The Tme Value of Moey 1
Iversemet Optos Year: 1624 Property Traded: Mahatta Islad Prce : $24.00, FV of $24 @ 6%: FV = $24 (1+0.06) 388 = $158.08 bllo Opto 1 0 1 2 3 4 5 t ($519.37) 0 0 0 0 $1,000 Opto 2 0 1 2 3 4 5 t ($1,000) 150 150 150 150 150 1,000 2
1) Future Value a sgle amout, a lump sum PV.. 1 = PV() = I 1 FV 1 = PV + PV = PV(1+) [ PV(1+)]()(1) = I 2 FV 2 = [PV(1+)] + [PV(1+)] FV 2 = PV+PV+PV+PV 2 = PV(1+I + + 2 ) = PV(1+ ) 2 FV 3 = PV(1+ ) 3.... FV = PV(1+ ) Theorem: If PV(a lump sum) s deposted a accout at a rate of terest,, compouded aually, the accout wll accumulate a total of FV = PV(1+ ) after years. FV s future value, edg amout, termal value. 3
Example: What would the future value of $125 be after 8 years at 8.5% compoud terest? N 8 I/YR 8.5% PV $125 PMT $0 FV $240.08 FV = PV(1+ ) =125(1+ 0.085) 8 = 125(1.920604) = $240.08 I = $240.08 125 = $115.08 FV PV ( 1 ) PV ( FVIF ), where ( FVIF ), s called future value terest factor at ad. FVIF 4
t where 1 < 2 < 3 < 4 < 5 < 6 Hgher => hgher FV, = costat Hgher => hgher FV, = costat 2) Preset Value sgle amout, a lump sum FV PV ( 1 ) PV ( FVIF ), PV FV (1 ) 1 FV ( (1 ) ) FV ( PVIF ) where ( PVIF ), s called preset value terest factor at ad., 5
Example: Suppose a U.S. treasury bod wll pay $2,500 fve years from ow. If the gog terest rate o 5-year treasury bods s 4.25%, how much s the bod worth today? N 5 I/YR 4.25% PMT $0 FV $2,500.00 PV $2,030.30 2500 =PV(1+0.0425) 5 => PV =2500 / (1.0425) 5 PV = 2500/1.231347 => PV = $2,030.30 Note: PVIF 6
where 1 < 2 < 3 < 4 < 5 t Hgher => lower PV, = costat Hgher => lower PV, = costat Example: Suppose the U.S. Treasury offers to sell you a bod for $747.25. No paymets wll be made utl the bod matures 5 years from ow, at whch tme t wll be redeemed for $1,000. What terest rate would you ear f you bought ths bod at the offer prce? N 5 PV $747.25 PMT $0 7
FV $1,000.00 I/YR 6.00% 747.25 =1000/(1+) 5 (1+) 5 = 1000/747.25 (1+) = (1.338240) 0.2 => = (1.338240) 0.2-1 = 1.060002 1 = 0.060002 = 6% Example: Last year Maso Corp's eargs per share were $2.50, ad ts growth rate durg the pror 5 years was 9.0% per year. If that growth rate were mataed, how may years would t take for Maso s EPS to double? I/YR 9.0% PV $2.50 PMT $0 FV $5.00 N 8.04 FV = PV(1+ ) 2 =1(1+0.09) =(1.09) L2 = l 1.09 0.693147 = 0.086178 = 0.693147 /0.086178 =8.04 years. Example: suppose two optos are avalable: Opto 1: Bak Oe rate = 12% for a 5 year vestmet. 8
Opto 2: Prce = $519.37, = 5, maturty value =$1000 Whch oe? 519.37 =1000/(1+) 5 (1+) 5 = 1000/519.37 (1+) = (1.925) 0.2 => = (1.925) 0.2-1 = 1.14 1 = 0.14 = 14% Example: Te years ago, Lev Ic. eared $0.50 per share. Its eargs ths year were $2.20. What was the growth rate Lev's eargs per share (EPS) over the 10-year perod? N 10 PV $0.50 PMT $0 FV $2.20 I/YR 15.97% 3) Auty: a auty s a set of perodc paymets (or recepts) of equal amout at fxed tervals for a specfed umber of perods(years). 9
PMT PMT PMT PMT PMT 0 1 2 3 4 5 t for example a typcal house mortgage repaymet schedule s a auty If the PMT are made at the ed of each perod, the auty s called a ordary auty or deferred auty. If the PMT are made at the begg of each perod, the auty s called a auty due. 4) Future Value of A Auty(S ) a) Future Value of A Ordary Auty: S (1 ) 1 ( ord.) PMT PMT ( FVIFA ), ( FVIFA ), K s called future value terest factor of auty at ad. Example: You wat to buy a ew sports car 3 years from ow, ad you pla to save $4,200 per year, begg oe year from today. You wll 10
