Chapter 4: Section 4-2 Annuities

Transcription

1 Chapter 4: Section 4-2 Annuities D. S. Malik Creighton University, Omaha, NE D. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 1 / 24

2 Annuities Suppose that we deposit $1000 every year for 5 years in an account that earns an interest of 6% per year compounded yearly. How much money will we have after 5 years? Note that $1000 is deposited every year. We have not specified whether the money is deposited at the end of the year or at the beginning of the year. Thus, there are two cases One in which the deposit is made at the end of the year Another in which the deposit is made at the beginning of the year. Both of these cases lead to a different accumulation. D. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 2 / 24

3 First suppose the deposit is made at the end of each year, see the following figure: $1000 Deposit ( ) 4 = Interest Interest Interest Interest $1000 Deposit ( ) 3 = Interest Interest Interest $1000 Deposit ( ) 2 = Interest Interest $1000 Deposit ( ) 1 = Interest $1000 Deposit Total = = D. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 3 / 24

4 Consider the case when the deposit is made at the beginning of the year, see the following figure: $1000 Deposit ( ) 5 = Interest Interest Interest Interest Interest Deposit 2 Deposit 3 Deposit 4 Deposit 5 $ ( ) 4 = Interest Interest Interest Interest $ ( ) 3 = Interest Interest Interest $ ( ) 2 = Interest Interest $ ( ) 1 = Interest Total = = D. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 4 / 24

5 Definition Let a and r be real numbers, and n be a nonnegative integer. Then the sequence of terms a, ar, ar 2, ar 3,..., ar n 1,... is called a geometric series. We call a the first term, r the common ratio, and ar n 1 the nth or general term of the series.. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 5 / 24

6 Example Consider the geometric series: , , , , Here a = , r = = Note that 1000 = a = ar 0, 1060 = = ar, = = ar 2, = = ar 3, = = ar 4.. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 6 / 24

7 Theorem Let a be the first term of a geometric series and r be the common ratio such that r = 1. Let n be a positive integer. Let S n denote the sum of the terms a, ar, ar 2, ar 3,..., ar n 1 of the geometric series, i.e., S n = a + ar + ar 2 + ar ar n 1. Then S n = a(r n 1), r = 1. r 1. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 7 / 24

8 Example Consider the previous geometric series: , , , , Here a = , r = = Then, S 5 = a(r 5 1) r 1 = 1000 ( ) = S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 8 / 24

9 Example Consider the following geometric series: , , , , We want to find the sum of this series: So = = 1000( ) = S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 9 / 24

10 Annuity Definition An annuity is a series of equal payments made at equal periods of time. (i) An ordinary annuity is an annuity in which a payment is made at the end of each period. (ii) An annuity due is an annuity in which a payment is made at the beginning of each period. Annuities Ordinary annuity Deposit is made at the end of each period Annuity due Deposit is made at the begining of each period. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 10 / 24

11 Theorem Suppose that R dollars per period, at the end of each period, is deposited m times a year for t years in an account earning an interest at the rate of r% per year. The total amount, S, accumulated at the end of t years is [ (1 + i) S = n ] 1 R, i where i = r m and n = mt. This amount S is called the future value of the ordinary annuity.. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 11 / 24

12 Ordinary Annuity Example Suppose that $750 is deposited at the end of each quarter for 10 years in an account earning 6% interest per year compounded quarterly. Then R = 750, r = 0.06, m = 4, t = 10, i = r m = 0.06 = 0.015, and n = mt = 4 10 = Thus, [ (1 + i) S = n ] [ 1 ( ) R = 40 ] i So the total amount accumulated at the end of 10 years is $40, S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 12 / 24

13 Annuity Due Theorem Suppose that R dollars per period, at the beginning of each period, is deposited m times a year for t years in an account earning an interest at the rate of r% per year. The total amount, S, accumulated at the end of t years is [ (1 + i) S = n+1 ] 1 R R, i where i = r m and n = mt. This amount S is called the future value of the annuity due.. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 13 / 24

14 Example Suppose that $300 is deposited at the beginning of each month for 20 years into an account that pays 9% interest per year compounded monthly. Then R = 300, r = 0.09, m = 12, t = 20, i = r m = 0.09 = , and n = mt = = Note that n + 1 = 241. Thus, S = [ (1 + i) n+1 ] [ 1 ( ) R R = ] i = So the total amount accumulated at the end of 20 years is $201, S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 14 / 24

