VALUATION OF FINANCIAL ASSETS

Transcription

1 P A R T T W O As a parter for Erst & Youg, a atioal accoutig ad cosultig firm, Do Erickso is i charge of the busiess valuatio practice for the firm s Southwest regio. Erickso s sigle job for the firm is valuig compaies, or what he calls a dedicated valuatio practice. He has atioal resposibilities for valuig certai idustries. I this role, he works with maagers of major compaies from across the atio. While he has valued busiesses of all kids ad sizes, he primarily focuses o certai idustries, such as oil ad gas firms ad major league sports frachises. For istace, he has valued a umber of major league sports clubs, icludig NFL, NBA, NHL, ad major league baseball teams. Erickso explais that, valuig a compay is a iterestig, eve itriguig, process. It requires havig a uderstadig of the outlook of the specific busiess ad the ature of the idustry, which allows us to predict multiple scearios of the future. More specifically, it is crucial to uderstad the two key value drivers: the compay s risk ad expected growth. If I were to ame oe variable that matters the most i estimatig a firm s value ad is the most difficult thig to determie, it would be our estimate of the firm s termial growth rate. Erickso cotiues, I valuig a busiess, we use several methods, the most importat beig a discouted cash flow approach. I am a true believer that the value of a compay is the preset value of the firm s expected future free cash flows. Whe asked about the discout rate used i fidig the preset value of a firm s cash flows, he says, I almost all cases, we use the effective market discout rate. I this approach, we estimate the discout rate for compaies that have recetly bee sold. Alteratively, we may compute a imputed discout rate for the idustry. I both cases, we will the adjust the discout rate for compay-specific risk. The greater the iheret risk, the higher the discout rate. Let me also say that it is essetial that a operatig maager be familiar with what determies firm value. Due to recet chages i accoutig rules for ay firm that is audited, accoutig ad fiace are becomig icreasigly itercoected. The chages will make it all the more importat for a maager to uderstad how much value is beig created, ot just at the firm level, but also at the divisio level. The chapters i this part of the text will provide you a basic uderstadig of valuatio cocepts ad procedures the same oes that Erickso uses i valuig a compay, or almost ay busiess asset for that matter. We will explai the role of risk, which Erickso idetifies as a matter of importace, ad the see how to value a compay s bods ad its stock. VALUATION OF FINANCIAL ASSETS

2 134 C H A P T E R 5

3 Chapter 5 The Time Value of Moey Compoud Iterest ad Future Value Preset Value Auities Auities Due Amortized Loas Compoud Iterest with Noaual Periods Preset Value of a Ueve Stream Perpetuities Fiace ad the Multiatioal Firm: The Time Value of Moey Learig Objectives After readig this chapter, you should be able to: 1. Explai the mechaics of compoudig, that is, how moey grows over time whe it is ivested. 2. Discuss the relatioship betwee compoudig ad brigig moey back to the preset. 3. Defie a ordiary auity ad calculate its compoud or future value. 4. Differetiate betwee a ordiary auity ad a auity due, ad determie the future ad preset value of a auity due. 5. Determie the future or preset value of a sum whe there are oaual compoudig periods. 6. Determie the preset value of a ueve stream of paymets. 7. Determie the preset value of a perpetuity. 8. Explai how the iteratioal settig complicates the time value of moey. I busiess, there is probably o other sigle cocept with more power or applicatios tha that of the time value of moey. I his ladmark book, A History of Iterest Rates, Sidey Homer oted that if $1,000 were ivested for 400 years at 8 percet iterest, it would grow to $23 quadrillio that would work out to approximately $5 millio per perso o earth. He was ot givig a pla to make the world rich, but effectively poitig out the power of the time value of moey. The time value of moey is certaily ot a ew cocept. Bejami Frakli had a good uderstadig of how it worked whe he left 1,000 each to Bosto ad Philadelphia. With the gift, he left istructios that the cities were to led the moey, chargig the goig iterest rate, to worthy appretices. The, after the moey had bee ivested i this way for 100 years, they were to use a portio of the ivestmet to build somethig of beefit to the city ad hold some back for the future. Two hudred years later, Frakli s Bosto gift resulted i the costructio of the Frakli Uio, has helped coutless medical studets with loas, ad still has over $3 millio left i the accout. Philadelphia, likewise, has reaped a sigificat reward from his gift. Bear i mid that all this has come from a gift of 2,000 with some serious help from the time value of moey. The power of the time value of moey ca also be illustrated through a story Adrew Tobias tells i his book Moey Agles. There he tells of a peasat who wis a chess touramet put o by the kig. The kig the asks the peasat what he would like as the prize. The peasat aswers that he would like for his village oe piece of grai to be placed o the first square of his chessboard, two pieces of grai o the secod square, four pieces o the third, eight o the fourth, ad so forth.the kig, thikig he was gettig off easy, pledged o his word of hoor that it would be doe. Ufortuately for the kig, by the time all 64 squares o the chessboard were filled, there were 18.5 millio trillio grais of wheat o the board the kerels were compoudig at a rate of 100 percet over the 64 squares of the chessboard. Needless to say, o oe i the village ever wet hugry, i fact, that is so much wheat that if the kerels were oe-quarter ich log (quite frakly, I have o idea how log a kerel of wheat is, but Adrew Tobias guess is oe-quarter ich), if laid ed to ed, they could stretch to the su ad back 391,320 times. Uderstadig the techiques of compoudig ad movig moey through time are critical to almost every busiess decisio. It will help you to uderstad such varied thigs as how stocks ad bods are valued, how to determie the value of a ew project, how much you should save for childre s educatio, ad how much your mortgage paymets will be. 135

4 I the ext five chapters, we focus o determiig the value of the firm ad the desirability of ivestmet proposals. A key cocept that uderlies this material is the time value of moey; that is, a dollar today is worth more tha a dollar received a year from ow. Ituitively this idea is easy to uderstad. We are all familiar with the cocept of iterest. This cocept illustrates what ecoomists call a opportuity cost of passig up the earig potetial of a dollar today. This opportuity cost is the time value of moey. I evaluatig ad comparig ivestmet proposals, we eed to examie how dollar values might accrue from acceptig these proposals. To do this, all dollar values must first be comparable; because a dollar received today is worth more tha a dollar received i the future, we must move all dollar flows back to the preset or out to a commo future date. A uderstadig of the time value of moey is essetial, therefore, to a uderstadig of fiacial maagemet, whether basic or advaced. I this chapter we develop the tools to icorporate Priciple 2: The Time Value of Moey A Dollar Received Today Is Worth More Tha a Dollar Received i the Future ito our calculatios. I comig chapters we use this cocept to measure value by brigig the beefits ad costs from a project back to the preset. O B J E C T I V E 1 compoud iterest Compoud Iterest ad Future Value Most of us ecouter the cocept of compoud iterest at a early age. Ayoe who has ever had a savigs accout or purchased a govermet savigs bod has received compoud iterest. Compoud iterest occurs whe iterest paid o the ivestmet durig the first period is added to the pricipal; the, durig the secod period, iterest is eared o this ew sum. For example, suppose we place $100 i a savigs accout that pays 6 percet iterest, compouded aually. How will our savigs grow? At the ed of the first year we have eared 6 percet, or $6 o our iitial deposit of $100, givig us a total of $106 i our savigs accout. The mathematical formula illustratig this pheomeo is FV1 = PV( 1 + i) (5-1) where FV 1 = the future value of the ivestmet at the ed of oe year i = the aual iterest (or discout) rate PV = the preset value, or origial amout ivested at the begiig of the first year I our example FV1 = PV( 1 + i) = $ 100( ) = $ 100( 1. 06) = $ 106 Carryig these calculatios oe period further, we fid that we ow ear the 6 percet iterest o a pricipal of $106, which meas we ear $6.36 i iterest durig the secod year. Why do we ear more iterest durig the secod year tha we did durig the first? Simply because we ow ear iterest o the sum of the origial pricipal, or preset value, ad the iterest we eared i the first year. I effect we are ow earig iterest o iterest; this is the cocept of compoud iterest. 136 Part Two: Valuatio of Fiacial Assets

5 Examiig the mathematical formula illustratig the earig of iterest i the secod year, we fid which, for our example, gives FV = FV ( + i ) FV 2 = $ 106( 1. 06) = $ (5-2) Lookig back at equatio (5-1), we ca see that FV 1, or $106, is actually equal to PV(1 + i), or $100(1 +.06). If we substitute these values ito equatio (5-2), we get FV2 = PV( 1 + i)( 1 + i) 2 = PV( 1 + i) (5-3) Carryig this forward ito the third year, we fid that we eter the year with $ ad we ear 6 percet, or $6.74 i iterest, givig us a total of $ i our savigs accout. Expressig this mathematically: FV3 = FV2( 1 + i) = $ ( 1. 06) = $ (5-4) If we substitute the value i equatio (5-3) for FV 2 ito equatio (5-4), we fid FV3 = PV( 1 + i)( 1 + i)( 1 + i) 3 = PV( 1 + i) (5-5) By ow a patter is becomig apparet. We ca geeralize this formula to illustrate the value of our ivestmet if it is compouded aually at a rate of i for years to be FV = PV( 1 + i) (5-6) where FV = the future value of the ivestmet at the ed of years = the umber of years durig which the compoudig occurs i = the aual iterest (or discout) rate PV = the preset value or origial amout ivested at the begiig of the first year Table 5-1 illustrates how this ivestmet of $100 would cotiue to grow for the first 10 years at a compoud iterest rate of 6 percet. Notice how the amout of iterest eared aually icreases each year. Agai, the reaso is that each year iterest is received o the sum of the origial ivestmet plus ay iterest eared i the past. Whe we examie the relatioship betwee the umber of years a iitial ivestmet is compouded for ad its future value as show graphically i Figure 5-1, we see that we ca icrease the future value of a ivestmet by either icreasig the Year Begiig Value Iterest Eared Edig Value 1 $ $ 6.00 $ Table 5-1 Illustratio of Compoud Iterest Calculatios Chapter 5: The Time Value of Moey 137

