6- Easy approach to compounding technique

Approach to compounding technique.

The video discusses the difference between linear function and exponential function. A given example for a deposit of$1000 with 6% interest yearly for a period of 3 years. How we can get the future values every year based on simple interest and also, how can we estimate the future value based on compound interest? The additional subject is how to derive the expression for 1$ compounded yearly with an interest of 100%.

The video has a closed caption in English.

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 a period of one year, then change the period of compounding to include fewer times than a year. If we have a linear function that 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*P₀*n), where n=1.

Assume that the value of the function at time t₁=y₁, and at time t₂=y₂ and at time t₃=y₃.

The difference between these ordinates is constant which means that y₁-y₀=y₂-y₁=y₃-y₂. To convert that linear function to an exponential function, we will make y₁/y₀=y₂/y₁=y₃/y₂, our start of making the exponential function will be at time t₁. From the linear function that represents the case of simple interest, we get the value of P₀ at a time t_o, and also the value of FV1 at the time t₁, the x-axis represents the time in years for both graphs.

The FV1 is the value of the final value at time t₁, while P₀ 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 P₀, 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 to get the value of Fv1 at time t1, where n represents the time in years. The future value FV₁ at time t₁=P₀+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 t₀ and FV1 at time t₁ 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 t₂ 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/P₀)*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 getting 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.