I = Prt. = P(1+i) n. A = Pe rt


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2 Chapte 6 Matheatcs of Fnance We wll look at the atheatcs of fnance. 6.1 Sple and Copound Inteest We wll look at two ways nteest calculated on oney. If pncpal pesent value) aount P nvested at nteest ate pe yea ove te t, sple nteest, I, s and total accuulated aount, A, s I = Pt A = P +I = P +Pt = P1+t). If s nteest peods pe yea, and n = t s total nube of nteest peods, total accuulated aount assung copound nteest s A = P 1+ ) t = P1+) n. If nteest ate copounded contnuously, total accuulated aount afte t yeas whee e = A = Pe t Execse 6.1 Sple and Copound Inteest) 1. Sple Inteest: A = P +Pt. a) If $ s nvested at 11% sple nteest, calculate ts value afte 8 yeas. A = P +Pt = +0.11)8) = 1116 / 1216 /
3 116 Chapte 6. Matheatcs of Fnance LECTURE NOTES 7) b) If $221 s nvested at % sple nteest, ts value afte 2.5 yeas s A = P +Pt = )2.5) = / 76.2 / 86.2 c) If $5 s nvested at 5% sple nteest, ts value afte 13.1 yeas s A = P +Pt = )13.1) = 3.8 / 7.3 / Copound Inteest: A = P 1+ ) t = P1+) n. a) If $321 s nvested at 2.5% nteest copounded quately, calculate ts value afte 7 yeas. A = P ) t ) 7) 1+ = = / / Calculato: /) 7) b) If $113 s nvested at 2.5% nteest copounded onthly, calculate ts value afte 3.7 yeas. A = ) t ) 12) = = / / Calculato: /12) ) c) If $121 s nvested at 3% annual nteest copounded daly assue 365 days pe yea), calculate ts value afte yeas. A = ) t ) 365) 1+ = = / / Calculato: /365) 365 ) d) If $ s nvested at 11% nteest copounded yealy o annually), calculate ts value afte 8 yeas. A = ) t ) 18) 1+ = = / / e) If $ s nvested at 11% nteest copounded onthly, calculate ts value afte 8 yeas. A = ) t ) 12)8 1+ = = / / Copound Inteest Contnuously): A = Pe t a) If $2000 s nvested at 7% nteest copounded contnuously, calculate ts value afte 3 yeas. A = Pe t = 2000e 0.073) = / / Calculato: 2000e ) b) If $00 s nvested at 6.5% nteest copounded contnuously, calculate ts value afte 3.5 yeas. A = Pe t = 00e ) = / / c) If $ s nvested at 11% nteest copounded contnuously, calculate ts value afte 8 yeas. A = Pe t = e 0.118) = / /
4 Secton 1. Sple and Copound Inteest LECTURE NOTES 7) 117 d) An aount $ nvested at 11% sple nteest $1316) s lesse / geate than $ nvested at 11% nteest copounded annually $ ) lesse / geate than $ nvested at 11% nteest copounded onthly $ ) lesse / geate than $ nvested at 11% nteest copounded contnuously $ ) afte 8 yeas.. Related questons. a) Inteest ate,?. If A =, P =, t = 10 yeas, nteest copounded yealy Snce A = P ) t, ) 110) 1+ then = 1+ 1 o 1+) 10 = o takng tenth oot of both sdes, 1+ = ) 1/10 ) 1/10 o = 1 0. / 0.39 / 0.7. Calculato: /) 0.1) 1. If A =, P =, t = 10 yeas, nteest copounded onthly Snce A = P ) t, ) 1210) ) = o = o takng 120th oot of both sdes, 1+ = ) 1/120 ) ) 1/ o = / 0.39 / 0.7. Calculato: 12 /) 1/120) 1) b) Nube of nteest peods, n = t?. If A =, P =, = nteest copounded yealy Snce A = P ) t, ) t 1+ = 1+ 1 o 1+) t = o takng natual logs of both sdes, ln1+) t = ln o tln1+) = ln o n = t = ln ln1.08 Calculato: ln/)/ ln1.08) 8 / 50 / 52.. If A =, P =, = nteest copounded onthly Snce A = P ) t, ) t ) t 1+ = o = o takng natual logs of both sdes, ln ) t = ln o ) 12tln = ln o n = 12t = ln ln1+ Calculato: ln/)/ ln1 + /12) c) Pncpal, P? 563 / 578 / If A =, t = 5 yeas, = nteest copounded yealy Snce A = P ) t, ) ) 1+ = P 1+ 1
