理財基礎-時間價值

貨幣的時間價值貨幣時間價值現(xiàn)值與終值年金和永續(xù)年金凈現(xiàn)值與內(nèi)部回報率貨幣的時間價值在金融理財中的應用與時間價值有關的術語時間軸PV即現(xiàn)值，也即今天的價值FV即終值，也即未來某個時間點的價值t表示終值和現(xiàn)值之間的這段時間r表示利率所有的定價問題都與PV、FV、t、r這四個變量有關，確定其中三個即能得出第四個。...0123tPVFV單期中的終值假設利率為5％，你準備拿出1萬元進行投資，一年后，你將得到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，算式為：￥10,500=￥10,000×(1+5%).

投資結束時獲得的價值被稱為終值（FV）單期中的終值單期中終值計算公式為：FV=PV×(1+r)

其中，PV是第0期的現(xiàn)金流，r是利率。FV=￥10,500年度01PV=￥10,000￥10,000×

1.05PV×(1+r)單期中的現(xiàn)值假設利率為5％，你想保證自己通過一年的投資得到1萬元，那么你的投資在當前應該為9,523.81元要得到一年后1萬元，在當前所必須的資金價值被稱為現(xiàn)值（PV）：￥10,000=￥9,523.81×(1+5%)單期中的現(xiàn)值單期中現(xiàn)值的計算公式為：其中，F(xiàn)V是在1時期的現(xiàn)金流，r是利率。FV=￥10,000年度01PV=￥9,523.81￥10,000/1.05FV/(1+r)多期中的終值計算多期中的終值公式：FV=PV×(1+r)T其中， PV是第0期的價值,r是利率,T

是投資時間。案例假設年利率為12%，今天投入5,000元6年后你將獲得多少錢？用單利計算是怎樣的？用復利計算是怎樣的？利率為12%時,用復利計算是: ￥5000(1+r)t

= ￥5000(1+12%)6 = ￥9869.11利率為12%時,用單利計算是：

￥5000(1+t

r)=￥5000(1+612%)

=￥8600復利和單利計算之間的差異即為：￥9869.11-￥8600=￥1269.11終值利率因子（復利終值系數(shù)）我們注意到￥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%)3一般說來，經(jīng)過t時期后，今天投入的1元的終值將是FVt

=￥1(1+r)t(1+r)t

是終值利率因子(FVIF),也稱為復利終值系數(shù)多期中的現(xiàn)值假如利率是15％，你想在5年后獲得2萬元，你需要在今天拿出多少錢進行投資?012345￥20,000PV已知終值（2萬），利率（8%)，投資時間（三年）那么現(xiàn)值可以這樣得到：

FVt =PVx(1+r)t

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

因此我們得到：年利率為r時，要計算t時期價值1元的投資的現(xiàn)值，可以用以下公式：

PV=1/(1+r

)t它被稱為現(xiàn)值利率因子(PVIF)，也稱為復利現(xiàn)值系數(shù)現(xiàn)值利率因子（復利現(xiàn)值系數(shù)）假設你三年后需要2萬元來支付研究生的學費，投資收益率是8%，今天你需要拿出多少錢來投資？期限不同，利率不同時1元的現(xiàn)值如何變化？多期中的終值假設劉先生購買了九州龍騰公司首次公開發(fā)售時的股票。該公司的當前分紅為每股1.10元，并預計能在未來5年中以每年40％的速度增長。問：5年后的股利為多少?FV=PV×(1+r)T￥5.92=￥1.10×(1+40%)5復利對終值的影響012345Q.假如你買彩票中獎100萬，將其存為10年期，年利率為6％的定期存款，按復利計算?；蛘撸銓⑵浣唤o表兄打理，10年中，每年按6％的單利計算。10年后，哪種方式獲利多？“利滾利”演示

