1.7 Effective and Equivalent Interest Rates

Introduction

How can you compare interest rates posted with different compounding? For example, let’s say you are considering the purchase of a new home, so for the past few weeks you have been shopping around for financing. You have spoken with many banks as well as onsite mortgage brokers in the show homes. With semi-annual compounding, the lowest rate you have come across is $6.6\mathrm{\%}$. In visiting another show home, you encounter a mortgage broker offering a mortgage for $6.57\mathrm{\%}$ compounded quarterly. You remember from your business math class that the compounding is an important component of an interest rate and wonder which one you should choose—$6.6\mathrm{\%}$ compounded semi-annually or $6.57\mathrm{\%}$ compounded quarterly.

When considering interest rates on loans, you clearly want the best rate. If all of your possible loans are compounded in the same manner, selecting the best interest rate is a matter of picking the lowest number. However, when interest rates are compounded differently the lowest number may in fact not be your best choice. For investments, on the other hand, you want to earn the most interest. However, the highest nominal rate may not be as good as it appears depending on the compounding frequency.

To compare interest rates fairly and select the best, they all have to be expressed with the same compounding frequency. This section explains the concept of an effective interest rate, and you will learn to convert interest rates from one compounding frequency to a different frequency.

Effective Interest Rates

The effective interest rate is the interest rate compounded annually that has the same future value of a given present value for a fixed term as an interest rate compounded at some other (non-annual) frequency. In other words, the amount of interest accrued at the effective interest rate once in an entire year exactly equals the amount of interest accrued at the periodic interest rate successively compounded the stated number of times in a year. For example, if you invest $\mathrm{\$}1,000$ at $10.25\mathrm{\%}$ effective (compounded annually) for one year you will have the same future value ($\mathrm{\$}1,102.50$) as investing the $\mathrm{\$}1,000$ for one year at $10\mathrm{\%}$ compounded semi-annually. Because $10.25\mathrm{\%}$ effective and $10\mathrm{\%}$ compounded semi-annually result in the same future value, these interest rates are called equivalent.

The corresponding effective interest rate ($f$) for a given nominal interest can be calculated using the following formula.

$\boxed{{\displaystyle 1.5}}$ Effective Interest Rate

$f\text{is the Effective Interest:}$ the corresponding effective interest rate.

$i\text{is the Periodic Interest Rate:}$ for the given nominal interest rate.

$m\text{is the Compounding Frequency:}$ for the given nominal interest rate.

_{Video: Nominal and Effective Rate Conversions by Joshua Emmanuel [5:50] (Transcript Available).}

Paths to Success

To compare interest rates that compound at different frequencies, you must convert the rates so that they compound at the same frequency. As in the above example, both interest rates were converted to their effective interest rates and then their effective rates are compared.

• When choosing an interest rate for a loan, you want to pay the least amount of interest. So, select the rate with the lowest effective interest rate.
• When choosing an interest rate for an investment, you want to receive the most interest. So, select the rate with the highest effective interest rate.

Equivalent Interest Rates

Equivalent interest rates are interest rates with different compounding frequencies that result in the same future value for the same given present value and term. For example, if you invest $\mathrm{\$}1,000$ at $10.25\mathrm{\%}$ effective (compounded annually) for one year you will have the same future value ($\mathrm{\$}1,102.50$) as investing the $\mathrm{\$}1,000$ for one year at $10\mathrm{\%}$ compounded semi-annually. Because $10.25\mathrm{\%}$ effective and $10\mathrm{\%}$ compounded semi-annually result in the same future value, these interest rates are equivalent.

The convert a given nominal interest ($j_1$) to an equivalent nominal interest ($j_2$) for a given nominal interest use the following formula.

$\boxed{{\displaystyle 1.6}}$ Equivalent Interest

$i_1\text{is the Periodic Interest Rate:}$ for the given Nominal Interest Rate $j_1$.

$m_1\text{is the Compounding Frequency:}$ for the given Nominal Interest Rate $j_1$.

$m_2\text{is the Compounding Frequency:}$ for the New Nominal Interest Rate $j_2$.

$i_2\text{is the Periodic Interest Rate:}$ for the New Nominal Interest Rate $j_2$.

_{Video: Nominal and Effective Rate Conversions by Joshua Emmanuel [5:50] (Transcript Available).}

Section 1.7 Exercises

Attribution

“1.7 Effective and Equivalent Interest Rates” from Financial Math – Math 1175 by Margaret Dancy is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.