The Time Value of Money

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1 Part 2 FOF12e_C03.qxd 8/13/04 3:39 PM Page 39 Valuatio 3 The Time Value of Moey Cotets Objectives The Iterest Rate After studyig Chapter 3, you should be able to: Simple Iterest Compoud Iterest Uderstad what is meat by the time value of moey. Uderstad the relatioship betwee preset ad future value. Semiaual ad Other Compoudig Periods Cotiuous Compoudig Effective Aual Iterest Rate Describe how the iterest rate ca be used to adjust the value of cash flows both forward ad backward to a sigle poit i time. Amortizig a Loa Summary Table of Key Compoud Iterest Formulas Calculate both the future ad preset value of: (a) a amout ivested today; (b) a stream of equal cash flows (a auity); ad (c) a stream of mixed cash flows. Summary Questios Distiguish betwee a ordiary auity ad a auity due. Self-Correctio Problems Problems Use iterest factor tables ad uderstad how they provide a shortcut to calculatig preset ad future values. Solutios to Self-Correctio Problems Selected Refereces Use iterest factor tables to fid a ukow iterest rate or growth rate whe the umber of time periods ad future ad preset values are kow. Build a amortizatio schedule for a istallmet-style loa. Sigle Amouts Auities Mixed Flows Compoudig More Tha Oce a Year 39

2 Part 2 Valuatio The chief value of moey lies i the fact that oe lives i a world i which it is overestimated. H. L. MENCKEN A Mecke Chrestomathy The Iterest Rate Iterest Moey paid (eared) for the use of moey. Which would you prefer $1,000 today or $1,000 te years from today? Commo sese tells us to take the $1,000 today because we recogize that there is a time value to moey. The immediate receipt of $1,000 provides us with the opportuity to put our moey to work ad ear iterest. I a world i which all cash flows are certai, the rate of iterest ca be used to express the time value of moey. As we will soo discover, the rate of iterest will allow us to adjust the value of cash flows, wheever they occur, to a particular poit i time. Give this ability, we will be able to aswer more difficult questios, such as: which should you prefer $1,000 today or $2,000 te years from today? To aswer this questio, it will be ecessary to positio time-adjusted cash flows at a sigle poit i time so that a fair compariso ca be made. If we allow for ucertaity surroudig cash flows to eter ito our aalysis, it will be ecessary to add a risk premium to the iterest rate as compesatio for ucertaity. I later chapters we will study how to deal with ucertaity (risk). But for ow, our focus is o the time value of moey ad the ways i which the rate of iterest ca be used to adjust the value of cash flows to a sigle poit i time. Most fiacial decisios, persoal as well as busiess, ivolve time value of moey cosideratios. I Chapter 1, we leared that the objective of maagemet should be to maximize shareholder wealth, ad that this depeds, i part, o the timig of cash flows. Not surprisigly, oe importat applicatio of the cocepts stressed i this chapter will be to value a stream of cash flows. Ideed, much of the developmet of this book depeds o your uderstadig of this chapter. You will ever really uderstad fiace util you uderstad the time value of moey. Although the discussio that follows caot avoid beig mathematical i ature, we focus o oly a hadful of formulas so that you ca more easily grasp the fudametals. We start with a discussio of simple iterest ad use this as a sprigboard to develop the cocept of compoud iterest. Also, to observe more easily the effect of compoud iterest, most of the examples i this chapter assume a 8 percet aual iterest rate. Take Note Before we begi, it is importat to soud a few otes of cautio. The examples i the chapter frequetly ivolve umbers that must be raised to the th power for example, (1.05) to the third power equals (1.05) 3 equals [(1.05) (1.05) (1.05)]. However, this operatio is easy to do with a calculator, ad tables are provided i which this calculatio has already bee doe for you. Although the tables provided are a useful aid, you caot rely o them for solvig every problem. Not every iterest rate or time period ca possibly be represeted i each table. Therefore you will eed to become familiar with the operatioal formulas o which the tables are based. (As a remider, the appropriate formula is icluded at the top of every table.) Those of you possessig a busiess calculator may feel the urge to bypass both the tables ad formulas ad head straight for the various fuctio keys desiged to deal with time value of moey problems. However, we urge you to master first the logic behid the procedures outlied i this chapter. Eve the best of calculators caot overcome a faulty sequece of steps programmed i by the user. 40

3 3 The Time Value of Moey Simple Iterest Simple iterest Iterest paid (eared) o oly the origial amout, or pricipal, borrowed (let). Simple iterest is iterest that is paid (eared) o oly the origial amout, or pricipal, borrowed (let). The dollar amout of simple iterest is a fuctio of three variables: the origial amout borrowed (let), or pricipal; the iterest rate per time period; ad the umber of time periods for which the pricipal is borrowed (let). The formula for calculatig simple iterest is SI = P 0 (i )() (3.1) where SI = simple iterest i dollars P 0 = pricipal, or origial amout borrowed (let) at time period 0 i = iterest rate per time period = umber of time periods For example, assume that you deposit $100 i a savigs accout payig 8 percet simple iterest ad keep it there for 10 years. At the ed of 10 years, the amout of iterest accumulated is determied as follows: $80 = $100(0.08)(10) Future value (termial value) The value at some future time of a preset amout of moey, or a series of paymets, evaluated at a give iterest rate. To solve for the future value (also kow as the termial value) of the accout at the ed of 10 years (FV 10 ), we add the iterest eared o the pricipal oly to the origial amout ivested. Therefore FV 10 = $100 + [$100(0.08)(10)] = $180 For ay simple iterest rate, the future value of a accout at the ed of periods is FV = P 0 + SI = P 0 + P 0 (i )() or, equivaletly, FV = P 0 [1 + (i )()] (3.2) Preset value The curret value of a future amout of moey, or a series of paymets, evaluated at a give iterest rate. Sometimes we eed to proceed i the opposite directio. That is, we kow the future value of a deposit at i percet for years, but we do t kow the pricipal origially ivested the accout s preset value (PV 0 = P 0 ). A rearragemet of Eq. (3.2), however, is all that is eeded. PV 0 = P 0 = FV /[1 + (i )()] (3.3) Now that you are familiar with the mechaics of simple iterest, it is perhaps a bit cruel to poit out that most situatios i fiace ivolvig the time value of moey do ot rely o simple iterest at all. Istead, compoud iterest is the orm; however, a uderstadig of simple iterest will help you appreciate (ad uderstad) compoud iterest all the more. Compoud Iterest Compoud iterest Iterest paid (eared) o ay previous iterest eared, as well as o the pricipal borrowed (let). The distictio betwee simple ad compoud iterest ca best be see by example. Table 3.1 illustrates the rather dramatic effect that compoud iterest has o a ivestmet s value over time whe compared with the effect of simple iterest. From the table it is clear to see why some people have called compoud iterest the greatest of huma ivetios. The otio of compoud iterest is crucial to uderstadig the mathematics of fiace. The term itself merely implies that iterest paid (eared) o a loa (a ivestmet) is 41

