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# Effective Interest Rate

Step-by-Step Guide to Understanding Effective Interest Rate (EIR)

Last Updated June 14, 2024

• The effective interest rate is the actual interest rate paid on a loan upon accounting for the effects of compounding across a given period.
• The EIR formula is: Effective Interest Rate = (1 + Nominal Rate ÷ Compounding Periods)^(Compounding Periods) – 1
• The effective interest rate represents the real cost of carrying debt or the actual return on an investment, considering the effects of compounding (“interest on interest”).
• The more frequent the compounding periods, the higher the effective interest rate — all else being equal.
• If the compounding frequency exceeds once per year, the effective interest rate is always higher than the stated annual interest rate.

## How to Calculate Effective Interest Rate

The effective interest rate—often used interchangeably with the terms effective annual rate (EAR) or annual equivalent rate (AER)—provides the actual annual interest rate paid on a loan or debt security since the impact of compounding within a given year is accounted for.

Conceptually, the effective interest rate facilitates the “apples-to-apples” comparison of securities with different compounding frequencies.

Unlike the nominal interest rate (or stated interest rate), the effective interest rate can contribute toward better informed financial decisions because the basis for comparison is more accurate.

The concept of compound interest—i.e. the mechanism of earning “interest on interest”—must not be neglected by borrowers (or lenders), particularly for securities with long-dated maturities.

To calculate the effective annual interest rate of a loan or financial security, two pieces of information are necessary:

• Nominal Interest Rate (i) ➝ Stated Interest Rate on Loan (or Financial Security)
• Number of Compounding Periods per Year (n) ➝ Loan Maturity × Compounding Frequency

The steps to calculate the effective interest rate are as follows:

• Step 1 ➝  Determine the Nominal Interest Rate (i) and Compounding Frequency
• Step 2 ➝ Divide the Nominal Interest Rate (i) by the Total Number of Compounding Periods (n)
• Step 3 ➝ Add One to the Resulting Figure (i ÷ n)
• Step 4 ➝ Raise the Resulting Figure to the Power of the Number of Compounding Periods (n)
• Step 5 ➝ Subtract the Resulting Figure by One
• Step 6 ➝ Multiply by 100 to Convert the Output from Decimal Notation to Percentage Form

## Effective Interest Rate Formula

The effective interest rate (EIR) formula converts the nominal rates into a comparable metric, even for debt securities that compound at different frequencies.

The formula to calculate the effective interest rate is as follows.

Effective Interest Rate (EIR) = [1 + (i ÷ n)]^n 1

Where:

• i ➝ Nominal Interest Rate
• n ➝ Number of Compounding Periods per Year

## How Does Compounding Impact Effective Interest Rate?

The higher the compounding frequency, the higher the effective interest rate — all else being equal.

The common compounding frequencies are the following:

• Annual Compounding ➝ n = 1x
• Semi-Annual Compounding ➝ n = 2x
• Quarterly Compounding ➝ n = 4x
• Monthly Compounding ➝ n = 12x
• Daily Compounding ➝ n = 365x

The pattern illustrated is that higher compounding frequency coincides with a higher effective interest rate (and vice versa).

The effective annual interest rate should form the basis for comparisons when analyzing the cost of borrowing—or cost of debt (kd)—for accurate decision-making.

## Effective vs. Nominal Interest Rate: What is the Difference?

The effective interest rate and nominal interest rate are two different methods of expressing the interest rate on a loan or financial security.

The nominal rate is the stated annual interest rate, while the effective rate reflects the actual interest rate after factoring in compounding effects.

• Nominal Interest Rate ➝ The nominal interest rate—or the stated interest rate—is the pricing rate before accounting for the effects of compounding. The nominal rate is quoted on an annual basis, causing the rate to be misleading because the pricing does not reflect the real cost if the compounding frequency is not on a yearly basis.
• Effective Interest Rate ➝ The effective interest rate accounts for the compounding periods within a given year, thereby reflecting the actual annual cost of borrowing (or the actual annual return on an investment).

