THE TIME VALUE OF MONEY


 Jessie Nicholson
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1 QRMC04 9/17/01 4:43 PM Page 51 CHAPTER FOUR THE TIME VALUE OF MONEY 4.1 INTRODUCTION AND FUTURE VALUE The perspective ad the orgaizatio of this chapter differs from that of chapters 2 ad 3 i that topics are arraged by fiace applicatio rather tha mathematics area. The mathematics tools preseted i chapters 2 ad 3 are applied i this chapter to closely examie the aalytical aspects uderlyig what might be the sigle most importat topic i fiace the time value of moey. I this chapter, we study how ivestors ad borrowers iteract to value ivestmets ad determie iterest rates o loas ad fixed icome securities. Iterest is paid by borrowers to leders for the use of leders moey. The level of iterest charged is typically stated as a percetage of the pricipal (the amout of the loa). Whe a loa matures, the pricipal must be repaid alog with ay upaid accumulated iterest. I a free market ecoomy, iterest rates are determied joitly by the supply of ad demad for moey. Thus, leders will usually attempt to impose as high a iterest rate as possible o the moey they led; borrowers will attempt to obtai the use of moey at the lowest iterest rates available to them. Competitio amog borrowers ad competitio amog leders will ted to lead iterest rates toward some competitive level. Factors affectig the levels of iterest rates will do so by affectig supply ad demad coditios for moey. Amog these factors are iflatio rates, loa risks, ivestor itertemporal moetary prefereces (how much idividuals ad istitutios prefer to have moey ow rather tha have to wait for it), govermet policies, ad the admiistrative costs of extedig credit. 4.2 SIMPLE INTEREST (Backgroud readig: sectios 2.4, 2.7, ad 4.1) Iterest is computed o a simple basis if it is paid oly o the pricipal of the loa. Compoud iterest is paid o accumulated loa iterest as well as o the pricipal. Thus, if a sum of moey (X 0 ) were borrowed at a aual iterest rate i ad repaid at the
2 QRMC04 9/17/01 4:43 PM Page The time value of moey ed of years with accumulated iterest accruig o a simple basis, the total sum repaid (FV or Future Value at the ed of year ) is determied as follows: FV = X 0 (1 + i) (4.1) The subscripts ad 0 merely desigate time; they do ot imply ay arithmetic fuctio. The product i whe multiplied by X 0 reflects the value of iterest paymets to be made o the loa; the value 1 accouts for the fact that the pricipal of the loa must be repaid. If the loa duratio icludes some fractio of a year, the value of will be fractioal; for example, if the loa duratio were oe year ad three moths, would be The total amout paid (or, the future value of the loa) will be a icreasig fuctio of the legth of time the loa is outstadig () ad the iterest rate (i) charged o the loa. For example, if a cosumer borrowed $1,000 at a iterest rate of 10% for oe year, his total repaymet would be $1,100, determied from equatio (4.1) as follows: FV 1 = $1,000( ) = $1, = $1,100. If the loa were to be repaid i two years, its future value would be determied as follows: FV 2 = $1,000( ) = $1, = $1,200. Cotiuig our example, if the loa were to be repaid i five years, its Future Value would be FV 5 = $1,000( ) = $1, = $1,500. The loger the duratio of a loa, the higher will be its future value. Thus, the loger leders must wait to have their moey repaid, the greater will be the total iterest paymets made by borrowers. 4.3 COMPOUND INTEREST (Backgroud readig: sectios 2.7, 3.1, ad 4.2) Iterest is computed o a compoud basis whe the borrower pays iterest o accumulated iterest as well as o the loa pricipal. If iterest o a give loa must accumulate for a full year before it is compouded, the future value of this loa is determied as follows: FV = X 0 (1 + i). (4.2) For example, if a idividual were to deposit $1,000 ito a savigs accout payig aually compouded iterest at a rate of 10% (here, the bak is borrowig moey), the future value of the accout after five years would be $1,610.51, determied by equatio (4.2) as follows:
3 QRMC04 9/17/01 4:43 PM Page 53 Fractioal period compoudig of iterest 53 FV 5 = $1,000( ) 5 = $1, = $1, = $1, Notice that this sum is greater tha the future value of the loa ($1,500) whe iterest is ot compouded. The compoud iterest formula ca be derived ituitively from the simple iterest formula. If iterest must accumulate for a full year before it is compouded, the the future value of the loa after oe year is $1,100, exactly the same sum as if iterest had bee computed o a simple basis: FV = X 0 (1 + i) = X 0 (1 + 1 i) = X 0 (1 + i) 1 = $1,000( ) = $1,100. (4.3) The future values of loas where iterest is compouded aually ad whe iterest is computed o a aual basis will be idetical oly whe equals oe. Sice the value of this loa is $1,100 after oe year ad iterest is to be compouded, iterest ad future value for the secod year will be computed o the ew balace of $1,100: FV 2 = X 0 (1 + 1 i)(1 + 1 i) = X 0 (1 + i)(1 + i) = X 0 (1 + i) 2, FV 2 = $1,000( )( ) = $1,000( ) 2 = $1,210. (4.4) This process ca be cotiued for five years: FV 5 = $1,000( )( )( )( )( ) = $1,000( ) 5 = $1, More geerally, the process ca be applied for a loa of ay maturity. Therefore: FV = X 0 (1 + i)(1 + i) (1 + i) = X 0 (1 + i), FV = $1,000( )( ) ( ) = $1,000( ). (4.5) 4.4 FRACTIONAL PERIOD COMPOUNDING OF INTEREST I the previous examples, iterest is compouded aually; that is, iterest must accumulate at the stated rate i for a etire year before it ca be compouded or recompouded. I may savigs accouts ad other ivestmets, iterest ca be compouded semiaually, quarterly, or eve daily. If iterest is to be compouded more tha oce per year (or oce every fractioal part of a year), the future value of such a ivestmet will be determied as follows: FV = X 0 (1 + i/m) m, (4.6) where iterest is compouded m times per year. The iterpretatio of this formula is fairly straightforward. For example, if m is 2, the iterest is compouded o a semiaual basis. The semiaual iterest rate is simply i/m or i/2. If the ivestmet is held