4 Part 2 Valuatio Table 3.1 Future value of $1 ivested for various time periods at a 8% aual iterest rate YEARS AT SIMPLE INTEREST AT COMPOUND INTEREST 2 $ 1.16 $ $4,838, periodically added to the pricipal. As a result, iterest is eared o iterest as well as the iitial pricipal. It is this iterest-o-iterest, or compoudig, effect that accouts for the dramatic differece betwee simple ad compoud iterest. As we will see, the cocept of compoud iterest ca be used to solve a wide variety of problems i fiace. Sigle Amouts Future (or Compoud) Value. To begi with, cosider a perso who deposits $100 ito a savigs accout. If the iterest rate is 8 percet, compouded aually, how much will the $100 be worth at the ed of a year? Settig up the problem, we solve for the future value (which i this case is also referred to as the compoud value) of the accout at the ed of the year (FV 1 ). FV 1 = P 0 (1 + i ) = $100(1.08) = $108 Iterestigly, this first-year value is the same umber that we would get if simple iterest were employed. But this is where the similarity eds. What if we leave $100 o deposit for two years? The $100 iitial deposit will have grow to $108 at the ed of the first year at 8 percet compoud aual iterest. Goig to the ed of the secod year, $108 becomes $116.64, as $8 i iterest is eared o the iitial $100, ad $0.64 is eared o the $8 i iterest credited to our accout at the ed of the first year. I other words, iterest is eared o previously eared iterest hece the ame compoud iterest. Therefore, the future value at the ed of the secod year is FV 2 = FV 1 (1 + i ) = P 0 (1 + i )(1 + i ) = P 0 (1 + i ) 2 = $108(1.08) = $100(1.08)(1.08) = $100(1.08) 2 = $ At the ed of three years, the accout would be worth or FV 3 = FV 2 (1 + i ) = FV 1 (1 + i )(1 + i ) = P 0 (1 + i ) 3 = $116.64(1.08) = $108(1.08)(1.08) = $100(1.08) 3 = $ I geeral, FV, the future (compoud) value of a deposit at the ed of periods, is FV = P 0 (1 + i ) (3.4) FV = P 0 (FVIF i, ) (3.5) where we let FVIF i, (i.e., the future value iterest factor at i% for periods) equal (1 + i). Table 3.2, showig the future values for our example problem at the ed of years 1 to 3 (ad beyod), illustrates the cocept of iterest beig eared o iterest. A calculator makes Eq. (3.4) very simple to use. I additio, tables have bee costructed for values of (1 + i) FVIF i, for wide rages of i ad. These tables, called (appropriately) 42

5 3 The Time Value of Moey Table 3.2 Illustratio of compoud iterest with $100 iitial deposit ad 8% aual iterest rate INTEREST EARNED BEGINNING DURING PERIOD ENDING YEAR AMOUNT (8% of begiig amout) AMOUNT (FV ) 1 $ $ 8.00 $ Table 3.3 Future value iterest factor of $1 at i % at the ed of periods (FVIF i, ) (FVIF i, ) = (1 + i ) INTEREST RATE (i ) PERIOD () 1% 3% 5% 8% 10% 15% , future value iterest factor (or termial value iterest factor) tables, are desiged to be used with Eq. (3.5). Table 3.3 is oe example coverig various iterest rates ragig from 1 to 15 percet. The Iterest Rate (i) headigs ad Period () desigatios o the table are similar to map coordiates. They help us locate the appropriate iterest factor. For example, the future value iterest factor at 8 percet for ie years (FVIF 8%,9 ) is located at the itersectio of the 8% colum with the 9-period row ad equals This figure meas that $1 ivested at 8 percet compoud iterest for ie years will retur roughly $2 cosistig of iitial pricipal plus accumulated iterest. (For a more complete table, see Table I i the Appedix at the ed of this book.) If we take the FVIFs for $1 i the 8% colum ad multiply them by $100, we get figures (aside from some roudig) that correspod to our calculatios for $100 i the fial colum of Table 3.2. Notice, too, that i rows correspodig to two or more years, the proportioal icrease i future value becomes greater as the iterest rate rises. A picture may help make this poit a little clearer. Therefore, i Figure 3.1 we graph the growth i future value for a $100 iitial deposit with iterest rates of 5, 10, ad 15 percet. As ca be see from the graph, the greater the iterest rate, the steeper the growth curve by which future value icreases. Also, the greater the umber of years durig which compoud iterest ca be eared, obviously the greater the future value. 43

6 Part 2 Valuatio Figure 3.1 Future values with $100 iitial deposit ad 5%, 10%, ad 15% compoud aual iterest rates TIP TIP O a umber of busiess professioal (certificatio) exams you will be provided with iterest factor tables ad be limited to usig oly basic, o-programmable, had-held calculators. So, for some of you, it makes added sese to get familiar with iterest factor tables ow. Compoud Growth. Although our cocer so far has bee with iterest rates, it is importat to realize that the cocept ivolved applies to compoud growth of ay sort for example, i gas prices, tuitio fees, corporate earigs, or divideds. Suppose that a corporatio s most recet divided was $10 per share but that we expect this divided to grow at a 10 percet compoud aual rate. For the ext five years we would expect divideds to look as show i the table. YEAR GROWTH FACTOR EXPECTED DIVIDEND/SHARE 1 (1.10) 1 $ (1.10) (1.10) (1.10) (1.10) Questio I 1790 Joh Jacob Astor bought approximately a acre of lad o the east side of Mahatta Islad for $58. Astor, who was cosidered a shrewd ivestor, made may such purchases. How much would his descedats have i 2005, if istead of buyig the lad, Astor had ivested the $58 at 5 percet compoud aual iterest? 44

