投资学10版习题答案15.docx

投资学10版习题答案15.docx

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投资学10版习题答案15

CHAPTER15:

THETERMSTRUCTUREOFINTERESTRATES

PROBLEMSETS.

1.Ingeneral,theforwardratecanbeviewedasthesumofthemarket’sexpectationofthefutureshortrateplusapotentialrisk（orliquidity）premium.Accordingtotheexpectationstheoryofthetermstructureofinterestrates,theliquiditypremiumiszerosothattheforwardrateisequaltothemarket’sexpectationofthefutureshortrate.Therefore,themarket’sexpectationoffutureshortrates（i.e.,forwardrates）canbederivedfromtheyieldcurve,andthereisnoriskpremiumforlongermaturities.

Theliquiditypreferencetheory,ontheotherhand,specifiesthattheliquiditypremiumispositivesothattheforwardrateisgreaterthanthemarket’sexpectationofthefutureshortrate.Thiscouldresultinanupwardslopingtermstructureevenifthemarketdoesnotanticipateanincreaseininterestrates.Theliquiditypreferencetheoryisbasedontheassumptionthatthefinancialmarketsaredominatedbyshort-terminvestorswhodemandapremiuminordertobeinducedtoinvestinlongmaturitysecurities.

2.True.Undertheexpectationshypothesis,therearenoriskpremiabuiltintobondprices.Theonlyreasonforlong-termyieldstoexceedshort-termyieldsisanexpectationofhighershort-termratesinthefuture.

3.Uncertain.Expectationsoflowerinflationwillusuallyleadtolowernominalinterestrates.Nevertheless,iftheliquiditypremiumissufficientlygreat,long-termyieldsmayexceedshort-termyieldsdespiteexpectationsoffallingshortrates.

4.Theliquiditytheoryholdsthatinvestorsdemandapremiumtocompensatethemforinterestrateexposureandthepremiumincreaseswithmaturity.Addthispremiumtoaflatcurveandtheresultisanupwardslopingyieldcurve.

5.Thepureexpectationstheory,alsoreferredtoastheunbiasedexpectationstheory,purportsthatforwardratesaresolelyafunctionofexpectedfuturespotrates.Underthepureexpectationstheory,ayieldcurvethatisupward（downward）sloping,meansthatshort-termratesareexpectedtorise（fall）.Aflatyieldcurveimpliesthatthemarketexpectsshort-termratestoremainconstant.

6.Theyieldcurveslopesupwardbecauseshort-termratesarelowerthanlong-termrates.Sincemarketratesaredeterminedbysupplyanddemand,itfollowsthatinvestors（demandside）expectratestobehigherinthefuturethaninthenear-term.

7.

Maturity

Price

YTM

ForwardRate

1

$943.40

6.00%

2

$898.47

5.50%

（1.0552/1.06）–1=5.0%

3

$847.62

5.67%

（1.05673/1.0552）–1=6.0%

4

$792.16

6.00%

（1.064/1.05673）–1=7.0%

8.Theexpectedpricepathofthe4-yearzerocouponbondisshownbelow.（Notethatwediscountthefacevaluebytheappropriatesequenceofforwardratesimpliedbythisyear’syieldcurve.）

BeginningofYear

ExpectedPrice

ExpectedRateofReturn

1

$792.16

（$839.69/$792.16）–1=6.00%

2

（$881.68/$839.69）–1=5.00%

3

（$934.58/$881.68）–1=6.00%

4

（$1,000.00/$934.58）–1=7.00%

9.Ifexpectationstheoryholds,thentheforwardrateequalstheshortrate,andtheone-yearinterestratethreeyearsfromnowwouldbe

10.a.A3-yearzerocouponbondwithfacevalue$100willselltodayatayieldof6%andapriceof:

$100/1.063=$83.96

Nextyear,thebondwillhaveatwo-yearmaturity,andthereforeayieldof6%（fromnextyear’sforecastedyieldcurve）.Thepricewillbe$89,resultinginaholdingperiodreturnof6%.

