理财基础-时间价值

货币的时间价值货币时间价值现值与终值年金和永续年金净现值与内部回报率货币的时间价值在金融理财中的应用与时间价值有关的术语时间轴PV即现值，也即今天的价值FV即终值，也即未来某个时间点的价值t表示终值和现值之间的这段时间r表示利率所有的定价问题都与PV、FV、t、r这四个变量有关，确定其中三个即能得出第四个。...0123tPVFV单期中的终值假设利率为5％，你准备拿出1万元进行投资，一年后，你将得到10,500元。¥500 利息收入(¥10,000×5%)¥10,000

本金投入(¥10,000×1)¥10,500 全部收入，算式为：¥10,500=¥10,000×(1+5%).

投资结束时获得的价值被称为终值（FV）单期中的终值单期中终值计算公式为：FV=PV×(1+r)

其中，PV是第0期的现金流，r是利率。FV=¥10,500年度01PV=¥10,000¥10,000×

1.05PV×(1+r)单期中的现值假设利率为5％，你想保证自己通过一年的投资得到1万元，那么你的投资在当前应该为9,523.81元要得到一年后1万元，在当前所必须的资金价值被称为现值（PV）：¥10,000=¥9,523.81×(1+5%)单期中的现值单期中现值的计算公式为：其中，FV是在1时期的现金流，r是利率。FV=¥10,000年度01PV=¥9,523.81¥10,000/1.05FV/(1+r)多期中的终值计算多期中的终值公式：FV=PV×(1+r)T其中， PV是第0期的价值,r是利率,T

是投资时间。案例假设年利率为12%，今天投入5,000元6年后你将获得多少钱？用单利计算是怎样的？用复利计算是怎样的？利率为12%时,用复利计算是: ¥5000(1+r)t

= ¥5000(1+12%)6 = ¥9869.11利率为12%时,用单利计算是：

¥5000(1+t

r)=¥5000(1+612%)

=¥8600复利和单利计算之间的差异即为：¥9869.11-¥8600=¥1269.11终值利率因子（复利终值系数）我们注意到¥110=¥100(1+10%)¥121=¥110(1+10%)=¥100(1+10%)(1+10%)=¥100(1+10%)2¥133.10 = ¥121(1+10%)=¥100(1+10%)(1+10%)(1+10%) = ¥100(1+10%)3一般说来，经过t时期后，今天投入的1元的终值将是FVt

=¥1(1+r)t(1+r)t

是终值利率因子(FVIF),也称为复利终值系数多期中的现值假如利率是15％，你想在5年后获得2万元，你需要在今天拿出多少钱进行投资?012345¥20,000PV已知终值（2万），利率（8%)，投资时间（三年）那么现值可以这样得到：

FVt =PVx(1+r)t

¥20,000=PVx(1+8%)3 PV =¥20,000/(1+8%)3 =¥15,876.64

因此我们得到：年利率为r时，要计算t时期价值1元的投资的现值，可以用以下公式：

PV=1/(1+r

)t它被称为现值利率因子(PVIF)，也称为复利现值系数现值利率因子（复利现值系数）假设你三年后需要2万元来支付研究生的学费，投资收益率是8%，今天你需要拿出多少钱来投资？期限不同，利率不同时1元的现值如何变化？多期中的终值假设刘先生购买了九州龙腾公司首次公开发售时的股票。该公司的当前分红为每股1.10元，并预计能在未来5年中以每年40％的速度增长。问：5年后的股利为多少?FV=PV×(1+r)T¥5.92=¥1.10×(1+40%)5复利对终值的影响012345Q.假如你买彩票中奖100万，将其存为10年期，年利率为6％的定期存款，按复利计算。或者，你将其交给表兄打理，10年中，每年按6％的单利计算。10年后，哪种方式获利多？“利滚利”演示

A.定期存款的终值是1,000,000x(1+6%)10=1,790,847.70

从表兄那里获得的终值是

1,000,000+1,000,000x

6%x10=¥1,600,000.00

复利（利滚利）引起的是将近191,000

元的资产增值。(当然我们还没有考虑到将这笔钱交给你表兄打理的风险）年利率为10％时的终值年度 年初值单利复利引起的利息增加总利息终值

1¥100.00¥10.00¥0.00¥10.00¥110.00 2110.00 10.001.0011.00121.00 3121.00 10.002.1012.10133.10 4133.1010.003.3113.31146.41 5146.4110.004.6414.64161.05

总计¥50.00¥11.05¥61.05

例题确定变量:

