1. Math 210 Finite Mathematics

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1 1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: 2. ath 210 FINAL EXA onday, December 7, :50 pm Cole Hall Savng oney Every onth Suppose you put dollars n the bank every month, for sx months. If the bank pays monthly nterest (6% annual nterest), your money grows lke ths: month formula calculated Total Sum A Summaton Formula To compute S = x 5 +x 4 +x 3 +x 2 +x 1 +1 use the followng trck: ultply S by x then subtract S. xs = x 6 +x 5 +x 4 +x 3 +x 2 +x 1 S= x 5 +x 4 +x 3 +x 2 +x 1 +1 xs S= x 6 1 All the other terms on the rght cancel! The rest s easy: xs S = x 6 1 = (x 1)S = x 6 1 = S = x6 1 x Fnal Balance After 6 months, you account wll be worth Factorng out an, ths sum s ( ) We want a formula for the sum nsde the parentheses. 6. Usng the Summaton Formula We saw that f you depost dollars n the bank for sx consecutve months, then the balance after sx months s S = ( ) Pluggng x = nto the Summaton Formula 1+x+x 2 + +x 5 = x6 1 x 1 gves S = =

2 2 7. Fnal Calculaton If you depost = 100 dollars every monthforsxmonths,atamonthlyrate of =.005, you wll have S = = dollars after 6 months..005 The 7.55 represents accumulated monthly nterest. 9. Long Term Savngs Suppose you saved = 100 dollars every month, for 45 years, at a monthly rate of =.005. Here = 100 =.005 n = = 540 After 45 years the account balance would be S = 100 = , 599 dollars. Your monthly contrbutons were = 54,000, so most of the growth n the account s due to accrued nterest. 11. Savng for Retrement Cont d Your sster, on the other hand, wats untl her 45th brthday to begn savng for her retrement. Lke you, she has $200 taken out of her paycheck every month nto a 7% IRA account. At age 65, who has saved more money: you or your sster? 8. onthly Savngs formula S = (1+)n 1 where = amount saved per month S = endng balance = r/12 = monthly nterest rate n = number of months 10. Savng for Retrement You and your sster have dfferent deas about savng for your retrement at age 65. At age 25 you start put asde 200 every month nto an IRA yeldng 7 percent nterest. After ten years, at age 35, you decde that famly oblgatons requre you to savethe$200forcollegemoneyforyour three kds. So you stop puttng money nto the account, but let t contnue to collect 7% nterest untl you retre at age Your Sster Let s look at your sster frst, snce her stuaton s easer to analyze. The monthly nterest rate s =.07/12 for n = 240 months. Snce = 200, the monthly savngs formula gves an endng balance after 20 years of (1+.07/12) = 104,185.07/12 more than doublng the = 48, 000 she has nvested.

3 3 13. Your Turn You on the other hand have nvested one half as much money as your sster: = 24,000. After ten years, when you are 35, your endng balance s computed by the onthly Savngs Formula (wth =.07/12 and n = 120 months): S = (1+.07/12) =.07/12 34, Now ths money earns compound nterest over the next 30 years or = 360 months, so that at age 65 your IRA account wll be worth (1+.07/12) 360 S = 280, You made 2.7 tmes as much as your sster, although you only nvested half as much money. 15. onthly Payment to obtan Balance S = (1+) n 1 S Ths formula comes from the onthly Savngs Formula S = (1+)n Advce oral: How much you accumulate for retrement depends upon three thngs: () when you start savng, () how much you manage to save, and () how much your nvestments return over the long run. Of the three, when you start savng turns out to be the most mportant. oral: Invest when you are young! 16. Savng for College Kaylee s parents want to put asde money every month so that ther daughter wll have $25,000 for college when she turns 18. At 5% annual nterest, how much do they need to save per month? Set the varables: amount saved per month: desred endng balance: 25, 000 monthly nterest rate: =.05/12 = number of months: = = ( ) ,000 = 71.60

4 4 17. Annutes A sequence of payments made at regular ntervals s called an annuty. Specal types of annutes: certan annuty the term s a fnte tme perod ordnary annuty eah payment s made at the end of a payment perod smple annuty the payment perod concdes wth the nterest converson perod The annutes we consder are () certan; () ordnary; () smple; and (v) the perodc payments are all the same sze. 19. Present Value of an Annuty The dea s smply to determne what prncpal P would you need to start wth n order to end up wth the balance S after n nterest perods, compounded per perod at an annual rate r? That s, we set the compund nterest formula equal to the future value of an annuty formula: P(1+) n = (1+)n 1 = P = (1+)n 1 (1+) n = P = 1 (1+) n 18. Future Value of an Annuty S = (1+)n 1 where = amount pad per nvestment perod S = endng balance = perodc nterest rate n = number of perods Note that the nterest perods need not be months, though frequently they are. 20. Annutes If you ever nhert (or wn from the lottery) a large sum of money, you may wshtosetupanannuty,typcallywth an nsurance company or fnancal nsttuton. An annuty works exactly lke a home mortgage, reversng the roles of lender and borrower. Wth a home mortgage, a fnancal nsttutongvesyoualargesumofmoney (to buy your house) n exchange for your monthly payments over the next n months. Wth an annuty, you gve the fnancal nsttuton a large sum of money and they agree to pay you the monthly (or annual) payments over a certan tme perod.

