Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

Definitions

Nominal interest rate is the annual interest rate (per year) for a certain compounding period. Nominal interest rate can be applied to the advertised or stated interest rate on a loan, without taking into account any fees or compounding of interest. The nominal interest rate can be calculated using the formula:

$$j=im$$,

where:

• $$i$$ is the periodic interest rate
• $$j$$ is the nominal/stated rate
• $$m$$ is the number of compounding periods

The effective interest rate (f), (or simply effective rate) is the annual interest rate compounded annually. It may be seen on a loan or financial product restated from the nominal interest rate and expressed as the equivalent interest rate if compound interest was payable annually in arrears. It can be calculated with the following formula:

$$f=(1+i)^m−1$$,

where:

• $$i$$ is the periodic interest rate
• $$f$$ is the effective rate
• $$m$$ is the number of compounding periods

Examples

Example 1

What is the nominal rate of interest on a company that has a 7.77% rate of effective interest annually (rounded to two decimal places)?

Solution

Since we're looking for the nominal rate of interest, we are determining $$j$$. First, we can identify the information that is given in the problem:

$$m = 12$$ (months per year), $$i=7.77\%=0.0777$$

We can use the effective interest rate formula, $$i=(1+\frac{j}{m})^m-1$$, substitute the above values and solve for $$r$$:

$$0.0777=(1+\frac{j}{12})^{12}-1$$

$$0.0777+1=(1+\frac{j}{12})^{12}$$

$$\sqrt[12]{1.0777}-1=\frac{j}{12}$$

$$j=12(\sqrt[12]{1.0777}-1) \approx 0.0751$$

So, the nominal rate of interest on a company that has a 7.77% effective rate (compounded annually per year) is $$7.51\%$$ compounded monthly per year.

Example 2

A credit card company charges 21% interest per year, compounded monthly. What effective annual interest rate (to two decimal places) does the company charge?

Solution

Since we're looking for the effective rate, we are determining $$f$$. First, we can identify the information that is given in the problem:

$$m = 12$$ (months per year), $$j=21\%=0.21$$

We can use the effective interest rate formula, $$i=(1+i)^m-1$$, substitute the above values and solve for $$f$$:

$$f=(1+\frac{0.21}{12})^{12}-1$$

$$f=(1+0.0175)^{12}-1$$

$$f \approx 0.2314$$

So, effective rate the company charges is $$23.14\%$$ compounded annually per year.