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

從表兄那里獲得的終值是

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

6%x10=￥1,600,000.00

復利（利滾利）引起的是將近191,000

元的資產(chǎn)增值。(當然我們還沒有考慮到將這筆錢交給你表兄打理的風險）年利率為10％時的終值年度 年初值單利復利引起的利息增加總利息終值

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

總計￥50.00￥11.05￥61.05

例題確定變量:

FV

=￥1,000,000

r=10％

t=65-21=44年 PV=?代入終值算式中并求解現(xiàn)值：

￥1,000,000=PV(1+10%)44

PV=￥1,000,000/(1+10%)44=

￥15,091.當然我們忽略了稅收和其他的復雜部分，但是現(xiàn)在你需要的是籌集15000元！假如你現(xiàn)在21歲，每年收益率10％，要想在65歲時成為百萬富翁，今天你要一次性拿出多少錢來投資？案例：確定利率富蘭克林死于1790年。他在自己的遺囑中寫道，他將分別向波士頓和費城捐贈1000元。捐款將于他死后200年贈出。1990年時，付給費城的捐款已經(jīng)變成200萬，而給波士頓的已達到450萬。請問兩者的年投資回報率各為多少？對于費城，有以下算式:

￥1,000=￥2,000,000/(1+r

)200 (1+r

)200=2,000.00

求解r，得到年投資回報率為3.87%.同理我們可以得到波士頓的年投資回報率為4.3%.72法則如果年利率為r%,你的投資將在大約72/r年后翻番。

例如，如果年收益率為6％，你的投資將于約12年后翻番。為什么要說“大約”？因為如果利率過高或過低，該法則不再適用。 假設r=72%FVIF(72,1)=1.7200,而非2.00

假設r=36% FVIF(36,2)=1.8496，而非2.00

可見，該法則只是一個近似估計。例題計算如下：

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%假設你現(xiàn)在拿出5,000元投資于一個年收益率為r的產(chǎn)品。10年后你將得到10,000元，那么r為多少？例題：普通股票的長期回報率據(jù)研究，1802－1997年間普通股票的年均收益率是8.4%.假設你的祖先在1802年對一個充分分散風險的投資組合進行了1000元的投資。1997年的時候，這個投資的價值是多少？1998年普通股票價值增長了28.59%,那么上述投資組合在1998年的價值是多少？t=195r=8.4%,FVIF(8.4,195)=6,771,892.09695所以該投資的價值應為:￥6,771,892,096.95!1998年末的投資價值為￥6,771,892,096.95(1+28.59%)=￥8,707,976,047.47!例題1.下列哪些說法是對的？如果r和t都大于0，終值利率因子FVIF(r,t)永遠都大于0.如果r和t都大于0，現(xiàn)值利率因子PVIF(r,t)永遠都大于0.2.判斷題:對于既定的r和t，PVIF(r,t)是FVIF(r,t)的倒數(shù).3.其他條件都不變，對于一個現(xiàn)金流來說，貼現(xiàn)率越高，其現(xiàn)值越高還是越低？兩個說法都正確。正確.PVIF(r,t)=1/FVIF(r,t)越低。對同一個現(xiàn)金流來說，貼現(xiàn)率越高，其現(xiàn)值越低。例題

現(xiàn)值 = ￥15,000

終值 = ￥200,000

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

解得：r ＝15.48%假設你的子女在18年后將接受大學教育，屆時需要學費20萬元。你現(xiàn)在有15,000元可以用于投資，問你需要怎樣的回報率？例題終值=4,250,000,t=20,r=8%

