期权期货与其他衍生产品第九版课后习题与答案Chapter.docx
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期权期货与其他衍生产品第九版课后习题与答案Chapter
CHAPTER24
CreditRisk
PracticeQuestions
Problem24.1.
Thespreadbetweentheyieldonathree-yearcorporatebondandtheyieldonasimilarrisk-freebondis50basispoints.Therecoveryrateis30%.Estimatetheaveragehazardrateperyearoverthethree-yearperiod.
Fromequation（24.2）theaveragehazardrateoverthethreeyearsis00050（103）00071or0.71%peryear.
Problem24.2.
SupposethatinProblem24.1thespreadbetweentheyieldonafive-yearbondissuedbythesamecompanyandtheyieldonasimilarrisk-freebondis60basispoints.Assumethesamerecoveryrateof30%.Estimatetheaveragehazardrateperyearoverthefive-yearperiod.
Whatdoyourresultsindicateabouttheaveragehazardrateinyears4and5?
Fromequation（24.2）theaveragehazardrateoverthefiveyearsis00060（103）00086or0.86%peryear.Usingtheresultsinthepreviousquestion,thehazardrateis0.71%peryearforthefirstthreeyearsand
000865000713
2
or1.07%peryearinyears4and5.
Problem24.3.
00107
Shouldresearchersusereal-worldorrisk-neutraldefaultprobabilitiesfora）calculatingcreditvalueatriskandb）adjustingthepriceofaderivativefordefaults?
Real-worldprobabilitiesofdefaultshouldbeusedforcalculatingcreditvalueatrisk.Risk-neutralprobabilitiesofdefaultshouldbeusedforadjustingthepriceofaderivativefordefault.
Problem24.4.
Howarerecoveryratesusuallydefined?
herecoveryrateforabondisthevalueofthebondimmediatelyaftertheissuerdefaultsasapercentofitsfacevalue.
Problem24.5.
Explainthedifferencebetweenanunconditionaldefaultprobabilitydensityandahazardrate.
Thehazardrate,h（t）
attime
tisdefinedsothath（t）
t
istheprobabilityofdefault
betweentimest
and
t
t
conditionalonnodefaultpriortotimet.Theunconditional
defaultprobabilitydensity
q（t）isdefinedsothatq（t）
t
istheprobabilityofdefault
betweentimest
and
t
t
asseenattimezero.
Problem24.6.
Verifya）thatthenumbersinthesecondcolumnofTable24.3areconsistentwiththenumbersinTable24.1andb）thatthenumbersinthefourthcolumnofTable24.4areconsistentwiththenumbersinTable24.3andarecoveryrateof40%.
ThefirstnumberinthesecondcolumnofTable24.3iscalculatedas
1ln（1000245）00003504
7
or0.04%peryearwhenrounded.Othernumbersinthecolumnarecalculatedsimilarly.ThenumbersinthefourthcolumnofTable24.4arethenumbersinthesecondcolumnofTable24.3multipliedbyoneminustheexpectedrecoveryrate.Inthiscasetheexpectedrecoveryrateis0.4.
Problem24.7.
Describehownettingworks.Abankalreadyhasonetransactionwithacounterpartyonitsbooks.Explainwhyanewtransactionbyabankwithacounterpartycanhavetheeffectof
increasingorreducingthebank’exposurecredittothecounterparty.
SupposecompanyAgoesbankruptwhenithasanumberofoutstandingcontractswithcompanyB.NettingmeansthatthecontractswithapositivevaluetoAarenettedagainstthosewithanegativevalueinordertodeterminehowmuch,ifanything,companyAowes
companyB.CompanyAisnotallowedto“cherrypick”bykeeping-valuethepositive
contractsanddefaultingonthenegative-valuecontracts.
Thenewtransactionwillincreasethebank’sexposuretothecounterpartyifthecontract
tendstohaveapositivevaluewhenevertheexistingcontracthasapositivevalueandanegativevaluewhenevertheexistingcontracthasanegativevalue.However,ifthenewtransactiontendstooffsettheexistingtransaction,itislikelytohavetheincrementaleffectofreducingcreditrisk.
Problem24.8.
“DVAcanimprovethebottomlinewhenabankisexperiencingfinancialdifficulty.”Explainwhythisstatementistrue.
Whenabankisexperiencingfinancialdifficulties,itscreditspreadislikelytoincrease.Thisincreasesqi*andDVAincreases.Thisisabenefittothebank:
thefactthatitismorelikelytodefaultmeansthatitsderivativesareworthless.
Problem24.9.
ExplainthedifferencebetweentheGaussiancopulamodelforthetimetodefaultandCreditMetricsasfarasthefollowingareconcerned:
a）thedefinitionofacreditlossandb）thewayinwhichdefaultcorrelationismodeled.
（a）IntheGaussiancopulamodelfortimetodefaultacreditlossisrecognizedonlywhenadefaultoccurs.InCreditMetricsitisrecognizedwhenthereisacreditdowngradeaswellaswhenthereisadefault.
（b）IntheGaussiancopulamodeloftimetodefault,thedefaultcorrelationarisesbecausethevalueofthefactorM.Thisdefinesthedefaultenvironmentoraveragedefault
rateintheeconomy.InCreditMetricsacopulamodelisappliedtocreditratingsmigrationandthisdeterminesthejointprobabilityofparticularchangesinthecreditratingsoftwocompanies.
Problem24.10.