depost your savgs a accout that pays 5.2% terest. How much wll you have just after you make the 3rd depost, 3 years from ow? N 3 I/YR 5.2% PV $0.00 PMT $4,200 FV $13,266.56 (1 0.052) S ( ord.) $4,200 0.052 $13,266.56 3 1 4,200(3.158704) Example: Jack a move star. Because of the falure of hs recet moves at the box offce, he feels hs career wll last oly 4 years. To prepare for hs future lack of popularty, he s plag to accumulate a sum of $5,864,636.00 at ed of year 4 to retre Florda. The accout that he s lookg at pays 6% terest compouded aually. How much paymet of equal amout Jack has to deposts that accout at the ed of each year order to accumulate $5,864,636.00? 11
(1 0.06) $5,864,636 $ PMT 0.06 PMT $1,340,605.90 4 1 PMT (4.374616) b) Future Value of A Auty Due S (1 ) 1 ( due) PMT (1 ) PMT ( FVIFA ), (1 ) Example: You wat to buy a ew sports car 3 years from ow, ad you pla to save $4,200 per year, begg mmedately. You wll make 3 deposts a accout that pays 5.2% terest. Uder these assumptos, how much wll you have 3 years from today? Make sure to adjust your calculator for BEG N 3 I/YR 5.2% PV $0.00 PMT $4,200 FV $13,956.42 12
(1 0.052) S ( due) 4,200 0.052 $13,956.42 3 1 (1 0.052) 4,200(3.322957) Why FV of auty due s hgher? 5) Preset Value of A Auty (A ) a) PV of a Ord. Auty 1 1 (1 ) A PMT PMT ( PVIFA), Where ( PVIFA ), s called preset value terest factor of auty at ad. Example: You have to make a set of paymets of equal amout of $2000 at the ed of each year for the ext 3 years. 0 1 2 3 2,000 2,000 2,000 How much moey you must depost a accout today order to be able to make these 13
paymets? Accout pay 5% terest compouded aually. A 1 $2,000 1 (1 0.05) 0.05 3 $2,000(2.7232) $5,446.40 ote: ( PVIFA ) 5%, 2. 7232 3 Example: You have a chace to buy a auty that pays $1,200 at the ed of each year for 3 years. You could ear 5.5% o your moey other vestmets wth equal rsk. What s the most you should pay for the auty? A N 3 I/YR 5.5% PMT $1,200 FV $0.00 PV $3,237.52 1 1 (1 0.055) $1,200 0.055 3 $1,200(2.697933) $3,237.52 14
b) PV of a Auty due. A 1 PMT 1 (1 ) (1 ) PMT ( PVIFA), (1 ) Example: You have a chace to buy a auty that pays $1,200 at the begg of each year for 3 years. You could ear 5.5% o your moey other vestmets wth equal rsk. What s the most you should pay for the auty? Make sure to adjust your calculator for BEG N 3 I/YR 5.5% PMT $1,200 FV $0.00 PV $3,415.58 1 1 3 (1 0.055) A $1,200 (1 0.055) 0.055 $1,200(2.697933)(1.055) $3,415.58 Why PV of auty due s hgher? 15
6) Perpetuty: Perpetuty s a auty that s expected to cotue forever (that s the lfe of PV perpetuty s fty.e. ). I ths case the ( ) preset value of perpetuty perpetuty s calculated as follows: PV perpetuty PMT Example: If PMT =$100 (perodc paymet) ad =8% the PV perpetuty PMT $100 0.08 $1,250 What happes to PV f creases (decreases)? Example: What s the preset value of a perpetuty that pays $250 per year f the approprate terest rate s 5%? I/YR 5.0% PMT $250 PV $5,000.00 16
PV perpetuty PMT $250 0.05 $5,000 7) Growg Auty: ths s a case whch all varables ordary auty formula are costat over tme except paymets(cfs) that grow at a costat rate per year over the lfe of the auty. PVA g PMT (1 1 g) (1 g) (1 ) g Graphcal presetato: PMT PMT(1+g) PMT(1+g) 2 PMT(1+g) 3 PMT(1+g) 4 0 1 2 3 4 Example: How much are you wllg to pay for ths gold mg compay? E(lfe) = 20 years, extract(producto)= 5000 ouces/year. Curret prce of gold = $300/ouce 17
