Stats & Discrete Math Annuities Notes

Transcription

1 Stats & Discrete Math Annuities Notes Many people have long term financial goals and limited means with which to accomplish them. Your goal might be to save $3,000 over the next four years for your college education, to save $10,000 over the next ten years for the down payment on a home, or to save $300,000 over the next forty years for your retirement. It seems incredible, but each of these goals can be achieved by saving only $50 a month (if interest rates are favorable)! All you need to do is start an annuity. 1

2 An is simply a sequence of equal, regular payments into an account in which each payment receives compound interest. Because most annuities involve relatively small periodic payments, they are affordable for the average person. Over longer periods of time, the payments themselves start to amount to a significant sum, but it is really the power of compound interest that makes annuities so amazing. A is an annuity that is set up to save for Christmas shopping. A Christmas Club participant makes regular equal deposits, and the deposits and the resulting interest are released to the participant in December when the money is needed. 2

3 Ex. 1: On August 12, Patty Leitner joined a Christmas Club through her bank. For the next three months, she would deposit $200 at the beginning of each month. The money would earn 8.75% interest compounded monthly, and on December 1, she could withdraw her money for shopping. Use the compound interest formula to find the future value of the account. First, we ll calculate the future value of the first payment (made on September 1). Use t = 3 months, since it will receive interest during September, October, and November. 3

4 Next, we ll calculate the value of the second payment (made on October 1). Use t = 2 months, since it will receive interest during October and November. 4

5 Finally, we can calculate the future value of the third payment (made on November 1). Use t = 1 month, since it will receive interest during November. 5

6 The total value of Patty s annuity is the sum of the three future values: Over three months, Patty deposited $600, so she earned in interest on her deposits. 6

7 An is one in which each payment is due at the beginning of its time period. Patty s annuity in the last example was an annuity due because the payments were due at the beginning of each month. An is an annuity for which each payment is due at the end of its time period. As the name implies, this form of annuity is more typical. As we will see in the next example, the difference is one of timing. 7

8 Ex. 2: Dan Bach also joined a Christmas Club through his bank. His was just like Patty s except that his payments were due at the end of each month, and his first payment was due September 30. Use the Compound Interest Formula to find the future value of the account. This is an ordinary annuity because the payments are due at the end of each month. Let s calculate the future value of the first payment (made on September 30). Use t = 8

9 To calculate the future value of the second payment (made on October 31), use t = The third payment (made on November 30) won t earn any interest since the annuity expires on December 1. Therefore, its value is just $200. 9

10 The total value of Dan s annuity is the sum of the three future values: Over three months, Dan deposited $600, so he earned in interest on his deposits. The difference between an ordinary annuity and an annuity due is strictly a timing difference because any ordinary annuity in effect will become an annuity due if you leave all funds in the account for one extra period. 10

11 The procedure used in the first two examples reflects what actually happens with annuities, and it works fine for a small number of payments. However, most annuities are long term, and the procedure would become tedious if we were computing the future value over many years. Because of this, long term annuities are calculated with their own formula. Ordinary Annuity Formula The future value FV of an ordinary annuity with payment size pymt, rate r, compounded n times per year over t years is: 11

12 Annuity Due Formula The future value FV of an annuity due with payment size pymt, rate r, compounded n times per year over t years is: 12

13 A is an annuity that is set up to save for retirement. Money is automatically deducted from the participant s paychecks until retirement, and the federal (and perhaps state) tax deduction is computed after the annuity payment has been deducted, resulting in significant tax savings. 13

14 Ex. 3: Tom and Betty decided that they should start saving for retirement, so they set up a taxdeferred annuity. They arranged to have $200 taken out of each of Tom s monthly checks, which will earn 8.75% interest. Because of the tax deferring effect of the TDA, Tom s takehome pay went down by only $115. Tom just had his thirtieth birthday, and his ordinary annuity will come to term when he is 65. a) Find the future value of the annuity. pymt =, r =, n =, and t = 14

15 b) Find Tom s contribution and the interest portion. The interest is almost six times as large as Tom s contribution! The magnitude of the earnings illustrates the amazing power of annuities and the effect of compound interest over a long period of time. 15