answer_question 1_Homework1

answer_question1_Homework1AnswerKeytoProblemSet1Econ121Instructor:ChaoWeiTotal:15pointsGradingguidelinesinGeneral:1.Befair,butbegenerous,givebene?tofdoubtwhenyoucan.2.Clearlymarktheerrors,withcommentswhennecessary.3.Ifthestudentwritesdowncorrectformulas,butthe?nalnumericalansweriswrong,hegetspartialcreditfortheformulas.However,ifsomeonedoesnotwritedownformulas,andthenumericalansweriswrong,hegetsnopartialcredit.4.Givefullcredittostudentswhodonotwritedownformulas,butprovidethecorrectanswer.5.Inquestion3,itispossiblethatonenumericalerrorcanleadtoanothernumericalerror.Forexample,ifthestudentgetsawronganswerforbondprices,hewillgetwronganswersontherateofreturnaswell.Inthecase,onlydeductpointsatoneplace.1.Supposeyouaretheagentforabaseballpitcher.Supposeheiso?eredthefollowingcontractbytheNewYorkYankees:asigningbonusof$3,000,000(tobereceivedimmediately),a?rstyear’ssalaryof$6,000,000(tobereceivedoneyearfromtoday),asecondyear’ssalaryof$7,000,000(tobereceivedtwoyearsfromtoday),andathirdyear’ssalaryof$8,000,000(tobereceivedthreeyearsfromtoday).Supposeheiso?eredthefollowingcontractsbytheSanFranciscoGiants:asigningbonusof$6,000,000,a?rstyear’ssalaryof$5,500,000,asecondyear’ssalaryof$6,000,000,andathirdyear’ssalaryof$6,000,000.Ifyoubelievetheinterestrateis10%,whicho?erwouldyouadvisethepitchertoaccept?Wouldyouradvicechangeifyoubelievedtherelevantdiscountratewere5%?(4points)Ifi=10%,thepresentvalueofYankee’so?eristhefollowing:PVY=3,000,000+6,000,0001.1+7,000,0001.1+8,000,0001.1←?0.5point=20,250,188.←?0.5point1ThepresentvalueofGiant’so?eris:PVG=6,000,000+5,500,0001.1+6,000,0001.1+6,000,0001.1←?0.5point=20,466,566.←?0.5point.So,basedonthepresentvaluecriterion,theGiantsaremakingthemorevaluableo?er.Withaninterestrateof5%,thepresentvalueofYankee’so?eristhefol-lowing:PVY=3,000,000+6,000,000+7,000,000+8,000,000←?0.5point=$21,974,193←?0.5pointThepresentvalueofGiant’so?erisPVG=6,000,000+5,500,000+6,000,000+6,000,000←?0.5point=21,863,298←?0.5pointSoYankee’so?erismorevaluable.2.Supposethatyourroommatewantstoborrowmoneyfromyou.Heo?ersthreerepaymentplans.AccordingtoplanA,youlendhim$800now,andherepaysyou$1000intwoyears.AccordingtoplanB,youlendhim$800now,andherepaysyou$500inoneyear,andanother$500intwoyears.AccordingtoPlanC,youlendhim$800now,andherepaysyou$900inoneyear.Whichplandoyouprefer?(orequivalently,whichplangivesyouthehighestyieldtomaturity?)(3points,1pointforeachplan)ForplanA,800=1000(1+i)2=?i=11.80%ForplanB,800=500(1+i)+500(1+i)=?i=16.26%ForplanC,800=900=?i=12.50%(7points)21.(a)DenotethepriceofconsolandtheyieldtomaturityatthebeginningoftheyearrespectivelyasPtandit,anddenotetheannualcouponpaymentasC:(1point)Pt=Ct=2000.2=1000DenotethepriceofconsolandtheyieldtomaturityattheendoftheyearrespectivelyasPt+1andit+1,anddenotetheannualcouponpaymentasC:(1point)Pt+1=Cit+1=2000.1=2021Thereturnfortheyear,Rt+1,canbewrittenas(1points.Intheequationsbelow,eitherthe?rstlineorthesecondlinequali?esfor0.5point,the?nalanswerquali?esfor0.5point.)Rt+1=CPt+Pt+1?PtPt=200+2021?1000=120%.(b)Thepriceofthecouponbondatthebeginningoftheyear,Pt,isequaltoitsfacevalue,1000.DenotethepriceofthecouponbondandtheyieldtomaturityattheendoftheyearrespectivelyasPt+1andit+1,anddenotetheannualcouponpaymentasC.Notethatattheendoftheyear,thecouponbondwillmatureinanotheryear.WecomputePt+1asthepresentvalueofthelastcouponpaymentandprincipal,bothdueatmatu-rity.(1points.Intheequationsbelow,eitherthe?rstlineorthesecondlinequali?esfor0.5points,the?nalanswer3quali?esfor0.5point.)Pt+1=Ct+1+Ft+1=200+1000≈1091Thereturnfortheyear,Rt+1,canbewrittenas(1points.Intheequationsbelow,eitherthe?rstlineorthesecondlinequal-i?esfor0.5points,the?nalanswerquali?esfor0.5point.)Rt+1=CPt+Pt+1?PtPt=200+1091?1000=29.1%(c)Thepriceofthecouponbondatthebeginningoftheyear,Pt,isequaltoitsfacevalue,1000.DenotethepriceofthecouponbondandtheyieldtomaturityattheendoftheyearrespectivelyasPt+1andit+1,anddenotetheannualcouponpaymentasC.Notethatattheendoftheyear,thecouponbondwillmatureinanothertwoyears.WecomputePt+1asthepresentvalueofthelasttwocouponpayments,duerespectivelyoneyearandtwoyearslaterandprincipaldueatmaturity.(1points.Intheequationsbelow,eitherthe?rstlineorthesecondlinequali?esfor0.5points,the?nalanswerquali?esfor0.5point.)Pt+1=Ct+1+C(1+it+1)2+F(1+it+1)2=200+200(1+0.1)2+1000(1+0.1)2≈1173.6Thereturnfortheyear,Rt+1,canbewrittenas(1points.Intheequationsbelow,eitherthe?rstlineorthesecondlinequal-i?esfor0.5points,the?nalanswerquali?esfor0.5point.4)Rt+1=Ct+Pt+1?Ptt=2001000+1173.6?10001000=37.4%4.Iftheinterestrateis10%,whatisthepresentvalueofasecuritythatpaysyou$1,100nextyear,$1,210theyearafter,and$1,331theyearafterthat?(1point)Thepresentvalueofthesecurityis:P=1100+1210(1+10%)2+1331(1+10%)3=3000.5