1 CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all the techiques used i fiace, oe is more importat tha the cocept of the time value of moey, or discouted cash flow (DCF) aalysis. The priciples of time value aalysis that are developed i this chapter have may applicatios, ragig from settig up schedules for payig off loas to decisios about whether to acquire ew equipmet. Future value ad preset value techiques ca be applied to a sigle cash flow (lump sum), ordiary auities, auities due, ad ueve cash flow streams. Future ad preset values ca be calculated usig a regular calculator or a calculator with fiacial fuctios. Whe compoudig occurs more frequetly tha oce a year, the effective rate of iterest is greater tha the quoted rate. OUTLINE The cash flow time lie is oe of the most importat tools i time value of moey aalysis. Cash flow time lies help to visualize what is happeig i a particular problem. Cash flows are placed directly below the tick marks, ad iterest rates are show directly above the time lie; ukow cash flows are idicated by questio marks. Thus, to fid the future value of $100 after 5 years at 5 percet iterest, the followig cash flow time lie ca be set up: Time: % Cash flows: -100 FV 5 =? A cash outflow is a paymet, or disbursemet, of cash for expeses, ivestmets, ad so o. A cash iflow is a receipt of cash from a ivestmet, a employer, or other sources. Compoudig is the process of determiig the value of a cash flow or series of cash flows some time i the future whe compoud iterest is applied. The future value is the amout to which a cash flow or series of cash flows will grow over a give period of time whe compouded at a give iterest rate. The future value ca be calculated as
2 40 FV = PV(1 + k), where PV = preset value, or begiig amout; k = iterest rate per period; ad = umber of periods ivolved i the aalysis. This equatio ca be solved i oe of two ways: umerically or with a fiacial calculator. For calculatios, assume the followig data that were preseted i the time lie above: preset value (PV) = $100, iterest rate (k) = 5%, ad umber of years () = 5. Compouded iterest is iterest eared o iterest. To solve umerically, use a regular calculator to fid 1 + k = 1.05 raised to the fifth power, which equals Multiply this figure by PV = $100 to get the fial aswer of FV 5 = $ With a fiacial calculator, the future value ca be foud by usig the time value of moey iput keys, where N = umber of periods, I = iterest rate per period, PV = preset value, PMT = auity paymet, ad FV = future value. By eterig N = 5, I = 5, PV = -100, ad PMT = 0, ad the pressig the FV key, the aswer is displayed. Some fiacial calculators require that all cash flows be desigated as either iflows or outflows, thus a outflow must be etered as a egative umber (for example, PV = -100 istead of PV = 100). Some calculators require you to press a Compute key before pressig the FV key. A graph of the compoudig process shows how ay sum grows over time at various iterest rates. The greater the rate of iterest, the faster is the rate of growth. The iterest rate is, i fact, a growth rate. The time value cocepts ca be applied to aythig that is growig. Fidig the preset value of a cash flow or series of cash flows is called discoutig, ad it is simply the reverse of compoudig. I geeral, the preset value is the value today of a future cash flow or series of cash flows. By solvig for PV i the future value equatio, the preset value, or discoutig, equatio ca be developed ad writte i several forms: PV = FV (1 k) FV 1 (1 k). To solve for the preset value of $ discouted back 5 years at a 5% opportuity cost rate, oe ca utilize either of the two solutio methods: Numerical solutio: Divide $ by 1.05 five times to get PV = $100. Fiacial calculator solutio: Eter N = 5, I = 5, PMT = 0, ad FV = , ad the
3 41 press the PV key to get PV = The opportuity cost rate is the rate of retur o the best available alterative ivestmet of equal risk. A graph of the discoutig process shows how the preset value of ay sum to be received i the future dimiishes ad approaches zero as the paymet date is exteded farther ito the future. At relatively high iterest rates, fuds due i the future are worth very little today, ad eve at a relatively low discout rate, the preset value of a sum due i the very distat future is quite small. The compoudig ad discoutig processes are reciprocals, or iverses, of oe aother. I additio, there are four variables i the time value of moey equatios: PV, FV, k, ad. If three of the four variables are kow, you ca fid the value of the fourth. If we are give PV, FV, ad, we ca determie k by substitutig the kow values ito either the preset value or future value equatios, ad the solvig for k. Thus, if you ca buy a security at a price of $78.35 which will pay you $100 after 5 years, what is the iterest rate eared o the ivestmet? Numerical solutio: Use a trial ad error process to reach the 5% value for k. This is a tedious ad iefficiet process. Alteratively, you could use algebra to solve the time value equatio. Fiacial calculator solutio: Eter N = 5, PV = , PMT = 0, ad FV = 100, the press the I key, ad I = 5 is displayed. Likewise, if we are give PV, FV, ad k, we ca determie by substitutig the kow values ito either the preset value or future value equatios, ad the solvig for. Thus, if you ca buy a security with a 5 percet iterest rate at a price of $78.35 today, how log will it take for your ivestmet to retur $100? Numerical solutio: Use a trial ad error process to reach the value of 5 for. This is a tedious ad iefficiet process. The equatio ca also be solved algebraically. Fiacial calculator solutio: Eter I = 5, PV = , PMT = 0, ad FV = 100, the press the N key, ad N = 5 is displayed. A auity is a series of equal paymets made at fixed itervals for a specified umber of periods. If the paymets occur at the ed of each period, as they typically do, the auity is a ordiary, or deferred, auity. If the paymets occur at the begiig of each period, it is called a auity due. The future value of a ordiary auity, FVA, is the total amout oe would have at the ed of the auity period if each paymet were ivested at a give iterest rate ad held to the ed of the auity period.
