Finding the effective annual interest rate of a loan or investment

Two accounts advertise the same rate but compound differently. The effective annual rate is what you actually earn, and it is the only fair way to compare them.

01 Worked Example Video

▸ VIDEO COMING SOON

A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Lamar's account

Lamar invested $4,300 in an account paying 2.6% compounded quarterly.

Find the effective annual interest rate, as a percentage. Type just the number.

Do not round intermediate computations; round to the nearest hundredth of a percent.

EAR =

Semiannual, and a bigger gap

An account pays 4.8% compounded semiannually.

Find the effective annual interest rate, as a percentage. Type just the number.

Round to the nearest hundredth of a percent.

EAR =

Reading a small difference

A savings account pays 3.15% compounded quarterly.

Find the effective annual interest rate, as a percentage. Type just the number.

Round to the nearest hundredth of a percent.

EAR =

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

grow $1 for one year → $\left(1 + \tfrac{r}{n}\right)^{n}$ the exponent is just n, because t = 1

$EAR = \left(1 + \tfrac{r}{n}\right)^{n} - 1$ subtract the original dollar, then read it as a percent

The effective annual rate answers one question: after a year of compounding, what percentage did the account actually grow by? Grow a dollar for a year, take away the dollar you started with, and what remains is the answer. It is always at least the stated rate, and larger whenever n > 1.

Common slips

(1) Forgot to subtract 1. 1.0263 is the growth factor, not the rate. The rate is 2.63%. (2) Reported a decimal. ALEKS asks for a percentage, so 0.0263 becomes 2.63. (3) Rounded too early. The answer is wanted to the nearest hundredth of a percent, so the factor needs six or more decimals before you subtract.

04 Practice — Step by Step

Two parts, as ALEKS asks them

Ines invests $6,000 at 3.9% compounded monthly.

Find the balance after one year, then the effective annual rate.

Step 1

The one-year balance

$(1.00325)^{12} = 1.0397047$. Multiply by $6,000 and round to the nearest cent.

Step 2

Turn the factor into a rate

Subtract 1 from 1.0397047 and write the result as a percentage, to the nearest hundredth.

EAR =

Step 3

Sanity check

The stated rate was 3.9%. Is the effective rate higher or lower? Type higher or lower.

Q10 Q12

Finding the effective annual interest rate of a loan or investment

Why anyone cares