# 1.5 Calculating the Interest Rate

## Learning Objectives

• Calculate the interest rate of a loan or investment

## 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

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

## Introduction

You need to calculate the nominal interest rate under many circumstances including (but not limited to) the following.

• Determining the interest rate on a single payment loan.
• Understanding what interest rate is needed to achieve a future savings goal.
• Calculating the interest rate that generated a specific amount of interest.
• Finding a fixed interest rate that is equivalent to a variable interest rate.

## Using a Financial Calculator

Although it is possible to find the interest rate by using the future value or present value formulas, it is much more practical to use a financial calculator. You use the financial calculator in the same way as described previously, but the only difference is that the unknown quantity is $I/Y$ (the nominal interest rate). You must still load the other six variables into the calculator and apply the cash flow sign conventions carefully.

## Using the TI BAII Plus Calculator to Find the Interest Rate for Compound Interest

Enter values for the known variables ($PV$, $FV$, $N$, $PMT$, $P/Y$ and $C/Y$) following the steps below and paying close attention to the cash flow sign convention for $PV$ and $FV$.

• For the main button keys in the $TVM$ row (i.e. $N$, $I/Y$, $PV$, $PMT$, $FV$), enter the number first and then press the corresponding button.
• For example, to enter $N=34$, enter $34$ on the calculator and then press $N$.
• For $P/Y$ and $C/Y$, press 2nd $I/Y$. At the $P/Y$ screen, enter the value for $P/Y$ and then press ENTER. Press the down arrow to access the $C/Y$ screen. At the $C/Y$ screen, enter the value for $C/Y$ and then press ENTER. Press $2\text{nd QUIT}$ (the CPT button) to exit the menu.
• For example, to enter $P/Y=4$ and $C/Y=4$, press $2nd \text{I/Y}$. At the $P/Y$ screen, enter $4$ and press ENTER. Press the down arrow. At the $C/Y$ screen, enter $4$ and press ENTER. Press $2\text{nd QUIT}$ to exit.

After all of the known quantities are loaded into the calculator, press CPT and then $I/Y$ to solve for the interest rate.

### Things to Watch Out For

When entering both $PV$ and $FV$ into the calculator, ensure proper application of cash flow sign convention to $PV$ and $FV$. One number must be negative and the other must be positive. An ERROR message will appear on the calculator display if $PV$ and $FV$ are entered with the same signs (i.e. both are negative or both are positive).

### Example 1.5.1

When Sandra borrowed $\7,100$ from Sanchez, she agreed to reimburse him $\8,615.19$ three years from now including interest compounded quarterly. What nominal quarterly compounded rate of interest is being charged?

Solution

The timeline for the loan is shown below.

 N $4 \times 3=12$ PV $7,100$ FV $-8,615.19$ PMT $0$ I/Y ? P/Y $4$ C/Y $4$

$\displaystyle{I/Y=6.5\%}$

Sanchez is charging an interest rate of $6.5\%$ compounded quarterly on the loan to Sandra.

### Example 1.5.2

Five years ago, Taryn placed $\15,000$ into an RRSP that earned $\6,799.42$ of interest compounded monthly. What was the nominal interest rate for the investment?

Solution

The timeline for the investment is shown below.

Step 1: Calculate the future value.

$\begin{eqnarray*} FV & = & PV+I \\ & = & 15,000+6,799.42 \\ & = & \21,799.42 \end{eqnarray*}$

Step 2: Calculate $I/Y$.

 N $12 \times 5=60$ PV $15,000$ FV $-21,799.42$ PMT $0$ I/Y ? P/Y $12$ C/Y $12$

$\displaystyle{I/Y=7.5\%}$

Step 3: Write as a statement.

Tarynâ€™s investment in his RRSP earned $7.5\%$ compounded monthly over the five years.

### Try It

1) Your company paid an invoice five months late. If the original invoice was for $\6,450$and the amount paid was $\6,948.48$, what monthly compounded interest rate is your supplier charging on late payments?

Solution
 N $12 \times \frac{5}{12}=5$ PV $6,450$ FV $-6,948.48$ PMT $0$ I/Y ? P/Y $12$ C/Y $12$

$\displaystyle{I/Y=18\%}$

### Try It

2) At what monthly compounded interest rate does it take five years for an investment to double?

Solution
 N $12 \times 5=60$ PV $-1$ FV $2$ PMT $0$ I/Y ? P/Y $12$ C/Y $12$

$\displaystyle{I/Y=13.94\%}$

### Section 1.5 Exercises

1. What is the interest rate compounded monthly if a $\101,000$ loan is repaid $10$ years later with a payment of $\191,981.42$?
Solution

$6.44\%$

2. You invested $\59,860.48$ five and half years ago. Today the investment is worth $\78,500$. What interest rate compounded semi-annually did your investment earn?
Solution

$4.99\%$

3. If a $\5,000$ investment grew to $\20,777.73$ in five years, what interest rate compounded daily did the investment earn?
Solution

$28.5\%$

4. In a civil lawsuit, a plaintiff was awarded damages of $\15,000$ plus $\4,621.61$ in interest for a period of $3\frac{1}{4}$ years. What quarterly compounded rate of interest was used in the settlement?
Solution

$8.35\%$

5. Muriel just received $\4,620.01$ including $\840.01$ of interest as payment in full for a sum of money that was loaned $2$ years and $11$ months ago. What monthly compounded rate of interest was charged on the loan?
Solution

$6.9\%$

6. At what monthly compounded interest rate does it take five years for an investment to double?
Solution

$13.94\%$

7. In $2003$, a home in Winnipeg was purchased for $\214,000$. In $2011$, the same home was appraised at $\450,000$. What annually compounded rate of growth does this reflect?
Solution

$9.74\%$

8. On October 1, 1975, the minimum wage in Manitoba was $\2.60$ per hour. It rose to $\10$ per hour by October 1, 2011. What is the annually compounded growth rate for minimum wage in Manitoba during this period?
Solution

$3.81\%$

9. Jean-Luc’s first month’s gross salary in June 1994 was $\800$. By June 2012 his monthly gross salary was $\1,969.23$ higher. What monthly compounded rate did his salary increase by over the period?
Solution

$6.92\%$