# The Theory of Interest

__Current Situation__

Interest can be viewed as the periodic money paid for a debt taken. Due to the 2008 economic crisis, the central banks of most developed countries has decided to lower their interest rate levels that is close to 0. This interest rate level is also called the nominal interest rate. Real interest rate is obtained by subtracting the nominal interest rate by inflation rate. Thus, effectively, most developed countries are having negative interest rates.

__Assumption
__However, this article, is talking about interest that one could get assuming certain conditions. Some of the conditions, though unrealistic but is necessary for ease of analysis, are (i) constant interest rate level at all times, (ii) nominal interest rate levels are positive, (iii) there is no default risk and (iv) interest is charged linearly.

__Concept needed for CFA level 1
__CFA level 1 tests candidate on the concept of interest, particularly, constant interest rate level throughout investment period.

Suppose you lend $1,000 to me, I am charged 3% linear interest every year and I have decided to repay $1,000 plus the accumulated interest by the end of 3 years. I believe the question that would interest you is the amount of money that you would get at the end of 3 years.

There are 2 portions which are (i) the principal portion and (ii) the interest portion. The following equation shows the amount which I owe you for both principal and interest portion.

Amount Owe

=Interest+Principal

=Interest_{1}+Interest2+Interest3+Principal

=$1,000×0.03×1.03^{2}+$1,000×0.03×1.03+$1,000×0.03+$1,000

=$1092.73

where stands for the interest at end of i^{th} year which accumulates interest till the end of 3^{rd} year.

The formula to compute such is

P(1+r)^{n}

where

- is the loan amount
- is the interest rate of the loan
- is the number of years where the loan exists

On the other hand, suppose I agreed to return $1,000 to you at the end of 3 years while being charged 3% for each year. The value of this agreement today would be simply

P/(1+r)^{n}

The moral of the story, $1 today doesn’t usually worth $1 in the future and vice versa.