FV

=¥1,000,000

r=10％

t=65-21=44年 PV=?代入终值算式中并求解现值：

¥1,000,000=PV(1+10%)44

PV=¥1,000,000/(1+10%)44=

¥15,091.当然我们忽略了税收和其他的复杂部分，但是现在你需要的是筹集15000元！假如你现在21岁，每年收益率10％，要想在65岁时成为百万富翁，今天你要一次性拿出多少钱来投资？案例：确定利率富兰克林死于1790年。他在自己的遗嘱中写道，他将分别向波士顿和费城捐赠1000元。捐款将于他死后200年赠出。1990年时，付给费城的捐款已经变成200万，而给波士顿的已达到450万。请问两者的年投资回报率各为多少？对于费城，有以下算式:

¥1,000=¥2,000,000/(1+r

)200 (1+r

)200=2,000.00

求解r，得到年投资回报率为3.87%.同理我们可以得到波士顿的年投资回报率为4.3%.72法则如果年利率为r%,你的投资将在大约72/r年后翻番。

例如，如果年收益率为6％，你的投资将于约12年后翻番。为什么要说“大约”？因为如果利率过高或过低，该法则不再适用。 假设r=72%FVIF(72,1)=1.7200,而非2.00

假设r=36% FVIF(36,2)=1.8496，而非2.00

可见，该法则只是一个近似估计。例题计算如下：

FV=10,000 PV=5,000 t=10 PV = FVt/(1+r)t ¥5000 = ¥10,000/(1+r)10求解r: (1+r)10=¥10,000/¥5,000=2.00

r=(2.00)1/10-1=0.0718=7.18%假设你现在拿出5,000元投资于一个年收益率为r的产品。10年后你将得到10,000元，那么r为多少？例题：普通股票的长期回报率据研究，1802－1997年间普通股票的年均收益率是8.4%.假设你的祖先在1802年对一个充分分散风险的投资组合进行了1000元的投资。1997年的时候，这个投资的价值是多少？1998年普通股票价值增长了28.59%,那么上述投资组合在1998年的价值是多少？t=195r=8.4%,FVIF(8.4,195)=6,771,892.09695所以该投资的价值应为:¥6,771,892,096.95!1998年末的投资价值为¥6,771,892,096.95(1+28.59%)=¥8,707,976,047.47!例题1.下列哪些说法是对的？如果r和t都大于0，终值利率因子FVIF(r,t)永远都大于0.如果r和t都大于0，现值利率因子PVIF(r,t)永远都大于0.2.判断题:对于既定的r和t，PVIF(r,t)是FVIF(r,t)的倒数.3.其他条件都不变，对于一个现金流来说，贴现率越高，其现值越高还是越低？两个说法都正确。正确.PVIF(r,t)=1/FVIF(r,t)越低。对同一个现金流来说，贴现率越高，其现值越低。例题

现值 = ¥15,000

终值 = ¥200,000

t=18r=?200,000 = 15,000FVIF(r,18) FVIF(r,18) =200,000/15,000=13.333...

解得：r ＝15.48%假设你的子女在18年后将接受大学教育，届时需要学费20万元。你现在有15,000元可以用于投资，问你需要怎样的回报率？例题终值=4,250,000,t=20,r=8%

现值=?现值=4,250,000PVIF(8,20) =¥911,829.8815某公司有一笔价值425万的债务需要在20年后偿还，假如年贴现率为8％，这笔债务的现值是多少？怎样求解等待期间？假如我现在投资5,000元于一个年收益率为10％的产品，我需要等待多久该投资才能增长到10,000?多少利率才合适？假设你的孩子12年后上大学时大学学费的总需求为¥50,000。你今天有¥5,000用来投资，你需要多高的投资回报率才能支付小孩上大学的学费？年金和永续年金年金（普通年金）在一定期限内，时间间隔相同、不间断、金额相等、方向相同的一系列现金流。永续年金在无限期内，时间间隔相同、不间断、金额相等、方向相同的一系列现金流。年金（Annuity）(期末)年金现值的公式为：01C2C3CTC(期末)年金终值的公式为：例题：年金的现值如果你采用分期付款方式购车，期限36个月，每月底支付400元，年利率为7%，那么你能购买一辆价值多少钱的汽车？01¥

4002¥4003¥40036¥400例题：如何计算等额支付C？问题：如果你想买一辆价值￥25,000的车，首付10％，其余部分银行按12％的年利率给你贷款60个月，你的月供是多少？回答：你将借贷的总额是90%￥25,000=￥22,500.这是贷款的现值，月利率为1％，连续计复利60次:

