《金融学第二版》讲义大纲及课后习题答案详解 第五章Word文档格式.docx
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《金融学第二版》讲义大纲及课后习题答案详解 第五章Word文档格式.docx
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5.5ShouldYouBuyorRent?
Summary
∙Inmakinglifetimesaving/consumptiondecisions:
(1)Dotheanalysisinrealterms(constantdollars)tosimplifythecalculationsandtoavoidhavingtoforecastinflation.
(2)Startbycomputingthepresentvalueofyourlifetimeresources.Thepresentvalueofyourlifetimespendingcannotexceedthisamount.
∙Socialsecurityoranyotherforcedsavingprogramwilloffsetvoluntarysaving.Itmayhaveapositiveoranegativeeffectonthepresentvalueofyourtotallifetimeresources.
∙Tax-deferredretirementaccountsareadvantageousbecausetheyallowyoutoearnabefore-taxrateofreturnuntilmoneyiswithdrawnfromtheaccount.Theyareadvantageousifyouareinthesametaxbracketbeforeandafteryouretire,andevenmoresoifyourtaxbracketislowerafteryouretire.
∙Gettingaprofessionaldegreecanbeevaluatedasaninvestmentinhumancapital.Assuch,itshouldbeundertakenifthepresentvalueofthebenefits(suchasincreaseinyourearnings)exceedsthepresentvalueofthecosts(suchastuitionandforgonesalary.)
∙Indecidingwhethertobuyorrentanapartmentoraconsumerdurable,choosethealternativewiththelowerpresentvalueofcosts
SolutionstoProblemsatEndofChapter
SavingforRetirement
1.Assumethatyouare40yearsoldandwishtoretireatage65.Youexpecttobeabletoaveragea6%annualrateofinterestonyoursavingsoveryourlifetime(bothpriortoretirementandafterretirement).Youwouldliketosaveenoughmoneytoprovide$8,000peryearbeginningatage66inretirementincometosupplementothersources(socialsecurity,pensionplans,etc.).Supposeyoudecidethattheextraincomeneedstobeprovidedforonly15years(uptoage80).AssumethatyourfirstcontributiontothesavingsplanwilltakeplaceoneyearfromNOW.
a.Howmuchmustyousaveeachyearbetweennowandretirementtoachieveyourgoal?
b.Iftherateofinflationturnsouttobe6%peryearbetweennowandretirement,howmuchwillyourfirst$8000withdrawalbeworthintermsoftoday’spurchasingpower?
SOLUTION:
a.Age4041656680
Time01252640
....
XX8,0008,000
Itisa2partcomputation.Firstcomputetheamountneededatage65tofinancethe$8,000peryearannuityfor15yearsstartingatage66:
n
i
PV
FV
PMT
Result
15
6
?
8000
PV=$77,697.99
Thencomputetheannuityrequiredfor25yearstoaccumulateto$77,697.99byage65:
25
77697.99
PMT=$1,416.18
b.Ataninflationrateof6%peryearthefirst$8000withdrawn26yearsfromnowwillbeworth:
26
PV=$1,758.48
intermsoftoday’spurchasingpower.
2.
Youaresavingforretirementandyoucomeacrossthefollowingtable.Itshowsthepercentageofyourcurrentsalarythatyoushouldsaveforyourretirementinordertoretirewithanannuityequalto70%ofyoursalaryifyouhavenotyetsavedanything.Itassumesthatyourannualsalarywillremainconstantinrealtermsuntilretirement,andthatyouwilllivefor25yearsafterretiring.Forinstance,ifyouhave35yearsleftbeforeyouretireandearn3.5%peryearonyourinvestments,thenyoushouldsave17.3%ofyourcurrentsalary.
TableA.SavingRateNeededtoAchieve70%ReplacementRate
RealInterestrate
Yearstoretirement
3.5%peryear
59.79%
35
17.30%
a.FillinthemissingnumberinTableA.
Themethodforcomputinghowmuchsavingisneededtoreachthedesiredtarget(70%)consistsoftwosteps:
Firstcomputetheamountyouneedtohaveaccumulatedinyourpersonalretirementaccountwhenyoureachtheretirementage.(We’lldothecalculationsaspercentagesofrealsalary)
3.5
.7
PV=11.537
Thencomputetheannualamountofsavingneededtoreachthatfuturevalue
11.537
PMT=.5979
PMT=.2962
PMT=.1730
Hence,you’llneedtosave
59.79%ifyouhave15yearstoretire
29.62%ifyouhave25yearstoretire
17.30%ifyouhave35yearstoretire
b.NowfillinTableB.Itassumesthatinsteadoftargetinga70%replacementrateofpreretirementincome,yourgoalistomaintainthesamelevelofconsumptionspendingbothbeforeandafterretirement.