5 118 Chapte 6. Matheatcs of Fnance LECTURE NOTES 7) o P = 1+) / / Calculato: ). If A =, t = 5 yeas, = nteest copounded onthly Snce A = P ) t, ) 125) 1+ = P o P = ) / / Calculato: 1+/12) 60) d) Othe. Two hunded dollas $200) s deposted onthly nto account payng 6.25% copounded onthly. Afte 3 yeas, accuulated aount s put nto 2yea cetfcate whch pays 8% copounded quately. Detene fnal accuulated aount. A = ) t ) 123) 1+ = / Calculato: /12) 36) A = 1+ ) t ) 2) = / Calculato: /) 8) 5. Usng the TI 83 Calculato: Copound Inteest. Detene the futue value of $ whch s nvested at 11% nteest whch s copounded onthly afte 8.3 yeas. Pess APPS ENTER FINANCE ENTER TVM Solve ENTER. Set the TVM Solve paaetes as N = 8.3, I% = 11, PV =, PMT = 0, FV = 0, P/Y = 1, C/Y = 12. Aow back to FV and then pess ALPHA ENTER. The answe FV = appeas. N stands fo nube of yeas. I% s the yealy nteest ate. PV stands fo pesent value and s typed n as a negatve nube because t s consdeed as an outflow of cash. PMT s the payent aount, whch, n ths case, does not apply and so s set to zeo. FV s futue value and s the vaable we ae tyng to detene n ths queston. P/Y s the nube of payent peods pe yea, whch, n ths case, does not apply and so s set to one. C/Y s the nube of copoundng peods pe yea. 6.2 Odnay Annutes We wll look at annutes a sequence of payents ade at egula te ntevals); oe specfcally, odnay annutes annuty whee nteest on payents copounded at sae te payent ade). If pncpal pesent value) aount P nvested at nteest ate pe yea ove te t, s nteest peods pe yea, and n = t s total nube of nteest peods, futue value of an odnay annuty, ) t 1+ [ 1 1+) n ] 1 A = p = p payents to a snkng fund, [ ] ) p = A ) t = A ) n 1
6 Secton 2. Odnay Annutes LECTURE NOTES 7) 119 odnay annuty foula, A = P 1+ ) ) t t +p 1+ 1 pesent value of an odnay annuty, 1 1+ P = p ) t = P1+) n +p = p [ ] 1 1+) n. [ 1+) n ] 1 Execse 6.2 Odnay Annutes) 1. Futue value of an annuty: A = p [ ] 1+ ) t 1 = p [ ] 1+) n 1 a) Futue value of 5 yea te annuty, $100 pad each quate, eanng nteest [ at 8.5% annually, ] copounded [ quately, ] s 1+ A = p ) t ) 1 = / Calculato: /) 20) 1)/5/) b) Futue value of 3 yea te annuty, $120 pad each onth, eanng nteest at 9.5% [ annually, ] copounded [ onthly, ] s 1+ A = p ) t ) 1 = / Calculato: /12) 36) 1)/0.095/12) c) Futue value of 3.2 yea te annuty, $105 pad each day, eanng nteest at 6.5% [ annually, ] copounded [ daly 365 days), ] s 1+ A = p ) t ) ) 1 = ,313.0 / 136, Calculato: /365) ) 1)/0.065/365) 2. Payents to snkng fund: p = A [ ) 1+ ) t 1 ] = A [ ] 1+) n 1 a) Lab of coputes eplaced n 3 yeas te fo antcpated futue) cost of $25,000 whee $25,000 accuulated ove 3 yea peod though equal nstallents ade at end of each onth. If yealy nteest ate s 8.5%, sze of[ each nstallent ] s [ ] p = A ) = / / ) t ) 1 Calculato: /12)/1 + 5/12) 36) 1)
