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  • Date Submitted: 12/05/2015 09:32 PM
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Compound Interest
1. Compound Interest
The simplest example of interest is a loan agreement two children might make:
“I will lend you a dollar, but every day you keep it, you owe me one more penny.”
In this example, the interest rate is 1%/day and the amount owed after t days is
A(t) = 1 + .01t
In this formula, the quantity .01t is the interest at time t. (In general, the interest
is the difference between what was borrowed and what is owed.)
Remark. In the above example, we can describe the interest rate as a percent
(1%) or as a numeric value ( .01). When we state an interest rate we will always
mean a numeric value, and not a percent, unless we indicate otherwise.
If, as above, the interest is proportional to time, then we say that the interest
is simple interest. Thus, if we borrow P at rate i simple interest, the amount owed
at time t is
A(t) = P + itP = (1 + it)P
Example 1. On Jan. 1 of a non-leap year, I borrow $5,000 at 3% simple
interest per year. How much do I owe on May 1? How much would I owe after 3
Solution. On May 1, I have had the money for 31 + 28 + 31 + 30 = 120 days,
which is 120/365th of a year. Hence, I owe

.03 5000 = 5049.32

After 3 years, I owe
(1 + 3(.03))5000 = 5450.00
Remark. In dealing with money, we will usually round our answers to the
nearest penny. When reporting interest rates we will round to at least three significant figures, e.g. 6.13%.
Remark. In computing interest, it is typically assumed that interest is earned
only on either the first day the account is open or the last day, but not on both.
Which day doesn’t matter in computing the interest. Thus, in Example 1, it is
correct not to count the interest earned on May 1.
The question of how many days are in a year is actually somewhat complicated.
The most obvious answer is that a year will have either 365 or 366 days, depending
on whether or not it is a leap year. It has to be remembered, however, that





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