# Compound Interest Rate Calculator

Created by Tibor Pál, PhD candidate
Reviewed by Dominik Czernia, PhD and Jack Bowater
Last updated: Jan 15, 2023

Use the compound interest rate calculator to compute the precise interest rate that is applied to an initial balance that reaches a certain surplus with a given compound frequency over a certain period.

If you read further, you can also get some insight into how compound interest rates work, and you can learn the compound interest rate formula, so you will know how to calculate it from scratch in the future.

## How does the compound interest rate calculator work?

Compound interest is the addition of interest to the existing balance (principal) of a loan or saving, which, together with the principal, becomes the base of the interest computation in the next period.

In other words, compounding interest means reinvesting the interest rather than paying it out, so that in the following period you earn interest on the principal sum plus the previously accumulated interest. Therefore, the more often the interest is added to (capitalized on) the principal amount, the faster your balance grows.

Using the definition above, the compound interest rate is the annual rate where the compounding frequency is taken into account.

Now, to help you understand the concept of compound interest even more, let's have a quick review of the parameters you need to provide to this compound interest rate calculator:

• Initial balance - the amount of money at the beginning of the specific term that constitutes the principal amount or, in other words, the present value (PV) of investment;

• Final balance - the amount of money at the end of the given period, or the future value (FV);

• Surplus - the difference between the initial balance and the final balance; it is the interest gain on your initial principal;

• Term (t) - the given interval of which interest compounds onto your principal; and

• Compounding frequency (m) - the period after which the interest will be calculated on the principal amount and then added to it (capitalized on it).

Note, that if you leave the initial and final balances unchanged, a higher the compounding frequency will require a lower interest rate. This is because a higher compounding frequency implies more substantial growth on your balance, which means you need a lower rate to reach the same amount of total interest.

You may choose to set the frequency as continuous, which is a theoretical limit of recurrence of interest capitalization. In this case, interest compounds every moment, so the accumulated interest reaches its maximum value. To understand the math behind this, check out our natural logarithm calculator, in particular the The natural logarithm and the common logarithm section.

After setting the above parameters, you will immediately receive your exact compound interest rate.

## Compound interest rate formula

The basic formula for computing the Future Value (FV) of your initial balance is the following:

$FV = PV \times \bigg(1 + \frac{r}{m}\bigg)^{mt}$

Based on this, the interest rate is equal to:

$r = m \times \bigg(\bigg(\frac{FV}{PV}\bigg)^{\frac{1}{mt}} - 1\bigg)$

If the compound frequency is continuous, the basic formula takes the following form, where $e$ stands for constant exponent:

$FV = PV \times e^{rt}$

After some algebra, the compound interest rate formula is:

$r = ln\bigg(\frac{FV}{PV}\bigg) \times \frac{1}{t}$

where:

• $PV$ - Present Value;
• $FV$ - Future Value;
• $t$ - term; and,
• $m$ - compound frequency in a given term;
• $r$ - interest rate.
Tibor Pál, PhD candidate
Initial balance
$Final balance$
Surplus
\$
Compounding frequency
monthly (12/Yr)
Term
yrs
mos
Compound interest rate
5.8923
%
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