Chapter 3. Time Value of Money

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1 Chaper 3 Time Value of Money A bird in he hand is worh wo in he bush A folklore saying Learning Oucomes Upon compleion of his chaper, you will be able: 1. To evaluae he significance of he ime value of money. 2. To undersand he facors ha influence he ime value of money. 3. To calculae he presen value of a fuure sum of money or a sream of fuure reurns. 4. To explain he inverse relaionship of he he presen value wih ineres rae (or discoun facor) and ime. 5. To demonsrae he calculaion of an annuiy and of a perpeuiy. Preview We saw in Chaper 1 and 2 ha invesmen decisions of individuals and businesses are forward-looking. In oher words, he oucomes of he decisions made oday (such as he decision o inves in a porfolio of securiies or an invesmen projec for a new plan or a new machinery, ec) will maerialize in some fuure ime. For example he benefis (such as reurns, dividend, capial gains, revenues, profis, ec) will be received in some fuure ime. Bu, hese fuure reurns are no worh he same as he money spen oday. This presens a problem for he raional invesor or raional firm ha has o compare he addiional coss of any decision/acion wih he addiional benefis accrued from ha decision/acion. I becomes clear ha here is a ime value of money dimension ha has o be aken ino consideraion in evaluaing such invesmen decisions. A Greek Cyprio folklore expression says: «Κάλλιον πέντε και στο χέρι, παρά δέκα και καρτέρει» which loosely ranslaed o English says Beer have five in he hand han en for which you have o wai. This of course is equivalen o he sandard English expression: A bird in he hand is worh wo in he bush We ake up in his chaper he concep of ime value of money and relaed conceps such as discouning, presen value, fuure value, and annuiies. We will focus mosly on undersanding he raionale and mehodology of discouning and calculaing he presen value of a fuure sum of money by using he compound and discoun ables as presened in Appendix 1 hrough Appendix 4 a he end of he book. There are, of course, nowadays many compuer programmes (e.g., Excel) and hand-held financial calculaors ha make i easy o calculae presen values.

2 Ineres Raes, Compounding and he Fuure Value of Money Though we will be mosly concerned wih discouning fuure sums and wih he presen value concep, he bes way o undersand hese conceps is o firs undersand he criical role of ineres raes in his process. Therefore, we sar wih he concep of compounding and he fuure value of sums of money. As he folklore saying above implies, people value having a sum of money oday more han having he same amoun in some fuure period. This is why when you borrow a sum of money oday, he lender (say he bank) requires you o pay back more han he iniial sum. Par of his is o cover he cos of service for providing he loan (including a profi), bu much has o do wih recovering he loss of purchasing power of he iniial sum, wha is he opporuniy cos of money, or he ime value of money. This ime value of money depends on he ineres rae, he percenage of addiional money you have o repay over and above he iniial money borrowed (he principal). Of course, we can look a hings from he reverse, from he perspecive ha you are no a borrower bu you are a saver. In an analogous way, you expec o receive in some fuure ime no only he sum of money iniially deposied, bu also an addiional sum which will compensae you for no using your money now, for foregoing he use of your money for curren consumpion. You require, in oher words, o be paid an ineres rae, he percenage of addiional money you require o receive over and above he iniial money saved (he principal). Example 3.1: Calculaion of he Fuure Value of a Sum Le s consider an example. Assume ha you pu 1,000 in a bank accoun a an ineres rae of 10% (i is wishful hinking, of course, ha you can earn 10% on a savings accoun bu le s amuse ourselves)! So, he iniial principal amoun oday (or presen value, PV, of he sum of money) is 1,000. How much would your accoun balance be in one year? Obviously he sum would be he iniial principal of 1,000 plus he ineres earned on his amoun a 10% (i.e., 100). The balance in oher words will be 1,100. Here is how we derive he answer: 1 PV PV ( r) 1, (0.10) 1, ,100 If you now decide o leave your money for a second year a he bank, and he ineres rae remains a 10%, he calculaions would be: 2 PV PV ( r) [ PV ( r)]( r) 1, (0.10) 1000(0.10)(0.10) 1, ,210 So, a he end of he second year you would have a sum of money equal o 1,210. Le s ake his process o a hird sage and assume ha you leave your money unouched for a hird year wih he same ineres rae. A he end of he hird year, your accoun balance will be: 3 = PV + PV(r) + [PV(r)](r) + {[PV(r)](r)}(r) = (0.10) +1000(0.10)(0.10)+ 1000(0.10)(0.10)(0.10) = 1, = 1,331 We can generalize he fuure value for a sum of money (or deposi in our example) o be received in periods in he fuure wih he following formula: PV ( 1 r) where: = he fuure value (he cumulaive balance of he bank accoun) PV = presen value (he iniial sum of money deposied in he bank) = he number of periods in he fuure over which he ineres income is earned r = he ineres rae on he deposis Le s verify ha his las formula when applied o an iniial sum of 1000 a an ineres rae of 10% will indeed give us afer 3 years he value of 1,331 found above:

