Math3340,Fall2017 Homework5answers JinLauQuestion1Currentyield=!""#!$ !"#$%$&# !"#$%&'

!"##$%& !"#$ !"#$%

Giventhatcurrentyieldis2%(APR),annualinterestpayment=18(2)=$36,Wehave,0.02= !"

!"##$%& !"#$ !"#$%

Pricesheoffertobuythisbond= !"!.!"

=180/$1800Then,theCapitalgainyield=(!"""

!)!!-1

Plugint=7years,=(!"""

!"##)!!–1

=-0.080540722So,Yieldtomaturity=Currentyield+Capitalgainyield=0.02+(-0,080540722)≈-0.0605/-6.05%Question2(a)NOTE:For(a)therearetwosetsofvalueswecoulduse.Forinstancein(i),Wecouldeitherusen=6,r=0.03,y=0.02ORn=12,r=0.015,y=0.01i.For2.0%yieldperyear,3%U.STreasurybondwith6yearstomaturity,Priceof1USTreasurybond:P=100(𝑟 !! !!! !!

!+ (1+ 𝑦)!!)

P=100((0.03) !! !!!.!" !!

!.!"+ (1+ 0.02)!!)

=100(1.056014309)=105.6014309/$1056.01So,Priceof10suchbonds=10(105.6014309)=1056.0143309/$10560.14ii.For2.2%yieldperyear,3%U.STreasurybondwith6yearstomaturity,Priceof1USTreasurybond:P=100(𝑟 !! !!! !!

!+ (1+ 𝑦)!!)

P=100((0.03) !! !!!.!"" !!

!.!""+ (1+ 0.022)!!)

=100(1.044510553)=104.4510553/$1044.51So,Priceof10suchbonds=10(104.4510553)=1044.510553/$10445.11iii.For2.5%yieldperyear,3%U.STreasurybondwith6yearstomaturity,Priceof1USTreasurybond:P=100(𝑟 !! !!! !!

!+ (1+ 𝑦)!!)

P=100((0.03) !! !!!.!"# !!

!.!"#+ (1+ 0.025)!!)

=100(1.027540627)=102.7540627/$1027.54So,Priceof10suchbonds=10(102.769828)=1027.540627/$10275.41(b)i.Currentyield= !""#!$ !"#$%$&# !"#$%&'

!"##$%& !"#$ !" !"#$% !"#$!!"#"

=(!.!")(!""")(!" !"#$)!"#$".!"

=0.028408714ii.Currentyield= !""#!$ !"#$%$&# !"#$%&'

!"##$%& !"#$ !" !"#$% !"#$!!"#"

=(!.!")(!""")(!" !"#$)!"##$.!!

=0.028721574iii.Currentyield= !""#!$ !"!"#"$! !"#$%&'

!"##$%& !"#$ !" !"#$% !"#$!!"#"

=(!.!")(!""")(!" !"#$)!"#$%.!"

=0.029195915(c)Forallthreecases:InterestpaymentstheTreasuryactuallypaytobondholdereachyear=(0.03)(100)(10bonds)=30/$300Question3UseFormulaytm= (!""

!)!! − 1+ !!

!"!

GivenP=104fora3%bondwith3yearstomaturity,T=3years,2I=0.03(1000)=30Yieldtomaturity= (!""

!"#)!! − 1+ !"

!"(!"#)

=0.987011517–1+0.028846154=0.015857671/1.59%

Toderivethemaximumpriceofthebond,welettheyieldtomaturity=0So,wehavetheEquation:0=(!""

!)!! − 1+ !!

!"!

(!""!)!! − 1+ !"

!"! = 0

Further simplifying, We get the equation: 𝑃!-109𝑃!+27P-27 Then,usingtheBisectionmethodwithinitialinterval(100,200).AfternumerousiterationsofthebisectionmethodinExcel,We have Maximum Price of the Bond = 108.752441 ≈108.75 Question 4 (i)Discountfactorformula: !

(!!!!)!! , Formulafor𝑆! =ln(1+𝑦!)

For,𝑡!=0.5and𝑦!=0.0125Discountfactor= !

(!!!.!"#$)!.!=0.99380799

Andspotrate,𝑆! =ln(1+0.0125)=0.01242252For,𝑡!=1.0and𝑦!=0.014DiscountFactor= !

!!!.!"#

=0.986193294Andspotrate,𝑆! =ln(1+0.014)=0.013902905For,𝑡!=1.5and𝑦!=0.015DiscountFactor= !

(!!!.!"#)!.!

=0.977914615Andspotrate,𝑆! =ln(1+0.015)=0.014888612For,𝑡!=2.0and𝑦!=0.02DiscountFactor= !

(!!!.!")!

=0.961168781Andspotrate,𝑆! =ln(1+0.02)=0.019802627For,𝑡!=2.5and𝑦!=0.022DiscountFactor= !

(!!!.!"")!.!

=0.947049677Andspotrate,𝑆! =ln(1+0.022)

=0.021761492For,𝑡!=3.0and𝑦!=0.025DiscountFactor= !

(!!!.!"#)!

=0.928599411Andspotrate,𝑆! =ln(1+0.025)=0.024692613(ii)Fora4%bondpayinginterestannuallywith3yearstomaturity,0$40$40$(1000+40)| | | |0 1 23PresentValueofthesecashflowswiththeyieldcurveprovided= !"(!!!.!"#)!

+ !"(!!!.!")!

+ !"""!!"(!!!.!"#)!

=1043.63787≈1043.64(iii)Fora4%bondpayinginterestsemiannuallywith3yearstomaturity,GraphofCashFlowpayments:0$20$20$20$20$20$(1000+20)| | | | | | |0 0.5 11.522.53PresentValueofthesecashflowswiththeyieldcurveprovided= !"

(!!!.!"#$)!!+ !"

(!!!.!"#)!+ !"

(!!!.!"#)!!+ !"(!!!.!")!

+ !"

(!!!.!"")!!+ !"""!!"

(!!!.!!")!

=19.8761598+19.72386588+19.5582923+19.22337562+18.94099355+947.171.3991=1044.494086≈1044.49Theanswerin(iii)isgreaterthantheanswerin(ii).Thisisbecausetheyieldcurvegivesasmallerdiscountfactorfordiscountingsemiannuallyincomparisontodiscountingannually.Thusthepresentvalueofthecashflowsforthesemiannualbond(iii)shouldbegreater.(iv)Use:P=100(𝑟 !! !!! !!

!+ (1+ 𝑦)!!)

SinceP=!"##.!"!"

=104.45forthe4%bondthatpaysinsemiannuallywith3yearstomaturity,

EquationforInternalRateofReturn:104.45=100((0.02) !! !!! !!

!+ (1+ 𝑦)!!)

OrifweletX=(1+y)andmultiplebothsidesoftheequationby𝑋!(X-1)ThentheEquationfortheInternalRateofReturnwouldbe:1.0445𝑋! − 1.0645𝑋! − 𝑋 + 1.02 = 0