Before going through these examples, you might want to read the section on how to use a calculator.
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 = ?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 = ?Notice that almost five dollars more interest will be earned if the interest is compounded monthly instead of quarterly.
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
IThe 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
INotice 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.
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 = $245.00Since 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:
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 = ?Compare the future value of $755.15 with the $738.00 that would have resulted from simple interest.
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 = ?
Suppose the interest rate on the savings bond in the previous problem is raised to 7.2% compounded monthly. What is the purchase price and how much interest is earned?
r = 0.072
ppy = 12
i = r/ppy = 0.072/12 = 0.006
t = 5.5
n = (t)(ppy) = (5.5)(12) = 66
P = ?
A = 500
I = ?Compare these results with the first example. With a higher interest rate, the amount of interest earned is greater and consequently the purchase price goes down.
Suppose the $500 bond earns 6% compounded monthly but matures in 7 years rather than 5.5 years. What is the purchase price and how much interest is earned?
r = 0.06
ppy = 12
i = r/ppy = 0.06/12 = 0.005
t = 7
n = (t)(ppy) = (7)(12) = 84
P = ?
A = 500
I = ?Compared with the first example, the amount of interest is greater because the investment had more time to earn interest. Consequently, the purchase price is lower.
What is the annual percentage yield corresponding to 6% compounded monthly?
r = 0.06
ppy = 12
i = r/ppy = 0.06/12 = 0.005
What is the annual percentage yield corresponding to 6.01% compounded quarterly?
r = 0.0601
ppy = 4
i = r/ppy = 0.0601/4 = 0.015025Notice that this APY is lower than that in the first example even though the nominal annual rate in this example is higher.
