Class Sessions 2, 3, and 4: The Time Value of Money
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1 Class Sessios 2, 3, ad 4: The Time Value of Moey Associated Readig: Text Chapter 3 ad your calculator s maual. Summary Moey is a promise by a Bak to pay to the Bearer o demad a sum of well, moey! Oe risk is that the Bak may make more promises tha it ca meet. (Weimar Republic). You ca move moey i Space Swap Federal Reserve promise (Dollars) for a Bak of Eglad promise (Pouds Sterlig). You ca move moey i Time o Ivestmet = Swap paymet o demad for paymet at a future date. Certificates of Deposit etc. o Borrrowig = Swap paymets i the future for a paymet o demad. The value of moey depeds o whe you are goig to get it hece the Time Value of Moey. $100 i your had today (preset value - PV) put i a savigs accout will, with iterest, tur ito $10x i a year s time (future value - FV). The promise of $10x dollars i a year s time is equivalet (i value) to $100 today. This sectio of the course illustrates the seve basic time value formulas ad applies them i almost every preset value ad future value situatio. Begiig with discoutig (pullig back i time) ad the compoudig (pushig forward i time) of a lump sum (or oe value). We ll the cosider auities. Almost everythig we do will be based upo these calculatios Adrew Hall Class Notes to FOMGT301 Sprig Page 8
2 simple iterest compoudig compoud iterest auity auity due effective aual iterest rate discoutig discout rate Key Terms Class Outlie Future Value of a Lump Sum: FV = PV ( 1 + r) (3.2) Preset Value of a Lump Sum: PV FV = (3.6) ( 1 + r) Class Applicatio: The Story of Newma ad the Moey Machie. While drivig his truck, Newma oticed somethig shiig o the side of the road. Upo ispectio, he held i his had a moey machie. As he touched the play butto, a beautiful soudig voice bega hello you lucky dog! Simply spi the wheel, ad read off your cash prize. Prizes of cash to be received i the future ca be traded for cash today {the moey will shoot out through the slot o the right.) Prizes for cash to be received today ca be traded for moey to be received i the future, with future amouts arrivig i the mail o the promised date. Ad remember that the moey machie comes with a 100% guaratee. Also ote that the moey machie will base all prizes off a aual iterest rate of 10%. Have a good day. Newma could hardly cotai himself as he said I ve bee waitig all my life for somethig like this, ad ow it s fially here. He laughed ad spu of the wheel Adrew Hall Class Notes to FOMGT301 Sprig Page 9
3 Moey Machie Questio Number 1: Suppose Newma s spi lads o the prize of $100 to be collected i exactly 2 years, but that Newma wats to istead have moey to sped today ad ot wait. The Moey Machie ca trade future cash for cash today usig a aual iterest rate of 10% -- but how much cash? How may thik the aswer is greater tha $100? Why? How may thik the aswer is less tha $100? Why? Aswer: Apply (3.6) to fid the equivalet preset value of $100 i 2 years: Preset Value = $100 (1.10) 2 = $ Adrew Hall Class Notes to FOMGT301 Sprig Page 10
4 MMQ 8: Although the wheel lads o the amout of $100 to be received immediately, Newma would rather wait 2 years. How much will he receive i the mail 2 years from today i exchage for his prize? Aswer: Sice the calculatio ivolves the future value of oe amout today, we ca apply formula (3.2) for lump sums: FV = $100 (1.1) 2 = $ Note that the aswer is ot $ Why? PRS Questio o Compoud Iterest: Remember we borrowed $1,000 from Toy Soprao at five percet per week compoudig weekly How may weeks will it be before the debt is over $10,000? Press 1 for 12 weeks? Press 2 for 24 weeks? Press 3 for 36 weeks? Press 4 for 48 weeks? Press 5 for 60 weeks? Note: Oce future values have bee coverted ito preset values, the preset values ca be added ad subtracted, so Adrew Hall Class Notes to FOMGT301 Sprig Page 11
5 MMQ 2: Suppose Newma s spi lads o the prize of $100 to be received three differet times, the first comig i oe year, the secod i 2 years, ad the third i 3 years. Of course, Newma has the optio of receivig the equivalet value of the prize today usig a aual iterest rate of 10% -- but how much cash? Aswer: PV = $100 (1.1) 1 + $100 (1.1) 2 + $100 (1.1) 3 = $ Alteratively, you ca use formula (3.8) sice the paymets are of equal amouts: 1 1 PV = (1 +.10) = $ Preset Value of a Auity: PVA 1 1 A r r(1 + r) = (3.8) where: PV is the preset value, r is the rate of iterest that is used to discout, is the future time period, ad FV is the future value (e.g. FV 4 meas that the amout FV will come i time period 4). Note: The expressio i brackets, 1 r r(1 + r) 1 is also kow as the preset value iterest factor for the auity, or PVIFA for short. {PVIFA r,} would be read as the preset value iterest factor for the auity with r iterest per period over periods. The preset values of auities ca be added ad subtracted. Receivig a paymet at the ed of the third year ad at the ed of the fourth year is the same as receivig a paymet at the ed of each of the ext 4 years less receivig a paymet at the ed of the ext 2 years: Adrew Hall Class Notes to FOMGT301 Sprig Page 12