7 3 The Time Value of Moey Aswer I Table I, i the Appedix at the ed of the book, we wo t fid the FVIF of $1 i 215 years at 5 percet. But, otice that we ca fid the FVIF of $1 i 50 years ad the FVIF of $1 i 15 years So what, you might ask. Beig a little creative, we ca express our problem as follows: 1 FV 215 = P 0 (1 + i) 215 = P 0 (1 + i) 50 (1 + i) 50 (1 + i) 50 (1 + i) 50 (1 + i) 15 = $ = $58 35, = $2,084, Give the curret price of lad i New York City, Astor s oe-acre purchase seems to have passed the test of time as a wise ivestmet. It is also iterestig to ote that with a little reasoig we ca get quite a bit of mileage out of eve a basic table. Discout rate (capitalizatio rate) Iterest rate used to covert future values to preset values. Similarly, we ca determie the future levels of other variables that are subject to compoud growth. This priciple will prove especially importat whe we cosider certai valuatio models for commo stock, which we do i the ext chapter. Preset (or Discouted) Value. We all realize that a dollar today is worth more tha a dollar to be received oe, two, or three years from ow. Calculatig the preset value of future cash flows allows us to place all cash flows o a curret footig so that comparisos ca be made i terms of today s dollars. A uderstadig of the preset value cocept should eable us to aswer a questio that was posed at the very begiig of this chapter: which should you prefer $1,000 today or $2,000 te years from today? 2 Assume that both sums are completely certai ad your opportuity cost of fuds is 8 percet per aum (i.e., you could borrow or led at 8 percet). The preset worth of $1,000 received today is easy it is worth $1,000. However, what is $2,000 received at the ed of 10 years worth to you today? We might begi by askig what amout (today) would grow to be $2,000 at the ed of 10 years at 8 percet compoud iterest. This amout is called the preset value of $2,000 payable i 10 years, discouted at 8 percet. I preset value problems such as this, the iterest rate is also kow as the discout rate (or capitalizatio rate). Fidig the preset value (or discoutig) is simply the reverse of compoudig. Therefore, let s first retrieve Eq. (3.4): Rearragig terms, we solve for preset value: FV = P 0 (1 + i) PV 0 = P 0 = FV /(1 + i) = FV [1/(1 + i) ] (3.6) Note that the term [1/(1 + i) ] is simply the reciprocal of the future value iterest factor at i% for periods (FVIF i, ). This reciprocal has its ow ame the preset value iterest factor at i% for periods (PVIF i, ) ad allows us to rewrite Eq. (3.6) as PV 0 = FV (PVIF i, ) (3.7) A preset value table cotaiig PVIFs for a wide rage of iterest rates ad time periods relieves us of makig the calculatios implied by Eq. (3.6) every time we have a preset value problem to solve. Table 3.4 is a abbreviated versio of oe such table. (Table II i the Appedix foud at the ed of the book is a more complete versio.) 1 We make use of oe of the rules goverig expoets. Specifically, A m+ = A m A 2 Alteratively, we could treat this as a future value problem. To do this, we would compare the future value of $1,000, compouded at 8 percet aual iterest for 10 years, to a future $2,

8 Part 2 Valuatio Table 3.4 Preset value iterest factor of $1 at i % for periods (PVIF i, ) (PVIF i, ) = 1/(1 + i ) INTEREST RATE (i) PERIOD () 1% 3% 5% 8% 10% 15% We ca ow make use of Eq. (3.7) ad Table 3.4 to solve for the preset value of $2,000 to be received at the ed of 10 years, discouted at 8 percet. I Table 3.4, the itersectio of the 8% colum with the 10-period row pipoits PVIF 8%, This tells us that $1 received 10 years from ow is worth roughly 46 cets to us today. Armed with this iformatio, we get PV 0 = FV 10 (PVIF 8%,10 ) = $2,000(0.463) = $926 Fially, if we compare this preset value amout ($926) with the promise of $1,000 to be received today, we should prefer to take the $1,000. I preset value terms we would be better off by $74 ($1,000 $926). Discoutig future cash flows turs out to be very much like the process of hadicappig. That is, we put future cash flows at a mathematically determied disadvatage relative to curret dollars. For example, i the problem just addressed, every future dollar was hadicapped to such a extet that each was worth oly about 46 cets. The greater the disadvatage assiged to a future cash flow, the smaller the correspodig preset value iterest factor (PVIF). Figure 3.2 illustrates how both time ad discout rate combie to affect preset value; the preset value of $100 received from 1 to 10 years i the future is graphed for discout rates of 5, 10, ad 15 percet. The graph shows that the preset value of $100 decreases by a decreasig rate the further i the future that it is to be received. The greater the iterest rate, of course, the lower the preset value but also the more proouced the curve. At a 15 percet discout rate, $100 to be received 10 years hece is worth oly $24.70 today or roughly 25 cets o the (future) dollar. Questio How do you determie the future value (preset value) of a ivestmet over a time spa that cotais a fractioal period (e.g., 1 1 /4 years)? Aswer Simple. All you do is alter the future value (preset value) formula to iclude the fractio i decimal form. Let s say that you ivest $1,000 i a savigs accout that compouds aually at 6 percet ad wat to withdraw your savigs i 15 moths (i.e., 1.25 years). Sice FV = P 0 (1 + i), you could withdraw the followig amout 15 moths from ow: FV 1.25 = $1,000( ) 1.25 = $1,

