理財(cái)基礎(chǔ)-時(shí)間價(jià)值

貨幣的時(shí)間價(jià)值貨幣時(shí)間價(jià)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(bào)率貨幣的時(shí)間價(jià)值在金融理財(cái)中的應(yīng)用與時(shí)間價(jià)值有關(guān)的術(shù)語(yǔ)時(shí)間軸PV即現(xiàn)值，也即今天的價(jià)值FV即終值，也即未來(lái)某個(gè)時(shí)間點(diǎn)的價(jià)值t表示終值和現(xiàn)值之間的這段時(shí)間r表示利率所有的定價(jià)問(wèn)題都與PV、FV、t、r這四個(gè)變量有關(guān)，確定其中三個(gè)即能得出第四個(gè)。...0123tPVFV單期中的終值假設(shè)利率為5％，你準(zhǔn)備拿出1萬(wàn)元進(jìn)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(jié)束時(shí)獲得的價(jià)值被稱(chēng)為終值（FV）單期中的終值單期中終值計(jì)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(shè)利率為5％，你想保證自己通過(guò)一年的投資得到1萬(wàn)元，那么你的投資在當(dāng)前應(yīng)該為9,523.81元要得到一年后1萬(wàn)元，在當(dāng)前所必須的資金價(jià)值被稱(chēng)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(jì)算公式為：其中，F(xiàn)V是在1時(shí)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(jì)算多期中的終值公式：FV=PV×(1+r)T其中， PV是第0期的價(jià)值,r是利率,T

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

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

￥5000(1+t

r)=￥5000(1+612%)

=￥8600復(fù)利和單利計(jì)算之間的差異即為：￥9869.11-￥8600=￥1269.11終值利率因子（復(fù)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(shuō)來(lái)，經(jīng)過(guò)t時(shí)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(chēng)為復(fù)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(wàn)元，你需要在今天拿出多少錢(qián)進(jìn)行投資?012345￥20,000PV已知終值（2萬(wàn)），利率（8%)，投資時(shí)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(shí)，要計(jì)算t時(shí)期價(jià)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(chēng)為現(xiàn)值利率因子(PVIF)，也稱(chēng)為復(fù)利現(xiàn)值系數(shù)現(xiàn)值利率因子（復(fù)利現(xiàn)值系數(shù)）假設(shè)你三年后需要2萬(wàn)元來(lái)支付研究生的學(xué)費(fèi)，投資收益率是8%，今天你需要拿出多少錢(qián)來(lái)投資？期限不同，利率不同時(shí)1元的現(xiàn)值如何變化？多期中的終值假設(shè)劉先生購(gòu)買(mǎi)了九州龍騰公司首次公開(kāi)發(fā)售時(shí)的股票。該公司的當(dāng)前分紅為每股1.10元，并預(yù)計(jì)能在未來(lái)5年中以每年40％的速度增長(zhǎng)。問(wèn)：5年后的股利為多少?FV=PV×(1+r)T￥5.92=￥1.10×(1+40%)5復(fù)利對(duì)終值的影響012345Q.假如你買(mǎi)彩票中獎(jiǎng)100萬(wàn)，將其存為10年期，年利率為6％的定期存款，按復(fù)利計(jì)算?；蛘?，你將其交給表兄打理，10年中，每年按6％的單利計(jì)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(fù)利（利滾利）引起的是將近191,000

元的資產(chǎn)增值。(當(dāng)然我們還沒(méi)有考慮到將這筆錢(qián)交給你表兄打理的風(fēng)險(xiǎn)）年利率為10％時(shí)的終值年度 年初值單利復(fù)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(jì)￥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(dāng)然我們忽略了稅收和其他的復(fù)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(shí)成為百萬(wàn)富翁，今天你要一次性拿出多少錢(qián)來(lái)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(xiě)道，他將分別向波士頓和費(fèi)城捐贈(zèng)1000元。捐款將于他死后200年贈(zèng)出。1990年時(shí)，付給費(fèi)城的捐款已經(jīng)變成200萬(wàn)，而給波士頓的已達(dá)到450萬(wàn)。請(qǐng)問(wèn)兩者的年投資回報(bào)率各為多少？對(duì)于費(fèi)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(bào)率為3.87%.同理我們可以得到波士頓的年投資回報(bào)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(shuō)“大約”？因?yàn)槿绻蔬^(guò)高或過(guò)低，該法則不再適用。 假設(shè)r=72%FVIF(72,1)=1.7200,而非2.00