6 PVA4 PVA2 = preset value of paymet at ed of the third year plus the preset value of paymet at the ed of the fourth year. MMQ 3: Suppose Newma s spi lads o the prize of $100 to be received three times, but this time the first paymet does ot begi util 2 years from today. This meas that the secod paymet comes i 3 years, ad the third paymet comes i 4 years. What amout today is equivalet to this prize (use a aual iterest rate r = 10%)? Aswer: We ca apply formula (3.6) three differet times ad add them up: PV = $100 (1.1) 2 + $100 (1.1) 3 + $100 (1.1) 4 = $ {$22.61 less} Alteratively, you ca use the auity formula (3.8) sice the paymets are of equal dollar amouts. However, because the auity formula assumes that the paymets begi i oe year, ad the first paymet here is i two years, we eed to adjust for the extra year s wait: 1 PV = (1 +.10) 3 (1.1) 1 = $ If you re havig trouble coceptualizig the calculatio above, try this: The auity formula will always give the value of the auity oe period prior to the start of the auity. If the auity begis i oe year, the the formula returs the preset value (oe period prior to oe year is the preset value.) If the auity begis i two years, the formula returs the value at the ed of year oe, ot year zero. Thus the extra adjustmet Adrew Hall Class Notes to FOMGT301 Sprig Page 13
7 MMQ 4: Suppose Newma s spi lads o the prize of $100 to be received three times, but this time the first paymet does ot begi util 18 years from today. This meas that the secod paymet comes i 19 years, ad the third paymet comes i 20 years. What amout today is equivalet to this prize ( r = 10%)? Aswer: We ca use the auity formula (3.8) sice the paymets are of equal dollar amouts. However, because the auity formula assumes that the paymets begi i oe year, ad the first paymet is i eightee years, we eed to adjust for the extra wait: 1 PV = (1 +.10) 3 (1.1) 17 = $ (1.1) 17 = $49.25 MMQ 5: Suppose Newma s spi lads o the prize of $100 to be received three times: the first i 1 year, the secod i two years, ad the third i four years (there is o year 3 paymet). What amout today is equivalet to this prize ( r = 10%)? Aswer: Tricky! Istead of addig up 3 separate PV s, try this: Assume for the momet that there was a $100 paymet i year 3. The calculate the PV of a 4 paymet auity PV = $100 {3.1699} = $ Now subtract the $100 paymet i year 3 that ever was: PV = 100 / (1.10) 3 = $ Adrew Hall Class Notes to FOMGT301 Sprig Page 14
8 PV = $ $75.13 = $ Adrew Hall Class Notes to FOMGT301 Sprig Page 15
9 MMQ 6: Newma spis the wheel ad it lads o the prize of $1 millio to be received immediately {ow we re talki}. Newma is scared however to get his hads o that much cash at oe time, ad cofesses this fear to the Moey Machie. Have o fear says the MM as I ca deliver the prize i whatever package you d prefer. How bout this: I ll give you $50,000 smakeroos today, the followed by a equal amout of moey to be received at the ed of each year for the ext 25 years. How s that soud Mr. Newma? Newma is overjoyed. Help him compute that equal amout usig a iterest rate of 10%. Aswer: First, we ll remove the $50,000 received immediately, leavig $950,000. The we ll use formula (3.8) to solve for the auity amout over 25 years that is equivalet to gettig $950,000 today: 1 $950,000 = Auity Amout (1 +.10) 25 = $104, MMQ 7: Leavig Newma aside, suppose you purchase a ew car for $21,500, put $5,000 dow, ad the fiace the balace over 36 moths at a iterest rate of 6%. What will be the size of the mothly car loa paymet? Aswer: Use the PVA formula to solve for the auity. You kow the amout of the loa is $16,500. You kow the term of the loa is 36 moths, ad sice you are solvig for the mothly paymet you eed to compute the mothly rate of iterest o the loa:.06 /12 =.005, which is oe half of oe percet. 1 1 $16,500 = A ( ) Adrew Hall Class Notes to FOMGT301 Sprig Page 16