9 3 The Time Value of Moey Figure 3.2 Preset values with $100 cash flow ad 5%, 10%, ad 15% compoud aual iterest rates Ukow Iterest (or Discout) Rate. Sometimes we are faced with a time-value-ofmoey situatio i which we kow both the future ad preset values, as well as the umber of time periods ivolved. What is ukow, however, is the compoud iterest rate (i) implicit i the situatio. Let s assume that, if you ivest $1,000 today, you will receive $3,000 i exactly 8 years. The compoud iterest (or discout) rate implicit i this situatio ca be foud by rearragig either a basic future value or preset value equatio. For example, makig use of future value Eq. (3.5), we have FV 8 = P 0 (FVIF i,8 ) $3,000 = $1,000(FVIF i,8 ) FVIF i,8 = $3,000/$1,000 = 3 Readig across the 8-period row i Table 3.3, we look for the future value iterest factor (FVIF) that comes closest to our calculated value of 3. I our table, that iterest factor is ad is foud i the 15% colum. Because is slightly larger tha 3, we coclude that the iterest rate implicit i the example situatio is actually slightly less tha 15 percet. For a more accurate aswer, we simply recogize that FVIF i,8 ca also be writte as (1 + i) 8, ad solve directly for i as follows: (1 + i) 8 = 3 (1 + i) = 3 1/8 = = i = (Note: Solvig for i, we first have to raise both sides of the equatio to the 1/8 or power. To raise 3 to the power, we use the [y x ] key o a hadheld calculator eterig 3, pressig the [y x ] key, eterig 0.125, ad fially pressig the [=] key.) 47

10 Part 2 Valuatio Ukow Number of Compoudig (or Discoutig) Periods. At times we may eed to kow how log it will take for a dollar amout ivested today to grow to a certai future value give a particular compoud rate of iterest. For example, how log would it take for a ivestmet of $1,000 to grow to $1,900 if we ivested it at a compoud aual iterest rate of 10 percet? Because we kow both the ivestmet s future ad preset value, the umber of compoudig (or discoutig) periods () ivolved i this ivestmet situatio ca be determied by rearragig either a basic future value or preset value equatio. Usig future value Eq. (3.5), we get FV = P 0 (FVIF 10%, ) $1,900 = $1,000(FVIF 10%, ) FVIF 10%, = $1,900/$1,000 = 1.9 Readig dow the 10% colum i Table 3.3, we look for the future value iterest factor (FVIF) i that colum that is closest to our calculated value. We fid that comes closest to 1.9, ad that this umber correspods to the 7-period row. Because is a little larger tha 1.9, we coclude that there are slightly less tha 7 aual compoudig periods implicit i the example situatio. For greater accuracy, simply rewrite FVIF 10%, as ( ), ad solve for as follows: ( ) = 1.9 (l 1.1) = l 1.9 = (l 1.9)/(l 1.1) = 6.73 years To solve for, which appeared i our rewritte equatio as a expoet, we employed a little trick. We took the atural logarithm (l) of both sides of our equatio. This allowed us to solve explicitly for. (Note: To divide (l 1.9) by (l 1.1), we use the [LN] key o a hadheld calculator as follows: eter 1.9 ; press the [LN] key; the press the [ ] key; ow eter 1.1 ; press the [LN] key oe more time; ad fially, press the [=] key.) Auity A series of equal paymets or receipts occurrig over a specified umber of periods. I a ordiary auity, paymets or receipts occur at the ed of each period; i a auity due, paymets or receipts occur at the begiig of each period. Auities Ordiary Auity. A auity is a series of equal paymets or receipts occurrig over a specified umber of periods. I a ordiary auity, paymets or receipts occur at the ed of each period. Figure 3.3 shows the cash-flow sequece for a ordiary auity o a time lie. Assume that Figure 3.3 represets your receivig $1,000 a year for three years. Now let s further assume that you deposit each aual receipt i a savigs accout earig 8 percet compoud aual iterest. How much moey will you have at the ed of three years? Figure 3.4 provides the aswer (the log way) usig oly the tools that we have discussed so far. Expressed algebraically, with FVA defied as the future (compoud) value of a auity, R the periodic receipt (or paymet), ad the legth of the auity, the formula for FVA is FVA = R(1 + i) 1 + R(1 + i) R(1 + i) 1 + R(1 + i) 0 = R[FVIF i, 1 + FVIF i, FVIF i,1 + FVIF i,0 ] As you ca see, FVA is simply equal to the periodic receipt (R) times the sum of the future value iterest factors at i percet for time periods 0 to 1. Luckily, we have two shorthad ways of statig this mathematically: or equivaletly, t FVA = R ( 1 + i) = R([( 1 + i) 1]/ i) t = 1 (3.8) FVA = R(FVIFA i, ) (3.9) where FVIFA i, stads for the future value iterest factor of a auity at i% for periods. 48

11 3 The Time Value of Moey Psst! Wat to Double Your Moey? The Rule of 72 Tells You How. Bill Veeck oce bought the Chicago White Sox baseball team frachise for $10 millio ad the sold it 5 years later for $20 millio. I short, he doubled his moey i 5 years. What compoud rate of retur did Veeck ear o his ivestmet? A quick way to hadle compoud iterest problems ivolvig doublig your moey makes use of the Rule of 72. This rule states that if the umber of years,, for which a ivestmet will be held is divided ito the value 72, we will get the approximate iterest rate, i, required for the ivestmet to double i value. I Veeck s case, the rule gives 72/ = i or 72/5 = 14.4% Alteratively, if Veeck had take his iitial ivestmet ad placed it i a savigs accout earig 6 percet compoud iterest, he would have had to wait approximately 12 years for his moey to have doubled: 72/i = or 72/6 = 12 years Ideed, for most iterest rates we ecouter, the Rule of 72 gives a good approximatio of the iterest rate or the umber of years required to double your moey. But the aswer is ot exact. For example, moey doublig i 5 years would have to ear at a percet compoud aual rate [( ) 5 = 2]; the Rule of 72 says 14.4 percet. Also, moey ivested at 6 percet iterest would actually require oly 11.9 years to double [( ) 11.9 = 2]; the Rule of 72 suggests 12. However, for ballpark-close moey-doublig approximatios that ca be doe i your head, the Rule of 72 comes i pretty hady. Figure 3.3 Time lie showig the cash-flow sequece for a ordiary auity of $1,000 per year for 3 years Figure 3.4 Time lie for calculatig the future (compoud) value of a (ordiary) auity [periodic receipt = R = $1,000; i = 8%; ad = 3 years] TIP TIP It is very helpful to begi solvig time value of moey problems by first drawig a time lie o which you positio the relevat cash flows. The time lie helps you focus o the problem ad reduces the chace for error. Whe we get to mixed cash flows, this will become eve more apparet. 49