4 42 Defiig FVA as the future value of a ordiary auity of years, ad PMT as the periodic paymet, we ca write FVA = PMT t 1 t ( 1 k) = PMT 1 t (1 k) 1 ( 1 k) = PMT. t 0 k Usig a fiacial calculator, eter N = 3, I = 5, PV = 0, ad PMT = The press the FV key, ad is displayed. For a auity due, each paymet is compouded for oe additioal period, so the future value of the etire auity is equal to the future value of a ordiary auity compouded for oe additioal period. Thus: (1 k) FVA (DUE) = PMT k 1 (1 k). Most fiacial calculators have a switch, or key, marked DUE or BEG that permits you to switch from ed-of-period paymets (a ordiary auity) to begiig-ofperiod paymets (a auity due). Switch your calculator to BEG mode, ad calculate as you would for a ordiary auity. Do ot forget to switch your calculator back to END mode whe you are fiished. The preset value of a ordiary auity, PVA, is the sigle (lump sum) paymet today that would be equivalet to the auity paymets spread over the auity period. It is the amout today that would permit withdrawals of a equal amout (PMT) at the ed (or begiig for a auity due) of each period for periods. Defiig PVA as the preset value of a ordiary auity of years ad PMT as the periodic paymet, we ca write PVA = PMT t1 1 (1 t k) = PMT 1 (1 k) k 1 1 (1 k) = PMT. k Usig a fiacial calculator, eter N = 3, I = 5, PMT = -100, ad FV = 0, ad the press the PV key, for a aswer of $ Oe especially importat applicatio of the auity cocept relates to loas with costat paymets, such as mortgages ad auto loas. With these amortized loas the amout borrowed is the preset value of a ordiary auity, ad the paymets costitute the auity stream. The preset value for a auity due is
5 43 PVA (DUE) = PMT 1 (1 k) k 1 (1 k). Usig a fiacial calculator, switch to the BEG mode, ad the eter N = 3, I = 5, PMT = -100, ad FV = 0, ad the press PV to get the aswer, $ Agai, do ot forget to switch your calculator back to END mode whe you are fiished. You ca solve for the iterest rate (rate of retur) eared o a auity. To solve umerically, you must use the trial-ad-error process ad plug i differet values for k i the auity equatio to solve for the iterest rate. You ca use the fiacial calculator by eterig the appropriate values for N, PMT, ad either FV or PV, ad the pressig I to solve for the iterest rate. You ca solve for the umber of periods (N) i a auity. To solve umerically, you must use the trial-ad-error process ad plug i differet values for N i the auity equatio to solve for the umber of periods. You ca use the fiacial calculator by eterig the appropriate values for I, PMT, ad either FV or PV, ad the pressig N to solve for the umber of periods. A perpetuity is a stream of equal paymets expected to cotiue forever. The preset value of a perpetuity is: PVP = Paymet Iterest rate PMT. k For example, if the iterest rate were 12 percet, a perpetuity of $1,000 a year would have a preset value of $1,000/0.12 = $8, A cosol is a perpetual bod issued by the British govermet to cosolidate past debts; i geeral, ay perpetual bod. The value of a perpetuity chages dramatically whe iterest rates chage. May fiacial decisios require the aalysis of ueve, or ocostat, cash flows rather tha a stream of fixed paymets such as a auity. A ueve cash flow stream is a series of cash flows i which the amout varies from oe period to the ext. The term paymet, PMT, desigates costat cash flows, while the term CF desigates cash flows i geeral, icludig ueve cash flows.