￥22,500=C

{1-1/(1+1%)60}/1% = C

{1-0.55045}/1% = C

44.955

C =22,500/44.955=每月￥500.50案例：年金的现值假如你今后3年的学费是每年¥20,000，每年年底支付。你如果今天将一笔钱存入年复利率为8％的银行帐户，这笔钱应该是多少才能正好支付你今后三年的学费？

PV=¥20,000/(1+8%)+¥20,000/(1+8%)2+¥20,000/(1+8%)3 = ¥18,518.52+¥17,146.77+¥15,876.65 = ¥51,541.94或直接用公式 PV= ¥20,000[1-1/(1+8%)3]/8% = ¥20,0002.577097 = ¥51,541.94案例：年金现值（续）假如上例中，银行年复利率仅为4％，那么你今天需要存多少钱？ PV=¥20,000/(1+4%)+¥20,000/(1+4%)2+¥20,000/(1+4%)3 = ¥19,230.77+¥18,491.12+¥17,779.93 = ¥55,501.82或 PV=¥20,000[1-1/(1+4%)3]/4% = ¥20,0002.775091 = ¥55,501.82如何求时间期间t?问题：假如你的信用卡帐单上的余额为￥2000，月利率为2％。如果你月还款的最低额为￥50，你需要多长时间才能将￥2000的账还清？回答：很长时间……

¥2000 = ¥50{1-1/(1+2%)t}/2% 0.80 = 1-1/(1+2%)t (1+2%)t

= 5.0

t = 81.3个月，大约6.78年！！！例题：如何求等额支付C?前面的例题中提到，一个21岁的年轻人今天投资¥15,091（10％的年复利率），可以在65岁时（44年后）获得¥100万元。假如你现在一次拿不出¥15,091，而想在今后44年中每年投资一笔等额款，直至65岁。这笔等额款为多少？ ¥1,000,000=C[(1+10%)44-1]/10%

C=¥1,000,000/652.6408=¥1,532.24

成为一个百万富翁也不是异想天开！！！例题：年金如果你现在已经40岁“高龄”了，才想起考虑养老问题，也想在65岁时成为百万富翁。如果你的投资的年复利率也为10％，从现在（年底）开始每年投资一笔等额款，直至65岁。这笔等额款为多少？¥100万元=C

[(1+10%)25-1]/10%

C=¥1,000,000/98.34706=¥10,168.07如果你的投资年复利率为20％，这笔等额款为¥100万元=C

[(1+20%)25-1]/20%

C=¥1,000,000/471.9811=¥2,118.73永续年金0…1C2C3C（期末）永续年金现值的公式为：例题：永续年金的现值假如某股票每年都分红15元，年利率为10％，那么它的价格是多少？

0…1¥152¥153¥15PV=¥15/10%=¥150期末年金与期初年金期末年金：利息收入，红利收入，房贷本息支付，储蓄等。期初年金：房租，养老金支出，生活费，教育金支出，保险等。01C2C3CTC01C2C3CTCT-1CT-1C期末年金与期初年金的关系期初年金现值等于期末年金现值的(1+r)倍,即:期初年金终值等于期末年金终值的(1+r)倍,即:净现值净现值(NPV):是指所有现金流（包括正现金流和负现金流在内）的现值之和。对于一个投资项目，如果NPV>0，表明该项目有利可图；相反地，如果NPV<0，表明该项目无利可图。例题：净现值对于一个投资项目，初始投资10,000元，共投资4年，各年的现金流如下所示：如果贴现率为5%，那么净现值为：01¥2,0002¥3,000¥10,0003¥4,0004¥5,000内部回报率内部回报率(IRR)：是指使净现值等于0的贴现率。

