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lecture 7

lecture 7 -...

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Unformatted text preview: wm———————————_——_———n——--------------——-——————————""""HIIlllllllllllIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII'IIIIIIIIIIIIIIIIIg-I 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 relate-1, 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‘ ﬁat-Irma“? * 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(1-os)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 : Zack-13*; {\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 ail-smaller 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 I ‘ . :3. no 5 P W ili+h5m52, i ’1 i ‘lh '» I ‘ .. vow: rials—J; Woman l _ i1 l U iriiij‘i iii—i 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 one-year 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,B-Fatt 2011 Z" — _ ...
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