- Approach to compounding technique.
- A solved example is given as an illustration of the process.
- How to get the value of Fv1?
- How to get the value of Fv2 for the compound interest of 6%?
- How to get the value of Fv3 for the compound interest of
- 6%?
- Plotting the future values of the solved example.
- How to get the future value of $1 after one year with interest =100%?

## Approach to compounding technique.

Compounding technique: If the interest is compounded, the interest earned at the end of the year will be added to the principal and will go on until the end of time. Future values are calculated by using this compounding interest.

As interest rates increase, compounding interest also increases, which means if you want a large sum of money, interest rates must be high. This is achieved by compounding in shorter periods.

We will start to talk about the compounding technique by how to convert the simple interest to compound interest after one year, then change the period of compounding to include fewer times than a year. If we have a linear function which is represented by a line with equation y=mx+c, where m is the slope and c is **the intersecting height** with the Y-axis The slope is constant.

The difference between ordinates is equal to (i*P_{0}*n), where n=1.

Assume that the value of the function at time t_{1}=y_{1}, and at time t_{2}=y_{2} and at time t_{3}=y_{3}.

**The difference between these ordinates is constant** which means that y_{1}-y_{0}=y_{2}-y_{1}=y_{3}-y_{2}. To convert that linear function to an exponential function, we will make y_{1}/y_{0}=y_{2}/y_{1}=y_{3}/y_{2}, our start of making the exponential function will be at time t_{1}. From the linear function that represents the case of simple interest, we get the value of P_{0} at a time t_{o}, and also the value of FV1 at the time t_{1}, the x-axis represents the time in years for both graphs.

The FV1 is the value of the final value at time t_{1}, while P_{0} is the starting present value at the time t_{o} . The left-side sketch represents the exponential function form, compound interest of the linear function keeping Fv1/Fv0=Fv2/Fv1=Fv3/Fv2.

### A solved example is given as an illustration of the process.

A solved example is given where we have P_{0}, and the present value of an investment deposited in a bank is $1000, at a time t_{o}. The interest rate is 6% as simple interest for each year.

### How to get the value of Fv1?

We are interested in getting the value of Fv1 at time t1, where n represents the time in years. The future value FV_{1} at time t_{1}=P_{0}+i *P0*i*(n)=1000+0.06*(1000*)*(1)=$1060. On the right side is the compounded graph of the same problem, too and P0 is still the present value at time t_{0} and FV1 at time t_{1} is the same with the estimated value of $1060.

### How to get the value of Fv2 for the compound interest of 6%?

The future value of the money at time t_{2} which is FV2 will be different from the one estimated for the simple interest rate. FV2 due to compounding can be estimated by multiplying (FV1/P_{0})*Fv1. The P_{o} value is $1000.

We have obtained the value of Fv1 as $1060. If we estimate the value of FV2 it will be=(1060/1000)*(1060)=$1123.60, please refer to the next image for more details.

### How to get the value of Fv3 for the compound interest of

### 6%?

The future value of the money at time t3 which is FV3 can be estimated by multiplying (FV2/P1*Fv2. If we estimate the value of Fv3, it will be (1123.6)^2/1060=$1191.0.

### Plotting the future values of the solved example.

The two graphs are drawn together for the invested money at a simple interest of 6% for three years, the above graph shows the investment of $1000, but for a compound interest of 6% compounded yearly.

The difference can be seen in that the slope is increased in the case of compound interest.

### How to get the future value of $1 after one year with interest =100%?

This is an application for the conversion from simple interest to compound interest, it is required to get a future value of $1, after one year based on a compound interest 100% compounded yearly.

The future value of the $1 after one year with 100%, as 100% compounded yearly can be found to be=1$+(100/100)*(1)*(1)=$2.00. The value matches Table 4.13 for the compounded value of $1.

This is the pdf file used in the illustration of this post.

For a useful external resource, Engineering Economy. Applying Theory to Practice.

In the next post, we will derive the expression for the different frequencies of the compound interest. The next post will introduce the various types of frequencies of compounding.