6 Figure 5-1 Future Value of $100 Iitially Deposited ad Compouded at 0, 5, ad 10 Percet % Future value (dollars) % 0% Year umber of years for which we let it compoud or by compoudig it at a higher iterest rate. We ca also see this from equatio (5-6) because a icrease i either i or while PV is held costat results i a icrease i FV. Keep i mid that future cash flows are assumed to occur at the ed of the time period durig which they accrue. For example, if a cash flow of $100 occurs i time period 5, it is assumed to occur at the ed of time period 5, which is also the begiig of time period 6. I additio, cash flows that occur i time t = 0 occur right ow; that is, they are already i preset dollars. If we place $1,000 i a savigs accout payig 5 percet iterest compouded aually, how much will our accout accrue to i 10 years? Substitutig PV = $1,000, i = 5 percet, ad = 10 years ito equatio (5-6), we get FV = PV( 1 + i) = $, 1 000( ) = $, 1 000( ) = $, (5-6a) Thus, at the ed of 10 years we will have $1, i our savigs accout. future-value iterest factor Time Value of Moey (TVM) Tables Solutio Because determiig future value ca be quite time-cosumig whe a ivestmet is held for a umber of years, the future-value iterest factor for i ad (FVIF i, ) defied as (1 + i), has bee compiled i the back of the book for various values of i ad. A abbreviated compoud iterest or future-value iterest factor table appears i Table 5-2, with a more comprehesive versio of this table appearig i Appedix B at the back of this book. Alteratively, the FVIF i, values could easily be determied usig a calculator. Note that the compoudig factors give i these tables represet the value of $1 compouded at rate i at the ed of the th year. Thus, to calculate the future value of a iitial ivestmet, we eed 138 Part Two: Valuatio of Fiacial Assets

7 N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Table 5-2 FVIF i,, or the Compoud Sum of $1 oly to determie the FVIF i, usig a calculator or the tables at the ed of the text ad multiply this times the iitial ivestmet. I effect, we ca rewrite equatio (5-6) as follows: FV = PV( FVIF ) i, (5-6b) If we ivest $500 i a bak where it will ear 8 percet compouded aually, how much will it be worth at the ed of 7 years? Lookig at Table 5-2 i the row = 7 ad colum i = 8 percet, we fid that FVIF 8%, 7 yr has a value of Substitutig this ito equatio (5-6b), we fid FV = PV( FVIF8%, 7 yr ) = $ 500( ) = $ 857 Thus, we will have $857 at the ed of 7 years. I the future we will fid several uses for equatio (5-6); ot oly ca we fid the future value of a ivestmet, but we ca also solve for PV, i, or. I ay case, you will be give three of the four variables ad will have to solve for the fourth. As you read through the chapter it is a good idea to solve the problems as they are preseted. If you just read the problems, the priciples behid them ofte do ot sik i. The material preseted i this chapter forms the basis for the rest of the course; therefore, a good commad of the cocepts uderlyig the time value of moey is extremely importat. Let s assume that the DaimlerChrysler Corporatio has guarateed that the price of a ew Jeep will always be $20,000, ad you d like to buy oe but curretly have oly $7,752. How may years will it take for your iitial ivestmet of $7,752 to grow to $20,000 if it is ivested at 9 percet compouded Chapter 5: The Time Value of Moey 139

8 aually? We ca use equatio (5-6b) to solve for this problem as well. Substitutig the kow values i equatio (5-6b), you fid FV = PV( FVIFi, ) $ 20, 000 = $ 7, 752( FVIF9%, yr ) $ 20, 000 $ 7, 752( FVIF9%, yr ) = $ 7, 752 $ 7, = FVIF9 yr Thus you are lookig for a value of 2.58 i the FVIF i, tables, ad you kow it must be i the 9% colum. To fiish solvig the problem, look dow the 9% colum for the value closest to You fid that it occurs i the = 11 row. Thus it will take 11 years for a iitial ivestmet of $7,752 to grow to $20,000 if it is ivested at 9 percet compouded aually. %, Now let s solve for the compoud aual growth rate, ad let s go back to that Jeep that always costs $20,000. I 10 years, you d really like to have $20,000 to buy a ew Jeep, but you oly have $11,167. At what rate must your $11,167 be compouded aually for it to grow to $20,000 i 10 years? Substitutig the kow variables ito equatio (5-6b), you get FV = PV( FVIFi, ) $ 20, 000 = $ 11167, ( FVIFi, 10 yr ) $ 20, 000 $ 11167, ( FVIFi, 10 yr ) = $ 11167, $ 11167, = FVIF i, 10 yr You kow to look i the = 10 row of the FVIF i, tables for a value of 1.791, ad you fid this i the i = 6% colum. Thus, if you wat your iitial ivestmet of $11,167 to grow to $20,000 i 10 years, you must ivest it at 6 percet. At what rate must $100 be compouded aually for it to grow to $ i 10 years? I this case we kow the iitial ivestmet, PV = $100; the future value of this ivestmet at the ed of years, FV = $179.10; ad the umber of years that the iitial ivestmet will compoud for, = 10 years. Substitutig ito equatio (5-6), we get FV = PV( 1 + i) $ = $ 100( 1 + i) = ( 1 + i) We kow to look i the = 10 row of the FVIF i, table for a value of 1.791, ad we fid this i the i = 6 percet colum. Thus, if we wat our iitial ivestmet of $100 to accrue to $ i 10 years, we must ivest it at 6 percet. Just how powerful is the time value of moey? Mahatta Islad was purchased by Peter Miuit from the Idias i 1624 for $24 i kickkacks ad jewelry. If at the ed of 1624 the Idias had ivested their $24 at 8 percet compouded aually, it would be worth over $111 trillio today (by the ed of 2003, 379 years later). That s certaily eough to buy back all of Mahatta i fact, with $111 trillio i the bak, the $80 billio to $90 billio you d have to pay to buy back all of Mahatta would seem like pocket chage. This story illustrates the icredible power of time i compoudig. There simply is o substitute for it Part Two: Valuatio of Fiacial Assets

9 Fiacial Calculator Solutio Time value of moey calculatios ca be made simple with the aid of a fiacial calculator. I solvig time value of moey problems with a fiacial calculator, you will be give three of four variables ad will have to solve for the fourth. Before presetig ay solutios usig a fiacial calculator, we itroduce the calculator s five most commo keys. (I most time value of moey problems, oly four of these keys are relevat.) These keys are N I/Y PV PMT FV where: N I/Y PV FV PMT Stores (or calculates) the total umber of paymets or compoudig periods. Stores (or calculates) the iterest or discout rate. Stores (or calculates) the preset value of a cash flow or series of cash flows. Stores (or calculates) the future value, that is, the dollar amout of a fial cash flow or the compoud value of a sigle flow or series of cash flows. Stores (or calculates) the dollar amout of each auity paymet deposited or received at the ed of each year. Taki It to the Net There are a umber of calculators o the Iteret aimed at ot oly movig moey through time, but also solvig other problems that ivolve the time value of moey. Oe great site for fiacial calculators is Kipliger Olie Calculators com/calc/calchome.html. Whe you use a fiacial calculator, remember that outflows geerally have to be etered as egative umbers. I geeral, each problem will have two cash flows: oe a outflow with a egative value, ad oe a iflow with a positive value. The idea is that you deposit moey i the bak at some poit i time (a outflow), ad at some other poit i time you take moey out of the bak (a iflow). Also, every calculator operates a bit differetly with respect to eterig variables. Needless to say, it is a good idea to familiarize yourself with exactly how your calculator fuctios. As previously stated, i ay problem you will be give three of four variables. These four variables will always iclude N ad I/Y; i additio, two out of the fial three variables PV, FV, ad PMT will also be icluded. To solve a time value of moey problem usig a fiacial calculator, all you eed to do is eter the appropriate umbers for three of the four variables ad the press the key of the fial variable to calculate its value. It is also a good idea to eter zero for ay of the five variables ot icluded i the problem i order to clear that variable. Now let s solve the previous example usig a fiacial calculator. We were tryig to fid at what rate $100 must be compouded aually for it to grow to $ i 10 years. The solutio usig a fiacial calculator would be as follows: Taki It to the Net Aother excellet site that provides fiacial calculators is Moey Advisors This site has, amog other thigs, a loa calculator, retiremet spedig calculator, preset value calculator, ad calculators for future value of a auity calculatio. CALCULATOR SOLUTION Data Iput Fuctio Key 10 N 100 PV FV 0 PMT Fuctio Key Aswer CPT I/Y 6.00% Ay of the problems i this chapter ca easily be solved usig a fiacial calculator; ad the solutios to may examples usig a Texas Istrumets BAII Plus fiacial calculator are provided i the margis. If you are usig the TI BAII Plus, make sure that you have selected both the END MODE ad oe paymet per Chapter 5: The Time Value of Moey 141

