Lecture 8 - ILnJe_V 9f ongx-Pafill - Annuities 0n dim...

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Unformatted text preview: ILnJe_V 9f ongx-Pafill - Annuities 0n dim 90.31 (“F JAM? Gains on” gquhulf - Perpetuities - Amortizing E‘Safismttl AWL F35 Brew 0.“ EMMA“? Wyiim-‘m Annuities Question: Why can’t we just worry about lump sum payments? grrkh problem solving) 1. Capital budgeting decisions 2. Mortgages 3. Planning for retirement a 0% annuity“:5 M1159, (W! at Pm: A f annuity due . I . gs at it a. w M We want to know how'to compute the P a whim aide-nave. (3P shew» [Tecture 8 7 ' —__J_____1_page 52 W loch nd FV of any annuity. Why? We are takin a series of cashflorvs the a ' nn 1 - . lum sum cash flow at the b inmn P orlénd and convertin them to an e trivalent of the a ment stream. To do this, we need to know the following: o PMT ULMuI-J’“ E’GCL naming Farmer _ Na: ,- number (it Petier mutt) a“ * - note that these definitions ' FVs and PVs. FIN 301 AB - Fall 2011 52 ll .. ‘9 — Richard T. Bliss, Babsm University and Tmy D. Nixon, Miami Univctsily Lecture 8 _ 1 page 53 [ WW _ N i ' i H r,l 1" PV of ordinary annuiijli=$PMT .1 PMT r/M This is the Present Value Annuity Formula (PVAF) We use the PVAF to compute the lump—sum equivalent at the b om the first cash flow, of the stream of annuity payments. eginning, i.e., TIMELINE: j____,___._.J__——s——~—*—; Example 9: You expect to receive $1 ,000 per year at the end of each of the next five years. The market interest rate is 12%. What is the PV gt these payments today? TIMELINE: 0W lit UL lic PMT=lmo 0 “c H; M: ' r= .11. N = I x555 I l- m _ _ 1—1/(1+rrM)” 2 mi 1 ] PVOfannutW-3PM‘[—HM :l ) "- 3801;]? NOTE: This is the lump-sum eguivalent (at t=0) to the five $1000 payments. We can get the exact same answerb discountin each of the 1 000 a ments individuall and Pv {5 payments of $1.000} = 100011.12 +1000n.122 + 100011.123 +1000r1.12‘ + 100011.125: 3‘00Ltfi £9 - Richard T. Bliss, Babson University and Terry D. Nixon. Miami University 53 FIN 391A,B - Fall 2011 ———--———— ---—-fi'—— "*2 . 7' A --‘. - in {- ,_ a I . . ‘ [EH 7' >1 ‘f‘i; {11016 ’ inflf \ Hfl't ,‘n i-: ‘ . .-L. t . 7 r; _ t .1. p- .1 ’ v- 7 u A: {-3.41 (4‘ r 4ft, .5:- P’y 3} Dirt! Lure a Example 10: Your roommate asks for a loan, offering to make weekly payments of $25 Example 11: You win the Ohio Lettery and get to choose between receiving $950,000 for one year. If the appropriate interest rate is 8.25% APR, compounded weekly. how today, or $100,000 at the end of each of the next 25 years ($2.5 miliion in totai much would you be willing to lend her today? Assume she wili make her first payment of payments). If the discount rate is 9.75%, which is the better option? $25 to you in one week. . v: m SIN TIMELINE: WW”! TiMELINE: im ago is m 15 - v ‘ s - - 1E 25 fl! of lump-sum: PMT z Ho} 950.600 M = 3'; PV of $100,000 payments: r = .0923 - PMT = (00,0!30 N = i x: 2151 l M = I _ l—I/(l+rlM}N I“ . 3L5 PVo annm =$P m = H -°-:-- f [y MT[ r/M ] w[ K 1“)“ Jztltrio 55’] r: 0‘175 £211) ' ST: N = l x1515 l . lvll(l+r/M)“' : ‘°°'°°° " W P V of annmty—3PMT': r [M J l E Practice: What amount would you lend if the payments are for three years? : 617$: (ta-g1 l 15 was; - sits g1) l .Lt‘ ‘ 3 WWI— M ” —1 FVofiénnuiMPdE%fj—] O bf ’81 KY r This is the Future Value of an Annuity Formma (FVAF) 9 _ e - Richard T. Bl'm, Babson University - Richard '1'. Elm. Babson University and Terry D. Nixm, Miami University FIN 301A,B - Fa|12011 54 FIN 301A,B - Fall 2011 i i Lectures—W _ —1::::@ 'nMEUNE: o~._h——b——~———+—r4#‘”* {a [o to i0 1 Example 12: What is the stream of five $1,000 payments from Example 9 worth at the end of the fifth year? 