# Effective Annual Interest Rate

Effective annual interest rate is the annual interest rate that when applied to the opening balance of a loan amount results in a future value that is the same as the future value arrived at through the multi-period compounding based on nominal interest rate (i.e. the stated interest rate).

A loan agreement states at least the following three things: the loan balance, the interest rate to be charged i.e. the nominal interest rate and the number of times in each year the interest shall be calculated and applied to the loan i.e. the number of compounding periods per year.

Let’s say you have a $100,000 loan on which you must pay 10% interest rate and the interest rate shall be calculated once a year. The outstanding balance of the loan after one year shall be $110,000 (=$100,000 × (1 + 10%)^{1}). It is straight forward because the interest is calculated and added to the loan at the end of the year. However, bankers are smart, and they would most likely recalculate your loan balance more than once in a year.

Now, let’s say the interest on the above loan is calculated semiannually and added to the loan. After the first six months, your loan balance will be $105,000 (=$100,000 × (1 + 5%)). Since the loan balance is being calculated after half-year, we have used the half-yearly rate of interest. After the second-half of the year, your loan balance will stand at $110,250 (=$105,000 × (1 + 5%)). If we apply 10% interest to the initial loan balance of $100,000, we get only $110,000 (=$100,000 × (1 + 10%)). You can see that if interest is calculated and applied more than one a year, the loan balance at the end of the year is higher than the balance we arrive at by simply applying the annual interest rate quoted by the bank to the initial loan balance. This is where the concept of effective interest rate applies. To give a complete picture, we need to calculate the annual rate that captures the magnifying effect of multiple compounding periods in one year. It equals 10.25% (=($110,250–$100,000)/$100,000).

## Formula

Effective Interest Rate = | 1 + Nominal Annual Interest Rate | ^{n} | – 1 | ||

n |

Where n is the number of compounding periods per year.

Let’s see how we arrived at this formula so you don’t have to memorize it.

We aim to find a single annual rate with one compounding per year that would give us the same future value of $1 as the nominal interest rate quoted by the bank over the multiple compounding periods. The left-hand side of the equation below captures the effect of effective annual interest rate and the right-hand side calculates future value using the nominal interest rate and number of compounding periods (n) per year.

$1 × (1 + Effective Interest Rate) = $1 × | 1 + Nominal Annual Interest Rate | ^{n} | ||

n |

Let’s remove $1 from both sides:

1 + Effective Interest Rate = | 1 + Nominal Annual Interest Rate | ^{n} | ||

n |

We just need to subtract 1 from both sides to get:

Effective Interest Rate = | 1 + Nominal Annual Interest Rate | ^{n} | – 1 | ||

n |

We can use EFFECT formula in Microsoft Excel to calculate effective interest rate. The formula syntax is EFFECT(nominal_rate, npery). Nominal rate is the stated annual rate quoted by the bank we discussed above and npery is the number of compounding periods per year. In case of the example above, you need to enter EFFECT(10%, 2) in the formula bar to get 10.25%.

BONUS: effective interest rate in case of continuous compounding is calculated using the following formula:

Effective interest rate (continuous compounding) = e^{i} – 1

Where e = 2.71828

## Example

Calculate effective interest rate for a loan with a nominal interest rate of 10% for (a) semiannual, (b) quarterly, (c) monthly and (d) daily and (e) continuous compounding.

__Solution__

Effective interest rate for semiannual compounding = (1 + 10%/2)^{2} – 1 = 10.25%

Effective interest rate for quarterly compounding = (1 + 10%/4)^{4} – 1 = 10.38%

Effective interest rate for monthly compounding = (1 + 10%/12)^{12} – 1 = 10.47%

Effective interest rate for daily compounding = (1 + 10%/365)^{365} – 1 = 10.5156%

Effective interest rate for continuous compounding = e^{0.1} – 1 = 2.71828^{0.1} – 1 = 10.5171%

Written by Obaidullah Jan, ACA, CFAhire me at