3 3 1,000(1 0.10) = 1,000(1.10) 3 = 1,331 The erm in parenhesis in he formula, ha is (1+r), is wha we call he fuure value ineres facor (IF) which we can easily find in ineres facor ables such as Appendix 3 for differen values of i and, ha is (IF r, ). To demonsrae how we find he IF from ineres facor ables, we reproduce in Table 3.1 a secion of Appendix 3 ha includes he relevan coordinaes, namely discoun or ineres rae (r = 10%) and relevan ime period ( = 3). Table 3.1: Fuure (or Compound) Value of 1: IF(, r ) = ( 1 r) r 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Noice ha by going o he 10% column and moving down o find he row = 3 (or alernaively go o o row 3 and move o he righ wnil we find he column for r = 10%) we find ha he IF 10, 3 = Remember ha his is he value represening he erm in he parenhesis in he formula above. Therefore, we muliply his value (1.3310) imes he presen value of he iniial deposi ( 1,000) o ge he fuure value: = 1,000 * = 1,331 Fuure Value of a Sream of Cash Flows In he even ha here is a series of differen sums of money deposied for differen imes in he fuure a differen ineres raes (a realisic scenario since banks do pay differen ineres raes for differen sums (small vs. large deposis), and for differen ime periods (long-erm vs. shor-erm deposis), hen we can generalize he fuure value formula o accoun for a sream of flows (or deposis in our example) for periods in he fuure as follows: where: n 1 PV ( 1 r) = he fuure value (he cumulaive balance of he bank accoun) PV = presen value (he differen sums of money deposied in he bank) = is he number of years he deposis are locked and ineres is earned r = he differen ineres raes for each ype of deposis, and Σ = is he summaion operaor (he Greek capial leer sigma ). Example: The Power of Compounding To undersand he power of compounding, consider ha you go oday, say a age 25, o your invesmen adviser, James Dimiriou of Axia Financial Advisers, and pu a sum of 10,000 in an invesmen accoun which gives you an average reurn of 10% per year. Then, wihou ouching ha money (he principal or he ineres income) for he nex 50 years (!?), a age 75 you will have accumulaed a forune of 1,173,900 (yes, over one million Euros!). Jus o impress you more, consider now ha (somehow!) James Dimiriou, your financial adviser, manages o secure for you a 20% annual reurn. Then, in 50 years your iniial 10,000 will be worh 91,004,000 (yes, over niney-one million Euros!).

