Topic 4: Time Value of Money (Copyright 2019 Joseph W. Trefzger) The basic factors for computing time value-adjusted dollar amounts are as follows:

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1 Topic 4: Time Value of Money (Copyight 2019 Joseph W. Tefzge) In this unit, we discuss time value of money (pesent and futue value) applications. These applications have many uses in both pesonal and business financial management. A typical use in pesonal money management is planning and saving fo etiement. A typical use in business financial management is the capital budgeting decision: detemining whethe it is appopiate to spend a lage amount of money to buy new long-lived equipment (the question addessed in Topic 6, Capital Budgeting Analysis). The basic factos fo computing time value-adjusted dolla amounts ae as follows: What textbooks call Futue Value of $1: What textbooks call Pesent Value of $1: Futue Value of a Level Odinay Annuity: 1 (1 + ) (1 + ) n n o, equivalently, ( 1 ( (1 + )n 1 ) 1 + )n Pesent Value of a Level Odinay Annuity: ( 1 ( )n In each of these fomulas, epesents the ate of etun o gowth that occus each time peiod (the specified peiod is often pesented as a full yea in intoductoy examples, but in actual financial tansactions we ae moe likely to see payments occuing evey half yea, quate of a yea, o month, and time intevals of days o even moe unusual peiods ae possible). [Note that the ate of etun is epesented vaiously in diffeent efeence souces as g (indicating a ate of gowth in size o value), i (inteest ate), k (the cost of capital in a business setting), o (ate of etun, the most geneal designation, which we might favo fo its vesatility).] The othe vaiable, n, epesents the numbe of those time peiods (yeas, months, etc.) ove which the peiodic ate of etun is eaned o paid. Befoe the ealy 1980s, when calculatos became cheap enough to be widely available, financial analysts computed time value-adjusted amounts with the help of pinted tables, which povided factos (computed with the fomulas above) fo a limited numbe of selected and n values. Today we have poweful calculatos and speadsheets to facilitate these computations. But while most textbooks attempt to teach time value via financial calculato function keys, the beginning student should be encouaged instead to wok diectly with the above factos, employing the exponent and logaithm keys on a financial o basic scientific calculato. Using shotcut computational techniques, such as financial calculatos function keys, when we do not yet fully undestand the basic ideas is akin to taking shotcut tavel outes when we do not yet know majo oads and landmaks in an aea: we ae apt to get lost, and even if we get to the ight destination we may not tuly undestand what we have done. Once we undestand the undelying logic, we can figue out which buttons to hit on a paticula financial calculato in a matte of minutes. The outline that follows pesents a tool box appoach to time value that diffes in many ways fom pesentations typically offeed in textbooks. I developed this appoach, ove many yeas of teaching, to bette addess questions students asked, and concepts with which they outinely encounteed difficulties, when woking with the standad time value of money (often abbeviated as TVM) tools. The tool box is a use-fiendly way of oganizing taditional time value tools; nothing in ou tool box is incompatible with taditional textbook coveage. Indeed, no one who tuly undestands the time value of money fom ealie couses needs to change anything in the appoach he o she has become accustomed to (including the use of financial calculato function keys, which ae legal, though discouaged, fo use on ou exams). But the tool box appoach pesented below has geneated much positive feedback fom students ove the yeas fo simplifying and demystifying time value of money computations and the undelying ideas. ) 1

2 I. UNDERSTANDING REQUIRED RATES OF RETURN (sometimes shown as g o i o k) In evey time value of money situation, we wok with an expected o equied peiodic ate of etun. (This etun is often called an inteest ate, teminology that we fequently employ fo convenience, although technically the etun eaned is inteest only if a lende povides money to a boowe. The ate would instead be a etun on equity if the money povide wee an owne athe than a lende.) How do we detemine, befoe the fact, the aveage annual o othe peiodic ate of etun that would leave us feeling popely compensated fo all the isks we encounte as money povides? And why is the expected/equied aveage peiodic ate of etun 4% in some instances and 19% in othes? Conside the case of a lende. We can think of an inteest ate in a building block fashion, embodying the bits and pieces that compensate fo the expected cost to the lende of beaing the vaious isks faced. We epesent a lende s peiodic (typically annual) equied ate of etun in a staightfowad additive fom as = * + IP + DRP + LP + MRP + FP o, combining the fist two tems to the ight of the equals sign: = RF + DRP + LP + MRP + FP [A moe technically coect, but cumbesome, multiplicative fom is epesented as (1 + ) = (1 + *) x (1 + IP) x (1 + DRP) x (1 + LP) x (1 + MRP) x (1 + FP)] The meaning of the symbols is as follows: = nominal inteest ate we actually obseve (pehaps the annual ate at which a bank offes to lend); * = eal isk-fee inteest ate (histoically measued in the 2 4% pe yea ange); IP = inflation pemium to cove the potential aveage peiodic loss in puchasing powe; RF = * + IP [o the multiplicative (1 + *)(1 + IP) 1] is the nominal isk-fee inteest ate that lendes chage the U.S. fedeal govenment when it boows money on a shot-tem basis; DRP = default isk pemium to cove the isk of not being epaid in full and in a timely manne (applicable to pivate boowes, and even to city and state govenments when they boow, but not to the U.S. govenment); LP = liquidity pemium to cove the isk of losing money by potentially having to sell an investment in a thinly-taded, illiquid maket; MRP = matuity isk pemium to cove the added isks that accompany long-tem financial commitments; FP = foeign exchange isk pemium to cove isks of uncetainty in the elative values of diffeent counties cuencies including concens ove inflation in othe egions that can accompany investments outside the county in whose cuency the investo nomally tansacts. The equation essentially tells us that the aveage annual ate of etun a money povide expects to ean consists of a basic isk-fee ate that would be anyone s minimum equied aveage yealy etun unde any conditions, plus additional amounts (pemiums) to cove vaious isks that apply to the cicumstances at hand. If a financial aangement coves multiple time peiods, the value used as expected o equied peiodic ate of etun in ou time value factos should be the aveage (technically the multiplicative geometic aveage) of the individual peiods etuns expected ove the entie time inteval in question. Example 1: We plan to lend money to the U.S. Govenment fo one yea. The annual eal isk-fee inteest ate is 2.5%, and we expect inflation to be 3% ove the next yea. When we lend to the fedeal govenment thee is no default o liquidity isk (the boowe is vey stong financially, and the maket is huge and vey active). Hee thee is no matuity isk (the loan is shot-tem) o foeign exchange isk pemium (both the boowe and lende ae domestic paties). What annual inteest ate should we chage on this 1-yea loan? 2