現(xiàn)值=?現(xiàn)值=4,250,000PVIF(8,20) =￥911,829.8815某公司有一筆價值425萬的債務需要在20年后償還，假如年貼現(xiàn)率為8％，這筆債務的現(xiàn)值是多少？怎樣求解等待期間？假如我現(xiàn)在投資5,000元于一個年收益率為10％的產(chǎn)品，我需要等待多久該投資才能增長到10,000?多少利率才合適？假設你的孩子12年后上大學時大學學費的總需求為￥50,000。你今天有￥5,000用來投資，你需要多高的投資回報率才能支付小孩上大學的學費？年金和永續(xù)年金年金（普通年金）在一定期限內(nèi)，時間間隔相同、不間斷、金額相等、方向相同的一系列現(xiàn)金流。永續(xù)年金在無限期內(nèi)，時間間隔相同、不間斷、金額相等、方向相同的一系列現(xiàn)金流。年金（Annuity）(期末)年金現(xiàn)值的公式為：01C2C3CTC(期末)年金終值的公式為：例題：年金的現(xiàn)值如果你采用分期付款方式購車，期限36個月，每月底支付400元，年利率為7%，那么你能購買一輛價值多少錢的汽車？01￥

4002￥4003￥40036￥400例題：如何計算等額支付C？問題：如果你想買一輛價值￥25,000的車，首付10％，其余部分銀行按12％的年利率給你貸款60個月，你的月供是多少？回答：你將借貸的總額是90%￥25,000=￥22,500.這是貸款的現(xiàn)值，月利率為1％，連續(xù)計復利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：年金的現(xiàn)值假如你今后3年的學費是每年￥20,000，每年年底支付。你如果今天將一筆錢存入年復利率為8％的銀行帳戶，這筆錢應該是多少才能正好支付你今后三年的學費？

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：年金現(xiàn)值（續(xù)）假如上例中，銀行年復利率僅為4％，那么你今天需要存多少錢？ 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如何求時間期間t?問題：假如你的信用卡帳單上的余額為￥2000，月利率為2％。如果你月還款的最低額為￥50，你需要多長時間才能將￥2000的賬還清？回答：很長時間……

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

= 5.0

t = 81.3個月，大約6.78年?。?！例題：如何求等額支付C?前面的例題中提到，一個21歲的年輕人今天投資￥15,091（10％的年復利率），可以在65歲時（44年后）獲得￥100萬元。假如你現(xiàn)在一次拿不出￥15,091，而想在今后44年中每年投資一筆等額款，直至65歲。這筆等額款為多少？ ￥1,000,000=C[(1+10%)44-1]/10%

C=￥1,000,000/652.6408=￥1,532.24

成為一個百萬富翁也不是異想天開！??！例題：年金如果你現(xiàn)在已經(jīng)40歲“高齡”了，才想起考慮養(yǎng)老問題，也想在65歲時成為百萬富翁。如果你的投資的年復利率也為10％，從現(xiàn)在（年底）開始每年投資一筆等額款，直至65歲。這筆等額款為多少？￥100萬元=C

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

C=￥1,000,000/98.34706=￥10,168.07如果你的投資年復利率為20％，這筆等額款為￥100萬元=C

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

C=￥1,000,000/471.9811=￥2,118.73永續(xù)年金0…1C2C3C（期末）永續(xù)年金現(xiàn)值的公式為：例題：永續(xù)年金的現(xiàn)值假如某股票每年都分紅15元，年利率為10％，那么它的價格是多少？

0…1￥152￥153￥15PV=￥15/10%=￥150期末年金與期初年金期末年金：利息收入，紅利收入，房貸本息支付，儲蓄等。期初年金：房租，養(yǎng)老金支出，生活費，教育金支出，保險等。01C2C3CTC01C2C3CTCT-1CT-1C期末年金與期初年金的關系期初年金現(xiàn)值等于期末年金現(xiàn)值的(1+r)倍,即:期初年金終值等于期末年金終值的(1+r)倍,即:凈現(xiàn)值凈現(xiàn)值(NPV):是指所有現(xiàn)金流（包括正現(xiàn)金流和負現(xiàn)金流在內(nèi)）的現(xiàn)值之和。對于一個投資項目，如果NPV>0，表明該項目有利可圖；相反地，如果NPV<0，表明該項目無利可圖。例題：凈現(xiàn)值對于一個投資項目，初始投資10,000元，共投資4年，各年的現(xiàn)金流如下所示：如果貼現(xiàn)率為5%，那么凈現(xiàn)值為：01￥2,0002￥3,000￥10,0003￥4,0004￥5,000內(nèi)部回報率內(nèi)部回報率(IRR)：是指使凈現(xiàn)值等于0的貼現(xiàn)率。