4 QRMC04 9/17/01 4:43 PM Page The time value of moey for periods, the it is held for 2 semiaual periods. Thus, we compute a semiaual iterest rate i/2 ad the umber of semiaual periods the ivestmet is held 2. If $1,000 were deposited ito a savigs accout payig iterest at a aual rate of 10% compouded semiaually, its future value after five years would be $1,628.90, determied as follows: FV 5 = $1,000( /2) 2 5 = $1,000(1.05) 10 = $1,000( ) = $1, Notice that the semiaual iterest rate is 5% ad that the accout is outstadig for te sixmoth periods. This sum of $1, exceeds the future value of the accout if iterest is compouded oly oce aually ($1,610.51). I fact, the more times per year iterest is compouded, the higher will be the future value of the accout. For example, if iterest o the same accout were compouded mothly (12 times per year), the accout s future value would be $1,645.31: FV 5 = $1,000( /12) 12 5 = $1,000( ) 60 = $1, The mothly iterest rate is ad the accout is ope for m or 60 moths. With daily compoudig, the accout s value would be $1,648.61: FV 5 = $1,000( /365) = $1, Therefore, as m icreases, future value icreases, as i table 4.1. However, this rate of icrease i future value becomes smaller with larger values for m; that is, the icreases i FV iduced by icreases i m evetually become quite small. Thus, the differece i the future values of two accouts where iterest is compouded hourly i oe ad every miute i the other may actually be rather trivial. Table 4.1 Future values ad aual percetage yields of accouts with iitial $10,000 deposits at 10% Years to Future value Future value Future value Future value Future value maturity, simple compouded compouded compouded compouded iterest ($) aually ($) mothly ($) daily ($) cotiuously ($) 1 11,000 11,000 11,047 11,052 11, ,000 12,100 12,204 12,214 12, ,000 13,310 13,481 13,498 13, ,000 14,641 14,894 14,917 14, ,000 16,105 16,453 16,486 16, ,000 25,937 27,070 27,179 27, ,000 67,275 73,281 73,870 73, , , , , , ,000 1,173,909 1,453,699 1,483,116 1,484,140 Aual Varies percetage with yield
5 QRMC04 9/17/01 4:43 PM Page 55 Fractioal period compoudig of iterest 55 APPLICATION 4.1: APY AND BANK ACCOUNT COMPARISONS Fiacial istitutios ofte have may ways of defiig the terms or rules associated with their loas, accouts, ad other ivestmets. Such large umbers of terms ad rules frequetly lead to cofusio amog ivestors ad cosumers, particularly whe tryig to compare their various alteratives. For this reaso, there exist several covetios which are iteded to stadardize the disclosure of these terms. For example, we have see i the previous two sectios the impact that chagig the compoudig itervals has o future value. Compariso betwee ivestmets is more complicated whe their umbers of compoudig itervals differ. To simplify the compariso betwee loas with varyig compoudig itervals, it is ofte useful to compute aual percetage yields, also kow as equivalet aual rates. The aual percetage yield (APY) represets the yield that, if compouded oce per year, will produce the same future value as the stated rate i compouded m times per year: 1 Thus, we ca compute APY as follows: m i FV = X 0 1+ X ( APY). m = 0 1+ m i APY = 1+. m 1 (4.7) Because the aual percetage yield simplifies compariso betwee accouts with differet compoudig itervals, U.S. baks are ormally required by law to disclose APYs alog with their stated iterest rates i their advertisemets solicitig bak accouts. Cosider a example where a savigs accout at bak X pays 6% iterest compouded daily ad a similar accout at bak Y pays 6 % iterest, compouded semi 1 4 aually. Which accout will pay more to a ivestor who leaves a $100 deposit for oe year? Based o equatio (4.6), we ca obtai the followig future values: FV X = $ = $ , 365. FV Y = $ = Thus, a accout payig a stated rate of 6% compouded daily yields a future value equivalet to a accout payig slightly more tha 6.18% compouded aually. A accout payig a stated rate of 6.25% compouded semiaually yields a future value equivalet to a accout payig slightly more tha 6.437% compouded aually If iterest is ot compouded, APY = (1 + i) 1.
6 QRMC04 9/17/01 4:43 PM Page The time value of moey Therefore, the accout i bak Y is preferred to that at bak X. We ca arrive at the same preferece rakig by examiig aual percetage yields: APY X = +., APYY = = = Because the accout at bak Y has the higher APY, it is preferred. The accout with the higher APY will produce a higher future value. However, it is ot ecessarily true that the accout with the highest stated rate also has the highest APY. A 1997 advertisemet i a New York ewspaper offered a fiveyear certificate of deposit accout payig iterest at a aual rate of 5.83%, compouded daily. The aual percetage yield (APY) o this accout was advertised at 6.00%. Give these details, the future value of $100 deposited ito this accout ca be computed to be $133.84: FV = $100( /365) = $ The APY of this accout is determied as follows: APY = ( /365) = The 6% APY advertised by the bak was approximately correct; such advertisemets are ofte rouded slightly. I ay case, the future value of this accout ca be determied with the 6.003% accout APY as follows: FV = $100( ) 5 = $ A $100 iitial deposit ito a fiveyear CD accout payig iterest at a aual rate of 5.85%, compouded quarterly, would have a future value of $133.69: FV = $100( /4) 4 5 = $ The APY of this accout is , determied as follows: APY = ( /4) 4 1 = Note that the future value ad the APY of the secod accout are lower tha those of the first accout eve though the stated iterest rate o the secod accout is higher. Compoudig ca have a sigificat effect o both future value ad APY. 4.5 CONTINUOUS COMPOUNDING OF INTEREST (Backgroud readig: sectios 2.5 ad 4.4) If iterest were to be compouded a ifiite umber of times per period, we would say that iterest is compouded cotiuously. However, we caot obtai a umerical