6 44 The preset value of a ueve cash flow stream is the sum of the PVs of the idividual cash flows of the stream. The PV is foud by applyig the followig geeral preset value equatio: 1 PV = CF t. t t1 (1 k) With a fiacial calculator, eter each cash flow (begiig with the t = 0 cash flow) ito the cash flow register, CF j, eter the appropriate iterest rate, ad the press the NPV key to obtai the PV of the cash flow stream. Be sure to clear the cash flow register before startig a ew problem. Similarly, the future value of a ueve cash flow stream, or termial value, is the sum of the FVs of the idividual cash flows of the stream. The FV ca be foud by applyig the followig geeral future value equatio: t FV = CF t (1 k). t1 Some calculators have a et future value (NFV) key which allows you to obtai the FV of a ueve cash flow stream. We geerally are more iterested i the preset value of a asset s cash flow stream tha i the future value because the preset value represets today s value, which we ca compare with the price of the asset. Oce we kow its preset value, we ca fid the future value of a ueve cash flow stream by treatig the preset value as a lump sum amout ad compoudig it to the future period. If oe kows the relevat cash flows, the effective iterest rate ca be calculated efficietly with a fiacial calculator. Eter each cash flow (begiig with the t = 0 cash flow) ito the cash flow register, CF j, ad the press the IRR key to obtai the iterest rate of a ueve cash flow stream. IRR stads for iteral rate of retur, which is the retur o a ivestmet. Aual compoudig is the arithmetic process of determiig the fial value of a cash flow or series of cash flows whe iterest is added oce a year. Semiaual, quarterly, ad other compoudig periods more frequet tha o a aual basis are ofte used i fiacial trasactios. Compoudig o a oaual basis requires a adjustmet to both the compoudig ad discoutig procedures discussed previously. Moreover, whe comparig securities with differet compoudig periods, they eed to be put o a commo basis. This requires distiguishig betwee the simple, or quoted, iterest rate ad the effective aual rate.
7 45 The simple, or quoted, iterest rate is the cotracted, or quoted, iterest rate that is used to calculate the iterest paid per period. The periodic rate is the iterest rate charged per period. Periodic rate = Stated aual iterest rate/number of periods per year. The aual percetage rate, APR, is the periodic rate times the umber of periods per year. The effective aual rate, EAR, is the rate that would have produced the fial compouded value uder aual compoudig. The effective aual rate is give by the followig formula: k SIMPLE Effective aual rate (EAR) = 1 1.0, m where k SIMPLE is the simple, or quoted, iterest rate (that is, the APR), ad m is the umber of compoudig periods (iterest paymets) per year. The EAR is useful i comparig securities with differet compoudig periods. m For example, to fid the effective aual rate if the simple rate is 6 percet ad semiaual compoudig is used, we have: EAR = ( /2) = 6.09%. For aual compoudig use the formula to fid the future value of a sigle paymet (lump sum): FV = PV(1 + k). Whe compoudig occurs more frequetly tha oce a year, use this formula: m k SIMPLE FV = PV1. m Here m is the umber of times per year compoudig occurs, ad is the umber of years. The amout to which $1,000 will grow after 5 years if quarterly compoudig is applied to a omial 8 percet iterest rate is foud as follows: FV = $1,000( /4) (4)(5) = $1,000(1.02) 20 = $1, Fiacial calculator solutio: Eter N = 20, I = 2, PV = -1000, ad PMT = 0, ad the press the FV key to fid FV = $1,
8 46 The preset value of a 5-year future ivestmet equal to $1,485.95, with a 8 percet omial iterest rate, compouded quarterly, is foud as follows: $1, PV PV(1 0.08/4) $1, (1.02) (4)(5) $1,000. Fiacial calculator solutio: Eter N = 20, I = 2, PMT = 0, ad FV = , ad the press the PV key to fid PV = -$1, I geeral, oaual compoudig ca be hadled oe of two ways. State everythig o a periodic rather tha o a aual basis. Thus, = 6 periods rather tha = 3 years ad k = 3% istead of k = 6% with semiaual compoudig. Fid the effective aual rate (EAR) with the equatio below ad the use the EAR as the rate over the give umber of years. k SIMPLE EAR = m A importat applicatio of compoud iterest ivolves amortized loas, which are paid off i equal istallmets over the life of the loa. m The amout of each paymet, PMT, is foud usig a fiacial calculator by eterig N (umber of years), I (iterest rate), PV (amout borrowed), ad FV = 0, ad the pressig the PMT key to fid the periodic paymet. Each paymet cosists partly of iterest ad partly of repaymet of the amout borrowed (pricipal). This breakdow is ofte developed i a loa amortizatio schedule. The iterest compoet is largest i the first period, ad it declies over the life of the loa as the outstadig balace of the loa decreases. The repaymet of pricipal is smallest i the first period, ad it icreases thereafter. The text discussio has ivolved three differet iterest rates. It is importat to uderstad their differeces. The simple, or quoted, rate, k SIMPLE, is the iterest rate quoted by borrowers ad leders. This quotatio must iclude the umber of compoudig periods per year. This rate is ever show o a time lie, ad it is ever used as a iput i a fiacial calculator uless compoudig occurs oly oce a year. k SIMPLE = Periodic rate m = Aual percetage rate = APR. The periodic rate, k PER, is the rate charged by a leder or paid by a borrower each iterest period. Periodic rate = k PER = k SIMPLE /m.
9 47 The periodic rate is used for calculatios i problems where two coditios hold: (1) paymets occur o a regular basis more frequetly tha oce a year, ad (2) a paymet is made o each compoudig (or discoutig) date. The APR, or aual percetage rate, represets the periodic rate stated o a aual basis without cosiderig iterest compoudig. The APR ever is used i actual calculatios; it is simply reported to borrowers. The effective aual rate, EAR, is the rate with which, uder aual compoudig, we would obtai the same result as if we had used a give periodic rate with m compoudig periods per year. EAR is foud as follows: k SIMPLE EAR = m m