EFFECTIVE RATE: WHAT IT IS, HOW IS IT CALCULATED, EXAMPLES - ECONOMY - 2025

The effective rate is the interest rate that is actually earned or paid on an investment, loan or other financial product, due to the result of the compounding in a given period of time. It is also called the effective interest rate, the effective annual interest rate, or the equivalent annual rate.

The effective rate is a way of reaffirming the annual interest rate so that the effects of compounding are taken into account. It is used to compare the annual interest between loans with different compounding periods (week, month, year, etc.).

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At the effective rate, the periodic rate is annualized using compounding. It is the standard in the European Union and in a large number of countries around the world.

The effective rate is an analogous concept also used for savings or investment products, such as a certificate of deposit. As any loan is an investment product for the lender, the term can be used to apply it to this transaction, changing the point of view.

What does it consist of?

The effective rate is an important concept in finance because it is used to compare different products, such as loans, lines of credit, or investment products like certificates of deposit, that calculate compound interest differently.

For example, if Investment A pays 10%, compounded monthly, and Investment B pays 10.1%, compounded semi-annually, the effective rate can be used to determine which investment will actually pay the most over the course of the year.

The effective rate is more accurate in financial terms, when taking into account the effects of compounding. That is, taking in each period that the interest is not calculated on the principal capital, but on the amount of the previous period, which includes the principal and interest.

This reasoning is easily understandable when savings are considered: interest is compounded every month and each month the saver earns interest on the previous period's interest.

As an effect of compounding, the interest earned during a year represents 26.82% of the initial amount, instead of 24%, which is the monthly interest rate of 2%, multiplied by 12.

How is it calculated?

The effective annual interest rate can be calculated by using the following formula:

Effective rate = (1 + (i / n)) ^ (n) - 1.

In this formula, i is equal to the established nominal annual interest rate, and n is equal to the number of compounding periods in the year, which is usually semi-annual, monthly, or daily.

The focus here is the contrast between the effective rate and i. If i, the annual interest rate, is 10%, then with a monthly compounding, where n is equal to the number of months in a year (12), the effective annual interest rate is 10.471%. The formula would appear as:

(1 + 10% / 12) ^ 12 - 1 = 10.471%.

Using the effective rate helps us understand how differently a loan or investment performs whether it is compounded semi-annually, monthly, daily, or any other time period.

Example

If we had $ 1,000 in a loan or investment that is compounded monthly, we would generate $ 104.71 of interest in one year (10.471% of $ 1,000), an amount greater than if we had the same loan or investment compounded annually.

Annual compounding would only generate $ 100 of interest (10% of $ 1,000), a difference of $ 4.71.

If the loan or investment were compounded daily (n = 365) instead of monthly (n = 12), the interest on that loan or investment would be $ 105.16.

As a general rule, the more periods or capitalizations (n) the investment or loan has, the higher the effective rate.

Difference with nominal rate

The nominal rate is the established annual rate, which is indicated by a financial instrument. This interest works according to the simple interest, without taking into account the compounding periods.

The effective rate is the one that distributes the compounding periods during a payment plan. It is used to compare the annual interest between loans with different compounding periods (week, month, quarterly, etc.).

The nominal rate is the periodic interest rate multiplied by the number of periods per year. For example, a nominal rate of 12%, based on a monthly compounding, means an interest rate of 1% per month.

In general, the nominal rate is less than the effective rate. The latter represents the true image of financial payments.

A nominal rate without a compounding frequency is not completely defined: you cannot specify an effective rate without knowing the compounding frequency and the nominal rate. The nominal rate is the calculation basis for deriving the effective rate.

Nominal interest rates are not comparable, unless their compounding periods are the same. Effective rates correct this by "converting" nominal rates to annual compound interest.

Examples

Investment A pays 10%, compounded monthly, and Investment B pays 10.1% compounded semi-annually.

The nominal interest rate is the rate established in the financial product. For investment A the nominal rate is 10%, and for investment B, 10.1%.

The effective rate is calculated by taking the nominal interest rate and adjusting it according to the number of compounding periods that the financial product will experience in the given period of time. The formula is:

Effective rate = (1 + (nominal rate / number of compounding periods)) ^ (number of compounding periods) - 1.

For investment A, this would be: 10.47% = (1 + (10% / 12)) ^ 12 - 1.

For investment B, it would be: 10.36% = (1 + (10.1% / 2)) ^ 2 - 1

Although Investment B has a higher nominal rate, its effective rate is lower than Investment A.

It is important to calculate the effective rate, because if $ 5,000,000 were invested in one of these investments, the wrong decision would cost more than $ 5,800 per year.

Capitalization limit

As the number of compounding periods increases, so does the effective rate. The results of different periods capitalized, with a nominal rate of 10% would be:

- Semiannual = 10,250%

- Quarterly = 10.381%

- Monthly = 10.471%

- Daily = 10.516%

There is a limit to the phenomenon of compounding. Even if compounding occurred an infinite number of times, the compounding limit would be reached. At 10%, the continuously compounded effective rate would be 10.517%.

This rate is calculated by raising the number "e" (approximately equal to 2.71828) to the power of the interest rate, and subtracting one. In this example it would be 2.171828 ^ (0,1) - 1.

References

1. Investopedia (2018). Effective Annual Interest Rate. Taken from: investopedia.com.
2. Investopedia (2018). The Effective Annual Interest Rate. Taken from: investopedia.com.
3. Wikipedia, the free encyclopedia (2018). Effective interest rate. Taken from: en.wikipedia.org.
4. CFI (2018). Effective Annual Rate. Taken from: corporatefinanceinstitute.com.
5. Elias (2018). What is the difference between effective interest rates and nominal interest rates? CSUN. Taken from: csun.edu.