7 QRMC04 9/17/01 4:43 PM Page 57 Auity future values 57 solutio for future value by merely substitutig i for m i equatio (4.6) calculators have o key. I the previous sectio, we saw that icreases i m cause the future value of a ivestmet to icrease. As m approaches ifiity, FV cotiues to icrease, however at decreasig rates. More precisely, as m approaches ifiity (m ), the future value of a ivestmet ca be defied as follows: FV = X 0 e i, (4.8) where e is the atural log whose value ca be approximated at 2.718, or derived as i sectio 2.5. If a ivestor were to deposit $1,000 ito a accout payig iterest at a rate of 10%, cotiuously compouded (or compouded a ifiite umber of times per year), the accout s future value would be approximately $1,648.64: FV 5 = $1,000 e 1 5 $1, = $1, The Future Value of this accout exceeds oly slightly the value of the accout if iterest were compouded daily. Also, ote that cotiuous compoudig simply meas that iterest is compouded a ifiite umber of times per time period. 4.6 ANNUITY FUTURE VALUES (Backgroud readig: sectios 2.8, 3.4, ad 4.3) A auity is a series of equal paymets made at equal itervals. Suppose that paymets are to be made ito a iterestbearig accout. The future value of that accout will be a fuctio of iterest accruig o prior deposits as well as the deposits themselves. A future value auity factor (fvaf ) is used to determie the future value of a auity. This auity is a series of equal paymets made at idetical itervals. The future value auity factor may be derived through the use of the geometric expasio procedure discussed i sectio 3.4. This techique is very useful for future value computatios whe a large umber of time periods are ivolved. The geometric expasio eables us to reduce a repetitive expressio requirig may calculatios to a expressio that ca be computed much more quickly. Suppose that we wish to determie the future value of a accout based o a paymet of X made at the ed of each year t for years, where that accout pays a aual iterest rate equal to i: FVA = X[(1 + i) 1 + (1 + i) (1 + i) 2 + (1 + i) 1 + 1]. (4.9) The paymet made at the ed of the first year will accumulate iterest for a total of 1 years, the paymet at the ed of the secod year will accumulate iterest for 2 years, ad so o. Clearly, determiig the future value of this accout with equatio (4.9) will be very timecosumig if is large. The first step i the geometric expasio to simplify equatio (4.9) is to multiply both of its sides by 1 + i:
8 QRMC04 9/17/01 4:43 PM Page The time value of moey FVA(1 + i) = X[(1 + i) + (1 + i) (1 + i) 3 + (1 + i) 2 + (1 + i)]. (4.10) The secod step i this geometric expasio is to subtract equatio (4.9) from equatio (4.10), to obtai: FVA(1 + i) FVA = X[(1 + i) 1]. (4.11) Notice that the subtractio led to the cacellatio of may terms, reducig the equatio that we wish to compute with to a much more maageable size. Fially, we rearrage terms i equatio (4.12) to obtai equatios (4.12) ad (4.13): FVA 1 + FVA i FVA = X[(1 + i) 1] = FVA i = X[(1 + i) 1], (4.12) FVA X i [( 1+ ) 1] =. i (4.13) Practicig derivatios such as this is a excellet way to uderstad the ituitio behid fiacial formulas. Uderstadig the derivatios is ecessary i order to be able to modify the formulas for a variety of more complex (ad realistic) scearios. Cosider a example applicable to may idividuals who ope Idividual Retiremet Accouts (I.R.A. s), from which they may withdraw whe they reach the age of 1 59 years. Cosider a idividual who makes a $2,000 cotributio to his I.R.A. at 2 the ed of each year for 20 years. All of his cotributios receive a 10% aual rate of iterest, compouded aually. What will be the total value of this accout, icludig accumulated iterest, at the ed of the 20year period? Equatio (4.13) ca be used to evaluate the future value of this auity, where X is the aual cotributio made at the ed of each year by the ivestor to his accout, i is the iterest rate o the accout, ad FVA is the future value of the auity. The future value of this idividual s I.R.A. is $114,550: 20 ( ) 1 FVA = $ 2000, = $ 114, This future value auity equatio ca be used wheever idetical periodic cotributios are made toward a accout. Sectio 4.8 will preset a discussio o determiig the preset value of such a series of cash flows. (The term preset value is also defied later i sectio 4.8.) Note that each of the above calculatios assumes that cash flows are paid at the ed of each period. If, istead, cash flows were realized at the begiig of each period, the auity would be referred to as a auity due. The auity due would geerate a extra year of iterest o each cash flow. Hece, the future value of a auity due is determied by simply multiplyig the future value auity formula by (1 + i): FVA, due = ( + i) ( + i) ( 1+ i) X ( 1 + i) = X. i i (4.14)
9 QRMC04 9/17/01 4:43 PM Page 59 Auity future values 59 From the above example, we fid that the future value of the idividual s I.R.A. is $126,005 if paymets to the I.R.A. are made at the begiig of each year: 21 ( ) ( ) FVA, due = $ 2, 000 = $ 126, APPLICATION 4.2: PLANNING FOR RETIREMENT (Backgroud readig: sectios 2.5, 3.1, ad 4.5) Suppose that a 23yearold accoutat wishes to retire as a millioaire based o her retiremet savigs accout. She iteds to ope ad cotribute to a taxdeferred 401k retiremet accout sposored by her employer each year util she retires with $1,000,000 i that accout. Would she meet her retiremet goal if she deposited $10,000 ito that accout at the ed of each year util she is 65 years of age? Assume that her accout will geerate a aual rate of iterest equal to 5% for each of the ext 42 years. Equatio (4.13) will be used to solve this problem: FV. = $ 10, = $ 1, 352, Now, suppose that she would like to retire as soo as possible with $1,000,000 i her accout. Assumig that othig else associated with her situatio chages, what is the earliest age at which she ca retire? Now, we will use equatio (4.13) to algebraically solve for, the umber of years that the accoutat must wait to retire: FV ( 1+ i) 1 ( +. ) = X = $ 10, i 005. ( 1+ i) 1 FV i = X, = ( 1+ i) 1, i X FV i FV i log + log( i), log log( i), X 1 = 1+ + X 1 1+ = log $ 1000,, log( ) = + log(. ).. $, + = log( 105. ) = Sice paymets are made at the ed of each year, the accoutat must wait 37 years whe she is 60 before she ca retire as a millioaire. Note that we were able to fid a closedform solutio (put o oe side aloe) usig simple algebra. I may time value problems, the exact placemet of the expoet will prevet us from obtaiig a solutio so easily.
