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CHAPTER6INTERNATIONALPARITYRELATIONSHIPS
SUGGESTEDANSWERSANDSOLUTIONSTOEND-OF-CHAPTER
QUESTIONSANDPROBLEMS
QUESTIONS
1.Giveafulldefinitionofarbitrage.
Answer:Arbitragecanbedefinedastheactofsimultaneouslybuyingandsellingthesameorequivalentassetsorcommoditiesforthepurposeofmakingcertain,guaranteedprofits.
2.Discusstheimplicationsoftheinterestrateparityfortheexchangeratedetermination.
Answer:Assumingthattheforwardexchangerateisroughlyanunbiasedpredictorofthefuturespotrate,IRPcanbewrittenas:
S=[(1+I)/(1+I)]E[SI].
£$t+1t
Theexchangerateisthusdeterminedbytherelativeinterestrates,andtheexpectedfuturespotrate,conditionalonalltheavailableinformation,I,asofthepresenttime.Onethuscansaythatexpectationis
t
self-fulfilling.Sincetheinformationsetwillbecontinuouslyupdatedasnewshitthemarket,theexchangeratewillexhibitahighlydynamic,randombehavior.
3.Explaintheconditionsunderwhichtheforwardexchangeratewillbeanunbiasedpredictorofthefuturespotexchangerate.
Answer:Theforwardexchangeratewillbeanunbiasedpredictorofthefuturespotrateif(I)theriskpremiumisinsignificantand(ii)foreignexchangemarketsareinformationallyefficient.
4.Explainthepurchasingpowerparity,boththeabsoluteandrelativeversions.Whatcausesthedeviationsfromthepurchasingpowerparity?
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--
Answer:Theabsoluteversionofpurchasingpowerparity(PPP):
S=P/P
$£.
Therelativeversionis:
e=-.
$£
PPPcanbeviolatediftherearebarrierstointernationaltradeorifpeopleindifferentcountrieshavedifferentconsumptiontaste.PPPisthelawofonepriceappliedtoastandardconsumptionbasket.
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--
8.Explaintherandomwalkmodelforexchangerateforecasting.Canitbeconsistentwiththetechnicalanalysis?
Answer:Therandomwalkmodelpredictsthatthecurrentexchangeratewillbethebestpredictorofthefutureexchangerate.Animplicationofthemodelisthatpasthistoryoftheexchangerateisofnovalueinpredictingfutureexchangerate.Themodelthusisinconsistentwiththetechnicalanalysiswhichtriestoutilizepasthistoryinpredictingthefutureexchangerate.
*9.Deriveandexplainthemonetaryapproachtoexchangeratedetermination.
Answer:ThemonetaryapproachisassociatedwiththeChicagoSchoolofEconomics.Itisbasedontwotenets:purchasingpowerparityandthequantitytheoryofmoney.Combingthesetwotheoriesallowsforstating,say,the$/£spotexchangerateas:
S($/£)=(M/M)(V/V)(y/y),
$£$££$
whereMdenotesthemoneysupply,Vthevelocityofmoney,andythenationalaggregateoutput.Thetheoryholdsthatwhatmattersinexchangeratedeterminationare:
Therelativemoneysupply,
Therelativevelocitiesofmonies,and
Therelativenationaloutputs.
10.CFAquestion:1997,Level3.
A.Explainthefollowingthreeconceptsofpurchasingpowerparity(PPP):
Thelawofoneprice.
AbsolutePPP.
RelativePPP.
B.EvaluatetheusefulnessofrelativePPPinpredictingmovementsinforeignexchangerateson:
Short-termbasis(forexample,threemonths)
Long-termbasis(forexample,sixyears)
Answer:
A.a.Thelawofoneprice(LOP)referstotheinternationalarbitrageconditionforthestandardconsumptionbasket.LOPrequiresthattheconsumptionbasketshouldbesellingforthesamepriceinagivencurrencyacrosscountries.
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--
A.b.AbsolutePPPholdsthatthepricelevelinacountryisequaltothepricelevelinanothercountrytimestheexchangeratebetweenthetwocountries.
A.c.RelativePPPholdsthattherateofexchangeratechangebetweenapairofcountriesisaboutequal
tothedifferenceininflationratesofthetwocountries.
B.a.PPPisnotusefulforpredictingexchangeratesontheshort-termbasismainlybecause
internationalcommodityarbitrageisatime-consumingprocess.
B.b.PPPisusefulforpredictingexchangeratesonthelong-termbasis.