4 Alernaively, you may decide o pu every monh 100 (or 1,200 per year) in a high-yield muual fund earning on average 10% per year. Assuming he same life span as before, when you are 75 years old you will have accumulaed in your accoun close o 1.4 million ( 1,396,680 o be exac). Moral of he sory: i is no early enough o sar saving for your reiremen, no maer how difficul i is for you (say a 25 years old) o ge in ha frame of mind. The Rule of 72 The Rule of 72 is a very simple and useful rule of humb ha is used o deermine he ime (number of years) i akes for an invesmen o double is value. To find his, we simply divide he number 72 by he relevan ineres rae. For example, if he ineres rae is 6%, using he rule of 72 we find ha an invesmen of 1,000 will double in value in 12 years: 72 / 6 = 12. In oher words, in 12 years he iniial 1,000 will accumulae o 2,000. Le s verify his using he more sophisicaed annual compounding formula. The fuure ineres rae facor for 12 years a 6% is Thus: 1,000 * = 2,012. This is indeed very close o he 2,000 found by he rule of humb. If he marke ineres rae is 10%, hen using he rule of 72, we find ha he invesmen will double in 7.2 years. This is indeed very close o he answer found using he annual compounding formula (where he ineres facor for 10% and 7.2 years) is Thus, he = 1,000 * = 1,990. Discouning and he Presen Value of a Sum The presen value is he inverse of he fuure value. We look a hings in reverse. We sar wih sums of money received in some fuure period and seek o calculae wha hese sums of money are worh oday. For example, how much do you need o pu in he bank oday a 10% ineres if you wan o have in one year a sum of 1,100? The obvious answer ha follows from our previous calculaions is 1,000. We can ask effecively he same quesion in a slighly differen way: How much is 1,100 ha you will have nex year worh oday if he going ineres rae is 10%? The answer of course is again 1,000. The formula for he presen value is derived direcly from he fuure value formula by solving for PV. Recall ha he fuure value formula is = PV(1+r). By solving his equaion for PV we ge: 1 PV (1 r) (1 r) where PV = presen value (he iniial sum deposied in he bank) = he fuure value (he cumulaive balance of he bank accoun) = he number of periods in he fuure over which ineres income is earned r = he ineres rae on he deposi Example 3.2: Calculaion of he Presen Value In erms of he bank accoun example, if you wan o accumulae in your accoun a sum of 1,331 over hree years, wihou wihdrawing any money, a an ineres rae of 10%, obviously is presen value (wha i is worh oday, or wha you need o iniially deposi in your accoun) is much less. Applying he PV formula, we ll find his sum o be jus 1,000. PV = 1,331 [1/(1+0.10) 3 ] = 1,000 Again, we can hink of he 1,000 as he presen sum of money (he principal) ha you need o pu in a bank earning 10% so ha you accumulae in hree years a sum of 1,331. This is he fuure value hinking.

5 The erm in square brackes in he PV formula (ha is, 1/(1+r) ) is he presen value ineres facor (PVIF), which we can easily find in ineres facor saisical ables such as Appendix 1 for differen values of i and, ha is (PVIF r, ). To demonsrae how we find he PVIF from ineres facor ables, we reproduce in Table 3.2 a secion of Appendix 1 ha includes he relevan coordinaes, namely discoun or ineres rae which is 10% (r = 10%) and relevan ime, he number of years in he fuure ha he sum of money will be received, ( = 3). 1 Table 3.2: Presen value of 1 ( =1 o 5 and r=1 o 10). PVIF(, r ) = (1 r) r 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Noice ha by going o he 10% column and moving down o find he row = 3 we find ha he PVIF 10, 3 = Remember ha his is he value represening he erm in he square brackes in he PV formula above, namely 1. Therefore, we muliply his value (0.7513) imes he fuure value o be (1 r) received in hree years ( 1,331) o ge he presen value: PV = 1,331 * = or 1,000 The generalized formula for he presen value for a series of expeced fuure cash flows or reurns from an invesmen (wheher an individual s porfolio invesmen or a firm s invesmen in a business projec) is given by: PV n 1 1 r where, as before, PV = presen value (he iniial sum deposied in he bank) = he fuure value (he cumulaive sum o be received in some fuure period) = is he number of periods in he fuure over which he ineres income is earned r = he ineres rae on he deposi, and Σ = is he summaion operaor (he Greek capial leer sigma ). Noice ha since he discoun rae r (which for pracical purposes is usually represened by he ineres rae) and he ime variable are in he denominaor of he formula, he presen value of a sum of money (or he sum of a sream of money) is inversely relaed o he ineres rae and o ime. In oher words, wih all oher hings consan, say a a given, as he ineres rae increases he presen value falls. Also, a a given ineres rae r, as ime increases he presen value ges smaller. These relaionships beween r and, on he one hand, and PV on he oher hand are shown in Figure 3.1. We see, for insance ha he PV of 100 o be received in 5 years a 5% ineres is abou 78 (since he PVIF 5,5 = as found in Appendix 1) and is represened by poin A in Figure 3.1, which lies on he r=5% curve a he level of 5 years on he horizonal axis (he upper curve). On he oher hand, poin B represens he PV of 100 sill a 5% bu when he sum of 100 is received in 20 year from now. The PV is shown o be jus below 40 ( 37.7 o be exac). Thus, holding he ineres rae consan a 5% (he upper curve) and allowing ime o vary we see he inverse relaionship beween PV and ime (). In oher words:

