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Unformatted text preview: wm———————————_——_———n——————————————""""HIIlllllllllllIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII'IIIIIIIIIIIIIIIIIgI Learjjﬁﬂ Lecture 7 e45
Lumgsum Future Value and present Value In this example, what are the values for the following variables?:
F U I P I c
Deﬁnitions: PV = {00 ‘00 1H I) fillOZ'S
*r’pv: ﬁle'seﬁrvaru‘ér relate1, r =  OS
Ika—y Future Value aw VF M = l
l or r a anm'linte—rest) rate (annual percentage rate or APR) N = ' X133—
M —+ number of compounding periods per year; the frequency of compounding
r") " I 7      —m~ Mamet 1"  (3' f  bl I f d th ﬁfth
Rigor rlM r» intrapenod, or periodic interest rate a I‘ ﬁatIrma“? * I I ﬂy W
N a total number ofﬁeriods] (M x number of years) % qu“ a “5 ’3'} ‘ I Example 1: You put $100 into the bank today at 8% annual interest. What is it worth in H I J" 6 years? TlMELINE: If._._t_ﬂr4—a: *9
‘Jx‘J‘t'f‘rT N034 ‘ 3N I
. _—, IOO we
; Future Value PV
! I I r = .
l I If we put $100 In the bank today and earn Interest of 5% per year (annual compounding). 08
' what do we have after two years? I M = i
TIMELINE: N = M l ‘lo
lac _.—+ “30L! 5'5) ——— 2' {OS tn? tuna: I la , ﬂ
FV= va [1 + rlM1N= :ook Hedi} : I. see First year: $100 x (1 + .05) = $1 x 1.05 = $105, or our original $100 Plus $5 interest. Second year: $105 x (1 + .05): $105 x 1.05 = 311025 What if we havewcompounding? :I? II
or, TIMELINE: I Mfg ;
.0 ‘
$100 x (1 + .05) x (1 + .05) = $100 x (1.05)2 = 511025
PV = :00
Fv~=2..=5%{$100}= $100 x(1os)2= $110.25, r= m
or’ M T. : N = U \{9 :ﬁq_
FV PV x [1 + rlM] This IS the Future Value Formula (FVF) N ' IN _ _ I 0 CE, I 2& III
FV= PVX[1+rlM = goo a :7 W where r is the interest rate per year and N is the total .
between the PV and W (in this caso Years). number Of Compound'”9 peliOdS in this example, each compounding period is 3 months, and there are 24 of them over
the 6 year period. 6  Richard T. Bliss, Babson University
I I and Terry D. Nixon, Miami Univexsity
9  Richard T. Bliss. Babson Univasru I
and Terry D. Nixon, Mimi University PM 301 AB — Fall 201‘! 44 l 45 FIN 301 AIB _ Fan 2011 ﬁg Ear«me v ' a i Lecture? 1:? A ‘m‘aﬂ e47
Example 2: You have $300 to invest for four years. Key Bank is offering 5% 3 Example 3: What is the value today of $3,000 received in two years? The interest rate
comeUnded annuaﬂyi while the Fifth—Third’s rate is 4.9% APR. bUt With daily i '3 6% compounded annually 0 ‘ "—77 W 7 iii
compounding. Where do you put your money? TIMELINE: I .mﬁ‘, A,“ 3ND I ‘93 I
KeyBank PV= a.” r: L; M=‘= rIM=f N= $4; FV: 3mg
. L" r =  0b
TIMELtNE: “H """mo'm‘F‘ M = i
N = In i =7.
Fv=var1+r/M}“= gipgatzgcyt W
~ I II .1 : ' 3 Q ,4 n
PV= FV +{1 + rIM]N = 0:, I : Zack13*;
{\1 + . .  ~ .0433
———_F'ﬁh‘Th"d PV = r = . M = 10g? TIM = N = 7m? Example 4: How much must you invest in the bank today if you want to have $1,500 ‘
o L «it after 4 years? Interest is 11% compounded annually; TIMELINE: Engmoi; ' TIMELINE: c—Wc—fﬁ ‘ 3W 7 ism
Fv=vat1+rrM]”= gmw Wot 19W? I" FV= E7300
3 “'3 3 131“? J? N= 4177\i2’wr PV=FV +[1+rIM]N= .— What if the bank pays 11% APR compounded monthly? 8 l U L's’f
e urvaient vaiue of cash . we are °°king for the L—ui—s —r4~—*"
T: ff d' thﬂoi: at a date mono the" recetpt) TIMELIN E1 ? fix
e process 0 in mg 9 of future ca . _ _ t
the PU formula from above: Sh ﬂows '5 called d’scoum‘w We begin With _
FV = 5"“
= N = i
FV PV x [1 + rfM] now rearrange this to solve for PV [VI _ is
W = + ~ . . N = am at
FV [1 + rlM] This IS the Present Value Formula (pVF)
f («r r ,2; PV=FV +[1+r/M}“= ism 1T «57 a ’ ~
‘7—
©  Richard T. Btiss, Babson University
and Terry D. Nixon, Miami University
©  Richard T. Bliss, Babson Unitas“)t FW 301 A 8 F31! 2011 “new 0. Nixon. Miamiummiw l 45  ' 47 FIN 301 AB — Fair 2011
:.‘.‘ _ _ a e48
Lecture? ﬂ , /’ /, Note that more frgguent comgounding means ailsmaller investment Edgy. In other words our money grows faster. Why? 7 El Practice: How much would you have to invest today if the bank paid 11% compounded daily? 955;}; §glving Emblems Where N or r an: unknown. Example 5: You have $6,000 today, but need $9,000 for the down payment on a new
sailboat. If you put the money in a bank that pays 12.5% compounded annually, how long before you are able to make the purchase? N TIMELINE: F"‘“"J');—_'wu—‘_—_H“‘*—~am]
but 90%;?