对于一个投资项目，如果r<IRR，表明该项目有利可图；相反地，如果r>IRR，表明该项目无利可图。其中r表示融资成本。例题：内部回报率对于一个投资项目，初始投资10,000元，共投资4年，各年的现金流如下所示：那么内部回报率为：01¥2,0002¥3,000¥10,0003¥4,0004¥5,000复利期间一年内对某金融资产计m次复利，T年后，你得到的价值是：例如，你将50元进行投资，年利率为12％，每半年计息一次，那么三年后你的投资价值变为：货币的时间价值在个人金融理财中的应用金融理财涉及一定时间跨度的成本和收益核算。无论是个人和家庭，都必须根据未来的预期收入，评估当前投资，因而不可避免地要对不同时期的金融资产进行价值比较。金融理财师在和客户讨论现金的流入（收入）和流出（支出）时，必须按照时间的顺序，列明现金流。计算现金流时，需要分析两个重要因素：一是时间间隔的长短，也就是时间上的联系；二是金额的高低，也就是价值上的联系。对现金流进行分析，是为客户进行财务策划的第一步，也是最基本的计算和分析方法。最典型的现金流计算包括：终值、现值、年金、不等额年金、永续年金和递延年金等各方面的计算。PV现值、FV终值、PMT年金、i利率、n期数，是运用财务计算器计算货币时间价值的五大变量。只要输入任何四个变量，就可以求出剩下的一个变量。输入数字时，如投资、存款、生活费用支出、房贷本息支出都是现金流出，输入符号为负；收入、赎回投资、借入本金都是现金流入，输入符号为正。在解决货币时间价值问题时，最好先画出现金流量与时间图。把理财目标实现的时间当作基准点，基准点之前我们通过累积资产来实现理财目标，是用现值（比如现有资产）或年金（比如每期储蓄）来求复利终值或年金终值。基准点之后可以理解为以借款来实现理财目标，之后再分期摊还，是用终值（比如预留遗产额）或年金（比如每期学费、每期生活费、每期房贷）来求复利现值或年金现值。如果前段现值与年金所累计的资产，等于后段终值与年金所算出的负债之时，就是理财目标可以实现的时间。而折现率的高低，则是决定何时资产会等于负债的关键因素。

用目标基准点法为客户进行理财规划基准点年份目标持续的年数离基准点的年数复利终值复利现值拟留遗产FV年金终值生息资产PV年储蓄PMT年金现值年支出PMT基准点购车：购车当年购屋：交屋当年子女教育：子女满18岁要上大学之年退休：打算退休当年

理财规划计算原理图解基准点转运站公路复利-点对点铁路年金-固定持续转运站之前累积资产用终值的观念铁路年金-固定持续公路复利-点对点转运站之后偿还负债用现值的观念退休-退休当年购屋-交屋当年子女教育-上大学当年Part1TheBasicsoftheTimeValueofMoneyChapter1

ExamplesExample1.1

Supposethatyouinvest$100,000todayinaninvestmentthatproducesa

returnof5%peryear.Whatwilltheinvestmentbeworthintwoyears?AnswerFV2=$100,000(1+0.05)2=$100,000(1.1025)=$110,250Example1.2

Supposeyouhaveachoicebetweentwoaccounts,AccountAandAccount

B.AccountAprovides5%interest,

compoundedannuallyandAccountB

provides5.25%

simpleinterest.Whichaccountprovidesthehighest

balance

attheendoffouryears?Whatisthedifferenceinthe

valuesofthetwoaccounts?AnswerAnswer:AccountAprovidesthehigherbalanceat

theendoffouryears.

Consideradepositof$10,000today(thoughitreally

doesn’tmatterwhatthebeginningbalanceis).AccountA:FV4=$10,000×(1+0.05)4=$12,155.06AccountB:FV4=$10,000+($10,000×0.0525×4)=$12,100.00Thedifference,$55.06,istheinterestoninterest.Example1.3Supposeyouinvest$20,000inanaccountthatpays12%interest,compounded

monthly.Howmuchdoyouhaveinthe

accountattheendof

5years?AnswerThenumberofperiodsis60:n=5years×12monthsperyear=60monthsandtherateperperiodis1%:i=Rateperperiod=12%÷12=1%Therefore,thefuturevalueis$36,333.93.Usingthemath,FV=$20,000(1+0.01)60=$20,000(1.8167)=$36,333.93Example1.4Supposeyouinvest$1,000todayinan

accountthatpays9%interest,

compounded

continuously.Whatwillbe

thevalueinthisaccountattheendof

tenyears?AnswerThefuturevalueis$2,459.60:FV=$1,000e0.09×10=$1,000e0.9=$1,000(2.4596)=$2,459.60Example1.5Supposeyouinvest$5,000inanaccount

thatearns10%interest.Howmuch

morewouldyouhaveafter20yearsifinterest

compoundscontinuously

insteadofcompoundedsemi-annually?AnswerYouwouldhave$1,745.34more:FV

continuously=$5,000e0.1×20=$5,000(7.3891)=$36,945.28FV

semiannually=$5,000(1+0.05)40=$5,000(7.0400)=$35,199.94Difference=$36,945.28−35,199.94=$1,745.34Chapter1

