风险管理与金融机构约翰第二版答案

1、SolutionstoFurtherProblemsRiskManagementandFinancialInstitutionsSecondEditionJohnC.HullChapter 1: Introduction1.15. Theimpactofinvestingwiinthefirstinvestmentandw2=1-wiinthesecondinvestmentisshowninthetablebelow.Therangeofpossiblerisk-returntrade-offsisshowninfigurebelow.w1w2pp0.01.012%20%0.20.811.2

2、%17.05%0.40.610.4%14.69%%13.22%%12.97%1.00.08.0%14.00%1.16. Inthiscasetheefficientfrontierisasshowninthefigurebelow.Thestandarddeviationofreturnscorrespondingtoanexpectedreturnof10%is9%.Thestandarddeviationofreturnscorrespondingtoanexpectedreturnof20%is39%.1.17.(a) Thebankcanbe99%c

3、ertainthatprofitwillbetterthan0.82.332or385%ofassets.Itthereforeneedsequityequalto3.85%ofassetstobe99%certainthatitwillhaveapositiveequityattheyearend.(b) Thebankcanbe99.9%certainthatprofitwillbegreaterthan0.83.092or-5.38%ofxassets.Itthereforeneedsequityequalto5.38%ofassetstobe99.9%certainthatitwill

4、haveapositiveequityattheyearend.1.18. Whentheexpectedreturnonthemarketis30%theexpectedreturnonaportfoliowithabetaof0.2is0.05+0.2(0.300.05)=0.02or-2%.TheactualreturnofT0%isworsethantheexpectedreturn.Theportfoliomanagerhasachievedanalphaof-8%!Chapter 2: Banks2.15. Thereisa99.9%chancethattheprofitwilln

5、otbeworsethan0.63.0902.0=$5.58million.Regulatorswillrequire$0.58millionofadditionalcapital.2.16. Depositinsurancemakesdepositorslessconcernedaboutthefinancialhealthofabank.Asaresult,banksmaybeabletotakemoreriskwithoutbeingindangeroflosingdeposits.Thisisanexampleofmoralhazard.(Theexistenceoftheinsura

6、ncechangesthebehaviorofthepartiesinvolvedwiththeresultthattheexpectedpayoutontheinsurancecontractishigher.)Regulatoryrequirementsthatbankskeepsufficientcapitalfortheriskstheyaretakingreducetheirincentivetotakerisks.Oneapproach(usedintheUS)toavoidingthemoralhazardproblemistomakethepremiumsthatbanksha

7、vetopayfordepositinsurancedependentonanassessmentoftheriskstheyaretaking.2.17. WhenrankedfromlowesttohighestthebiddersareG,D,EandF,A,C,H,and.BIndividualsG,D,E,andFbidfor170,000sharesintotal.IndividualAbidforafurther60,000shares.ThepricepaidbytheinvestorsisthereforethepricebidbyA(i.e.,$50).Individual

8、sG,D,E,andFgetthewholeamountofthesharestheybidfor.IndividualAgets40,000shares.2.18.Ifitsucceedsinsellingall10millionsharesinabesteffortsarrangement,itsfeewillbe$2million.Ifitisabletosellthesharesfor$10.20,thiswillalsobeitsprofitinafirmcommitmentarrangement.Thedecisionislikelytohingeona)anestimateoft

9、heprobabilityofsellingthesharesformorethan$10.20andb)theinvestmentbanksappetiteforrisk.Forexample,ifthebankis95%certainthatitwillbeabletosellthesharesformorethan$10.20,itislikelytochooseafirmcommitment.Butifassessestheprobabilityofthistobeonly50%or60%itislikelytochooseabesteffortsarrangement.Chapter

10、 3: InsuranceCompaniesandPensionFunds3.16. (SpreadsheetProvided).Theunconditionalprobabilityofthemandyinginyearsone,two,andthreecanbecalculatedfromTable3.1asfollows:Year1:0.011858Year2:(10.011858)0.0位966=0.012812Year3:(10.011858)(10.012966)0.014123=0.013775Theexpectedpayoutsattimes0.5,1.5,2.5arether