假設(shè)r=36% FVIF(36,2)=1.8496，而非2.00

可見(jiàn)，該法則只是一個(gè)近似估計(jì)。例題計(jì)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(shè)你現(xiàn)在拿出5,000元投資于一個(gè)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(zhǎng)期回報(bào)率據(jù)研究，1802－1997年間普通股票的年均收益率是8.4%.假設(shè)你的祖先在1802年對(duì)一個(gè)充分分散風(fēng)險(xiǎn)的投資組合進(jìn)行了1000元的投資。1997年的時(shí)候，這個(gè)投資的價(jià)值是多少？1998年普通股票價(jià)值增長(zhǎng)了28.59%,那么上述投資組合在1998年的價(jià)值是多少？t=195r=8.4%,FVIF(8.4,195)=6,771,892.09695所以該投資的價(jià)值應(yīng)為:￥6,771,892,096.95!1998年末的投資價(jià)值為￥6,771,892,096.95(1+28.59%)=￥8,707,976,047.47!例題1.下列哪些說(shuō)法是對(duì)的？如果r和t都大于0，終值利率因子FVIF(r,t)永遠(yuǎn)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(yuǎn)都大于0.2.判斷題:對(duì)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(duì)于一個(gè)現(xiàn)金流來(lái)說(shuō)，貼現(xiàn)率越高，其現(xiàn)值越高還是越低？?jī)蓚€(gè)說(shuō)法都正確。正確.PVIF(r,t)=1/FVIF(r,t)越低。對(duì)同一個(gè)現(xiàn)金流來(lái)說(shuō)，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(shè)你的子女在18年后將接受大學(xué)教育，屆時(shí)需要學(xué)費(fèi)20萬(wàn)元。你現(xiàn)在有15,000元可以用于投資，問(wèn)你需要怎樣的回報(bào)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(jià)值425萬(wàn)的債務(wù)需要在20年后償還，假如年貼現(xiàn)率為8％，這筆債務(wù)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(gè)年收益率為10％的產(chǎn)品，我需要等待多久該投資才能增長(zhǎng)到10,000?多少利率才合適？假設(shè)你的孩子12年后上大學(xué)時(shí)大學(xué)學(xué)費(fèi)的總需求為￥50,000。你今天有￥5,000用來(lái)投資，你需要多高的投資回報(bào)率才能支付小孩上大學(xué)的學(xué)費(fèi)？年金和永續(xù)年金年金（普通年金）在一定期限內(nèi)，時(shí)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(wú)限期內(nèi)，時(shí)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(gòu)車(chē)，期限36個(gè)月，每月底支付400元，年利率為7%，那么你能購(gòu)買(mǎi)一輛價(jià)值多少錢(qián)的汽車(chē)？01￥

4002￥4003￥40036￥400例題：如何計(jì)算等額支付C？問(wèn)題：如果你想買(mǎi)一輛價(jià)值￥25,000的車(chē)，首付10％，其余部分銀行按12％的年利率給你貸款60個(gè)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(jì)復(fù)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(xué)費(fèi)是每年￥20,000，每年年底支付。你如果今天將一筆錢(qián)存入年復(fù)利率為8％的銀行帳戶(hù)，這筆錢(qián)應(yīng)該是多少才能正好支付你今后三年的學(xué)費(fèi)？

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(fù)利率僅為4％，那么你今天需要存多少錢(qián)？ 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(shí)間期間t?問(wèn)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(zhǎng)時(shí)間才能將￥2000的賬還清？回答：很長(zhǎng)時(shí)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(gè)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(gè)21歲的年輕人今天投資￥15,091（10％的年復(fù)利率），可以在65歲時(shí)（44年后）獲得￥100萬(wàn)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(gè)百萬(wàn)富翁也不是異想天開(kāi)?。。±}：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(wèn)題，也想在65歲時(shí)成為百萬(wàn)富翁。如果你的投資的年復(fù)利率也為10％，從現(xiàn)在（年底）開(kāi)始每年投資一筆等額款，直至65歲。這筆等額款為多少？￥100萬(wàn)元=C

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

C=￥1,000,000/98.34706=￥10,168.07如果你的投資年復(fù)利率為20％，這筆等額款為￥100萬(wàn)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(jià)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(chǔ)蓄等。期初年金：房租，養(yǎng)老金支出，生活費(fèi)，教育金支出，保險(xiǎn)等。01C2C3CTC01C2C3CTCT-1CT-1C期末年金與期初年金的關(guān)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(fù)現(xiàn)金流在內(nèi)）的現(xiàn)值之和。對(duì)于一個(gè)投資項(xiàng)目，如果NPV>0，表明該項(xiàng)目有利可圖；相反地，如果NPV<0，表明該項(xiàng)目無(wú)利可圖。例題：凈現(xiàn)值對(duì)于一個(gè)投資項(xiàng)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(bào)率內(nèi)部回報(bào)率(IRR)：是指使凈現(xiàn)值等于0的貼現(xiàn)率。