Problems1.1Ifyouinvest$10,000inanaccountthatpays4%interest,compounded

quarterly,howmuchwillbeintheaccountattheendoffive

yearsifyoumakenowithdrawals?AnswerPV=$10,000n=5×4=10quartersi=4%÷4=1%perqtr.FV=$10,000(1+0.01)20=$10,000×1.22019=$12,201.901.2Ifyouinvest$2,000inanaccountthatpays12%peryear,compounded

monthly,howmuchwillbeintheaccountattheendofsix

yearsifyoudonotmakeany

withdrawals?AnswerPV=$2,000n=6×12=72monthsi=12%÷12=1%permonthFV=$2,000(1+0.01)72=$2,000×2.0471=$4,094.201.3Supposeyouinvest$3,000inanaccountthatpaysinterestatthe

rateof8%peryear,compoundedsemi-annually.Howmuchwillyou

haveintheaccountattheendoffiveyearsifyoudonotmakeany

withdrawals?AnswerPV=$3,000n=5×2=106-monthperiodsi=8%÷2=4%FV=$3,000(1+0.04)10=$3,000×1.48024=$4,440.721.4Supposeyouinvest$100for20yearsinanaccountthatpays2%per

year,

compounded

quarterly.a.Howmuchwillyouhaveintheaccountattheendof20years?b.Howmuchinterestoninterestwillbeintheaccountattheendof

20years?AnswerPV=$100n=20×4=80i=2%÷4=0.5%FV=$100×(1+0.005)80=$100×1.49034=$149.03FVwithsimpleinterest=$100+($100×0.02×20)=$140Interestoninterest=$9.031.5Ifyoudeposit$100inanaccountthatpays4%interest,compounded

annually,whatisthebalanceintheaccountattheendofthreeyears

ifyouwithdrawonlytheinterest

ontheinteresteachyear?AnswerPV=$100i=4%n=3Theproblemrequiresthefuturevalueifthereissimpleinterest,FV=$100+12=$112.ThefuturevaluewithcompoundingisFV=$100×(1+0.04)3=$112.49.Withdrawalsarethedifferencebetweenthefuturevaluewithcompounding

andthefuturevaluewithsimpleinterest=$112.49−$112.00=$Supposeyouinvest€100todayinaninvestmentthatyields5%per

year,compoundedannually.Howmuchwillyouhaveintheaccount

attheendofsixyears?AnswerPV=€100i=5%n=6yearsFV=€100(1+0.05)6=€100×1.3401=€134.011.7Whichinvestmentof$10,000willprovidethelargervalueafterfour

years:a.InvestmentAearns5%interest,

compoundedsemiannually.b.InvestmentBearns4.8%

interest,

compoundedcontinuously.AnswerInvestmentAprovidesthelargerbalance:A:PV=$10,000;I=5%÷2=2.5%;n=4×2=8;FV=$12,184.03B:PV=$10,000;factor=e4×0.048;FV=$10,000×1.2116705=$12,116.711.8Whatwillbethevalueinanaccountattheendof12yearsifyou

deposit$100todayandtheaccountearns6%interest,

compounded

annually?AnswerPV=$100n=12i=6%FV=$100(1+0.06)12=$100×2012197=$201.221.9Whatwillbethevalueinanaccountattheendofsixyearsifyou

deposit$100todayandtheaccountearns12%interest,compounded

annually?AnswerPV=$100n=6i=12%FV=$100(1+0.12)6=$100×1.97382=$197.381.10Whatwillbethevalueinanaccountattheendof10yearsifyou

deposit$1,000todayandtheaccountearns7%interest,compounded

continuously?AnswerPV=$1,000n=10i=7%FV=$1,000e10

×0.07=$1,000×2.01375=$2,013.75Chapter2ExamplesExample2.1Supposethatyouwishtohave$20,000savedbytheendofsixyears.And

