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CHAPTER2HOWTOCALCULATEPRESENTVALUESANSWERSTOPROBLEMSETS1IFTHEDISCOUNTFACTORIS507,THEN507X11261ESTTIME01052DFX139125THEREFORE,DF125/139899ESTTIME01053PV374/109917220ESTTIME01054PV432/115137/1152797/11533761045241,003ESTTIME01055FV100X115830590ESTTIME01056NPV1,548138/091467COSTTODAYPLUSTHEPRESENTVALUEOFTHEPERPETUITYESTTIME01057PV4/140440ESTTIME01058APV1/1010BSINCETHEPERPETUITYWILLBEWORTH10INYEAR7,ANDSINCETHATISROUGHLYDOUBLETHEPRESENTVALUE,THEAPPROXIMATEPVEQUALS5YOUMUSTTAKETHEPRESENTVALUEOFYEARS17ANDSUBTRACTFROMTHETOTALPRESENTVALUEOFTHEPERPETUITYPV1/10/110710/25APPROXIMATELYCAPERPETUITYPAYING1STARTINGNOWWOULDBEWORTH10,WHEREASAPERPETUITYSTARTINGINYEAR8WOULDBEWORTHROUGHLY5THEDIFFERENCEBETWEENTHESECASHFLOWSISTHEREFOREAPPROXIMATELY5PV1055APPROXIMATELYDPVC/RG10,000/1005200,000ESTTIME06109APV10,000/10557,83526ASSUMINGTHECOSTOFTHECARDOESNOTAPPRECIATEOVERTHOSEFIVEYEARSBTHESIXYEARANNUITYFACTOR1/0081/008X10864623YOUNEEDTOSETASIDE12,000SIXYEARANNUITYFACTOR12,000462355,475CATTHEENDOFSIXYEARSYOUWOULDHAVE108660,47655,4757,935ESTTIME061010AFV1,000E12X51,000E61,82212BPV5E12X85E961914MILLIONCPVC1/R1/RERT2,0001/121/12E12X1513,912ESTTIME010511AFV10,000,000X106412,624,770BFV10,000,000X106/124X1212,704,892CFV10,000,000XE4X0612,712,492ESTTIME010512APV100/101109053BPV100/113102946CPV100/12515352DPV100/112100/1122100/112324018ESTTIME010513AR1010501050095R1DFB089R222CAF2DF1DF2090508191724DPVOFANANNUITYCANNUITYFACTORATRFORTYEARSHERE246510AF3AF32465EAF3DF1DF2DF3AF2DF324651724DF3DF30741ESTTIME061014THEPRESENTVALUEOFTHE10YEARSTREAMOFCASHINFLOWSIS68,7391404170,PV0THUSNPV800,000886,7396686,73966ATTHEENDOFFIVEYEARS,THEFACTORYSVALUEWILLBETHEPRESENTVALUEOFTHEFIVEREMAINING170,000CASHFLOWS6583,271404170,PVESTTIME01051523,6915150,268,10,29,1,08,0,7,5,38,2CNPV976543210TTESTTIME010516ALETSTSALARYINYEART301TT1854,PV3760,258030,BPVSALARYX00538,03313FUTUREVALUE38,03313X10830382,71430C38,901080C1R1RCPV220T382,714,0ESTTIME061017PERIODPRESENTVALUE0400,000001100,000/11289,285712200,000/1122159,438783300,000/1123213,53407TOTALNPV62,25856ESTTIME010518WECANBREAKTHISDOWNINTOSEVERALDIFFERENTCASHFLOWS,SUCHTHATTHESUMOFTHESESEPARATECASHFLOWSISTHETOTALCASHFLOWTHEN,THESUMOFTHEPRESENTVALUESOFTHESEPARATECASHFLOWSISTHEPRESENTVALUEOFTHEENTIREPROJECTALLDOLLARFIGURESAREINMILLIONSCOSTOFTHESHIPIS8MILLIONPV8MILLIONREVENUEIS5MILLIONPERYEAR,ANDOPERATINGEXPENSESARE4MILLIONTHUS,OPERATINGCASHFLOWIS1MILLIONPERYEARFOR15YEARSMILONMILON859108015PVMAJORREFITSCOST2MILLIONEACHANDWILLOCCURATTIMEST5ANDT10PV2MILLION/10852MILLION/108102288MILLIONSALEFORSCRAPBRINGSINREVENUEOF15MILLIONATT15PV15MILLION/108150473MILLIONADDINGTHESEPRESENTVALUESGIVESTHEPRESENTVALUEOFTHEENTIREPROJECTNPV8MILLION8559MILLION2288MILLION0473MILLIONNPV1256MILLIONESTTIME061019APV100,000BPV180,000/1125102,13683CPV11,400/01295,000D4107,352201219,0PVEPV6,500/01200592,85714PRIZEDISTHEMOSTVALUABLEBECAUSEITHASTHEHIGHESTPRESENTVALUEESTTIME010520MRBASSETISBUYINGASECURITYWORTH20,000NOW,WHICHISITSPRESENTVALUETHEUNKNOWNISTHEANNUALPAYMENTUSINGTHEPRESENTVALUEOFANANNUITYFORMULA,WEHAVE2,6539010808,2C1,RR1CPV2TESTTIME010521ASSUMETHEZHANGSWILLPUTASIDETHESAMEAMOUNTEACHYEARONEAPPROACHTOSOLVINGTHISPROBLEMISTOFINDTHEPRESENTVALUEOFTHECOSTOFTHEBOATANDTHENEQUATETHATTOTHEPRESENTVALUEOFTHEMONEYSAVEDFROMTHISEQUATION,WECANSOLVEFORTHEAMOUNTTOBEPUTASIDEEACHYEARPVBOAT20,000/110512,418PVSAVINGSANNUALSAVINGS5100BECAUSEPVSAVINGSMUSTEQUALPVBOATANNUALSAVINGS48,2510ANNUALSAVINGS3,760012,485ANOTHERAPPROACHISTOUSETHEFUTUREVALUEOFANANNUITYFORMULA2,5SAVINGAUL10ANNUALSAVINGS3,276ESTTIME061022THEFACTTHATKANGAROOAUTOSISOFFERING“FREECREDIT”TELLSUSWHATTHECASHPAYMENTSAREITDOESNOTCHANGETHEFACTTHATMONEYHASTIMEVALUEA10ANNUALRATEOFINTERESTISEQUIVALENTTOAMONTHLYRATEOF083RMONTHLYRANNUAL/12010/1200083083THEPRESENTVALUEOFTHEPAYMENTSTOKANGAROOAUTOSIS8,93108308131,03ACARFROMTURTLEMOTORSCOSTS9,000CASHTHEREFORE,KANGAROOAUTOSOFFERSTHEBETTERDEAL,IE,THELOWERPRESENTVALUEOFCOSTESTTIME010523THENPVSAREAT517,68058,13,70,NPV2AT104,7,AT1516,0858,13,70,2THEFIGUREBELOWSHOWSTHATTHEPROJECTHASZERONPVATABOUT135ASACHECK,NPVAT135IS17835870,13,70,NPV2ESTTIME061024ATHISISTHEUSUALPERPETUITY,ANDHENCE1,428570RCPVBTHISISWORTHTHEPVOFSTREAMAPLUSTHEIMMEDIATEPAYMENTOF100PV1001,428571,52857CTHECONTINUOUSLYCOMPOUNDEDEQUIVALENTTOA7ANNUALLYCOMPOUNDEDRATEISAPPROXIMATELY677,BECAUSELN10700677ORE0067710700THUS1,47006RCPVNOTETHATTHEPATTERNOFPAYMENTSINPARTBISMOREVALUABLETHANTHEPATTERNOFPAYMENTSINPARTCITISPREFERABLETORECEIVECASHFLOWSATTHESTARTOFEVERYYEARTHANTOSPREADTHERECEIPTOFCASHEVENLYOVERTHEYEARWITHTHEFORMERPATTERNOFPAYMENT,YOURECEIVETHECASHMOREQUICKLYESTTIME061025APV1BILLION/008125BILLIONBPV1BILLION/008004250BILLIONCBILON9811080BILON1PV2DTHECONTINUOUSLYCOMPOUNDEDEQUIVALENTTOAN8ANNUALLYCOMPOUNDEDRATEISAPPROXIMATELY77,BECAUSELN10800770ORE0077010800THUSBILON1023E0711BILONPV072THISRESULTISGREATERTHANTHEANSWERINPARTCBECAUSETHEENDOWMENTISNOWEARNINGINTERESTDURINGTHEENTIREYEARESTTIME061026WITHANNUALCOMPOUNDINGFV100115201,63665WITHCONTINUOUSCOMPOUNDINGFV100E015202,00855ESTTIME010527ONEWAYTOAPPROACHTHISPROBLEMISTOSOLVEFORTHEPRESENTVALUEOF1100PERYEARFOR10YEARS,AND2100PERYEARINPERPETUITY,WITHTHEFIRSTCASHFLOWATYEAR11IFTHISISAFAIRDEAL,THESEPRESENTVALUESMUSTBEEQUAL,ANDTHUSWECANSOLVEFORTHEINTERESTRATERTHEPRESENTVALUEOF100PERYEARFOR10YEARSIS10RR10PVTHEPRESENTVALUE,ASOFYEAR10,OF100PERYEARFOREVER,WITHTHEFIRSTPAYMENTINYEAR11,ISPV10100/RATT0,THEPRESENTVALUEOFPV10ISR10RPVEQUATINGTHESETWOEXPRESSIONSFORPRESENTVALUE,WEHAVER10RR1R100USINGTRIALANDERRORORALGEBRAICSOLUTION,WEFINDTHATR718ESTTIME061028ASSUMETHEAMOUNTINVESTEDISONEDOLLARLETAREPRESENTTHEINVESTMENTAT12,COMPOUNDEDANNUALLYLETBREPRESENTTHEINVESTMENTAT117,COMPOUNDEDSEMIANNUALLYLETCREPRESENTTHEINVESTMENTAT115,COMPOUNDEDCONTINUOUSLYAFTERONEYEARFVA11012111200FVB1100585211204FVC1E0115111219AFTERFIVEYEARSFVA11012517623FVB11005851017657FVC1E0115517771AFTERTWENTYYEARSFVA110122096463FVB11005854097193FVC1E01152099742THEPREFERREDINVESTMENTISCESTTIME061029BECAUSETHECASHFLOWSOCCUREVERYSIXMONTHS,WEFIRSTNEEDTOCALCULATETHEEQUIVALENTSEMIANNUALRATETHUS,1081R/22R7846SEMIANNUALLYCOMPOUNDEDAPRTHEREFORETHERATEFORSIXMONTHSIS7846/2,OR3923147,8603910392110,PVESTTIME061030AEACHINSTALLMENTIS9,420,713/19495,8274,761210808495,27PV9BIFERCISWILLINGTOPAY42MILLION,THEN19RR495,274,20USINGEXCELORAFINANCIALCALCULATOR,WEFINDTHATR981ESTTIME061031A3402,67180870,PVBYEARBEGINNINGOFYEARBALANCEYEARENDINTERESTONBALANCETOTALYEARENDPAYMENTAMORTIZATIONOFLOANENDOFYEARBALANCE1402,2647332,1811870,0000037,81882364,445912364,4459129,1556770,0000040,84433323,601583323,6015825,8881370,0000044,11187279,489714279,4897122,3591870,0000047,64082231,848885231,8488818,5479170,0000051,45209180,396796180,3967914,4317470,0000055,56826124,828547124,828549,9862870,0000060,0137264,81482864,814825,1851970,0000064,81481001ESTTIME061032THISISANANNUITYPROBLEMWITHTHEPRESENTVALUEOFTHEANNUITYEQUALTO2MILLIONASOFYOURRETIREMENTDATE,ANDTHEINTERESTRATEEQUALTO8WITH15TIMEPERIODSTHUS,YOURANNUALLEVELOFEXPENDITURECISDETERMINEDASFOLLOWS23,659108082,C1,0R1RCPV55TWITHANINFLATIONRATEOF4PERYEAR,WEWILLSTILLACCUMULATE2MILLIONASOFOURRETIREMENTDATEHOWEVER,BECAUSEWEWANTTOSPENDACONSTANTAMOUNTPERYEARINREALTERMSR,CONSTANTFORALLT,THENOMINALAMOUNTCTMUSTINCREASEEACHYEARFOREACHYEARTRCT/1INFLATIONRATETTHEREFOREPVALLCTPVALLR1INFLATIONRATET2,000,0002,0081408140R512R0963009273056772,000,000R1123902,000,000R177,952ALTERNATIVELY,CONSIDERTHATTHEREALRATEISTHEN,REDOING038461THESTEPSABOVEUSINGTHEREALRATEGIVESAREALCASHFLOWEQUALTO7,9521038460384612,C15THUSC1177,952104185,070,C2192,473,ETCESTTIME111533A9430,2581050515,PV2BTHEANNUALLYCOMPOUNDEDRATEIS55,SOTHESEMIANNUALRATEIS10551/2100271271SINCETHEPAYMENTSNOWARRIVESIXMONTHSEARLIERTHANPREVIOUSLYPV430,9258910271442,60398ESTTIME061034INTHREEYEARS,THEBALANCEINTHEMUTUALFUNDWILLBEFV1,000,000103531,108,718THEMONTHLYSHORTFALLWILLBE15,0007,5001,5006,000ANNUALWITHDRAWALSFROMTHEMUTUALFUNDWILLBE6,0001272,000ASSUMETHEFIRSTANNUALWITHDRAWALOCCURSTHREEYEARSFROMTODAY,WHENTHEBALANCEINTHEMUTUALFUNDWILLBE1,108,718TREATINGTHEWITHDRAWALSASANANNUITYDUE,WESOLVEFORTASFOLLOWSR1RR1CPVT103503572,0,08TUSINGEXCELORAFINANCIALCALCULATOR,WEFINDTHATT2138YEARSESTTIME061035APV2/1216667MILLIONBPVMILLION931420102CPV2/120322222MILLIONDPVMILLION06182013021ESTTIME061036AFIRSTWEMUSTDETERMINETHE20YEARANNUITYFACTORATA6INTERESTRATE20YEARANNUITYFACTOR1/061/0610620114699ONCEWEHAVETHEANNUITYFACTOR,WECANDETERMINETHEMORTGAGEPAYMENTMORTGAGEPAYMENT200,000/11469917,43691BYEARBEGINNINGBALANCEYEARENDINTERESTTOTALYEARENDPAYMENTAMORTIZATIONOFLOANENDOFYEARBALANCE1200,0000012,0000017,436915,43691194,563092194,5630911,6737917,436915,76313188,799963188,7999611,3280017,436916,10891182,691054182,6910510,9614617,436916,47545176,215605176,2156010,5729417,436916,86398169,351636169,3516310,1611017,436917,27581162,075817162,075819,7245517,436917,71236154,363458154,363459,2618117,436918,17510146,188349146,188348,7713017,436918,66561137,5227310137,522738,2513617,436919,18555128,3371911128,337197,7002317,436919,73668118,6005112118,600517,1160317,4369110,32088108,2796213108,279626,4967817,4369110,9401397,339491497,339495,8403717,4369111,5965485,742951585,742955,1445817,4369112,2923373,450611673,450614,4070417,4369113,0298760,420741760,420743,6252417,4369113,8116746,609071846,609072,7965417,4369114,6403731,968711931,968711,9181217,4369115,5187916,449922016,449929869917,4369116,44992000CNEARLY69OFTHEI
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