10 year (P/Y = 1). This sets the paymet coditios to a maximum of oe paymet per period occurrig at the ed of the period. Oe fial poit, you will otice that solutios usig the preset-value tables versus solutios usig a calculator may vary slightly a result of roudig errors i the tables. For further explaatio of the TI BAII Plus, see Appedix A at the ed of the book. The cocepts of compoud iterest ad preset value follow us through the remaider of this book. Not oly do they allow us to determie the future value of ay ivestmet, but also they allow us to brig the beefits ad costs from ew ivestmet proposals back to the preset ad thereby determie the value of the ivestmet i today s dollars. Obviously, the choice of the iterest rate plays a critical role i how much a ivestmet grows, but do small chages i the iterest rate have much of a impact o future values? To aswer this questio, let s look back at Peter Miuit s purchase of Mahatta. If the Idias had ivested their $24 at 10 percet rather tha 8 percet compouded aually at the ed of 1624, they would have over $117 quadrillio by the ed of 2003 (379 years later). That is 117 followed by 15 zeros, or $117,000,000,000,000,000. Actually, that is eough to buy back ot oly Mahatta Islad, but the etire world ad still have plety left over! Now let s assume a lower iterest rate say 6 percet. I that case the $24 would have oly grow to a mere $93.6 billio less tha oe oe-hudredth of what it grew to at 8 percet, ad less tha oe millioth of what it would have grow to at 10 percet. With today s real estate prices, you d have a tough time buyig Mahatta, ad if you did, you probably could t pay your taxes! To say the least, the iterest rate is extremely importat i ivestig. Spreadsheet Solutio Without questio, i the real world most calculatios ivolvig movig moey through time will be carried out with the help of a spreadsheet. Although there are several competig spreadsheets, the most popular oe is Microsoft Excel. Just as with the keystroke calculatios o a fiacial calculator, a spreadsheet ca make easy work of most commo fiacial calculatios. Listed here are some of the most commo fuctios used with Excel whe movig moey through time: Calculatio Preset Value Future Value Paymet Number of Periods Iterest Rate where: rate umber of periods paymet future value preset value type guess Formula = PV (rate, umber of periods, paymet, future value, type) = FV (rate, umber of periods, paymet, preset value, type) = PMT (rate, umber of periods, preset value, future value, type) = NPER (rate, paymet, preset value, future value, type) = RATE (umber of periods, paymet, preset value, future value, type, guess) = i, the iterest rate or discout rate =, the umber of years or periods = PMT, the auity paymet deposited or received at the ed of each period = FV, the future value of the ivestmet at the ed of periods or years = PV, the preset value of the future sum of moey = whe the paymet is made, (0 if omitted) 0 = at ed of period 1 = at begiig of period = a startig poit whe calculatig the iterest rate; if omitted, the calculatios begi with a value of 0.1 or 10% 142 Part Two: Valuatio of Fiacial Assets

11 Just like with a fiacial calculator, the outflows have to be etered as egative umbers. I geeral, each problem will have two cash flows: oe positive ad oe egative. The idea is that you deposit moey at some poit i time (a outflow or egative value), ad at some poit later i time, you withdraw your moey (a iflow or positive value). For example, let s look back o the example o page139. Etered values i cell d13: =FV(d7,d8,d9,-d10,d11) Etered values i cell d15: =RATE(d9,d10,-d11,d12,d13,d14) 1. Priciple 2 states that a dollar received today is worth more tha a dollar received i the future ; explai this statemet. 2. How does compoud iterest differ from simple iterest? 3. Explai the formula FV = PV(1 + i) Preset Value Up to this poit we have bee movig moey forward i time; that is, we kow how much we have to begi with ad are tryig to determie how much that sum will grow i a certai umber of years whe compouded at a specific rate. We are ow goig to look at the reverse questio: What is the value i today s dollars of a sum of moey to be received i the future? The aswer to this questio will help us deter- 2 O B J E C T I V E Chapter 5: The Time Value of Moey 143

12 preset value mie the desirability of ivestmet projects i Chapters 9 ad 10. I this case we are movig future moey back to the preset. We will determie the preset value of a lump sum, which i simple terms is the curret value of a future paymet. I fact, we will be doig othig other tha iverse compoudig. The differeces i these techiques come about merely from the ivestor s poit of view. I compoudig, we talked about the compoud iterest rate ad the iitial ivestmet; i determiig the preset value, we will talk about the discout rate ad preset value of future cash flows. Determiatio of the discout rate is the subject of Chapter 11 ad ca be defied as the rate of retur available o a ivestmet of equal risk to what is beig discouted. Other tha that, the techique ad the termiology remai the same, ad the mathematics are simply reversed. I equatio (5-6) we were attemptig to determie the future value of a iitial ivestmet. We ow wat to determie the iitial ivestmet or preset value. By dividig both sides of equatio (5-6) by (1 + i), we get PV = 1 FV ( 1 + i) (5-7) CALCULATOR SOLUTION Data Iput Fuctio Key 10 N 6 I/Y 500 FV 0 PMT Fuctio Key Aswer CPT PV where FV = the future value of the ivestmet at the ed of years = the umber of years util the paymet will be received i = the aual discout (or iterest) rate PV = the preset value of the future sum of moey Because the mathematical procedure for determiig the preset value is exactly the iverse of determiig the future value, we also fid that the relatioships amog, i, ad PV are just the opposite of those we observed i future value. The preset value of a future sum of moey is iversely related to both the umber of years util the paymet will be received ad the discout rate. This relatioship is show i Figure 5-2. Although the preset value equatio [equatio (5-7)] is used extesively i evaluatig ew ivestmet proposals, it should be stressed that the preset value equatio is actually the same as the future value or compoudig equatio [equatio (5-6)], where it is solved for PV. Figure 5-2 Preset Value of $100 to Be Received at a Future Date ad Discouted Back to the Preset at 0, 5, ad 10 Percet Preset value (dollars) % 5% 10% Year Part Two: Valuatio of Fiacial Assets

13 What is the preset value of $500 to be received 10 years from today if our discout rate is 6 percet? Substitutig FV 10 = $500, = 10, ad i = 6 percet ito equatio (5-7), we fid 1 PV = $ 500 ( ) 1 = $ = $ 500 (. 558) = $ 279 Thus, the preset value of the $500 to be received i 10 years is $ To aid i the computatio of preset values, the preset-value iterest factor for i ad (PVIF i, ), defied as [1/(1 + i) ], has bee compiled for various combiatios of i ad ad appears i Appedix C at the back of this book. A abbreviated versio of Appedix C appears i Table 5-3. A close examiatio shows that the values i Table 5-3 are merely the iverse of those foud i Table 5-2 ad Appedix B. This, of course, is as it should be because the values i Appedix B are (1 + i) ad those i Appedix C are [1(1 + i) ]. Now, to determie the preset value of a sum of moey to be received at some future date, we eed oly determie the value of the appropriate PVIF i,, either by usig a calculator or cosultig the tables, ad multiply it by the future value. I effect we ca use our ew otatio ad rewrite equatio (5-7) as follows: PV = FV ( PVIF ) (5-7a) You re o vacatio i a rather remote part of Florida ad see a advertisemet statig that if you take a sales tour of some codomiiums you will be give $100 just for takig the tour. However, the $100 that you get is i the form of a savigs bod that will ot pay you the $100 for 10 years. What is the preset value of $100 to be received 10 years from today if your discout rate is 6 percet? By lookig at the = 10 row ad i = 6% colum of Table 5-3, you fid the PVIF 6%, 10 yr is.558. Substitutig FV 10 = $100 ad PVIF 6%, 10 yr =.558 ito equatio (5-7a), you fid i, preset-value iterest factor CALCULATOR SOLUTION Data Iput Fuctio Key 10 N 6 I/Y 100 FV 0 PMT Fuctio Key Aswer CPT PV PV = $ 100( PVIF6 %, 10 yr ) = $ 100(. 558) = $ Thus, the value i today s dollars of that $100 savigs bod is oly $ N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Table 5-3 PVIF i,, or the Preset Value of $1 Chapter 5: The Time Value of Moey 145

14 Dealig with Multiple, Ueve Cash Flows Agai, we oly have oe preset-value future-value equatio; that is, equatios (5-6) ad (5-7) are idetical. We have itroduced them as separate equatios to simplify our calculatios; i oe case we are determiig the value i future dollars, ad i the other case the value i today s dollars. I either case the reaso is the same: to compare values o alterative ivestmets ad to recogize that the value of a dollar received today is ot the same as that of a dollar received at some future date. We must measure the dollar values i dollars of the same time period. For example, if we looked at these projects oe that promised $1,000 i 1 year, oe that promised $1,500 i 5 years, ad oe that promised $2,500 i 10 years the cocept of preset value allows us to brig their flows back to the preset ad make those projects comparable. Moreover, because all preset values are comparable (they are all measured i dollars of the same time period), we ca add ad subtract the preset value of iflows ad outflows to determie the preset value of a ivestmet. Let s ow look at a example of a ivestmet that has two cash flows i differet time periods ad determie the preset value of this ivestmet. What is the preset value of a ivestmet that yields $500 to be received i 5 years ad $1,000 to be received i 10 years if the discout rate is 4 percet? Substitutig the values of = 5, i = 4 percet, ad FV 5 = $500; ad = 10, i = 4 percet, ad FV 10 = $1,000 ito equatio (5-7) ad addig these values together, we fid 1 PV = $ 500 $, ( 1. 04) ( ) = $ 500( PVIF4%, 5 yr ) + $ 1, 000( PVIF4%, 10 yr) = $ 500(. 822) + $ 1, 000(. 676) = $ $ 676 = $, Agai, preset values are comparable because they are measured i the same time period s dollars. 1. What is the relatioship betwee the preset value equatio (5-7) ad the future value, or compoudig, equatio (5-6)? 2. Why is the preset value of a future sum always less tha that sum s future value? O B J E C T I V E 3 auity ordiary auities compoud auities Auities A auity is a series of equal dollar paymets for a specified umber of years. Whe we talk about auities, we are referrig to ordiary auities uless otherwise oted. With a ordiary auity the paymets occur at the ed of each period. Because auities occur frequetly i fiace for example, as bod iterest paymets we treat them specially. Although compoudig ad determiig the preset value of a auity ca be dealt with usig the methods we have just described, these processes ca be time-cosumig, especially for larger auities. Thus, we have modified the formulas to deal directly with auities. Compoud Auities A compoud auity ivolves depositig or ivestig a equal sum of moey at the ed of each year for a certai umber of years ad allowig it to grow. Perhaps we are savig moey for educatio, a ew car, or a vacatio home. I ay case we wat to kow how much our savigs will have grow by some poit i the future. 146 Part Two: Valuatio of Fiacial Assets

