# 1.7 Effective and Equivalent Interest Rates

## Learning Objectives

• Calculate effective interest rates
• Calculate equivalent interest rates

## Formula & Symbol Hub

#### Symbols Used

• $f$ or $EFF$ = Effective interest rate
• $FV$ = Future value or maturity value
• $i$ = Periodic interest rate
• $j$ or $I/Y$ = Nominal interest rate per year
• $m$ or $C/Y$ = Number of compounds per year or compounding frequency
• $n$ or $N$ = Total number of compound periods for the term
• $PV$ = Present value of principal

#### Formulas Used

• ##### Formula 1.1 – Total Number of Compounds

$n=m \times \mbox{time in years}$

• ##### Formula 1.2 – Periodic Interest Rate

$i=\frac{j}{m}$

• ##### Formula 1.3 – Future Value

$FV=PV \times (1+i)^n$

• ##### Formula 1.4 – Present Value

$PV=FV \times (1+i)^{-n}$

• ##### Formula 1.5 – Effective Interest Rate

$f=(1+i)^m-1$

• ##### Formula 1.6 – Equivalent Interest Rate

$i_2=(1+i_1)^{\frac{m_1}{m_2}}-1$

## 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 $6.6\%$. 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 ($f$) for a given nominal interest can be calculated using the following formula.

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

$\Large{\color{red}{f}}=(1+{\color{blue}{i}})^{\color{green}{m}}-1$

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

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

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

### Example 1.7.1

Convert $10\%$ compounded semi-annually to its corresponding effective interest rate.

Solution

Step 1: Calculate the periodic interest rate.

$\begin{eqnarray*} i & = & \frac{j}{m} \\ & = & \frac{10\%}{2} \\ & = & 5\% \\ & = & 0.05 \end{eqnarray*}$

Step 2: Calculate the effective interest rate.

$\begin{eqnarray*} f & = & (1+i)^m-1 \\ & = & (1+0.05)^2-1 \\ & = & 0.1025 \\ & = & 10.25\% \end{eqnarray*}$

## Using the TI BAII Plus Calculator to Find and Effective Interest Rate

The TI BAII Plus calculator has a built-in effective interest rate converter called ICONV. The interest rate converter involves three variables.

 Variable Meaning NOM Nominal interest rate, I/Y. C/Y Compounding frequency, C/Y. EFF Effective interest rate.

To find the effective interest rate:

1. Press $2\text{nd ICONV}$ (the number $2$ key).
2. At the NOM screen, enter the nominal interest rate $I/Y$ and press ENTER. The interest rate is entered as a percent (without the percent sign). For example $10\%$ is entered as $10$.
3. Press the up arrow.
4. At the $C/Y$ screen, enter the compounding frequency for the nominal interest rate entered in the previous step and press ENTER.
5. Press the up arrow.
6. At the EFF screen, press CPT to calculate the effective interest rate.

The interest converter allows you to solve for any of these three variables, not just the effective interest rate.

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

### Example 1.7.2

Convert $10\%$ compounded semi-annually to its corresponding effective interest rate.

Solution
 NOM $10$ I/Y $2$ EFF ?
1. Press $2\text{nd ICONV}$.
2. At the NOM screen, enter $10$ and press ENTER.
3. Press the up arrow.
4. At the $C/Y$ screen, enter $2$ and press ENTER.
5. Press the up arrow.
6. At the EFF screen, press CPT.

$EFF=10.25\%$

### Example 1.7.3

You are offered loans at two different interest rates: $6.6\%$ compounded semi-annually and $6.57\%$ compounded quarterly. Which loan should you accept?

Solution

Step 1: Convert $6.6\%$ compounded semi-annually to its effective interest rate.

 NOM $6.6$ I/Y $2$ EFF ?

$EFF=6.71%$

Step 2: Convert $6.75\%$ compounded quarterly to its effective interest rate.

 NOM $6.57$ I/Y $4$ EFF ?

$EFF=6.73%$

Step 3: Write as a statement.

You should select the loan at $6.6\%$ compounded semi-annually because it has the lower effective interest rate.

## 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.

### Try It

1) If your investment earns $5.5\%$ compounded monthly, what is the effective rate of interest?

Solution
 NOM $5.5$ I/Y $12$ EFF ?

$EFF=5.64%$

### Try It

2) 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 $8.4\%$ compounded monthly. At your last stop, the credit union agent says that by taking out a car loan with them, you would be charged $8.65\%$ effective. Should you go with the bank loan or the credit union loan?

Solution

Convert the bank’s rate to it’s effective rate:

 NOM $8.4$ I/Y $12$ EFF ?

$EFF=8.73%$

The credit union’s $8.65\%$ effective is the better choice because it is a lower effective rate than the bank’s.