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PROBLEMS
1.SupposethatthetreasurerofIBMhasanextracashreserveof$100,000,000toinvestforsixmonths.Thesix-monthinterestrateis8percentperannumintheUnitedStatesand6percentperannuminGermany.Currently,thespotexchangerateis€1.01perdollarandthesix-monthforwardexchangerateis€0.99perdollar.ThetreasurerofIBMdoesnotwishtobearanyexchangerisk.Whereshouldhe/sheinvesttomaximizethereturn?
Themarketconditionsaresummarizedasfollows:
I=4%;i
$
€
=3.5%;S=€1.01/$;F=€0.99/$.
If$100,000,000isinvestedintheU.S.,thematurityvalueinsixmonthswillbe
$104,000,000=$100,000,000(1+.04).
Alternatively,$100,000,000canbeconvertedintoeurosandinvestedattheGermaninterestrate,withtheeuromaturityvaluesoldforward.Inthiscasethedollarmaturityvaluewillbe
$105,590,909=($100,000,000x1.01)(1+.035)(1/0.99)
Clearly,itisbettertoinvest$100,000,000inGermanywithexchangeriskhedging.
2.WhileyouwerevisitingLondon,youpurchasedaJaguarfor£35,000,payableinthreemonths.YouhaveenoughcashatyourbankinNewYorkCity,whichpays0.35%interestpermonth,compoundingmonthly,topayforthecar.Currently,thespotexchangerateis$1.45/£andthethree-monthforwardexchangerateis$1.40/£.InLondon,themoneymarketinterestrateis2.0%forathree-monthinvestment.TherearetwoalternativewaysofpayingforyourJaguar.
KeepthefundsatyourbankintheU.S.andbuy£35,000forward.
BuyacertainpoundamountspottodayandinvesttheamountintheU.K.forthreemonthssothatthematurityvaluebecomesequalto£35,000.
Evaluateeachpaymentmethod.Whichmethodwouldyouprefer?Why?
Solution:Theproblemsituationissummarizedasfollows:
A/P=£35,000payableinthreemonths
i=0.35%/month,compoundingmonthly
NY
i=2.0%forthreemonths
LD
S=$1.45/£;F=$1.40/£.
--
--
Optiona:
Whenyoubuy£35,000forward,youwillneed$49,000inthreemonthstofulfilltheforwardcontract.Thepresentvalueof$49,000iscomputedasfollows:
$49,000/(1.0035)3
=$48,489.
Thus,thecostofJaguarasoftodayis$48,489.
Optionb:
Thepresentvalueof£35,000is£34,314=£35,000/(1.02).Tobuy£34,314today,itwillcost$49,755=34,314x1.45.ThusthecostofJaguarasoftodayis$49,755.
Youshoulddefinitelychoosetouse“optiona”,andsave$1,266,whichisthedifferencebetween$49,755and$48489.
3.Currently,thespotexchangerateis$1.50/£andthethree-monthforwardexchangerateis$1.52/£Thethree-monthinterestrateis8.0%perannumintheU.S.and5.8%perannumintheU.K.Assumethatyoucanborrowasmuchas$1,500,000or£1,000,000.
Determinewhethertheinterestrateparityiscurrentlyholding.
IftheIRPisnotholding,howwouldyoucarryoutcoveredinterestarbitrage?Showallthestepsanddeterminethearbitrageprofit.
ExplainhowtheIRPwillberestoredasaresultofcoveredarbitrageactivities.
Solution:Let’ssummarizethegivendatafirst:
S=$1.5/£;F=$1.52/£;I=2.0%;I=1.45%
$£
Credit=$1,500,000or£1,000,000.
a.(1+I)=1.02
$
(1+I)(F/S)=(1.0145)(1.52/1.50)=1.0280
£
Thus,IRPisnotholdingexactly.
b.(1)Borrow$1,500,000;repaymentwillbe$1,530,000.
Buy£1,000,000spotusing$1,500,000.
Invest£1,000,000atthepoundinterestrateof1.45%;
maturityvaluewillbe£1,014,500.
(4)Sell£1,014,500forwardfor$1,542,040
Arbitrageprofitwillbe$12,040
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--
c.Followingthearbitragetransactionsdescribedabove,
Thedollarinterestratewillrise;
Thepoundinterestratewillfall;
Thespotexchangeratewillrise;
Theforwardexchangeratewillfall.
TheseadjustmentswillcontinueuntilIRPholds.
4.Supposethatthecurrentspotexchangerateis€0.80/$andthethree-monthforwardexchangerateis€0.7813/$.Thethree-monthinterestrateis5.6percentperannumintheUnitedStatesand5.40percentperannuminFrance.Assumethatyoucanborrowupto$1,000,000or€800,000.
Showhowtorealizeacertainprofitviacoveredinterestarbitrage,assumingthatyouwanttorealizeprofitintermsofU.S.dollars.Alsodeterminethesizeofyourarbitrageprofit.