12 Part 2 Valuatio Table 3.5 Future value iterest factor of a (ordiary) auity of $1 per period at i % for periods (FVIFA i, ) (PVIFA ) = ( 1 + i ) = [( 1 + i ) 1]/ i i, t = 1 t INTEREST RATE (i ) PERIOD () 1% 3% 5% 8% 10% 15% Figure 3.5 Time lie for calculatig the preset (discouted) value of a (ordiary) auity [periodic receipt = R = $1,000; i = 8%; ad = 3 years] A abbreviated listig of FVIFAs appears i Table 3.5. A more complete listig appears i Table III i the Appedix at the ed of this book. Makig use of Table 3.5 to solve the problem described i Figure 3.4, we get FVA 3 = $1,000(FVIFA 8%,3 ) = $1,000(3.246) = $3,246 This aswer is idetical to that show i Figure 3.4. (Note: Use of a table rather tha a formula subjects us to some slight roudig error. Had we used Eq. (3.8), our aswer would have bee 40 cets more. Therefore, whe extreme accuracy is called for, use formulas rather tha tables.) Retur for the momet to Figure 3.3. Oly ow let s assume the cash flows of $1,000 a year for three years represet withdrawals from a savigs accout earig 8 percet compoud aual iterest. How much moey would you have to place i the accout right ow (time period 0) such that you would ed up with a zero balace after the last $1,000 withdrawal? Figure 3.5 shows the log way to fid the aswer. As ca be see from Figure 3.5, solvig for the preset value of a auity boils dow to determiig the sum of a series of idividual preset values. Therefore, we ca write the geeral formula for the preset value of a (ordiary) auity for periods (PVA ) as PVA = R[1/(1 + i) 1 ] + R[1/(1 + i) 2 ] R[1/(1 + i) ] = R[PVIF i,1 + PVIF i, PVIF i, ] 50

13 3 The Time Value of Moey Table 3.6 Preset value iterest factor of a (ordiary) auity of $1 per period at i % for periods (PVIFA i, ) (PVIFA ) = 1/ ( 1 + i ) = ( 1 [/( i )])/ i i, t = 1 t INTEREST RATE (i) PERIOD () 1% 3% 5% 8% 10% 15% Notice that our formula reduces to PVA beig equal to the periodic receipt (R) times the sum of the preset value iterest factors at i percet for time periods 1 to. Mathematically, this is equivalet to t PVA = R 11 /( + i) = R[( 1 [ 1/(1 + i) ])/ i] t = 1 ad ca be expressed eve more simply as (3.10) PVA = R(PVIFA i, ) (3.11) where PVIFA i, stads for the preset value iterest factor of a (ordiary) auity at i percet for periods. Table IV i the Appedix at the ed of this book holds PVIFAs for a wide rage of values for i ad, ad Table 3.6 cotais excerpts from it. We ca make use of Table 3.6 to solve for the preset value of the $1,000 auity for three years at 8 percet show i Figure 3.5. The PVIFA 8%,3 is foud from the table to be (Notice this figure is othig more tha the sum of the first three umbers uder the 8% colum i Table 3.4, which gives PVIFs.) Employig Eq. (3.11), we get PVA 3 = $1,000(PVIFA 8%,3 ) = $1,000(2.577) = $2,577 Ukow Iterest (or Discout) Rate. A rearragemet of the basic future value (preset value) of a auity equatio ca be used to solve for the compoud iterest (discout) rate implicit i a auity if we kow: (1) the auity s future (preset) value, (2) the periodic paymet or receipt, ad (3) the umber of periods ivolved. Suppose that you eed to have at least $9,500 at the ed of 8 years i order to sed your parets o a luxury cruise. To accumulate this sum, you have decided to deposit $1,000 at the ed of each of the ext 8 years i a bak savigs accout. If the bak compouds iterest aually, what miimum compoud aual iterest rate must the bak offer for your savigs pla to work? To solve for the compoud aual iterest rate (i) implicit i this auity problem, we make use of future value of a auity Eq. (3.9) as follows: FVA 8 = R (FVIFA i,8 ) $9,500 = $1,000(FVIFA i,8 ) FVIFA i,8 = $9,500/$1,000 = 9.5 Readig across the 8-period row i Table 3.5, we look for the future value iterest factor of a auity (FVIFA) that comes closest to our calculated value of 9.5. I our table, that iterest factor is ad is foud i the 5% colum. Because is slightly larger tha 9.5, we 51