3 = appoximately 2.5% + 3% = 5.5%, o moe technically (1 + ) = (1.025) x (1.03) = , so =.05575, o 5.575% Thus we would expect to lend to the fedeal govenment (buy T-Bills ) fo one yea in etun fo appoximately a 5½% inteest ate. This 5½% ate is called a nominal ate; it is the one we actually expect to obseve in the maketplace. But we do not eally feel as though we will be gaining 5½% in wealth, since 3% of that 5½% constitutes compensation fo expected inflation (lost puchasing powe). Ou expected annual eal ate of etun (ou inceased ability to buy actual goods and sevices) theefoe is appoximately 2.5%. Of couse, if inflation tuns out to be less than 3% fo the yea ou eal etun, measued afte the fact, will have been moe than 3%. If inflation tuns out to be moe than 3% ou eal annual etun, measued afte the fact, will have been less than 3%. We efe to this elationship that the nominal isk-fee inteest ate (fee of default, liquidity, etc. isks) consists of the eal isk-fee ate plus an inflation pemium as the Fishe effect, afte the late economist Iving Fishe. Example 2: We plan to lend to a pivate individual fo five yeas. Again the eal isk-fee ate is 2.5% pe yea, and we expect the aveage annual ate of inflation to be 3% ove the next five yeas. We also assign an annual default isk pemium of 2%, an annual liquidity isk pemium of 1%, and an annual matuity isk pemium of.5%. Thee is no foeign exchange isk pemium (since we ae lending to anothe domestic paty). What annual inteest ate should we chage on this loan? = appoximately 2.5% + 3% + 2% + 1% +.5% = 9.0%, o moe pecisely (1 + ) = (1.025) x (1.03) x (1.02) x (1.01) x (1.005) = , so = , o % When we do time value of money computations, we might think of the above method as the basis fo detemining the equied aveage annual ate of etun. The main point to ecognize fo now is that an investo who faces moe isk should expect to ean a highe aveage peiodic etun (and will be unhappy afte the fact if the ealized etun was not high enough to have compensated fo the identified isks). Of couse, if thee is moe isk, then thee is moe of a chance that the investo s equiement will not be met, that the actual (afte-the-fact) ate of etun will tun out to be less than the equied o expected (befoe-the-fact) etun. So in ou time value coveage, when a poblem states a 4% aveage annual equied ate of etun, we should assume that the povide of money in that situation peceives little isk of not being epaid pomptly, o of losing puchasing powe though inflation. A situation specifying a 19% equied aveage annual ate of etun suggests a money povide who feas seious potential advese effects fom inflation, default, liquidity, matuity, foeign exchange, o othe poblems. A few points to conside: One way to think about isk and etun is that if we make isky investments, we know it is likely that some of those investments will not tun out well. So we spead ou money acoss a mix of isky investments (a potfolio, as we will call it in a late discussion), hoping that the highe etuns eaned on the ones that do pay off will compensate fo losses on the ones that inevitably will not pay off. Fo example, let s say we have $500,000 to lend. We could lend it all to a financially stong paty in etun fo a 5% ageed-on annual inteest ate, and if thee is essentially no chance the boowe will default (fail to pay some o all of what is owed when it is due) then ou expected annual ate of etun is 5%. O we could make 100 loans of $5,000 each to less financially stong paties, chaging each 36% inteest pe yea ( payday loans could be an example). If expeience-based analysis indicates that 40 boowes will pay as the contact specifies, 30 will pay back pincipal and half of the ageedon inteest (18% annual ate of etun), 16 will epay pincipal but no inteest (0% annual ate of 3

4 etun), and 14 will pay no inteest o pincipal (negative 100% annual etun), then ou expected annual etun is.40 (.36) +.30 (.18) +.16 (0) +.14 ( 1.00) = =.058, o 5.8%. This expected annual ate of etun is slightly highe than the safe investment povides, to compensate fo the highe peceived isk (and fo any added administative cost of dealing with many small, iskie loans to the extent that these costs ae not coveed by loan application fees). A lende must chage an inteest ate to ewad himself fo giving up the chance to consume today. The amount chaged depends on how poductive the lende s oppotunities would be if he put the money diectly to use in the economy. The peiodic pecentage measue of this poductivity, which economists call the maginal poductivity of capital, can be a useful way to think about *. If we could use ou money to buy machines that would poduce goods whose eal value would incease ou wealth by an aveage of 3% each yea, then we would not lend money to any paty even the U.S. fedeal govenment, and even if we did not expect any inflation fo less than * = 3% pe yea. In a simila manne, money uses that have bette oppotunities (because they ae moe efficient, pehaps) ae able, and willing, to pay highe inteest ates than ae less efficient uses. Fo example, if efficient Use A can ean a 15% aveage annual etun on its investments, it will pay up to 15% pe yea to boow money. Inefficient Use B aveages only 7% annual investment eanings, and thus can affod to pay only up to 7% pe yea fo boowed money. Use B, offeing to pay only 7% inteest, will find no willing money povides, and will be diven out of the maketplace by its inefficiency. Why is a multiplicative appoach to constucting an expected annual inteest ate moe technically coect than the additive appoach? Conside a case with * =.03, IP =.02, and all othe pemiums ae 0 (just fo computational simplicity). The additive appoach tells us that a lende should chage a one-yea inteest ate of =.05, o 5%, in ode to get back 3% moe than the puchasing powe of the pincipal lent. If we lend an amount that would buy 100 pounds of flou today, we want to get back enough money in one yea to buy 103 pounds of flou at that time. Now assume that thee is, in fact, 2% inflation in the subsequent yea. Someone who lent $100, which initially would have paid fo 100 pounds of flou costing $1 each, gets back $105. Since flou has isen in pice by 2% with inflation, to $1.02 pe pound, the $105 the lende gets back pays fo only $105 $1.02 = pounds. To be able to buy a full 103 pounds at $1.02 each, the lende should have gotten back $ an amount that he would have eceived if the inteest ate had been quoted as (1 + *)(1 + IP) 1 = (1.02)(1.03) 1 = =.0506, o 5.06%. But the additive method povides a close appoximation, and makes it easie fo people to undestand the building block explanation of inteest ates. Because the additive appoach gives a easonable appoximation, and pehaps especially because input figues like the aveage annual inflation ate o the appopiate amount to chage fo possible default ae subject to guess wok and eo, we typically view the additive appoach as an acceptable tool fo explaining and examining ates of etun. With the undestanding of expected aveage peiodic ates of etun developed in this section, we should find the following discussion of time value concepts and computations to be moe eal-wold elevant. II. PROBLEMS WITHOUT SERIES OF EQUAL (OR RELATED) PAYMENTS We can view any basic time value of money poblem as falling into one of two categoies: those that involve seies of equal o elated payments [annuity], and those that do not [we might coin the tem non-annuity ]. Indeed, the fist step in using ou tool box to do time value analysis is to identify whethe we ae dealing with an annuity o non-annuity case. We ask the question: is thee a seies of epeated 4

