You take out a $1,000 loan on January 1st and pay it back on June 30 that same year. The interest on the loan is 12% APR, compounded quarterly. What is the effective interest rate for the period in which the balance is outstanding?

Round your answer to a percentage with two decimal places and do not use the “%” sign, i.e. if the answer is 1.23% then write 1.23

Question

You take out a $1,000 loan on January 1st and pay it back on June 30 that same year. The interest on the loan is 12% APR, compounded quarterly. What is the effective interest rate for the period in which the balance is outstanding?

Round your answer to a percentage with two decimal places and do not use the “%” sign, i.e. if the answer is 1.23% then write 1.23

BlackTom AI · Accepted Answer

We start by identifying the key loan terms and the period of interest.
The loan amount is 1000, the annual percentage rate (APR) is 12%, and interest is compounded quarterly, which means the interest rate per quarter is 12% / 4 = 3% (or 0.03 in decimal form).
The balance is outstanding for six months, which spans exactly two quarters.
To find the effective interest rate over this 6-month period, we compound the quarterly rates: (1 + 0.03)^2 = 1.03 × 1.03 = 1.0609.
Subtracting 1 gives the effective rate for the period: 1.0609 − 1 = 0.0609, which is 6.09% when expressed as a percentage.
Since the instruction specifies writing the answer without a percent sign, the numeric value to report is 6.09.
A quick check of the logic: compounding once per quarter at 3% for two quarters yields a growth factor of 1.0609, confirming a 6.09% effective rate over the six-month window.
If you wanted to verify using a formula, you’d compute the quarterly rate r_q = 0.12/4 = 0.03, then the two-period effective rate as (1 + r_q)^2 − 1, which simplifies to 1.0609 − 1 = 0.0609, matching the result above.

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