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Interest Calculation
Introduction to interest calculation formulas that determine how money balances deposited into an interest earning account grow over time.
Derivation of Continuous Compounding
Derivation of continuous compounding shows how compound interest converges to an exponential function as compounding intervals tend to zero.
Annuity Calculation
Introduction to fixed interest annuity calculation with formulas for present and future value as well as usage examples.
Derivation of Annuity Formulas
Easy to understand derivation of annuity formulas for the calculation of fixed interest series of constant payments and perpetuities.
Time Value of Money
The financial concept of time value of money considers the changes of the value of monetary balances with time.

# Effective Annual Rate (EAR)

The effective annual rate (EAR), also called annual equivalent rate (AER) or effective interest rate (EIR), is a conversion of interest rates to an equivalent rate with annual compounding. So for investments compounding interest multiple times per year the EAR is the rate that results in the same total of interest payments as if compounding had happened just once per year.

In order to understand the principle underlying the calculation of effective rates it may be helpful to read up on compound interest.

The effective annual rate (EAR) assumes interest on loans or investments accumulating over the course of one calendar year. The EAR then results from an equivalence of simple and compound interest:

(1 + EAR) = (1 + \frac{r}{m})^m
\begin{aligned}
EAR &: \mathrm{effective \space annual \space rate}\\
r &: \mathrm{nominal \space interest \space rate}\\
m &: \mathrm{compounding \space frequency}
\end{aligned}

Subtracting one from both sides of the above equation provides the definition of the EAR:

EAR = (1 + \frac{r}{m})^m -1

For a nominal rate of 4% Table 2 gives an overview about how the EAR rises when the compounding frequency goes up:

The EAR is meant to help consumers to compare the actual interest charges or earnings of loans and investments. Because both nominal interest and compounding frequency determine the real cost, offers stating just a nominal rate may be confusing. For instance, a nominal rate of 4.05% with annual compounding generates lower interest charges than a nominal rate of 4.00% with monthly compounding. What Table 2 says is that a mortgage at a nominal rate of 4% with monthly payments actually costs 4.07415%.

## References

Consumer Credit, European Commission

Published: November 24, 2022
Updated: May 6, 2023

Financial Algebra
Financial Algebra