CHAPTER 4: NET PRESENT VALUE


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1 EMBA 807 Corporate Fiace Dr. Rodey Boehe CHAPTER 4: NET PRESENT VALUE (Assiged probles are, 2, 7, 8,, 6, 23, 25, 28, 29, 3, 33, 36, 4, 42, 46, 50, ad 52) The title of this chapter ay be Net Preset Value, but it is essetially a itroductio (or review if you have had a prior fiace course) of the Tie Value of Moey cocepts. Nevertheless, the aterial covered i this chapter is iportat ad essetial i order to uderstad the cotet of this course. Studets eterig this course have a high variace of fiacial kowledge. Soe have take fiace courses ad others are takig their first fiace course. As a result, these otes are writte to accoodate all. Oe period odels (for ow, periods are oe year uits): Exaple : You have $000 cash ow (t=0) therefore, the Preset Value or PV 0 = $000. Let the cost of capital or iterest rate be r=% per year. What is the Future Value or of this aout exactly oe year fro today at t=? Here we say that = year. = PV 0 [ + r] = 000[ + 0.] = $0.00 Practical iterpretatio: $000 i a bak accout earig r=% will ear $0 iterest over oe year. Fial aout i accout is $0.00. Exaple 2: You wat to have exactly $000 i your accout exactly oe year fro today. The accout ears a iterest rate r=9% per year. How uch ust you have i this accout today (t=0) i order to have $000 at t=. We thus have = $000. Give r=9% ad = year, we ust fie PV 0. Fro above: = PV 0 [ + r], rearrage to obtai PV 0 = /[ + r] PV 0 = /[ + r] = 000/[ ] = $97.43 (this is kow as discoutig) Practical iterpretatio: If $97.43 ears r=9% for oe year, it grows to $000. Multiperiod exaples (uber of aual periods is greater tha ): We have these followig basic forulas: = PV 0 [ + r] (copoudig) PV 0 = /[ + r] (discoutig) r = [ /PV 0 ] / = l( /PV 0 )/l(+r) Note that all forulas o this page (sigle ad ultiperiod) pertai to lup su aouts, i.e., oe aout today at t=0 ad oe aout at a future tie t=, where is easured i years, ad r is a aual effective rate of iterest. Page
2 EMBA 807 Corporate Fiace Dr. Rodey Boehe Exaple 3: You deposit $500 ito a accout today that ears r=% aual iterest. The oey is left i the accout for =5 years. What aout 5 is i the accout 5 years fro today? = PV 0 [ + r] 5 = 500[ + 0.] 5 = 500[.685] = $ Note: 5 = PV 0 [ + r][ + r][ + r][ + r][ + r] = PV 0 [ + r] 5 Exaple 4: You ow a bod that pays oe future cash flow of $00,000 exactly 20 years fro today. The applicable aual iterest rate over this horizo is r=6.75%. What is this bod or fiacial clai worth today (t=0)? Here, 20 = $00,000 ad = 20 years. Fid PV 0 by discoutig the 20 back to today (t=0). PV 0 = /[ + r] = 20 /[ ] 20 = 00,000/ = $27,079.6 Note: $27,079.6 deposited ito accout for 20 years at r=6.75% will grow to $00,000 at t=20 years. Exaple 5: A stock idex is at 7900 today. It was at 800 exactly 5 years ago. What was the aualized rate of retur for this 5year horizo? Here, 5 = 7900, PV 0 = 800, ad, of course, = 5 years. Fid r r = [ 5 /PV 0 ] /5 = [7900/800] /5 = or % Note: aualized retur ad geoetric average (Chapter 9) are the sae ad are used to report ivestet perforace (utual fuds report this i their prospectus). Also ote that the Dow Joes Idustrial Average or DJIA was 800 ad 7900 i August 982 ad August 997, respectively. Therefore, the aualized rate of retur was 6.493% for this 5year period. You ca say that we are dealig with PV 982 ad 997 i this exaple. Calculator ethod of solvig Exaple 5: Eter the followig; = 7900 PV = 800, the PV ust be of the opposite sig of the future value, otherwise you will receive a error essage. Thik of the cash flows as havig opposite flows; you pay 800 ad later receive (it really does ot atter which oe is positive or egative here, just as log as you eter the ito the calculator as opposite sigs. N = 5 P/Y = iterest copoudig period per year (the calculator default is P/Y = 2). Solve for I = 6.493% Oce you set P/Y, it stays there util you later aually chage the settig or reove the batteries. Page 2