6 when =5 ==> PV= 78.4, when =10 ==> PV= 61.4, and when =20 ==> PV= 37.7 Figure 3.1: Presen Value of a Sum of 100 for Differren r and PV of 100) A 40 B r = 20% r = 10% r = 5%, years If we now hold ime consan, say a =5 ( is on he horizonal axis), when r=10 (represened by he middle curve) he PV is abou 62 (PVIF 5,10 = 0.621), whereas when r=20 (he lower curve) he PV is abou 40 (PVIF 5, 20 = 0.402). You can verify his by looking a he secion of Appendix 1 as shown in Table 3.2. The presen value ineres facor (PVIF) becomes smaller for a given ime period as he ineres rae increases. For example, by looking a row =3 in Table 3.2, we see ha he PVIF of 100 will sar a 97.2 for 1% ineres rae (poin A) and i becomes smaller as we move o he righ, in oher words for higher and higher ineres raes. For example, a 8% he PV of 100 is abou 80 ( 79.4 o be exac) as shown by poin B, while a ineres rae of 20% he PV of 100 (a =3) is 57.9 (poin C). We show his inverse relaionship beween PV of 100 and ineres rae in Figure 3.2 (wih =3), while in Figure 3.3 we presen his inverse relaionship for differen ime periods ( = 1, 5, 10, 20). Figure 3.2: Presen Value of 100 wih = 3 and r = from 1% o 20%

7 Figure 3.3: Presen Value of 100 wih = 1-20 years and r = 1% o 15% Wih regard o he iming of receiving he fuure reurns, as deermine by he value of, i can also be verified from he secion of Appendix 1 shown in Table 3.2 ha a a given ineres rae as ime increases he PVIF ges smaller. For example, looking a r = 10 (las column), he PVIF is for = 1, and i becomes smaller and smaller as increases. For example, for = 5, he PVIF becomes Inuiively, his means ha he longer ino he fuure (greaer ) a sum of money is o be received, he smaller will be is presen worh (value). Case Sudy 3.1: Toal Reurns of a Sock Porfolio An individual wans o inves 10,000 in a porfolio of socks lised on he Cyprus Sock Exchange. He expecs ha here would be a sream of dividend flows from he firs year o he fifh year as follows: 500, 600, 750, 1000, and 750. The invesor believes ha he would be able o sell he porfolio a he end of he fifh year for 11,000. Assume an ineres rae of 10%. Quesions: Is ha a good invesmen? Should he invesor go ahead? A firs sigh one may hasily decide ha since here is some income and a he end he invesor recovers his iniial invesmen plus here is a profi of 1,000, he may be inclined o go ahead wih he invesmen? Should he? To answer such quesions one would need o find ou he PV of he fuure dividend flows and he cashingou sum of he invesmen of 11,000? The calculaions of he PV of each dividend sream over he five years and he sum colleced from selling he porfolio are presened in he las column of Table 3.3. Noice ha he numbers of he hird column are he presen value ineres facors (PVIF) found in Appendix 1. So, he PV of he invesmen ( 9,492.45) is less han he sum of money ha he invesor would be puing down ( 10,000). The invesor should no inves in his porfolio, which anyhow carries a lo of risk as far as he realizaion of hose expeced reurns. He is beer off invesing in he risk-free (or a leas lower-risk) money marke earning 10%. Table 3.3: Presen Value of a Firm s Profi Sream Year Annual Dividends PVIF PV of each Dividends , Cash ou 11, Presen Value of Invesmen = ΣPV i = 9, Case Sudy 3.2: Value of a Firm In preparing he sraegic plan of Omega Bank, he direcor of Planning and Developmen and he Financial Conroller of he bank expec ha he sream of profis over he nex 5 years will be as shown in