10 QRMC04 9/17/01 4:43 PM Page The time value of moey 4.7 DISCOUNTING AND PRESENT VALUE (Backgroud readig: sectio 4.3) Cash flows realized at the preset time have a greater value to ivestors tha cash flows realized later, for the followig reasos: 1 Iflatio. The purchasig power of moey teds to declie over time. 2 Risk. We ever kow with certaity whether we will actually realize the cash flow that we are expectig. 3 The optio to either sped moey ow or defer spedig it is likely to be worth more tha beig forced to defer spedig the moey. The purpose of the PresetValue model is to express the value of a future cash flow i terms of cash flows at preset. Thus, the PresetValue model is used to compute how much a ivestor would pay ow for the expectatio of some cash flow to be received i years. The preset value of this cash flow would be a fuctio of iflatio, the legth of wait before the cash flow is received (), the riskiess associated with the cash flow, ad the time value a ivestor associates with moey (how much he eeds moey ow as opposed to later). Perhaps the easiest way to accout for these factors whe evaluatig a future cash flow is to discout it i the followig maer: CF PV = ( 1 + k), (4.15) where CF is the cash flow to be received i year, k is a appropriate discout rate accoutig for risk, iflatio, ad the ivestor s time value associated with moey, ad PV is the preset value of that cash flow. The discout rate eables us to evaluate a future cash flow i terms of cash flows realized today. Thus, the maximum a ratioal ivestor would be willig to pay for a ivestmet yieldig a $9,000 cash flow i six years assumig a discout rate of 15% would be $3,891, determied as follows: $ 9, 000 PV = ( ) 6 $ 9, 000 = = $ 3, I the above example, we simply assumed a 15% discout rate. Realistically, perhaps the easiest value to substitute for k is the curret iterest or retur rate o loas or other ivestmets of similar duratio ad riskiess. However, this marketdetermied iterest rate may ot cosider the idividual ivestor s time prefereces for moey. Furthermore, the ivestor may fid difficulty i locatig a loa (or other ivestmet) of similar duratio ad riskiess. For these reasos, more scietific methods for determiig appropriate discout rates will be discussed later. I ay case, the discout rate should accout for iflatio, the riskiess of the ivestmet, ad the ivestor s time value for moey.
11 QRMC04 9/17/01 4:43 PM Page 61 The preset value of a series of cash flows 61 Derivig the presetvalue formula The presetvalue formula ca be derived easily from the compoud iterest formula. Assume that a ivestor wishes to deposit a sum of moey ito a savigs accout payig iterest at a rate of 15%, compouded aually. If the ivestor wishes to withdraw from his accout $9,000 i six years, how much must he deposit ow? This aswer ca be determied by solvig the compoud iterest formula for X 0 : FV FV = X ( 1 + i), X = ( 1 + i) $ 9, 000 = ( ) $ 9, 000 = = $ 3, Therefore, the ivestor must deposit $3, ow i order to withdraw $9,000 i six years at 15%. Notice that the presetvalue equatio (4.15) is almost idetical to the compoud iterest formula where we solve for the pricipal (X 0 ): CF PV = FV X ( + k), 0 = 1 ( 1+ i). Mathematically, these formulas are the same; however, there are some differeces i their ecoomic iterpretatios. I the iterest formulas, iterest rates are determied by market supply ad demad coditios, whereas discout rates are idividually determied by ivestors themselves (although their calculatios may be iflueced by market iterest rates). I the presetvalue formula, we wish to determie how much some future cash flow is worth ow; i the iterest formula above, we wish to determie how much moey must be deposited ow to attai some give future value. 4.8 THE PRESENT VALUE OF A SERIES OF CASH FLOWS (Backgroud readig: sectios 2.8 ad 4.7) Suppose that a ivestor eeds to evaluate a series of cash flows. She eeds oly to discout each separately ad the sum the preset values of each of the idividual cash flows. Thus, the preset value of a series of cash flows CF t received i time period t ca be determied by the followig expressio: CFt PV = t ( + k). 1 t= 1 (4.16) For example, if a ivestmet were expected to yield aual cash flows of $200 for each of the ext five years, assumig a discout rate of 5%, its preset value would be $865.90: PV = ( ) ( ) ( ) ( ) ( ) = $
12 QRMC04 9/17/01 4:43 PM Page The time value of moey Therefore, the maximum price a idividual should pay for this ivestmet is $865.90, eve though the cash flows yielded by the ivestmet total $1,000. Because the idividual must wait up to five years before receivig the $1,000, the ivestmet is worth oly $ Use of the presetvalue series formula does ot require that cash flows CF t i each year be idetical, as does the auity model preseted i the ext sectio. 4.9 ANNUITY PRESENT VALUES (Backgroud readig: sectios 3.4, 4.6, ad 4.8) The expressio for determiig the preset value of a series of cash flows ca be quite cumbersome, particularly whe the paymets exted over a log period of time. This formula requires that cash flows be discouted separately ad the summed. Whe is large, this task may be rather timecosumig. If the aual cash flows are idetical ad are to be discouted at the same rate, a auity formula ca be a useful timesavig device. The same problem as discussed i the previous sectio ca be solved usig the followig auity formula: PV A CF 1 = 1 k ( 1 + ) k, (4.17) where CF is the level of the aual cash flow geerated by the auity (or series). Use of this formula does require that all of the aual cash flows be idetical. Thus, the preset value of the cash flows i the problem discussed i the previous