15 Actually, we ca fid the aswer by usig equatio (5-6), our compoudig equatio, ad compoudig each of the idividual deposits to its future value. For example, if to provide for a college educatio we are goig to deposit $500 at the ed of each year for the ext 5 years i a bak where it will ear 6 percet iterest, how much will we have at the ed of 5 years? Compoudig each of these values usig equatio (5-6), we fid that we will have $2, at the ed of 5 years FV 5 = $ 500( ) + $ 500( ) + $ 500( ) + $ 500( ) + $ 500 = $ 500( ) + $ 500( ) + $ 500( ) + $ 500( ) + $ 500 = $ $ $ $ $ = $ 2, Taki It to the Net USA Today also has a selectio of olie calculators mcfrot.htm aimed at helpig you with all kids of persoal fiace questios ivolvig the time value of moey. From examiig the mathematics ivolved ad the graph of the movemet of moey through time i Table 5-4, we ca see that this procedure ca be geeralized to 1 t FV = PMT + i ( 1 ) t = 0 (5-8) where FV = the future value of the auity at the ed of the th year PMT = the auity paymet deposited or received at the ed of each year i = the aual iterest (or discout) rate = the umber of years for which the auity will last To aid i compoudig auities, the future-value iterest factor for a auity 1 for i ad (FVIFA i, ), defied as ( 1 + ), is provided i Appedix D for various combiatios of ad i; a abbreviated versio is show i Table 5-5 o the i t 0 t = ext page. Aother useful aalytical relatioship for FV is future-value iterest factor for a auity FV = PMT[( 1 + i) 1]/ i. (5-8a) Usig this ew otatio, we ca rewrite equatio (5-8) as follows: FV = PMT( FVIFA ) (5-8b) Reexamiig the previous example, i which we determied the value after 5 years of $500 deposited i the bak at 6 percet at the ed of each of the ext 5 years, we would look i the i = 6 percet colum ad = 5 row ad fid the value of the FVIFA 6%, 5 yr to be Substitutig this value ito equatio (5-8b), we get FV 5 = $ 500( ) = $ 2, i, CALCULATOR SOLUTION Data Iput Fuctio Key 5 N 6 I/Y 0 PV 500 PMT Fuctio Key CPT FV Aswer 2, This is the same aswer we obtaied earlier usig equatio (5-6). Year Dollar Deposits at Ed of Year $ Future value of the auity $2, Table 5-4 Illustratio of a 5-year $500 Auity Compouded at 6 Percet Chapter 5: The Time Value of Moey 147

16 Table 5-5 FVIFA i,, or the Sum of a Auity of $1 for Years N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% CALCULATOR SOLUTION Data Iput Fuctio Key 8 N 6 I/Y 10,000 FV 0 PV Fuctio Key CPT PMT Aswer 1, Rather tha ask how much we will accumulate if we deposit a equal sum i a savigs accout each year, a more commo questio is how much we must deposit each year to accumulate a certai amout of savigs. This problem frequetly occurs with respect to savig for large expeditures ad pesio fudig obligatios. For example, we may kow that we eed $10,000 for educatio i 8 years; how much must we deposit i the bak at the ed of each year at 6 percet iterest to have the college moey ready? I this case we kow the values of, i, ad FV i equatio (5-8); what we do ot kow is the value of PMT. Substitutig these example values i equatio (5-8), we fid 8 1 $ 10, 000 = PMT ( ) t 0 t = $ 10, 000 = PMT( FVIFA6%, 8 yr) $ 10, 000 = PMT( ) $ 10, 000 = PMT $ $, = PMT Thus, we must deposit $1, i the bak at the ed of each year for 8 years at 6 percet iterest to accumulate $10,000 at the ed of 8 years. CALCULATOR SOLUTION Data Iput Fuctio Key 10 N 8 I/Y 5,000 FV 0 PV Fuctio Key Aswer CPT PMT How much must we deposit i a 8 percet savigs accout at the ed of each year to accumulate $5,000 at the ed of 10 years? Substitutig the values FV 10 = $5,000, = 10, ad i = 8 percet ito equatio (5-8), we fid 10 1 t $ 5, 000 = PMT ( ) ( ) 0 = PMT FVIFA8%, 10 yr t = $ 5, 000 = PMT( ) $ 5, 000 = PMT $ = PMT Thus, we must deposit $ per year for 10 years at 8 percet to accumulate $5, Part Two: Valuatio of Fiacial Assets

17 A timelie ofte makes it easier to uderstad time value of moey problems. By visually plottig the flow of moey, you ca better determie which formula to use. Arrows placed above the lie are iflows, whereas arrows below the lie represet outflows. Oe thig is certai: Timelies reduce errors. Preset Value of a Auity Pesio fuds, isurace obligatios, ad iterest received from bods all ivolve auities. To compare them, we eed to kow the preset value of each. Although we ca fid this by usig the preset-value table i Appedix C, this ca be timecosumig, particularly whe the auity lasts for several years. For example, if we wish to kow what $500 received at the ed of each of the ext 5 years is worth to us give the appropriate discout rate of 6 percet, we ca simply substitute the appropriate values ito equatio (5-7), such that 1 PV = $ 500 $ 500 $ 500 ( 1. 06) 2 3 ( 1. 06) ( ) $ 500 $ ( 1. 06) ( ) = $ 500(. 943) + $ 500(. 890) + $ 500(. 840) + $ 500(. 792) + $ 500(. 747) = $ 2106, Thus, the preset value of this auity is $2, From examiig the mathematics ivolved ad the graph of the movemet of these fuds through time i Table 5-6, we see that we are simply summig up the preset value of each cash flow. Thus, this procedure ca be geeralized to PV = PMT 1 (5-9) i t t + = 1 ( 1 ) where PMT = the auity paymet deposited or received at the ed of each year i = the aual discout (or iterest) rate PV = the preset value of the future auity = the umber of years for which the auity will last To simplify the process of determiig the preset value of a auity, the preset-value iterest factor for a auity for i ad (PVIFA i, ), defied as 1 has bee compiled for various combiatios of i ad i Appedix E, 1 ( 1 + ) = i t t with a abbreviated versio provided i Table 5-7. Aother useful aalytical relatioship for PV is PV = PMT [ 1 1/( 1+ i) ]/ i. (5-9a) preset-value iterest factor for a auity Year Dollars received at the ed of year $ Preset value of the auity $2, Table 5-6 Illustratio of a 5-Year $500 Auity Discouted to the Preset at 6 Percet Chapter 5: The Time Value of Moey 149

18 Table 5-7 PVIFA i,, or the Preset Value of a Auity of $1 N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Usig this ew otatio we ca rewrite equatio (5-9) as follows: PV = PMT( PVIFA i, ) (5-9b) Solvig the previous example to fid the preset value of $500 received at the ed of each of the ext 5 years discouted back to the preset at 6 percet, we look i the i = 6 percet colum ad = 5 row ad fid the PVIFA 6%, 5 yr to be Substitutig the appropriate values ito equatio (5-9b), we fid PV = = $ 500( ) $ 2, 106 This, of course, is the same aswer we calculated whe we idividually discouted each cash flow to the preset. The reaso is that we really oly have oe table; the Table 5-7 value for a -year auity for ay discout rate i is merely the sum of the first values i Table 5-3. We ca see this by comparig the value i the preset-valueof-a-auity table (Table 5-8) for i = 8 percet ad = 6 years, which is 4.623, with the sum of the values i the i = 8 percet colum ad = 1,..., 6 rows of the presetvalue table (Table 5-3), which is equal to 4.623, as show i Table 5-8. CALCULATOR SOLUTION Data Iput Fuctio Key 10 N 5 I/Y 1,000 PMT 0 FV Fuctio Key CPT PV Aswer 7, What is the preset value of a 10-year $1,000 auity discouted back to the preset at 5 percet? Substitutig = 10 years, i = 5 percet, ad PMT = $1,000 ito equatio (5-9), we fid PV = $, t = 1 Determiig the value for the PVIFA 5%, 10 yr from Table 5-7, row = 10, colum i = 5 percet, ad substitutig it i, we get PV = = $, 1 000( ) $ 7, 722 Thus, the preset value of this auity is $7, t ( ) PVIFA = $, 1 000( 5 %, 10 yr ) As with our other compoudig ad preset-value tables, give ay three of the four ukows i equatio (5-9), we ca solve for the fourth. I the case of the presetvalue-of-a-auity table, we may be iterested i solvig for PMT, if we kow i,, ad PV. The fiacial iterpretatio of this actio would be: How much ca be withdraw, perhaps as a pesio or to make loa paymets, from a accout that 150 Part Two: Valuatio of Fiacial Assets

19 Oe dollar received at the ed of year Preset value Preset value of the auity Table 5-8 Preset Value of a 6-Year Auity Discouted at 8 Percet ears i percet compouded aually for each of the ext years if we wish to have othig left at the ed of years? For example, if we have $5,000 i a accout earig 8 percet iterest, how large a auity ca we draw out each year if we wat othig left at the ed of 5 years? I this case the preset value, PV, of the auity is $5,000, = 5 years, i = 8 percet, ad PMT is ukow. Substitutig this ito equatio (5-9), we fid $ 5, 000 = PMT( ) $, = PMT Thus, this accout will fall to zero at the ed of 5 years if we withdraw $1, at the ed of each year. CALCULATOR SOLUTION Data Iput Fuctio Key 5 N 8 I/Y 5,000 PV 0 FV Fuctio Key CPT PMT Aswer 1, Could you determie the future value of a 3-year auity usig the formula for the future value of a sigle cash flow? How? 2. What is the PVIFA 10%, 3 yr? Now add up the values for the PVIF 10%, yr. For = 1, 2, ad 3. What is this value? Why do these values have the relatioship they do? Auities Due Because auities due are really just ordiary auities i which all the auity paymets have bee shifted forward by 1 year, compoudig them ad determiig their preset value is actually quite simple. Remember, with a auity due, each auity paymet occurs at the begiig of each period rather tha at the ed of the period. Let s first look at how this affects our compoudig calculatios. Because a auity due merely shifts the paymets from the ed of the year to the begiig of the year, we ow compoud the cash flows for oe additioal year. Therefore, the compoud sum of a auity due is simply FV ( auity due) = PMT( FVIFA )( 1 + i) (5-10) For example, earlier we calculated the value of a 5-year ordiary auity of $500 ivested i the bak at 6 percet to be $2, If we ow assume this to be a 5-year auity due, its future value icreases from $2, to $2, i, 4 O B J E C T I V E auity due Taki It to the Net The FiaCeter Web site has a great selectio of Iteret calculators. I fact, may of the calculators used o the Iteret actually have bee developed by the FiaCeter. FV = $ 500( FVIFA %, 5 yr )( ) = $ 500( 5, 637)( 1. 06) = $ 2, Likewise, with the preset value of a auity due, we simply receive each cash flow 1 year earlier that is, we receive it at the begiig of each year rather tha at the ed of each year. Thus, because each cash flow is received 1 year earlier, it is discouted back for oe less period. To determie the preset value of a auity due, Chapter 5: The Time Value of Moey 151