Assumethatyouwanttorealizeprofitintermsofeuros.Showthecoveredarbitrageprocessanddeterminethearbitrageprofitineuros.
Solution:
a.(1+i)=1.014<(F/S)(1+i
$
makearbitrageprofit.
€
)=1.053.Thus,onehastoborrowdollarsandinvestineurosto
Borrow$1,000,000andrepay$1,014,000inthreemonths.
Sell$1,000,000spotfor€1,060,000.
Invest€1,060,000attheeurointerestrateof1.35%forthreemonthsandreceive€1,074,310atmaturity.
Sell€1,074,310forwardfor$1,053,245.
Arbitrageprofit=$1,053,245-$1,014,000=$39,245.
b.Followthefirstthreestepsabove.Butthelaststep,involvingexchangeriskhedging,willbedifferent.
5.Buy$1,014,000forwardfor€1,034,280.
Arbitrageprofit=€1,074,310-€1,034,280=€40,030
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--
5.IntheissueofOctober23,1999,theEconomistreportsthattheinterestrateperannumis5.93%inthe
UnitedStatesand70.0%inTurkey.WhydoyouthinktheinterestrateissohighinTurkey?Basedonthereportedinterestrates,howwouldyoupredictthechangeoftheexchangeratebetweentheU.S.dollar
andtheTurkishlira?
Solution:AhighTurkishinterestratemustreflectahighexpectedinflationinTurkey.AccordingtointernationalFishereffect(IFE),wehave
E(e)=i-i
$Lira
=5.93%-70.0%=-64.07%
TheTurkishlirathusisexpectedtodepreciateagainsttheU.S.dollarbyabout64%.
6.AsofNovember1,1999,theexchangeratebetweentheBrazilianrealandU.S.dollarisR$1.95/$.TheconsensusforecastfortheU.S.andBrazilinflationratesforthenext1-yearperiodis2.6%and20.0%,respectively.HowwouldyouforecasttheexchangeratetobeataroundNovember1,2000?
Solution:SincetheinflationrateisquitehighinBrazil,wemayusethepurchasingpowerparitytoforecasttheexchangerate.
E(e)=E()-E()
$R$
=2.6%-20.0%
=-17.4%
E(S)=S(1+E(e))
To
=(R$1.95/$)(1+0.174)
=R$2.29/$
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--
7.(CFAquestion)OmniAdvisors,aninternationalpensionfundmanager,usestheconceptsofpurchasingpowerparity(PPP)andtheInternationalFisherEffect(IFE)toforecastspotexchangerates.Omnigathersthefinancialinformationasfollows:
asepricelevel
urrentU.S.pricelevel105CurrentSouthAfricanpricelevel111
Baserandspotexchangerate$0.175Currentrandspotexchangerate$0.158ExpectedannualU.S.inflation7%ExpectedannualSouthAfricaninflation5%
ExpectedU.S.one-yearinterestrate10%ExpectedSouthAfricanone-yearinterestrate8%
100
Calculatethefollowingexchangerates(ZARandUSDrefertotheSouthAfricanandU.S.dollar,respectively).
ThecurrentZARspotrateinUSDthatwouldhavebeenforecastbyPPP.
UsingtheIFE,theexpectedZARspotrateinUSDoneyearfromnow.
UsingPPP,theexpectedZARspotrateinUSDfouryearsfromnow.
Solution:
ZARspotrateunderPPP=[1.05/1.11](0.175)=$0.1655/rand.
ExpectedZARspotrate=[1.10/1.08](0.158)=$0.1609/rand.
c.ExpectedZARunderPPP=[(1.07)
4/(1.05)
4](0.158)=$0.1704/rand.
8.Supposethatthecurrentspotexchangerateis€1.50/₤andtheone-yearforwardexchangerateis€1.60/₤.Theone-yearinterestrateis5.4%ineurosand5.2%inpounds.Youcanborrowatmost€1,000,000ortheequivalentpoundamount,i.e.,₤666,667,atthecurrentspotexchangerate.
Showhowyoucanrealizeaguaranteedprofitfromcoveredinterestarbitrage.Assumethatyouareaeuro-basedinvestor.Alsodeterminethesizeofthearbitrageprofit.
Discusshowtheinterestrateparitymayberestoredasaresultoftheabove
transactions.
c.Supposeyouareapound-basedinvestor.Showthecoveredarbitrageprocessand
determinethepoundprofitamount.
--
--
Solution:
a.First,notethat(1+i)=1.054islessthan(F/S)(1+i)=(1.60/1.50)(1.052)=1.1221.
€€
Youshouldthusborrowineurosandlendinpounds.