3 EMBA 807 Corporate Fiace Dr. Rodey Boehe Streas of ueve cash flows: The followig tie lie illustrates five idividual cash flows, begiig ow at t=0 ad edig at t=4 years. Let the applicable iterest rate r=0% per year. t=0 t= t=2 t=3 t=4 CF 0 = 50 CF = 5 CF 2 = 20 CF 3 = 20 CF 4 = 5 Exaple 6: Calculate the curret Preset Value or PV 0 of this strea of cash flows. Especially ote that this cash flow strea resebles a busiess project that costs $50 ow ad produces four future cash flows. We calculate the PV 0 of each of the idividual five cash flows ad the su the to obtai the PV 0 of this ueve cash flow strea. PV 0 = CF 0 /(+r) 0 + CF /(+r) + CF 2 /(+r) 2 + CF 3 /(+r) 3 + CF 4 /(+r) 4 PV 0 = 50/(+0.) 0 + 5/(+0.) + 20/(+0.) /(+0.) 3 + 5/(+0.) 4 PV 0 = =$ If this were a busiess project or real ivestet, the the Net Preset Value or NPV would be $5.44. The iterpretatio is that the future cash iflows are $5.44 ore valuable tha the $50 cash outflow required today at t=0, i.e., you sped $50 today to create a asset that is istatly worth $ Calculator ethod: This is a ajor short cut. The procedure varies across calculators but resebles the followig: C0 = 50 C = 5 C2 = 20 C3 = 20 C4 = 5 The eter I = 0%. Use the NPV key or fuctio to obtai $ Exaple 7: Calculate the Future Value or 4 of the followig strea of cash flows. Let the applicable iterest rate r=7% per year. t=0 t= t=2 t=3 t=4 CF 0 = 90 CF = 60 CF 2 = 50 CF 3 = 70 CF 4 = 80 Page 3
4 EMBA 807 Corporate Fiace Dr. Rodey Boehe Oe practical applicatio of this exaple is to ake five deposits ito a ivestet accout. The first deposit is ade today (t=0) ad the fifth ad fial deposit is ade four years fro today at t=4. We just treat this proble as five separate ivestets ito a accout. I other words, we calculate the 4 of each of these five cash flows. The first deposit at t=0 ears aually copouded iterest for = 4 years, the secod deposit at t= ears aually copouded iterest for = 3 years, ad so o. The fifth ad fial deposit is ade at t=4 ad obviously ears zero iterest. 4 = CF 0 (+r) 4 + CF (+r) 3 + CF 2 (+r) 2 + CF 3 (+r) + CF 4 (+r) 0 4 = 90(+0.07) (+0.07) (+0.07) (+0.07) + 80(+0.07) 0 4 = = $ Calculator ethod: First fid the PV 0 of the strea of cash flows. Next, take this PV 0 ad copoud it up to t=4 at r=7%: 4 = PV 0 [+r] 4. Calculators do ot have ay fuctio that coputes the of a cash flow strea. Perpetuities: A perpetuity is a ifiite strea of costat cash flows, occurrig at equal itervals. The followig tie lie illustrates a perpetuity of $5 per year. t=0 t= t=2 t=3 t=4 CF = 5 CF 2 = 5 CF 3 = 5 CF 4 = 5 The forula for the Preset Value of a perpetuity is: PV 0 = CF/r. This equatio calculates the PV of a ifiite strea of equal cash flows. Exaple 8: Calculate the curret Preset Value or PV 0 of the above perpetuity. Let the applicable iterest rate r=6% per year. PV 0 = CF/r = 5/0.06 = $ Therefore, a share of preferred stock that pays a divided of $5 per year should sell for $83.33 if the required rate of retur is r=6% per year. Page 4