20 we merely eed to fid the preset value of a ordiary auity ad multiply that by (1 + i), which i effect cacels out 1 year s discoutig. PV( auity due) = PMT( PVIFA )( 1 + i) (5-11) Reexamiig the earlier example i which we calculated the preset value of a 5-year ordiary auity of $500 give a appropriate discout rate of 6 percet, we ow fid that if it is a auity due rather tha a ordiary auity, the preset value icreases from $2,106 to $2,232.36, PV = $ 500( PVIFA6 %, 5 yr )( ) = $ 500( )( 1. 06) = $ 2, The result of all this is that both the future ad preset values of a auity due are larger tha those of a ordiary auity because i each case all paymets are received earlier. Thus, whe compoudig a auity due, it compouds for oe additioal year, whereas whe discoutig a auity due, the cash flows are discouted for oe less year. Although auities due are used with some frequecy i accoutig, their usage is quite limited i fiace. Therefore, i the remaider of this text, wheever the term auity is used, you should assume that we are referrig to a ordiary auity. The Virgiia State Lottery rus like most other state lotteries: You must select 6 out of 44 umbers correctly i order to wi the jackpot. If you come close, there are some sigificatly lesser prizes, which we igore for ow. For each millio dollars i the lottery jackpot you receive $50,000 per year for 20 years, ad your chace of wiig is 1 i 7.1 millio. A recet advertisemet for the Virgiia State Lottery wet as follows: Okay, you got two kids of people. You ve got the kid who play Lotto all the time, ad the kid who play Lotto some of the time. You kow, like oly o a Saturday whe they stop i at the store o the corer for some peaut butter cups ad diet soda ad the jackpot happes to be really big. I mea, my fried Ned? He s like Hey, it s oly $2 millio this week. Well, hellloooo, aybody home? I mea, I do t kow about you, but I would t mid havig a measly 2 mill comig my way.... What is the preset value of these paymets? The aswer to this questio depeds o what assumptio you make about the time value of moey. I this case, let s assume that your required rate of retur o a ivestmet with this level of risk is 10 percet. Keepig i mid that the Lotto is a auity due that is, o a $2 millio lottery you would get $100,000 immediately ad $100,000 at the ed of each of the ext 19 years. Thus, the preset value of this 20-year auity due discouted back to preset at 10 percet becomes PVauity due = PMT( PVIFAi%, yr )( 1 + i) = $ 100, 000( PVIFA10%, 20 yr )( ) = $ 100, 000( )( 1. 10) = $ 851, 400( 1. 10) = $ 936, 540. Thus, the preset value of the $2 millio Lotto jackpot is less tha $1 millio if 10 percet is the appropriate discout rate. Moreover, because the chace of wiig is oly 1 i 7.1 millio, the expected value of each dollar ivested i the lottery is oly (1/7.1 millio ) ($936,540) = That is, for every dollar you sped o the lottery you should expect to get, o average, about 13 cets back ot a particularly good deal. Although this igores the mior paymets for comig close, it also igores taxes. I this case, it looks like my fried Ned is doig the right thig by stayig clear of the lottery. Obviously, the mai value of the lottery is etertaimet. Ufortuately, without a uderstadig of the time value of moey, it ca soud like a good ivestmet. i, 152 Part Two: Valuatio of Fiacial Assets

21 1. How does a auity due differ from a ordiary auity? 2. Why are both the future ad preset values greater for a auity? Amortized Loas This procedure of solvig for PMT, the auity paymet value whe i,, ad PV are kow, is also used to determie what paymets are associated with payig off a loa i equal istallmets over time. Loas that are paid off this way, i equal periodic paymets, are called amortized loas. For example, suppose a firm wats to purchase a piece of machiery. To do this, it borrows $6,000 to be repaid i four equal paymets at the ed of each of the ext 4 years, ad the iterest rate that is paid to the leder is 15 percet o the outstadig portio of the loa. To determie what the aual paymets associated with the repaymet of this debt will be, we simply use equatio (5-9) ad solve for the value of PMT, the aual auity. Agai we kow three of the four values i that equatio: PV, i, ad. PV, the preset value of the future auity, is $6,000; i, the aual iterest rate, is 15 percet; ad, the umber of years for which the auity will last, is 4 years. PMT, the auity paymet received (by the leder ad paid by the firm) at the ed of each year, is ukow. Substitutig these values ito equatio (5-9), we fid 4 1 $ 6, 000 = PMT t 1 ( ) t = $ 6, 000 = PMT( PVIFA15%, 4 yr ) $ 6, 000 = PMT( ) $ 2, = PMT To repay the pricipal ad iterest o the outstadig loa i 4 years, the aual paymets would be $2, The breakdow of iterest ad pricipal paymets is give i the loa amortizatio schedule i Table 5-9, with very mior roudig error. As you ca see, the iterest paymet declies each year as the loa outstadig declies. amortized loa CALCULATOR SOLUTION Data Iput Fuctio Key 4 N 15 I/Y 6,000 PV 0 FV Fuctio Key CPT PMT Aswer 2, What is a amortized loa? Outstadig Repaymet of Loa Balace Iterest Portio the Pricipal After the of the Portio of Auity Year Auity Auity a the Auity b Paymet 1 $2, $ $1, $4, , , , , , , , , Table 5-9 Loa Amortizatio Schedule Ivolvig a $6,000 Loa at 15 Percet to Be Repaid i 4 Years a The iterest portio of the auity is calculated by multiplyig the outstadig loa balace at the begiig of the year by the iterest rate of 15 percet. Thus, for year 1 it was $6, = $900.00, for year 2 it was $4, = $719.76, ad so o. b Repaymet of the pricipal portio of the auity was calculated by subtractig the iterest portio of the auity (colum 2) from the auity (colum 1). Chapter 5: The Time Value of Moey 153

22 Spreadsheets: The Loa Amortizatio Problem Now let s look at a loa amortizatio problem i which the paymet occurs mothly usig a spreadsheet. To buy a ew house, you take out a 25-year mortgage for $100,000. What will your mothly iterest rate paymets be if the iterest rate o your mortgage is 8 percet? Etered values i cell d16: =PMT((8/12)%, d9,d10,d11,d12) You ca also use Excel to calculate the iterest ad pricipal portio of ay loa amortizatio paymet. You ca do this usig the followig Excel fuctios: Calculatio Formula Iterest portio of paymet = IPMT (rate, period, umber of periods, preset value, future value, type) Pricipal portio of paymet = PPMT (rate, period, umber of periods, preset value, future value, type) where period refers to the umber of a idividual periodic paymet. Etered values i cell c12: =IPMT((8/12)%,48,d5,d6, d7,d8) Etered values i cell c18: =PPMT((8/12)%,48,d5,d6, d7,d8) O B J E C T I V E 5 Compoud Iterest with Noaual Periods Util ow we have assumed that the compoudig or discoutig period is always aual; however, it eed ot be, as evideced by savigs ad loa associatios ad commercial baks that compoud o a quarterly, daily, ad i some cases cotiu- 154 Part Two: Valuatio of Fiacial Assets

23 ous basis. Fortuately, this adjustmet of the compoudig period follows the same format as that used for aual compoudig. If we ivest our moey for 5 years at 8 percet iterest compouded semiaually, we are really ivestig our moey for 10 six-moth periods durig which we receive 4 percet iterest each period. If it is compouded quarterly, we receive 2 percet iterest per period for 20 three-moth periods. This process ca easily be geeralized, givig us the followig formula for fidig the future value of a ivestmet for which iterest is compouded i oaual periods: FV = PV 1 + i m m (5-12) where FV = the future value of the ivestmet at the ed of years = the umber of years durig which the compoudig occurs i = aual iterest (or discout) rate PV = the preset value or origial amout ivested at the begiig of the first year m = the umber of times compoudig occurs durig the year We ca see the value of itrayear compoudig by examiig Table Because iterest is eared o iterest more frequetly as the legth of the compoudig period declies, there is a iverse relatioship betwee the legth of the compoudig period ad the effective aual iterest rate. If we place $100 i a savigs accout that yields 12 percet compouded quarterly, what will our ivestmet grow to at the ed of 5 years? Substitutig = 5, m = 4, i = 12 percet, ad PV = $100 ito equatio (5-12), we fid FV 5 = 100. $ = $ 100( ) = $ 100( ) = $ Thus, we will have $ at the ed of 5 years. Notice that the calculator solutio is slightly differet because of roudig errors i the tables, ad it also takes o a egative value. CALCULATOR SOLUTION Data Iput Fuctio Key 20 N 3 I/Y 100 PV 0 PMT Fuctio Key Aswer CPT FV FOR 1 YEAR AT i PERCENT i = 2% 5% 10% 15% Compouded aually $ $ $ $ Compouded semiaually Compouded quarterly Compouded mothly Compouded weekly (52) Compouded daily (365) FOR 10 YEARS AT i PERCENT i = 2% 5% 10% 15% Compouded aually $ $ $ $ Compouded semiaually Compouded quarterly Compouded mothly Compouded weekly (52) Compouded daily (365) Table 5-10 The Value of $100 Compouded at Various Itervals Chapter 5: The Time Value of Moey 155

