytomakethesepayments,youwanttomakethreeequalpaymentsintoaninvestmentaccount,thefirsttobemadeoneyearfromtoday.Assuminga10%rateofreturn,whatistheamountofthesethreeThefirststepinthistypeofproblemistodeterminetheamountofmoneythatmustbeavailableatthebeginningofYear4(t=3)inordertosatisfythepaymentrequirements.BGNmode:N=5;I/Y=10;PMT=-1,000;FV=0;CPT:DeterminetheamountofthethreeENDmode:N=3;I/Y=10;PV=0;FV=-4,169.87;CPT:R5TimeValueofAboutDefinition:Aperpetuityisafinancialinstrumentsthatpaysafixedamountofmoneyatsetintervalsoveraninfiniteperiodoftime. AAA (1 AAA 1r(1r)2(1 1r(1(2) r r【梦轩考 专业提供CFAFRM全 R6DiscountedCashFlowDiscountedCashFlowNPV&2.䇑㇇HPYˈEAYㅹ᚟⳺⦷ˈԕ৺ᆳԜ᚟ӂѻ䰤Ⲵ䖜3.Money-weightedreturn&Time-weightedR6R6DiscountedCashFlowNP CFNP CFCF(1r11CF(1rN22...CF(1rCFNNttt(1r 0 (1 (1 ... (1Nt (1WhenNPV=0,thediscountIRRmethodassumestheproject’scashflowswillbereinvestedattheIRR.MultiplesolutionsProblemoftheIRRcalculation(#signchanges)R6DiscountedCashFlowProjectDecisionSingleprojectNPVmethod:AcceptitifIRRmethod:AcceptitifIRR>r(requiredrateofTwoProjectsIndependentSimilartoSingleprojectsMutuallyExclusiveNPVmethod:ChoosetheonewithhigherIRRmethod:ChoosetheonewithhigherNPVandIRRmethodsmaywitheach 【梦轩考 专业提供CFAFRM CalabashCrabHouseisconsideringaninvestmentinkitchen-upgradeprojectswiththefollowingcashflows:AssumingCalabashhasa12.5percentcostofcapital,whichofthefollowinginvestmentdecisionshastheleastjustification?Accept:ProjectBbecausethenetpresentvalue(NPV)ishigherthanthatofProjectProjectAbecausetheIRRishigherthanthecostofProjectAbecausetheinternalrateofreturn(IRR)ishigherthanthatofProjectB.Correctanswer:Define:theholdingperiodreturnissimplythepercentagechangeinthevalueofaninvestmentovertheperioditishold.R6DiscountedCashFlowR6Example:DiscountedCashFlowProjectProjectInitialYearYearYearYearrr( tP1P0R6DiscountedCashFlow(1+BEY)22 (1HPY)365/t【梦轩考 专业提供CFAFRM全程+讲R6DiscountedCashFlowTheHPYistheactualreturnaninvestorwillreceiveifthemoneymarketinstrumentishelduntilmaturity.TheEAYistheannualizedHPYonthebasisofa365-dayyearandincorporatestheeffectsofcompounding.TherMMistheannualizedyieldthatisbasedonpriceanda360-dayyearanddosenotaccountfortheeffectsofcompounding–itassumessimpleinterest.R6Example:DiscountedCashFlowJanePeeblespurchasedaT-billthatmaturesin200daysfor$97,500.Thefacevalueofthebillis$100,000.Whatisthemoneymarketyieldonthebill?Correctanswer:R6Example:DiscountedCashFlowA175-dayT-billhasaneffectiveannualyieldof3.80%.Itsbankdiscountyieldisclosestto:Answer:Wouldaclientmakingadditionsorwithdrawalsoffundsmostlikelyaffecttheirportfolio’s:Correctanswer:Thetime-weightedreturnisnotaffectedbycashwithdrawalsoradditiontotheportfolio,themoney-weightedreturnmeasureWouldaclientmakingadditionsorwithdrawalsoffundsmostlikelyaffecttheirportfolio’s:Correctanswer:Thetime-weightedreturnisnotaffectedbycashwithdrawalsoradditiontotheportfolio,themoney-weightedreturnmeasurewouldbeaffectedbyclientadditionsorwithdrawals,ifaclientaddsfundsatafavorabletimethemoney-weightedreturnwillbeelevated.R6Example:DiscountedCashFlowR6DiscountedCashFlowMoney-weightedandtime-weightedRateoftime-weightedreturnᦼᨑᾲᘥ৺ޜᔿᾲᘥ˖Time-weightedrateofreturnmeasurescompound僔৺ޜᔿFirstly,computetheHPR;then,compute(1+HPR)foreachsubperiodtoobtainatotalreturnfortheentiremeasurementperiod[eg.(1+HPR1)*(1+HPR2)…(1+HPRn)].money-weightedreturnᦼᨑᾲᘥ৺ޜᔿᾲᘥ˖theIRRbasedonthecashflowsrelatedtothe↕僔৺ޜᔿ˖Firstly,determinethetimingofeachcashthen,usingthecalculationtocomputeIRR,orusinggeometricሏᯩ⌅˖㇇˗⌘᚟㇇timeweightedreturn᚟ྲHPRн⭘ᔰR6Example:DiscountedCashFlowAssumeaninvestorbuysashareofstockfor$100att=0andattheendofthenextyear(t=1),shebuysanadditionalsharefor$120.AttheendofYear2,theinvestorsellsbothsharesfor$130each.Attheendofeachyearintheholdingperiod,thestockpaida$2.00persharedividend.Whatisthemoney-weightedrateofWhatistheannualtime-weightedrateofMoney-weightedrateofreturn(IRR)Time-weightedrateofreturn(geometricmeanreturn)=Time-weightedMoney-weighted【梦轩考 专业提供CFAFRM全 R7StatisticalConceptsandMarketStatisticalTypesofmeasurementMeasuresofcentralMADਃVar䇑㇇ԕ৺∄Chebyshev’sCV&SharpSkewness&R7StatisticalConceptsandMarketDescriptiveSummarizetheimportantcharacteristicsoflargedataInferentialMakeforecasts,estimates,orjudgmentsaboutalargesetofdataonthebasisofthestatisticalcharacteristicsofasmallerset(asample)R7StatisticalConceptsandMarketTypesofmeasurementNominaldistinguishingtwodifferentthings,noorder,onlyhasexample:assigningthenumber1toamunicipalbondfund,thenumber2toacorporatebondfund.Ordinalscales(>,makingthingsinorder,butthedifferencearenotexample:therankingof1,000smallcapgrowthstocksbymaybedonebyassigningthenumber1tothe100bestperformingIntervalscales(>,<,+,-subtractisexample:Ratioscales(>,<,+,-,*,withoriginalexample:money,ifyouhavezerodollars,youhavenopurchasingbutifyouhave$4.00,youhavetwiceasmuchpurchasingpowerasawith$2.00.Absolute 33Absolute 335–0–5–10R7StatisticalConceptsandMarketAmeasureusedtodescribeacharacteristicofapopulationisreferredtoasaparameter.Inthesamemannerthataparametermaybeusedtodescribeacharacteristicofapopulation,asamplestatisticisusedtomeasureacharacteristicofasample.R7StatisticalConceptsandMarketRelativeTherelativefrequencyiscalculatedbydividingtheabsolutefrequencyofeachturnintervalbythetotalnumberofobservations.FrequencyAfrequencydistributionisatabularpresentationofstatisticaldatathataidstheysisoflargedatasets.Cumulativefrequency/CumulativeRelativeCouldbecalculatedbysummingtheabsoluteorrelativefrequenciesstartingatthelowestintervalandprogressingthroughthehighest.R7StatisticalConceptsandMarketFrequencyR7StatisticalConceptsandMarket【梦轩考 