5 EMBA 807 Corporate Fiace Dr. Rodey Boehe Growig perpetuity or Gordo Costat Growth Model: This refers to a cash flow strea that grows at a costat aual growth rate g. The followig tie lie illustrates such a cash flow strea. t=0 t= t=2 t=3 t=4 CF CF 2 = CF (+g) CF 3 = CF (+g) 2 CF 4 = CF (+g) 3 Here, CF 3 = CF (+g)(+g) = CF (+g) 2 The costat growth odel forula is: PV 0 = CF /(rg) Note the tiig differece betwee the cash flow ad the PV i the odel! The cash flow CF is always oe period ahead of the PV 0 you are calculatig. The rate of retur r ust be greater tha the aual cash flow growth rate g i order to use this forula. The growth rate g ca be egative, as well. The costat growth odel is a extreely iportat forula ad will be used extesively i this course. This tie value of oey forula or tool is used i early every stock valuatio exaple. Note that if there is a cash flow CF 0 that has ot yet bee paid out, the use: PV 0 = CF 0 + CF /(rg) Exaple 9: Trilliu is a ature corporatio. 2 Trilliu coo stock has just paid a cash flow of CF 0 = $2.00 per share. 3 A aalyst estiates that Trilliu s cash flows will grow at a costat aual growth rate of g=5.5% per year forever. Based o the risk of this stock, ivestors expect or require a aual rate of retur of r=0% per year. Based o this estiate, what is this stock s curret value? We ust use: PV 0 = CF /(rg). Note that CF 0 has bee paid out ad is o loger part of Trilliu or it s stock price. The stock s value should be the Preset Value of all future expected cash flows. CF 0 is o loger a future cash flow. We eed CF, but we are oly give CF 0, r, ad g. However, CF is easily foud: CF = CF 0 (+g) = 2(+0.055) = $2. per share. Now we ca fid PV 0. PV 0 = CF /(rg) = 2./( ) =2./0.045 = $46.89 per share 2 Mature firs are the oly firs that ca be cosidered to have a costat growth rate fro today forward. However, whe we cover stock valuatio i Chapter 5, we state that all firs ust ature soeday. 3 Firs pay out cash i the for of () divideds ad (2) stock repurchases. For stock valuatio, what atters is how uch ca be paid out. How it is paid should ot atter, as far as deteriig what the fir is worth. Page 5
6 EMBA 807 Corporate Fiace Dr. Rodey Boehe A additioal ote cocerig the costat growth odel before we ove o. Say that a fir will pay CF 2020 i the year What will the fir be worth i 209, just after the year 209 cash flow is paid out? Aswer: it will be worth PV 209 = CF 2020 /(rg). The oe year tiig differece betwee the PV ad CF ever chage with this odel! Auities: A auity is a fiite strea of equal cash flows, occurrig at equal itervals. It does ot atter if the payets are beig paid out or received, just as log as the payets are all i the sae directio. The followig is a exaple of a sixyear (=6) ordiary auity cosistig of six $50 payets. t=0 t= t=2 t=3 t=4 t=5 t=6 C = 50 C = 50 C = 50 C = 50 C = 50 C = 50 Sice this is a cash flow strea, the PV 0 ad 6 ca easily be foud by usig the procedure covered earlier for ueve cash flows. This ca be log ad tedious. However, sice these auity payets are ot ueve, we have shortcuts for PV 0 ad. The followig two forulas are for the ad PV 0 of a ordiary auity. Note: with a ordiary auity, the first cash flow is always at t=. PV 0 = C r r ( r) + = C r ( + r) r Exaple 0: Let r=0% per year. Calculate the Preset Value or PV 0 of the above six year auity. Essetially, how uch would you pay today (t=0) i order to receive this six year auity of cash flows? 4 PV 0 = C r = 50 r( + r) ( + 0.) 6 = $27.76 Log ethod: PV 0 = 50/(+0.) + 50/(+0.) /(+0.) /(+0.) /(+0.) /(+0.) 6 = $27.76 Calculator ethod: PMT = 50 4 You ca also thik of the PV 0 as how uch you should deposit today i order to withdraw $50 for each of the ext six years, so that the accout is epty after the sixth or last $50 withdrawal. Page 6