24 1. Why does the future value of a give amout icrease whe iterest is compouded oaually as opposed to aually? 2. How do you adjust the preset- ad future-value formulas whe iterest is compouded mothly? O B J E C T I V E 6 Preset Value of a Ueve Stream Although some projects will ivolve a sigle cash flow ad some auities, may projects will ivolve ueve cash flows over several years. Chapter 9, which examies ivestmets i fixed assets, presets this situatio repeatedly. There we will be comparig ot oly the preset value of cash flows betwee projects but also the cash iflows ad outflows withi a particular project, tryig to determie that project s preset value. However, this will ot be difficult because the preset value of ay cash flow is measured i today s dollars ad thus ca be compared, through additio for iflows ad subtractio for outflows, to the preset value of ay other cash flow also measured i today s dollars. For example, if we wished to fid the preset value of the followig cash flows Year Cash Flow Year Cash Flow 1 $ give a 6-percet discout rate, we would merely discout the flows back to the preset ad total them by addig i the positive flows ad subtractig the egative oes. However, this problem is complicated by the auity of $500 that rus from years 4 through 10. To accommodate this, we ca first discout the auity back to the begiig of period 4 (or ed of period 3) by multiplyig it by the value of PVIFA 6%, 7 yr ad get its preset value at that poit i time. We the multiply this value times the PVIF 6%, 3 yr i order to brig this sigle cash flow (which is the preset value of the 7-year auity) back to the preset. I effect we discout twice, first back to the ed of period 3, the back to the preset. This is show graphically i Table 5-11 ad umerically i Table Thus, the preset value of this ueve stream of cash flows is $2, Table 5-11 Illustratio of a Example of Preset Value of a Ueve Stream Ivolvig Oe Auity Discouted to the Preset at 6 Percet Year Dollars received at ed of year $ $2,791 2, Total preset value $2, Part Two: Valuatio of Fiacial Assets

25 Table 5-12 Determiatio of the Preset Value of a Example with Ueve Stream Ivolvig Oe Auity Discouted to the Preset at 6 Percet 1. Preset value of $500 received at the ed of 1 year = $500(.943) = $ Preset value of $200 received at the ed of 2 years = $200(.890) = Preset value of a $400 outflow at the ed of 3 years = 400(.840) = (a) Value at the ed of year 3 ad a $500 auity, years 4 through 10 = $500(5.582) = $2, (b) Preset value of $2, received at the ed of year 3 = $2,791(.840) = 2, Total preset value = $2, What is the preset value of a ivestmet ivolvig $200 received at the ed of years 1 through 5, a $300 cash outflow at the ed of year 6, ad $500 received at the ed of years 7 through 10, give a 5-percet discout rate? Here we have two auities, oe that ca be discouted directly back to the preset by multiplyig it by the value of the PVIFA 5%, 5 yr ad oe that must be discouted twice to brig it back to the preset. This secod auity, which is a 4-year auity, must first be discouted back to the begiig of period 7 (or ed of period 6) by multiplyig it by the value of the PVIFA 5%, 4 yr. The the preset value of this auity at the ed of period 6 (which ca be viewed as a sigle cash flow) must be discouted back to the preset by multiplyig it by the value of the PVIF 5%, 6 yr. To arrive at the total preset value of this ivestmet, we subtract the preset value of the $300 cash outflow at the ed of year 6 from the sum of the preset value of the two auities. Table 5-13 shows this graphically; Table 5-14 gives the calculatios. Thus, the preset value of this series of cash flows is $1, Remember, oce the cash flows from a ivestmet have bee brought back to the preset, they ca be combied by addig ad subtractig to determie the project s total preset value. 1. If you wated to calculate the preset value of a ivestmet that produced cash flows of $100 received at the ed of year 1 ad $700 at the ed of year 2, how would you do it? Year Dollars received at ed of year $ $1,773 1, Table 5-13 Illustratio of a Example of the Preset Value of a Ueve Stream Ivolvig Two Auities Discouted to the Preset at 5 Percet Total preset value $1, Chapter 5: The Time Value of Moey 157

26 Table 5-14 Determiatio of the Preset Value of a Example With Ueve Stream Ivolvig Two Auities Discouted to the Preset at 5 Percet 1. Preset value of first auity, years 1 through 5 = $200(4.329) $ Preset value of $300 cash outflow = $300(.746) = (a) Value at ed of year 6 of secod auity, years 7 through 10 = $500(3.546) = $1, (b) Preset value of $1, received at the ed of year 6 = $1,773.00(.746) = 1, Total preset value = $1, O B J E C T I V E 7 perpetuity Perpetuities A perpetuity is a auity that cotiues forever; that is, every year from its establishmet this ivestmet pays the same dollar amout. A example of a perpetuity is preferred stock that pays a costat dollar divided ifiitely. Determiig the preset value of a perpetuity is delightfully simple; we merely eed to divide the costat flow by the discout rate. For example, the preset value of a $100 perpetuity discouted back to the preset at 5 percet is $100/.05 = $2,000. Thus, the equatio represetig the preset value of a perpetuity is PP PV = (5-13) i where PV = the preset value of the perpetuity PP = the costat dollar amout provided by the perpetuity i = the aual iterest (or discout) rate What is the preset value of a $500 perpetuity discouted back to the preset at 8 percet? Substitutig PP = $500 ad i =.08 ito equatio (5-11), we fid $ 500 PV = = $ 6, Thus, the preset value of this perpetuity is $6, What is a perpetuity? 2. Whe the i, the aual iterest (or discout) rate, icreases, what happes to the preset value of a perpetuity? Why? O B J E C T I V E 8 The Multiatioal Firm: The Time Value of Moey From Priciple 1: The Risk Retur Trade-off We Wo t Take o Additioal Risk Uless We Expect to Be Compesated with Additioal Retur, we foud that ivestors demad a retur for delayig cosumptio, as well as a additioal retur for takig o added risk. The discout rate that we use to move moey through time should reflect this retur for delayig cosumptio; ad as the Fisher effect showed i Chapter 2, this discout rate should reflect aticipated iflatio. I the Uited States, aticipated iflatio is quite low, although it does ted to fluctuate over time. Elsewhere i the world, however, the iflatio rate is difficult to predict because it ca be dramatically high ad udergo huge fluctuatios. Let s look at Argetia, keepig i mid that similar examples aboud i Cetral ad South America ad Easter Europe. At the begiig of 1992, Argetia itroduced the fifth currecy i 22 years, the ew peso. The austral, the currecy that was replaced, was itroduced i Jue 1985 ad was iitially equal i value to $1.25 U.S. currecy. Five ad a half years later, it took 100,000 australs to equal $1. Iflatio had reached the poit at which the stack of moey eeded to buy a cady bar was bigger ad weighed more tha the cady bar itself, ad may workers received their weeks wages 158 Part Two: Valuatio of Fiacial Assets

27 i grocery bags. Needless to say, if we were to move australs through time, we would have to use a extremely high iterest or discout rate. Ufortuately, i coutries sufferig from hyperiflatio, iflatio rates ted to fluctuate dramatically, ad this makes estimatig the expected iflatio rate eve more difficult. For example, i 1989 the iflatio rate i Argetia was 4,924 percet; i 1990 it dropped to 1,344 percet; i 1991 it was oly 84 percet; i 1992, oly 18 percet; ad i 2000 it was close to zero. However, as iflatio i Argetia dropped, iflatio i Brazil heated up, goig from 426 percet i 1991 to 1,094 percet i By 2000, the iflatio rate i Brazil had dropped to 7%. However, i 2000 the iflatio rate i Uzbekista, our ally i the war agaist terrorists based i Afghaista, reached 1,568 percet. Fially, at the extreme, i 1993 i Serbia the iflatio rate reached 360,000,000,000,000,000 percet. The bottom lie o all this is that because of the dramatic fluctuatios i iflatio that ca take place i the iteratioal settig, choosig the appropriate discout rate of movig moey through time is a extremely difficult process. 1. How does the iteratioal settig complicate the choice of the appropriate iterest rate to use whe discoutig cash flows back to the preset? Summary To make decisios, fiacial maagers must compare the costs ad beefits of alteratives that do ot occur durig the same time period. Whether to make profitable ivestmets or to take Table 5-15 Summary of Time Value of Moey Equatios a Calculatio Future value of a sigle paymet Preset value of a sigle paymet Equatio FV = PV( 1 + i) = PV( FVIFi, ) 1 PV = FV FV PVIF i ( 1 + i) = (, ) 1 O B J E C T I V E 2 O B J E C T I V E Future value of a auity Preset value of a auity Future value of a auity due 1 t FV = PMT + i PMT FVIFAi ( 1 ) = (, ) t = 0 PV = PMT 1 PMT PVIFA t i t + i = = 1 ( 1 ) (, ) FV (auity due) = PMT( FVIFAi, )( 1 + i) 3 O B J E C T I V E 4 O B J E C T I V E Preset value of a auity due PV(auity due) = PMT( PVIFAi, )( 1 + i) Future value of a sigle paymet with oaual compoudig Preset value of a perpetuity m i FV = PV 1 + m PV = PP i 5 O B J E C T I V E 7 O B J E C T I V E Notatios: FV = the future value of the ivestmet at the ed of years = the umber of years util paymet will be received or durig which compoudig occurs i = the aual iterest or discout rate PV = the preset value of the future sum of moey m = the umber of times compoudig occurs durig the year PMT = the auity paymet deposited or received at the ed of each year PP = the costat dollar amout provided by the perpetuity a Related tables appear i Appedixes B through E at the ed of the book. 159