8 Table 3.4. They also expec ha a he end of he sixh year he asses and good will of he bank would be worh 2 billion. Assume ha he ineres rae is 6% and i is expeced o remain a ha level for he whole period. Table 3.4: Expeced Fuure Profis of Omega Bank Year Annual Profi Sream (in million) ,000 Quesions 1. Acceping ha he above profi and ineres rae scenario is rue and logical, if InerBank is ineresed o buy Omega Bank how much would i pay oday? 2. InerBank believes ha he rue cos of capial will be10% over he nex 6 years. How much would i be willing o offer o buy Omega Bank? Answer o Quesion 1: To answer his quesion, we need o calculae he sum of he presen values of each sream of profis over he five-year period as well as he presen value of he value of asses and good will expeced in he sixh year. These calculaions of he presen values are shown in he las column of Table 3.5. Noice ha he numbers of he hird column are he presen value ineres facors for 6% (PVIF, 6 ), which are found in Appendix 1 in he back of he Book. Table 3.5: Presen Value of Expeced Fuure Profis of Omega Bank Year Annual Profi Sream (in million) PVIF (r = 6%) PV of each Profi Sream ( million) , Adding he values in he las column would give us he presen value of he firm as: Value of Omega Bank = ΣPV i = million (or abou 2 billion) Thus, acceping ha he expeced profis and he ineres rae4s are rue and logical, he Inerbank would have o pay 2 billion o buy Omega Bank. Answer o Quesion 2: To answer his quesion, essenially we repea he process followed in Quesion 1, bu now he PVIFs will be for 10%. For pracice, he answer is lef o you. (Hin: since he discoun rae is higher, inuiively you should find ha he PV is smaller han before. Recall ha he PV is inversely relaed o he discoun rae! Example 3.3: Excel Spreadshee Pracice for Finding he Presen Value Task: Find he presen value of 10,000 o be received a he end of 5 years a 10% per year. Procedure: In he Formula funcion fx, ener =-PV( and you will ge he following promp from Excel (noice he negaive sign in fron of PV and he opening parenhesis afer PV): = PV(rae;nper;pm;fv;ype) where: rae (r) is he ineres rae per period (wih he % sign) nper () is he number of year in he fuure when paymen will be received

9 pm is he periodic paymen in he case of an annuiy. (Ener 0) fv is he fuure value, or a cash flow ype (omi i or se i o 0). OPTION 1: Ener he values direcly ino he formula (fx) separaed by semi colons. Noice he minus sign in fron of he formula and he % sign for r: OPTION 2: Ener cell numbers ino he formula (fx): Presen Value of an Annuiy Le s firs define an annuiy. An annuiy is a fixed sum of money received every year for a specified number of years. For example, he ineres paymens you receive on a bond (he coupon) is an annuiy (o be discussed in Chaper 6). If you win a scholarship of 20,000 per year for four years, hen his is an annuiy as well. If you are a beneficiary of a rus eniling you o receive a series of equal annual paymens of 5,000 for he following 5 years, hen his is also an annuiy. For sure, due o he opporuniy cos of money (he ime value of money) he presen value of hose five equal paymens of 5,000 is cerainly less han 25,000 (he nominal sum of 5,000 received for 5 years). Of course, you know addiionally ha he presen value would depend on he ineres rae, indeed he opporuniy cos of money. Bu, wha is he acual presen value of he 5,000 annuiy from he rus? There are wo ways o find he soluion. Indirec Mehod: The firs way is simply o use he sandard PV formula for a series of paymens, in which case each is 5,000 which is hen discouned by he appropriae PVIF for he specific and r. We know ha = 5 (years). Le s assume ha he ineres rae is 5% (r = 0.05). So from he PV ineres facor ables of Appendix 1, reproduced in Table 3.6 for your convenience, we simply go o he 5% column and hen go down ha column for years 1 hrough 5.