PV = to?
FV = ﬁ‘
r = '.
M :
r/M = ' «r l
N : can plug in the variables and solve for 
“Mfg”; MW OK
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P W ili+h5m52,
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l _ i1 l
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1 Fix “ _ i. 'P(' F14 { x
FIN 301A,B ~Fali2011 f .l" i.‘ h“ e  Richard r Bliss, Babson University
and Terry D. Nixon, Miami Univewu MW; EMMA fairway;le What if the interest is 12.5% APR compounded monthlﬂ Will the time be shorter or
longer? . .iu
PV=b000 Fv=9iooo raps: M=i2. rrM=l;_ N: WEE] have
MW : [n LESS)
.I‘LS ‘ 33'11? P243045 a: 3,1571%
[HU+“T£) 7t Mau’ci'lﬁ What if you want to buy e boat in 34/2 years? How much must the bank pay with
annual compounding 7 ‘ to make this possible? PV= 600° TIMELINE:
FV: 6000
r:
M: l
N .2: _ x, = _ 3 75 is P“: v
Now we need to rearrange to solve for r. dig}
FV "N _" £92.? [/35 ,1 r "L Fir
"{lwi i in U475 lug ‘ ﬂirt
w '(w 7)
Note that this gives us the annual interest rate. L _ Elly/n I
M ' PU “
’ Fv '
him) l M l, t :n \f
i ‘41 :5; A: 1,)
Y3 I1 \' :'“'.'_ i . _ g [HERN
 ll 1‘3 D  Richard T. Bliss, Baboon University
and Terry D. Nixon, Miami Univmity 49 FIN 301 A,B — Fall 2011 _— .1 Lecture? a; a 51
APR mncLEA a Example 8: You are comparing oneyear investment options. Bank A pays 5.15% APR
iﬁ—m “ compounded monthly, Bank 8, 5.2% APR compounded quarterly.
 ' ‘ 9
APR stands for Annual Percentage Rate which is often the interest rate quoted by Whig '3 meg???“ 93C“ Option
lenders. (5 5“ .11!
What is the EAR of each Option? P U‘l‘ Hoff )m—l 1 5:917:3371 @: rate per period x number of compounding periods per year = rlM X M =© Note that APR is the same as rin our prior problems. .032 \1
‘ (H I). J—l : Joggzgfgﬂ
To make comparisonsrwhen we have different interest rates or compounding frequency. Wh'Ch do you prefer'
we use EAR or theﬁfﬁectﬁW lfhtei;
17> is W Eh!
EAR tells us how much actual cash interest is aid or received. This is what we reall
M QOTE: Once we compute EARS! we have a valid basis fgr comparisonE ' : 2 b t‘ EARS for the two banks.
EAR = (1 + r“mu _ 1 1 El Pragjlce Answer Example ycompu Ing keg S? S ‘“ o .
P APR for annual compounding? [3 3' 2‘!
BAR . may? at as t, M Example 6: The quoted interest rate on your credit ' 
compounding. What is the eﬁ‘ective rate? card IS “25% (APR) With montth Practice: What i s the EAR if Sam offers 12% APR With monthly compounding?
1 ‘1. 10$, e  Richard T. Bliss, Babson umva
. and Tiny D. Nixon, Miami University
©  Richard ‘1‘. Bliss, Babson Univﬂsf'l'
and Terry D, Nixon, Miami Univeﬁﬁl' _
FlN 301A,B  Fall 2011 50 51 FIN 301 A,BFatt 2011
Z" — _ ...
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 Annual Percentage Rate, Terry D. Nixon, Richard T. Bliss

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