對(duì)于一個(gè)投資項(xiàng)目，如果r<IRR，表明該項(xiàng)目有利可圖；相反地，如果r>IRR，表明該項(xiàng)目無(wú)利可圖。其中r表示融資成本。例題：內(nèi)部回報(bào)率對(duì)于一個(gè)投資項(xiàng)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(bào)率為：01￥2,0002￥3,000￥10,0003￥4,0004￥5,000復(fù)利期間一年內(nèi)對(duì)某金融資產(chǎn)計(jì)m次復(fù)利，T年后，你得到的價(jià)值是：例如，你將50元進(jìn)行投資，年利率為12％，每半年計(jì)息一次，那么三年后你的投資價(jià)值變?yōu)椋贺泿诺臅r(shí)間價(jià)值在個(gè)人金融理財(cái)中的應(yīng)用金融理財(cái)涉及一定時(shí)間跨度的成本和收益核算。無(wú)論是個(gè)人和家庭，都必須根據(jù)未來(lái)的預(yù)期收入，評(píng)估當(dāng)前投資，因而不可避免地要對(duì)不同時(shí)期的金融資產(chǎn)進(jìn)行價(jià)值比較。金融理財(cái)師在和客戶(hù)討論現(xiàn)金的流入（收入）和流出（支出）時(shí)，必須按照時(shí)間的順序，列明現(xiàn)金流。計(jì)算現(xiàn)金流時(shí)，需要分析兩個(gè)重要因素：一是時(shí)間間隔的長(zhǎng)短，也就是時(shí)間上的聯(lián)系；二是金額的高低，也就是價(jià)值上的聯(lián)系。對(duì)現(xiàn)金流進(jìn)行分析，是為客戶(hù)進(jìn)行財(cái)務(wù)策劃的第一步，也是最基本的計(jì)算和分析方法。最典型的現(xiàn)金流計(jì)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(jì)算。PV現(xiàn)值、FV終值、PMT年金、i利率、n期數(shù)，是運(yùn)用財(cái)務(wù)計(jì)算器計(jì)算貨幣時(shí)間價(jià)值的五大變量。只要輸入任何四個(gè)變量，就可以求出剩下的一個(gè)變量。輸入數(shù)字時(shí)，如投資、存款、生活費(fèi)用支出、房貸本息支出都是現(xiàn)金流出，輸入符號(hào)為負(fù)；收入、贖回投資、借入本金都是現(xiàn)金流入，輸入符號(hào)為正。在解決貨幣時(shí)間價(jià)值問(wèn)題時(shí)，最好先畫(huà)出現(xiàn)金流量與時(shí)間圖。把理財(cái)目標(biāo)實(shí)現(xiàn)的時(shí)間當(dāng)作基準(zhǔn)點(diǎn)，基準(zhǔn)點(diǎn)之前我們通過(guò)累積資產(chǎn)來(lái)實(shí)現(xiàn)理財(cái)目標(biāo)，是用現(xiàn)值（比如現(xiàn)有資產(chǎn)）或年金（比如每期儲(chǔ)蓄）來(lái)求復(fù)利終值或年金終值?；鶞?zhǔn)點(diǎn)之后可以理解為以借款來(lái)實(shí)現(xiàn)理財(cái)目標(biāo)，之后再分期攤還，是用終值（比如預(yù)留遺產(chǎn)額）或年金（比如每期學(xué)費(fèi)、每期生活費(fèi)、每期房貸）來(lái)求復(fù)利現(xiàn)值或年金現(xiàn)值。如果前段現(xiàn)值與年金所累計(jì)的資產(chǎn)，等于后段終值與年金所算出的負(fù)債之時(shí)，就是理財(cái)目標(biāo)可以實(shí)現(xiàn)的時(shí)間。而折現(xiàn)率的高低，則是決定何時(shí)資產(chǎn)會(huì)等于負(fù)債的關(guān)鍵因素。

用目標(biāo)基準(zhǔn)點(diǎn)法為客戶(hù)進(jìn)行理財(cái)規(guī)劃基準(zhǔn)點(diǎn)年份目標(biāo)持續(xù)的年數(shù)離基準(zhǔn)點(diǎn)的年數(shù)復(fù)利終值復(fù)利現(xiàn)值擬留遺產(chǎn)FV年金終值生息資產(chǎn)PV年儲(chǔ)蓄PMT年金現(xiàn)值年支出PMT基準(zhǔn)點(diǎn)購(gòu)車(chē)：購(gòu)車(chē)當(dāng)年購(gòu)屋：交屋當(dāng)年子女教育：子女滿(mǎn)18歲要上大學(xué)之年退休：打算退休當(dāng)年

理財(cái)規(guī)劃計(jì)算原理圖解基準(zhǔn)點(diǎn)轉(zhuǎn)運(yùn)站公路復(fù)利-點(diǎn)對(duì)點(diǎn)鐵路年金-固定持續(xù)轉(zhuǎn)運(yùn)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(fù)利-點(diǎn)對(duì)點(diǎn)轉(zhuǎn)運(yùn)站之后償還負(fù)債用現(xiàn)值的觀念退休-退休當(dāng)年購(gòu)屋-交屋當(dāng)年子女教育-上大學(xué)當(dāng)年P(guān)art1TheBasicsoftheTimeValueofMoneyChapter1

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