9.6 Effective and Equivalent Interest Rates
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%. In the fine print, it indicates the rate is 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.
To compare interest rates fairly and select the best, they all have to be expressed with equal compounding. 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.
The Formula
[latex]\colorbox{LightGray}{Formula 9.6}\; \color{BlueViolet}{\text{Interest Rate Conversion:}\;i_{\text{New}}=(1+i_{\text{Old}})^{\frac{C/Y_{\text{Old}}}{C/Y _{\text{New}}}}-1 }[/latex]
How It Works
Follow these steps to calculate effective interest rates:
Step 1: Identify the known variables including the original nominal interest rate (I/Y) and original compounding frequency (C/YOld). Set the C/YNew = 1.
Step 2: Calculate iOld using Formula 9.1.
[latex]i_{\text{Old}}=\frac{I/Y}{C/Y_{\text{Old}}}[/latex]
Step 3: Apply the formula for iNew to convert to the effective interest rate.
[latex]i_{\text{New}}=(1+i_{\text{Old}})^{\frac{C/Y_{\text{Old}}}{C/Y _{\text{New}}}}-1[/latex]
Note: With a compounding frequency of 1, this makes iNew = I/Y compounded annually.
Comparing the interest rates of 6.6% compounded semi-annually and 6.57% compounded quarterly requires you to express both rates in the same units. Therefore, you could convert both nominal interest rates to effective rates.
| Steps | 6.6% compounded semi-annually | 6.57% compounded quarterly |
|---|---|---|
| Step 1 | [latex]I/Y=6.6[/latex] | [latex]I/Y=6.57[/latex] |
| Step 2 | [latex]i_{\text{Old}}=6.6[/latex] | [latex]i_{\text{Old}}=6.57[/latex] |
| Step 3 | [latex]\begin{align} i_{\text{New}}&=(1+0.033)^{\frac{2}{1}}−1\\ &=6.7089 \end{align}[/latex] | [latex]\begin{align} i_{\text{New}}&=(1+0.016425)^{\frac{4}{1}}−1\\ &=6.7336\end{align}[/latex] |
The rate of 6.6% compounded semi-annually is effectively charging 6.7089%, while the rate of 6.57% compounded quarterly is effectively charging 6.7336%. The better mortgage rate is 6.6% compounded semi-annually, as it results in annually lower interest charges.
Your BAII Plus Calculator
The Texas Instruments BAII Plus calculator has a built-in effective interest rate converter called ICONV located on the second shelf above the number 2 key. To access it, press 2nd ICONV. You access three input variables using your [latex]\uparrow[/latex] or [latex]\downarrow[/latex] scroll buttons. Use this function to solve for any of the three variables, not just the effective rate.
| Variable | Description | Algebraic Symbol |
|---|---|---|
| NOM | Nominal Interest Rate | [latex]I/Y[/latex] |
| EFF | Effective Interest Rate | [latex]i_{\text{New}}\; \text{(annually compounded)}[/latex] |
| C/Y | Compound Frequency | [latex]C/Y_{\text{Old}}[/latex] |
To use this function, enter two of the three variables by keying in each piece of data and pressing ENTER to store it. When you are ready to solve for the unknown variable, scroll to bring it up on your display and press CPT. For example, use this sequence to find the effective rate equivalent to the nominal rate of 6.6% compounded semi-annually:
2nd ICONV, 6.6 Enter [latex]\uparrow[/latex], 2 Enter [latex]\downarrow[/latex], CPT
Answer: 6.7089
Concept Check:
Example 9.6.1: Understanding your Investment
If your investment earns 5.5% compounded monthly, what is the effective rate of interest?
Solution:
Step 1: Given information:
I/Y = 5.5; C/YOld = monthly = 12; C/YNew = 1;
Step 2: Calculate iOld.
[latex]\begin{align} i_{\text{Old}}&=\frac{I/Y}{C/Y_{\text{Old}}}\\ &=\frac{5.5\%}{12}\\ &=0.458\overline{3}\%\; \text{or}\;0.00458\overline{3} \end{align}[/latex]
Step 3: Calculate iNew.