SupposethattheLIBOR/swapcurveisflatat6%withcontinuouscompoundingandafive-yearbondwithacouponof5%（paidsemiannually）sellsfor90.00.Howwouldanassetswaponthebondbestructured?
Whatistheassetswapspreadthatwouldbecalculatedinthissituation?
Supposethattheprincipalis$100.Theassetswapisstructuredsothatthe$10ispaidinitially.Afterthat$2.50ispaideverysixmonths.InreturnLIBORplusaspreadisreceivedontheprincipalof$100.Thepresentvalueofthefixedpaymentsis
1025e0060525e0061卐250065100e00651053579
ThespreadoverLIBORmustthereforehaveapresentvalueof5.3579.Thepresentvalueof$1receivedeverysixmonthsforfiveyearsis8.5105.Thespreadreceivedeverysixmonthsmustthereforebe5357985105$06296.Theassetswapspreadistherefore
20629612592%perannum.
Problem24.11.
Showthatthevalueofacoupon-bearingcorporatebondisthesumofthevaluesofitsconstituentzero-couponbondswhentheamountclaimedintheeventofdefaultistheno-defaultvalueofthebond,butthatthisisnotsowhentheclaimamountisthefacevalueofthebondplusaccruedinterest.
Whentheclaimamountistheno-defaultvalue,thelossforacorporatebondarisingfromadefaultattimetis
?
v（t）（1R）B
wherev（t）isthediscountfactorfortimetandBistheno-defaultvalueofthebondattimet.Supposethatthezero-couponbondscomprisingthecorporatebondhaveno-default
valuesattimetofZ1,Z2,
Zn,respectively.Thelossfromtheithzero-couponbond
arisingfromadefaultattime
tis
?
v（t）（1
R）Zi
Thetotallossfromallthezero-couponbondsis
n
垐
v（t）（1
R）B
v（t）（1R）Zi
i
Thisshowsthatthelossarisingfromadefaultattimet
isthesameforthecorporatebond
asfortheportfolioofitsconstituentzero-couponbonds.Itfollowsthatthevalueofthecorporatebondisthesameasthevalueofitsconstituentzero-couponbonds.
Whentheclaimamountisthefacevalueplusaccruedinterest,thelossforacorporatebondarisingfromadefaultattimetis
?
v（t）Bv（t）R[La（t）]
whereListhefacevalueanda（t）istheaccruedinterestattimet.Ingeneralthisisnotthesameasthelossfromthesumofthelossesontheconstituentzero-couponbonds.
Problem24.12.
Afour-yearcorporatebondprovidesacouponof4%peryearpayablesemiannuallyandhasayieldof5%expressedwithcontinuouscompounding.Therisk-freeyieldcurveisflatat3%withcontinuouscompounding.Assumethatdefaultscantakeplaceattheendofeachyear（immediatelybeforeacouponorprincipalpaymentandtherecoveryrateis30%.Estimatetherisk-neutraldefaultprobabilityontheassumptionthatitisthesameeachyear.
DefineQastherisk-freerate.Thecalculationsareasfollows
Time
Def.
Recovery
Risk-free
LossGiven
Discount
PVofExpected
（yrs）
Prob.
Amount（$）
Value（$）
Default（$）
Factor
Loss（$）
1.0
Q
30
104.78
74.78
0.9704
7257Q
2.0
Q
30
103.88
73.88
0.9418
6958Q
3.0
Q
30
102.96
72.96
0.9139
6668Q
4.0
Q
30
102.00
72.00
0.8869
6386Q
Total
27269Q
Thebondpaysacouponof2everysixmonthsandhasacontinuouslycompoundedyieldof5%peryear.Itsmarketpriceis96.19.Therisk-freevalueofthebondisobtainedbydiscountingthepromisedcashflowsat3%.Itis103.66.Thetotallossfromdefaultsshould
thereforebeequatedto10366
9619
746.ThevalueofQimpliedbythebondpriceis
thereforegivenby27269Q
746.or
Q00274.Theimpliedprobabilityofdefaultis
2.74%peryear.
Problem24.13.
Acompanyhasissued3-and5-yearbondswithacouponof4%perannumpayableannually.Theyieldsonthebonds（expressedwithcontinuouscompounding）are4.5%and4.75%,respectively.Risk-freeratesare3.5%withcontinuouscompoundingforallmaturities.Therecoveryrateis40%.Defaultscantakeplacehalfwaythrougheachyear.Therisk-neutraldefaultratesperyearareQ1foryears1to3andQ2foryears4and5.EstimateQ1and
Q2.
Thetableforthefirstbondis
Time
Def.
Recovery
Risk-free
LossGiven
Discount
PVof
（yrs）
Prob.
Amount（$）
Value（$）
Default（$）
Factor
Expected
Loss（$）
0.5
Q1
40
103.01
63.01
0.9827
6192Q1
1.5
Q1
40
102.61
62.61
0.9489
59
41Q1
2.5
Q1
40
102.20
62.20
0.9162
56
98Q1
Total
17831Q1
Themarketpriceofthebondis98.35andtherisk-freevalueis101.23.ItfollowsthatQ1isgivenby
17831Q1101239835
sothatQ100161.
Thetableforthesecondbondis
Time
Def.
Recovery
Risk-free
Loss
Discount
PVofExpected
（yrs）
Prob.
Amount
Value（$）
Given
Factor
Loss（$）
（$）
Default
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