Class Notes for Maagerial Fiace These otes are a compilatio from:. Class Notes Supplemet to Moder Corporate Fiace Theory ad Practice by Doald R. Chambers ad Nelso J. Lacy. I gratefully ackowledge the permissio of Hayde-McNeil Publishig Ic ad professors Chambers ad Lacey to use ad reproduce these materials. 2. Material by the author. Copyright i each is reserved to the author. Time Value of Moey Associated Readig: Text Chapters 4 ad 5 (Chapter 5 o Auities ad Perpetuities!) Excel Fuctio Referece i Excel Help o the followig Excel fuctios: =PV(rate, per, pmt, [fv],[type]) =FV(rate, per, pmt, [pv],[type]) =PMT(rate, per, pv, [fv],[type]) =RATE() The Time Value of Moey with Excel Hadouts I ad II. Summary Moey is a promise by a Bak to pay to the Bearer o demad a sum of well, moey! A cotract betwee the Bearer ad the Bak. Every fiacial istrumet is a cotract defiig rights to movemets of cash or cash flows i time or over time. Oe risk is that the Bak may make more promises tha it ca meet. (Weimar Republic). The resultig iflatio reduces the value of moey i terms of goods ad services. 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. 2007. Adrew Hall Class Notes to Maagerial Fiace. Page
Iterest paid o ivestmet of cash rewards ivestor for risk of iflatio ad risk of ot beig repaid. The value of moey depeds o whe you are goig to get it hece the Time Value of Moey. $00 i your had today (preset value - PV) put i a savigs accout will, with iterest, tur ito $0x i a year s time (future value - FV). The promise of $0x dollars i a year s time is equivalet (i value) to $00 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. Sice a fiacial istrumet is a cotract settig up rights to cash flows over time, the ability to place a value, i the preset, o cash flows i the future will give us the ability to value fiacial istrumets. Almost everythig we do i this course will be based upo or rooted i these calculatios. simple iterest compoudig compoud iterest auity auity due effective aual iterest rate discoutig discout rate Key Terms 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 2
Class Outlie A. Simple Compoudig Future Value from Preset Value Future Value of a Lump Sum after time period: FV = PV ( + r) (4.) Future Value of a Lump Sum after time periods: FV = PV ( + r) (4.2) B. Simple Compoudig Preset Value from Future Value Preset Value of a Lump Sum: (4.6) PV = FV r) ( + The Story of Newma ad the Moey Machie. The Cotext for all the Moey Machie Questios or MMQ 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 00% guaratee. Also ote that the moey machie will base all prizes off a aual iterest rate of 0%. 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. 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 3
MMQ : Suppose Newma s spi lads o the prize of $00 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 0% -- but how much cash? How may thik the aswer is greater tha $00? Why? How may thik the aswer is less tha $00? Why? Aswer: Apply (4.6) to fid the equivalet preset value of $00 i 2 years: Preset Value = $00 (.0) 2 = $82.64 MMQ 2: Although the wheel lads o the amout of $00 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 (4.2) for lump sums: FV = $00 (.) 2 = $2.00 Note that the aswer is ot $20.00. Why? C. The Rule of 72 How log will it take for your moey to double at a give iterest rate? Take the iterest rate ad divide it ito 72 ad you will get a approximate aswer. Applyig the Rule of 72 A Questio o Compoud Iterest: Remember we borrowed $,000 from Toy Soprao at five percet per week compoudig weekly How may weeks will it be before the debt is over $0,000?. 2 weeks? 2. 24 weeks? 3. 36 weeks? 4. 48 weeks? 5. 60 weeks? 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 4
D. More Frequet Compoudig: FV r = PV + m m * (4.3) where m is the umber of times per year iterest is compouded. MMQ 3: Newma spis the wheel, ad receives $0,000 immediately. Alteratively, he ca leave it with the machie for 2 years to ear iterest of 0% compouded semi-aually. How much will he have? Aswer: Apply formula (4.3): FV 2 = $0,000 { + (.0/2)} 4 = $2,55.06 MMQ 4: Retur to MMQ 3. Same problem, except that 0% iterest compouds daily. How much will he have? Aswer: Apply formula (4.3): FV 2 = $0,000 { + (.0/365} 730 = $2,24.69 E. 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 = 3,536,000). Suppose that iterest could compoud for every fractio of a secod? The limit, as m goes to ifiity, reduces equatio (4.3) to its cotiuous form: Cotiuous Compoudig: FV = r* PV e (4.4) 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 5
Where e is the base umber for atural logarithms, approximated by the value 2.7828. 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 (4.4) as: = r* FV PV 2.7828 MMQ 5: Retur to MMQ 3. Same problem, except that 0% iterest compouds cotiuously. How much will he have? Aswer: Apply formula (4.4): FV 2 = $00 (e.0 2 ) = $22.4 F. Solvig for the iterest rate: FV r = (4.0) PV MMQ 6: Let s go back to MMQ, where we leared that $82.64 is equivalet to receivig $00 i two years at a particular iterest rate. Solve for that rate. 00 2 Aswer: r =.0 82.64 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 6
