Finite Math Chapter 10: Study Guide and Solution to Problems

Transcription

1 Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount you depost n the account. Balance or Compound Amount Amount to whch the prncpal grows. Interest Rate Per Perod If the nterest s compounded m tmes per year and the annual nterest rate s r, = r m. Basc Formulas Compound Interest Queston In the followng formula: what s F? What s n? Solve for P n equaton 10.3 to get: F = (1 + ) n P, (10.1) The followng formula: defnes the effectve rate. 1 (1 + ) n F. r eff = (1 + ) m 1, (10.2) Queston What s the practcal use of r eff? What s m here? Smple Interest F = (1 + nr)p, (10.3) Queston What s the dfference between compounded and smple nterest?

2 10.2 Annutes Queston What s an annuty? Increasng Annuty Queston What s a ncreasng annuty? Gve an example. The rent, denoted by R, of a ncreasng annuty s the value of the perodc depost. The future value of an decreasng annuty s gven by: F = (1 + )n 1 R, (10.4) Queston Solve for R n equaton What s n? Decreasng Annuty Queston What s a decreasng annuty? Gve an example. The present value of an decreasng annuty s gven by: 1 (1 + ) n R, (10.5) Queston Solve 10.5 to get a formula for the rent of an decreasng annuty Amortzaton of Loans An amortzaton s the process of payng a loan. Queston Whch type of annuty s an amortzaton, decreasng or ncreasng? Problems Problem 1 A famly obtans a $75, 000 house loan for 30 years at 8 % nterest compounded monthly. a) Fnd the monthly payments. b) What s the total amount pad over the 30 years? c) What s the total amount of nterest pad? Comment on the relatve szes of the loan, the total amount of nterest pad, and the total pad over the 30 years. d) What s the unpad balance after 10 years of payments? After 20 years? 25? (Hnt: The remanng payments consttute a decreasng annuty.) e) In d) you computed the unpad balance after 20 years. How much of the frst payment n the 21st year goes toward prncpal and how much toward nterest? Soluton: Ths s a decreasng annuty.

3 a) =.08 =, n = 30 = 360, $75, (1 + ) n $75, 000 = $550.32, 1 (1 + ) 360 b) Total amount pad = number of perods rent = 360 $ = 198, c) Total amount of nterest pad= Total amount pad - prncpal = $(198, , 000) = $3, The amount of nterest pad s almost twce the value of the prncpal. d) After 10 years there are 20 years left, so the number of perods remanng s 20 = (1 + ) n 1 (1 + ) 240 $ = $65, After 20 years there are 10 years left, so the number of perods remanng s 10 = 0. 1 (1 + ) n 1 (1 + ) 0 $ = $45, After 25 years there are 5 years left, so the number of perods remanng s 5 = (1 + ) n 1 (1 + ) 60 $ = $27, e) The prncpal after 119 perods are left ( that s, after the frst payment n the 21st year, ) s: 1 (1 + ) n 1 (1 + ) 1 $ = $45, so that from the rent of that month (namely $550.32) only 45, , = $ goes to pay the prncpal. The rest = goes to pay nterest. Problem 2 a) Raul borrows $, 000 to buy a car. He pays 6.3 % nterest compounded monthly and the loan s for two years. Fnd the monthly payments. b) Raul knows he can get 8.8% nterest (compounded monthly) from a money market account, and so plans to make the payments you computed n part a) va automatc deducton from ths account. He plans to put enough n ths account so that the monthly car payments wll exactly exhaust ths account over these two years. How much should he depost? c) Comment on the result of b). Soluton: a) =.0063 = , n = 2 = 24, $, 000 b) =.008 = (1 + ) n $, 000 = $ ( ) 24 1 (1 + ) n c) Raul saves $300 on hs bke by usng ths strategy. 1 ( ) 24 $ = $11,

4 Problem 3 Tm and Ann bought a house wth a down-payment of $10, 000 and an $80, 000 loan. The loan was for 25 years at a 9% nterest rate. Closng costs amounted to an addtonal 1.5%. Two years later they were transferred and sold the house for what they pad for t, that s $90, 000. The real estate agent charged a 6% fee for sellng the house. Fnd the average monthly cost of the house takng nto consderaton the monthly payment, the costs of buyng and sellng and the equty bult up over 2 years. Soluton: Cost of buyng the house$90, = $1, 350. Also =.09 =.0075, n = 25 = 300, so that 1 (1 + ) n.0075 $80, 000 = $ ( ) 300 s the monthly cost. Cost of sellng the house = 90, 000 6% = $5, 400. Prncpal after 2 years (23 = 276 perods left) Equty: 1 (1 + ) n 1 ( ) 276 $ = $78, , , = 11, 868. Total monthly ncome from sellng the house: T m= ( Equty -Cost of sellng the house-cost of buyng the house )/24. 11, 868 5, 400 1, 350 T = = Monthly Cost = $ = $

5 Questons 4 and 5 should be solved usng Excel. Problem 4 a) Suppose you wsh to borrow $300, 000 to buy a house. As you anxously awat the call from the mortgage broker, you decde to calculate the monthly payments on your mortgage for varous dfferent nterest rates. Calculate the monthly payment on a 30-year fxed-rate loan, f the nterest rate s 5%, 5.25%, 5.5%, 5.75%, 6%, 6.25%, 6.5%, 6.75% and 7%. The nterest s compounded monthly. b) Calculate the payment f you took out a 15-year fxed-rate loan at the same rates. c) Consder the case of a 30-year fxed-rate at 6.25%. Suppose you earn a Chrstmas bonus every year, and you choose to spend ths on makng an extra mortgage payment (just thnk of doublng the last payment of the year.) How much faster wll you loan be pad off and how much money wll you save? (Hnt, make a spreadsheet usng 360 cells contanng the payment each month, keepng n mnd that one payment n wll be hgher. Fnd the prncpal left after each month, and see when t gets to zero.) Soluton: The answer for c) s 3 years and eght months earler. Ths takes some thnkng!!