'HMEUNE: ' L i t 5 PMT: iOoO = l r= .11 N= Ix§=s N _ h :11 5 FVofAnnuiry=$PMT[(.H_riflf#] ___ [000 Li+ ) _1 '3 r 1 NOTE : We can at the same answer b movin each of th of ear five indeua" - e 1000 a ments to the end FV {5 payments of $1,000} = 1000 (1.12)‘ +1000(1_12)3 +1ooo(1_12)2 + moo 7 (142)+1ooo= (0351 .Eyg or by taking the result from Exampte 9 and moving it into the future as a [ump_sum FV= PV x{1+ r/M]N = 3wqau +3}_,?__B3§2__%S 0 § m _+H‘___+ 30M 3 © - Richard T. Bliss, Babson Univcfiity and Terry D. Nixon, Miami Universit)l FIN 301 AB - Fail 2011 55 IIIIllIllllllllllllllllll-II-uu..._____ I LmlureB—afi‘ my? *3 PLEASE, PLEASE know the foltowingi: There are three critical rules to remember in all annuity calculations: 1. When we compute the PV of an annuity, we find the equivalent lump sum payment one period before the first annuity cashflow. 2. When we compute the FV of an annuity, we find the equivalent lump sum payment at the point of the last annuity oashflow. 3. Once an annuity stream is converted to a lump—sum, moving that lump-sum is equivalent to moving the entire annuity. Example 13: What is the value three years from today of a $50 end—of-month payment received for three years if the annual interest rate is 13%? nMEUNE ,$_j;_____‘__,fl_.._a” f0 5‘“: PMT = 50 m = ll r=. J; [[M = i?— N = 1113:? .v _ ' +-_|z,__ 3i. FVofAnnuily=$PMT[a-Ffil#] = 50 : In} "[17 hi) Q: What can we do if the first annuity payment is not one period after we want the PV? © - Richard T. Bliss, Babson Univmity and Terry D. Nixon, Miami University 57 FM 301 AB - Fatt 2011 ,r' ¥ JESSE 55 Lecture 8 j _ _ ‘w Example 15: You make 24 monthly payments of $200 (beginning at the end of this month) into a savings account with 10% annual interest compounded monthly. What is the value of these payments 4 months after the last payment? Assume the payments o t l n are reinvested every month til this time, TIMELINE: W M 2“. 2g X [150 [m TIMELINE: i—F————-—»—*—-——-——-—-t PMT= \‘LSO m 260 7 = r PMT: ZOO r= oh}; : P— N: “([0er r: [91 r ILS‘O I‘- " . to "'M : '[olll PVofannuigzgpm[M = X {H.953} J HM (n EEK/l) N: “filzm _ .fLQ lg. N — -‘_ _ . FVofAnnuibzz$Pm M z 10 O Li-r 12) r F “3 Ylgfivg Is this where we want the FV? if not, what do we need to do? {U0 ) i4“; -’+ muting elf-fer He up“ ,9th 29L 1? l—__f_..r-{——~+___j 3369.38" .Io 61301.38(H’ “Ti— : Stu;an - I © - Richard T. Bliss, Babson University © ‘(liziTChard TI Bliss: university and Terry D. Nixon, Miami University an - .5 m? , , c") 0- NIXOI'I, Miamr U am FIN 301 A,B—Fait 2011 m t! 59 FIN 301A.B « Fat12011 mm~_ __ "I — A you do? us today if the Interest rate is 17%, what should PV of 0 tion 1 TlMELlNE Wmfl ‘ 3’0 S70 80 Some [‘ " (H 4,1)!- C9} 0‘ m Tam _ N “'— W: = $000 (‘4' {:1 )2 } f/M 47%: 157,997)? E a) How much would you have to put in the bank today (lump—sum) to be able to finance this dream vacation? fl 0 1. $7,347.27. Suppose that instead of a lump—sum, you wish to make 60 monthly payments (beginning one month from today) to fund the trip. How much would each monthly payment have to be? rm gain monthly hat the interest rate is 6.3%, compounded - ml C}: e . st . -. '1- t H TlMELlNE. ~w Annuity Due ‘5 s- ' . . Y FM” 2 gmg Recall that an annuity due is an annuity where the cash flows occur at the beginning of : FL each period. r = . s _ “5 Example 18: You expect to receive five annual payments of $1,000 startth today. The r/M — (MS {H1 market interest rate is 12%. What is the PV of these payments today? 