[latex]\begin{align} i_{\text{New}}&=(1+i_{\text{Old}})^{\frac{C/Y_{\text{Old}}}{C/Y _{\text{New}}}}-1\\ &=(1+0.00458\overline{3})^{\frac{12}{1}}-1\\ &=0.056408\;\text{or}\;5.6408\% \end{align}[/latex]
Calculator instructions:
2nd ICONV
| NOM | C/Y | EFF |
|---|---|---|
| 5.5 | 12 | Answer: 5.640786 |
You are effectively earning 5.6408% interest per year.
Example 9.6.2: Your Car Loan
As you search for a car loan, all banks have quoted you monthly compounded rates (which are typical for car loans), with the lowest being 8.4%. At your last stop, the credit union agent says that by taking out a car loan with them, you would effectively be charged 8.65%. Should you go with the bank loan or the credit union loan?
Solution:
Step 1: Given information:
iNew = 8.65% effective rate; C/YOld = monthly = 12; C/YNew = 1
(Note: In this case the iNew is known, so the process is reversed to arrive at the I/Y).
Step 2: Using the formula for iNew, rearrange and solve for iOld.
[latex]\begin{align} i_{\text{New}}&=(1+i_{\text{Old}})^{\frac{C/Y_{\text{Old}}}{C/Y _{\text{New}}}}-1\\ 0.0865&=(1+i_{\text{Old}})^{\frac{12}{1}}-1\\ 1.0865&=(1+i_{\text{Old}})^{12}\\ 1.0865^{\frac{1}{12}}&=1+i_{\text{Old}}\\ 1.006937&=1+i_{\text{Old}}\\ i_{\text{Old}}&=0.006937 \end{align}[/latex]
Step 3: Solve for the nominal rate, I/Y.
[latex]\begin{align} i_{\text{Old}}&=\frac{I/Y}{C/Y_{\text{Old}}}\\ 0.006937&=\frac{I/Y}{12}\\ I/Y&=0.083249\%\; \text{or}\;8.3249 \end{align}[/latex]
Calculator instructions:
2nd ICONV
| NOM | C/Y | EFF |
|---|---|---|
| Answer: 8.324896 | 12 | 8.65 |
The offer of 8.65% effectively from the credit union is equivalent to 8.3249% compounded monthly. If the lowest rate from the banks is 8.4% compounded monthly, the credit union offer is the better choice.
Equivalent Interest Rates
At times you must convert a nominal interest rate to another nominal interest rate that is not an effective rate. This brings up the concept of equivalent interest rates, which are interest rates with different compounding that produce the same effective rate and therefore are equal to each other. After one year, two equivalent rates have the same future value.
How It Works
To convert nominal interest rates you need no new formula. Instead, you make minor changes to the effective interest rate procedure and add an extra step. Follow these steps to calculate any equivalent interest rate:
Step 1: Identify the given nominal interest rate (I/Y) and compounding frequency
(C/YOld). Also identify the new compounding frequency (C/YNew).
Step 2: Calculate the original periodic interest rate (iOld) using the formula
[latex]i_{\text{Old}}=\frac{I/Y}{C/Y_{\text{Old}}}[/latex]
Step 3: Calculate the new periodic interest rate (iNew) using the formula
[latex]i_{\text{New}}=(1+i_{\text{Old}})^{\frac{C/Y_{\text{Old}}}{C/Y _{\text{New}}}}-1[/latex]
Step 4: Use the formula [latex]i_{\text{New}}=\frac{I/Y}{C/Y_{\text{New}}}[/latex], rearrange and solve for the new converted nominal rate I/Y.
Your BAII Plus Calculator
Converting nominal rates on the BAII Plus calculator takes two steps:
Step 1: Convert the original nominal rate and compounding to an effective rate. Input NOM (this is the given nominal rate I/Y) and the corresponding old C/Y, then compute the EFF.
Step 2: Input the new C/Y and compute the new converted nominal rate NOM.
Example 9.6.3: Comparing Mortgage Rates
Revisiting the mortgage rates from the section opener, compare the 6.6% compounded semi-annually rate to the 6.57% compounded quarterly rate by converting one compounding to another.
Solution:
It is arbitrary which interest rate you convert. In this case, choose to convert the 6.57% compounded quarterly rate to the equivalent nominal rate compounded semi-annually.
Step 1: Given information:
I/Y = 6.57%; C/YOld = quarterly = 4; Convert to C/YNew = semi-annually = 2
Step 2: Calculate iOld.
