If you are paying an APR of 6%, what is your monthly interest rate i?
Solution: The APR (Annual Percentage Rate) is the percentage charged annually for interest. Since the APR is a percentage it must be multiplied by .01 to convert the percentage to an interest rate in decimal form. In most problems in finance, the interest is compounded and paid quarterly, monthly, daily or some other period, so the annual interest must be divided by the number of periods in a year to get the interest rate per period.
The formula to convert the APR to the periodic interest rate i is
i = (APR * .01) / m where m is number of periods in a year.
For this example,
i = (6 * .01) / 12 or .005
Problem: How many monthly payments n are in a four year loan
Solution: If we are given the term in years, the total number of periods n is calculated by using the following formula where m is the number of periods in a year
n = term * m
So, for a 4 year loan, the number of monthly payments n is 4 * 12 or 48.
In finance, the interest is usually compounded, but what is compounding?
Compound interest is interest computed on both principal and interest and the resulting principle and interest is the compound amount.
Compounding is accomplished by applying a rule that the previous amount is multiplied by the factor (1 + i), where i is the interest rate. If one dollar is compounded, we get one dollar plus the interest, $1 + $i, which is $1 times the factor (1 + i). We have thus shown that the rule holds for period one. By applying the same rule to the previous result, for n periods, then the final amount compounded becomes
The symbol ^ in the formula indicates an exponent, enabling exponential growth when the number of periods n is large.
On the other hand, the formula for simple interest per unit dollar is $1(1 + i*n) enabling only straight line growth.
The following table compares the amounts resulting from compounding using the formula, $(1 + i)^n versus simple using the formula, $(1 + i*n). At the end of the first month, there is no difference. The compound amount after the second month is 1.0201, which is only .0001 greater than what we would get for simple interest. This small difference is the interest earned on interest. From the table we see that the interest earned by compounding increases exponentially at ten years (120 months) and at 20 years (240 months), while the simple amount follows a straight line.
Now, if we double the interest rate to 2% for the 20 years, then the compound amount is about ten times greater than what we would get with 1% when compounded and 20 times what we get for simple interest at the same 2% interest.
n i $(1 + i)^n $(1 + i*n) Difference
1 1% 1.01 1.01 None
2 1% 1.0201 1.02 .0001
120 1% 3.30 2.20 1.01
240 1% 10.89 3.40 7.49
240 2% 115.89 5.80 110.09