28 O B J E C T I V E 6 O B J E C T I V E 8 advatage of favorable iterest rates, fiacial decisio makig requires a uderstadig of the time value of moey. Maagers who use the time value of moey i all of their fiacial calculatios assure themselves of more logical decisios. The time value process first makes all dollar values comparable; because moey has a time value, it moves all dollar flows either back to the preset or out to a commo future date. All time value formulas preseted i this chapter actually stem from the sigle compoudig formula FV = PV(1 + i). The formulas are used to deal simply with commo fiacial situatios, for example, discoutig sigle flows, compoudig auities, ad discoutig auities. Table 5-15 provides a summary of these calculatios. Because of the dramatic fluctuatios i iflatio that ca take place i the iteratioal settig, choosig the appropriate discout rate of movig moey through time is a extremely difficult process. Key Terms amortized loa, 153 auity, 146 auity due, 151 compoud auity, 146 compoud iterest, 136 future-value iterest factor (FVIF i, ), 138 future-value iterest factor for a auity (FVIFA i, ), 147 ordiary auity, 146 perpetuity, 158 preset value, 144 preset-value iterest factor (PVIF i, ), 145 preset-value iterest factor for a auity (PVIFA i, ), 149 Study Questios 5-1. What is the time value of moey? Why is it so importat? 5-2. The process of discoutig ad compoudig are related. Explai this relatioship How would a icrease i the iterest rate (i) or a decrease i the holdig period () affect the future value (FV ) of a sum of moey? Explai why Suppose you were cosiderig depositig your savigs i oe of three baks, all of which pay 5 percet iterest; bak A compouds aually, bak B compouds semiaually, ad bak C compouds daily. Which bak would you choose? Why? 5-5. What is the relatioship betwee the PVIF i, (Table 5-3) ad the PVIFA i, (Table 5-7)? What is the PVIFA 10%, 10 yr? Add up the values of the PVIF 10%,, for = 1,..., 10. What is this value? Why do these values have the relatioship they do? 5-6. What is a auity? Give some examples of auities. Distiguish betwee a auity ad a perpetuity. Self-Test Problems ST-1. You place $25,000 i a savigs accout payig a aual compoud iterest of 8 percet for 3 years ad the move it ito a savigs accout that pays 10 percet iterest compouded aually. How much will your moey have grow at the ed of 6 years? ST-2. You purchase a boat for $35,000 ad pay $5,000 dow ad agree to pay the rest over the ext 10 years i 10 equal aual ed-of-the-year paymets that iclude pricipal paymets plus 13 percet compoud iterest o the upaid balace. What will be the amout of each paymet? Study Problems 5-1. (Compoud Iterest) To what amout will the followig ivestmets accumulate? a. $5,000 ivested for 10 years at 10 percet compouded aually b. $8,000 ivested for 7 years at 8 percet compouded aually c. $775 ivested for 12 years at 12 percet compouded aually d. $21,000 ivested for 5 years at 5 percet compouded aually 5-2. (Compoud Value Solvig for ) How may years will the followig take? a. $500 to grow to $1, if ivested at 5 percet compouded aually b. $35 to grow to $53.87 if ivested at 9 percet compouded aually c. $100 to grow to $ if ivested at 20 percet compouded aually d. $53 to grow to $78.76 if ivested at 2 percet compouded aually 160 Part Two: Valuatio of Fiacial Assets

29 5-3. (Compoud Value Solvig for i) At what aual rate would the followig have to be ivested? a. $500 to grow to $1, i 12 years b. $300 to grow to $ i 7 years c. $50 to grow to $ i 20 years d. $200 to grow to $ i 5 years 5-4. (Preset Value) What is the preset value of the followig future amouts? a. $800 to be received 10 years from ow discouted back to the preset at 10 percet b. $300 to be received 5 years from ow discouted back to the preset at 5 percet c. $1,000 to be received 8 years from ow discouted back to the preset at 3 percet d. $1,000 to be received 8 years from ow discouted back to the preset at 20 percet 5-5. (Compoud Auity) What is the accumulated sum of each of the followig streams of paymets? a. $500 a year for 10 years compouded aually at 5 percet b. $100 a year for 5 years compouded aually at 10 percet c. $35 a year for 7 years compouded aually at 7 percet d. $25 a year for 3 years compouded aually at 2 percet 5-6. (Preset Value of a Auity) What is the preset value of the followig auities? a. $2,500 a year for 10 years discouted back to the preset at 7 percet b. $70 a year for 3 years discouted back to the preset at 3 percet c. $280 a year for 7 years discouted back to the preset at 6 percet d. $500 a year for 10 years discouted back to the preset at 10 percet 5-7. (Compoud Value) Staford Simmos, who recetly sold his Porsche, placed $10,000 i a savigs accout payig aual compoud iterest of 6 percet. a. Calculate the amout of moey that will have accrued if he leaves the moey i the bak for 1, 5, ad 15 years. b. If he moves his moey ito a accout that pays 8 percet or oe that pays 10 percet, rework part (a) usig these ew iterest rates. c. What coclusios ca you draw about the relatioship betwee iterest rates, time, ad future sums from the calculatios you have completed i this problem (Compoud Iterest with Noaual Periods) Calculate the amout of moey that will be i each of the followig accouts at the ed of the give deposit period. Compoudig Period Aual (Compouded Deposit Amout Iterest Every Period Accout Deposited Rate Moths) (Years) Theodore Loga III $ 1,000 10% Verell Coles 95, Thomas Elliott 8, Waye Robiso 120, Eugee Chug 30, Kelly Craves 15, (Compoud Iterest with Noaual Periods) a. Calculate the future sum of $5,000, give that it will be held i the bak 5 years at a aual iterest rate of 6 percet. b. Recalculate part (a) usig compoudig periods that are (1) semiaual ad (2) bimothly. c. Recalculate parts (a) ad (b) for a 12-percet aual iterest rate. d. Recalculate part (a) usig a time horizo of 12 years (aual iterest rate is still 6 percet). e. With respect to the effect of chages i the stated iterest rate ad holdig periods o future sums i parts (c) ad (d), what coclusios do you draw whe you compare these figures with the aswers foud i parts (a) ad (b)? (Solvig for i with Auities) Nicki Johso, a sophomore mechaical egieerig studet, receives a call from a isurace aget, who believes that Nicki is a older woma ready to retire from teachig. He talks to her about several auities that she could buy that would guaratee her a aual fixed icome. The auities are as follows: Chapter 5: The Time Value of Moey 161

30 Iitial Paymet ito Duratio Auity Amout of Moey of Auity Auity (at t = 0) Received per Year (Years) A $50,000 $8, B $60,000 $7, C $70,000 $8, If Nicki could ear 11 percet o her moey by placig it i a savigs accout, should she place it istead i ay of the auities? Which oes, if ay? Why? (Future Value) Sales of a ew fiace book were 15,000 copies this year ad were expected to icrease by 20 percet per year. What are expected sales durig each of the ext 3 years? Graph this sales tred ad explai (Future Value) Barry Bods hit 73 home rus i If his home-ru output grew at a rate of 10 percet per year, what would it have bee over the followig 5 years? (Loa Amortizatio) Mr. Bill S. Presto, Esq., purchased a ew house for $80,000. He paid $20,000 dow ad agreed to pay the rest over the ext 25 years i 25 equal aual ed-of-year paymets that iclude pricipal paymets plus 9 percet compoud iterest o the upaid balace. What will these equal paymets be? (Solvig for PMT of a Auity) To pay for your child s educatio, you wish to have accumulated $15,000 at the ed of 15 years. To do this you pla o depositig a equal amout ito the bak at the ed of each year. If the bak is willig to pay 6 percet compouded aually, how much must you deposit each year to obtai your goal? (Solvig for i i Compoud Iterest) If you were offered $1, te years from ow i retur for a ivestmet of $500 curretly, what aual rate of iterest would you ear if you took the offer? (Future Value of a Auity) I 10 years you are plaig o retirig ad buyig a house i Oviedo, Florida. The house you are lookig at curretly costs $100,000 ad is expected to icrease i value each year at a rate of 5 percet. Assumig you ca ear 10 percet aually o your ivestmets, how much must you ivest at the ed of each of the ext 10 years to be able to buy your dream home whe you retire? (Compoud Value) The Aggarwal Corporatio eeds to save $10 millio to retire a $10 millio mortgage that matures i 10 years. To retire this mortgage, the compay plas to put a fixed amout ito a accout at the ed of each year for 10 years, with the first paymet occurrig at the ed of 1 year. The Aggarwal Corporatio expects to ear 9 percet aually o the moey i this accout. What equal aual cotributio must it make to this accout to accumulate the $10 millio i 10 years? (Compoud Iterest with Noaual Periods) After examiig the various persoal loa rates available to you, you fid that you ca borrow fuds from a fiace compay at 12 percet compouded mothly or from a bak at 13 percet compouded aually. Which alterative is more attractive? (Preset Value of a Ueve Stream of Paymets) You are give three ivestmet alteratives to aalyze. The cash flows from these three ivestmets are as follows: INVESTMENT Ed of Year A B C 1 $10,000 $10, , , , ,000 $10, ,000 50, , , , ,000 10, Part Two: Valuatio of Fiacial Assets