专业提供CFAFRM全 +讲R7StatisticalConceptsandMarketmidpointofeachintervalisplottedonthehorizontalaxis,midpointofeachintervalisplottedonthehorizontalaxis,andtheabsolutefrequencyforthatintervalisplottedontheverticalaxis.Histogramisgraphicalfrequencydistribution876543210NXi n n wiXi(w1X1w2i)NGN inn(1/XiiThearithmeticmeanTheweightedmeanThegeometricmeanTheharmonicmeanharmonicmean<=geometricmean<=arithmeticR7StatisticalConceptsandMarketR7Example:StatisticalConceptsandWhichisthemostHarmonic ArithmeticGeometric Correctanswer:【梦轩考 专业提供CFAFRM全 R7Example:StatisticalConceptsandMarket ystobtainsthefollowingannualratesofreturnforamutualReturnAnswer:Thefund’sannualholdingperiodreturnisclosestAnswer:R7Example:StatisticalConceptsandMarketAhypotheticalinvestmentinasinglestockinitiallycosts$100.oneyearlater,thestockistradingat$200.Attheendofthesecondyear,thestockpricefallsbacktotheoriginalpurchasepriceof$100.Nodividendarepaidduringthetwo-yearperiod.Calculatethearithmeticandgeometricmeanannualreturns.ReturninYear1=200/100-1=100%ReturninYear2=100/200-1=-50%Arithmeticmean=(100%-50%)/2=25%Geometricmean=(2.0h0.5)1/2–1=Thegeometricmeanreturnof0%accurayreflectsthattheendingvalueoftheinvestmentinYear2equalsthestartingvalueinYear1.Thecompoundrateofreturnontheinvestmentis0%.Thearithmeticmeanreturnreflectstheaverageoftheone-yearreturns.R7StatisticalConceptsandMarketTheuseofarithmeticmeanandgeometricmeanwhendetermininginvestmentreturnsThearithmeticmeanisthestatisticallybestestimatorofthenextyear’sreturnsgivenonlythethreeyearsofreturn Sincepastannualreturnsarecompoundedeachperiod,thegeometricmeanofpastannualreturnsistheappropriatemeasureofpastN inN()iForN inN()iFor i1 Nn( Xi i1 nR7StatisticalConceptsandMarketQuartileThethirdquartile:75%,orthree-fourthsoftheobservationsfallbelowthatvalue.CalculationLy=(n+1)y/100,LyistheObservers˖8101213151717181923N=11ˈLy=(11+1)*75%=9,i.e.the9thnumberisThethirdquartiles=R7R7StatisticalConceptsandMarketAbsolutedispersion:istheamountofvariabilitypresentwithoutcomparisontoanyreferencepointorben RangeRange umvalue–minimumR7StatisticalConceptsandMarketForanysetofobservations(samplesorpopulation),theproportionofthevaluesthatliewithinkstandarddeviationsofthemeanisatleast11/k2,wherekisanyconstantgreaterthan1.ሩԫօа㓴㿲⍻ˈњփ㩭᚟᚟ઘതkњḷ߶ᐞѻ᚟Ⲵᾲ⦷нሿҾ1/k2,ሩԫ᚟k>1DŽThisrelationshipappliesregardlessoftheshapeofthe【梦轩考 专业提供CFAFRM全 R7Example:StatisticalConceptsandAssumeasampleofbeerpricesisnegativelyskewed.Approximaywhatpercentageofthedistributionlieswithinplusorminus2.40standarddeviationsofthemean?Correctanswer:R7R7StatisticalConceptsandMarketCoefficientofvariationmeasurestheamountofdispersioninadistributionrelativetothedistribution’smean.(relativedispersion)CV=CV=sxXThesharpratiomeasuresexcessreturnperunitofR7StatisticalConceptsandMarketMeanMedian Positiveskewed˖Mode<median<mean,havingarightfatAreturndistributionwithpositiveskewhasfrequentsmalllossesandafewextremeNegativeskewed˖Mode>media>mean,havingaleftfatAreturndistributionwithnegativeskewhasfrequentsmallgainsandafewInvestorsshouldbeattractedbyapositiveskewbecausethemeanreturnfallsabove Sample (X

