1.5 Calculating the Interest Rate

Learning Objectives

  • Calculate the interest rate of a loan or investment

Formula & Symbol Hub

Symbols Used

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

Formulas Used

  • Formula 1.1 – Total Number of Compounds

[latex]n=m \times \mbox{time in years}[/latex]

  • Formula 1.2 – Periodic Interest Rate

[latex]i=\frac{j}{m}[/latex]

  • Formula 1.3 – Future Value

[latex]FV=PV \times (1+i)^n[/latex]

  • Formula 1.4 – Present Value

[latex]PV=FV \times (1+i)^{-n}[/latex]

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 [latex]I/Y[/latex] (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 ([latex]PV[/latex], [latex]FV[/latex], [latex]N[/latex], [latex]PMT[/latex], [latex]P/Y[/latex] and [latex]C/Y[/latex]) following the steps below and paying close attention to the cash flow sign convention for [latex]PV[/latex] and [latex]FV[/latex].

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

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

Things to Watch Out For

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

Example 1.5.1

When Sandra borrowed [latex]\$7,100[/latex] from Sanchez, she agreed to reimburse him [latex]\$8,615.19[/latex] 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.

Timeline: PV = $7,100 at Today. FV = $8,615.19 at 3 years. Unknown % quarterly.
Figure 1.5.1
Image Description

The image shows a horizontal, light blue line representing a loan timeline with labels corresponding to values discussed in the rest of the example.

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

[latex]\displaystyle{I/Y=6.5\%}[/latex]

Sanchez is charging an interest rate of [latex]6.5\%[/latex] compounded quarterly on the loan to Sandra.

Example 1.5.2

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

Solution

The timeline for the investment is shown below.

Timeline: PV = $15,100 at 5 years ago. FV = $15,000 +$6,799.42 = $21,799.42 at Today. Unknown % monthly.
Figure 1.5.2
Image Description

The image shows a horizontal, light blue line representing an investment timeline with labels corresponding to values discussed in the rest of the example.

Step 1: Calculate the future value.

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

Step 2: Calculate [latex]I/Y[/latex].

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

[latex]\displaystyle{I/Y=7.5\%}[/latex]

Step 3: Write as a statement.

Taryn’s investment in his RRSP earned [latex]7.5\%[/latex] compounded monthly over the five years.

Try It

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

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

[latex]\displaystyle{I/Y=18\%}[/latex]

Try It

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

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

[latex]\displaystyle{I/Y=13.94\%}[/latex]

Section 1.5 Exercises

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

    [latex]6.44\%[/latex]

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

    [latex]4.99\%[/latex]

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

    [latex]28.5\%[/latex]

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

    [latex]8.35\%[/latex]

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

    [latex]6.9\%[/latex]

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

    [latex]13.94\%[/latex]

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

    [latex]9.74\%[/latex]

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

    [latex]3.81\%[/latex]

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

    [latex]6.92\%[/latex]

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