sectio is $865.90, determied as follows: $ 200 PV A = ( ) 5 = $ 4, 000( ) = $ As becomes larger, this formula becomes more useful relative to the presetvalue series formula discussed i the previous sectio. However, the auity formula requires that all cash flows be idetical ad be paid at the ed of each year. The presetvalue auity formula ca be derived easily from the perpetuity formula discussed i sectio 4.11, or from the geometric expasio procedure described later i this sectio. Note that each of the above calculatios assumes that cash flows are paid at the ed of each period. If, istead, cash flows were realized at the begiig of each period, the auity would be referred to as a auity due. Each cash flow geerated by the auity due would, i effect, be received oe year earlier tha if cash flows were realized at the ed of each year. Hece, the preset value of a auity due is determied by simply multiplyig the presetvalue auity formula by (1 + k): PVA due CF 1 = 1 k ( 1 + k) ( 1 + k). (4.18)
13 QRMC04 9/17/01 4:43 PM Page 63 Auity preset values 63 The preset value of the fiveyear auity due discouted at 5% is determied as follows: PVA due 200 = ( +. ) = $4,000[ ](1.05) = ( ) Derivig the presetvalue auity formula The preset value auity factor (pvaf ) may be derived through use of the geometric expasio (see sectio 3.4). Cosider the case where we wish to determie the preset value of a ivestmet based o a cash flow of CF made at the ed of each year t for years, where the appropriate discout rate is k: PV A = CF ( + k) ( + k) ( + k) (A) Thus, the paymet made at the ed of the first year is discouted for oe year, the paymet at the ed of the secod year is discouted for two years, ad so o. Clearly, determiig the preset value of this accout will be very timecosumig if is large. The first step of the geometric expasio is to multiply both sides of (A) by (1 + k): 1 PVA( 1+ k) = CF 1+ ( 1 + k) ( 1 k) + 1. (B) The secod step i the geometric expasio is to subtract equatio (A) from equatio (B), to obtai: 1 PVA( 1+ k) PVA= CF 1 ( 1 + ) k, (C) which simplifies to 1 PVA( 1+ k 1) = PVA() k = CF 1 ( 1 + ) k. (D) Notice that the subtractio led to the cacellatio of may terms, reducig the equatio that we wish to compute to a much more maageable size. Fially, we cacel the oes o the left side ad divide both sides of equatio (D) by k, to obtai: PV A CF 1 = 1 k ( 1 + ) k. (4.17)
14 QRMC04 9/17/01 4:43 PM Page The time value of moey APPLICATION 4.3: PLANNING FOR RETIREMENT, PART II (Backgroud readig: applicatio 4.2 ad sectio 4.9) Suppose that the 23yearold accoutat from applicatio 4.2 wishes to retire as a millioaire based o her retiremet savigs accout, but eeds to kow what the preset value of that milliodollar accout is. If the accout is ope for the full 37 years, its future value will be $1,016,282, based o equatio (4.13). Based o a discout rate of 5% ad assumig that the accout is ope for 37 years, its preset value is easily determied from equatio (4.15) as follows: CF $ 1, 016, 282 PV = = = $ ,. 07. ( 1 + k) 37 ( ) I presetvalue terms, this milliodollar accout is obviously worth much less tha $1,000,000. However, what is the preset value of the aual series $10,000 deposits that she will make to that accout? Agai, based o a 5% discout rate, we determie this preset value with equatio (4.17) as follows: $ 10, PV A = ( ) = $ 167, Notice that the preset value of cotributios that she makes to the accout is idetical to the preset value of what she will be able to retire with. 37 APPLICATION 4.4: VALUING A BOND Because the preset value of a series of cash flows is simply the sum of the preset values of the cash flows, the auity formula ca be combied with other presetvalue formulas to evaluate ivestmets. Cosider, for example, a 7% coupo bod makig aual iterest paymets for ie years. If this bod has a $1,000 face (or par) value, ad its cash flows are discouted at 6%, its cash flows will be $70 i each of the ie years plus $1,000 i the teth year. The preset value of the bod s cash flows ca be determied as follows: $ 70 PV = ( ) $ 1, ( ) 9 9 $ 1, 000 = $ 1166,. 67( ) = $ = $1, Thus, the value of a bod is simply the sum of the preset values of the cash flow streams resultig from iterest paymets ad from pricipal repaymet.
15 QRMC04 9/17/01 4:43 PM Page 65 Amortizatio 65 Now, let us revise the above example to value aother 7% coupo bod. This bod will make semiaual (twice yearly) iterest paymets for ie years. If this bod has a $1,000 face (or par) value, ad its cash flows are discouted at the stated aual rate of 6%, its value ca be determied as follows: $ 35 PV = ( ) $ 1, ( ) = $ = $1, $ 1, 000 = $ 1166,. 67( ) Agai, the value of the bod is the sum of the preset values of the cash flow streams resultig from iterest paymets ad from the pricipal repaymet. However, the semiaual discout rate equals 3% ad paymets are made to bodholders i each of 18 semiaual periods AMORTIZATION (Backgroud readig: sectio 4.9) At the begiig of this chapter, we derived the cocept of preset value from that of future value. Amortizatio is essetially a topic relatig to iterest, but the presetvalue auity model preseted i this chapter is crucial to its developmet. Amortizatio is the paymet structure associated with a loa. That is, the amortizatio schedule of a loa is its paymet schedule. Cosider the auity model from equatio (4.17): PV A CF 1 = 1 k ( 1 + ) k. (4.17) Typically, whe a loa is amortized, the loa repaymets will be made i equal amouts; that is, each aual or mothly paymet will be idetical. At the ed of the repaymet period, the balace (amout of pricipal remaiig) o the loa will be zero. Thus, each paymet made by the borrower is applied to the pricipal repaymet as well as to iterest. A bak ledig moey will require that the sum of the preset values of its repaymets be at least as large as the sum of moey it loas. Therefore, if the bak loas a sum of moey equal to PV for years at a iterest rate of i, the amout of the aual loa repaymet will be CF: 1 CF = [ PVA k] 1 ( 1 + ) k. (4.18) For example, if a bak were to exted a $865,895 five year mortgage to a corporatio at a iterest rate of 5%, the corporatio s aual paymet o the mortgage would be $200,000, determied by equatio (4.18):
