# Present Value Methodology

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1 Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer s wealh is heir curren endowmen plus he fuure endowmen discouned back o he presen by he rae of ineres (rae a which presen and fuure consumpion can be exchanged). Why do his? Purpose of comparison apples o apples (emporal) comparison wih muliple l agens or apples o apples comparison of invesmen/consumpion / i opporuniies Uniform mehod for valuing presen and fuure sreams of consumpion in order for appropriae decision making by consumer/producer Useful concep for valuing muliple period invesmens and pricing financial insrumens Calculaing Presen Value Presen value calculaions are he reverse of compound growh calculaions: Suppose V 0 = a value oday (ime 0) r = fixed ineres rae (annual) T = amoun of ime (years) o fuure period The value in T years we calculae as: V T = V 0 (1+r) T (Fuure Value) 1

2 Example A \$30,000 Cerificae of Deposi wih 5% annual ineres in 10 years will be worh: V T = V T = 000 *(1 05) 10 0 (1 + r) 30, ) = = \$48, Noe: Compuaion is easy o do in Excel = 30,000 *( )^10 In reverse: Presen Value V 0 = V T /(1+r) T (Presen Value) The presen value amoun is he fuure value discouned (divided) by he compounded rae of ineres Example: A \$48, Cerificae of Deposi received 10 years from now is worh oday: V 0 = \$48,866.84/(1+0.05) 10 = \$30,000 Exam Review Be able o calculae presen and fuure values For any hree of four variables: (V 0, r, T, V T ) you should be able o deermine he value of he fourh variable. How do changes o r and T impac V 0 and V T? 2

3 Example: Rule of 70 Q: How many years, T, will i ake for an iniial invesmen of V 0 o double if he annual ineres rae is r? A: Solve V 0 (1 + r) T = 2V 0 => (1 + r) T = 2 => Tln(1 + r) = ln(2) => T = ln(2)/ln(1+r) = 0.69/ln(1 + r) 0.70/r for r no oo big Presen Value of Fuure Cash Flows A cash flow is a sequence of daed cash amouns received (+) or paid (-): C 0, C 1,, C T Cash amouns received are posiive; whereas, cash amouns paid are negaive The presen value of a cash flow is he sum of he presen values for each elemen of he cash flow Discoun facors: Ineremporal Price of \$1 wih consan ineres rae r 1/(1+r) = price of \$1 o be received 1 year from oday 1/(1+r) 2 = price of \$1 o be received 2 years from oday 1/(1+r) T = price of \$1 o be received T years from oday 3

4 Presen Value of a Cash Flow {C 0, C 1, C 2, C T } represens a sequence of cash flows where paymen C i is received a ime i. Le r = he ineres or discoun rae. Q: Wha is he presen value of his cash flow? A: The presen value of he sequence of cash flows is he sum of he presen values: PV = C 0 + C 1 /(1+r) + C 2 /(1+r) C T /(1+r) T Summaion Noaion PV = T = C 0 (1 + r ) = C + 0 T = 1 C (1 + r) Example You receive he following cash paymens: ime 0: -\$10,000 (Your iniial invesmen) ime 1: \$4,000 ime 2: \$4,000 ime 3: \$4,000 The discoun rae = 0.08 (or 8%) PV = -\$10,000 + \$4,000/(1+0.08) + \$4,000/(1+0.08) 2 + \$4,000/(1+0.08) 3 = -\$10,000 + \$3, \$3, \$3, = \$ See econ422presenvalueproblems.xls for Excel calculaions 4

5 PV Calculaions in Excel Excel funcion NPV: NPV(rae, value1, value2,, value29) Rae = per period fixed ineres rae value1 = cash flow in period 1 value 2 = cash flow in period 2 value 29 = cash flow in 29 h period Noe: NPV funcion does no ake accoun of iniial period cash flow! Presen Value Calculaion Shor-cus PERPETUITY: A perpeuiy pays an amoun C saring nex period and pays his same consan amoun C in each period forever: C 1 = C, C 2 = C, C 3 = C, C 4 = C,. PV(Perpeuiy) C C C (1 + r) (1 + r) (1 + r) 1 2 = C C = = = C = 1 (1 + r) = 1 (1 + r) = 1 (1 + r) 1 PV of Perpeuiy Based on he infinie sum propery, we can wrie PV as: PV = Iniial Term/[1 Common Raio] = C/(1 + r)/[1 - (1/(1 + r))] = C/r Iniial Term = C/(1 + r) Common Raio = 1/(1 + r) 5