Expected rate of prce crease=3%/year Our cost of captal (dscout rate) (retur o alteratve vestmet)=10% The preset ower s askg for $14,100,000. Is ths compay a barga? (1 0.03) 1 (1 0.10) ($300)(5,000)1.03 PVA g 0.10 0.03 $16,145,980 What s your decso? Example: Your father ow has $1,000,000 vested a accout that pays 9.00%. He expects flato to average 3%, ad he wats to make aual costat dollar (real) begg-ofyear wthdrawals over each of the ext 20 years ad ed up wth a zero balace after the 20th year. How large wll hs tal wthdrawal (ad thus costat dollar (real) wthdrawals) be? 20 20 $1,000,000 PMT (1 (1 0.03) 1 (1 0.09) 0.03) 0.09 0.03 20 20 PMT = $85,951.76 18
Calculator: NOM 9.00% Ital sum1,000,000 Iflato 3.00% Years 20 r = [(1 + NOM )/(1 + P%)] 1 r = 5.825243% PMT = $85,951.76 (where NOM = omal terest rate, r = real terest rate, ad P%= flato rate 8) Preset Value of a stream of uequal Paymets ( PV uqualpmt): PV uequalpmt PV PMT 1 t t t t 1 t 1 (1 ) where PMT t s paymet to be made at the ed of perod t ad PMT t PMT t+1 for some or all t Example: At a rate of 6.25%, what s the preset value of the followg cash flow stream? $0 at Tme 0; $75 at the ed of Year 1; $225 at the ed of Year 2; $0 at the ed of Year 3; ad $300 at the ed of Year 4? I/YR = 6.25% 19
0 1 2 3 4 CFs: $0 $75 $225 $0 $300 PV: $0 $71 $199 $0 $235 PV uequlapmt = 0/(1.0625) 0 + 75/(1.0625) 1 + 225/(1.0625) 2 +0/(1.0625) 3 + 300/(1.0625) 4 = =0.0 + 70.59 + 199.31+ 0.0 + 235.40=$505.30 Calculator soluto: 6.25 I/YR 0 CFj 75 CFj 225 CFj 0 CFj 300 CFj Yellow key : NPV 505.2956 Example: What s PV of the followg uequal CFs @ 12% CF 1 = $1, CF 2 = $2,000, CF 3 = $2,000, CF 4 = $2,000, CF 5 = $0, CF 6 = -$2,000 PV uequlapmt = 1/(1.12) + 2000/(1.12) 2 + 2000/(1.12) 3 +2000/(1.12) 4 + 0/(1.12) 5 + (-2000)/(1.12) 6 =0.89 + 1594.40 + 1423.60+ + 1271.0 + 0-1013.2 = $3276.69 Example 3 20
Suppose PV of the followg uequal cash flow stream s $5,979.04 at 12% terest rate: CF 1 = $1000, CF 2 = $?, CF 3 = $2,000, CF 4 = $2,000, how much s CF 2? 1 PVuequalPMT PVt PMTt t (1 t 1 t 1 ) 1000 x 2000 5,979.04 1 2 3 (1.12) (1.12) (1.12) 2000 (1.12) 4 5979.04 = 892.86 + x/(1.12) 2 +1423.56+1271.04 => x/(1.12) 2 =2391.58 => x =$2999.00 = $3000 9) Future Value of a Stream of uequal Paymets ( FV uequalpmt): FV uequalpmt t FVt PMTt (1 ) t1 t1 Example 1; 21
Suppose a frm plas to depost $2,000 today ad $1,500 oe year from ow a bak accout wth o other future depost or wthdrawal. The bak pays 10% terest compouded aually. What s the future value of the accout at the ed of 4 years? FV 2,000(1 uequalpmt 4924.70 0.10) 0(1 0.10) 0(1 0.10) t1 42 44 40 PMT t (1 ) 1,500(1 0(1 0.10) t 0.10) 43 41 2928.20 1996.50 Example: You just graduated, ad you pla to work for 10 years ad the to leave for the Australa. You fgure you ca save $1,000 a year for the frst 5 years ad $2,000 a year for the ext 5 years. These savgs cash flows wll start oe year from ow. I addto, your famly has just gve you a $5,000 graduato gft. If you put the gft ow, ad your future savgs whe they start, to a accout whch pays 8 percet compouded aually, what wll your facal "stake" be whe you leave for Australa 10 years from ow? 22