supposeyoudepositfundstodayinanaccountthatpays3%interest,

compoundedannually.Howmuchmustyoudeposittodaytomeetyourgoal?AnswerFV=$20,000;n=6;i=3%.Solveforthepresentvalue,PV:PV=$20,000÷(1+0.03)6=$20,000÷1.1941=$16,749.69Example2.2Supposethatyouwishtohave$1millionfortyyearsfromnow.Ifyoudepositfundstodayinanaccountthatpays5%interest,compoundedannually,whatamountmustyoudeposittodaytoreachyourgoal?AnswerFV=$1,000,000n=40i=5%ThepresentvalueisPV=$1,000,000÷(1+0.05)40=$142,045.68Example2.3Howmuchwouldyouhavetodeposittodayinanaccountthatpays4%annualinterest,compoundedquarterly,ifyouwishtohaveabalanceof$100,000attheendof10years?AnswerFV=$100,000i=4%÷4=1%n=10×4=40quartersPV=$100,000÷(1+0.01)40=$100,000(0.6717)=$67,165.31Example2.4Howmuchwouldyouhavetodeposittodayinanaccountthatpays4%annualinterest,compoundedcontinuously,ifyouwishtohaveabalanceof$100,000attheendof10years?AnswerFV=$100,000i=4%n=10yearsPV=$100,000÷e0.04×10=$100,000(0.67032)=$67,032Example2.5Supposeyouhavetwoinvestmentopportunitiesthatpromise$1millionin20years:InvestmentA:Areturnof6%peryear,compoundedmonthly.InvestmentB:Areturnof5.8%peryear,compoundedcontinuously.Whichinvestmentrequiresalargerinvestmenttodaytoreachyourgoal?AnswerPVA=$1,000,000÷(1+(0.06/12))240=$302,096.14PVB=$1,000,000÷e0.058×20=$313,486.18InvestmentBrequiresalargerinvestmenttodaytoreachthegoal.Example2.6Supposeafriendwantstoborrowsomemoneyfromyouandiswillingtopayyouback$5,000twoyearsfromnowandthen$7,000fouryearsfromtoday.Ifyouropportunitycostoffundsis5%(thatis,whatyoucouldhaveearnedonthemoneyinaninvestmentwithsimilarrisktoaloantoyourfriend),howmuchareyouwillingtolendyourfriend?AnswerExample2.7Howlongdoesittaketodoubleyourmoneyiftheinterestrateis5%peryear,compoundedannually?AnswerInputs:PV=$1;FV=$2;i=5%Solvingforn:n=(ln2−ln1)÷ln1.05=(0.6931−0)÷0.0488=14.202915yearsExample2.8Howlongdoesittaketotripleyourmoneyiftheinterestrateis5%peryear,compoundedannually?AnswerInputs:PV=$1;FV=$3;i=5%Solvingforn:n=(ln3−ln1)÷ln1.05=(1.0986−0)÷0.0488=22.5123years23yearsExample2.9Howlongdoesittaketodoubleyourmoneyiftheinterestrateis12%peryear,compoundedquarterly?AnswerInputs:PV=$1;FV=$2;i=12%÷4=3%Solvingforn:n=(ln2−ln1)÷ln1.03=(0.6931−0)÷0.0296=23.4155quarters⇒24quarters=6yearsProblems2.1Completethefollowing,solvingforthepresentvalue,PV:AnswerYouaregiventhreeinputs:FV,i,andN,andarerequiredtosolveforPV:FutureValueInterestRateNumberofPeriodsPresentValueA$10,0005%5$7,835.26B
563,0004%20