6 PV(Perpeuiy) = C/(1 + r) + C/(1 + r) 2 + C/(1 + r) C/(1 + r) +... Le a = C/(1 + r) = iniial erm x = 1/(1 + r) = common raio Rewriing: PV = a (1 + x + x 2 + x 3 + ) (1.) Pos muliplying by x: PVx = a(x + x 2 + x 3 + ) (2.) Subracing (2.) from (1.): PV(1 - x) = a PV = a/(1 - x) PV(1-1/(1 + r)) = C/(1 + r) Muliplying hrough by (1 + r): PV = C/r Example The preferred sock of a secure company will pay he owner of he sock \$100/year forever, saring nex year. Q: If he ineres rae is 5%, wha is he share worh? A: The share should be worh he value o you as an invesor oday of he fuure sream of cash flows. This share of preferred sock is an example of a perpeuiy, such ha PV(preferred sock) = \$100/0.05 = \$2,000 Example Coninued Q: Wha if he ineres rae is 10%? PV(preferred sock) = \$100/0.10 = \$1,000 Noice: Tha when he ineres rae doubled, he presen value of he preferred sock decreased by ½. 6

7 Example Coninued The preferred sock of a secure company will pay he owner of he sock \$100/year forever, saring his year. Q: If he ineres rae is 5%, wha is he share worh? A: The share should be worh he value o you as an invesor oday of he fuure sream of cash flows (perpeuiy componen) plus he \$100 received his year. PV(preferred sock) = \$100 + \$100/0.05 = \$100 + \$2,000 = \$2,100 GROWING PERPETUITY Suppose he cash flow sars a amoun C a ime 1, bu grows a a rae of g hereafer, coninuing forever: C 1 = C, C 2 = C (1+g), C 3 = C(1+g) 2, C 4 = C(1+g) 3, 2 1 C C (1 + g ) C (1 + g ) C (1 + g ) PV(Perpeuiy) = (1 + r) (1 + r) (1 + r) (1 + r) 1 (1 + g) = C (1 + r) = 1 GROWING PERPETUITY Based on he infinie sum propery, we can wrie his as: PV = Iniial Term/[1 Common Raio] = C/(1 + r)/[1- ((1 + g)/(1 + r))] = C/(r - g) Noe: This formula requires r > g. 7

8 Example Your nex year s cash flow or parenal sipend will be \$10,000. Your parens have generously agreed o increase he yearly amoun o accoun for increases in cos of living as indexed by he rae of inflaion. Your parens have esablished a rus vehicle such ha afer heir deah you will coninue o receive his cash flow, so effecively his will coninue forever. Assume he rae of inflaion is 3%. Assume he marke ineres rae is 8%. Q: Wha is he value o you oday of his parenal suppor? Answer This is a growing perpeuiy wih C = \$10,000, 000 r = 0.08, 08 g = Therefore, PV = \$10,000/( ) = \$200,000 FINITE ANNUITY A finie annuiy will pay a consan amoun C saring nex period hrough period T, so ha here are T oal paymens (e.g., financial vehicle ha makes finie number of paymens based on deah of owner or join deah or erm cerain number of paymens, ec.) C 1 = C, C 2 = C, C 3 = C, C 4 = C,. C T = C PV(Finie Annuiy) C C C = r + r + r 2 T (1 ) (1 ) (1 ) T T C 1 = = C (1 + r) (1 + r) = 1 = 1 8