G. Solvig for the time period: Rearrage formula (4.2) ad take the atural logarithm of both sides of the equatio to get: *l (+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. MMQ 7: The wheel lads o $00,000 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 $65,000. If Newma takes the optio of leavig the moey with the Moey Machie, ad if the moey ears 0% 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 $00,000 grow to $65,000? * l(.0) = l {$65,000 $00,000} * l(.0) = l {.65} * 0.0953 = 0.500775 = 0.500775 0.0953 = 5.2547 years. Ufortuately his dream will take over 5 years to realize. 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 7
H. Time Lies Draw a lie mark of some poits at regular itervals. Startig with zero fill i times from the left had side to the right had side Add values at the poit i time at which they occur: Calculate their preset values: PV A = 30 ( + r ) 5 B PV = 40 ( + r) 7 Note: Oce future values have bee coverted ito preset values, the preset values ca be added ad subtracted, so PV = PV A + PV B 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 8
I. Simple Auities MMQ 8: Suppose Newma s spi lads o the prize of $00 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 0% -- but how much cash? Aswer: PV = $00 (.) + $00 (.) 2 + $00 (.) 3 = $248.69 Alteratively, you ca use formula (4.8) sice the paymets are of equal amouts: PV = 00 3.0.0( +.0) = $248.69 Preset Value of a Simple Auity: PVA A r r( + r) = (4.8) 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, r r( + r) 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: 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 9
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 9: Suppose Newma s spi lads o the prize of $00 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 = 0%)? Aswer: We ca apply formula (4.6) three differet times ad add them up: PV = $00 (.) 2 + $00 (.) 3 + $00 (.) 4 = $226.08 {$22.6 less} Alteratively, you ca use the auity formula (4.8) 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: PV = 00.0.0( +.0) 3 (.) = $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 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 0
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. MMQ 0: Suppose Newma s spi lads o the prize of $00 to be received three times, but this time the first paymet does ot begi util 8 years from today. This meas that the secod paymet comes i 9 years, ad the third paymet comes i 20 years. What amout today is equivalet to this prize ( r = 0%)? Aswer: We ca use the auity formula (4.8) 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: PV = 00.0.0( +.0) 3 (.) 7 = $248.69 (.) 7 = $49.25 MMQ : Suppose Newma s spi lads o the prize of $00 to be received three times: the first i 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 = 0%)? Aswer: Tricky! Istead of addig up 3 separate PV s, try this: Assume for the momet that there was a $00 paymet i year 4. The calculate the PV of a 4 paymet auity PV = $00 {4.699} = $36.99 Now subtract the $00 paymet i year 3 that ever was: PV = 00 / (.0) 3 = $75.3 2007. Adrew Hall Class Notes to Maagerial Fiace. Page
$36.99 - $75.3 = $24.86 MMQ 2: Newma spis the wheel ad it lads o the prize of $ 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 24 years. How s that soud Mr. Newma? Newma is overjoyed. Help him compute that equal amout usig a iterest rate of 0%. Aswer: First, we ll remove the $50,000 received immediately, leavig $950,000. The we ll use formula (4.8) to solve for the auity amout over 25 years that is equivalet to gettig $950,000 today: $950,000 = Auity Amout.0.0( +.0) 25 = $04,660.00 J. Loa Repaymets MMQ 3: Leavig Newma aside, suppose you purchase a ew car for $2,500, 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 $6,500. 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 /2 =.005, which is oe half of oe percet. $6,500 = A 36.005.005( +.005) A = 6,500 / 32.870, 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 2
K. Preset Value of a Auity Due: A Auity Due is oe where the first paymet happes at the begiig of each year rather that at the ed of each year. Oe optio is to treat the paymets as oe today A 0 separated from ad added to a Simple Auity over - paymets. PVA DUE = A 0 + A r r ( + r) A secod optio is to treat it as a Simple Auity which has to be traslated forward by oe year. PVA Due = A * ( + r) r r( + r) MMQ 4: Suppose Newma s spi lads o the prize of $00 to be received three times, but this time the first paymet is to be paid today. This meas that the secod paymet comes i year, ad the third paymet comes i 2 years. What amout today is equivalet to this prize (use a aual iterest rate r = 0%)? Aswer: We ca 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 3