7 EMBA 807 Corporate Fiace Dr. Rodey Boehe I = 0% N = 6 P/Y = Solve for PV = (PV will be the opposite sig of the payet) Exaple : Let r=0% per year. Calculate the Future Value or 6 of the above sixyear auity. Essetially, if you ake six $50 deposits ito a accout, the first ad sixth (last) payets beig ade at t= ad t=6 years, respectively, how uch oey will be i the accout i six years (t=6)? ( r) + = C = r r ( 0.) = $ Calculator ethod: PMT = 50 (sig does ot really atter here, but will be of opposite sig) I = 0% N = 6 P/Y = Solve for = (positive here sice PMT was ade egative) Growig auities: Basically, this is a fiite series of cash flows that grows at a costat aual rate. While the Gordo costat growth odel discussed earlier is ifiite, the growig auity is fiite. This cocept is great for ivestet plaig. The PV 0 ad equatios are: PV 0 t=0 t= t=2 t=3 t=4 = C r g CF CF 2 = CF (+g) + g + r + g = C + r g r +, where C is the CF o the above tielie ( r) CF 3 = CF (+g) 2 CF 4 = CF (+g) 3 Exaple 2: Let r=% per year. You deposit $2000 ito a accout exactly oe year fro today at t=. The aout you deposit will grow by g=8% each year. You will ake 30 yearly deposits, the fial deposit is 30 years fro today at t=30. How uch oey will be i the accout at t=30 years? Page 7
8 EMBA 807 Corporate Fiace Dr. Rodey Boehe = 2000 ( 0. ) 30 = $855,309.3 While this sees like a large aout, reeber that this is a aout 30 years ito the future, ad it does ot have the purchasig power of $855,309 today. Deferred auities: (highly proe to errors ad istakes!) Whe the first payet of a auity begis ore tha oe period fro ow, we call it a deferred auity. Let us cosider the auity o the followig tie lie. We have a auity of =4 payets of $0,000 each. The first payet is 37 years fro today, ad the 4th ad fial payet is 50 years fro today. Calculate the curret Preset Value or PV 0 of this deferred auity. There are actually several ethods or procedures that will work here. I preset the ost coo ethod below. Step : Jup ahead to t=36 or 36 years fro ow. Fro the perspective of t=36 years, we would be lookig at just a =4 year ordiary auity with the first payet occurrig oe year or period ahead of that poit i tie. Calculate the Preset Value of this 4 year auity at t=36 years. Let r=8% per year. PV 36 t=0 t= t=36 t=37 t=49 t=50 = C = 0,000 r r( + r) ( ) CF = 0,000 4 CF = 0,000 = $82, A good aalogy is the followig: Thirty six years fro ow (at t=36), you will eed to have $82, i a accout if you wat to ake 4 future withdrawals of $0,000 each, the first aout reoved i oe year ad the last (4th) aout reoved i 4 years (t=50). Step 2: The Preset Value PV 36 fro Step becoes 36 i this secod step. This lup su $82, aout fro Step ust be discouted back exactly 36 years at r=8% per year i order to calculate PV 0. PV 0 = 36 /[ + r] 36 = 82,442.37/[ ] 36 = $ CF = 0,000 Page 8
9 EMBA 807 Corporate Fiace Dr. Rodey Boehe Other tha aual iterest copoudig periods: Prior to this poit i these otes, iterest rates were quoted as effective aual rates, ad copoudig periods were aual, e.g., r=9% per year. However, hoe ad car loas calculate the iterest othly. Credit cards calculate the iterest daily. May cotracts use iterest rates that ot calculated aually. What if you deposit $000 today ito a accout that pays r=4% iterest every six oths or seiaually? After six oths you will have $000[+0.04] = $040 i the accout. After oe year (two seiaual periods) you will have $000[+0.04][+0.04] = $08.60 i the accout. It is ucoo to see seiaual iterest rates quoted. However, we do kow that this accout pays 4% iterest every six oths, ad the aout of oey i this accout does effectively grow by r=4% every six oths. The covetio is to quote rates as aual, eve though the copoudig periods ay ot be aual. Here