256,945.85C$5,0005.5%3$4,258.072.2Supposeyouwanttohave$0.5millionsavedbythetimeyoureachage30,andsupposethatyouare20yearsoldtoday.Ifyoucanearn5%onyourfunds,howmuchwouldyouhavetoinvesttodaytoreachyourgoal?AnswerFV=$500,000;i=5%;n=10PV=$500,000(1÷(1+0.05)10)=$500,000×0.6139=$306,959.632.3HowmuchwouldIhavetodepositinanaccounttodaythatpays12%interest,compoundedquarterly,sothatIhaveabalanceof$20,000intheaccountattheendof10years?AnswerFV=$20,000;i=12%÷4=3%;n=10×4=40quartersPV=$6,131.142.4SupposeIwanttobeabletowithdraw$5,000attheendoffiveyearsandwithdraw$6,000attheendofsixyears,leavingazerobalanceintheaccountafterthelastwithdrawal.IfIcanearn5%onmybalances,howmuchmustIdeposittodaytosatisfymywithdrawalsneeds?AnswerTherearetwodifferentfuturevalues.Treatastwoseparatepresentvalues,thencombine.FV=$5,000;n=5,i=5%PV=$3,917.63FV=$6,000;n=6,i=5%PV=$4,477.29PVofthetwofuturevalues=$3,917.63+4,477.29=$8,394.92Or,youcanusetheNPVfunctioninafinancialcalculator:IntheTI-83/84,thecashflowsare{0,0,0,0,5000,6000}IntheHP10B,thecashflowsare0,0,0,0,0,5000,60002.5Usinganinterestrateof5%peryear,whatisthevaluetodayofthefollowingcashflows:AnswerPV=$8,638.38+8,227.02=$16,865.40Note:IntheTI-83/84calculator,cashflowlistis{0,0,10000,10000}.2.6Whichofthefollowingserieshasthehighestpresentvalue,assuminganannualinterestrateof5%?AnswerB.ValueofA=€432ValueofB=€449ValueofC=€4482.7Whatisthepresentvalueof$500tobereceivedintwoyearsiftheinterestrateis4%peryearand:Compoundsdaily?Compoundscontinuously?AnswerThetwopresentvaluesdifferslightly:PV=$500÷(1+(0.04/365))730=$461.5602PV=$500÷e0.08=$461.55822.8Whatisthepresentvalueof£5milliontobereceivedin10yearsifinterestis12%compoundedmonthly?AnswerPV=£5,000,000÷(1+0.01)120=£1,514,9742.9Whatisthepresentvalueof$6,000tobereceivedin10yearsifinterestis6%,compoundedcontinuously?AnswerPV=$6,000÷e0.6=$3,292.872.10Whatisthepresentvalueof$10,000tobereceivedinthreeyearsiftheinterestrateis5%?AnswerPV=$10,000÷(1+0.05)3=$8,638.38Chapter3ExamplesExample3.1Supposeyoudeposit$100today,$200oneyearfromtoday,and$300twoyearsfromtoday,inanaccountthatpays10%interest,compoundedannually.Whatisthebalanceintheaccountattheendoftwoyears?AnswerFV=[$100×(1.10)2]+[$200×1.10]+$300=$641Example3.2Considerthreecashflows:AnswerQ1:Iftheinterestrateis5%,whatisthevalueofthesecashflowsattheendof20X1?AnswerQ2:Whatisthevalueofthesecashflowsattheendof20X3?Example3.3Supposeyoubuyashareofstockthathasa$2dividend,paidattheendofeachyear.Ifyouexpectthedividendtobeconstantandpaideachyear,forever,whatareyouwillingtopayforthisshareofstockiftheopportunitycostoffundsconsideringtheriskofthestock,is8%?AnswerPV=$2/0.08=$25Example3.4Youobservethatashareofstockiscurrentlysellingfor$30pershare.Ifthisstockhasaconstantdividendof$3peryear,paidattheendofeachyear,forever,whatistherequiredrateofreturnonthisstock?Answeri=$3/$30=10%Example3.5Whatisthevalueofaninvestmentthatprovidescashflowsof$2,000attheendofeachyearforthenextfouryearsifyouhavedeterminedthattheappropriatediscountrateonthisinvestmentis6%?Answer