11、efore$59,290.00,$64,061.25,and$68,872.91.Thesehaveapresentvalueof$175,598.60.ThesurvivalprobabilityofthemanisYear0:1Year1:10.011858=0.988142Year2:10.0118580.012812=0.97533Thepresentvalueofthepremiumsreceivedperdollarofpremiumistherefore2.797986.Theminimumpremiumisor$62,77(a) Thelossesinmill

12、ionsofdollarsarenormallydistributedwithmean150andstandarddeviation50.Thepayoutfromthereinsurancecontractisthereforenormallydistributedwithmean90andstandarddeviation30.Assumingthatthereinsurancecompanyfeelsitcandiversifyawaytherisk,theminimumcostofreinsuranceisor$85.71million.(Thisassumesthattheinter

13、estrateiscompoundedannually.)(b) Theprobabilitythatlosseswillbegreaterthan$200millionistheprobabilitythatanormallydistributedvariableisgreaterthanonestandarddeviationabovethemean.Thisis0.1587.Theexpectedpayoffinmillionsofdollarsistherefore0.1587100=15.87andthexvalueofthecontractisor$15.11million.3.1

14、8. Thevalueofabondincreaseswheninterestratesfall.Thevalueofthebondportfolioshouldthereforeincrease.However,alowerdiscountratewillbeusedindeterminingthevalueofthepensionfundliabilities.Thiswillincreasethevalueoftheliabilities.Theneteffectonthepensionplanislikelytobenegative.Thisisbecausetheinterestra

15、tedecreaseaffects100%oftheliabilitiesandonly40%oftheassets.3.19. (SpreadsheetProvided)Thesalaryoftheemployeemakesnodifferencetotheanswer.(Thisisbecauseithastheeffectofscalingallnumbersupordown.)Ifweassumetheinitialsalaryis$100,000andthattherealgrowthrateof2%isannuallycompounded,thefinalsalaryattheen

16、dof45yearsis$239,005.31.ThespreadsheetisusedinconjunctionwithSolvertoshowthattherequiredcontributionrateis25.02%(employeeplusemployer).Thevalueofthecontributiongrowsto$2,420,354.51bytheendofthe45yearworkinglife.(Thisassumesthattherealreturnof1.5%isannuallycompounded.)Thisvaluereducestozerooverthefol

17、lowing18yearsundertheassumptionsmade.ThiscalculationconfirmsthepointmadeinSection3.12thatdefinedbenefitplansrequirehighercontributionratesthatthosethatexistinpractice.Chapter 4: MutualFundsandHedgeFunds4.15. Theinvestorpaystaxondividendsof$200and$300inyear2009and2010,respectively.Theinvestoralsohast

18、opaytaxonrealizedcapitalgainsbythefund.Thismeanstaxwillbepaidoncapitalgainsof$500and$300inyear2009and2010,respectivelyTheresultofallthisisthatthebasisforthesharesincreasesfrom$50to$63.Thesaleat$59inyear2011leadstoacapitallossof$4pershareor$400intotal.4.16. Theinvestorsoverallreturnis1.080.92仅12088-1

19、=0.0207or2.07%forthefouryears.4.17. Theoverallreturnontheinvestmentsistheaverageof5%,1%,10%,15%,and20%or8.2%.Thehedgefundfeesare2%,2.2%,4%,5%,and6%.Theseaverage3.84%.Thereturnsearnedbythefundoffundsafterhedgefundfeesaretherefore7%,1.2%,6%,10%,and14%.Theseaverage4.36%.Thefundoffundsfeeis1%+0.436%or1.