9 Finie Annuiy Formula Resul: PV (Finie Annuiy) = C*(1/r) [1 1/(1 + r) T ] = C*PVA(r, T) where PVA(r, T) = (1/r) [1 1/(1+r) T ] = PV of annuiy ha pays \$1 for T years Value of Finie Annuiy = Difference Beween Two Perpeuiies Consider he Finie Annuiy cash flow: C 1 = C, C 2 = C, C 3 = C, C 4 = C,. C T = C Suppose you wan o deermine he presen value of his fuure sream of cash. Recall a perpeuiy cash flow (#1): C 1 = C, C 2 = C, C 3 = C, C 4 = C, C T = C, C T+1 = C, From our formula, he value oday of his perpeuiy = C/r Consider a second perpeuiy (#2) saring a ime T+1: C T+1 = C, C T+2 = C, C T+3 = C, The value oday of his perpeuiy saring a T+1: = C/r [1/(1+r) T ] (why?) Noe: The Annuiy = Perpeuiy #1 Perpeuiy #2 = C/r C/r [1/(1+r) T ] = C/r [1-1/(1+r) T ] Alernaive Derivaion PV(Finie Annuiy) = C/(1+r) + C/(1+r) 2 + C/(1+r) C/(1+r) T-1 Le a = C/(1+r) x = 1/(1+r) Rewriing: PV = a (1 + x + x 2 + x 3 + +x T-1 ) (1.) Muliplying py by x: PVx = a(x + x 2 + x 3 + +x T ) (2.) Subracing (2.) from (1.): PV(1-x) = a (1-x T ) PV = a (1-x T ) /(1-x) PV = C/(1+r)[(1-1/(1+r) T )/(1-1/(1+r)] Muliplying he (1+r) in he denominaor hru: PV (Finie Annuiy) = C/r [1 1/(1+r) T ] 9

10 Example Find he value of a 5 year car loan wih annual paymens of \$3,600 per year saring nex year (i.e, 5 paymens of \$3,600 in he fuure). The cos of capial or opporuniy cos of capial is 6%. PV = \$3,600*PVA(5, 6%) = \$3,600*(1/0.06)[1 1/(1.06) 5 ] = \$15, Example Coninued Suppose you had also made a down-paymen for he car of \$5,000 o lower your monhly loan paymens. The oal cos/value of he car you purchased is hen: PV(down paymen) + PV(loan annuiy) = \$5,000 + \$15, = \$20, Compuing Presen Value of Finie Annuiies in Excel Excel funcion PV: PV(Rae, Nper, Pm, Fv, Type) Rae = per period ineres rae Nper = number of annuiy paymens Fv = cash balance afer las paymen Type = 1 if paymens sar in firs period; 0 if paymens sar in iniial period 10

11 Example Borrow \$200,000 o buy a house. Annual ineres rae = 10% Loan is o be paid back in 30 years Q: Wha is he annual paymen? PV = \$200,000 = C*PVA(0.10, 30) => C = \$200,000/PVA(0.10, 30) PVA(0.10, 30) = (1/0.10)[1 1/(1.10) 30 ] = => C = \$200,000/9.427 = \$21, Compuing Paymens from Finie Annuiies in Excel Excel funcion PMT: PMT(Rae, Nper, Pv, Fv, Type) Rae = per period ineres rae Nper = number of annuiy paymens Pv = iniial presen value of annuiy Fv = fuure value afer las paymen Type = 1 if paymens are due a he beginning of he period; 0 if paymens are due a he end of he period Example You win he \$5 million loery! 25 annual insallmens of \$200,000 saring nex year Q: Wha is he PV of winnings if r = 10%? PV = \$200,000 * PVA(0.10, 25) PVA = (1/0.10)[1 1/(1.10) 25 ] = => PV= \$200,000 * ( ) = \$1,815,408 < \$5M! 11

12 Fuure Value of an Annuiy Inves \$C every year, saring nex year, for T years a a fixed rae r How much will invesmen be worh in year T? Trick: FVA(r,T) = PVA(r,T)*(1+r) (1 r) T = (1/r) [1-1/(1+r) T ] *(1+r) T = (1/r)[(1+r) T 1] Therefore FV = C*FVA(r, T) where FVA(r, T) = FV of \$1 invesed every year for T years a rae r Example Save \$1,000 per year, saring nex year, for 35 years in IRA Annual rae = 7% Q: How much will you have saved in 35 years? FV = \$1,000*FVA(0.07, 35) FVA(0.07, 35) = (1/0.07)*[(1.07) 35 1] = => FV = \$1,000*( ) = \$138, Compuing Fuure Value of Finie Annuiies in Excel Excel funcion FV: FV(Rae, Nper, Pm, Pv, Type) Rae = per period ineres rae Nper = number of annuiy paymens Pm = paymen made each period Pv = presen value of fuure paymens Type = 1 if paymens sar in firs period; 0 if paymens sar in iniial period 12