i our exaple, the r=4% effective seiaual rate is thus ultiplied by the =2 seiaual periods per year, ad the iterest rate will be quoted as 8% aual oial, copouded seiaually. For this particular exaple we state: () r o =8% ad (2) =2. r o is what we call the oial or stated iterest rate, ad is the uber of iterest copoudig periods per year. Caveat: while the 8% aual oial rate is typically quoted, it is ot the effective aual iterest rate. Note that: r o / = 8%/2 = 4%. This 4% is the effective seiaual iterest rate. The effective aual iterest rate is (+0.04)(+0.04) = or 8.6%, sice here $ grows to $.086 i oe year. Note that the followig two equatios relate the Preset ad Future lup su values, where is the uber of copoudig periods per year, ad is the uber of years. ro = PV0 + ad PV0 = ro + Exaple 3: You deposit $0,000 today ito a accout that pays 6% aual oial iterest, copouded quarterly. How uch oey will be i the accout exactly 0 years fro today if o further deposits are ade? Page 9
10 EMBA 807 Corporate Fiace Dr. Rodey Boehe The accout will ear r o / = 6%/4 =.5% iterest every 3 oths for = (0)(4) = 40 quarterly periods. 0 4 ro 0.06 = PV0 + = 0,000 + = $8, Exaple 4: You wat to have $20,000 i a accout 5 years fro today. The accout pays 6% oial iterest, copouded quarterly. How uch should you deposit today i order to have $20,000 i 5 years (=20 quarterly payets)? PV 0 = = ro + 20, = $4,849.4 Covertig oial to effective rates: I Exaples 3 ad 4, the iterest rate was r o of 6% aual oial, copouded quarterly. The effective quarterly iterest rate is just r o / = 6%/4 =.5% per quarter. The forula to covert oial to effective rates is give as: Effective aual rate (EAR) ro = + I the case of Exaples 3 ad 4, calculate the actual effective aual rate of iterest: Effective aual rate 4 ro 0.06 = + = 4 + = or 6.364% Also, to covert the aual effective rate to a aual oial rate, give : / r = [( r + ) ] o eff Exaple 5: A credit card states that the aual iterest rate charged is 8% APR (aual percetage rate) ad that iterest charges are calculated daily. APR is a oial rate. I this exaple, the iterest rate beig quoted is 8% aual oial, copouded daily. What is the aual effective rate of iterest? Effective aual rate 365 ro 0.8 = + = = or 9.764% Exaple 6: $20,000 is borrowed today to fiace a car purchase. The car loa is for 48 oths. The loa cotract states the iterest rate as 0% APR, ad the loa s iterest charges ad payets are ade othly. The aout of each Page 0
11 EMBA 807 Corporate Fiace Dr. Rodey Boehe othly payet is idetical. Calculate the effective aual iterest rate ad the othly payet o this loa. The iterest rate quoted here is 0% aual oial, copouded othly. Effective aual rate 2 ro 0.0 = + = 2 + = or 0.473% Note: here the effective othly iterest rate is r o / = 0%/2 = % per oth. The 48 oth car loa is a 48 period auity, cosistig of 48 othly payets or periods. Note the followig auity forula (it is siilar to what we covered earlier i the itroductio to auities). PV0 = C ro r r o o + The aout borrowed today, $20,000, is the PV 0. 20,000 C ( + 0. ) = 20,000 = C[ ] C = $ Calculator ethod: PV = 20,000 I = 0% (the oial aual rate of iterest) N = 48 (there are 48 periods or payets i this auity s life) P/Y = 2 (the calculator takes the i o =0% ad divides it by or P/Y=2 so it ca solve the auity usig a othly effective rate of % per oth) Solve for PMT = (egative here sice PV was ade positive) Alterative calculator ethod: PV = 20,000 I = % (the effective othly rate of iterest) N = 48 (there are 48 periods or payets i this auity s life) P/Y = Solve for PMT = Page
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