16 QRMC04 9/17/01 4:43 PM Page The time value of moey Table 4.2 The amortizatio schedule of a $865,895 loa with equal aual paymets for five years at 5% Year Pricipal ($) Paymet ($) Iterest ($) Paymet to pricipal ($) 1 865, ,000 43, , , ,000 35, , , ,000 27, , , ,000 18, , , ,000 9, ,476 The loa is fully repaid by the ed of the fifth year. The pricipal represets the balace at the begiig of the give year. The paymet is made at the ed of the give year, ad icludes oe year of iterest accruig o the pricipal from the begiig of that year. The remaiig part of the paymet is paymet to the pricipal. This paymet to the pricipal is deducted from the pricipal or balace as of the begiig of the followig year. 1 CF = [$ 865, ] 1 ( ) = $ 200, 000. Thus, each year, the corporatio will pay $200,000 toward both the loa pricipal ad iterest obligatios. The amouts attributed to each are give i table 4.2. Notice that as paymets are applied toward the pricipal, the pricipal declies; correspodigly, the iterest paymets declie. Noetheless, total aual paymets are idetical util the pricipal dimiishes to zero i the fifth year. 5 APPLICATION 4.5: DETERMINING THE MORTGAGE PAYMENT A family has purchased a home with $30,000 dow ad a $300,000 mortgage. The mortgage will be amortized over 30 years with equal mothly paymets. The iterest rate o the mortgage will be 9% per year. Based o this data, we would like to determie the mothly mortgage paymet ad compile a amortizatio table decomposig each of the mothly paymets ito iterest ad paymet toward priciple. First, we will express aual data as mothly data. Three hudred ad sixty (12 30) moths will elapse before the mortgage is fully paid, ad the mothly iterest rate will be , or 9% divided by 12. Give this mothly data, mothly mortgage paymets are determied as follows: Paymet = $ 300, ( ) 360 $,.. = Table 4.3 depicts the amortizatio schedule for this mortgage.
17 QRMC04 9/17/01 4:43 PM Page 67 Perpetuity models 67 Table 4.3 The amortizatio schedule of a $300,000 loa with equal mothly paymets for 30 years at 9% iterest per aum (0.0075% per moth) Moth Begiigof Total Paymet o Paymet o moth pricipal ($) paymet ($) iterest ($) pricipal ($) 1 300, , , , , , , , , , , , , , , , , , , , , , , , Studets should be able to work through the figures o this table startig from the upper lefthad corer, the workig to the left, the dow. I this particular example, because is large (360), use of a computerized spreadsheet will make computatios substatially more efficiet PERPETUITY MODELS (Backgroud readig: sectio 4.9) As the value of approaches ifiity i the auity formula, the value of the righthad side term i the brackets, 1 ( 1 + k), approaches zero. That is, the cash flows associated with the auity are paid each year for a period approachig forever. Therefore, as approaches ifiity, the value of the ifiite time horizo auity approaches CF PVA = [ 1 0]; k PV CF k P =. (4.19) The auity formula discussed i sectio 4.9 ca be derived ituitively by use of figure 4.1. First, cosider a perpetuity as a series of cash flows begiig at time period oe (oe year from ow) ad extedig idefiitely ito perpetuity. Cosider a secod perpetuity with cash flows begiig i time period ad extedig idefiitely ito perpetuity. If a ivestor is to receive a year auity, the secod perpetuity represets those cash flows from the first perpetuity that he will ot receive. Thus, the
18 QRMC04 9/17/01 4:43 PM Page The time value of moey Preset value of perpetuity begiig i oe year = CF k Time 1 Preset value of year auity: PVA Preset value of perpetuity begiig i year ( + 1): (CF/k) (1 + k) Figure 4.1 Derivig auity preset value from perpetuity preset values. The preset value of a perpetuity begiig i oe year mius the preset value of a secod perpetuity begiig i year ( + 1) equals the preset value of a year auity. Thus, PVA = CF/k (CF/k) (1 + k) = CF/k [1 1/(1 + k) ]. differece betwee the preset values of the first ad secod perpetuities represets the value of the auity that he will receive. Note that the secod perpetuity is discouted a secod time, sice its cash flows do ot begi util year : PV A CF CF/ k CF 1 = = 1 k ( 1 + k) k ( 1 + k) The perpetuity model is useful i the evaluatio of a umber of ivestmets. Ay ivestmet with a idefiite or perpetual life expectacy ca be evaluated with the perpetuity model. For example, the preset value of a stock, if its divided paymets are projected to be stable, will be equal to the amout of the aual divided (cash flow) geerated by the stock divided by a appropriate discout rate. I Europea fiacial markets, a umber of perpetual bods have bee traded for several ceturies. I may regios i the Uited States, groud rets (perpetual leases o lad) are traded. The proper evaluatio of these ad may other ivestmets requires the use of perpetuity models. The maximum price a ivestor would be willig to pay for a perpetual bod geeratig a aual cash flow of $200, each discouted at a rate of 5%, ca be determied from equatio (4.19): PV = $ 200 P. =$ 4 005, SINGLESTAGE GROWTH MODELS (Backgroud readig: sectios 4.9 ad 4.11) If the cash flow associated with a ivestmet were expected to grow at a costat aual rate of g, the amout of the cash flow geerated by that ivestmet i year t would be