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

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

$\Large{\color{red}{i_2}}=(1+{\color{blue}{i_1}})^{\frac{\color{green}{m_1}}{\color{purple}{m_2}}-1}$

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

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

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

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

### Key Takeaways

Because Formula 1.5 only gives the periodic interest rate for the new nominal interest rate, the nominal interest rate must be calculated from the periodic interest rate:

$\displaystyle{j_2=m_2 \times i_2}$

## Using the TI BAII Plus Calculator to Find an Equivalent Interest Rate

To find an equivalent interest rate:

1. Press 2nd ICONV (the number $2$ key).
2. At the NOM screen, enter the nominal interest rate $I/Y$ for the old interest rate and press ENTER. The interest rate is entered as a percent (without the percent sign). For example $10\%$ is entered as $10$.
3. Press the up arrow.
4. At the $C/Y$ screen, enter the compounding frequency for the old interest rate entered in the previous step and press $ENTER$.
5. Press the up arrow.
6. At the $EFF$ screen, press $CPT$ to calculate the effective interest rate.
7. Press the down arrow.
8. At the $C/Y$ screen, enter the compounding frequency for the new interest rate and press $ENTER$.
9. Press the down arrow.
10. At the $NOM$ scree, press $CPT$ to calculate the new interest rate.

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

### Note

Two equivalent interest rates have the same effective rate. The calculator uses this fact to find an equivalent rate. The first step to finding an equivalent rate on the calculator requires the calculation of the effective rate for the old interest rate. Using this effective rate, the calculator finds the new interest rate.

### Example 1.7.4

What rate compounded quarterly is equivalent to $8\%$ compounded monthly?

Solution

Step 1: Convert $8\%$ compounded monthly to its effective rate.

 NOM $8.8$ I/Y $12$ EFF ?

$EFF=8.2999...\%$

Step 2: Convert the effective rate from the previous step to the new interest rate compounding quarterly.

Make sure that you keep all of the decimal places from the effective interest rate found in the previous step to avoid any round off error in calculating the new interest rate.

 NOM ? I/Y $4$ EFF $8.2999....$

$NOM=8.05\%$

### Things to Watch Out For

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 that you want to convert from and the new compounding that you want to convert to. A little extra time spent on double-checking these values helps avoid mistakes.

### Example 1.7.5

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: Convert $7.7\%$ compounded quarterly to its effective rate.

 NOM $7.7$ I/Y $4$ EFF ?

$EFF=7.9252...\%$

Step 2: Convert the effective rate in the previous step to the new interest rate compounding semi-annually.

 NOM ? I/Y $2$ EFF $7.9252...$

$NOM=7.77\%$

Step 3: Write as a statement.

$7.7\%$ compounded quarterly is equivalent to $7.77\%$ 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.

### Try It

3) What nominal interest rate compounded semi-annually would result in the same financial position as $9.5\%$ compounded monthly?

Solution
 NOM $9.5$ $\color{blue}{9.69}$ C/Y $12$ $2$ EFF $\color{blue}{9.9247...}$ $9.9247...$

### Try It

4) You are offered three different investment opportunities at three different interest rates: $6.2\%$ compounded semi-annually, $6.19\%$ compounded quarterly, and $6.07\%$ compounded daily. Which investment should you pick?

Solution

Convert $6.09\%$ compounded quarterly to a rate compounded semi-annually.

 NOM $6.19$ $\color{blue}{6.24}$ C/Y $4$ $2$ EFF $\color{blue}{6.335...}$ $6.335...$

$6.19\%$ compounded quarterly is equivalent to $6.24\%$ compounded semi-annually.

Convert $6.07\%$ compounded daily to a rate compounded semi-annually.

 NOM $6.07$ $\color{blue}{6.16}$ C/Y $365$ $2$ EFF $\color{blue}{6.257...}$ $6.257...$

$6.07\%$ compounded daily is equivalent to $6.16\%$ compounded semi-annually.

Select the investment at $6.19\%$ compounded quarterly.

Note: There are other ways to solve this problem. For example, all of the interest rates could be converted to rates that compound quarterly, or all the interest rates could be converted to their effective rates.

### Section 1.7 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.
Solution

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

2. Convert $10\%$ effective to the equivalent rate compounded semi-annually.
Solution

$9.76\%$

3. Convert $12\%$ effective to the equivalent rate compounded monthly.
Solution

$11.39\%$

4. Convert $8\%$ effective to the equivalent rate compounded quarterly.
Solution

$7.77\%$

5. Convert $6\%$ compounded monthly to the equivalent rate compounded semi-annually.
Solution

$6.08\%$

6. Convert $4.5\%$ compounded quarterly to the equivalent rate compounded monthly.
Solution

$4.48\%$

7. Covert $8\%$ compounded semi-annually to the equivalent rate compounded quarterly.
Solution

$7.92\%$

8. The HBC credit card has a nominal interest rate of $26.44669\%$ compounded monthly. What effective rate is being charged?
Solution

$29.9\%$

9. What is the effective rate on a credit card that charges interest of $0.049315\%$ per day?
Solution

$19.72\%$

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?
Solution

$3.832\%$ 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.
Solution
 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?
Solution

$9\%$

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?
Solution

$19.89\%$

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 $3$½ 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.
Solution
 Investor Rate $3$ $8.098\%$ $1$ $8.085\%$ $2$ $8.084\%$ $5$ $8.08\%$ $4$ $8.057\%$