5 payments that ae equally spaced in time and equal o elated in amount? If thee is no seies, just the compaison of two values at diffeent points in time, then we have a non-annuity situation. A. What Textbooks Call Futue Value of a Single Dolla Amount In the simplest time value application thee is no seies of payments, and the unknown to solve fo is the dolla amount we will have at a futue date; textbooks typically call this non-annuity situation futue value of a single dolla amount. Example: we deposit $100 in the bank today. If ou account balance eans a 6% compounded aveage annual inteest ate ( compounded because we do not spend the inteest as we ean it, but athe leave it in the account and then ean inteest on the inteest we often say that we compound an initial amount to an ending o futue amount), how much money will be in the account afte thee yeas? We deposit $100 out of pocket only one time; thee is no seies of equal o elated payments. Time value of money suggests that the passage of time plays an impotant ole in ou analysis. While the time lines some textbook authos favo may not be essential tools fo undestanding time value, it is impotant fo us to ecognize that evey time value situation plays out ove a span of time that begins on the fist day of yea (o othe peiod) 1 and ends on the last day of yea (o othe peiod) n. Yeas ove which financial aangements occu could involve any dates, e.g. May 17 of yea 1 to May 16 of yea 8, but hee we will illustate with Januay 1 to Decembe 31 peiods because they ceate less mental clutte. Duing yea 1 we ean 6% on ou $100 deposit (which is $6), so by the end of yea 1 we have $100 + $6, o $100 x 1.06 = $106. Duing yea 2 we ean 6% on ou $106 balance (oiginal deposit plus fist yea s inteest), so by the end of yea 2 we have $106 x 1.06 = $ Finally, duing yea 3 we ean 6% on the $ balance, so by the end of yea 3 we have $ x 1.06 = $ Y Y 1/1-01-Y Y 2/1-01-Y Y 3 Deposit $100 Have $ Have $ Have $119 We can compute the answe moe diectly by simply compounding the oiginal $100 balance at a 6% ate fo thee peiods. [Multiplication is chaacteized by the associative popety: instead of multiplying the $100 deposit by 1.06, and then multiplying that poduct by 1.06, and then multiplying by 1.06 again, we can simply multiply the $100 deposit by 1.06 x 1.06 x 1.06, o (1.06) 3, which textbooks call the futue value of $1 facto.] With ou time value tool box we handle the computations fo any non-annuity poblem (with no seies of equal o elated payments into o out of an account, such that thee ae just two single dolla values compaed at diffeent points in time) using the following equation: Beginning Amount (1 + ) n = Ending Amount, o BAMT (1 + ) n = EAMT $100 (1.06) 3 = EAMT $100 x = $119 What if the expected aveage annual ate of etun wee instead 10%? Then the answe would be $100 (1) 3 = EAMT $100 x = $133 Note that if is highe, the computed ending amount (what textbooks call the futue value) in a poblem like this one is highe. It should be intuitively clea that if we can ean a highe aveage peiodic ate of etun, then we will have accumulated a highe total by the end of a specified numbe of time peiods. B. What Textbooks Call Pesent Value of a Single Dolla Amount A slightly moe complicated time value application is the non-annuity case in which the unknown to solve fo is the amount we must have initially so that we can each a desied balance by a specified late 5

