4.7 Effective and Equivalent Interest Rates

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%. In visiting another show home, you encounter a mortgage broker offering a mortgage for 6.57% 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% compounded semi-annually or 6.57% 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 $1,000 at 10.25% effective (compounded annually) for one year you will have the same future value ($1,102.50) as investing the $1,000 for one year at 10% compounded semi-annually. Because 10.25% effective and 10% compounded semi-annually result in the same future value, these interest rates are called equivalent.

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

[latex]\displaystyle{f=(1+i)^m-1}[/latex]

where

• [latex]f[/latex] is the effective interest.
• [latex]i[/latex] is the periodic interest rate for the given nominal interest rate.
• [latex]m[/latex] is the compounding frequency for the given nominal interest rate.

_{Nominal and Effective Rate Conversions by Joshua Emmanuel [5:50] (transcript available).}

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 $1,000 at 10.25% effective (compounded annually) for one year you will have the same future value ($1,102.50) as investing the $1,000 for one year at 10% compounded semi-annually. Because 10.25% effective and 10% compounded semi-annually result in the same future value, these interest rates are equivalent.

The convert a given nominal interest ([latex]j_1[/latex]) to an equivalent nominal interest ([latex]j_2[/latex]) for a given nominal interest use the following formula.

[latex]\displaystyle{i_2=(1+i_1)^{\frac{m_1}{m_2}}-1}[/latex]

where

• [latex]i_1[/latex] is the periodic interest rate for the given nominal interest rate [latex]j_1[/latex].
• [latex]m_1[/latex] is the compounding frequency for the given nominal interest rate [latex]j_1[/latex].
• [latex]m_2[/latex] is the compounding frequency for the new nominal interest rate [latex]j_2[/latex].
• [latex]i_2[/latex] is the periodic interest rate for the new nominal interest rate [latex]j_2[/latex].

Because this formula only gives the periodic interest rate for the new nominal interest rate, the nominal interest rate must be calculated from the periodic interest rate:

[latex]\displaystyle{j_2=m_2 \times i_2}[/latex]

_{Nominal and Effective Rate Conversions by Joshua Emmanuel [5:50] (transcript available).}

Exercises

1. For each of the following, find their effective rates.
   1. 4.75% compounded quarterly.
   2. 7.2% compounded monthly.
   3. 3.95% compounded semi-annually.
   Click to see Answer

   a. 4.84%, b. 7.44%, c. 3.99%

    

2. Convert 10% effective to the equivalent rate compounded semi-annually.
   
   Click to see Answer

   9.76%

    

3. Convert 12% effective to the equivalent rate compounded monthly.
   
   Click to see Answer

   11.39%

    

4. Convert 8% effective to the equivalent rate compounded quarterly.
   
   Click to see Answer

   7.77%

    

5. Convert 6% compounded monthly to the equivalent rate compounded semi-annually.
   
   Click to see Answer

   6.08%

    

6. Convert 4.5% compounded quarterly to the equivalent rate compounded monthly.
   
   Click to see Answer

   4.48%

    

7. Covert 8% compounded semi-annually to the equivalent rate compounded quarterly.
   
   Click to see Answer

   7.92%

    

8. The HBC credit card has a nominal interest rate of 26.44669% compounded monthly. What effective rate is being charged?
   
   Click to see Answer

   29.9%

    

9. What is the effective rate on a credit card that charges interest of 0.049315% per day?
   
   Click to see Answer

   19.72%

    

10. RBC offers two different investment options to its clients. The first option is compounded monthly while the latter option is compounded quarterly. If RBC wants both options to have an effective rate of 3.9%, what nominal rates should it set for each option?
    
    Click to see Answer

    3.832% compounded monthly, 3.844 compounded quarterly

     

11. Louisa is shopping around for a loan. TD Canada Trust has offered her 8.3% compounded monthly, Conexus Credit Union has offered 8.34% compounded quarterly, and ING Direct has offered 8.45% compounded semi-annually. Rank the three offers and show calculations to support your answer.
    
    Click to see Answer

    Rank	Company	Rate
    1	ING	8.629%
    2	TD	8.623%
    3	Conexus	8.605%

     

12. Your three-year monthly compounded investment just matured and you received $9,712.72, of which $2,212.72 was interest. What effective rate of interest did you earn?
    
    Click to see Answer

    9%

     

13. The TD Emerald Visa card wants to increase its effective rate by 1%. If its current interest rate is 19.067014% compounded daily, what new daily compounded rate should it advertise?
    
    Click to see Answer

    19.89%

     

14. Five investors reported the following results. Rank the effective rates of return realized by each investor from highest to
    lowest and show your work.
    • Investor 1: $4,000 principal earned $1,459.10 of interest compounded monthly over four years.
    • Investor 2: $9,929.85 maturity value including $1,429.85 of interest compounded quarterly for two years.
    • Investor 3: $14,750 principal maturing at $19,370.83 after 3½ years of semi-annually compounded interest.
    • Investor 4: $3,194.32 of interest earned on a $6,750 principal after five years of daily compounding.
    • Investor 5: $5,321.65 maturity value including $1,421.65 of interest compounded annually over four years.
    Click to see Answer

    Investor	Rate
    3	8.098%
    1	8.085%
    2	8.084%
    5	8.08%
    4	8.057%

Attribution

“Chapter 9 Summary” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.