X (

XS ]()(n1)(n ሏᯩ⌅ṩᦞ᧿䘠Ⲵ᚟᚟ᱟPositivelyskewed䘈ᱟNegativeṩᦞᐢ⸕ ᓖˈ䘹᚟䜭ᴹଚӋ⢩⛩R7StatisticalConceptsandMarket【梦轩考 R7StatisticalConceptsandMarketLeptokurticvs.Itdealswithwhetherornotadistributionismoreorless“peaked”thananormalExcesskurtosis=samplekurtosis–NormalSampleExcessSample n(n (XiX (XiX i i1 (n1)(n2)(n 㘳ሏ᚟⌅˖ṩᦞ᧿䘠Ⲵ᚟᚟ᱟleptokurtic䘈ṩᦞᐢ⸕Ⲵጠᓖˈ䘹᚟䜭ᴹଚӋ⢩⛩ਟ᚟㘳䈅ѝՊਃskewਸᒦ㘳Ṩ㔬ਸ⸕R7StatisticalConceptsandMarketFatAleptokurticreturndistributionhasmorefrequentextremelylargedeviationsfromthemeanthananormaldistribution.R7Example:StatisticalConceptsandMarketABTheyststatedthatthedistributionnormaldistributionandthatthedistributleftsideofthedistribution.IstheyPorfolioforPortfolionforPost’sioAismorepeakedthanartfolioBhasalongtailontheentcorrectwithrespectto:oSolution:⳷㹗⺞⭤⭞⺞㦆㵧㏎Priori【梦轩考 专业提供CFAFRM全 ⳷㹗⺞⭤⭞⺞㦆㵧㏎PrioriR8ProbabilityProbabilityTwodefiningpropertiesofEmpirical,subjective,andprioriOddsfororMultiplicationruleandadditionDependentandindependentCovariance&Expectedvalue,variance,andstandarddeviationofarandomvariableandofreturnsonaportfolioBayes’R8ProbabilityBasicRandomvariableisuncertaineisanobservedvalueofarandomMutuallyexclusiveevents—cannotbothhappenatthesameExhaustiveevents—includeall TwoDefiningPropertiesof0≤P(E)≤P(E1)+P(E2)+……+R8ProbabilityR8Probability⳷㹗⺞㦆⭤⭞ㅌ Basedonintuitionorsubjective【梦轩考 专业提供CFAFRM全 R8ProbabilityEmpiricalprobability㓿ᾲeg.Historically,theDowJonesIndustrialAveragehasclosedhigherthanthepreviousclosetwooutofeverythreetradingdays.Therefore,theprobabilityoftheDowgoinguptomorrowistwo-thirds,or66.7%.Prioriprobabilityݸᾲeg.Yesterday,24ofthe30DJIAstocksincreasedinvalue.Thus,if1of30stocksisselectedatrandom,thereisan80%(24/30)probabilitythatitsvalueincreasedyesterdaySubjectiveprobabilityѫᾲwillclosehighertomorrowisR8ProbabilityOddsforanP(E)/(1-OddsagainstanLastyear,theaveragesalaryincreaseforPoultryResearchAssistantswas2.5percent.Ofthe10,000PoultryResearchAssistants,2,000receivedraisesinexcessofthisamount.TheoddsthataPoultryResearchAssistantreceivedasalaryincreaseinexcessof2.5percentare:1to2toCorrectanswer:R8ProbabilityUnconditionalProbability(marginalprobability):Conditionalprobability:【梦轩考 专业提供CFAFRM全 