FV1 S ( ord.) (1 0.08) 1,000 0.08 5 1 (1.08) 5 $8,619.96 FV 2 S ( ord.) (1 0.08) 2,000 0.08 5 1 $11,733.20 FV 3 5000(1 0.08) 10 $10,794.62 FV = 8,619.96 + 11,733.20 + 10,794.62 =31,147.78 OR FV = $1,000(FVIFA 8%,10 )+ $1,000(FVIFA 8%,5 ) + $5,000(FVIF 8%,10 ) = $1,000((1.08 10-1)/.08) + $1,000((1.08 5-1)/.08) + $5,000(1.08 10 ) = $1,000(14.487) + $1,000(5.866) + $5,000(2.1589) = $14,487 + $5,866 + $10,794.50 = $31,148. 23
10) Determg Iterest Rate: You just wo the state lottery, ad you have a choce betwee recevg $3,500,000 today or a 10-year auty of $500,000, wth the frst paymet comg oe year from today. What rate of retur s bult to the auty? N 10 PV $3,500,000 PMT $500,000 FV $0.00 I/YR 7.07% Example: FV = $1,610.50 PV = $1000 ad = 5 so or FV = PV (1 + ) => 1,610.50 = 1000 (1 + ) 5 (1 +) 5 = 1,610.50 = 1.610 hece or 1 + = (1.610) 1/5 = (1.610) 0.20 = 1.10 = 1.10 1 =.10 = 10% 24
Suppose you sg a loa cotract that calls for aual paymet of $2,545.16 over 25 years (at the ed of each year) ad the amout that you borrow s $25,000. The questo s what terest rate the bak s chargg you? the -Always remember the amout that you borrow, ad you wat to repay t a set of equal perodc paymets over a gve umber of years, s preset value of a auty Here: Preset value of ths auty (A ) s $25,000, = 25 ad PMT = $2545.16 so we use the followg formula: é ê1- A = PMT ê ê ëê 1 (1+ ) ù ú ú ú ûú é ê1-25, 000 = $2545.16ê ê ëê 1 (1+) 25 ù ú ú ú ûú 25
Calculator soluto: N 25 PV $25,000 PMT $2,545.16 FV $0.00 I/YR 9.00% 11) Semaual ad other Compoudg Perod: So far we have assumed that terest s compouded oce a year, however, f terest s compouded more tha oce a year; we have to adjust all the formulas that we have dscussed so for to reflect ths fact. For example: If PV dollars s deposted a accout that pays percet terest compouded m tmes a year, the accout wll cota a total of FV PV ( 1 ) m dollars after years. m Example: What s the future value of $1000 at 8% compouded quarterly after 6 years? Here PV = 1000, = 8%, = 6, ad m = 4 (4 quarters oe year) so 26
FV $1000(1 0.08 ) 4 (4)(6) $1,608.44 Example: What s the future value of $1,500 after 5 years f the approprate terest rate s 6%, compouded semaually? Years 5 Perods/Yr 2 Nom. I/YR 6.0% ===================== N = Perods 10 PMT $0 I = I/Perod 3.0% PV $1,500 FV $2,015.87 We ca make the above type of adjustmets for all the formulas that we dscussed case of terest beg compouded aually. For stace, to calculate future value of a ordary auty wth a lfe of years whch terest s compouded m tmes a year( PMTs are made m tmes a year), we use the followg adjusted formula: S PMT (1 ) m m m 1 27
Example: Suppose you depost $1000 every moth a accout that pays 8% for 35 years. How much wll your accout cota after 35 years? Here, m = 12, = 35, = 8%, ad PMT = $1000. The future value of ths auty wll be: (1 ) S m PMT m 0.08 (1 ) $1,000 12 0.08 12 $2,293,880 m (12)(35) 1 1 (1,000)(2,293.88) For preset value of auty, the adjusted formula s: 28
A 1 PMT 1 (1 m ) m m Example: Suppose you wat to buy a car that s prced $110,000, you put $60,000 dow paymet ad borrow the rest at 7%. You wat to pay off the loa 5 years. What s your mothly car paymet? Amout borrowed = $110,000 $ 60,000= $50,000. $50,000 1 PMT 1 0.07 (1 ) 12 0.07 12 (5)(12) the, $50,000 = PMT (50.501996) so 29
