INTEREST

By Richard Gooden

-Principal: The amount of money initially invested.

-Accumulated Value: Denoted by A(t), the amount of money that the principal has grown to after any given time

period.

– -Interest: the money that has been gained on the principal or the difference between the accumulated value and the principal.

-Formula: Interest=Accumulated Value – Principal.

-i is defined as the effective rate of interest.-We define effective rate of interest in the nth year as:

in= [A(n)-A(n-1)]/A(n-1).-There are two types of interest:

(1) Simple Interest and (2) Compound Interest.

--Simple Interest: 1 +it (where t>=0)***This is just something to be aware of:

-Example- If we were to borrow 1000 dollars at fifteen percent simple interest for 16 days, we would assume exact simple interest, which means at the end

of the 16 days, we would owe:1000[1 + (16/365)(.15)]=1006.58.

-However, under the Banker’s rule, we would owe:

1000[1 + (16/360)(.15)]=1006.67 because they use 360 days instead of

365 days.

-Compound Interest- This is when you earn interest on interest. Your accumulated value is re-invested.

Formula: A(t)= (1 + i)t (where t >=0)

-Present Value and Future Value:

In life, compound interest is assumed unless directly stated.

In mathematics, one dollar now is not worth one dollar in three years. Similarly, one

dollar now is not worth one dollar a year ago.

Example 1: How much is 2000 dollars worth in four years from now

assuming an effective interest rate of three percent?

Solution: 2000(1 + .03)4=2251.02.

Example 2: How much is 2000 dollars worth four years ago, assuming an

effective rate of interest of three percent?

Solution: 2000(1.03)-4=1776.97.

So as we can see from the two examples, when moving forward in time, (1 + i) is

raised to the positive power of time. However, when moving back in time, (1 + i) is raised to the negative power

of time.

-Nominal Rates:

Definition: It is an interest rate that is convertible over a period other than a year.

i(m) is a nominal rate that is converted m times a year, which would make the effective rate of

interest, i={[1 + (i(m)/m)]m-1}.

Example: If you have a credit card that charges eighteen percent compounded monthly, they are charging more than eighteen percent, though the credit card

holder may not know it. Let’s see!

Solution: i(12)=.18, because it is eighteen percent compounded monthly. So, when

we calculate the effective rate of interest, i, we have

[1 + (.18/12)]12-1=.1956 or 19.56% (not 18% as stated).

Therefore a general rule is the more times your rate is compounded, the more debt you will

be in at the end of the period.