13 Finie Growing Annuiies Similar o how we amended he Perpeuiy formula for Growing Perpeuiies, we can amend he Annuiy formula o accoun for a Growing Annuiy. The cash flow for a finie growing annuiy pays an amoun C, saring nex period, wih he cash flow growing hereafer a a rae of g, hrough period T: PV = C/(1+r) + C(1+g)/(1+r) 2 + C(1+g) 2 /(1+r) C(1+g) T-1 /(1+r) T = Σ C(1+g) -1 /(1+r) for = 1,, T ` = C/(r-g) [1- (1+g) T /(1+r) T ] Class Example An asse generaes a cash flow ha is \$1 nex year, bu is expeced o grow a 5% per year indefiniely. Suppose he relevan discoun rae is 7%. Q: Afer receiving he hird paymen, wha can you expec o sell he asse for? Q: Wha is he presen value of he asse you held? Compounding Frequency Cash flows can occur annually (once per annum), semi-annually (wice per annum), quarerly (four imes per annum), monhly (welve imes per annum), daily (365 imes per annum), ec. Based on he cash flows, he formulas for compounding and discouning can be adjused accordingly: General formula: For saed annual ineres rae r compounded for T years n imes per year: FV =V 0 * [1 + r/n] nt 13

14 Compounding Frequency Effecive Annual Rae (annual rae ha gives he same FV wih compounding n imes per year): [1 + r EAR ] T = [1 + r/n] nt => r EAR = [1 + r/n] n -1 Example Inves \$1,000 for 1 year Annual rae (APR) r = 10% Semi-annual compounding: semi-annual rae = 0.10/2 = 0.05 FV = \$1,000*(1 + r/2) 2*1 = \$1,000*(1.05) 2 = \$ Noe: \$1,000*( ) 2 = \$1,000*(1 + 2*(0.05) + (0.05) 05) 2 ) = \$1,000 + \$100 + \$2.5 = principal + simple ineres + ineres on ineres Effecive annual rae: (1 + r EAR ) = (1 + APR/2) 2 => r EAR = (1.05) 2 1 = or 10.25% Example: The Difference In Compounding Annual rae of Ineres 5% T = 1 Year Compounding Times One plus Frequency Per Annum Effecive Rae Yearly Semi-Annual Quarerly Monhly Daily Hourly 8, By he minue 525, By he second 31,536,

15 Example Take ou (borrow) \$300, year fixed rae morgage Annual rae = 8%, monhly rae = 0.08/12 = *12 = 360 monhly paymens Q: Wha is he monhly paymen? PV = \$300,000 = C*PVA(0.08/12, 360) PVA(0.0067, 360) = => C = \$300,000/ = \$2, Noe: oal amoun paid over 30 years is 360*\$2, = \$792,468 Example Consider previous 30 year morgage Suppose he day afer he morgage is issued, he annual rae on new morgages shoos up o 15% Q: How much is he old morgage worh? PV = \$2,201*PVA(0.15/12, 360) PVA(0.15/12, 360) = => PV = \$2,201* = \$174,092 < \$300,000! Coninuous Compounding Increasing he frequency of compounding o coninuously: lim n [1 + r/n] nt = (2.718) rt = e rt Effecive Annual Rae: [1 + r EAR ] T = e rt => r EAR = e r -1 15

16 Example r = annual (simple) ineres rae = 10%, T = 1 year FV of 1\$ wih annual compounding: FV = \$1(1+r) = \$1.10 FV of 1\$ wih coninuous compounding: FV = \$1*e r = = \$ Effecive annual rae 1 + r EAR = => r EAR = = % Furher Insigh on Coninuous Compounding Example: Inves \$V 0 for 1 year wih annual rae r and coninuous compounding V V 1 ln = r V 0 ln V ln V = r r 1 1 r V1 = V 0e = e V Tes/Pracical Tips General formula will always work by may be edious Shor-cus exis if you can recognize hem Use shor-cus! Break down complicaed problems ino simple pieces 16

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