# EAR Calculator

*“Financial and Insurance Formulas“*(2006)

The **EAR calculator**, or **effective annual rate calculator**, is a handy tool created for computing the **effective annual rate** of an investment or a loan.

Read further and learn *what is effective annual rate*, *how to calculate EAR* with the *effective annual interest rate formula* and *why is it so important to compute*.

## What is effective annual rate?

The **effective annual rate (EAR or EFF%)** is a **type of annual rate that corresponds to the same** **future value as compounding at the periodic rate for m times a year**.

For example, if you have a loan that has a 12 percent annual interest rate, but interest is calculated and added to your balance monthly, your periodic interest rate is 1 percent (`12% / 12 = 1%`

) and compounding happens twelve times (`m = 12`

). You may see that **the more often the bank adds and recalculates the interest on your balance** (or the higher the compounding frequency), **the higher the** **finance charge is after a year**.

The **effective annual rate aims to adjust the annual nominal rate by the effect of compounding**, which **shows you the actual growth of the balance for a year**.

Let's approach the problem from a different angle. Assume you have 1,000 dollars to invest and would like to know the future value after a year with different compounding frequencies.

The below table represents the effect of increased frequency of compounding on a 1,000 dollars balance with a 12 percent annual interest rate. Because **interest is earned on interest more often, you should expect higher future values (FV) the more frequently compounding occurs**. However, you may notice that the percentage growth of future values is gradually smaller. The largest increase in FV (and in EFF%) happens when compounding goes from annual to semi-annual, but the impact is relatively small when moving from monthly to daily compounding.

Note, that **compounding can happen even continuously**, which is the **theoretical limit of the process of compounding**. In this case, **interest compounds every possible moment**, so **the accumulated interest reaches its maximum value**. To understand the math behind this, check out our natural logarithm calculator and, in particular, the *The natural logarithm and the common logarithm* section.

Compounding frequency | Annual rate | Periods in year | EAR or EFF% | Future value of $1,000 | Increase in FV |
---|---|---|---|---|---|

Annual | 12% | 1 | 12.0000% | 1,120.00 | |

Semi-annual | 12% | 2 | 12.3600% | 1,123.60 | 0.3214% |

Quarterly | 12% | 4 | 12.5509% | 1,125.51 | 0.1700% |

Monthly | 12% | 12 | 12.6825% | 1,126.83 | 0.1173% |

Daily | 12% | 365 | 12.7475% | 1,127.47 | 0.0568% |

Continuous | 12% | ∞ | 12.7497% | 1,127.50 | 0.0027% |

In the present calculator, for daily frequency, we apply the average number of days in a year (365.242).

## The effective annual interest rate formula - How to calculate EAR?

The **EAR formula in finance** takes the following general form:

**EAR = (1 + r / m) ^{m} − 1**

Where:

`EAR`

- the effective annual interest rate, or*effective rate*;`r`

- the annual interest rate, which is the nominal interest rate in percent, also called*stated*or*quoted rate*; and`m`

- compounding periods, which is the number of times compounding occurs in a year, or, in other words, the period after which the interest will be calculated on the principal amount and then added to it (capitalized on it).

If the compound frequency is continuous, you need to apply another equation:

`EAR = e^^m^^ − 1`

Where `e`

stands for constant of the exponent.

## How to find effective annual rate, and why is it important?

Probably the best way to demonstrate the advantage of the effective annual rate is to go through a real-world example. Let's say you need 10,000 dollars to contribute to your savings to buy a car. You have two options:

- Take an auto loan with an annual rate of
`12%`

, compounded monthly, or - Debit your credit card that charges
`1%`

per month, compounded daily.

Would you be better off taking the car loan or using your credit card?

To answer this question, you must express the cost of each alternative as an effective annual rate.

So, how to calculate effective annual rate in this case? By applying the effective annual rate formula, the two scenarios result in the following EARs:

**EAR = (1 + 0.12 / 12)**^{12}− 1 = 12.6825%**EAR = (1 + 0.12 / 365.242)**^{365.242}− 1 = 12.7475%

Therefore, it is more expensive to turn to your bank card compared to the bank loan.

As you can see by now, **converting the nominal annual rate into EAR provides an excellent way to compare the effective costs of different loans or return rates on different investments when compounding differs**, as in our example of the credit card versus a bank loan.

## How to apply the EAR calculator?

To operate this effective annual rate calculator, you need to set the below parameters, and you will receive the results instantly.

**Annual interest rate**- the nominal interest in a year;**Periodic rate**- the charged rate by a lender or paid by a borrower each period. In our effective annual rate calculator, the period refers to the compounding frequency, which coincides with the payment period;**Compounding frequency**- the number of times compounding occurs in a year; and**Effective Annual Rate (EAR)**.

In the *advance mode*, you can reach the following additions that you can use to compute future or present value with the given EAR:

**Term**;**Initial balance**; and**Final balance**.