Compound Interest Formula

There are generally two types of compound interest used.

1. Periodic Compounding - Under this method, the interest rate is applied at intervals and generated. This interest is added to the principal. Periods here would mean annually, bi-annually, monthly, or weekly.
   
2. Continuous compounding: This method is based on a natural logarithm and calculates the interest at the minimum possible interval. The interest is then returned to the principal. The constant rate of growth can be equated to the Euler number, which was derived from physics. The Euler number is a well-known irrational number with a precision of over 1 trillion digits. It is represented by the letter E. 

Periodic Compound Interest Formula Overview

There are two formulas you can use to calculate compound interest, depending on what result you wish to find out. You can find out the following:

1. The total value of the deposit.
2. The total compound interest earned.
   

Value of the Deposit

Formulas can be a deterrent to many. If you aren't savvy with math, your eyes turn away from these codes or just skip them altogether. But once it's explained, it's pretty simple to understand. To calculate the total value of your deposit, the formula is as follows:

P (1+ i/n)^nt

P = Principal invested.

i = Nominal Rate of Interest.

n = Compounding Frequency or number of compounding periods in a year.

t = Time, meaning the length of time the interest is applicable, generally in years.

Simply put, you calculate the interest rate divided by the number of times in a year the compound interest is generated. For instance, if your bank compounds interest quarterly, there are 4 quarters in a year, so n = 4. This result must be multiplied to the power of the deposit period.

For example, if your deposit is for 10 years, t = 10. This whole result should be multiplied by the principal you invested. The result generated will equal the total accumulated value of your deposit.

You can find out how much your deposit is worth currently after accumulating interest.

Total Compound Interest Earned

To find out how much interest was earned, you can use the following formula for Compound Interest

P[(1+ i/n))^nt-1]

Compound Interest Equation and Calculation

To understand the compound interest equation further, we can break it down in simpler terms. If you decide to invest in a fixed deposit with compound interest, this is how you will earn interest every year.

To collapse this formula, we can pull out factors of (1+i). Simply substitute iP with (1+i) to get the following:

Formula for Annual Compound Interest

To calculate the compound interest for a number of years together, we need to multiply P(1+i) to the power of the number of years of the deposit. So we end up with this formula:

P (1+ i/n)ⁿ

This formula can be used to calculate compound interest that is compounded annually. This means you receive interest only once a year. It is added to your principal, and you continue to earn interest on the new amount.

Half-Yearly, Quarterly, Monthly Compound Interest Formula

If you are earning interest multiple times in a year, you need to factor in this number into the equation. So the formula generated is:

P (1+ i/n)^nt

This formula can also be used for instances where the interest is compounded once every two years. In this case, n = 0.5, as each year is calculated as half.

Examples of Compound Interest

For example, Rs. 10,000 is invested in a fixed deposit for 10 years. The interest is compounded every quarter which means 4 times in a year. The interest paid by the bank is 5%. To find out your nominal rate of interest, you need to divide 5 by 100 which equals 0.05. Now, we look at the formula and substitute the letters with the relevant numbers.

Calculating the Total Value of the Deposit

P (1+ i/n)^nt

Step 1: 10,000 (1+0.05/4)^4x10

Step 2: 10,000(1+0.0125)⁴⁰

Step 3: 10,000 (1.0125)⁴⁰

Step 4: 10,000 (1.64361946349)

Step 5: 16436.1946349

We can round of this total to Rs. 16,436.19. So the compound interest earned after 10 years is Rs. 6,436.19.

Calculating the Interest Earned

We can also arrive at this figure using the formula for compound interest earned. We can substitute the numbers for letters as seen below:

P[(1+ i/n)^nt -1]

Step 1: 10,000 [(1+0.05/4)^4x10 -1]

Step 2: 10,000 [(1+0.0125)⁴⁰-1]

Step 3: 10,000 [(1.0125)⁴⁰-1]

Step 4: 10,000 [(1.64361946349) -1]

Step 5: 10,000 (0.664361946349

Step 5: 6436.1946349

We can now add this interest earned to the principal amount to find out the value of the deposit. The maturity value will be Rs. 16,436.19.

The earnings through compound interest can be demonstrated with the following graph.

Simple Interest Formula vs. Compound Interest Formula

To demonstrate the difference between simple interest and compound interest, let's take for example two fixed deposits. Both deposits are of Rs. 10,000 for 10 years. The interest offered on Deposit 1 is 5% compound interest. The interest offered on Deposit 2 is 5% simple interest. The interest is calculated annually on both deposits.

From the graph above, we can see clearly the higher earnings through compound interest compared to simple interest. The difference is not too much upto the 4th year. This is because the interest accumulated over the years is added to the principal, thus making the principal significantly higher. From Year 5, there is a major difference in the interest earned. At the end of 10 years, Deposit 1 earns Rs. 6,288.95, while Deposit 2 earns Rs. 5,000. The difference between the two is Rs. 1,288.95.

Compound Interest with Monthly Contributions

If you have a fixed deposit where you can make monthly contributions, compounding interest is a little bit more complicated. For the sum invested during compounding, interest will accrue for the original investment amount plus the monthly contributions. Fixed deposits are very rare, but they are very good investments with great returns. 

For example, Rs. 10,000 is the initial fixed deposit amount. The investor deposits Rs. 1,000 every month for 5 years. If the interest is compounded annually, then the interest will be as follows:

Through this table, we can see that the interest earned is accumulated every year and added to the principal amount. The total money contributed by the investor is Rs. 10,000 initially, followed by Rs. 1,000 every month or Rs. 12,000 every year. The investor made a total contribution of Rs. 10,000 + Rs. 60,000. At the end of 5 years, the value of his deposit is Rs. 82,385.77. The total compound interest earned is Rs. 12,385.77.

The Benefits of Compound Interest

Compound interest is your biggest friend when it comes to deposits and investments. Working in favor of investments, you stand to gain much more from the interest payable. But compound interest will be your worst enemy when it is calculated on your loan or other debt. You will end up paying significantly more interest on your loan. In terms of fixed deposits, compound interest is a great way of earning more on your investment. You earn much higher returns with compound interest on long term deposits. Compounding interest monthly, quarterly and half-yearly can spike your interest even higher. The benefits of compound interest can be listed as follows:

1. Reinvestment - The interest earned will be reinvested into the same deposit.
2. Higher value of the deposit - Compound interest leads to a higher value of the deposit. Upon maturity, your deposit will be more than a deposit with simple interest.
3. Long-term savings - Compound interest deposits encourage long-term savings as the return on investment is much higher after 10 years or more.
4. Increased Earnings - Options of compounding monthly, quarterly, and half-yearly increase the interest earned.

Where Compound Interest Applies in Finance

Compound interest is used for both debit and credit aspects of the financial world. Listed below are some of the investments and credit options that use compound interest.

Investments

1. Savings Accounts
2. Fixed Deposits
3. Recurring Deposits
4. Other Certificates of Deposits
5. Reinvested Dividend Stocks
6. Retirement Funds

Debt

1. Loans
2. Credit Cards
3. Mortgages

When it is used in case of deposits and investments, we stand to benefit. On the other hand, when compound interest is charged on loans and debt, the banks and lenders stand to gain.