10 1 Table 3.6: Presen Value of 1 ( =1 o 5 and r=1 o 10). PVIF(, r ) = (1 r) r 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% So, using he indirec mehod, he presen value of he annuiy is found as follows: PV n 5,000 5,000 5,000 5,000 5,000 ( r (1.05) (1.05) (1.05) (1.05) 1 ) = (5000*0.9524) + (5000*0.9070) + (5000*0.8638) +(5000*0.8227) +(5000*0.7835) = = 21,647 You immediaely recognize, of course, ha he numbers in parenheses (in oher words, , , ) ha he 5,000 is muliplied wih are he PVIF for a 5% discoun rae. Direc Mehod: The second way is more direc and less ime-consuming han he mehod jus shown. We use a specially consruced presen value able for annuiies ha basically adds he PVIF sequenially for each addiional year, since he sum of money received is equal for all years. Essenially he formula for he PV of an annuiy is expressed as: PVA n r The summaion of he PVIFs is wha we call he presen value ineres facor for an annuiy (PVIFA). The ask of finding he presen value of an annuiy hen is simplified significanly since all we have o do is o find he PVIFA from he appropriae ables and hen muliply ha by he annuiy, he sum of money received each year ( 5,000 in our example). We reproduce in Table 3.7 a secion of Appendix 2 for he presen value ineres facors for an annuiy, PVIFA(, r ). Table 3.7: Presen Value of 1 Annuiy ( =1-5 and r=1-10). 1(1 r) PVIFA,r r r 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Le us use he PV formula for an annuiy and validae ha we will indeed ge he same answer as in he indirec way. We find from he above able ha he presen value ineres facor for an annuiy for 5 years a 5% ( PVIFA 5,5 ) is So, PVA n = 5,000 * = 21,647 (same as before). r

11 Case Sudy 3.3: Buying vs. Leasing Many car dealers offer you he choice o buy or o lease cars. Assume ha you are looking ino he wo opions. The local dealer of BMW sells he new model for 20,000 or leases i for 2,500 per year for 5 years and a he end of he five-year period you can buy he car for 15,000. Assume ha he ineres rae on car loans is 10% and i is expeced o remain a ha level for he nex five years. Quesion: Is buying or leasing he bes opion for you? To answer his quesion, we need o calculae he sum of he PV of he five insallmens and he PV of he selling price a he end of he five-year period. Since we have five equal insallmens, essenially we are faced wih an annuiy. From Appendix 2 we find ha he PVIFA for = 5 and r = 10% is , while he PVIF in =5 and r =10% is So we have he following calculaions: PV of leasing insallmens = 2,500* = 9,477 PV of selling price = 15,000* = 9,314 Adding hese presen values we find ha he PV of he leasing opion is 18,791. Comparing his wih he purchase price of 20,000 we see ha leasing is he preferred (less expensive) opion. Case Sudy 3.4: Winning he Loo Assume you win 1million in he Loo. For simpliciy le s also assume ha here is no axaion so you ge he full amoun. The usual pracice of he Loery Auhoriies is o appropriae he winnings over a number of years by advancing he winner an equal annual amoun. Le s say ha he Loo Auhoriies in Cyprus pay he winners in en equal insallmens. In your case, since you won 1million, you will be receiving 100,000 every year for he following 10 years. When you go o he Loo Auhoriies o presen your winning icke, he auhoriies presen you wih a proposal: Accep heir sandard policy of paying he winnings wih equal insallmens over 10 years, or pay you a lump sum now of 700,000. By giving you his opion, he Loo Auhoriies, of course, know very well ha money in he fuure is worh less han money oday. Quesions: 1. Which of he wo proposiions is more valuable a 5%? A 10%? 2. If you were o make a couner-offer which lump-sum would you recommend? Why his amoun? 3. Is here a lump-sum ha would basically make you indifferen as o which offer o accep? Presen Value of Perpeuiy A perpeuiy is a special case of an annuiy when he equal sum of money is received forever, in oher words in perpeuiy. Though i may be difficul o imagine siuaions where a sum of money is received indefiniely, here are siuaions of special ypes of bonds (perpeual bonds, known in he UK as consuls, which do no repay he principal a any ime), or special preferred sock which are issued wihou a mauriy (called perpeual preferred sock), or russ ha do pay a fixed amoun of money indefiniely o beneficiaries and heir heirs. In hose siuaions he presen value of hese perpeual flows can be calculaed using a simple formula: PV Perpeuiy (1 r) 1 (1 r) 2 (1 r) 3... (1 r) PV Perpeuiy r