31 Assumig a 20-percet discout rate, fid the preset value of each ivestmet (Preset Value) The Kumar Corporatio is plaig o issuig bods that pay o iterest but ca be coverted ito $1,000 at maturity, 7 years from their purchase. To price these bods competitively with other bods of equal risk, it is determied that they should yield 10 percet, compouded aually. At what price should the Kumar Corporatio sell these bods? (Perpetuities) What is the preset value of the followig? a. A $300 perpetuity discouted back to the preset at 8 percet b. A $1,000 perpetuity discouted back to the preset at 12 percet c. A $100 perpetuity discouted back to the preset at 9 percet d. A $95 perpetuity discouted back to the preset at 5 percet (Solvig for with Noaual Periods) About how may years would it take for your ivestmet to grow fourfold if it were ivested at 16 percet compouded semiaually? (Complex Preset Value) How much do you have to deposit today so that begiig 11 years from ow you ca withdraw $10,000 a year for the ext 5 years (periods 11 through 15) plus a additioal amout of $20,000 i that last year (period 15)? Assume a iterest rate of 6 percet (Loa Amortizatio) O December 31, Beth Klemkosky bought a yacht for $50,000, payig $10,000 dow ad agreeig to pay the balace i 10 equal aual ed-of-year istallmets that iclude both the pricipal ad 10 percet iterest o the decliig balace. How big would the aual paymets be? (Solvig for i of a Auity) You led a fried $30,000, which your fried will repay i 5 equal aual ed-of-year paymets of $10,000, with the first paymet to be received 1 year from ow. What rate of retur does your loa receive? (Solvig for i i Compoud Iterest) You led a fried $10,000, for which your fried will repay you $27,027 at the ed of 5 years. What iterest rate are you chargig your fried? (Loa Amortizatio) A firm borrows $25,000 from the bak at 12 percet compouded aually to purchase some ew machiery. This loa is to be repaid i equal aual istallmets at the ed of each year over the ext 5 years. How much will each aual paymet be? (Preset Value Compariso) You are offered $1,000 today, $10,000 i 12 years, or $25,000 i 25 years. Assumig that you ca ear 11 percet o your moey, which should you choose? (Compoud Auity) You pla o buyig some property i Florida 5 years from today. To do this you estimate that you will eed $20,000 at that time for the purchase. You would like to accumulate these fuds by makig equal aual deposits i your savigs accout, which pays 12 percet aually. If you make your first deposit at the ed of this year ad you would like your accout to reach $20,000 whe the fial deposit is made, what will be the amout of your deposits? (Complex Preset Value) You would like to have $50,000 i 15 years. To accumulate this amout you pla to deposit each year a equal sum i the bak, which will ear 7 percet iterest compouded aually. Your first paymet will be made at the ed of the year. a. How much must you deposit aually to accumulate this amout? b. If you decide to make a large lump-sum deposit today istead of the aual deposits, how large should this lump-sum deposit be? (Assume you ca ear 7 percet o this deposit.) c. At the ed of 5 years you will receive $10,000 ad deposit this i the bak toward your goal of $50,000 at the ed of 15 years. I additio to this deposit, how much must you deposit i equal aual deposits to reach your goal? (Agai assume you ca ear 7 percet o this deposit.) (Comprehesive Preset Value) You are tryig to pla for retiremet i 10 years, ad curretly you have $100,000 i a savigs accout ad $300,000 i stocks. I additio you pla o addig to your savigs by depositig $10,000 per year i your savigs accout at the ed of each of the ext 5 years ad the $20,000 per year at the ed of each year for the fial 5 years util retiremet. Chapter 5: The Time Value of Moey 163

32 a. Assumig your savigs accout returs 7 percet compouded aually ad your ivestmet i stocks will retur 12 percet compouded aually, how much will you have at the ed of 10 years? (Igore taxes.) b. If you expect to live for 20 years after you retire, ad at retiremet you deposit all of your savigs i a bak accout payig 10 percet, how much ca you withdraw each year after retiremet (20 equal withdrawals begiig oe year after you retire) to ed up with a zero balace at death? (Loa Amortizatio) O December 31, So-Na Che borrowed $100,000, agreeig to repay this sum i 20 equal aual ed-of-year istallmets that iclude both the pricipal ad 15 percet iterest o the decliig balace. How large will the aual paymets be? (Loa Amortizatio) To buy a ew house you must borrow $150,000. To do this you take out a $150,000, 30-year, 10-percet mortgage. Your mortgage paymets, which are made at the ed of each year (oe paymet each year), iclude both pricipal ad 10-percet iterest o the decliig balace. How large will your aual paymets be? (Preset Value) The state lottery s millio-dollar payout provides for $1 millio to be paid over 19 years i 20 paymets of $50,000. The first $50,000 paymet is made immediately, ad the 19 remaiig $50,000 paymets occur at the ed of each of the ext 19 years. If 10 percet is the appropriate discout rate, what is the preset value of this stream of cash flows? If 20 percet is the appropriate discout rate, what is the preset value of the cash flows? (Solvig for i i Compoud Iterest Fiacial Calculator Needed) I September 1963, the first issue of the comic book X-MEN was issued. The origial price for that issue was $.12. By September 2002, 39 years later, the value of this comic book had rise to $7,500. What aual rate of iterest would you have eared if you had bought the comic i 1963 ad sold it i 2002? (Comprehesive Preset Value) You have just iherited a large sum of moey, ad you are tryig to determie how much you should save for retiremet ad how much you ca sped ow. For retiremet, you will deposit today (Jauary 1, 2003) a lump sum i a bak accout payig 10 percet compouded aually. You do t pla o touchig this deposit util you retire i 5 years (Jauary 1, 2008), ad you pla o livig for 20 additioal years ad the droppig dead o December 31, Durig your retiremet you would like to receive icome of $50,000 per year to be received the first day of each year, with the first paymet o Jauary 1, 2008, ad the last paymet o Jauary 1, Complicatig this objective is your desire to have oe fial 3-year flig durig which time you d like to track dow all the livig origial members of Leave It to Beaver ad The Brady Buch ad get their autographs. To fiace this you wat to receive $250,000 o Jauary 1, 2023, ad othig o Jauary 1, 2024, ad Jauary 1, 2025, as you will be o the road. I additio, after you pass o (Jauary 1, 2028), you would like to have a total of $100,000 to leave to your childre. a. How much must you deposit i the bak at 10 percet o Jauary 1, 2003, to achieve your goal? (Use a timelie to aswer this questio.) b. What kids of problems are associated with this aalysis ad its assumptios? (Spreadsheet Problem) If you ivest $900 i a bak i which it will ear 8 percet compouded aually, how much will it be worth at the ed of 7 years? Use a spreadsheet to do your calculatios (Spreadsheet Problem) I 20 years you d like to have $250,000 to buy a vacatio home, but you oly have $30,000. At what rate must your $30,000 be compouded aually for it to grow to $250,000 i 20 years? Use a spreadsheet to calculate your aswer (Spreadsheet Problem) To buy a ew house you take out a 25-year mortgage for $300,000. What will your mothly iterest rate paymets be if the iterest rate o your mortgage is 8 percet? Use a spreadsheet to calculate a aswer. Now, calculate the portio of the 48th mothly paymet that goes toward iterest ad pricipal. Comprehesive Problem For your job as the busiess reporter for a local ewspaper, you are give the task of puttig together a series of articles that explais the power of the time value of moey to your readers. Your editor would like you to address several specific questios i additio to demostratig 164 Part Two: Valuatio of Fiacial Assets

33 for the readership the use of time value of moey techiques by applyig them to several problems. What would be your respose to the followig memoradum from your editor: TO: Busiess Reporter FROM: Perry White, Editor, Daily Plaet RE: Upcomig Series o the Importace ad Power of the Time Value of Moey I your upcomig series o the time value of moey, I would like to make sure you cover several specific poits. I additio, before you begi this assigmet, I wat to make sure we are all readig from the same script, as accuracy has always bee the corerstoe of the Daily Plaet. I this regard, I d like a respose to the followig questios before we proceed: a. What is the relatioship betwee discoutig ad compoudig? b. What is the relatioship betwee the PVIF i, ad PVIFA i,? c. 1. What will $5,000 ivested for 10 years at 8 percet compouded aually grow to? 2. How may years will it take $400 to grow to $1,671 if it is ivested at 10 percet compouded aually? 3. At what rate would $1,000 have to be ivested to grow to $4,046 i 10 years? d. Calculate the future sum of $1,000, give that it will be held i the bak for 5 years ad ear 10 percet compouded semiaually. e. What is a auity due? How does this differ from a ordiary auity? f. What is the preset value of a ordiary auity of $1,000 per year for 7 years discouted back to the preset at 10 percet? What would be the preset value if it were a auity due? g. What is the future value of a ordiary auity of $1,000 per year for 7 years compouded at 10 percet? What would be the future value if it were a auity due? h. You have just borrowed $100,000, ad you agree to pay it back over the ext 25 years i 25 equal ed-of-year aual paymets that iclude the pricipal paymets plus 10 percet compoud iterest o the upaid balace. What will be the size of these paymets? i. What is the preset value of a $1,000 perpetuity discouted back to the preset at 8 percet? j. What is the preset value of a $1,000 auity for 10 years, with the first paymet occurrig at the ed of year 10 (that is, te $1,000 paymets occurrig at the ed of year 10 through year 19), give a appropriate discout rate of 10 percet? k. Give a 10-percet discout rate, what is the preset value of a perpetuity of $1,000 per year of the first paymet does ot begi util the ed of year 10? Self-Test Solutios SS-1. This is a compoud iterest problem i which you must first fid the future value of $25,000 growig at 8 percet compouded aually for 3 years ad the allow that future value to grow for a additioal 3 years at 10 percet. First, the value of the $25,000 after 3 years growig at 8 percet is FV3 = PV( 1 + i) FV3 = $ 25, 000( ) FV3 = $ 25, 000( ) FV = $ 31, 500 Thus, after 3 years you have $31,500. Now this amout is allowed to grow for 3 years at 10 percet. Pluggig this ito equatio (5-6), with PV = $31,500, i = 10 percet, ad = 3 years, we solve for FV 3. FV FV FV Thus, after 6 years the $25,000 will have grow to $41, = $ 31, 500( ) = $ 31, 500( ) = $ 41, SS-2. This loa amortizatio problem is actually just a preset-value-of-a-auity problem i which we kow the values of i,, ad PV ad are solvig for PMT. I this case the value of i is 13 percet, is 10 years, ad PV is $30,000. Substitutig these values ito equatio (5-9) we fid 10 1 $ 30, 000 = PMT t 1 ( ) t = $ 30, 000 = PMT( ) $ 5, = PMT 3 3 Chapter 5: The Time Value of Moey 165

34 166 Part Two: Valuatio of Fiacial Assets