5 5 21. The Lottery Typcally lottery wnnngs are always pad as annutes these days. Suppose you wn one mllon dollars n a contest. The rules of the contest state that you wll be pad 40,000 per year over the next 25 years. If the nterest rate s r =.065, how much does the company sponsorng the contest need to pay to set up the annuty? Soluton: Compute the present value of an annuty whch pays 40,000 per year at a rate.065 over 25 years. Note that payments are annual, not monthly. 23. Amortzaton of Loans Suppose you purchase a home or car by borrowng a prncpal of P dollars. ThenP maybethoughtofasthepresentvalue of an annuty and the value of n the formula P = 1 (1+) n s the amount you need to pay every month n order to pay off the loan n n payments. Solvng for n terms of P: = 1 (1+) P n 25. Calculator Tps You need to use parentheses around 1+ You need parentheses around the entre denomnator You need to use the negatve button (not subtract) on the exponent n You can store = r/12 n the memory of your calculator The onthly Payback Equaton then becomes ( RCL E 1 ( 1+RCL E ) ) n P 22. Lottery Contnued Varables n the Present Value Formula annual nterest rate : = r =.065 number of payments (n years) : n = 25 annual payment : = 40,000 Present Value Calculaton P = 1 (1.065) = 487, So t costs the company less than 488 thousand dollars to gve away a mllon dollars. 24. onthly Payback formula = 1 (1+) n P where = monthly payment P = prncpal = r/12 = monthly nterest rate n = number of months 26. Amortzaton To amortze a loan means to set asde money regularly for future payment of the debt. The general formula for the amortzaton of a loan s = 1 (1+) P n where = payment per perod P = prncpal debt = perodc nterest rate = r/number of perods per year n = number of perods Typcally, the perods are months.

6 6 27. Buyng a Car You want to buy a Dodge Neon for $12,500. Whch s the better deal: $1000 cash back and a 3 year bank loan at 8.5% nterest or the promotonal 1.9% loan from Dodge for the entre $12,500? 29. Opton 2. Low Interest For the 1.9% nterest opton, you need to borrow prncpal borrowed : P = 12,500 annual nterest rate : r =.019 monthly nterest rate : = r/12 = number of months : n = 12 3 = 36 The monthly payment for ths opton s = = Prncpal n terms of onthly Payment P = 1 (1+) n Ths s just the formula for the present value of an annuty. 28. Opton 1. Cash Back For the cash back opton, you need to borrow prncpal borrowed : P = 11,500 annual nterest rate : r =.085 monthly nterest rate : = r/12 = number of months : n = 12 3 = 36 The monthly payment for ths opton s = = Comparson Opton 1. Payment s $ Opton 2. Payment s $ Concluson: Opton 2 (wth the lower nterest) s slghtly better. It saves about $5.54 per month. Over 36 months, ths savngs amounts to almost $ Buyng a House You wsh to buy a house on a 30 year fxed mortgage. The nterest rate s a fxed 8% (per year). The largest mortgage you can afford s $1000 per month. What s the largest prncpal you can afford f the nterest rate s 8 percent 4.25 percent

7 7 33. Buyng a House Hgh Interest Varables: annual nterest rate : r =.08 monthly nterest rate : = r/12 = number of months : n = = 360 monthly payment : = 1000 Largest Prncpal You Can Borrow P = 1 ( ) = 136, Buyng a House Comparson Interest Prncpal 8% 136, % 203, Note: the amount that you wll pay one thousand dollars per month for 30 years s the same for both cases. 37. Reducng the Interest Total Interest = 360 P = , = 223, If you pad the same prncpal 136, off n a 15 year loan at the same nterest rate 8%, what would the monthly payments be? =.08/12 n = 180 P = 136, = , ( ) 180 = Buyng a House Low Interest Varables: annual nterest rate : r =.0425 monthly nterest rate : = r/12 = number of months : n = = 360 monthly payment : = 1000 Largest Prncpal You Can Borrow P = 1 ( ) = 203, Interest Pad 30 - year loan year prncpal nterest payoff frst mo last Interest Pad 15-year loan year prncpal nterest pay off frst mo last Total Interest = 180 P = , = 98,147.80

8 year loan 39. Comparson onthly Payment : = 1000 Number of months : n = 360 Total pad : 360,000 Total nterest : 223, year loan onthly Payment : = Number of months : n = 180 Total pad : 234,432 Total nterest : 98, A Calculator Error If $750 s the largest monthly house payment you can afford, what s the largest prncple you can borrow from the bank over 15 years at 7.4% nterest? The axmum Prncple Formula says P = 1 ( /12) ) /12 A student from a prevous semester typed (1 ( ) 180) = and got the answer $ What went wrong? Does ths answer seem reasonable? 43. Credt Cards Dsadvantages usually hgh nterest rate encourages excess spendng balance can keep growng large proporton spent on nterest a contnued lfe of debt 40. Advce The 15-year loan saves over $125,000 n nterest. oral: Go for the shorter loan f you can afford t. 42. Credt Cards Advantages convenent necessary for auto rental phone orders records and recepts elmnates worry of cash 44. Credt Card Interest You owe $2000 on your Vsa Card The nterest rate s 18% You are requred to pay 2% of the balance per month The monthly nterest s =.015 On a balance of $2000, you wll pay = 30 Your mnmal monthly payment: $40.00

9 9 45. Credt Card Interest Cont d The nterest you pay: $30.00 Amount pad on balance: $10.00 Gettng an A n ths class: prceless Notce that 3/4 of your payment s gong drectly to nterest. At ths rate, you wll pay $8000 for the $2000 worth of merchandse you bought. 46. How to get out of credt card debt You owe $2000 to Vsa. The nterest rate s 18% or 1.5% per month. You would lke to pay ths off n two years. Frst, cut up the card wth scssors or at least promse not to use t for two years. Now use the onthly Payback Formula: prncpal borrowed : P = 2,000 annual nterest rate : r =.18 monthly nterest rate : = r/12 =.015 number of months : n = 12 2 = Gettng out of credt card debt Calculate the monthly payment:.015 = = Send Vsa a check for $99.85 every month for two years and you wll be out of debt. Total Payments: = Total Interest: =

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