19 QRMC04 9/17/01 4:43 PM Page 69 Siglestage growth models 69 CF t = CF 1 (1 + g) t 1, (4.20) where CF 1 is the cash flow geerated by the ivestmet i year oe. Thus, if a stock payig a divided of $100 i year oe were expected to icrease its divided paymet by 10% each year thereafter, the divided paymet i the fourth year would be $133.10: CF 4 = CF 1 ( ) 4 1 Similarly, the cash flow geerated by the ivestmet i the followig year (t + 1) will be CF t+1 = CF 1 (1 + g) t. The stock s divided i the fifth year will be $146.41: CF 4+1 = CF 1 ( ) 4 = $ If the stock had a ifiite life expectacy (as most stocks might be expected to), ad its divided paymets were discouted at a rate of 13%, the value of the stock would be determied by PV gp $ 100 $ 100 = = = $ 3, This expressio is ofte called the Gordo Stock Pricig Model. It assumes that the cash flows (divideds) associated with the stock are kow i the first period ad will grow at a costat compoud rate i subsequet periods. More geerally, this growig perpetuity expressio ca be writte as follows: PV CF1 = k g gp. (4.21) The growig perpetuity expressio simply subtracts the growth rate from the discout rate; the growth i cash flows helps to cover the time value of moey. This formula for evaluatig growig perpetuities ca be used oly whe k > g. If g > K, either the growth rate or discout rate has probably bee calculated improperly. Otherwise, the ivestmet would have a ifiite value (eve though the formula would geerate a egative value). The formula (4.22) for evaluatig growig auities ca be derived ituitively from the growig perpetuity model. I figure 4.2, the differece betwee the preset value of a growig perpetuity with cash flows begiig i time period is deducted from the preset value of a perpetuity with cash flows begiig i year oe, resultig i the preset value of a year growig auity. Notice that the amout of the cash flow geerated by the growig auity i year ( + 1) is CF(1 + g). This is the first of the
20 QRMC04 9/17/01 4:43 PM Page The time value of moey Preset value of growig perpetuity begiig i oe year: CF 1 (k g) Time 1 Preset value of year growig auity: PV GA Preset value of growig perpetuity begiig i year ( + 1): [CF 1(1 + g) /(k g)] (1 + k) Figure 4.2 Derivig growig auity preset value from growig perpetuity pereset value. The preset value of a growig perpetuity begiig i oe year mius the preset value of a secod growig perpetuity begiig i year ( + 1) equals the preset value of a year growig auity: CF 1 /(k g) [CF 1 /(k g)] (1 + k) ; PV GA = [CF 1 /(k g)][1 (1 + g) ] (1 + k). cash flows ot geerated by the growig auity; it is geerated after the auity is sold or termiated. Because the cash flow is growig at the rate g, the iitial amout of the cash flow geerated by the secod perpetuity is exceeded by the iitial cash flow of the perpetuity begiig i year oe: PV GA = CF1 g k g ( 1 + ) 1. ( 1 + k) (4.22) Cash flows geerated by may ivestmets will grow at the rate of iflatio. For example, cosider a project udertake by a corporatio whose cash flow i year oe is expected to be $10,000. If cash flows were expected to grow at the iflatio rate of 6% each year util year six, the termiate, the project s preset value would be $48,320.35, assumig a discout rate of 11%: 6 $ 10, 000 ( +. ) PV GA = = $, (. ) = $, ( ) Cash flows are geerated by this ivestmet through the ed of the sixth year. No cash flow was geerated i the seveth year. Verify that the amout of cash flow that would have bee geerated by the ivestmet i the seveth year if it had cotiued to grow would have bee $10,000(1.06) 6 = $14,185. APPLICATION 4.6: STOCK VALUATION MODELS Cosider a stock whose aual divided ext year is projected to be $50. This paymet is expected to grow at a aual rate of 5% i subsequet years. A ivestor has
21 QRMC04 9/17/01 4:43 PM Page 71 Siglestage growth models 71 determied that the appropriate discout rate for this stock is 10%. The curret value of this stock is $1,000, determied by the growig perpetuity model: PV gp $ 50 = = $ 1, This model is ofte referred to as the Gordo Stock Pricig Model. It may seem that this model assumes that the stock will be held by the ivestor forever. But what if the ivestor iteds to sell the stock i five years? Its value would be determied by the sum of the preset values of cash flows the ivestor does expect to receive: PV GA DIV1 ( 1 + g) = 1 k g ( 1 + k), where P is the price the ivestor expects to receive whe he sells the stock i year ; ad DIV 1 is the divided paymet the ivestor expects to receive i year oe. The preset value of the divideds the ivestor expects to receive is $207.53: PV GA $ 50 ( +. ) = ( ) 5 5 = $ The sellig price of the stock i year five will be a fuctio of the divided paymets that the prospective purchaser expects to receive begiig i year six. Thus, i year five, the prospective purchaser will pay $1, for the stock, based o his iitial divided paymet of $63.81, determied by the followig equatios: DIV 6 = DIV 1 ( ) 6 1 = $63.81, stock value i year five = 63.81/( ) = $1, The preset value of the $1, that the ivestor will receive whe he sells the stock at the ed of the fifth year is $792.47: $ 1, PV = = $ ( 1+ 0.) 1 The total stock value will be the sum of the preset values of the divideds received by the ivestor ad his cash flows received from the sale of the stock. Thus, the curret value of the stock is $ plus $792.47, or $1,000. This is exactly the same sum determied by the growig perpetuity model earlier; therefore, the growig perpetuity model ca be used to evaluate a stock eve whe the ivestor expects to sell it.