14 Part 2 Valuatio coclude that the iterest rate implicit i the example situatio is actually slightly less tha 5 percet. (For a more accurate aswer, you would eed to rely o trial-ad-error testig of differet iterest rates, iterpolatio, or a fiacial calculator.) Ukow Periodic Paymet (or Receipt). Whe dealig with auities, oe frequetly ecouters situatios i which either the future (or preset) value of the auity, the iterest rate, ad the umber of periodic paymets (or receipts) are kow. What eeds to be determied, however, is the size of each equal paymet or receipt. I a busiess settig, we most frequetly ecouter the eed to determie periodic auity paymets i sikig fud (i.e., buildig up a fud through equal-dollar paymets) ad loa amortizatio (i.e., extiguishig a loa through equal-dollar paymets) problems. Rearragemet of either the basic preset or future value auity equatio is ecessary to solve for the periodic paymet or receipt implicit i a auity. Because we devote a etire sectio at the ed of this chapter to the importat topic of loa amortizatio, we will illustrate how to calculate the periodic paymet with a sikig fud problem. How much must oe deposit each year ed i a savigs accout earig 5 percet compoud aual iterest to accumulate $10,000 at the ed of 8 years? We compute the paymet (R) goig ito the savigs accout each year with the help of future value of a auity Eq. (3.9). I additio, we use Table 3.5 to fid the value correspodig to FVIFA 5%,8 ad proceed as follows: FVA 8 = R(FVIFA 5%,8 ) $10,000 = R(9.549) R = $10,000/9.549 = $1, Therefore, by makig eight year-ed deposits of $1, each ito a savigs accout earig 5 percet compoud aual iterest, we will build up a sum totalig $10,000 at the ed of 8 years. Perpetuity A ordiary auity whose paymets or receipts cotiue forever. Perpetuity. A perpetuity is a ordiary auity whose paymets or receipts cotiue forever. The ability to determie the preset value of this special type of auity will be required whe we value perpetual bods ad preferred stock i the ext chapter. A look back to PVA Eq. (3.10) should help us to make short work of this type of task. Replacig i Eq. (3.10) with the value ifiity ( ) gives us PVA = R [(1 [1/(1 + i) ])/i] (3.12) Because the bracketed term [1/(1 + i) ] approaches zero, we ca rewrite Eq. (3.12) as PVA = R [(1 0)/i] = R (1/i) or simply PVA = R /i (3.13) Thus the preset value of a perpetuity is simply the periodic receipt (paymet) divided by the iterest rate per period. For example, if $100 is received each year forever ad the iterest rate is 8 percet, the preset value of this perpetuity is $1,250 (that is, $100/0.08). Auity Due. I cotrast to a ordiary auity, where paymets or receipts occur at the ed of each period, a auity due calls for a series of equal paymets occurrig at the begiig of each period. Luckily, oly a slight modificatio to the procedures already outlied for the treatmet of ordiary auities will allow us to solve auity due problems. Figure 3.6 compares ad cotrasts the calculatio for the future value of a $1,000 ordiary auity for three years at 8 percet (FVA 3 ) with that of the future value of a $1,000 auity due for three years at 8 percet (FVAD 3 ). Notice that the cash flows for the ordiary auity are perceived to occur at the ed of periods 1, 2, ad 3, ad those for the auity due are perceived to occur at the begiig of periods 2, 3, ad 4. 52

15 3 The Time Value of Moey Figure 3.6 Time lies for calculatig the future (compoud) value of a (ordiary) auity ad a auity due [periodic receipt = R = $1,000; i = 8%; ad = 3 years] Notice that the future value of the three-year auity due is simply equal to the future value of a comparable three-year ordiary auity compouded for oe more period. Thus the future value of a auity due at i percet for periods (FVAD ) is determied as FVAD = R(FVIFA i, )(1 + i) (3.14) Take Note Whether a cash flow appears to occur at the begiig or ed of a period ofte depeds o your perspective, however. (I a similar vei, is midight the ed of oe day or the begiig of the ext?) Therefore, the real key to distiguishig betwee the future value of a ordiary auity ad a auity due is the poit at which the future value is calculated. For a ordiary auity, future value is calculated as of the last cash flow. For a auity due, future value is calculated as of oe period after the last cash flow. The determiatio of the preset value of a auity due at i percet for periods (PVAD ) is best uderstood by example. Figure 3.7 illustrates the calculatios ecessary to determie both the preset value of a $1,000 ordiary auity at 8 percet for three years (PVA 3 ) ad the preset value of a $1,000 auity due at 8 percet for three years (PVAD 3 ). As ca be see i Figure 3.7, the preset value of a three-year auity due is equal to the preset value of a two-year ordiary auity plus oe odiscouted periodic receipt or paymet. This ca be geeralized as follows: PVAD = R(PVIFA i, 1 ) + R = R(PVIFA i, 1 + 1) (3.15) 53

16 Part 2 Valuatio Figure 3.7 Time lies for calculatig the preset (discouted) value of a (ordiary) auity ad a auity due [periodic receipt = R = $1,000; i = 8%; ad = 3 years] Alteratively, we could view the preset value of a auity due as the preset value of a ordiary auity that had bee brought back oe period too far. That is, we wat the preset value oe period later tha the ordiary auity approach provides. Therefore, we could calculate the preset value of a -period auity ad the compoud it oe period forward. The geeral formula for this approach to determiig PVAD is PVAD = (1 + i)(r)(pvifa i, ) (3.16) Figure 3.7 proves by example that both approaches to determiig PVAD work equally well. However, the use of Eq. (3.15) seems to be the more obvious approach. The time-lie approach take i Figure 3.7 also helps us recogize the major differeces betwee the preset value of a ordiary auity ad a auity due. Take Note I solvig for the preset value of a ordiary auity, we cosider the cash flows as occurrig at the ed of periods (i our Figure 3.7 example, the ed of periods 1, 2, ad 3) ad calculate the preset value as of oe period before the first cash flow. Determiatio of the preset value of a auity due calls for us to cosider the cash flows as occurrig at the begiig of periods (i our example, the begiig of periods 1, 2, ad 3) ad to calculate the preset value as of the first cash flow. 54

