Example 1. = A. Given an investment of $3,000 at 5% compounded quarterly for 6 years, find the interest earned and the future value. Prepare a table showing the growth of the account balance and illustrate that growth with a chart. r = 0.05 ppy = 4 i = r/ppy = 0.05/4 = 0.0125 t = 6 n = (t)(ppy) = (6)(4) = 24 P = 3,000 A = ? I = ? Calculator Solution Compare the$1,042.05 interest earned to the $900 that would have been earned with simple interest. Notice that both the future value and the interest given by the formulas is off by one cent. B. For this same investment, suppose the interest is compounded monthly instead of quarterly. Find the interest earned and the future value. r = 0.05 ppy = 12 i = r/ppy = 0.05/12 t = 6 n = (t)(ppy) = (6)(12) = 72 P = 3,000 A = ? I = ? Calculator Solution Notice that almost five dollars more interest will be earned if the interest is compounded monthly instead of quarterly. ## Example 2 A. Find the present value of an investment if the future value is$1,000. The investment pays 4.5% compounded semiannually for seven years.

r = 0.045
ppy = 2
i = r/ppy = 0.045/2 = 0.0225
t = 7
n = (t)(ppy) = (7)(2) = 14
P = ?
A = 1,000
I

Calculator Solution

The present value for the corresponding simple interest problem was $760.46. Remember that with compound interest more interest is earned because the interest is periodically added to the balance. Consequently, the interest itself earns interest. Since more interest is being earned, it requires less of an investment to achieve the same future value. B. Suppose the interest is compounded daily instead of semiannually. Find the present value. r = 0.045 ppy = 365 i = r/ppy = 0.045/365 t = 7 n = (t)(ppy) = (7)(365) = 2555 P = ? A = 1,000 I Calculator Solution Notice that the present value is somewhat lower than in the example above. Since the interest is paid more frequently (daily instead of semiannually) the total interest paid is greater which lowers the present value even more. The change, however, is much less dramatic than going from simple interest to interest compounded semiannually. ## Example 3 The interest on a 4.5 year investment paying 3.6% compounded monthly was$245. How much was invested and what was the future value?

r = 0.036
ppy = 12
i = r/ppy = 0.003
t = 4.5
n = (t)(ppy) = (4.5)(12) = 54
P = ?
A = ?
I = $245.00 Since we know neither P nor A, we cannot use either the future value formula nor the present value formula directly to answer this question. However, with a little algebra we can derive a formula that will give us the present value. The only thing we know is that the interest is$245 so let’s start with the interest formula:

Substitute the future value formula for A:

Factor P out of the two terms on the right hand side of the equation:

Divide both sides by

to solve for P:

Substituting in the known values for I, i, and n, we obtain the following:

Calculator Solution

What is the future value of an investment of $600 at 2.3% compounded daily for 10 years? r = 0.023 ppy = 365 i = r/ppy = 0.023/365 t = 10 n = (t)(ppy) = (10)(365) = 3650 P = 600 A = ? I = ? Calculator Solution Compare the future value of$755.15 with the $738.00 that would have resulted from simple interest. ## Example 5 What is the purchase price of a$500 savings bond that earns 6% compounded monthly and matures in 5.5 years? How much interest is earned?

r = 0.06
ppy = 12
i = r/ppy = 0.06/12 = 0.005
t = 5.5
n = (t)(ppy) = (5.5)(12) = 66
P = ?
A = 500
I = ?

Calculator Solution