b.Theforwardratesbasedontoday’syieldcurveareasfollows:

Year

ForwardRate

2

（1.052/1.04）–1=6.01%

3

（1.063/1.052）–1=8.03%

Usingtheforwardrates,theforecastfortheyieldcurvenextyearis:

Maturity

YTM

1

6.01%

2

（1.0601×1.0803）1/2–1=7.02%

ThemarketforecastisforahigherYTMon2-yearbondsthanyourforecast.Thus,themarketpredictsalowerpriceandhigherrateofreturn.

11.a.

b.Theyieldtomaturityisthesolutionforyinthefollowingequation:

[Usingafinancialcalculator,entern=2;FV=100;PMT=9;PV=–101.86;Computei]YTM=7.958%

c.Theforwardratefornextyear,derivedfromthezero-couponyieldcurve,isthesolutionforf2inthefollowingequation:

f2=0.0901=9.01%.

Therefore,usinganexpectedratefornextyearofr2=9.01%,wefindthattheforecastbondpriceis:

d.Iftheliquiditypremiumis1%thentheforecastinterestrateis:

E（r2）=f2–liquiditypremium=9.01%–1.00%=8.01%

Theforecastofthebondpriceis:

12.a.Thecurrentbondpriceis:

（$85×0.94340）+（$85×0.87352）+（$1,085×0.81637）=$1,040.20

Thispriceimpliesayieldtomaturityof6.97%,asshownbythefollowing:

[$85×Annuityfactor（6.97%,3）]+[$1,000×PVfactor（6.97%,3）]=$1,040.17

b.Ifoneyearfromnowy=8%,thenthebondpricewillbe:

[$85×Annuityfactor（8%,2）]+[$1,000×PVfactor（8%,2）]=$1,008.92

Theholdingperiodrateofreturnis:

[$85+（$1,008.92–$1,040.20）]/$1,040.20=0.0516=5.16%

13.

Year

ForwardRate

PVof$1receivedatperiodend

1

5%

$1/1.05=$0.9524

2

7

1/（1.051.07）=$0.8901

3

8

1/（1.051.071.08）=$0.8241

a.Price=（$60×0.9524）+（$60×0.8901）+（$1,060×0.8241）=$984.14

b.Tofindtheyieldtomaturity,solveforyinthefollowingequation:

$984.10=[$60×Annuityfactor（y,3）]+[$1,000×PVfactor（y,3）]

Thiscanbesolvedusingafinancialcalculatortoshowthaty=6.60%:

PV=-$984.10;N=3;FV=$1,000;PMT=$60.SolveforI=6.60%.

c.

Period

PaymentReceivedatEndofPeriod:

WillGrowby

aFactorof:

ToaFuture

Valueof:

1

$60.00

1.071.08

$69.34

2

60.00

1.08

64.80

3

1,060.00

1.00

1,060.00

$1,194.14

$984.10（1+yrealized）3=$1,194.14

1+yrealized=

yrealized=6.66%

Alternatively,PV=-$984.10;N=3;FV=$1,194.14;PMT=$0.SolveforI=6.66%.

d.Nextyear,thepriceofthebondwillbe:

[$60×Annuityfactor（7%,2）]+[$1,000×PVfactor（7%,2）]=$981.92

Therefore,therewillbeacapitallossequalto:

$984.10–$981.92=$2.18

Theholdingperiodreturnis:

14.a.Thereturnontheone-yearzero-couponbondwillbe6.1%.

