Chapter 8.4 Annuities (Future value; different compounding periods)

Annuity A series of payments of investments made at regular intervals. A simple annuity is an annuity in which the payments coincide with the compounding period, or conversion period. An ordinary annuity is an annuity in which the payments are made at the end of each interval. Unless otherwise stated, each annuity in this chapter is a simple, ordinary annuity.

Using Future Value Formula 𝐹𝑉=𝑅× 1+𝑖 𝑛 −1 𝑖 R = Amount being invested at the end of every compounding period i = Compound Interest / number of compounding periods n = Amount of years until investment is completed × number of compounding periods

Example 1 An investment of $1000 is made every 6 months (semi-annually) that earns 4.8% compound interest for 20 years What is the future value of the annuity?

FV = 𝑅× 1+𝑖 𝑛 −1 𝑖 R = $1000 i = 0.048 2 = 0.024 n = 20 × 2 = 40 The future value of the annuity at the end of 20 years is $65927.08

Example 2 Bob receives a loan of $900,400 for the purchase of his house. He wants to make regular monthly payments over the next 15 years to pay off the loan. The bank is charging Bob 5.7% a compounded monthly. What monthly payments must bob make?

FV = 𝑅× 1+𝑖 𝑛 −1 𝑖 i = 0.057 12 = 0.0475 n = 15 × 12 = 180 a = $900,400 R =900400 ÷ 1+0.0475 180 −1 0.0475 R = 900400(0.0475) 1.0475 180 −1 R= 42769 4242.514543 R= 10.08104971 Bob needs to make monthly payments of $10.08

Using Geometric Series Formula a = Amount invested at the end of every compounding period r = 1+i i = compound rate/compound periods per year n = number of years used to invest × compound periods per year

Example 3 Jillian wants to invest $1000 for 20 years, at a rate of 4.8% compounded semi-annually. What will be the sum of all regular payments and interest earned?

Sn= 𝑎 𝑟 𝑛 −1 𝑟−1 a = $1000 r = 1+ 0.048 2 = 1.024 n = 20 × 2 = 40 The sum of the annuity at the end of 20 years will be $65927.08

Example 4 Mike wants to have a sum of $227410.08 and invests $850 at a rate of 5.2% compounded monthly. How long will Mike have to invest?

Sn= 𝑎 𝑟 𝑛 −1 𝑟−1 a = $850 Sn = $227410.08 r = 1+ 0.052 12 =1.043 $227410.08= 850 1.043 𝑛 −1 1.043−1 227410.08(0.043) = 850 1.043 𝑛 −1 9778.63344= 850 1.043 𝑛 11.50427464= 1.043 𝑛 𝑛= log 11.50427464 log 1.043

… Cont’d n = 58.0202005 N= 29 years Mike needs to invest for 29 years

Key Ideas The future of an annuity is the sum of all regular payments and interest earned The future value can be written as a geometric series (eg. FV = R+R(1+i)+R(1+i)2+R(1+i)3 The formula for the sum of a geometric series can be used to determine the future value of an annuity

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