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CHAPTER 1

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

years?

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

1+

120

.03 5000 = 5049.32

365

dollars.

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

1

2

1. COMPOUND INTEREST

accounting...

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

years?

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

1+

120

.03 5000 = 5049.32

365

dollars.

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

1

2

1. COMPOUND INTEREST

accounting...

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