L. Future Values of Simple Auities. Whe savig for some future expediture we ofte start a pla by which we ivest a regular amout each time period at some iterest rate ad we may wat to kow how much moey we will have after time periods. Assume that we start by puttig moey ito a accout at the ed of each time period the we have a Simple Auity. ( + r) Future Values of Simple Auities: FVA = A (4.9) r Note: The expressio i brackets, ( ) + r 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. MMQ 5: Suppose Newma s spi lads o the prize of $00 to be received twice, the first at the ed of the first year ad the secod at the ed of the secod year. Newma wats to sped this prize i two years time. How much will he have available to sped if he leaves the cash with the Moey machie util the ed of year two (use a aual iterest rate r = 0%)? Aswer: We ca M. Future Values of Auities Due More ofte whe savig, we start right away, puttig moey ito the accout at the begiig of each period ad we have a Auity Due. DUE Future Values of Auities Due: FVA = A { ( + r) } r * ( + r) 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 4
MMQ 6: Suppose Newma s spi lads o the prize of $00 to be received twice, the first today ad the secod at the ed of the first year. Newma wats to sped this prize i two years time. How much will he have available to sped if he leaves the cash with the Moey machie util the ed of year two (use a aual iterest rate r = 0%)? Aswer: We ca N. Perpetuities Perpetuities are auities where the paymets cotiue ito perpetuity there is ot ed to the paymets. Based o a Simple Auity = PVA A r r r what happes as gets bigger? ( + ) r( r) = 0 Ifiity + So, the Preset Value of a Simple Perpetuity is: PVP = A (4.2) r A is the cash flow at the ed of the first time period! MMQ 7: Suppose Newma s spi lads o the prize of $00 to be received every year for ever at the ed of every year. Newma wats to sped this prize ow. If he trades the prize for cash today how much will the Moey machie pay him (use a aual iterest rate r = 0%)? Aswer: We ca 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 5
Based o a Auity Due PVA DUE = A Well, ( ) r ( + r) = + A r ( + r) 0 = A PVP DUE A + 0 r r what happes as gets bigger? 0 so the Preset Value of a Perpetuity Due is Ifiity A is the cash flow at the ed of the first time period! MMQ 8: Suppose Newma s spi lads o the prize of $00 to be received every year startig ow. Newma wats to sped this prize ow. If he trades the prize for cash today how much will the Moey machie pay him (use a aual iterest rate r = 0%)? Aswer: We ca O. Growig Perpetuities Because of iflatio the omial value of cash flows teds to grow over time. For istace a salary may icrease by 4 percet a year ad so you might expect to receive 4 per cet more i each year. Assume your salary will cotiue ito perpetuity, the to kow how much it would be worth to you today required the calculatio of a growig perpetuity. Fortuately the calculatio is a simple oe PVCGP r C = g (4.3) 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 6
Where C is the value of the perpetual paymet at the ed of the first time period ad g is the rate at which the paymet is to grow expressed as a decimal. MMQ 9: Suppose Newma s spi lads o the prize of $00 to be received i oe year s time ad every year forever thereafter growig costatly at five per cet per aum. Newma wats to sped this prize ow. If he trades the prize for cash today how much will the Moey machie pay him (use a aual iterest rate r = 0%)? Aswer: We ca We will use perpetuities ad growig perpetuities to help i the valuatio of stocks ad bods i the ext module of the course. P. Moey today is worth more tha moey tomorrow. Chapter 4, Number 25: You ve wo the followig lottery prize: Prize #: Receive $2,000 i oe year, or Prize #2: Receive $ 500 today ad $,500 i oe year. 25. 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 #: PV = $2,000 / (.07) = $,869.6 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 7
25.2 Suppose that the aual iterest rate is 5%. Which is the best prize ad by how much i preset dollars? 25.3 Suppose the aual iterest rate is 0%. Which is the best prize ad by how much i preset dollars? Prize #: PV = $2,000 / (.0) = $,88.8 Prize #2: PV = $500 + $,500 / (.0) = $,864.64 Prize #2 is better by $45.45 25.4 What do these aswers tell us about the iterest rate i Time Value of Moey (TVM) problems? 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 8
25.5 Suppose that for prize # 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 #: PV = $2,000 / (.07) 2 = $,746.88 Prize #2: PV = $500 + $,500 / (.07) 2 = $,80.6 Prize #2 is better by $64.28 25.6 A bit more challegig problem: For prize #2, solve for a ew Year amout that would make you idifferet betwee the two prizes. $,869.6 = 500 + $x / (.07) $,369.6 = $x / (.07) $x = $,465.00 2007. Adrew Hall Class Notes to Maagerial Fiace. Page 9