22 QRMC04 9/17/01 4:43 PM Page The time value of moey 4.13 MULTIPLESTAGE GROWTH MODELS (Backgroud readig: sectio 4.12) The Gordo Stock Pricig Model may be urealistic i may scearios, i that it assumes that oe growth rate applies to the firm s cash flows ad that this growth rate exteds forever. Multiplestage growth models eable the user to allow for differet growth rates i differet periods. For example, a growth compay might geerate cash flows that are expected to grow at a high rate i the short term ad the declie as the firm matures. The multistage growth model ca accommodate this patter. Suppose, for example, that a ivestor has the opportuity to ivest i a stock curretly sellig for $100 per share. The stock is expected to pay a $3 divided ext year (at the ed of year 1). I each subsequet year util the seveth year, the aual divided is expected to grow at a rate of 20%. Startig i the eighth year, the aual divided will grow at a aual rate of 3% forever. All cash flows are to be discouted at a aual rate of 10%. Should the stock be purchased at its curret price? The followig TwoStage Growth Model ca be used to evaluate this stock: 1 ( 1+ g1 ) P0 = DIV1 k g 1 ( k g1 )( 1 + k) 1 DIV1( 1+ g1) ( 1+ g2) +. ( k g2)( 1 + k) (4.23) Note that this model begis with a year auity at growth rate g 1 ad accommodates the ew growth rate g 2 i the growig perpetuity that follows. The perpetuity is discouted a secod time because it is deferred; it does ot commece paymets util year. Substitutig values from the problem statemet yields the followig: ( ) ( )( ) 7 P 0 = $ $ 31 ( ) ( ) + = ( )( ) Sice the $100 purchase price of the stock exceeds its value, the stock should ot be purchased. The followig represets a ThreeStage Growth Model which is based o a growig auity, a deferred growig auity, ad a deferred growig perpetuity: ( 1) 1 ( 1+ g1 ) P0 = DIV1 k g 1 ( k g1 )( 1 + k) + DIV 1 ( 1) ( 1+ g1 ) ( 1+ g2) ( 1+ g1 ) ( 1+ g2) ( 1) ( 1 + k) ( k g2) ( k g2)( 1 + k) ( ) ( 2) ( 1) + 1 ( 1) 1 ( 2) ( 1) DIV1( 1+ g1) ( 1+ g2) ( 1+ g3) +. ( 2) ( k g )( 1 + k) 3 ( 1) ( 2)
23 QRMC04 9/17/01 4:43 PM Page 73 Exercises 73 There are three stages here, the first edig at time (1), the secod edig at time (2), ad the third extedig ito perpetuity. It may be a useful exercise to closely examie this expressio to determie why it is structured i this maer. Try to determie why the growth rates ad discout rates are structured as they are. Be certai to first be comfortable with the PresetValue Growig Auity ad Perpetuity Models ad the TwoStage Growth Model. EXERCISES 4.1. The Ruth Compay borrowed $21,000 at a aual iterest rate of 9%. What is the future value of this loa assumig iterest is accumulated o a simple basis? 4.2. The Cobb Compay has issued te millio dollars i 10% coupo bods maturig i five years. Iterest paymets o these bods will be made semiaually. (a) (b) (c) How much are Cobb s semiaual iterest paymets? What will be the total paymet made by Cobb o the bods i each of the first four years? What will be the total paymet made by Cobb o the bods i the fifth year? 4.3. I have the opportuity to deposit $10,000 ito my savigs accout today, which pays iterest at a aual rate of 5.5%, compouded daily. What will be the edig balace of my accout i five years if I make o additioal deposits or withdrawals? 4.4. What would be the future value of the loa i problem 4.1 if iterest were compouded: (a) (b) (c) (d) (e) aually? semiaually? mothly? daily? cotiuously? 4.5. A cosumer has the opportuity to deposit $10,000 ito his savigs accout today, which pays iterest at a aual rate of 5.5%, compouded daily. What will be the edig balace of his accout i five years if he makes o additioal deposits or withdrawals?
24 QRMC04 9/17/01 4:43 PM Page The time value of moey 4.6. The Speaker Compay has the opportuity to purchase a fiveyear $1,000 certificate of deposit (CD) payig iterest at a aual rate of 12%, compouded aually. The compay will ot withdraw early ay of the moey i its CD accout. Will this accout have a greater future value tha a fiveyear $1,000 CD payig a aual iterest rate of 10%, compouded daily? 4.7. The Waer Compay eeds to set aside a sum of moey today for the purpose of purchasig, for $10,000, a ew machie i three years. Moey used to fiace this purchase will be placed i a savigs accout payig iterest at a rate of 8%. How much moey must be placed i this accout ow to assure the Waer compay $10,000 i three years if iterest is compouded yearly? 4.8. A give savigs accout pays iterest at a aual rate of 3% compouded quarterly. Fid the aual percetage yield (APY) for this accout. 4.9.* Assumig o withdrawals or additioal deposits, how much time is required for $1,000 to double if placed i a savigs accout payig a aual iterest rate of 10% if iterest were: (a) (b) (c) (d) computed o a simple basis? compouded aually? compouded mothly? compouded cotiuously? What is the preset value of a security promisig to pay $10,000 i five years if its associated discout rate is: (a) 20%? (b) 10%? (c) 1%? (d) 0%? What is the preset value of a security to be discouted at a 10% rate promisig to pay $10,000 i: (a) (b) (c) (d) (e) 20 years? te years? oe year? six moths? 73 days? The Gehrig Compay is cosiderig a ivestmet that will result i a $2,000 cash flow i oe year, a $3,000 cash flow i two years, ad a $7,000 cash
25 QRMC04 9/17/01 4:43 PM Page 75 Exercises 75 flow i three years. What is the preset value of this ivestmet if all cash flows are to be discouted at a 8% rate? Should Gehrig Compay maagemet be willig to pay $10,000 for this ivestmet? The Cramde Compay has the opportuity to pay $30,000 for a security which promises to pay $6,000 i each of the ext ie years. Would this be a wise ivestmet if the appropriate discout rate were: (a) 5%? (b) 10%? (c) 20%? The Larse Compay is sellig preferred stock which is expected to pay a $50 aual divided per share. What is the preset value of divideds associated with each share of stock if the appropriate discout rate were 8% ad its life expectacy were ifiite? The Dikis Compay has purchased a machie whose output will result i a $5,000 cash flow i its first year of operatio. This cash flow is projected to grow at the aual 10% rate of iflatio over each of the ext te years. What will be the cash flow geerated by this machie i: (a) (b) (c) (d) its secod year of operatio? its third year of operatio? its fifth year of operatio? its teth year of operatio? The Wager Compay is cosiderig the purchase of a asset that will result i a $5,000 cash flow i its first year of operatio. Aual cash flows are projected to grow at the 10% aual rate of iflatio i subsequet years. The life expectacy of this asset is seve years, ad the appropriate discout rate for all cash flows is 12%. What is the maximum price that Wager should be willig to pay for this asset? What is the preset value of a stock whose $100 divided paymet ext year is projected to grow at a aual rate of 5%? Assume a ifiite life expectacy ad a 12% discout rate Which of the followig series of cash flows has the highest preset value at a 5% discout rate: (a) (b) (c) (d) $500,000 ow? $100,000 per year for eight years? $60,000 per year for 20 years? $30,000 each year forever?
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