Video:What Is the Formula for Finding Compound Interest?

Transcript:What Is the Formula for Finding Compound Interest?

Hi, I'm Zoya Popova for About.com, and today I'm going to show you how to calculate compound interest. Unlike simple interest, compound interest is calculated on both the amount of the principal and the interest previous accumulated.

Compound Interest Formula

So if P is the principal, and r is the interest rate, at the end of year one, the total amount of debt accumulated will be the principal plus the interest calculated on the principal: A(1) = P + Pr = P(1+r).

At the end of year two, the total amount of debt accumulated is the amount from last year, A1, plus the interest calculated on that amount: A(2) = A(1) x (1+r). But because A(1) is itself P(1+r), A(2) = P x (1+r) x (1+r)=P x (1+r)^2. Similarly, at the end of year three, A(3) = A(2) x (1+r) = P x (1+r)^3. So the general formula for compound interest is A(t) (the amount of debt accumulated by the end of year t) equals P x (1+r)^t.

Example of Compount Interest

So let's use this formula in an example. P = 500r = 10%t = 4. If you borrow $500 at 10% annual interest rate, and the interest is compounded each year, at the end of 4 years, you will owe:500 x (1+0.10)^4 = 732.05. And by subtracting the principal from this amount, you can calculate the amount you paid in interest: 732.05 – 500 = 232.05. Notice that if you divide total interest paid over 4 years by the number of years, you will find that you've been paying, on average, about $58 per year: 232.05/4 = 58.01.

This equals roughly 11.6% of your principal. This is obviously much more than you would have to pay in a simple interest scenario, which is why lenders of financial assets prefer compound interest over simple interest.

Compound Interest with Different Intervals

Another tricky part with compound interest is that interest can be compounded at different intervals. For instance, if in our previous example, P = $500r = 10% per year t = 4 years, interest were to be compounded on a monthly basis, that means that at the end of the 1st month, the lender would calculate the interest owed using 1/12 of the annual interest rate.

Practice the Interest Formula

Then, the lender would add that amount to the principle, and use the sum to calculate interest for the 2nd month:1st month: 500 + 500x0.10/12 = 500 x (1+0.10/12). 2nd month: 500 x (1+0.10/12) x (1+0.10/12) = 500 x (1+0.10/12)^2. This would go on for 48 months, until 4 years are up. 48th month: 500 x (1+0.10/12)^48.

This brings us to the formula which is used to calculate compound interest when compoundings have a frequency of more than once a year: A = P x (1 + r/n)^tn, where A is the total amount of debt accumulated by the end of year t, r is the annual interest rate, n is the number of compoundings a year, and t is the number of years in the credit period. And this is the formula for compound interest.

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