6 date; textbooks typically call this situation pesent value of a single dolla amount. It may seem moe complicated because computing pesent value has us thinking in evese, in some sense. But the pocess is staightfowad if we apply ou tool box and ecognize the situation as a non-annuity poblem with the beginning amount as the unknown to solve fo. Example: we want ou bank balance to have gown to $119 afte thee yeas. If the account eans a compounded etun aveaging 6% pe yea, how much do we need to have on deposit today? (Again we ae depositing money out of ou pocket only one time.) It should be evident that this poblem is the evese of the 6% example above, and that the answe should be $100. If depositing $100 today leaves us with $119 in thee yeas, then if we want to have $119 in thee yeas we need to stat with $100 today; as seen ealie: 1-01-Y Y 1/1-01-Y Y 2/1-01-Y Y 3 Deposit $100 Have $ Have $ Have $119 Recall that with ou tool box we set up any poblem involving an initial deposit/investment, an aveage peiodic ate of gowth o etun, and an ending amount (no epeated payment seies) with the equation: BAMT (1 + ) n = EAMT With one equation, we can find a unique solution only if thee is just one unknown fom among beginning amount,, n, and ending amount. In the ealie example, the $100 beginning amount was known, as wee the aveage peiodic ate of etun (6% pe yea) and the numbe of time peiods (thee yeas), so we solved fo the ending amount ($119). In this second example, we know the $119 ending amount, peiodic ate (6%), and numbe of time peiods (thee yeas), so we solve fo the beginning amount: BAMT (1 + ) n = EAMT so BAMT = EAMT (1 + ) n BAMT = $119 (1.06) 3 = $ o, in the multiplicative fom moe taditionally used, BAMT = $119 ( )3 = $ That is, by tadition we often multiply the ending amount by what textbooks call the pesent value of $1 facto (the ecipocal of the futue value of $1 facto) instead of dividing by the futue value of $1 facto, although those opeations povide the same diect esult (just as dividing something by 2 is the same as multiplying by ½). Multiplying an ending amount by the pesent value of $1 facto is often efeed to as discounting a futue amount to an initial value. It is vey useful in investment analysis to detemine a pesent value (how much we must deposit o invest today to be able to eceive a specified amount of money in the futue o, fom a diffeent pespective, how much we ae willing to pay today that is, what it is woth to us now to have the ight to ealize a paticula anticipated value at some futue date). How much must we open an account with today if we hope to have $200 in five yeas (o, in a slightly diffeent context, what ae we willing to pay today fo the ight to eceive $200 in five yeas) if the expected aveage annual ate of etun is 7%? BAMT (1 + ) n = EAMT BAMT (1.07) 5 = $200 BAMT = $200 (1.07) 5 = $ o BAMT = $200 ( )5 = $

7 What if the appopiate aveage annual ate of etun wee instead 10%? Then the answe would be BAMT = $200 (1) 5 = $ o BAMT = $200 ( 1 1 )5 = $ If is highe, the computed beginning amount (so-called pesent value of a single dolla amount) is lowe (wheeas when computing the ending amount, the so-called futue value of a single dolla amount case, a highe leads to a highe computed value). Thee ae two ways to think about this idea. One is that if we can ean a highe aveage yealy etun, then we can deposit less today and still each ou $200 goal ove five yeas. The othe is that if we equie a highe aveage peiodic ate of etun because we feel that the investment entails highe isk, then we ae not willing to hand ove as many dollas today, in etun fo an expected $200 (o othe amount) at a specified futue date, as we would be if less isk wee involved. But by setting up any non-annuity poblem with ou BAMT (1 + ) n = EAMT equation we can easily compute coectly without woying about how changing input values should affect the unknown being solved fo. C. Unknown Rate of Retun Sometimes the unknown we wish to solve fo in a time value poblem is not a dolla amount, but athe the aveage peiodic ate of etun that a commitment of money geneates. If an investment involves not a seies of equal o elated deposits o withdawals, but athe only an initial dolla commitment and an ending value (again, what ou tool box identifies as a non-annuity situation), we can solve diectly fo the unknown ate. If we put $100 in the bank today, and then find thee yeas late that ou balance has gown to $119, what aveage annual compounded ate of etun has been eaned? BAMT (1 + ) n = EAMT $100 (1 + ) 3 = $119 (1 + ) 3 = $119 $100 = One way to solve fo at this stage would be to use tial and eo, tying diffeent values until the equation s left-hand side equals about A moe efficient method is to eliminate the exponent by taking an appopiate oot. Since we have an equation, we can do the same thing to both sides and still have an equation. Hee, if we take the cube oot of each side, we solve as: 3 (1 + ) 3 3 = o /3 o (1 + ) = 1.06, so =.06, o 6% (fom ou initial examples in pats A and B above, we knew that had to be 6%). Unlike in annuity situations we will examine late, in which solving fo equies tial and eo, in a non-annuity case we can solve diectly fo an unknown because that ate appeas only once in the facto that connects the beginning dolla figue with the ending amount to which it compounds, and thus we need only take the oot coesponding to the numbe of peiods ove which the gowth occus. If we bought a painting fo $1,350,000 ten yeas ago, and then sold it today fo $3,000,000, what has been the aveage annual ate of incease in the at wok s value? BAMT (1 + ) n = EAMT $1,350,000 (1 + ) 10 = $3,000,000 (1 + ) 10 = $3,000,000 $1,350,000 = Solve by taking the tenth oot of each side: 10 (1 + ) = o /10 o (1 + ) = , so = , o % 7

8 This latte example illustates thee ideas woth noting. Fist is that a collectible item is a common example of a non-annuity situation. Second is that we have a eason fo showing the non-annuity equation in ou tool box as Beginning Amount (1 + ) n = Ending Amount athe than textbooks typical Pesent Value (1 + ) n = Futue Value. In the example above, $3,000,000 might seem to be the pesent value since it exists today, but that amount is what the at investo gets at the ending date. Indeed, a time value stoy could all have occued in the past (e.g., someone deposited $10,000 in 1992, added nothing to the account out of pocket ove the yeas, had $26,500 by 2017 all in the past and wants to detemine the aveage annual inteest ate that was eaned). The simple substitution of BAMT fo PV and EAMT fo FV thus can help pevent consideable confusion. Finally, distinguishing between pesent value of a single dolla amount (solving fo the beginning amount) and futue value of a single dolla amount (solving fo the ending amount), as textbooks typically show, is also needlessly confusing since the unknown we solve fo makes no diffeence in how a poblem is stuctued (a point that becomes moe evident when we ae solving fo o n). The unknown value solved fo affects only the last algeba step, not the boade thought pocess and analysis. With the BAMT (1 + ) n = EAMT equation we can see that all we ae doing is solving, with simple algeba, fo whateve is unknown in any non-annuity poblem. D. Unknown Numbe of Time Peiods A final unknown we might solve fo in a non-annuity case is the numbe of time peiods needed fo an initial dolla amount to gow to a specified ending amount. Let s say we put $4,000 into the bank today, and then add nothing moe to the account out of pocket (thee is not a seies of $4,000 deposits). If we ean a 7% aveage compounded annual etun, how long will it take fo ou $4,000 to gow to $12,000? BAMT (1 + ) n = EAMT $4,000 (1.07) n = $12,000 (1.07) n = 3 One way to solve fo n would be with tial and eo, tying diffeent n values until the equation s lefthand side equals about 3. A moe efficient method is to use logaithms (a logaithm is the exponent to which we aise a base value to each a specified quantity). We can do the same thing to both sides of the equation like taking the natual logaithm ln (fo which financial and scientific calculatos have builtin tables) of each side and still have an equation. Taking the ln of each side yields the esult ln [(1.07) n ] = ln 3 Because a logaithm is an exponent, when we take the log of something that has an exponent the exponent s value factos out as a multiplie: n x ln 1.07 = ln 3 Fom a table of natual logaithms, accessed with a financial o scientific calculato s LN key, we find that ln 1.07 = , and ln 3 = (We could also use the LOG key fo base-10 common logaithms, but financial calculatos typically offe only natual logaithm tables. The natual log is the powe to which we aise the iational numbe e, which equals about , to each a tageted value. e is computed as the limit of (1 + 1 n )n as n appoaches. To get 3 we would have to aise to a powe just geate than 1; e = 3, so ln 3 = Since an exponent of 0 applied to any value equals 1, to get 1.07 we would have to aise to a powe a little geate than 0; e = 1.07, so ln 1.07 = ) Thus we solve as n x ln 1.07 = ln 3 n x = n = =