15 Sinking Fund In a business environment, the owner(s) always foresee certain expenses. For example, a moving company should buy a certain number of trucks after a certain period. A warehouse might need to add a new building after, say five years. Parents would like to accumulate a certain amount for their children s college education. In cases such as these, we usually have a pretty good idea about the money we will need and the time we have to accumulate that money. However, we may not have a large amount to make one deposit to accumulate the desired amount. So we would like to periodically make a deposit into an account that will pay some interest. The objective is to determine how much money must be deposited during each period to meet our goal. D. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 15 / 24

16 Example Steve and Priya estimate that they will need $70, 000 in 14 years when their daughter begins college. They are not risk takers and want their investment to be safe. So they decide to open an account in a local bank. The account pays 6.50% interest per year compounded quarterly. Steve and Priya want to know the money that must be deposited at the end of each quarter into this account to accumulate $70, 000. Note that Steve and Priya are making a series of deposits, at the end of each period, in an account that compounds interest quarterly for 14 years. So this is an ordinary annuity in which the following are given: The future value of the ordinary annuity, which is $70, 000, The yearly interest rate, which is 6.50%, The compounding period, which is quarterly, and The number of years, which is t = 14. We want to find R. Now the future value, S, of the ordinary annuity is given by. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 16 / 24

17 Example We have R =?, r = 0.065, m = 4, t = 14, and S = 70, 000. Now [ (1 + i) S = n ] 1 R. i Solve it for R to get R = Si ( ) (1 + i) n = 1 ( ) 56 1 = Hence, Steve and Priya must deposit $ at the end of each quarter for 14 years to accumulate $ S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 17 / 24

18 Theorem ( Periodic Payments for a Sinking Fund) Suppose that a sinking fund of S dollars pays r% interest per year compounded m times a year for t years. Let i = r m and n = mt. Then the amount, R, of each deposit (or payment) at the end of each period is given by R = Si (1 + i) n 1.. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 18 / 24

19 Example The owners of the Fast Printing Company are expecting to replace an expensive copier after 7 years. The replacement cost is $30, 000. With the help of their bank, they set up a sinking fund that pays 8.5% interest per year compounded bimonthly, (i.e., 6 times a year). Let us determine their bimonthly payment. Here S = 30, 000, r = 0.085, m = 6, t = 7. We want to find R. Now i = r m = and n = mt = 6 7 = 42. Thus, R = Si (1 + i) n , 000 ( 6 ) = ( = ) 42 1 Hence, the owners of the Fast Printing Company should deposit $ bimonthly for 7 years into the account.. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 19 / 24

20 Present Value of a Future Ordinary Annuity Theorem Suppose that an annuity consists of payments of R dollars, at the end of each period, paying r% interest per year compounded m times a year for t years. Let i = r m and n = mt. Then the present value, P, of this annuity is given by [ ] 1 (1 + i) n P = R. i. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 20 / 24

21 Example We find the present value of the following annuity: Payments of $750 per quarter, interest rate of 6.75% per year compounded quarterly, for 15 years. Here R = 750, r = , m = 4, and t = 15. Now i = r m = = and n = mt = 4 15 = 60. Thus, [ ] [ ] 1 (1 + i) n 1 ( ) 60 P = R = 750 = i Hence, the present value of the given annuity is $28, S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 21 / 24

22 Exercise: Samantha deposits $100 at the end of every other month into an account that pays 4.2% interest per year compounded bimonthly. How much money will she have after 2 years to buy a computer when she begins college? Solution: We have R = 100, r = 0.042, m = 6, and t = 2. So i = r m = = and n = mt = 6 2 = 12. Thus, [ (1 + i) S = n ] [ 1 R = 100 i ( ) Samantha will have $1, after 2 years to buy a computer. ] = D. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 22 / 24

23 Exercise: Find the future value of the annuity due if the payment per period and the interest rate is as follows: R = $650; 7.9% interest compounded monthly for 13 years. Solution: We have R = 650, r = 0.079, m = 12, and t = 13. So i = r m = and n = mt = = 156. Thus, [ (1 + i) S = n+1 ] 1 R R i [ ( ) ] 12 1 = Hence, the future value is $177, D. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 23 / 24

24 Exercise: To stay competitive, owners of a local rental company will need 14 new trucks after 6 years. The cost of one truck is $75, 000. They plan to make bimonthly payments into a sinking fund at 6.55% interest compounded bimonthly. How much money should be deposited every other month into the fund so that 14 new trucks can be bought after 6 years? Solution: The total amount that we need in the sinking fund is S = = Also r = , m = 6, and t = 6. So i = r m = and n = mt = 6 6 = 36. Thus, Si R = (1 + i) n 1 = ( ) 6 ( ( ) 36 = Hence, the amount of bimonthly payment is $23, D. S. Malik Creighton University, Omaha, NE () Chapter 4: Section 4-2 Annuities 24 / 24