Understanding how the exponent in the compound interest formula
dramatically increases the future value of the investment is critical
to the field of banking, both for the banker and the client.
Negative Exponents The preceding examples reviewed positive exponents; what happens if the exponent is negative?
A base raised to a negative power, n, is simply the reciprocal of the base raised to the positive power, n. Consider the following example.
Imagine that you would like to retire in 20 years with $1 million in
your retirement account. Assume that you can earn a 12 percent annual
rate of return on this account, compounded quarterly. How much will
you have to invest today as a onetime contribution to be able to
retire in 20 years with $1 million?
The amount you would have to invest today is called the present
value. To solve for the present value, you would use the formula for
the present value of an investment with compound interest:
PV = Present value
C_{N} = Cash flow at time period N
r = periodic rate of interest, which is calculated as
N = number of periods, which is equal to the number of years multiplied by the number of periods per year
Using the information from the example above, identify the values for each variable:
PV= Present value
C_{N} = $1 million
r = (12% divided by 4 periods per year)
N = 80 (4 periods multiplied by 20 years)
Substitute the values of C_{N}, r, and N in the formula, and compute the present value.
You would have to invest $93,977.10 now at 12 percent compounded quarterly to have $1,000,000 in 20 years.
Roots
Fractional exponents also can be expressed as roots.
To understand how roots operate, try solving the following equation:
First, raise both sides of the equation to the power (the reciprocal of the original exponent, 4). This will isolate the variable y.
Then, express as a root, and determine the value of y.
The fourth root of 81 is 3. In other words, 3 multiplied by itself 4 times is equal to 81.
Exponential fractions that have numbers other than 1 in the numerator can also be expressed as roots.
Try this example. What is the cube root of 8 squared? There are two
approaches you could use to solve this problem. Both are shown below in
parallel.
The cube root of 8 squared is 4.
As you may have guessed, fractional exponents are used in business
as well. Imagine that you plan to buy a house in nine months. The
down payment is $25,000. You currently have $20,000 invested in a
certificate of deposit that compounds annually at an interest rate of
12 percent. Will your $20,000 grow sufficiently over the nine months
to cover the $25,000 you need at the time of closing, or will you have
to save additional money in the meantime?
To answer this question, use the future value formula presented below.
FV = C_{0} × (1 + r)^{N}
FV = Future value
C_{0} = initial investment
r = periodic rate of interest, which is calculated as
N = number of periods
Using the information in the example above, determine the values for each variable or constant.
FV = Future value
C_{0} = $20,000
r = (12% compounded annually)
N = (1 period for of a year)
Substitute the values of C_{0}, r, and N in the equation, and compute the future value.
In nine months, the value of $20,000 invested at 12 percent interest
compounded annually will be $21,774.27. You will have to save
additional money prior to closing to reach your $25,000 down payment.