17 3 The Time Value of Moey Mixed Flows May time value of moey problems that we face ivolve either a sigle cash flow or a sigle auity. Istead, we may ecouter a mixed (or ueve) patter of cash flows. Questio Assume that you are faced with the followig problem o a exam (arghh!), perhaps. What is the preset value of $5,000 to be received aually at the ed of years 1 ad 2, followed by $6,000 aually at the ed of years 3 ad 4, ad cocludig with a fial paymet of $1,000 at the ed of year 5, all discouted at 5 percet? The first step i solvig the questio above, or ay similar problem, is to draw a time lie, positio the cash flows, ad draw arrows idicatig the directio ad positio to which you are goig to adjust the flows. Secod, make the ecessary calculatios as idicated by your diagram. (You may thik that drawig a picture of what eeds to be doe is somewhat childlike. However, cosider that most successful home builders work from blueprits why should t you?) Figure 3.8 illustrates that mixed flow problems ca always be solved by adjustig each flow idividually ad the summig the results. This is time-cosumig, but it works. Ofte we ca recogize certai patters withi mixed cash flows that allow us to take some calculatio shortcuts. Thus the problem that we have bee workig o could be solved i a umber of alterative ways. Oe such alterative is show i Figure 3.9. Notice how our twostep procedure cotiues to lead us to the correct solutio: Take Note Step 1: Draw a time lie, positio cash flows, ad draw arrows to idicate directio ad positio of adjustmets. Step 2: Perform calculatios as idicated by your diagram. A wide variety of mixed (ueve) cash-flow problems could be illustrated. To appreciate this variety ad to master the skills ecessary to determie solutios, be sure to do the problems at the ed of this chapter. Do t be too bothered if you make some mistakes at first. Time value of moey problems ca be tricky. Masterig this material is a little bit like learig to ride a bicycle. You expect to fall ad get bruised a bit util you pick up the ecessary skills. But practice makes perfect. The Magic of Compoud Iterest Each year, o your birthday, you ivest $2,000 i a tax-free retiremet ivestmet accout. By age 65 you will have accumulated: COMPOUND ANNUAL STARTING AGE INTEREST RATE (i) % $ 425,487 $222,870 $109,730 $46, , , ,212 54, ,437, , ,694 63, ,716, , ,668 74,560 From the table, it looks like the time to start savig is ow! 55

18 Part 2 Valuatio Figure 3.8 (Alterative 1) Time lie for calculatig the preset (discouted) value of mixed cash flows [FV 1 = FV 2 = $5,000; FV 3 = FV 4 = $6,000; FV 5 = $1,000; i = 5%; ad = 5 years] Figure 3.9 (Alterative 2) Time lie for calculatig the preset (discouted) value of mixed cash flows [FV 1 = FV 2 = $5,000; FV 3 = FV 4 = $6,000; FV 5 = $1,000; i = 5%; ad = 5 years] 56

19 3 The Time Value of Moey Compoudig More Tha Oce a Year Nomial (stated) iterest rate A rate of iterest quoted for a year that has ot bee adjusted for frequecy of compoudig. If iterest is compouded more tha oce a year, the effective iterest rate will be higher tha the omial rate. Semiaual ad Other Compoudig Periods Future (or Compoud) Value. Up to ow, we have assumed that iterest is paid aually. It is easiest to get a basic uderstadig of the time value of moey with this assumptio. Now, however, it is time to cosider the relatioship betwee future value ad iterest rates for differet compoudig periods. To begi, suppose that iterest is paid semiaually. If you the deposit $100 i a savigs accout at a omial, or stated, 8 percet aual iterest rate, the future value at the ed of six moths would be FV 0.5 = $100(1 + [0.08/2]) = $104 I other words, at the ed of oe half-year you would receive 4 percet i iterest, ot 8 percet. At the ed of a year the future value of the deposit would be FV 1 = $100(1 + [0.08/2]) 2 = $ This amout compares with $108 if iterest is paid oly oce a year. The $0.16 differece is caused by iterest beig eared i the secod six moths o the $4 i iterest paid at the ed of the first six moths. The more times durig the year that iterest is paid, the greater the future value at the ed of a give year. The geeral formula for solvig for the future value at the ed of years where iterest is paid m times a year is FV = PV 0 (1 + [i/m]) m (3.17) To illustrate, suppose that ow iterest is paid quarterly ad that you wish to kow the future value of $100 at the ed of oe year where the stated aual rate is 8 percet. The future value would be FV 1 = $100(1 + [0.08/4]) (4)(1) = $100( ) 4 = $ which, of course, is higher tha it would be with either semiaual or aual compoudig. The future value at the ed of three years for the example with quarterly compoudig is FV 3 = $100(1 + [0.08/4]) (4)(3) = $100( ) 12 = $ compared with a future value with semiaual compoudig of ad with aual compoudig of FV 3 = $100(1 + [0.08/2]) (2)(3) = $100( ) 6 = $ FV 3 = $100(1 + [0.08/1]) (1)(3) = $100( ) 3 = $ Thus, the more frequetly iterest is paid each year, the greater the future value. Whe m i Eq. (3.17) approaches ifiity, we achieve cotiuous compoudig. Shortly, we will take a special look at cotiuous compoudig ad discoutig. Preset (or Discouted) Value. Whe iterest is compouded more tha oce a year, the formula for calculatig preset value must be revised alog the same lies as for the calculatio of future value. Istead of dividig the future cash flow by (1 + i) as we do whe aual compoudig is ivolved, we determie the preset value by 57

20 Part 2 Valuatio PV 0 = FV /(1 + [i /m]) m (3.18) where, as before, FV is the future cash flow to be received at the ed of year, m is the umber of times a year iterest is compouded, ad i is the discout rate. We ca use Eq. (3.18), for example, to calculate the preset value of $100 to be received at the ed of year 3 for a omial discout rate of 8 percet compouded quarterly: PV 0 = $100/(1 + [0.08/4]) (4)(3) = $100/( ) 12 = $78.85 If the discout rate is compouded oly aually, we have PV 0 = $100/( ) 3 = $79.38 Thus, the fewer times a year that the omial discout rate is compouded, the greater the preset value. This relatioship is just the opposite of that for future values. Cotiuous Compoudig I practice, iterest is sometimes compouded cotiuously. Therefore it is useful to cosider how this works. Recall that the geeral formula for solvig for the future value at the ed of year, Eq. (3.17), is FV = PV 0 (1 + [i /m]) m As m, the umber of times a year that iterest is compouded, approaches ifiity ( ), we get cotiuous compoudig, ad the term (1 + [i/m]) m approaches e i, where e is approximately Therefore the future value at the ed of years of a iitial deposit of PV 0 where iterest is compouded cotiuously at a rate of i percet is FV = PV 0 (e) i (3.19) For our earlier example problem, the future value of a $100 deposit at the ed of three years with cotiuous compoudig at 8 percet would be FV 3 = $100(e) (0.08)(3) = $100( ) (0.24) = $ This compares with a future value with aual compoudig of FV 3 = $100( ) 3 = $ Cotiuous compoudig results i the maximum possible future value at the ed of periods for a give omial rate of iterest. By the same toke, whe iterest is compouded cotiuously, the formula for the preset value of a cash flow received at the ed of year is PV 0 = FV /(e) i (3.20) Thus the preset value of $1,000 to be received at the ed of 10 years with a discout rate of 20 percet, compouded cotiuously, is PV 0 = $1,000/(e) (0.20)(10) = $1,000/( ) 2 = $ We see the that preset value calculatios ivolvig cotiuous compoudig are merely the reciprocals of future value calculatios. Also, although cotiuous compoudig results i the maximum possible future value, it results i the miimum possible preset value. 58