R8ProbabilityJointprobability:MultiplicationP(AB)=P(A|B)hP(B)=P(B|A)hP(A)IfAandBaremutuallyexclusiveevents,then:P(AB)=P(A|B)=P(B|A)=0ProbabilitythatatleastoneoftwoeventswillAdditionP(AorB)=P(A)+P(B)-IfAandBaremutuallyexclusiveevents,then:P(AorB)=P(A)+P(B)R8ProbabilityTheoccurrenceofAhasnoinfluenceofontheoccurrenceofP(A|B)=P(A)orP(AorB)=P(A)+P(B)-IndependenceandMutuallyExclusivearequiteIfexclusive,mustnotCauseexclusivemeansifAoccur,Bcannotoccur,AinfluentsR8Example:ProbabilityP(A)=0.5,P(B)=0.5,oddforconcurrentAandBis3/5,therelationshipbetweenAandB?MutuallyCorrectanswer:P(AB)=(3/5)/(1+3/5),P(A/B)=P(AB)/P(B)=3/4,P(A/B)нㅹҾP(A)【梦轩考 专业提供CFAFRM全 R8ProbabilityForunconditionalprobabilityofeventP(A P(AS1)P(S1 P(AS2)P(S2 ...P(ASN)P(SNwherethesetofeventsS1,S2,...SN ismutuallyexclusiveandExpectedvalue:E(X P(Xi xi*P(xi x1* x2*P(x2 xn*P(xnN P( EX iR8Example:ProbabilityAnystgatheredthefollowinginformation:theprobabilityofeconomyprosperityis75%,theprobabilityofeconomyrecessionis25%.Foracompany,whentheeconomyisprosperity,thereis10%ofprobabilitythatitsEPSis$2.0and90%ofprobabilitythattheEPSis$4.0.However,whentheeconomyisrecession,thereis25%ofprobabilitythattheEPSis$2.0and75%ofprobabilitythattheEPSis$4.0.Whatisthevarianceofthiscompany’sEPS,whentheeconomyisrecession?Correctanswer:WhentheeconomyE(EPS)=25%*2+75%*4=Var(EPS)=25%*(2-3.5)2+75%*(4-3.5)2=R8Probability Prob.Of Prob.Of E(EPS)18%1.8 42%1.7 24%1.316%1.01.51Thejointprobabilityofreturns,forsecuritiesAandB,Thejointprobabilityofreturns,forsecuritiesAandB,areasThecovarianceofthereturnsbetweensecuritiesAandBisclosest1224Correctanswer:R8Example:ProbabilityR8ProbabilityCovariancemeasureshowonerandomvariablemoveswithanotherrandomThecovarianceofRAwithitselfisequaltothevarianceofCovariancerangesfromnegativeinfinitytopositiveCOV(X,X)E[(XE(X))(X 2 E[(X-E(X))(Y- CorrelationmeasuresthelinearrelationshipbetweentworandomCorrelationhasnounits,rangesfrom–1to+1,standardizationofUnderstandthedifferencebetweencorrelationandIfρ=0,thisR8Example:ProbabilityThecovarianceofreturnsfortwomusthaveavaluebetween-1.0andmusthaveavalueequaltotheweightedaverageofthestandarddeviationsofthereturnsofthetwostockswillbepositiveiftheactualreturnsonbothstocksareconsistentlybelowtheirexpectedreturnsatthesametimeCorrectanswer:JointProbabilityFunctionofSecurityAandSecurityBReturns(Entriesarejointprobabilities)ReturnonsecurityReturnonsecurityReturnonsecurity0Returnonsecurity0Anindividualwantstoinvest$100,000andisconsideringthefollowingTheexpectedcorrelationofreturnsforthetwostocksis+0.5.Iftheinvestorinvests$40,000inStockAand$60,000inStockB,theexpectedstandarddeviationofreturnsontheportfoliowillbe:equaltolessthangreaterthan20.4%becausethecorrelationcoefficientisgreaterthanCorrectanswer:Anindividualwantstoinvest$100,000andisconsideringthefollowingTheexpectedcorrelationofreturnsforthetwostocksis+0.5.Iftheinvestorinvests$40,000inStockAand$60,000inStockB,theexpectedstandarddeviationofreturnsontheportfoliowillbe:equaltolessthangreaterthan20.4%becausethecorrelationcoefficientisgreaterthanCorrectanswer:R8Example:ProbabilityR8ProbabilityExpectedreturn,varianceandstandarddeviationofanE(rp