10 A = 16,500 / , A = Future Values of Auities: FVA = (1 + r) A r 1 (3.9) Note: The expressio i brackets, (1 ) + r 1 r, is also kow as the future value iterest factor for the auity, or FVIFA for short. {FVIFA r,} would be read as the future value iterest factor for the auity with r iterest per period over periods. Pre-Class Problem For Next Class. Pre-Class Problem: Chapter 3, Number 25: You ve wo the followig lottery prize: Prize #1: Receivig $2,000 i oe year, or Prize #2: Receivig $ 500 today, ad $1,500 i oe year. Which is the best prize assumig a aual iterest rate of 7%. What is the differece i preset dollars betwee the two prizes? Aswer: Prize #1: PV = $2,000 / (1.07) 1 = $1, Prize #2: PV = $500 + $1,500 / (1.07) 1 = $1, Prize #2 is better by $32.71 preset dollars Adrew Hall Class Notes to FOMGT301 Sprig Page 17
11 Let s re-work this problem uder the followig circumstaces: 1. Suppose that the aual iterest rate is 5%. Which is the best prize ad by how much i preset dollars? Prize #1: PV = $2,000 / (1.05) 1 = $1, Prize #2: PV = $500 + $1,500 / (1.05) 1 = $1, Prize #2 is better by $ Suppose the aual iterest rate is 10%. Which is the best prize ad by how much i preset dollars? Prize #1: PV = $2,000 / (1.10) 1 = $1, Prize #2: PV = $500 + $1,500 / (1.10) 1 = $1, Prize #2 is better by $45.45 What do these aswers tell us about the iterest rate i Time Value of Moey (TVM) problems? 3. Suppose that for prize #1 the wait is 2 years (r = 7%). How does this chage the problem? What does this teach us about the time period i TVM problems? Prize #1: PV = $2,000 / (1.07) 2 = $1, Prize #2: PV = $500 + $1,500 / (1.07) 2 = $1, Prize #2 is better by $ Adrew Hall Class Notes to FOMGT301 Sprig Page 18
12 4. A bit more challegig problem: For prize #2, solve for a ew Year 1 amout that would make you idifferet betwee the two prizes. $1, = $x / (1.07) 1 $1, = $x / (1.07) 1 $x = $1, III. Other Time Value of Moey Applicatios A. More Frequet Compoudig: FV m r = PV 1 + (3.3) m where m is the umber of times per year iterest is compouded. MMQ 11: Remember Newma? He spis the wheel, ad receives $10,000 immediately. Alteratively, he ca leave it with the machie for 2 years to ear iterest of 10% compouded semi-aually. How much will he have? Aswer: Apply formula (3.3): FV 2 = $10,000 {1 + (.10/2)} 4 = $12, MMQ 12: Retur to MMQ 11. Same problem, except that 10% iterest compouds daily. How much will he have? Aswer: Apply formula (3.3): FV 2 = $10,000 {1 + (.10/365} 730 = $12, Adrew Hall Class Notes to FOMGT301 Sprig Page 19
13 B. More Frequet Compoudig: Cotiuous Compoudig: Suppose m becomes very large, say compoudig iterest every hour of every day (hours = 8,760), or every miute of every day (miutes = 525,600) or eve every secod of every day (secods = 31,536,000). Suppose that iterest could compoud for every fractio of a secod? The limit, as m goes to ifiity, reduces equatio (3.3) to its cotiuous form: Cotiuous Compoudig: FV = PV {e r } (3.4) Where e is the base umber for atural logarithms, approximated by the value Calculator Check! Do you have the e key? If yes, the lear how to use it. If ot, the you ll eed to thik of Formula (3.4) as: FV = PV { r } MMQ 13: Retur to MMQ 11. Same problem, except that 10% iterest compouds cotiuously. How much will he have? Aswer: Apply formula (3.4): FV 2 = $100 (e.10 2 ) = $ Adrew Hall Class Notes to FOMGT301 Sprig Page 20
14 C. Solvig for the iterest rate: FV r = 1 (3.10) PV 1 MMQ 14: Let s go back to MMQ 1, where we leared that $82.64 is equivalet to receivig $100 i two years at a particular iterest rate. Solve for that rate Aswer: r = D. Solvig for the time period: Rearrage formula (3.2) ad take the atural logarithm of both sides of the equatio to get: *l (1+r) = l (FV PV) Calculator Check: Do you have the atural log key? If you do, lear how to use it? If you do t, the fid a table with atural logs Adrew Hall Class Notes to FOMGT301 Sprig Page 21
15 MMQ 15: Our last time with Newma (I promise). The wheel lads o $100,000 to be received immediately. But what Newma really wats i life is to ope a Fat-Free Yogurt Store, ad he kows that to purchase such a store will take $165,000. If Newma takes the optio of leavig the moey with the Moey Machie, ad if the moey ears 10% aually, how log will he have to wait? Aswer: Note that i this problem you already kow the PV, the FV, ad the iterest rate. The questio is, how log will it take for $100,000 grow to $165,000? * l(1.10) = l {$165,000 $100,000} * l(1.10) = l {1.65} * = = = years. Ufortuately his dream will take over 5 years to realize Adrew Hall Class Notes to FOMGT301 Sprig Page 22
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