 # 6a- Easy approach to frequencies of compounding.

## Frequencies of compounding.

In the next video, a description of the value of the valance of \$1 invested based on the interest of 100% compounded semi-annually & quarterly& monthly, and finally compounded daily. The previous post includes the discussion of the future value of \$1 that is compounded 100% yearly.

The video has a closed caption in English.

You can click on any picture to enlarge then press the small arrow at the right to review all the other images as a slide show.

The compounding frequency is the number of times per year (or rarely, another unit of time) the accumulated interest is paid out or capitalized (credited to the account), on a regular basis. The frequency could be yearly, half-yearly, quarterly, monthly, weekly, daily, or continuously (or not at all, until maturity). Quoted from the definition of compound interest.

## The balance of \$1 for the interest of 100%-compounded semi-annual.

This is the first type of frequencies of compounding which is compounding semi-annually.

After one year based on compound interest, 100% compounded yearly. The value was \$2.00, we want to find the balance value-based of 100% compounded-semi annually. The first step is to get the future value at time t1=0.50 year=1 semi-annual. As we can see from the graph that the Fv-1=(1+2)*0.50=\$1.50.

For the value of FV-2 after one year. We get the multiplication factor for the value at t=2 semi-annual, which is=(1.50)*1.50)/1.00=\$2.25.

The compounding starts after the first semi-annual and the slope gets increased based on the new ratio. This is the process of changing from linear to exponential function at t=0.50 year.

Recall that the Fv equation=P0*(1+i)^n. In this case p0=\$1, new i/n=(100/100)/2=50%. the power raised is (i*t)=2*1=2.00. The future value obtained is matching the value in table 4.13. the FV-2=(1.5*1.5)/1=\$2.25.

## The balance of \$1 for the interest of 100%-compounded quarterly.

This is the second type of frequencies of compounding which is compounding quarterly.

From the last post, we have estimated the future value of \$1, after one year based on a compound interest 100% compounded yearly. The value was \$2.00, we want to find the balance value-based of 100% compounded-quarterly.

For the value of FV-2 after one year. We get the multiplication factor for the value at t=1 quarter of a year, which is=1.25..

The compounding starts after the first quarter and the slope gets increased based on the new ratio. This is the process of changing from linear to exponential function at t=0.25 years.

Recall that the Fv equation=P0*(1+i)^n. In this case P0=\$1, new i/n=(100/100)/4=25%. The power raised is (i*t)=4*1=4. The value obtained is matching the value in table 4.13. the FV-2=(1.25*1.25)/2=\$2.4414.

## The balance of \$1 for the interest of 100%-compounded monthly.

This is the third type of frequencies of compounding which is compounding monthly.

From the last post, we have estimated the future value of \$1, after one year based on a compound interest 100% compounded yearly. The value was \$2.00, we want to find the balance value based of 100% compounded-monthly.

For the value of FV-2 after one year. We get the multiplication factor for the value at t=1 month of a year, which is=1.083333

The compounding starts after the first month and the slope gets increased based on the new ratio. This is the process of changing from linear to exponential function at t=(1/12) year.

Recall that the Fv equation=P0*(1+i)^n. In this case P0=\$1, new i/n=(100/100)/12=8.3333%. The power raised is (i*t)=12*1=12. The value obtained is matching the value in table 4.13. the FV-2=(1.08333*1.08833)/1=\$2.4414

## The balance of \$1 for the interest of 100%-compounded daily.

This is the fourth type of frequencies of compounding which is compounding daily.

From the last post, we have estimated the future value of \$1, after one year based on a compound interest 100% compounded yearly. The value was \$2.00, we want to find the balance value based on 100% compounded-monthly.

For the value of FV-2 after one year. We get the multiplication factor for the value at t=1 day of a year, which is=1.0027397.

The compounding starts after the first day and the slope gets increased based on the new ratio. This is the process of changing from linear to exponential function at t=(1/365) year.

Recall that the Fv equation=P0*(1+i)^n. In this case P0=\$1, new i/n=(100/100)/365=2.739726*(10^-3). The power raised is (i*t)=365*1=365.
The value obtained is matching the value in table 4.13. the FV-2=(1.0027397.*1.0027397)/1=\$2.714567.

## The second video I used for the illustration.

This is a short video, in which I have explained in brief the different kinds of compounding. A solved example is given. the video has a subtitle and a closed caption in English.

For a given interest of 6%, for interest to be compounded annually, it will be the same value as 6%.
For different frequencies like semi-annually, quarterly, the interest value will change. the nest slide image shows these values.

When the number of years exceeds one year, it will be raised as power and to be multiplied by I.

These are some examples of the different interest rates, with different, n values of years.

## Solved example 3.7 to estimate the future value for a given deposit.

This is solved example 3.7, for which it is required to estimate the future value for a given Po=\$500, with 6% interest compounded quarterly, for n=3 years.

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

In the next post, we will introduce more solved problems for the frequencies of compounding.

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

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