20、436%leaving2.924%fortheinvestor.Thereturnearnedisthereforedividedasshowninthetablebelow.Thisexampleexplainswhyfundsoffundshavedeclinedinpopularity.Returnearnedbyhedgefunds8.200%Feestohedgefunds3.840%Feestofundoffunds1.436%Returntoinvestor2.924%4.18. Theplotisshowninthechartbelow.Ifthehedgefundreturn

21、isnegative,thepensionfundreturnis2%lessthanthehedgefundreturn.Ifitispositive,thepensionfundreturnislessthanthehedgefundreturnby2%plus20%ofthereturn.Chapter 5: FinancialInstruments5.30. Thereisamargincallwhenmorethan$1,000islostfromthemarginaccount.Thishappenswhenthefuturespriceofwheatrisesbymorethan

22、1,000/5,000=0.20.Thereisamargincallwhenthefuturespriceofwheatrisesabove270cents.Anamount,$1,500,canbewithdrawnfromthemarginaccountwhenthefuturespriceofwheatfallsby1,500/5,000=0.30.Thewithdrawalcantakeplacewhenthefuturespricefallsto220cents.5.31. Theinvestmentincalloptionsentailshigherrisksbutcanlead

23、tohigherreturns.Ifthestockpricestaysat$94,aninvestorwhobuyscalloptionsloses$9,400whereasaninvestorwhobuyssharesneithergainsnorlosesanything.Ifthestockpricerisesto$120,theinvestorwhobuyscalloptionsgains2000X12095)9400=$40,600Aninvestorwhobuyssharesgains100演2094)=$2,600Thestrategiesareequallyprofitabl

24、eifthestockpricerisestoaleveSl,where100XS94)=2000(S95)9400orS=100Theoptionstrategyisthereforemoreprofitableifthestockpricerisesabove$100.5.32. SupposeSt is the price of oil at the bonds maturity. In additionStota$n1d0a0rd0Othielbondpays:ST<$25:0$40>ST>$2:170(ST25)ST>$40:2,550Thisisthepay

25、offfrom170calloptionsonoilwithastrikepriceof25lessthepayofffrom170calloptionsonoilwithastrikepriceof40.Thebondisthereforeequivalenttoaregularbondplusalongpositionin170calloptionsonoilwithastrikepriceof$25plusashortpositionin170calloptionsonoilwithastrikepriceof$40.Theinvestorhaswhatistermedabullspre

26、adonoil.5.33. Thearbitrageurcouldborrowmoneytobuy100ouncesofgoldtodayandshortfuturescontractson100ouncesofgoldfordeliveryinoneyear.Thismeansthatgoldispurchasedfor$500perounceandsoldfor$700perounce.Thereturn(40%perannum)isfargreaterthanthe10%costoftheborrowedfunds.Thisissuchaprofitableopportunitythat

27、thearbitrageurshouldbuyasmanyouncesofgoldaspossibleandshortfuturescontractsonthesamenumberofounces.Unfortunately,arbitrageopportunitiesasprofitableasthisrarely,ifever,ariseinpractice.5.34.(a) Byenteringintoathree-yearswapwhereitreceives6.21%andpaysLIBORthecompanyearns5.71%forthreeyears.(b) Byenterin

28、gintoafive-yearswapwhereitreceives6.47%andpaysLIBORthecompanyearns5.97%forfiveyears.(c) Byenteringintoaswapwhereitreceives6.83%andpaysLIBORfortenyearsthecompanyearns6.33%fortenyears.5.35. ThepositionisthesameasaEuropeancalltobuytheassetfKoronthedate.5.36.(a)WhentheCPrateis6.5%andTreasuryratesare6%wi

29、thsemiannualcompounding,theCMT%is6%andanExcelspreadsheetcanbeusedtoshowthatthepriceofa30-yearbondwitha6.25%couponisabout103.46.ThespreadiszeroandtheratepaidbyP&Gis5.75%.(b)WhentheCPrateis7.5%andTreasuryratesare7%withsemiannualcompounding,theCMT%is7%andthepriceofa30-yearbondwitha6.25%couponisabou