wiE(Riin2 wwcov(R,Ri i1jExpectedStandardDeviationofABR8ProbabilityBayes’ P(A|B P(B|A)*P(P(B P(S|R) P(R|Si)P(Si) P(R)【梦轩考 专业提供CFAFRM全 R8Probability䇺ᯝᴹᴹ ᚟⋑Ӫ⋑䇺᚟ᴹ 䇺᚟⋑ R8ProbabilityMultiplication n1hn2h……hn nLabeling(or n1! nk r)! n R9CommonProbabilityCommonProbabilityPropertiesofdiscretedistributionandcontinuousUniformrandomvariableandabinomialrandomThekeypropertiesofthenormalStandardizearandomConfidenceintervalforanormallydistributedrandomLognormalSafety-firstMonteCarlo【梦轩考 专业提供CFAFRM全程+讲R9CommonProbabilityProbabilityDescribetheprobabilitiesofallthepossible esforarandomDiscreteandcontinuousrandomDiscreterandomvariables:thenumberofpossible escanbecounted,andforeachpossible e,thereisameasurableandpositiveprobability.Continuousvariables:thenumberofpossible esisinfinite,eveniflowerandupperboundsexist.P(x)=0eventhoughxcanPR9CommonProbabilityProbability Fordiscreterandom0≤p(x)≤Probabilitydensityfunction(p.d.f):ForcontinuousrandomvariableCumulativeprobabilityfunction(c.p.f):ProbabilityProbabilitydensity0R9CommonProbabilityBinomialBernoullirandom P(Y=0)=1-BinomialrandomvariableˈBinomialBernoullirandom P(Y=0)=1-Binomialrandomvariableˈtheprobabilityofxsuccessesinnp(x P( x)nxpxp)nExpectationsandR9CommonProbabilityR9Example:CommonProbabilityWhichofthefollowingstatementsaboutprobabilitydistributionsForaprobabilitydistributionforthenumberofdaystheairpollutionisaboveaspecifiedlevel,p(x)=0whenxcannotoccur,orp(x)>0whenitcan.Foraprobabilitydistributionforthespecificlevelofairpollutiononagivenday,p(x)=0evenifxcanoccur.AcumulativedistributionfunctiongivestheprobabilitythatarandomvariabletakesavalueequaltoorgreaterthanagivenCorrectanswer:Acumulativedistributionfunctiongivestheprobabilitythatarandomvariabletakesavalueequaltoorlessthanagivennumber:P(X≤x),orF(X).R9CommonProbabilityDiscreteAdiscreteuniformrandomvariableisoneforwhichtheprobabilitiesforallpossible esforadiscreterandomvariableareequal.Forexample,considerthediscreteuniformprobabilitydistributiondefinedasX={1,2,3,4,5},p(x)=0.2.Here,theprobabilityforeach eisequalto0.2[i.e.,Bernoullirandomvariablepp(1-Binomialrandomvariablenp(1-【梦轩考 专业提供CFAFRM全 R9CommonProbabilityContinuousUniform----isdefinedoverarangethatspansbetweensomelowerlimit,a,andupperlimit,b,whichserveastheparametersofthedistribution.PropertiesofContinuousuniformForalla≤x1<x2P(X<aorX>b)=P(x x2 (x x1)/(b R9Example:CommonProbabilityWhichofthefollowingstatementsaboutprobabilitydistributionsisAcontinuousuniformdistributionhasalowerlimitbutnoupperAcumulativedistributionfunctiondefinestheprobabilitythatarandomvariableisgreaterthanagivenvalue.Abinomialdistributioncountsthenumberofsuccessesthatoccurinafixednumberofindependenttrialsthathavemutuallyexclusive(i.e.yesorno) Correctanswer:Arandomvariablewithafinitenumberofequallylikely