1 NE. 0 l 1 "S 4 C N— “<qu TlMELl . W ® (in 1% it“ (y) a? '7 © Richard I. 33.53, 331m, U (Q ~ Richard T. Bliss. Bahson University FIN 301 A3 _ F8“ 2011 and Ten-y D Nixon. Miami and Terry D. Nixon, Miami University 60 _t k" 51 FIN 301 AB - Fall 2011 — 4 m, i o u o ' ll The eaSIest way to value an annunty due is to View it as two pieces: What is the PV of the second option? TIMELINE: L-—-—‘--——*-———-—-__i l j G ' _ gm W0 1% W37 9i 9k - — ~ - Slg an ordinary four payment, $1,000 annwty starting one year from today. and if?) I ' (2) a Single, lump-sum payment of $1,000 today. PV of four payment ordinary annuity: is: » i7 . I—l/(Hr/M)” I 7‘ ~~ - PVOfanan=$PAJT 7““_fl ._. l-. i V . ‘ z r/M [Owl me)" Which do you select? _ t M - I _1 - . “1 1.1.3.54“ .LFm‘rf.: ’ ‘7'“ I L :7, a, 1. _.:r'f : 303—135 ‘ 4 0 i 1 3 '44 _ ' W Perpetmties PV of lump-sum today: “k w. l I . 1 i . _ «——.——- e etwt — mm “FIG l [00% 15+ up] ‘ “wt—Ill], p no y an M at 90% 10‘] 641W How can we compute the PV of an infinite stream of payments? — ll / M N P V0fannui¢=$P1W[wr—)—:| r/M What happens to this calculation as the number of payments increases, i.e., as N —> or, ? we going to compare the two choices? ODTKNE th CE) 5.- sfmi We. on Lil‘djfit ' _ ‘ , or the annuaty payment dwided by the periodic interest rate. What IS the PV of the first option? TiMELlNE- PVOfPerPE’“‘Uf", , M . % “fl i 7 ‘ 1 £99233: : Llwflr‘fi .50 “m In 0 {- © - Richard T. Bliss, Babson University Clde T_ Bliss, 3mm University and Terry D. Nixon. Miami Univcrsity 0- i ' ' 11‘ i FIN 301 AB — Fall 2011 “"1’ “mm Mmm U avers ry 63 FIN 301 AB — Fall 2011 Lecture 3 i _ 7 “gt—*— a 86* Lemma r I Example 20: What would you be willing to payt d f '3‘! Ma 5 . . o ay or an infinite stream of - . . . _ ? MW»; 3 00 payments beginning one year from tmay if the interest rate is 12%? annual What if the first payment Is received in nine months. m we tg 2 290.. 3,, 2... TIMELINE: PsFfl—RHH h ] . L H q ) —7‘Ess*li.% P '\_sw m .4 .0 " MT = $00 The PV of a gergetuity formula assumes the first payment occurs at the end of the geriod ’ = -‘ 1" (same as an ordinagr annuity). M = l Growing Pergetuities Pyof . PMT 53.1; _ i . . . 3 9 perpenmy: = “1 ‘ ' i What it the a ment recelved Is not constant, but rather grows at a constant rate (g). HM [-I.) MW} py We use the formula for the PV of a growing perpetuity: PMT 1+! r-g where g is the constant rate of growth that begins at time t. P Vof a growing perpetuity=P V, = Would your answer change if th . ' 3-11.94 LIVC cons-trust 4%“)? “.341: ($th e first payment occurs today? {. ,5 r I” r fitmth Lg ) begin. a w a .11 _ _ grid w {t- if y L H- T) 4%“ "m NOTE: We wili onl consider annual a merits and rowth rates for a rowm l gerpetuiy. El Practice- What is the PV - ' . tOdat‘llofanl f ' . . l_1ht—:hinterest rate is 12% APR compoundneggiasrttream 0f $250 quafiedy payments if Example 21: You are considering the purchase of a British consul bond which Just n ree months? ‘ erly and the first payment is received Yesterday paid interest of $75. The next interest payment Will be in one year. The \ 'fiferest payments grow at a constant rate of 7%. and the appropriate interest (discount) 155' a ’ rate is 10%. What would you pay for this bond today, i.e., what is the PV of the future ‘_ ""“ ‘= (BEER 7 vi} cash flows. U! 1 3 If]: 1 _‘_ ,, W TIMELlNE: I.._.——-——F——§ j 186711;” hat if the first pa ment is ' 1 Lu J .15 x V received today? : Sol: 1 es“? #4 53—53 3 +750 123%} 1,1. What will be the amount of the next interest payment? [if m i' i 5 - 1 ’1 it — .— F " .