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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Mathematics of Finance 5 Compound Interest Annuities Amortization and Sinking Funds Arithmetic and Geometric Progressions

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Simple Interest Interest: I = Prt Accumulated amount:A = P(1 + rt) Simple Interest - interest that is compounded on the original principal only.

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Ex. $800 is invested for 9 years in an account that pays 12% annual simple interest. How much interest is earned? What is the accumulated amount in the account? P = $800, r = 12%, and t = 9 years Interest: I = Prt = (800)(0.12)(9) = 864 or $864 Accumulated amount = principal + interest = 800 + 864 = 1664 or $1664

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Compound Interest A = Accumulated amount after n periods P = Principal r = Nominal interest rate per year m = Number of conversion periods per year t = Term (number of years) Compound Interest – interest is added to the original principal and then earns interest at the same rate.

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Ex. Find the accumulated amount A, if $4000 is invested at 3% for 6 years, compounded monthly. P = $4000, r = 3%, t = 6, and m = 12 So or $4787.79

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Effective Rate of Interest where r eff = Effective rate of interest r = Nominal interest rate per year m = Number of conversion periods per year Effective Rate – the simple interest rate that would produce the same accumulated amount in 1 year as the nominal rate compounded m times per year.

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Ex. Find the effective rate that corresponds to a nominal rate of 6% compounded quarterly. r = 6% and m = 4 So about 6.136% per year.

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Present Value (Compound Interest) Present Value (principal) – the amount required now to reach the desired future value.

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Ex. Jackson invested a sum of money 10 years ago in an account that paid interest at a rate of 8% compounded monthly. His investment has grown to $5682.28. How much was his original investment? A = $5682.28, r = 8%, t = 10, and m = 12 or $2560

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Annuity Annuity – a sequence of payments made at regular time intervals. Ordinary Annuity – payments made at the end of each payment period. Simple Annuity – payment period coincides with the interest conversion period.

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Future Value of an Annuity The future value S of an annuity of n payments of R dollars each, paid at the end of each investment period into an account that earns interest at the rate of i per period is

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Ex. Find the amount of an ordinary annuity of 36 monthly payments of $250 that earns interest at a rate of 9% per year compounded monthly. R = 250, n = 36 and or $10,288.18

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Present Value of an Annuity The present value P of an annuity of n payments of R dollars each, paid at the end of each investment period into an account that earns interest at the rate of i per period is

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Ex. Paige’s parents loaned her the money to buy a car. They required that she pay $150 per month, for 60 months, with interest charged at 2% per year compounded monthly on the unpaid balance. What was the original amount that Paige borrowed? or $8557.85

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Amortization Formula The periodic payment R on a loan of P dollars to be amortized over n periods with interest charged at a rate of i per period is

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Ex. The Kastners borrowed $83,000 from a credit union to finance the purchase of a house. The credit union charges interest at a rate of 7.75% per year on the unpaid balance, with interest computations made at the end of each month. The Kastners have agreed to repay the loan in equal monthly installments over 30 years. How much should each payment be if the loan is to be amortized at the end of the term? P = 83000, n = (30)(12) = 360, and Continued

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc So a monthly installment of $594.62

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Ex. A bank has determined that the Radlers can afford monthly house payments of at most $750. The bank charges interest at a rate of 8% per year on the unpaid balance, with interest computations made at the end of each month. If the loan is to be amortized in equal monthly installments over 15 years, what is the maximum amount that the Radlers can borrow from the bank? R = 750, n = (15)(12) = 180, and Continued

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc So they can borrow up to about $78480.44

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Sinking Fund Payment The periodic payment R required to accumulate S dollars over n periods with interest charged at a rate of i per period is

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Ex. Max has decided to set up a sinking fund for the purpose of purchasing a new car in 4 years. He estimates that he will need $25,000. If the fund earns 8.5% interest per year compounded semi-annually, determine the size of each (equal) semi-annual installment that Max should pay into the fund. S = 25000, n = 4(2) = 8, and Continued

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc So semi-annual payments of about $2899.91

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Arithmetic Progressions Arithmetic progression – a sequence of numbers in which each term after the first is obtained by adding a constant d (common difference) to the preceding term. Ex. 1, 8, 15, 22, 29, … First termCommon difference: d = 7

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Arithmetic Progression The nth term with the first term a and common difference d is given by The sum of the first n terms with the first term a and common difference d is given by

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Ex. Given the arithmetic progression 1, 8, 15, 22, 29, … find the 10 th term and the sum of the first 10 terms. 10 th term: a = 1, d = 7, and n = 10. = 64 Sum: = 325

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Geometric Progressions Geometric progression – a sequence of numbers in which each term after the first is obtained by multiplying the preceding term by constant r (common ratio). Ex. 9, 3, 1, 1/3,… Common ratio: r = 1/3

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Geometric Progression The nth term with the first term a and common ratio r is given by The sum of the first n terms with the first term a and common ratio r is given by

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Ex. Given the geometric progression 4, 12, 36, 108, … find the 8 th term and the sum of the first 8 terms. 8 th term: a = 4, r = 3, and n = 8. = 8748 Sum: = 13120

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