Becausethesecashflowsarethesameamountandoccuratregularintervalsoftime,wecansolvethisusinganordinaryannuity,whichmeanswecanusethecalculatororspreadsheetshortcutinvolvingthePMT—theperiodic,evencashflow.Wearegiventhefollowingdatainputs:PMT=$2,000N=4i=6%Solvingforthepresentvalueofanannuity,thevalueofthisinvestment

is$6,930.21.Problems

3.1Whatisthevalueattheendof2009ofthefollowingseriesofcashflowsifthediscountrateis5%?AnswerPV=[$1,000×0.952381]+[$3,000×0.8638]PV=$952.38+2,591.51=$3,543.89Usingcalculator:$3,543.893.2Whatisthevalueattheendof2012ofthefollowingseriesofcashflowsiftheinterestrateis5%?AnswerPV=[$1,000×1.1025]+[$3,000×1]PV=$1,102.50+3,000=$4,102.503.3Whatisthevaluetodayofapromisedseriesofcashflowsof$6,000attheendofeachofthenextfiveyears?Usea10%discountrate.AnswerUsingacalculator,PMT=$6,000;N=5;i=10%;PV=$22,744.7213.4Whatisthevaluetodayofthefollowingseriesofcashflowsifthediscountrateis10%?AnswerPV=£7,513.148+£6,830.13=£14,343.2783.5Supposeyoudeposit$1,000inanaccountattheendofeachyearforthreeyears.Iftheaccountearns5%interestperyear,whatisthebalanceintheaccountattheendofthreeyears?AnswerCF=$1,000N=3i=5%FV=$1,000×FVannuityfactor=$3,152.503.6Calculatethepresentvalueofafour-payment$1,000ordinaryannuityiftheinterestrateis5%.AnswerCF=$1,000N=4i=5%PV=$1,000×(PVannuityfactor)=$3,5463.7Supposeyoudeposit$1,000eachyearforthreeyearsinanaccountthatpays5%interest,compoundedannually.Ifyoumakethedepositsatthebeginningoftheyear,whatisthebalanceintheaccountattheendofthreeyears?AnswerFV=$1,000×3.1525×1.05=$1,000×3.3101=$3,310.10Usingthecalculationfunction:PMT=1,000;N=3;I=5;solveforFV3.8Supposeyouwin$7millionPowerballlottery.Youreceiveyourlotterywinningsin20equalannualinstallments,withthefirstinstallmentpaidimmediately.Ifyoucouldinvestthefundstoyield5%peryear,whatisthesmallestlumpsumthatyouwouldbewillingtotaketodayinexchangeforyour20installments?AnswerCFt=$350,000N=20i=5%PV=$350,000×12.4622×1.05=$350,000(13.0853)=$4,579,8623.9Consideranannuityconsistingofthreepaymentsof$4,000each.Iftheinterestrateis5%peryear,whatisthepresentvalueofthisas:a.Anordinaryannuity?b.Anannuitydue?c.Adeferredannuity,deferredtwoperiods?AnswerThevaluesshoulddifferbyafactorof1+0.05or1.05:a.Endmode:PMT=$4,000;n=3;i=5%PV0=$10,892.99b.Begmode:PMT=$4,000;n=3;i=5%PV0=$11,437.64c.PMT=$4,000;n=3;i=5%PV1=$10,892.99Thendiscounttothepresent,oneperiodPV0=$10,374.283.10Yourbrokerhasproposedthatyoupay$50,000todayforanannuityof$5,000peryearforfifteenyears.Ifyouropportunitycostoffundsis6%andthereturnsfromthisinvestmentaretax-free,isthisagooddeal?AnswerNo,becausethevalueoftheannuityislessthan$50,000.Given:PMT=$5,000;N=15;i=6%.PV=$48,561.24Chapter4ExamplesExample4.1Supposeabankoffersyoulendingratesat6%APR,withinterestcompoundedmonthly.Whatisthecompoundingperiod?Answer:Amonth.Whatistheratepercompoundingperiod?AnswerTheAPRis6%andthereare12compoundperiodsinayear.Therefore,6%÷12=0.5%Tocheckourwork,0.5%×12=6%.Example4.2Supposeyourcreditcardstatesthatinterestonunpaidbalancesis24%APR,withinterestcompoundedmonthly.Whatistheinterestratepermonthforthiscreditcard?AnswerTheAPRis24%andthereare12monthsinayear.Therefore,theratepermonthis24%÷12=2%.Example4.3Supposeabankoffersyoulendingratesat6%APR,withinterestcompoundedmonthly.Whatistheeffectiverateofinterestonthislending?AnswerTheEARis6.168%:EAR=(1+0.005)12−1=6.168%.Example4.4Supposeyourcreditcardstatesthatinterestonunpaidbalancesis24%APR,withinterestcompoundedmonthly.Whatistheeffectiveannualrateofinterestonunpaidbalances?AnswerTheEARis26.824%:EAR=(1+0.02)12−1=26.824%.Example4.5TheABCCreditCardCompanyoffersyouacreditcardwithanAPRof19.5%.Ifinterestcompoundsdaily,whatistheeffectiveannualrateofinterestonthiscreditcard?AnswerYouknowthefollowing:APR=19.5%;n=365andi=0.195÷365=0.00534247.TheEARistherefore21.525%:EAR=(1+0.00534247)365−1=0.21525or21.525%.Example4.6

Apaydayloanisashort-termloanwithveryhighinterestrates.Inatypicalpaydayloan,ifyouwanttoborrow$100youwriteacheckfor$125.Thelenderholdsontoyourcheckduringtheloanperiod.Attheendoftheloanperiod,usually10to14days,thelenderdepositsyourcheck.Ifyouwanttoextendyourloan,youpaytheminimumof$25cashandthenenterintoanewcontracttopay.Ifyoudonotpayofftheloanorpaythefeetorollovertheloan,thelenderwilldeposityourcheckandyouriskbeingchargedwithwritingbadchecks.1.WhatistheAPRforthispaydayloan?2.WhatistheEARforthispaydayloan?Answer1APR=0.25(365/14)=651.79%.Answer2EAR=(1+0.25)365/14−1=3,351.86%.Example4.7Whichofthefollowingtermsrepresentsthelowestcostofcreditonaneffectiveannualinterestratebasis?A:10%A