Borrow€1,000,000andpromisetorepay€1,054,000inoneyear.
Buy₤666,667spotfor€1,000,000.
Invest₤666,667atthepoundinterestrateof5.2%;thematurityvaluewillbe₤701,334.
Tohedgeexchangerisk,sellthematurityvalue₤701,334forwardinexchangefor€1,122,134.Thearbitrageprofitwillbethedifferencebetween€1,122,134and€1,054,000,i.e.,€68,134.
b.Asaresultoftheabovearbitragetransactions,theeurointerestratewillrise,thepound
interestratewillfall.Inaddition,thespotexchangerate(eurosperpound)willriseandtheforwardratewillfall.Theseadjustmentswillcontinueuntiltheinterestrateparityisrestored.
c.Thepound-basedinvestorwillcarryoutthesametransactions1),2),and3)ina.Buttohedge,he/shewillbuy€1,054,000forwardinexchangefor₤658,750.Thearbitrageprofitwillthenbe₤42,584=₤701,334-₤658,750.
9.Duetotheintegratednatureoftheircapitalmarkets,investorsinboththeU.S.andU.K.requirethesamerealinterestrate,2.5%,ontheirlending.Thereisaconsensusincapitalmarketsthattheannualinflationrateislikelytobe3.5%intheU.S.and1.5%intheU.K.forthenextthreeyears.Thespotexchangerateiscurrently$1.50/£
ComputethenominalinterestrateperannuminboththeU.S.andU.K.,assumingthattheFishereffectholds.
Whatisyourexpectedfuturespotdollar-poundexchangerateinthreeyearsfromnow?
Canyouinfertheforwarddollar-poundexchangerateforone-yearmaturity?
Solution.
a.NominalrateinUS=(1+ρ)(1+E(π))–1=(1.025)(1.035)–1=0.0609or6.09%.
$
NominalrateinUK=(1+ρ)(1+E(π))–1=(1.025)(1.015)–1=0.0404or4.04%.
₤
b.E(S)=[(1.0609)3/(1.0404)
T
3](1.50)=$1.5904/₤.
c.F=[1.0609/1.0404](1.50)=$1.5296/₤.
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--
MiniCase:TurkishLiraandthePurchasingPowerParity
VeritasEmergingMarketFundspecializesininvestinginemergingstockmarketsoftheworld.Mr.HenryMobaus,anexperiencedhandininternationalinvestmentandyourboss,iscurrentlyinterestedinTurkishstockmarkets.HethinksthatTurkeywilleventuallybeinvitedtonegotiateitsmembershipintheEuropeanUnion.Ifthishappens,itwillboostthestockpricesinTurkey.But,atthesametime,heisquiteconcernedwiththevolatileexchangeratesoftheTurkishcurrency.HewouldliketounderstandwhatdrivestheTurkishexchangerates.SincetheinflationrateismuchhigherinTurkeythanintheU.S.,hethinksthatthepurchasingpowerparitymaybeholdingatleasttosomeextent.Asaresearchassistantforhim,youwereassignedtocheckthisout.Inotherwords,youhavetostudyandprepareareportonthefollowingquestion:DoesthepurchasingpowerparityholdfortheTurkishlira-U.S.dollarexchangerate?Amongotherthings,Mr.Mobauswouldlikeyoutodothefollowing:
Plotthepastexchangeratechangesagainstthedifferentialinflationratesbetween
TurkeyandtheU.S.forthelastfouryears.
Regresstherateofexchangeratechangesontheinflationratedifferentialtoestimate
theinterceptandtheslopecoefficient,andinterprettheregressionresults.
Datasource:YoumaydownloadtheconsumerpriceindexdatafortheU.S.andTurkeyfromthefollowingwebsite:,“hotfile”(Excelformat).Youmaydownloadtheexchangeratedatafromthewebsite:.
Solution:
a.Inthecurrentsolution,weusethemonthlydatafromJanuary1999–December2002.
--
)
/
f
ˆ
f
t
ˆ
ˆ
--
Turkeyvs.U.S.
e
(
$
L
T
o
e
g
n
a
h
C
0.15
0.10
0.05
β1.47
o
e
a
R
0.00
-0.05
αˆ0.011
-0.0500.050.10.15
Inf_Turkey-Inf_US
b.Weregressexchangeratechanges(e)ontheinflationratedifferentialandestimatethe
intercept(α)andslopecoefficient(β):
eαˆβ(Inf_Turkey-Inf_US)ε
tt
αˆ0.011(t-0.649)
β1.472(t3.095)
Theestimatedinterceptisinsignificantlydifferentfromzero,whereastheslopecoefficientispositiveandsignificantlydifferentfromzero.
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