30、t90.65.Thespreadisthereforemax0,(98.57/5.7890.65)/100or28.64%.TheratepaidbyP&Gis35.39%.5.37. Thetraderhastoprovide60%ofthepriceofthestockor$2,400.Thereisamargincallwhenthemarginaccountbalanceasapercentofthevalueofthesharesfallsbelow30%.WhenthesharepriceisSthemarginaccountbalanceis2400+200(S20)an

31、dXhevalueofthepositionis200S.Thereisamargincallwhen2400+200(S20)<0.3200Sor140S<1600orS<11.43thatis,whenthestockpriceislessthan$11.43.Chapter 6: HowTradersManageTheirExposures6.15. Withthenotationofthetext,theincreaseinthevalueoftheportfolioisThisis0.55032+254=325Theresultshouldbeanincreasei

32、nthevalueoftheportfolioof$325.6.16. Theprice,delta,gamma,vega,theta,andrhooftheoptionare3.7008,0.6274,0.050,0.1135,0.00596,and0.1512.Whenthestockpriceincreasesto30.1,theoptionpriceincreasesto3.7638.Thechangeintheoptionpriceis3.76383.7008=0.0630.Deltapredictsachangeintheoptionpriceof0.62740.1=0.0627w

33、hichisveryclose.Whenthestockpriceincreasesto30.1,deltaincreasesto0.6324.Thesizeoftheincreaseindeltais0.63240.6274=0.005.Gammapredictsanincreaseof0.0500.1=0.005whichis(tothreedecimalplaces)thesame.Whenthevolatilityincreasesfrom25%to26%,theoptionpriceincreasesby0.1136from3.7008to3.8144.Thisisconsisten

34、twiththevegavalueof0.1135.Whenthetimetomaturityischangedfrom1to11/365theoptionpricereducesby0.006from3.7008to3.6948.Thisisconsistentwithathetaof0.00596.Finallywhentheinterestrateincreasesfrom5%to6%thevalueoftheoptionincreasesby0.1527from3.7008to3.8535.Thisisconsistentwitharhoof0.1512.6.17. Thedeltao

35、ftheportfoliois1,0000505000.802,000(0.40)5000.70=450Thegammaoftheportfoliois1,0002.25000.62,0001.3<5001.8=6,000Thevegaoftheportfoliois1,000%85000.22,0000.7<5001.4=4,000(a) Alongpositionin4,000tradedoptionswillgiveagamma-neutralportfoliosincethelongpositionhasagammaof4,0001.5=+6,000.Thedeltaoft

36、hewholeportfolio(includingtradedoptions)isthen:4,00006450=1,950Hence,inadditiontothe4,000tradedoptions,ashortpositionin£1,950isnecessarysothattheportfolioisbothgammaanddeltaneutral.(b) Alongpositionin5,000tradedoptionswillgiveavega-neutralportfoliosincethelongpositionhasavegaof5,0000.8=+4,000.T

37、hedeltaofthewholeportfolio(includingtradedoptions)isthen5,00006450=2,550Hence,inadditiontothe5,000tradedoptions,ashortpositionin£2,550isnecessarysothattheportfolioisbothvegaanddeltaneutral.6.18. Letw1bethepositioninthefirsttradedoptionandw2bethepositioninthesecondtradedoption.Werequire:6,000=1.

38、5w1+0.5w24,000=0.8w1+0.6w2Thesolutiontotheseequationscaneasilybeseentobwe1=3,200,w2=2,400.Thewholeportfoliothenhasadeltaof450+3,2000.6+2,4000.仅=1,710Thereforetheportfoliocanbemadedelta,gammaandveganeutralbytakingalongpositionin3,200ofthefirsttradedoption,alongpositionin2,400ofthesecondtradedoptionan

39、dashortpositionin£1,710.6.19. (SpreadsheetProvided)Considerthefirstweek.Theportfolioconsistsofashortpositionin100,000optionsandalongpositionin52,200shares.Thevalueoftheoptionchangesfrom$240,053atthebeginningoftheweekto$188,760attheendoftheweekforagainof$51,293.Thevalueoftheshareschangefrom52,20