PMT $50,000 / moth 50.501996 $990.06 / moth Cotuous compoudg: a case whch terest rate s compouded cotuously. Future value of a accout that pays terest compouded cotuously s calculated as: FV PV ( e) where e = 2.71828 Example : suppose PV =$1,000 ad accout pays 6% terest compouded cotuously. What s the FV of accout at the ed of year 10? FV 10(0.06) $1,000(2.71828) $1,822.12 A ote: the more ofte terest s compouded, the more terest there wll be eared. Usg followg example, compoudg ca be exteded to varous (more frequet) compoudg perod: semaual, quarterly, mothly, daly etc. PV =$1,000, = 6%, =10 years Compouded terest 30
Not at all(smple terest 1,000 x0.06x10=$600 yearly 0.06 10 I 1,000(1 ) 1,000 $790.85 1 sem-aually 0.06 20 I 1,000(1 ) 1,000 $806.11 2 quarterly 0.06 40 I 1,000(1 ) 1,000 $814. 02 4 0.06 1,000(1 ) 12 mothly 120 I 1,000 $819. 40 daly exercse hourly exercse Cotuously 1,822.12-1,000=822.12 Look at the patter: creasg the frequecy of compoudg makes smaller ad smaller dfferece the amout of terest eared. 12) Nomal or Stated Iterest rate: Stated(omal) terest rate s rate of terest stated the loa cotract. Let $1,000 be compouded mothly at a 12% for oe year. FV 1000(1 0.12 ) 12 12 $1126.83 31
What terest rate would gve us the same future value f terest s compouded aually for oe year? That s what s equvalet aual terest rate? 1,126.83 = 1,000(1+x) => (1+x) = 1.12683 x = 0.1268 = 12.68% 12.68% s called Effect Aual Rate(EAR) That s aual terest rate that would geerate the same future value uder more frequet compoudg(mothly the above example). I geeral oe ca prove that: EAR om m ( 1 ) 1 m usg for above example: EAR (1 0.12 ) 12 12 1 0.1268 12.68% For cotuous compoudg: EAR e om 1 32
f om = 10% ad terest s compouded cotuously, the EAR = (2.71828) 0.10 1 = 0.1052 = 10.52% We do eed EAR because dfferet vestmet (deposts, bods, stock etc) use dfferet compoudg perod. If we wat to compare securtes wth dfferet compoudg perods, we eed to put them s a commo bass,.e. returs o yearly bass. For example: State Bak CD rate = 6.5%, aual compoudg Natoal Bak: MMDA rate = 6.0%, daly compoudg If two baks are equally rsky, whch bak do you choose? EAR CD = 6.5% (why?) EAR MMDA = [1 + (0.06/365) 365 ] - 1 = 6.18% Example 33
East Coast Bak offers to led you $25,000 at a omal rate of 7.5%, compouded mothly. The loa (prcpal plus terest) must be repad at the ed of the year. Mdwest Bak also offers to led you the $25,000, but t wll charge a aual rate of 8.3%, wth o terest due utl the ed of the year. What s the dfferece the effectve aual rates charged by the two baks? Nomal rate, East Coast Bak 7.5% Nomal rate, Mdwest Bak 8.3% Perods/yr, East Coast 12 Perods/yr, Mdwest 1 EFF% East Coast 7.76% EFF% Mdwest 8.30% Dfferece 0.54% EAR East Coast bak = [1 + (0.075/12) 12 ] - 1 = 7.763 Amortzed Loa: s a loa whch s repad equal perodc amouts. Suppose a small frm takes a 5-year- loa of $40,000 from a bak to be repad 5 years equal paymet at the ed of each year. The bak charges 8% terest o outstadg balace each year, prepare loa amortzato schedule for ths frm: Frst, we got to fd PMT for each year: 34
$40,000 1 1 (5) (1 0.08) PMT PMT (3.99271) 0.08 PMT = $10,018.26/ year Amortzato Schedule year paymet terest repaymet of prcpal remag balace 1 10,018.26 3,200.00 6,818.26 33,181.74 2 10,018.26 2,654.54 7,363.72 25,818.02 3 10,018.26 2,065.44 7,952.82 17,865.20 4 10,018.26 1,429.22 8,589.04 9,276.16 5 10,018.26 742.10 9,276.16 0.00 5 50,091.30 10,091.30 40,000 Cost to the borrower: $50,091.30 $40,000 =$10,091.30 35