— ‘- git? ‘ q u " XJ g z E-O :15 © R. ha! 9 - Richard T, 13555, Babsm University ‘ ’° dT— Bl‘ , n- i and-Ferry o. Nixon, Miami University FIN 301 AB - Fall 201 1 and T”? D— 3:115:33 64 ' 55 FM 301 AB - Fat? 2011 PK:‘MT MI I Lecture 3 K m 7 77 "-275.99 I57 I r‘jgf , or in this case, 35% : 9° u- rmg .f ._ : 2t 5" a ‘ . _ _ 0*"7 7 i“ We «(Wm Practice: MBG Enterprises paid a dividend yesterday of $1.00. Next year’s dividend Note how this compa is expected to be $1.10 and the growth rate will be constant foreveri If the required PV res to the P V 01‘ a can t .1 y, .. rate oi return forMBG Enterprises is 17%. what should it's price per share be today? 0 = PMTI if = an; S ant$80,25 pe t . a . r i' ' rot i to; 2 T = (‘02 so rpe Ulty: Ll tr-i ————"' 2:57” Cle art __T_ = m (F '0 Example . 2 : dlvidend' Tie You have be $2.50, Afle Campany will p m. Example 23: Presley Enterprises, run by the former Mrs. Michael Jackson, pays a ' constant annual dividend of $2.00. However, a psychic tells Lisa Marie that in four years, (discount . hat current] ‘ ) rate '8 20%, what is {2: 1"ng by 5%“:“7‘f r years from {3:063 “Dim? a her father will be discovered working in a 7—11 outside of Detroit. This wilt greatly Me of this tregmrgéea forever tfiifiai’an? It Wm increase sales and starting in year 5. the dividend will grow at an annual rate of 8%. 'r res - i L 3 g p Yments 9 \I I “m- . . . if the appropriate discount rate Is 13%, what IS the current price per share of Presley Damn}??? '- Enterprises before the psychic's prediction? What 13 the first a} W 99%.“ starts payment of {h 9C 03 TIMELINE: a Z F’Vt‘L : “LE; ) tam grOWth pemetmm ‘Z‘iSJ ~09 =S7.So 2- 3 “7‘00 Where are we 0 #28 : f5 3 3/ n the t‘mfi‘hne ft . L") - \l r (7‘ er “1‘3 Calcula I Assuming that the psychic sails out to the Star, what will the current price he once the We need p ' V0 , 3 ha 0 we must make the fol: information hits the newsstand? (Assume that investors believe the psychic.) “m ° 0W: f at i 1 v - 2 t 9 Calculat' . TIMELINE: "i? , t +‘ ‘T Ion. ’ I 1 3 $— - {I -——7c>-D 9 I ‘ V .1 S A i "J "" remit-e -. abuse“, :__ 5 ‘- C‘ 'f' ‘z g a .- jg 1' “I, F: 1' nu, : 2"“0 -——A -7 _L__T_l_ J ‘ i ‘ ‘r ‘ ‘lu'i ' a A 5% £0 1 " _..-'—-—-. ~.. -"'-‘~ __ , ,_,_;...._.— l-— ' - 7 i ‘ ‘ x—‘H—H . + ’73—“ l ~ J - 511‘“ 2189i; Utter?“ [‘"i i j J i 3 ' v 1 J 0 “9 no ‘3 I 4 ‘55 W {—1 ,_.1 Ft . aid- ', 4 _ NwIAB-Faumu ‘ L 3 ‘+ 5‘ 6; in U1» “Hm "06 1 W. © ‘Riduad I Bliss B _ _ L03 _ . ‘ $7 1’ © - Richard T. BliSS, University . FIN 301 AB - Fal12011 " " "’-- ; and Ten-y!) Nita; Mmmi m : ,_ and Terry D. N1xon,MraanmvcrsIty 4P “W 6?- Fui was a Falt 2011 — ‘_ 7‘ fl sa @3——— WW I Time Value of Money - Part lll i l l How can we compute the amount of interest and principal for each loan payment? A RTLZATION TABLE -_ equal Payments overi ' - Principal Reduction Balance (C) - (d) (b) - (e) ’gwqw x .07 -: 2w; Points to note: “will x -07 :Il‘rtg M = i 1. The cash paid out each year is the same. 2. The Ending Balance after the last payment is $0. N = W5 :6 3. The total Principal Reduction is $5,000, the original amount borrowed. 4. The Total Payments is the same as the total interest paid plus the total principal repaid. Computing the Outstanding Balance on an Amortizing Loan How can we determine the balance on the loan at any point in time? Pfiflz—L‘ "‘m _ ‘ ‘+ Li/(I+r/M)"" ‘ W 401 airflow: a mum able r/M i“ T673 .1b x 0.0'] ach of the t corn ute the amount of each a Zfiffgnnual a merits Th“ 1: In addition, we can use the following short—cut: r 5“" amorfi - ' '5 Omtula ca 2m man. 11 be used to FIN 301 AB — Fall 2011 @ 31-13mm T. alias. Babsoa University TC"? D. Nixon, Miami Universin © — Richard T. Btiss, Eabson University and Tm D. Nixon. Miami University 59 FIN 391 AB — Falt 2011 ...
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Lecture 8 - ILnJe_V 9f ongx-Pafill - Annuities 0n dim...

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