3.3 Future value of an Annuity;Sinking Funds   An annuity is any

3.3 Future value of an Annuity;Sinking Funds 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: 0.08 100 100 1 100 1.02 (1.02) 100(1.02)(1.02)(1.02) 4 100 100(1.02) 100(1.02) 2 100(1.02) 3 3rd qtr 2nd qtr 1st qtr

General formula for future value of an annuity Here, R is the periodic payment, i is the interest rate per period and n is the total number of periods. S is the future value of the annuity: 1 i S R i n 1

Notational changes FV future value (amount) PMT periodic payment FV i interest rate per period n total number of payments (1 i ) n 1 PMT i

Example Suppose a 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 (1 i ) 1 Solution: FV PMT i 0.065 4(15) 1 1 4 100,336.68 FV 1000 0.065 4

Amount of interest earned How much interest was earned over the 15 year period? Solution: Each periodic payment was 1000. Over 15 years, 15(4) 60 payments made for a total of 60,000. Total amount in account after 15years is 100,336.68. Therefore, amount of accrued interest is 100,336.68- 60,000 40,336.68.

Graphical Display grow th of periodic payments of 1000 at 6.5% Display: 3000 This graph interest compounded quarterly for 15 years (60 total payments) displays the growth of each periodic payment over 2500 time 2000 1500 1000 500 0 0 20 40 60 80

Balance in the Account at the end of each period This graph displays the balance in the account at the end of each quarter: 120000 this graph displays the balance of the account at the end of each quarter 100000 80000 60000 40000 20000 0 0 20 40 60 80

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.

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: (1 i )n 1 FV PMT i i FV PMT n (1 i ) 1

Sample problem How much must Harry save each month in order to buy a new car in three years if the interest rate is 6% compounded monthly? i FV PMT n (1 i ) 1 0.06 12 pmt 305.06 12000 0.06 36 1 1 12

Approximating interest rates Mr. Ray has deposited 150 per month into an ordinary annuity. After 14 years, the annuity is worth 85,000. What annual rate compounded monthly has this annuity earned during the 14 year period? Solution: Use the FV formula: Here FV 85,000, PMT 150 and n, number of payments is 14(12) 168. Substitute these values into the formula. Solution is approximated graphically.

Graphical solution Solution: (1 i ) n 1 FV PMT i (1 i )14(12) 1 85, 000 150 i 85, 000 (1 i)168 1 150 i By determining the point of intersection of the two graphs using a graphing calculator, we obtain an approximate solution of 0.013 or 1.3% rate of return. (1 x)168 1 85, 000 y 566.67 x 150