Lesson Simple and Compund Interest

Source code of 'Simple and Compund Interest'

This Lesson (Simple and Compund Interest) was created by by bam878s(77)

: View Source, Show
About bam878s: I am a recent graduate with a B.S. Degree in Mathematics.

I've noted many interest-related problems posted on this site and thought that I would write some basic notions in interest theory.

We will first start off with a very basic principle 1 invested at a certain interest rate i.
Let's try to visualize what will happen over the course this principle is invested.
Beginning First year:  put 1 into the account.
End First Year:  1 * i is computed.
Now, what will happen next will be dependent on "simple" or "compound" interest.

--------------------------Simple Interest----------------------------------------------------------

Simple interest is just basically the assumption that the amount of interest calculated at the end of each period n is constant for each period.  For example, if we start out with 1 and the beginning of the year with a 5% annual interest rate we will have the following amount after each year.
end year 1: 1.05
end year 2: 1.10
end year 3: 1.15
.
.
.
end year n: 1 + in
         The function a(n) = 1 + in is called the accumulation function.  It is so-called because it is how we calcualte a sum at the end of a given period of time. 
To Illustrate, say we start with 5,000 and invest it at an annual interest rate of 5% for 10 years.
How much do we have at the end of year 10?
We use the accumulation function a(n)
5,000 * a(n) = amount at the end of year 10.
so 5,000 *  (1 + .05(10)) = 5,000 (1.5) = 7500.
Simple interest is usually not real-world applicable, but in practice it is important to make note of it.
In the real world, most accounts are credited with compound interest, which I will discuss next.

-----------------------------Compound Interest-----------------------------------------------------

Compund interest is the assumption that the interest at the end of the year is reinvested automatically into the account.  To Illustrate what is going on, I'll start with an amount 1 invested at an annual interest rate i.
Begiining of year 1: 1
End of year 1: 1 + 1(i) = 1(1+i)
End of year 2: 1(1+i) + i{1(1+i)) = 1(1+i)[1+i]
.
.
.
End of year n: 1(1+i)(1+i)(1+i)...(1+i) = {{{1*(1+i)^n}}}
        The function a(n) = {{{(1+i)^n}}} is called the accumulation function for compound interest.
Note the amound of interest is not the same for all periods.  The principle thus grows at an exponential rate.  Therefore, the principle will gain more value over time at a compund rate than at a simple rate.  After the first year, the simple interest and compund interest will produce the same results.  However, over a longer period compund produces a larger accumulated value.  
The word compund basically means that interest is being credited.  You may have heard the terms "compounded semiannually" or "compunded quarterly".  This basically means that interest is credited twice a year or 4 times a year on the account.  This gives rise to my next discussion on nominal rates of interest.

--------------------------Nominal Rates of Interest----------------------------------------------

We will first make some notation for this section.  The point is to simplify and not to confuse.  
The symbol for a nominal rate of interest compounded m times per period is {{{i^((m))}}}.  This is basically the yearly rate.  To find the effective interest rate per {{{m^th}}} period we take {{{(i^((m)))/m}}}.  
      For instance let's say we have an account that has an annual interest rate of 8% or .08.  Also, let's assume that this account credits interest at a semiannual rate.  To find this semiannual rate all we do is take {{{.08/2}}} = .04.  That is, 4% is the effective rate of interst per compund period.
To use this with the above accumulation function
(1 + {{{(i^((m)))/m}}}) ^ n*m
where n = number of years and m is the compund rate (i.e. semiannualy = 2, quarterly = 4, monthly = 12).

In my next lesson, I will take on the notion of discounting.  In this lesson, we took money at a certain point of time and calculated its "future value".  Well, discounting is the opposite of this process.  Basically we take money at a point in the future and "discount" it to find the value in the present.  More to come soon!