A Formula for Annuities

Size: px

Start display at page:

Download "A Formula for Annuities"

Milo Cunningham
6 years ago
Views:

1 A Formula for Annuities We ve seen that, with a bit of work, an annuity can be priced by summing geometric sequence. If we apply the geometric sum to a general annuity, we get a formula for annuities: p denotes the periodic interest rate (APR/(number of times interest compounded per year)) L is the future value of the last payment T is the total number of payments F is the future value of the annuity ( (1 + p) T ) F = L p (You need to know HOW and WHEN to use this formula, but don t worry about memorizing it. It will be given to you on the exam.) Paul Koester () MA 111, More on Deferred Annuities February 29, / 9

2 Annuity Example You save $ at the end of each month for 2 years. The account pays 2.4% APR compounded monthly. How much will you have saved by the end of the 2 years? Solution: Get in the habit of making educated guesses before doing lengthy calculations. There are lots places to make algebra and arithmetic mistakes in these problems, so its good to have an estimate of the final answer. Which is the only reasonable option? Paul Koester () MA 111, More on Deferred Annuities February 29, / 9

3 Some Estimates to Keep in Mind You make 24 investments of $200.00, thus you have invested $ The investment has a positive interest rate, so the future value better be greater than $ Suppose the entire $ was invested for the full two years. At the end of the two years you would have ( ) 24 = $ In actuality, most of the $ earns interest for only part of that time, so the future value of this annuity cannot exceed $ Paul Koester () MA 111, More on Deferred Annuities February 29, / 9

4 The precise answer You save $ at the end of each month for 2 years. The account pays 2.4% APR compounded monthly. How much will you have saved by the end of the 2 years? Solution: The periodic rate is p = 0.024/12 = The number of payments is 12 2 = 24. L = (L will not always be the value of the payments.) Thus, ( (1.002) 24 ) F = = $ Paul Koester () MA 111, More on Deferred Annuities February 29, / 9

5 Different L You save $ at the beginning of each month for 2 years. The account pays 2.4% APR compounded monthly. How much will you have saved by the end of the 2 years? Paul Koester () MA 111, More on Deferred Annuities February 29, / 9

6 Finding the precise value You save $ at the beginning of each month for 2 years. The account pays 2.4% APR compounded monthly. How much will you have saved by the end of the 2 years? Solution: The periodic rate is p = 0.024/12 = The number of payments is 12 2 = 24. The last payment is made at the beginning of the last month, so the last $ earns interest for one month. Thus, ( L = ) 1 = $ ( (1.002) 24 ) F = = $ Paul Koester () MA 111, More on Deferred Annuities February 29, / 9

7 A more interesting example Your rich Aunt Helga passed away last month. In her will, she left you an annuity which will pay you $40, at the end of each year for the next 10 years. You see an add on TV for a lending company that offers to buy annuities, so you decide to sell your inheritance so that you can buy a house for $350, today. How much should you expect to receive for the annuity, and will you be able to afford the house? Solution: The price of the annuity depends on the current market interest rate. Suppose the money can be invested at 2% APR compounded annually. The annuity payments are made at the end of each year, so the value of the annuity ten years from now is ( (1.02) 10 ) F = 40, 000 = $437, Paul Koester () MA 111, More on Deferred Annuities February 29, / 9

8 A more interesting example ( (1.02) 10 ) F = 40, 000 = $437, This is how much the annuity will be worth 10 years from now, but you are interested in selling it TODAY. To find the fair sale price, we use the compound interest techniques from last week: r = 0.02, t = 10, F = and we wish to find P : = P (1.02) 10 So P = = $ Thus, you can sell the annuity, buy the house, and still have $9, left over. Paul Koester () MA 111, More on Deferred Annuities February 29, / 9

9 An Important Lesson In order to compare two investment options, it is essential that we compare them at the SAME POINT IN TIME. In the previous example, we needed to compare the future value of an annuity 10 years into the future and the value of a house today. It is meaningless to compare these two values directly. Instead, we used the compound interest formula to move one of the values to a different point in time. In practice, it doesn t matter which point in time we choose. We just need to be sure that all of the options are considered at a common time. Paul Koester () MA 111, More on Deferred Annuities February 29, / 9

Simple Interest (for One Year)

Simple Interest (for One Year) Suppose you invest $1500.00 at 3.22% interest per year. How much will you have at the end of one year? Solution: 3.22% interest means that over the course of one year, one