對于一個投資項目，如果r<IRR，表明該項目有利可圖；相反地，如果r>IRR，表明該項目無利可圖。其中r表示融資成本。例題：內(nèi)部回報率對于一個投資項目，初始投資10,000元，共投資4年，各年的現(xiàn)金流如下所示：那么內(nèi)部回報率為：01￥2,0002￥3,000￥10,0003￥4,0004￥5,000復利期間一年內(nèi)對某金融資產(chǎn)計m次復利，T年后，你得到的價值是：例如，你將50元進行投資，年利率為12％，每半年計息一次，那么三年后你的投資價值變?yōu)椋贺泿诺臅r間價值在個人金融理財中的應用金融理財涉及一定時間跨度的成本和收益核算。無論是個人和家庭，都必須根據(jù)未來的預期收入，評估當前投資，因而不可避免地要對不同時期的金融資產(chǎn)進行價值比較。金融理財師在和客戶討論現(xiàn)金的流入（收入）和流出（支出）時，必須按照時間的順序，列明現(xiàn)金流。計算現(xiàn)金流時，需要分析兩個重要因素：一是時間間隔的長短，也就是時間上的聯(lián)系；二是金額的高低，也就是價值上的聯(lián)系。對現(xiàn)金流進行分析，是為客戶進行財務策劃的第一步，也是最基本的計算和分析方法。最典型的現(xiàn)金流計算包括：終值、現(xiàn)值、年金、不等額年金、永續(xù)年金和遞延年金等各方面的計算。PV現(xiàn)值、FV終值、PMT年金、i利率、n期數(shù)，是運用財務計算器計算貨幣時間價值的五大變量。只要輸入任何四個變量，就可以求出剩下的一個變量。輸入數(shù)字時，如投資、存款、生活費用支出、房貸本息支出都是現(xiàn)金流出，輸入符號為負；收入、贖回投資、借入本金都是現(xiàn)金流入，輸入符號為正。在解決貨幣時間價值問題時，最好先畫出現(xiàn)金流量與時間圖。把理財目標實現(xiàn)的時間當作基準點，基準點之前我們通過累積資產(chǎn)來實現(xiàn)理財目標，是用現(xiàn)值（比如現(xiàn)有資產(chǎn)）或年金（比如每期儲蓄）來求復利終值或年金終值?；鶞庶c之后可以理解為以借款來實現(xiàn)理財目標，之后再分期攤還，是用終值（比如預留遺產(chǎn)額）或年金（比如每期學費、每期生活費、每期房貸）來求復利現(xiàn)值或年金現(xiàn)值。如果前段現(xiàn)值與年金所累計的資產(chǎn)，等于后段終值與年金所算出的負債之時，就是理財目標可以實現(xiàn)的時間。而折現(xiàn)率的高低，則是決定何時資產(chǎn)會等于負債的關鍵因素。

用目標基準點法為客戶進行理財規(guī)劃基準點年份目標持續(xù)的年數(shù)離基準點的年數(shù)復利終值復利現(xiàn)值擬留遺產(chǎn)FV年金終值生息資產(chǎn)PV年儲蓄PMT年金現(xiàn)值年支出PMT基準點購車：購車當年購屋：交屋當年子女教育：子女滿18歲要上大學之年退休：打算退休當年

理財規(guī)劃計算原理圖解基準點轉運站公路復利-點對點鐵路年金-固定持續(xù)轉運站之前累積資產(chǎn)用終值的觀念鐵路年金-固定持續(xù)公路復利-點對點轉運站之后償還負債用現(xiàn)值的觀念退休-退休當年購屋-交屋當年子女教育-上大學當年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