12 Example 3.4: Presen Value of a Perpeuiy As an example, consider ha you are he privileged beneficiary of a rus lef o you by a rich (very rich!) uncle ha eniles you o receive 100,000 annually forever, in oher words for he res of your life and he life of your heirs! Wha is he value oday of ha newfound forune? Well, now you have a mehod o calculae i by simply applying he above simple formula for he presen value of perpeuiy: PV Perpeuiy r ,000,000 No bad o discover one morning ha you are a millionaire (over your life ime)! Sudy Quesions Muliple Choice Quesions 1. If you pu 5,000 in a bank accoun a 5% ineres, how much will you have in his accoun in hree years? A. 5,5125 B. 5,7880 C. 6,1080 D. 5, If you pu 500 every year in a bank accoun a 4% ineres, approximaely how much will you have in his accoun in 10 years? A. 6,000 B. 5,000 C. 7,000 D. 10, The erm (1 + i) n is called A. PVIF B. IF C. PVIFA D. IFA 4. Wih coninuous compounding a 10 percen for 30 years, he fuure value of an iniial invesmen of 2,000 is approximaely: A. 34,898 B. 40,171 C. 164,521 D. 328,282

13 5. You open a savings accoun ha pays 4.5% annually. How much mus you deposi each year in order o have 50,000 in five years? A. 8,321 B. 9,629 C. 8,636 D. 9,140 E. 6, An annuiy pays 4,000 a year for he nex 20 years. The ineres raes is 8% over his period. The presen value of he annuiy is approximaely: A. 32,562 B. 40,322 C. 39,272 D. 80, The presen value of a fuure sum of money is lower A. as ineres raes increase and as he ime of he paymen increases. B. as ineres raes increase and as he ime of he paymen decreases. B. as ineres raes decrease and as he ime of he paymen increases. D. as ineres raes decrease and as he ime of he paymen decreases. 8. You expec o receive in hree years an amoun of 5,000. If he ineres rae in he meanime increases, he presen value of ha fuure amoun o you would A. rise B. fall C. no change D. (a) and (b) are possible 9. Wha is he value of a perpeuiy of 400 a year if he required reurn is 10%? A. 3,500 B. 4,000 C. 14,500 D. 3,200 E. 40, Andreas, George and Chrisina each receive 5,000 from heir grandfaher. Andreas pus he 5,000 in a bank for 20 years a 4% ineres, George pus his 5,000 in a muual fund for 15 years a 6% reurn, while Chrisina invess her 5,000 in he sock marke for 10 years earning 10% oal reurn. Which one will accumulae he mos money in heir invesmen accoun a he end of heir chosen invesmen period? A. Andreas B. George C. Chrisina D. All hree accumulae he same amoun. 11. Wha is he PV of 8,000 o be paid a he end of hree years if he ineres rae is 11%? A. 5,850 B. 4,872 C. 6,725 D. 1,842 E. 1, Assume ha he ineres rae is 5%. Which of he following cash-flows will you prefer? Year 1 Year 2 Year 3 Year 4 A

14 B C D. Any of he above, since hey each sum o 1, The "Rule of 72" says ha if you earn 8% per year, your money will double in: A. 12 years B. 6 years C. 8 years D. 9 years E. 72 years 18. If he ineres rae is 5 percen, which of he following has he greaes presen value? A paid in wo years B. 500 paid in one year plus 500 paid in wo years C. 300 paid oday plus 400 paid in one year plus 400 paid in wo years D oday 15. As ineres raes go up, he presen value of a sream of fixed cash flows A. goes down B. goes up C. says he same D. canno ell 16. Which of he following is he correc expression for finding he presen value of a 1000 paymen 5 years from oday if he ineres rae is 4 percen? A /(1.05) 4 B (1.04) 5 C 1000 /(1.04) 5 D /(1.05) 4 E. None of he above is correc 17. You win 200,000 on he Loery. You can receive he enire amoun now or in en equal paymens of 25,000 per year saring one year from oday. Wha should you do if he ineres is 5% and is expeced o remain sable over he period? A. Take he opion wih insallmens of 25,000. B. Take he 200,000 up fron. C. Take he opion wih insallmens of 25,000 as long as you expec he ineres rae o increase over 5% in he following years D. Canno be deermined wih he informaion given. Essays, Problems and Applicaions 1.Wha is he PVIF for i = 8% and n = 10? 2. Wha is he PVIFA (annuiy) for i = 8% and n = 10? 3. Suppose you make an invesmen of 10,000. The firs year he invesmen reurns 15%, he second year i reurns 2%, and he hird year in reurns 10%. How much would his invesmen be worh a he end of hree years, assuming no wihdrawals are made? 4. You have jus won 5,000 playing he loery. You are going o save his for your reiremen in 30 years. If your invesmen yields 12% (and all of i is reinvesed in your reiremen accoun), how much will you have accumulaed (saved) for your reiremen.