9 So it takes slightly moe than 16 yeas fo an initial $4,000 deposit to gow to $12,000 if we ean a 7% aveage annual inteest ate. Check: $4,000 (1.07) = $12,000. E. Seies of Payments That Ae Not Equal (o Related) 1. Payments Coesponding to Lage Dolla Amount That Will Not Exist Intact Until a Futue Date Recall that we always begin by asking the question: is thee a seies of epeated payments that ae equally spaced in time, and equal o elated in amount? If thee is no seies, just the compaison of two values at diffeent points in time, then we have a simple o pue non-annuity situation. But what if thee is a seies of payments that ae equally spaced in time, but not equal o elated in amount? Then we still ae dealing with a non-annuity case but must wok seveal non-annuity poblems and sum the individual solutions to compute the coect oveall answe. With yea-to-yea payments ( cash flows ) that ae not equal, and not elated (e.g., changing by a constant pecentage), thee is no choice but to use a peiod-by-peiod appoach; we can not goup the payments fo computational puposes. Let s say we plan to deposit $2,000 at the end of yea 1; $2,100 at the end of yea 2; $2,800 at the end of yea 3; and $3,200 at the end of yea 4 (so $2,000; $2,100; $2,800; and $3,200 ae a seies of BAMTs that will occu at diffeent times). [We could analyze fom the view of the peson making deposits o the bank getting these amounts fom the account holde; the numbes look the same fom eithe side of the tansaction.] A time line shows: 1-01-Y Y 1/1-01-Y Y 2/1-01-Y Y 3/1-01-Y Y 4 Plan Stats Deposit $2,000 Deposit $2,100 Deposit $2,800 Deposit $3,200 With $0 Plan Ends If we can ean a 10% aveage annual inteest ate on the account s gowing balance, the total we will have by the end of yea 4 can be computed with fou applications of ou BAMT (1 + ) n = EAMT equation: $2,000 x (1) 3 = $2,000 x = $2, as EAMT deposit #1 will each by end of yea 4 $2,100 x (1) 2 = $2,100 x = $2, as EAMT deposit #2 will each by end of yea 4 $2,800 x (1) 1 = $2,800 x = $3, as EAMT deposit #3 will each by end of yea 4 $3,200 x (1) 0 = $3,200 x = $3, as EAMT deposit #4 will each by end of yea 4 fo a combined total ending balance, sometimes called the net futue value, of ($2, $2, $3, $3,200.00) = $11, We could think of this plan as involving fou sepaate accounts, but it could all occu within a single account, the pogess of which is shown in the following table: Beginning Plus 10% Total Plus End-of-Y. Ending Yea Balance Inteest Accumulated Deposit Balance 1 $0 $0 $0 $2, $ 2, $2, $ $2, $2, $ 4, $4, $ $4, $2, $ 7, $7, $ $8, $3, $11, Now let s assume we instead will deposit those dolla amounts at the beginning of each espective yea: 1-01-Y Y 1/1-01-Y Y 2/1-01-Y Y 3/1-01-Y Y 4 Plan Stats Deposit $2,100 Deposit $2,800 Deposit $3,200 Plan Ends Deposit $2,000 Again assuming we can ean a 10% aveage annual ate on the account s gowing balance, the total we will have by the end of yea 4 is again computed with fou applications of BAMT (1 + ) n = EAMT: 9