Thepriceofthe4-yearzerotodayis:

$1,000/1.0644=$780.25

Nextyear,iftheyieldcurveisunchanged,today’s4-yearzerocouponbondwillhavea3-yearmaturity,aYTMof6.3%,andthereforethepricewillbe:

$1,000/1.0633=$832.53

Theresultingone-yearrateofreturnwillbe:

6.70%

Therefore,inthiscase,thelonger-termbondisexpectedtoprovidethehigherreturnbecauseitsYTMisexpectedtodeclineduringtheholdingperiod.

b.Ifyoubelieveintheexpectationshypothesis,youwouldnotexpectthattheyieldcurvenextyearwillbethesameastoday’scurve.Theupwardslopeintoday'scurvewouldbeevidencethatexpectedshortratesarerisingandthattheyieldcurvewillshiftupward,reducingtheholdingperiodreturnonthefour-yearbond.Undertheexpectationshypothesis,allbondshaveequalexpectedholdingperiodreturns.Therefore,youwouldpredictthattheHPRforthe4-yearbondwouldbe6.1%,thesameasforthe1-yearbond.

15.Thepriceofthecouponbond,basedonitsyieldtomaturity,is:

[$120×Annuityfactor（5.8%,2）]+[$1,000×PVfactor（5.8%,2）]=$1,113.99

Ifthecouponswerestrippedandsoldseparatelyaszeros,then,basedontheyieldtomaturityofzeroswithmaturitiesofoneandtwoyears,respectively,thecouponpaymentscouldbesoldseparatelyfor:

Thearbitragestrategyistobuyzeroswithfacevaluesof$120and$1,120,andrespectivematuritiesofoneyearandtwoyears,andsimultaneouslysellthecouponbond.Theprofitequals$2.91oneachbond.

16.a.Theone-yearzero-couponbondhasayieldtomaturityof6%,asshownbelow:

y1=0.06000=6.000%

Theyieldonthetwo-yearzerois8.472%,asshownbelow:

y2=0.08472=8.472%

Thepriceofthecouponbondis:

Therefore:

yieldtomaturityforthecouponbond=8.333%

[Onafinancialcalculator,enter:

n=2;PV=–106.51;FV=100;PMT=12]

b.

c.Expectedprice

（Notethatnextyear,thecouponbondwillhaveonepaymentleft.）

Expectedholdingperiodreturn=

Thisholdingperiodreturnisthesameasthereturnontheone-yearzero.

d.Ifthereisaliquiditypremium,then:

E（r2）

E（Price）=

E（HPR）>6%

17.a.Weobtainforwardratesfromthefollowingtable:

Maturity

YTM

ForwardRate

Price（forpartsc,d）

1year

10%

$1,000/1.10=$909.09

2years

11%

（1.112/1.10）–1=12.01%

$1,000/1.112=$811.62

3years

12%

（1.123/1.112）–1=14.03%

$1,000/1.123=$711.78

b.Weobtainnextyear’spricesandyieldsbydiscountingeachzero’sfacevalueattheforwardratesfornextyearthatwederivedinpart（a）:

Maturity

Price

YTM

1year

$1,000/1.1201=$892.78

12.01%

2years

$1,000/（1.1201×1.1403）=$782.93

13.02%

Notethatthisyear’supwardslopingyieldcurveimplies,accordingtotheexpectationshypothesis,ashiftupwardinnextyear’scurve.

c.Nextyear,the2-yearzerowillbea1-yearzero,andwillthereforesellatapriceof:

$1,000/1.1201=$892.78

Similarly,thecurrent3-yearzerowillbea2-yearzeroandwillsellfor:

$782.93

Expectedtotalrateofreturn:

2-yearbond:

3-yearbond:

d.Thecurrentpriceofthebondshouldequalthevalueofeachpaymenttimesthepresentvalueof$1tobereceivedatthe“maturity”ofthatpayment.Thepresentvalueschedulecanbetakendirectlyfromthepricesofzero-couponbondscalculatedabove.

Currentprice=（$120×0.90909）+（$120×0.81162）+（$1,120×0.71178）

=$109.0908+$97.3944+$797.1936=$1,003.68

Similarly,theexpectedpricesofzerosoneyearfromnowcanbeused