[latex]\begin{align} i_{\text{Old}}&=\frac{I/Y}{C/Y_{\text{Old}}}\\ &=\frac{6.57\%}{4}\\ &=1.6425\%\\ &=0.016425 \end{align}[/latex]
Step 3: Calculate iNew.
[latex]\begin{align} i_{\text{New}}&=(1+i_{\text{Old}})^{\frac{C/Y_{\text{Old}}}{C/Y _{\text{New}}}}-1\\ &=(1+0.016425)^{\frac{4}{2}}-1\\ &=0.033119 \end{align}[/latex]
Step 4: Solve for the new converted nominal rate I/Y.
[latex]\begin{align} i_{\text{New}}&=\frac{I/Y}{C/Y_{\text{New}}}\\ 0.033229&=\frac{I/Y}{2}\\ I/Y&=0.06624\;\text{or}\; 6.624\% \end{align}[/latex]
Thus, 6.57% compounded quarterly is equivalent to 6.624% compounded semi-annually. Pick the mortgage rate of 6.6% compounded semi-annually since it is the lowest rate available.
Calculator instructions:
2nd ICONV
| Step | NOM | C/Y | EFF |
|---|---|---|---|
| 1 | 6.57 | 4 | Answer: 6.733648 |
| 2 | Answer: 6.623956 | 2 | 6.733648 |
Use this sequence:
2nd ICONV, 6.57 Enter [latex]\uparrow[/latex], 4 Enter [latex]\uparrow[/latex], CPT [latex]\downarrow[/latex], 2 Enter [latex]\downarrow[/latex], CPT
Answer: 6.623956
When converting interest rates, the most common source of error lies in confusing the two values of the compounding frequency, or C/Y. When working through the steps, clearly distinguish between the old compounding (C/YOld) that you want to convert from and the new compounding (C/YNew) that you want to convert to. A little extra time spent on double-checking these values helps avoid mistakes.
Example 9.6.4: Which Investment to Choose?
You are looking at three different investments bearing interest rates of 7.75% compounded semi-annually, 7.7% compounded quarterly, and 7.76% compounded semi-annually. Which investment offers the highest interest rate?
Solution:
Notice that two of the three interest rates are compounded semi-annually while only one is compounded quarterly. Although you could convert all three to effective rates (requiring three calculations), it is easier to convert the quarterly compounded rate to a semi-annually compounded rate. Then all rates are compounded semi-annually and are therefore comparable.
Step 1: Given information:
I/Y = 7.7%; C/YOld = quarterly =4; C/YNew = semi-annually = 2
Step 2: Calculate iOld.
[latex]\begin{align} i_{\text{Old}}&=\frac{I/Y}{C/Y_{\text{Old}}}\\ &=\frac{7.7\%}{4}\\ &=1.925\%\\ &=0.01925 \end{align}[/latex]
Step 3: Calculate iNew.
[latex]\begin{align} i_{\text{New}}&=(1+i_{\text{Old}})^{\frac{C/Y_{\text{Old}}}{C/Y _{\text{New}}}}-1\\ &=(1+0.01925)^{\frac{4}{2}}-1\\ &=0.038870 \end{align}[/latex]
Step 4: Solve for the new converted nominal rate I/Y.
[latex]\begin{align} i_{\text{New}}&=\frac{I/Y}{C/Y_{\text{New}}}\\ 0.038870&=\frac{I/Y}{2}\\ I/Y&=0.077741\;\text{or}\; 7.7741\% \end{align}[/latex]
The quarterly compounded rate of 7.7% is equivalent to 7.7741% compounded semi-annually. In comparison to the semi-annually compounded rates of 7.75% and 7.76%, the 7.7% quarterly rate is the highest interest rate for the investment.
Calculator instructions:
| Step | NOM | C/Y | EFF |
|---|---|---|---|
| 1 | 7.7 | 4 | Answer: 7.925204 |
| 2 | Answer: 7.774112 | 2 | 7.925204 |
Concept Check:
Exercises
In each of the exercises that follow, try them on your own. Full solutions are available should you get stuck.
- The HBC credit card has a nominal interest rate of 26.44669% compounded monthly. What effective rate is being charged? (Answer: 29.9% effectively)
- 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. (Answer: TD Canada Trust: 8.6231% effectively; CONEXUS Credit Union: 8.6045% effectively; ING Direct: 8.6285% effectively)
- 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? (Answer: 19.89% compounded daily)