10 $2,000 x (1) 4 = $2,000 x = $2, as EAMT deposit #1 will each by end of yea 4 $2,100 x (1) 3 = $2,100 x = $2,795 as EAMT deposit #2 will each by end of yea 4 $2,800 x (1) 2 = $2,800 x = $3, as EAMT deposit #3 will each by end of yea 4 $3,200 x (1) 1 = $3,200 x = $3, as EAMT deposit #4 will each by end of yea 4 fo a combined total value at the futue matuity date of ($2, $2,795 + $3, $3,520.00) = $12, Again we could think of this plan as involving fou sepaate accounts, but it all could occu within a single account, pogess fo which is shown in the table below. Notice that the final answe of $12, is meely the $11, answe found fo the same dolla amounts if paid in at the end of each yea, multiplied by (1 + ), hee (1): $11, x 1 = $12, The stuctues of the two seies ae identical, except that if payments occu at the beginning of each peiod inteest is applied to the emaining, gowing balance one diffeential numbe of times (hee, one moe) ove the plan s life. Beginning Plus Beg.-of-Y. Total Plus 10% Ending Yea Balance Deposit Accumulated Inteest Balance 1 $0 $2, $ 2, $ $ 2, $2, $2, $ 4, $ $ 4, $4, $2, $ 7, $ $ 8, $8, $3, $11, $1, $12, Payments Coesponding to Lage Dolla Amount That Exists Intact In the Pesent Now conside a case in which we find the total amount needed initially to fund a seies of withdawals by adding a seies of BAMT figues. (We could be analyzing fom the pespective of the individual making withdawals o the bank that is paying these amounts to the account holde; the numbes look the same fom eithe side of the tansaction.) Let s say we would like to withdaw $2,000 fom ou bank account at the end of yea 1; $2,100 at the end of yea 2; $2,800 at the end of yea 3; and $3,200 at the end of yea 4 (so $2,000; $2,100; $2,800; and $3,200 ae a seies of EAMTs that will occu at diffeent times). If inteest aveaging 10% pe yea is eceived by the account holde/paid by the bank on the emaining balance, how much must be on deposit today to fund the plan? A time line pespective would show: 1-01-Y Y 1/1-01-Y Y 2/1-01-Y Y 3/1-01-Y Y 4 Plan Stats Withdaw $2,000 Withdaw $2,100 Withdaw $2,800 Withdaw $3,200 With??? Plan Ends We can compute the answe (shown as??? above) with fou applications of ou BAMT (1 + ) n = EAMT equation [notice that dividing by (1 + ) n is equivalent to multiplying by its ecipocal]: $2,000 (1) 1 = $2,000 x ( 1 1 )1 = $2,000 x = $1, as BAMT that funds withdawal #1 $2,100 (1) 2 = $2,100 x ( 1 1 )2 = $2,100 x = $1, as BAMT that funds withdawal #2 $2,800 (1) 3 = $2,800 x ( 1 1 )3 = $2,800 x = $2, as BAMT that funds withdawal #3 $3,200 (1) 4 = $3,200 x ( 1 1 )4 = $3,200 x = $2, as BAMT that funds withdawal #4 fo a combined total beginning balance needed of ($1, $1, $2, $2, ) = $7, (This amount is sometimes called the net pesent value, though we will use that tem pimaily in connection with moe complex investment situations, such as capital budgeting analysis in Topic 6.) We could think of this plan as involving fou sepaate accounts, but it could as easily occu within just one account, with yea-by-yea figues as demonstated in the following table: 10

11 Beginning Plus 10% Total Minus End-of-Y. Ending Yea Balance Inteest Available Withdawal Balance 1 $7, $ $8, $2, $6, $6, $ $7, $2, $5, $5, $ $5, $2, $2, $2, $ $3, $3, $0 Now let s assume we instead ae taking those dolla amounts out at the beginning of each espective yea: 1-01-Y Y 1/1-01-Y Y 2/1-01-Y Y 3/1-01-Y Y 4 Plan Stats Withdaw $2,100 Withdaw $2,800 Withdaw $3,200 Plan Officially With??? $0 Balance Remains Ends Withdaw $2,000 Once again we can compute the unknown answe (shown as??? above) with fou applications of ou BAMT (1 + ) n = EAMT equation [dividing by (1 + ) n is equivalent to multiplying by the ecipocal]: $2,000 (1) 0 = $2,000 x ( 1 1 )0 = $2,000 x = $2, as BAMT that funds withdawal #1 $2,100 (1) 1 = $2,100 x ( 1 1 )1 = $2,100 x = $1, as BAMT that funds withdawal #2 $2,800 (1) 2 = $2,800 x ( 1 1 )2 = $2,800 x = $2, as BAMT that funds withdawal #3 $3,200 (1) 3 = $3,200 x ( 1 1 )3 = $3,200 x = $2, as BAMT that funds withdawal #4 fo a combined total beginning balance needed of ($2, $1, $2, $2,404.21) = $8, a lage answe than in the coesponding case with end-of-yea withdawals; moe money is needed up-font to fund a payment steam that stats ight away. Again we could envision this plan as involving fou sepaate accounts, but it could as easily occu within just one account, with yea-by-yea figues as demonstated in the table below. Just as when a payment steam coesponds to a lage dolla amount that will not exist intact until a futue date, the $8, final answe hee fo this beginning-ofyea payments case is meely the $7, answe found fo the same dolla amounts if taken at each yea s end multiplied by (1 + ): $7, x 1 = $8, The two seies stuctues ae identical, except that if payments occu at the beginning of each peiod inteest is applied to the emaining, declining balance one diffeential numbe of times (hee, one less) ove the life of the plan. Beginning Minus Beg.-of-Y. Total Plus 10% Ending Yea Balance Withdawal Available Inteest Balance 1 $8, $2, $6, $ $7, $7, $2, $5, $ $5, $5, $2, $2, $ $2, $2, $3, $0 $0 $0 III. FUTURE AND PRESENT VALUES OF LEVEL ANNUITIES A. Intoduction to Annuities Each example in pats A D of the pio section involves the compaison of a single initial deposit o investment with a single dolla value ealized afte some numbe of time peiods has passed. Of couse, many a financial situation instead involves a seies of payments into, o withdawals fom, an account that is accompanied by an aveage peiodic pecentage etun o cost. Pat E of the pio section deals with 11