esisbestdescribedbya:BinomialBernoulliDiscreteuniformCorrectanswer:R9Example:CommonProbabilityArecentstudyindicatedthat60%ofallbusinesseshaveafaxmachine.Fromthebinominalprobabilitydistributiontable,theprobabilitythatexactlyfourbusinesseswillhaveafaxmachineinarandomselectionofsixbusinessesis:Correctanswer:Assumethat40%ofcandidateswhositfortheCFAexaminationpassitthefirsttime.Ofarandomsampleof15candidateswhoaresittingfortheexamforthefirsttime,whatistheexpectednumberofcandidatesthatwillpass?Correctanswer:【梦轩考 专业提供CFAFRM全 R9Example:CommonProbabilityAnysthasrecentlydeterminedthatonly60percentofallU.S.pensionfundshaveholdingsinhedgefunds.Inevaluatingthisprobability,arandomsampleof50U.S.pensionfundsistaken.ThenumberofU.S.pensionfundsinthesampleof50thathavehedgefundsintheirportfoliowouldmostaccuraybedescribedas:AbinomialrandomABernoullirandomAcontinuousrandomCorrectanswer:AnenergyystforecaststhatthepriceperbarrelofcrudeoilfiveyearsfromnowwillrangebetweenUSD$75andUSD$105.Assumingacontinuousuniformdistribution,theprobabilitythatthepricewillbelessthanUSD$80fiveyearsfromnowisclosestto:Correctanswer:R9CommonProbabilityTrackingerroristhedifferencebetweenthetotalreturnonaportfolioandthetotalreturnonthebenarkagainstwhichitsperformanceismeasured.R9CommonProbabilityTheshapeofthedensityxX~N(µ,Symmetricaldistribution:skewness=0;Alinearcombinationofnormallydistributedrandomvariablesisalsonormallydistributed.Thetailsgetthinandgotozerobutextendinfiniy,asympotic䘁R9CommonProbability【梦轩考 专业提供CFAFRM全R9CommonProbabilityTheconfidence68%confidenceinterval[,]90%confidenceinterval[1.651.6595%confidenceinterval[1.961.9699%confidenceinterval[2.582.58U2.58σU1.96σU U+1σU+1.96σUR9Example:CommonProbability ystdeterminedthatapproxima y99percentoftheobservationsofdailysalesforacompanywerewithintheintervalfrom$230,000to$480,000andthatdailysalesforthecompanywerenormallydistributed.Themeandailysalesandstandarddeviationofdailysales,respectively,forthecompanywereclosestto:Meandaily Standarddeviationofdaily Correctanswer:R9CommonProbabilityStandardnormalN(0,1)orStandardization:ifX~N(µ,σ²), X ~N(0,1)F(-z)1- 1【梦轩考 专业提供CFAFRM全 R9CommonProbabilityR9Example:CommonProbabilityAstudyofhedgefundinvestorsfoundthattheirannualhousehold esarenormallydistributedwithameanof$175,000andastandarddeviationof$25,000.F(1)=0.8413,F(2)=0.9772,F(3)=0.9987Thepercentofhedgefundinvestorsthat eslessthan$100,000isclosestThepercentofhedgefundinvestorsthat esgreaterthan$225,000isclosestThepercentofhedgefundinvestorsthat esgreaterthan$150,000isclosestR9CommonProba