F.3 - Annuities and Sinking Funds

Transcription

1 F.3 - Annuities and Sinking Funds Math Blake Boudreaux Department of Mathematics Texas A&M University March 22, 2018 Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12

2 Objectives Know how to compute annuities Know how to compute sinking funds (Technically optional) learn more TVM solver applications in TI-83/84. Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12

3 Definitions In this section we transition from buy and hold -type payments to periodic payments. An annuity is a sequence of equal payments made at equal time periods. An ordinary annuity is one in which the payments are made at the end of the time periods of compounding. The term of an annuity is the time from the beginning of the first period to the end of the last period. The total amount in the account, including interest, at the ned of the term of an annuity is called the future value of the annuity. Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12

4 Example Suppose we make deposits of $1000 at the end of each year into a savings account that earns 8% per year. Determine how much is in the account at the end of the sixth year. Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12

5 Theorem (Summation Formula) For any positive integer n and any real numbers a and b, with b 1. S = a + ab + ab 2 + ab ab n 1 = a bn 1 b 1. Definition (Future Value of an Ordinary Annuity) The future value FV of an ordinary annuity of n payments of PMT dollars paid at the end of each period into an account that earns interest at the rate of i per period is FV = PMT (1 + i)n 1. i Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12

6 Example Since you ve been born, your grandparents have deposited $140 per quarter into an account which earned about 3.5% per year compounded quarterly. a. How much money was in the account when you turned 18? b. How much interest did the account earn? Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12

7 Practice In (a) through (d), find the future values of each of the ordinary annuities at the given annual rate r compounded as indicated. The payments are made to coincide with the periods of compounding. a. PMT = 1200, r = 0.08, compounded annually for 10 years. b. PMT = 600, r = 0.08, compounded semiannually for 10 years. c. PMT = 100, r = 0.12, compounded monthly for 20 years. d. PMT = 100, r = 0.12, compounded weekly for 40 years. Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12

8 Definition Often one knows that at some future date a certain amount FV of money will be needed. One can then establish an account for accumulating funds to meet this future need. This is known as a sinking fund. This question now becomes how much money PMT should be put away periodically so that the deposits plus interest will be FV at the needed point in time. Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12

9 Definition Definition (Sinking Fund Payment) The periodic payment PMT that is required to accumulated the sum FV over n periods of time with interest at the rate of i per period is PMT = FV 1 (1 + i) n 1. For example, suppose you wish to accumulate a retirement fund of $600, 000. How much should you deposit each month into your retirement account, which pays interest at a rate of 6.5% per year compounded monthly, to reach your goal when you retire 40 years from now? Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12

10 Example You forego your 3 lattes per week and contribute the $50 per month you save into a high-risk fund which earn 7.5% per year compounded monthly. a. How much money will you have in the account when you retire in 45 years? b. How much interest will you have earned total on the account? c. How much interest did the account earn during the 8th month of the 42nd year? Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12

11 Example Lauren plans to deposit $4000 into a bank account at the beginning of next month and $250 per month into the same account at the end of that month and at the end of each subsequent month for the next 5 years. If her bank pays interest at a rate of 5% per year compounded monthly, how much will Lauren have in her account at the end of 5 years? Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12

12 Example Suppose you buy a $300, 000 house with a 30 year mortgage having an interest rate of 6% compounded monthly. 5 years go by and interest rates have dropped to 4%. A bank calls you and asks if you d like to refinance the current balance of your loan into a new 30 year mortgage. Should you refinance? Blake Boudreaux (TAMU) F.3 - Annuities March 22, / 12