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. 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 8
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. 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 9
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 = $82.64 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 10
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 = $121.00 Note that the aswer is ot $120.00. 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 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 11
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 = $248.69 Alteratively, you ca use formula (3.8) sice the paymets are of equal amouts: 1 1 PV = 100 3.10.10(1 +.10) = $248.69 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: 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 12
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 = $226.08 {$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 = 100.10 1.10(1 +.10) 3 (1.1) 1 = $226.08 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. 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 13
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 = 100.10 1.10(1 +.10) 3 (1.1) 17 = $248.69 (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} = $316.99 Now subtract the $100 paymet i year 3 that ever was: PV = 100 / (1.10) 3 = $75.13 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 14
PV = $316.99 - $75.13 = $241.86 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 15
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.10 1.10(1 +.10) 25 = $104,660.00 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 36.005.005(1 +.005) 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 16
A = 16,500 / 32.8710, A = 501.96 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 = $1,869.16 Prize #2: PV = $500 + $1,500 / (1.07) 1 = $1,901.87 Prize #2 is better by $32.71 preset dollars. 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 17
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 = $1,904.76 Prize #2: PV = $500 + $1,500 / (1.05) 1 = $1,928.57 Prize #2 is better by $23.81 2. 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 = $1,818.18 Prize #2: PV = $500 + $1,500 / (1.10) 1 = $1,863.64 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 = $1,746.88 Prize #2: PV = $500 + $1,500 / (1.07) 2 = $1,810.16 Prize #2 is better by $63.28 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 18
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. $1,869.16 = 500 + $x / (1.07) 1 $1,369.16 = $x / (1.07) 1 $x = $1,465.00 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 = $12,155.06 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 = $12,213.69 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 19
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 2.71828. 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 {2.71828 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 ) = $122.14 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 20
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. 1 100 2 Aswer: r = 1.10 82.64 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. 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 21
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} * 0.09531 = 0.500775 = 0.500775 0.09531 = 5.25417 years. Ufortuately his dream will take over 5 years to realize. 2004. Adrew Hall Class Notes to FOMGT301 Sprig 2004. Page 22