7 120 Chapte 6. Matheatcs of Fnance LECTURE NOTES 7) b) Quately annuty equed futue) snkng fund of $30,000, needed afte 5 yeas, [ f yealy nteest ] ate [ s 7.5%, s ] p = A ) = / ) t ) 1 Calculato: /)/ /) 20) 1) 3. Odnay annuty foula: A = P 1+ ) t +p [ 1+ ) t 1 a) If $ s nvested now at 11% nteest copounded quately and also $120 s added each quate, calculate value of nvestent afte 8 yeas. A = [ ] ) t p ) t 1 = [ ] ) 8) ) / Calculato: /) 32) /) 32) 1)/0.11/) b) If $600 s nvested now at 1% nteest copounded seannually and also $100 s added evey sx onths, calculate value of nvestent afte 5 yeas. A = [ ] ) t p ) t 1 = 600 [ ] ) 25) ) 23) / Calculato: /2) 10) /2) 10) 1)/0.01/2) [ ] 1 1+ ) t. Pesent value of an annuty: P = p ] = p [ ] 1 1+) n a) Pesent value of 5 yea te annuty, $100 pad each quate, eanng 8.5% yealy[ nteest, copounded ] [ quately, s ] 1 1+ P = p ) t 1 1+ = ) / Calculato: /) 20))/5/) b) Pesent value of 3 yea te annuty, $120 pad onthly, eanng 9.5% yealy[ nteest, copounded ] [ onthly, s ] 1 1+ P = p ) t 1 1+ = ) / Calculato: /12) 36))/0.095/12) c) Pesent value of 7 yea te annuty, $97 pad onthly, eanng 9.5% yealy nteest, [ copounded ] daly [ 365 days), s ] 1 1+ P = p ) t 1 1+ = ) 3657) ,006 / 181, Calculato: /365) 365 7))/0.095/365)
8 Secton 3. Consue Loans and APR LECTURE NOTES 7) Consue Loans and APR Annual pecentage ate APR) o effectve nteest ate allows consues to copae dffeent nteest ates. Assung pncpal pesent value) aount s P, nteest ate s pe yea, s nteest peods pe yea, APR s APR = 1+ ) 1 = 1+) 1 aotzaton, aount of payents to ete a load, [ ] ) p = P 1 ) t = P ) n aotzaton, nube of payents to ete a load, ) p ) ln p P n = ) ln ) = ln p p P 1+ ln1+) Aotzaton table o aotzaton schedule s also dscussed. Execse 6.3 Consue Loans and APR) 1. APR = 1+ ) 1 = 1+) 1. Whch s lage: 10% copounded onthly o 10.2% copounded quately? a) Afte 1 yea, $1 nvested 10% copounded onthly, A = P ) t ) 121) 1+ = / / Calculato: /12) 12) so nteest eaned n one yea s ths aount subtact $1, APR = 1+ ) / / o 10.7% Calculato: /12) 12) 1 b) Afte 1 yea, $1 nvested 10.2% copounded quately, A = P ) t ) 1) 1+ = / / Calculato: /) ) so nteest eaned n one yea, APR = 1+ ) 1 = / / o 10.60% Calculato: /) ) 1 c) Consequently, 10% copounded onthly APR: 10.7%) s less / oe than 10.2% copounded quately APR: 10.60%). [ ] 2. Aotzaton, aount of payents to ete a loan: p = P ) 1 1+ ) t
9 122 Chapte 6. Matheatcs of Fnance LECTURE NOTES 7) a) Ca loan of $25,000 epad onthly ove 3 yea peod, yealy nteest 8.5%. [ Aount of each ] nstallent [ ] p = P ) = ) t / )3 Calculato: /12)/ /12) 36)) b) House loan of $125,000 epad quately ove 20 yea peod, yealy nteest 7.5%. [ Aount of each ] nstallent [ ] p = P ) = ) t / )20 Calculato: /)/ /) 80)) c) Aotzaton table. Loan of $5,000 epad quately ove 1.5 yea peod, yealy nteest 8.5%. Aount [ of each nstallent ] [ ] p = P ) 1 1+ ) t = )1.5 Calculato: /)/ /) 6)) / payent aount nteest pncpal balance To begn, nteest = = , then pncpal = = 790., and balance = = , then nteest = and so on. d) Tue / False. Aotzaton detenes sequence of payents annuty) equvalent to pesent lup su, wheeas snkng fund detenes annuty equvalent to futue lup su. 3. Aotzaton, nube of payents to ete a loan: n = ln ) p p P ) ln1+ ) = ln p P) ln1+) a) Nube of quately $1000 payents ) to epay ca loan of $25,000, 8.5%: ) n = ln 1000 p ln p P ) = / o 37 payents ln1+ ) ln1+ 5 Calculato: ln1000/ /))/ ln1 + 5/) b) Nube of onthly $1000 payents ) to epay house loan of $125,000, 7%: ) n = ln 1000 p ln p P ) = / o 225 payents ln1+ ) ln Calculato: ln1000/ /12))/ ln /12)
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