40、049=$2,557,800to52,20048.12=$2,511,864foralossof$45,936.Thenetgainis51,29345,936=$5,357.Thegammaandtheta(peryear)oftheportfolioare6,554.4and430,533sothatequation(6.2)predictsthegainas430,533/2+0.56,554.4(48.1249)2=5,742Theresultsforall20weeksareshowninthefollowingtable.WeekActualGain($)PredictedGain

41、($)15,3575,74225,6896,093319,74221,08441,9411,57253,7063,65269,3209,19176,2495,93689,4919,25999618701023,38018,992111,6432,497122,6451,3561311,36510,923142,8763,3421512,93612,302167,5668,815173,8802,763186,7646,899194,2955,205204,8044,805Chapter 7: InterestRateRisk7.15. Thebankhasanasset-liabilitymi

42、smatchof$25billion.Theprofitaftertaxiscurrently12%of$2billionor$0.24billion.IfinterestratesrisebyX%thebank'sbefore-taxloss(inbillionsofdollars)is250.01X<=0.25X.Aftertaxesthislossbecomes$0.70.25X=x0.175X.Thebank'sreturnonequitywouldbereducedtozerowhen0.175=0.24orX=1.37.A1.37%riseinrateswou

43、ldthereforereducethereturnonequitytozero.1.17.(a) ThedurationofPortfolioAisSincethisisalsothedurationofPortfolioB,thetwoportfoliosdohavethesameduration.(b) ThevalueofPortfolioAis2000e0.1x+6000e0.1汹0=4,016.95Whenyieldsincreaseby10basispointsitsvaluebecomes2000e0.101x+6000e0.101汹0=3,993.18Thepercentag

44、edecreaseinvalueis23.774,016.95100 = 0.59ThevalueofPortfolioBis5000e0.1-5=2,757.81Whenyieldsincreaseby10basispointsitsvaluebecomes5000e0.101595=2,741.45ThepercentagedecreaseinvalueisThepercentagechangesinthevaluesofthetwoportfoliosfora10basispointincreaseinyieldsarethereforethesame.(c) Whenyieldsinc

45、reaseby5%thevalueofPortfolioAbecomes2000e0.15>1+6000e0.15>10=3,060.20andthevalueofPortfolioBbecomes5000户55.95=2,048.15Thepercentagereductionsinthevaluesofthetwoportfoliosare:PortfolioA:100 = 23.82956.754,016.95PortfolioB:709.662,757.81100 = 25.734.18. ForPortfolioAtheconvexityisForportfolioBth

46、econvexityis5.952or35.4025Thepercentagechangeinthetwoportfoliospredictedbythedurationmeasureisthesameandequalto5.95.05=0.2975or-29.75%.However,theconvexitymeasurepredictsthatthepercentagechangeinthefirstportfoliowillbe5.95 0.05+0.555M00052=0.228andthatforthesecondportfolioitwillbe5.96 0.05=0.535M025

47、0052=0.253Durationdoesnotexplainthedifferencebetweenthepercentagechanges.Convexityexplainspartofthedifference.5%issuchabigshiftintheyieldcurvethateventheuseoftheconvexityrelationshipdoesnotgiveaccurateresults.Betterresultswouldbeobtainedifameasureinvolvingthethirdpartialderivativewithrespecttoaparal

48、lelshift,aswellasthefirstandsecond,wasconsidered.1.18. Theproportionalchangeinthevalueoftheportfolioresultingfromthespecifiedshiftis(2.09e+1.68e+0.67e+0.26e0.55e1.83e)=28.3eTheshiftisthesameasaparallelshiftof6andarotationofe.(Therotationisofthesamemagnitudeasthatconsideredinthetextbutintheoppositedi