12 seies of unequal, unelated payments, in which we must use a peiod-by-peiod bute foce appoach to computing; thee is no altenative to woking a goup of sepaate non-annuity poblems and summing thei answes. But what if thee is a seies of equally spaced expected payments, made into o taken out of some account o plan, that ae equal in amount, o elated in that they change by a constant pecentage fom peiod to peiod? We call a situation in which money is paid into o withdawn fom a plan epeatedly, at equally spaced time intevals and with equal o elated payments, an annuity. (It is not incoect to label a seies of equally spaced but unelated payments as a type of annuity, but in ou coveage we eseve the tem annuity fo cases in which we can goup multiple payments in a single computation. And dolla amounts that ae equal o elated but not spaced equally in time, such as $200 deposited in an account today, $200 six weeks late, $200 eight months late, and $200 two yeas late, can not be gouped computationally.) In any annuity example we ae dealing with a seies of equal o elated cash flows paid o eceived, an aveage expected peiodic ate of etun o cost, a numbe of time peiods n, and a lump sum of money that is equivalent, in time value-adjusted tems, to the seies of cash flows. An annuity poblem tuns out to be just the sum of a goup of non-annuity poblems, as seen in Pat E above, that we can teat as a goup fo computing puposes because of the distibutive popety. We identify the situation as pesent value o futue value of an annuity by looking at when the lump sum exists intact. If the lump sum, be it known o unknown, exists intact today (e.g., a loan epayment poblem the egula payments elate to a big amount the lende hands ove now), we have a pesent value of an annuity situation. If the lump sum will not exist intact until some futue date (e.g., saving up fo etiement the egula deposits and the inteest eaned will not have gown to the maximum amount until the day the save eties), it is a futue value of an annuity situation. As noted, annuities expected cash flow steams can be equal in amount ( level annuities, the most common examples in most textbook applications) o elated (changing by a constant pecentage fom peiod to peiod). An odinay annuity (also called annuity in aeas) is a seies of elated cash flows, each occuing at the end of the yea o othe specified time peiod. An annuity due (also called annuity in advance) is a seies of elated cash flows, each occuing at the beginning of the yea o othe specified time peiod. We can encounte odinay annuity and annuity due situations in both pesent value and futue value of annuity applications. Howeve, while useful odinay annuity and annuity due examples exist fo both futue and pesent value of annuity scenaios, it sometimes can be awkwad to think of the PV of an annuity due (which would involve an initial cash flow on the day the lump sum is paid o eceived, such as boowing money and then immediately making the fist loan payment) o FV of an odinay annuity (an example would be opening a etiement savings account on pape today but then waiting a yea to make the fist deposit). Finally, always emembe that thee ae two sides to evey tansaction o seies of tansactions, and what one side pays in is what the othe side effectively eceives o takes out and the numbes ae the same fo both paties to the tansaction. So do not ovesimplify by saying: if someone pays into a plan it is an FV of annuity situation, while withdawing fom a plan is a PV of annuity situation because the stuctue of the numbes is what mattes. In a etiement savings plan that gows with equal o elated deposits ove time (an FV of annuity case), it is equally valid to view the seies of deposits and the inteest applied to the gowing balance fom the pespective of the save (who makes the deposits and eceives the inteest) o that of the bank (which eceives the deposits and pays the inteest), and the lage futue total is the same amount egadless of whethe we think of it as being popety of the save, o owed to the save by the bank. In a plan fo depleting an endowed account with equal o elated payments ove time (the ich uncle poblem, a PV of annuity case), the pespective of the account holde (who takes withdawals fom the lage initial endowment and eans inteest on the declining balance) is no moe o less elevant than that of the bank (which, in effect, makes payments to the account holde fom the same lage initial balance and owes the inteest). All dolla amounts ae identical fo the paying and eceiving paties. We identify the type of poblem by its stuctue, not by the side of a tansaction we might envision being on. 12

13 B. A Geneal Oveview of the Annuity Idea 1. Futue Value of an Annuity a. Level Payments Think of a level (all payments ae equal) odinay annuity situation, in which we plan to deposit $2,300 at the end of each of yeas 1 though 4. A vey udimentay time line shows 1-01-Y Y 1/1-01-Y Y 2/1-01-Y Y 3/1-01-Y Y 4 Plan Stats Deposit $2,300 Deposit $2,300 Deposit $2,300 Deposit $2,300 With $0 Plan Ends If we can ean a 10% aveage annual inteest ate on the account s gowing balance, one way to solve fo the total we will have by the end of yea 4 is with fou applications of ou BAMT (1 + ) n = EAMT nonannuity equation, just as we did with fou unelated deposits at the end of the pio section: $2,300 x (1) 3 = $2,300 x = $3, as amount deposit #1 will each by end of yea 4 $2,300 x (1) 2 = $2,300 x = $2, as amount deposit #2 will each by end of yea 4 $2,300 x (1) 1 = $2,300 x = $2, as amount deposit #3 will each by end of yea 4 $2,300 x (1) 0 = $2,300 x = $2, as amount deposit #4 will each by end of yea 4 fo a combined total ending balance of ($3, $2, $2, $2,300.00) = $10, We could think of making annual deposits into fou sepaate accounts, but all could be made into the same account, with a gowing balance fo this odinay annuity shown as follows: Beginning Plus 10% Total Plus End-of-Y. Ending Yea Balance Inteest Accumulated Deposit Balance 1 $0 $0 $0 $2, $ 2, $2, $ $2, $2, $ 4, $4, $ $5, $2, $ 7, $7, $ $8, $2, $10, It is pefectly acceptable to compute the combined ending balance fo a seies of equal payments just as we would fo a seies of unelated payments, in the manne shown above. But thee is a moe expedient appoach to eaching this total when the payments ae equal. Note that in the above example we have [$2,300 x (1) 3 ] + [$2,300 x (1) 2 ] + [$2,300 x (1) 1 ] + [$2,300 x (1) 0 ] which, by the distibutive popety (the mathematical basis fo all annuity computations), can be estated as $2,300 [(1) 3 + (1) 2 + (1) 1 + (1) 0 ] = $2,300 ( (1)4 1 ) = $2,300 x = $10, (fotunately, a smat mathematician long ago figued out that [(1 + ) n (1 + ) 1 + (1 + ) 0 ] = ( (1+)n 1 )). That is, the futue value of a level odinay annuity facto is just the sum of the so-called futue value of $1 factos fo the same peiodic ate and numbe of time peiods. (See section A of the Appendix to this outline fo moe detail behind the mathematics of the FV of a level annuity facto.) A level annuity theefoe is just a seies of payments that we can teat fo computational puposes as a team, because they ae equal in value so the distibutive popety applies, athe than using the bute foce (seies of non- 13