TableB.SavingtoMaintainConstantLifetimeConsumptionSpending
RealInterestRate
1
PV=16.482
16.482
PMT=.8541
PMT=.4232
PMT=.2472
85.41%ifyouhave15yearstoretire
42.31%ifyouhave25yearstoretire
24.72%ifyouhave35yearstoretire
3.Youaresavingforretirementandyoucomeacrossthefollowingthetable.Itshowstheincreaseintheannualbenefityoucanreceiveinretirementperdollarthatyouincreaseyourannualretirementsavingintheyearsbeforeretirement.Itassumesthatyouwilllivefor20yearsafterretiring.Forinstance,ifyouhave30yearsleftbeforeyouretireandearnaninterestrateof3%peryear,thenyouwillobtainanincreaseof$3.20inyourannualretirementbenefitforevery$1peryearincreaseinannualsaving.Fillinthemissingtablevalues.
Interestrate
20
30
0.0%
$1.00
$1.25
$1.50
0.5%
$1.10
$1.40
$1.70
1.0%
$1.22
$1.57
1.5%
$1.35
$1.75
$2.19
2.0%
$1.49
$1.96
$2.48
3.0%
$1.81
$3.20
3.5%
$2.74
$3.63
4.0%
$3.06
$4.13
4.5%
$2.41
$3.43
$4.69
5.0%
$2.65
$3.83
Firstcalculatethefuturevalueofeveryadditional$1yousaveforeachofthemissingvaluesinthetable:
FV=$34.7849
3
FV=$36.4593
FV=$28.2797
5
FV=$66.4388
Thencalculatetheincreaseintheannualbenefityou’llreceivebycalculatingthecorrespondingannuities:
$34.7849
PMT=$1.93
$36.4593
PMT=$2.45
$28.2797
PMT=$1.99
$66.4388
PMT=$5.33
4.GeorgeThriftlessis45yearsold,earns$50,000peryear,andexpectsthathisfutureearningswillkeeppacewithinflation,butwillnotexceedinflation.Hehasnotyetsavedanythingtowardshisretirement.Hiscompanydoesnotofferanypensionplan.GeorgepaysSocialSecuritytaxesequalto7.5%ofhissalary,andheassumesthatwhenheretiresatage65,hewillreceive$12,000peryearininflation-adjustedSocialSecuritybenefitsfortherestofhislife.Hislifeexpectancyisage85.
GeorgebuysabookonretirementplanningthatrecommendssavingenoughsothatwhenprivatesavingsandSocialSecurityarecombined,hecanreplace80%ofhispreretirementsalary.Georgebuysafinancialcalculatorandgoesthroughthefollowingcalculations:
First,hecomputestheamounthewillneedtoreceiveineachyearofretirementtoreplace80%ofhissalary:
.8x$50,000=$40,000.
Sinceheexpectstoreceive$12,000peryearinSocialSecuritybenefits,hecalculatesthathewillhavetoprovidetheother$28,000peryearfromhisownretirementfund.
Usingthe8%interestrateonlong-termdefault-freebonds,Georgecomputestheamounthewillneedtohaveatage65as$274,908(thepresentvalueof$28,000for20yearsat8%peryear).Thenhecomputestheamounthewillhavetosaveineachofthenext20yearstoreachthatfutureaccumulationas$6,007(theannualpaymentthatwillproduceafuturevalueof$274,908ataninterestrateof8%peryear).Georgefeelsprettyconfidentthathecansave12%ofhissalary(i.e.,$6,007/$50,000)inordertoinsureacomfortableretirement.
a.Iftheexpectedlong-termrealinterestrateis3%peryear,approximatelywhatisthelong-termexpectedrateofinflation?
b.HasGeorgecorrectlytakenaccountofinflationinhiscalculations?
Ifnot,howwouldyoucorrecthim?
c.HowmuchshouldGeorgesaveineachofthenext20years(untilage65)ifhewantstomaintainaconstantlevelofconsumptionovertheremaining40yearsofhislife(fromage45toage85)?
Ignoreincometaxes.
a.Thelong-termexpectedrateofinflationcanbeapproximatedbysubtractingtheexpectedrealrateofinterest(3%peryear)fromthelong-termnominalinterestrate(8%peryear).Theexpectedrateofinflationisthereforeapproximately5%peryear.
Theexactvaluecanbecalculatedusing:
1+nom=(1+real)(1+inf)
hence,theinflationrateis4.854%peryear.
b.Georgehasusedthenominalinterestratetodiscountrealcashflows.Asaresult,hehasseriouslyunderestimatedhowmuchsavinghemustdotoachievean80%replacementrate.Thetimelineforthisproblemis:
Age4546656685
Time01202140
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