49、rection).Thetotaldurationoftheportfoliois0.2andsothepercentagechangeintheportfolioarisingfromtheparallelshiftis0.26e=1.2e.Thepercentagechangeintheportfoliovaluearisingfromtherotationis27e(ThisisthesameasthenumbercalculatedattheendofSection7.6butwiththeoppositesign.)Thetotalpercentagechangeistherefor

50、e28e3ascalculatedfromthepartialdurations.1.19. Thedeltawithrespecttothefirstfactoris0.21-+0.26(3)+0.32(的+0.35240.365+0.367+0.368=7.85Similarly,thedeltaswithrespecttothesecondandthirdfactorsare1.18and1.24,respectively.Therelativeimportanceofthefactorscanbeseenbymultiplyingthefactorexposurebythefactor

51、standarddeviation.Thesecondfactorisabout(1.186.05)/(7.8517M9><=5.2%asimportantasthefirstfactor.Thethirdfactorisabout(1.243.10)/(1.186.05)=53.8%asimportantasthesecondfactor.Chapter 8: ValueatRisk8.12.(a) Alossof$1millionextendsfromthe94percentilepointofthelossdistributiontothe96percentilepoint.

52、The95%VaRistherefore$1million.(b) Theexpectedshortfallforoneoftheinvestmentsistheexpectedlossconditionalthatthelossisinthe5percenttail.Giventhatweareinthetailthereisa20%chancethanthelossis$1millionandan80%chancethatthelossis$10million.Theexpectedlossistherefore$8.2million.Thisistheexpectedshortfall.

53、(c) Foraportfolioconsistingofthetwoinvestmentsthereisa0.040.04=0.0016chance4hatthelossis$20million;thereisa20.04(0:02=>0.0016chancethatthelossis$11million;thereisa20.04>0<94=0.0752chancethatthelossis$9million;thereisa0.020.02=x0.0004chancethatthelossis$2million;thereisa20.20.94=0.0376chance

54、thatthelossiszero;thereisa0.940.94=0港836chancethattheprofitis$2million.Itfollowsthatthe95%VaRis$9million.(d) Theexpectedshortfallfortheportfolioconsistingofthetwoinvestmentsistheexpectedlossconditionalthatthelossisinthe5%tail.Giventhatweareinthetail,thereisa0.0016/0.05=0.032chanceofalossof$20million

55、,a0.0016/0.05=0.032chanceofalossof$11million;anda0.936chanceofalossof$9million.Theexpectedlossistherefore$9.416.(e) VaRdoesnotsatisfythesubadditivityconditionbecause9>1+1.However,expectedshortfalldoesbecause9.416<8.2+8.2.(f) 3.Thecorrectmultiplierforthevarianceis10+290.12+28X0.122+27X:0.123+.+

56、20.129=10.417TheestimateofVaRshouldbeincreasedto.10.417/.10=2.229(g) 4.Inthiscasep=0.01,m=15,n=1000.Kupiecsteststatisticis2ln0.999985X0.0115+2ln(115/1000)985X15/1000)15=2.19Thisislessthan3.84.Weshouldnotthereforerejectthemodel.Chapter 9: Volatility9.18. Thecalculationsareshowninthetablebelow.andanes

57、timateofstandarddeviationofweeklyreturnsis:Thevolatilityperannumistherefore0.0288452=0.2079or20.79%.Thestandarderroroftheestimateor3.9%perannum.Week,iClosingStockPrice($)PriceRelative=S/S-1DailyReturn,ui=ln(S/S-1)130.2232.01.059600.05789331.10.97188-0.02853430.10.96785-0.03268530.21.003320.00332630.

58、31.003310.00331730.61.009900.00985833.01.078430.07551932.90.99697-0.003031033.01.003040.003031133.51.015150.015041233.51.000000.000001333.71.005970.005951433.50.99407-0.005951533.20.99104-0.009009.19.Theproportionalchangeinthepriceofgoldis2/300=0.00667.(a)UsingtheEWMAmodelthevarianceisupdatedto0.940.0132+0.060.006672=0.00016153sothatt