14 annuity poblems) appoach necessay when the payments ae unelated. (The futue value of an annuity is just a net futue value, if we wish to use that tem, in a case with equal o elated payments.) Thee inteconnected chaacteistics ae seen that help in identifying evey FV of an annuity situation: The lump sum value that coesponds to the steam of equal o elated payments will not be intact until a futue date; we stat with nothing and end up with a lage dolla amount. (The classic example of a FV of an annuity is saving up fo etiement, with the lage nest egg building up ove time and not being complete until the futue date when the save will etie.) The pincipal amount to which the inteest o othe ate of etun is applied gows lage (and theefoe the amounts of inteest do, as well) with each successive yea o othe time peiod. The FV of an annuity facto is lage than the numbe of payment peiods. Hee the futue value of an annuity facto is lage than the 4 payments: if we make fou equal deposits into a 10% savings account, the eventual total will be times each deposit. We end up with moe than the amounts deposited because inteest is applied on top of the fou deposits made. What if instead we plan to deposit $2,300 at the beginning of each of yeas 1 though 4, and expect to ean a 10% aveage inteest ate pe yea on the gowing balance? The payments timing would be 1-01-Y Y 1/1-01-Y Y 2/1-01-Y Y 3/1-01-Y Y 4 Plan Stats Deposit $2,300 Deposit $2,300 Deposit $2,300 Plan Ends Deposit $2,300 Of couse we can solve fo the total we will have in this annuity due situation by the end of yea 4 with fou applications of the BAMT (1 + ) n = EAMT equation ou tool box specifies fo non-annuity cases: $2,300 x (1) 4 = $2,300 x = $3, as amount deposit #1 will each by end of yea 4 $2,300 x (1) 3 = $2,300 x = $3, as amount deposit #2 will each by end of yea 4 $2,300 x (1) 2 = $2,300 x = $2, as amount deposit #3 will each by end of yea 4 $2,300 x (1) 1 = $2,300 x = $2, as amount deposit #4 will each by end of yea 4 fo a combined total ending balance of ($3, $3, $2, $2,530.00) = $11, We can follow the yea-by-yea activity in a single account holding these deposits with the table: Beginning Plus Beg.-of-Y. Total Plus 10% Ending Yea Balance Deposit Accumulated Inteest Balance 1 $0 $2, $ 2, $ $ 2, $2, $2, $ 4, $ $ 5, $5, $2, $ 7, $ $ 8, $8, $2, $10, $1, $11, Note that in this example we have [$2,300 x (1) 4 ] + [$2,300 x (1) 3 ] + [$2,300 x (1) 2 ] + [$2,300 x (1) 1 ] which, by the distibutive popety, can be estated as $2,300 [(1) 4 + (1) 3 + (1) 2 + (1) 1 ] = $2,300 [(1) 3 + (1) 2 + (1) 1 + (1) 0 ] x (1) = $2,300 [( (1)4 1 ) (1)] = $2,300 x 551 = $11,

15 (If deposits ae made at the beginning of each yea, then inteest is eaned duing yea 1, causing the final balance to be 1 times as lage as it would be if thee wee yea-end deposits, in which case no inteest would be eaned in the fist yea. The exponent in the FV of an odinay annuity facto is the numbe of cash flows, but in that facto the numbe of inteest applications is one less than the cash flow count.) The futue value of a level annuity facto theefoe is ( (1 + )n 1 ) fo cases with end-of-peiod cash flows (FV of a level odinay annuity) and [( (1 + )n 1 ) (1 + )] fo cases with beginning-of-peiod cash flows (FV of a level annuity due). b. Payments Changing by a Constant Peiodic Pecentage [FIL 240 Students: Just skim, not equied] Now conside a case in which the fist of fou deposits we plan to make at the ends of each of yeas 1 though 4 is $2,300, and then the amounts ae to incease by 2% pe yea. Theefoe the deposits will be $2,300 (1.02) 0 = $2,300; $2,300 (1.02) 1 = $2,346; $2,300 (1.02) 2 = $2,392.92; and $2,300 (1.02) 3 = $2, If a 10% aveage annual inteest ate can be eaned on the account s gowing balance, we again can solve fo the total we will have by the end of yea 4 with fou applications of ou BAMT (1 + ) n = EAMT non-annuity equation, just as we did with fou unelated deposits at the end of the pio section: $2, x (1) 3 = $2, x = $3, as amount deposit #1 will each by end of yea 4 $2, x (1) 2 = $2, x = $2, as amount deposit #2 will each by end of yea 4 $2, x (1) 1 = $2, x = $2, as amount deposit #3 will each by end of yea 4 $2, x (1) 0 = $2, x = $2, as amount deposit #4 will each by end of yea 4 fo a combined total ending balance of ($3, $2, $2, $2,440.78) = $10, If the fou deposits all wee made into the same account, the gowing balance would show as follows: Beginning Plus 10% Total Plus End-of-Y. Ending Yea Balance Inteest Accumulated Deposit Balance 1 $0 $0 $0 $2, $ 2, $2, $ $2, $2, $ 4, $4, $ $5, $2, $ 7, $7, $ $8, $2, $10, Recall that when we say we can combine, fo computational puposes, a seies of payments that ae equally spaced apat in time and equal o elated in amount, o elated means changing by a constant pecentage fom peiod to peiod. In this example, with the 2% constant peiodic change in cash eceived o paid out, a moe expedient appoach is to note that in the above example we have [$2,300 x (1.02) 0 (1) 3 ] + [$2,300 x (1.02) 1 (1) 2 ] + [$2,300 x (1.02) 2 (1) 1 ] + [$2,300 x (1.02) 3 (1) 0 ] which, by the distibutive popety, can be estated as $2,300 [(1.02) 0 (1) 3 + (1.02) 1 (1) 2 + (1.02) 2 (1) 1 + (1.02) 3 (1) 0 ] = $2,300 ( (1)4 (1.02) 4 ) = $2,300 x = $10, (see sections B and I of the Appendix to this outline fo moe detail on the mathematics behind the FV of a changing annuity facto). What if we instead make the deposits descibed above at the beginning of each of yeas 1 though 4, still expecting to ean a 10% aveage inteest ate o othe ate of etun pe yea on the gowing balance? The total expected by the end of yea 4 would be 15