15 5. Lionel Messi has received wo offers, one from Barcelona and one from Mancheser Unied for playing soccer. He wans o selec he bes offer, based on consideraions of money alone. The offer from Barca is for 10m o be received as follows: 2m a year for 5 years. The offer from MU is for 12m o be received as follows: 1.5m a year for four years and 6m o be received in year 5. Wha would you advise Lionel Messi o do? Assume hree ineres rae scenaria: 5%, 8% and 10%. 6. You are a financial planner and one of your cliens is abou o reire. He has he opion from his pension scheme o receive upon reiremen a lump sum of 200,000 or an annuiy of 25,000 for en years. Wha would you advise your clien o do (in oher words, which is worh more in presen value erms), if he ineres rae is 5% for he whole period? Assume ha no axes are paid in eiher case. 7. Assume ha you win 1,000,000 in he loery and you have he choice o ge 500,000 oday or accep o receive 50,000 per year for he nex 20 years. You believe ha he ineres rae will remain fairly consan a 5%. Which is he beer choice for you? 8. You inves your money in a bank accoun a a nominal annual rae of 7.2%, compounded annually. How many years will i ake for you o double your money? 9. Your long-forgoen uncle from he Unied Saes, Uncle Tom, died and lef you 100,000 (!!). Bu here is a cach! The condiion in his will saes ha for he nex 10 years you canno spend he money o buy goods (your favourie spors car, for example), bu insead o pu he money in an invesmen accoun. Le s say ha he invesmen earns 10% over he nex 10 years. How much will you have a he end of 10 years? 10. You are weny-five years old. You decide o sar puing 1,000 per year ino a reiremen accoun and will coninue o do so unil you are sixy-five, for a oal of 40 years. Assume ha you inves in fixed income securiies and he accoun earns 5% annually. How much will you have in he accoun upon your reiremen? 11. Your bes friend Andreas (age 25) is a smoker and spends 5 a day on cigarees. You convince him (somehow!?) o qui his bad habi and pu ha money (which amouns o 5*365 days = 1825 per year) insead in an invesmen accoun earning him 10% per year. How much will he have in he accoun a age 65? Do you hink he will be hankful for convincing him o qui smoking? 12. You deposi 10,000 in a bank accoun ha yields 6 percen compound ineres. a) How much will you accumulae in your accoun in 5 years? b) Using he compound formula, in how many years will your invesmen double? c) Using he Rule of 72, in how many years will your invesmen double? 13. In 2000 he average uiion for one year a European Universiy was 2,000. Ten years laer, in 2010, he average cos is abou 10,000. Wha is he growh rae in uiion over he 10-year period? Noe: Figures are hypoheical! 14. Sunrise Resors Ld is hinking o build a new luxury hoel complex in Limassol, Cyprus complee wih marina, spa, and oher modern faciliies and is esimaed o cos 100 million oday. The Company s financial manager esimaes ha he firm will have an income of 20 million per year for he following 10 years. If he expeced ineres over his period will remain consan a 5%, should Sunmiles Resors underake o build he hoel complex? 15. Venus Airways is considering puing hree differen roues on is scheduled flighs each of which will cos i 10 million for addiional airplanes and crew. Roue A will generae 12 million in revenue a he end of one year. Roue B will generae 15 million in revenue a he end of wo years. Roue C will generae 18 million in revenue a he end of hree years. Which opion should he firm choose?

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