The Time Value of Money

Transcription

1 The Tme Value of Moey 1

2 Iversemet Optos Year: 1624 Property Traded: Mahatta Islad Prce : $24.00, FV of 6%: FV = $24 (1+0.06) 388 = $ bllo Opto t ($519.37) $1,000 Opto t ($1,000) ,000 2

3 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 ) = PV(1+ ) 2 FV 3 = PV(1+ ) 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

4 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 $ FV = PV(1+ ) =125( ) 8 = 125( ) = $ I = $ = $ FV PV ( 1 ) PV ( FVIF ), where ( FVIF ), s called future value terest factor at ad. FVIF 4

5 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

6 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, PV $2, =PV( ) 5 => PV =2500 / (1.0425) 5 PV = 2500/ => PV = $2, Note: PVIF 6

7 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 $ 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 $ PMT $0 7

8 FV $1, I/YR 6.00% =1000/(1+) 5 (1+) 5 = 1000/ (1+) = ( ) 0.2 => = ( ) = = = 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 = = / =8.04 years. Example: suppose two optos are avalable: Opto 1: Bak Oe rate = 12% for a 5 year vestmet. 8

9 Opto 2: Prce = $519.37, = 5, maturty value =$1000 Whch oe? =1000/(1+) 5 (1+) 5 = 1000/ (1+) = (1.925) 0.2 => = (1.925) = = 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

10 PMT PMT PMT PMT PMT 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

11 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, ( ) S ( ord.) $4, $13, ,200( ) 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, 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

12 (1 0.06) $5,864,636 $ PMT 0.06 PMT $1,340, PMT ( ) 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,

13 ( ) S ( due) 4, $13, ( ) 4,200( ) 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 ,000 2,000 2,000 How much moey you must depost a accout today order to be able to make these 13

14 paymets? Accout pay 5% terest compouded aually. A 1 $2,000 1 (1 0.05) $2,000(2.7232) $5, ote: ( PVIFA ) 5%, 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, ( ) $1, $1,200( ) $3,

15 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, ( ) A $1,200 ( ) $1,200( )(1.055) $3, Why PV of auty due s hgher? 15

16 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 $ $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,

17 PV perpetuty PMT $ $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) 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

18 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 $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? $1,000,000 PMT (1 (1 0.03) 1 (1 0.09) 0.03) PMT = $85,

19 Calculator: NOM 9.00% Ital sum1,000,000 Iflato 3.00% Years 20 r = [(1 + NOM )/(1 + P%)] 1 r = % PMT = $85, (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

20 CFs: $0 $75 $225 $0 $300 PV: $0 $71 $199 $0 $235 PV uequlapmt = 0/(1.0625) /(1.0625) /(1.0625) 2 +0/(1.0625) /(1.0625) 4 = = =$ Calculator soluto: 6.25 I/YR 0 CFj 75 CFj 225 CFj 0 CFj 300 CFj Yellow key : NPV Example: What s PV of the followg uequal 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) /(1.12) /(1.12) /(1.12) 4 + 0/(1.12) 5 + (-2000)/(1.12) 6 = = $ Example 3 20

21 Suppose PV of the followg uequal cash flow stream s $5, at 12% terest rate: CF 1 = $1000, CF 2 = $?, CF 3 = $2,000, CF 4 = $2,000, how much s CF 2? 1 PVuequalPMT PVt PMTt t (1 t 1 t 1 ) 1000 x , (1.12) (1.12) (1.12) 2000 (1.12) = x/(1.12) => x/(1.12) 2 = => x =$ = $3000 9) Future Value of a Stream of uequal Paymets ( FV uequalpmt): FV uequalpmt t FVt PMTt (1 ) t1 t1 Example 1; 21

22 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 ) 0(1 0.10) 0(1 0.10) t PMT t (1 ) 1,500(1 0(1 0.10) t 0.10) 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

23 FV1 S ( ord.) (1 0.08) 1, (1.08) 5 $8, FV 2 S ( ord.) (1 0.08) 2, $11, FV (1 0.08) 10 $10, FV = 8, , , =31, OR FV = $1,000(FVIFA 8%,10 )+ $1,000(FVIFA 8%,5 ) + $5,000(FVIF 8%,10 ) = $1,000(( )/.08) + $1,000(( )/.08) + $5,000( ) = $1,000(14.487) + $1,000(5.866) + $5,000(2.1589) = $14,487 + $5,866 + $10, = $31,

24 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, PV = $1000 ad = 5 so or FV = PV (1 + ) => 1, = 1000 (1 + ) 5 (1 +) 5 = 1, = hece or 1 + = (1.610) 1/5 = (1.610) 0.20 = 1.10 = =.10 = 10% 24

25 Suppose you sg a loa cotract that calls for aual paymet of $2, 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 = $ so we use the followg formula: é ê1- A = PMT ê ê ëê 1 (1+ ) ù ú ú ú ûú é ê1-25, 000 = $ ê ê ëê 1 (1+) 25 ù ú ú ú ûú 25

26 Calculator soluto: N 25 PV $25,000 PMT $2, 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

27 FV $1000( ) 4 (4)(6) $1, 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, 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

28 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, $2,293,880 m (12)(35) 1 1 (1,000)(2,293.88) For preset value of auty, the adjusted formula s: 28

29 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 ) (5)(12) the, $50,000 = PMT ( ) so 29

30 PMT $50,000 / moth $ / 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 = 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( ) $1, 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

31 Not at all(smple terest 1,000 x0.06x10=$600 yearly I 1,000(1 ) 1,000 $ sem-aually I 1,000(1 ) 1,000 $ quarterly I 1,000(1 ) 1,000 $ ,000(1 ) 12 mothly 120 I 1,000 $ daly exercse hourly exercse Cotuously 1, ,000= 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( ) $

32 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, = 1,000(1+x) => (1+x) = x = = 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 ( ) % For cotuous compoudg: EAR e om 1 32

33 f om = 10% ad terest s compouded cotuously, the EAR = ( ) = = 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

34 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 = 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

35 $40, (5) (1 0.08) PMT PMT ( ) 0.08 PMT = $10,018.26/ year Amortzato Schedule year paymet terest repaymet of prcpal remag balace 1 10, , , , , , , , , , , , , , , , , , , , ,000 Cost to the borrower: $50, $40,000 =$10,