21 3 The Time Value of Moey Questio Whe a bak quotes you a aual percetage yield (APY) o a savigs accout or certificate of deposit, what does that mea? Aswer Based o a cogressioal act, the Federal Reserve requires that baks ad thrifts adopt a stadardized method of calculatig the effective iterest rates they pay o cosumer accouts. It is called the aual percetage yield (APY). The APY is meat to elimiate cofusio caused whe savigs istitutios apply differet methods of compoudig ad use various terms, such as effective yield, aual yield, ad effective rate. The APY is similar to the effective aual iterest rate. The APY calculatio, however, is based o the actual umber of days for which the moey is deposited i a accout i a 365-day year (366 days i a leap year). I a similar vei, the Truth-i-Ledig Act madates that all fiacial istitutios report the effective iterest rate o ay loa. This rate is called the aual percetage rate (APR). However, the fiacial istitutios are ot required to report the true effective aual iterest rate as the APR. Istead, they may report a ocompouded versio of the effective aual iterest rate. For example, assume that a bak makes a loa for less tha a year, or iterest is to be compouded more frequetly tha aually. The bak would determie a effective periodic iterest rate based o usable fuds (i.e., the amout of fuds the borrower ca actually use) ad the simply multiply this rate by the umber of such periods i a year. The result is the APR. Effective aual iterest rate The actual rate of iterest eared (paid) after adjustig the omial rate for factors such as the umber of compoudig periods per year. Effective Aual Iterest Rate Differet ivestmets may provide returs based o various compoudig periods. If we wat to compare alterative ivestmets that have differet compoudig periods, we eed to state their iterest o some commo, or stadardized, basis. This leads us to make a distictio betwee omial, or stated, iterest ad the effective aual iterest rate. The effective aual iterest rate is the iterest rate compouded aually that provides the same aual iterest as the omial rate does whe compouded m times per year. By defiitio the, (1 + effective aual iterest rate) = (1 + [i /m]) (m)(1) Therefore, give the omial rate i ad the umber of compoudig periods per year m, we ca solve for the effective aual iterest rate as follows: 3 effective aual iterest rate = (1 + [i /m]) m 1 (3.21) For example, if a savigs pla offered a omial iterest rate of 8 percet compouded quarterly o a oe-year ivestmet, the effective aual iterest rate would be (1 + [0.08/4]) 4 1 = ( ) 4 1 = Oly if iterest had bee compouded aually would the effective aual iterest rate have equaled the omial rate of 8 percet. Table 3.7 cotais a umber of future values at the ed of oe year for $1,000 earig a omial rate of 8 percet for several differet compoudig periods. The table illustrates that the more umerous the compoudig periods, the greater the future value of (ad iterest eared o) the deposit, ad the greater the effective aual iterest rate. 3 The special case formula for effective aual iterest rate whe there is cotiuous compoudig is as follows: effective aual iterest rate = (e) i 1 59

22 Part 2 Valuatio Table 3.7 Effects of differet compoudig periods o future values of $1,000 ivested at a 8% omial iterest rate INITIAL COMPOUNDING FUTURE VALUE AT EFFECTIVE ANNUAL AMOUNT PERIODS END OF 1 YEAR INTEREST RATE* $1,000 Aually $1, % 1,000 Semiaually 1, ,000 Quarterly 1, ,000 Mothly 1, ,000 Daily (365 days) 1, ,000 Cotiuously 1, *Note: $1,000 ivested for a year at these rates compouded aually would provide the same future values as those foud i Colum 3. Amortizig a Loa A importat use of preset value cocepts is i determiig the paymets required for a istallmet-type loa. The distiguishig feature of this loa is that it is repaid i equal periodic paymets that iclude both iterest ad pricipal. These paymets ca be made mothly, quarterly, semiaually, or aually. Istallmet paymets are prevalet i mortgage loas, auto loas, cosumer loas, ad certai busiess loas. To illustrate with the simplest case of aual paymets, suppose you borrow $22,000 at 12 percet compoud aual iterest to be repaid over the ext six years. Equal istallmet paymets are required at the ed of each year. I additio, these paymets must be sufficiet i amout to repay the $22,000 together with providig the leder with a 12 percet retur. To determie the aual paymet, R, we set up the problem as follows: Amortizatio schedule A table showig the repaymet schedule of iterest ad pricipal ecessary to pay off a loa by maturity. 6 t $22,000 = R 1/ ( ) t = 1 = R(PVIFA 12%,6 ) I Table IV i the Appedix at the ed of the book, we fid that the discout factor for a sixyear auity with a 12 percet iterest rate is Solvig for R i the problem above, we have $22,000 = R(4.111) R = $22,000/4.111 = $5,351 Thus aual paymets of $5,351 will completely amortize (extiguish) a $22,000 loa i six years. Each paymet cosists partly of iterest ad partly of pricipal repaymet. The amortizatio schedule is show i Table 3.8. We see that aual iterest is determied by Table 3.8 Amortizatio schedule for illustrated loa (1) (2) (3) (4) END ANNUAL PRINCIPAL PRINCIPAL AMOUNT OF INSTALLMENT INTEREST PAYMENT OWING AT YEAR END YEAR PAYMENT (4) t (1) (2) (4) t 1 (3) 0 $22,000 1 $ 5,351 $ 2,640 $ 2,711 19, ,351 2,315 3,036 16, ,351 1,951 3,400 12, ,351 1,542 3,809 9, ,351 1,085 4,266 4, , ,778 0 $32,106 $10,106 $22,000 60

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