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Unformatted text preview: ILnJe_V 9f ongxPaﬁll
 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 aidenave. (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 ULMuIJ’“ 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 ]
PVOfannutW3PM‘[—HM :l ) " 3801;]?
NOTE: This is the lumpsum 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 lumpsum:
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$: (tag1
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  Fa12011 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 IIIIllIllllllllllllllllllIIuu..._____ 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 lumpsum is
equivalent to moving the entire annuity. Example 13: What is the value three years from today of a $50 end—ofmonth payment
received for three years if the annual interest rate is 13%? nMEUNE ,$_j;_____‘__,ﬂ_.._a”
f0 5‘“:
PMT = 50
m = ll
r=. J;
[[M = i?—
N = 1113:?
.v _ ' +_z,__ 3i.
FVofAnnuily=$PMT[aFﬁl#] = 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 Lir 12) r
F “3 Ylgﬁvg Is this where we want the FV? if not, what do we need to do? {U0 ) i4“; ’+ muting elffer 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 Teny 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, lumpsum 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 lumpsum 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 = $00
The PV of a gergetuity formula assumes the ﬁrst payment occurs at the end of the geriod
’ = ‘ 1" (same as an ordinagr annuity).
M = l
Growing Pergetuities
Pyof . PMT 53.1; _ i . . . 3 9
perpenmy: = “1 ‘ ' i What it the a ment recelved Is not constant, but rather grows at a constant rate (g).
HM [I.) MW} py We use the formula for the PV of a growing perpetuity:
PMT 1+! rg
where g is the constant rate of growth that begins at time t. P Vof a growing perpetuity=P V, = Would your answer change if th . ' 311.94 LIVC construst
4%“)? “.341: ($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 EO :15
© R. ha! 9  Richard T, 13555, Babsm University
‘ ’° dT— Bl‘ , n i andFerry 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 tri ————"' 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 ""
remite . 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
_ NwIABFaumu ‘ 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 Teny!) 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 @ 3113mm 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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