Lesson 4: Annuities: Future Value Part A - Definitions/Introduction Annuity: a series of equal payments or investments made at regular intervals of time : Simple annuity: an annuity in which the payments coincide with the compounding period Ordinary annuity: an annuity in which the payments are made at the end of each interval Note: We will deal with simple, ordinary annuities. Part B Deriving the Formula Chie has decided to save $1 000 each year from her part-time job. She does this for 5 years and invests the amount each year at 9%/a compounded annually. How much will she have saved at the end of the 5% year. Draw a time line. Now 1 year 2 years 3 years 4years 5 years f f f f t i deposit deposit deposit deposit deposit $1 000 $1 000 $1 000 $1 000 $1 000 As =1000 9% compounded annually A, =1000(1.09) A; =1000(1.09)? A, =1000(1.09)° A, =1000(1.09)* At the end of the 5% year, the amount of money, including interest, accumulated is -given by: S5 =1000 +1000(1.09) +1000(1 .09)2 +1000(1 .09)3 +1000(1 .09)4 ——p —a+ar+ar®+ar®+ar® a(r" —1) This series is.a geometric series and the formula S,, = can be used. Therefore, 1000 (1.09° —1) ° 1.09-1 = 5984
Example 2: Gabriel wants to buy a new TV priced at $2 500 plus taxes. He plans to buy it in 16 months and will make a payment into his savings account at the end of each month. The account earns 3%/a compounded monthly. a) Determine the amount of each payment. 2500-16=156.25 126.25(1+0.03)=$160.94 b) Determine how much interest Gabriel will have earned? 156.25°0.03+16 =$74.88
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