The effective interest rate is higher than the nominal rate, unless the compounding frequency is on an annual basis.

Why? The nominal interest rates neglect the effects of compounding, while the effective interest rates take into account the impact of compounding periods.

The nominal rate of different securities is not suited for comparability unless the compounding period is set equivalent to an annualized basis. Otherwise, the insights derived from the analysis will be misleading.

The nominal interest rate is the stated interest rate on lending agreements as lenders are incentivized to show the lower interest rate to entice the potential borrower to accept the proposal.

From a borrower’s perspective, the effective interest rate is the actual cost of borrowing a loan or financial security or reflects the rate at which interest is earned from the viewpoint of the lender.

Considering the fact that the effective annual rate can be higher than the nominal rate from the effects of compounding, the decision to neglect that factor can be a costly mistake for the borrower.

###### Effective Annual Interest Rate (EAR) on Savings Account

The effective annual interest rate (EAR) of a savings account or money market account is the actual return.

The effective annual rate accounts for compounding, so the bank that structures and markets the product to consumers is incentivized to advertise the EAR in an effort to make the yield appear more attractive than the stated interest rate (or nominal interest rate).

From the perspective of a bank or institutional lender, a savings account or loan might be advertised with both a nominal interest rate and an effective annual interest rate. But the more marketable rate between the two (i.e. the lower interest rate), is usually the one focused on.

## Effective Interest Rate Calculator — Excel Template

We’ll now move to a modeling exercise, which you can access by filling out the form below.

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## Effective Interest Rate Calculation Example

Suppose we’re tasked with comparing the effective interest rates of four different loans, each with the same nominal interest rate (6.0%) but different compounding frequencies.

The four loans—which we’ll mark from “Loan A” to “Loan D”—are each priced at a 6.0% stated interest rate by the lender, but the compounding frequency will increase in ascending order from an annual basis to a monthly basis.

Loan A — Financing Assumptions

• Nominal Interest Rate (%) = 6.0%
• Compounding Frequency = Annual (1x)

Loan B — Financing Assumptions

• Nominal Interest Rate (%) = 6.0%
• Compounding Frequency = Semi-Annual (2x)

Loan C — Financing Assumptions

• Nominal Interest Rate (%) = 6.0%
• Compounding Frequency = Quarterly (4x)

Loan D — Financing Assumptions

• Nominal Interest Rate (%) = 6.0%
• Compounding Frequency = Monthly (12x)

For Loan A, the effective annual interest rate is 6.0%, as expected given the initial assumption (i.e., simple interest rate).

• Effective Interest Rate (EIR), Loan A = [1 + (6.0% ÷ 1)]^1 – 1 = 6.00%

On the other hand, the effective annual interest rate for Loan B is 6.1%.

• Effective Interest Rate (EIR), Loan B = [1 + (6.0% ÷ 2)]^2 – 1 = 6.09%

The effective annual interest rate for Loan A is 6.0%, whereas the effective annual interest rate for Loan B is approximately 6.2%, demonstrating the impact of compounding on the actual interest rate.

Given the relationship between the effective interest rate and compounding frequency, the effective interest rates for the remaining two loans should follow the trend of increasing.

• Effective Interest Rate (EIR), Loan C = [1 + (6.0% ÷ 4)]^4 – 1 = 6.14%
• Effective Interest Rate (EIR), Loan D = [1 + (6.0% ÷ 12)]^12 – 1 = 6.17%

The percent differential between the nominal and effective rates widens if the compounding frequency were to increase (and vice versa).

In closing, while the difference in effective interest rate might seem marginal, the incremental uptick in the interest rates for the loans can significantly influence the interest burden incurred by a borrower over long maturities (and periodic income earned by the lender), in particular for larger-sized loans.

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