Chapter 3 Mathematics of Finance

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1 Chapter 3 Mathematics of Finance Section 3 Future Value of an Annuity; Sinking Funds

2 Learning Objectives for Section 3.3 Future Value of an Annuity; Sinking Funds The student will be able to compute the future value of an annuity. The student will be able to solve problems involving sinking funds. The student will be able to approximate interest rates of annuities. 2

3 Table of Content Future Value of an Annuity Sinking Funds Approximating Interest Rates 3

4 Terms ordinary annuity future value sinking funds 4

5 Definition of Annuity An annuity is any sequence of equal periodic payments. An ordinary annuity is one in which payments are made at the end of each time interval. If for example, $100 is deposited into an account every quarter (3 months) at an interest rate of 8% per year, the following sequence illustrates the growth of money in the account after one year: (1.02) 100(1.02)(1.02)(1.02) (1.02) 100(1.02) 100(1.02) rd qtr 2 nd quarter 1 st quarter This amount was just put in at the end of the 4th quarter, so it has earned no interest. 5

6 Future Value of an Annuity Deposit of $100 every 6 months into an account that pays 6% compounded semiannually, over 3 years. How much money will be after the last deposit? 6

7 Future Value of an Annuity Let s look at it in terms of a time line using A = P(1 + i) n. 1 yr 2 yr 3 yr Years Number of periods $100 $100 $100 $100 $100 $100 Deposit $100(1.03) $100(1.03) 2 $100(1.03) 3 Future Value $100(1.03) 4 $100(1.03) 5 7

8 Future Value of an Annuity 1 Total amount in the account after six deposit: S = (1.03) + 100(1.03)^ (1.03)^3 + 00(1.03)^ (1.03)^5 2 Multiply each side by 1.03: 1.03S = 100(1.03) + 100(1.03) (1.03) (1.03) (1.03) (1.03) 6 Subtract Equation 1 from Equation S - S = 100(1.03) S = 100[(1.03) 6-1] 8

9 Future Value of an Annuity 9

10 General Formula for Future Value of an Annuity where FV = future value (amount) PMT = periodic payment i = rate per period FV PMT 1 i n 1 i n = number of payments (periods) Note: Payments are made at the end of each period. 10

11 Example Suppose a $1000 payment is made at the end of each quarter and the money in the account is compounded quarterly at 6.5% interest for 15 years. How much is in the account after the 15 year period? 11

12 Example Suppose a $1000 payment is made at the end of each quarter and the money in the account is compounded quarterly at 6.5% interest for 15 years. How much is in the account after the 15 year period? n Solution: (1 i) 1 FV PMT i FV 4(15) ,

13 Graphical Display 13

14 Balance in the Account at the End of Each Period 14

15 Amount of Interest Earned How much interest was earned over the 15 year period? 15

16 Amount of Interest Earned Solution How much interest was earned over the 15 year period? Solution: Each periodic payment was $1000. Over 15 years, 15(4)=60 payments were made for a total of $60,000. Total amount in account after 15 years is $100, Therefore, amount of accrued interest is $100, $60,000 = $40,

17 Example of Future Value of an Ordinary Annuity Example 1 Interest: Deposits = 20(2,000) = $40,000 Interest = value deposits = 96, ,000 = $56,

18 Example of Future Value of an Ordinary Annuity Balance Sheet The Table in next display is called a balance sheet. Taking a closer look we can see that the first line is a special case because the payment is made at the of the period and no interest is earned. Eacg subsequent line is computed as follows: Payment + Interest + Old Balance = New Balance The amounts at the bottom of each column agree with the results obtained with the formula of the Future Value of an Ordinary Annuity. 18

19 Example of Future Value of an Ordinary Annuity Balance Sheet A B C D Period Payment Interest Balance 1 2, , , , , , , , , , ,000 1, , ,000 1, , ,000 1, , ,000 1, , ,000 2, , ,000 2, , ,000 2, , ,000 3, , ,000 3, , ,000 4, , ,000 4, , ,000 5, , ,000 6, , ,000 6, , ,000 7, $ 96, TOTALS 40,000 $ 56,

20 Sinking Fund Definition: Any account that is established for accumulating funds to meet future obligations or debts is called a sinking fund. The sinking fund payment is defined to be the amount that must be deposited into an account periodically to have a given future amount. 20

21 Sinking Fund Payment Formula To derive the sinking fund payment formula, we use algebraic techniques to rewrite the formula for the future value of an annuity and solve for the variable PMT: FV FV n (1 i) 1 PMT i i PMT n (1 i) 1 21

22 Sinking Fund Sample Problem 1 How much must Harry save each month in order to buy a new car for $12,000 in three years if the interest rate is 6% compounded monthly? 22

23 Sinking Fund Sample Problem 1 Solution How much must Harry save each month in order to buy a new car for $12,000 in three years if the interest rate is 6% compounded monthly? Solution: FV i PMT n (1 i) pmt

24 Sinking Fund Sample Problem 2 The parents of a newborn child decide that on each of the child s birthday up to the 17 th year, they will deposit $PMT in an account that pays 6% compound annually. The money is to be used for college expenses. What should the annual deposit ($PMT) be in order for the amount in the account to be $80,000 after the 17 th deposit? 24

25 Sinking Fund Sample Problem 2 Solution 25

26 Sinking Fund Example 2 Computing the Payment A company estimates that it will have to replace a piece of equipment at a cost of $800,000 in 5 years. To have the money available in 5 years, a sinking fund is established by making equally monthly payments into an account paying 6.6% compounded monthly. A. How much should each payment be? B. How much interest is earned during the last year? 26

27 Sinking Fund Example 2 Computing the Payment 27

28 Sinking Fund Example 2 Computing the Payment (continued) 28

29 Example of Sinking Fund Problem (1) Example 2 Computing the Payment (continued) Solution: B. (Cont.) Interest: 800, , = 181, Growth in the 5yr 12 x 11, = 135, Payments during the 5yr 181, , = $46, Interest during the 5yr 29

30 Sinking Fund Example 3 Growth in an IRA Jane deposits $2,000 annually into an IRA that earns 6.85% compounded annually. Due to a change in employment, these deposits stops after 10 years, but the account continues to earn until Jane retires 25 years after the last deposit was made. How much is in the account when Jane retires? 30

31 Sinking Fund Example 3 Growth in an IRA 31

32 Sinking Fund Problem 2 (cont.) Example 3 Growth in an IRA (cont.): Now we use the compound interest formula (A = P(1+i)^n) with P = $27,437.89; i = , and n = 25 to find the amount at the moment of retirement: A = P(1+i)^n = 27,437.89(1.0685)^25 = $143,

33 Chapter 3 Mathematics of Finance Section 3 Future Value of an Annuity; Sinking Funds END Last Update: April 1/2013

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