Question 11
A three-year loan has interest charged at 6.2% p.a., compounding monthly.
What is the effective annual rate of interest for the loan?
Not quite. You gave the nominal rate, 6.2% p.a., as the effective rate (no compounding).
Not quite. You compounded quarterly: $i = \dfrac{0.062}{4} = 0.0155$, $(1.0155)^{4} - 1 = 0.0634565 \approx 6.35\%$.
That is it. Step through the hints below to see why.
Not quite. You used the 36 months of the loan as $k$: $(1.00516667)^{36} - 1 = 0.203846 \approx 20.38\%$.
The line is the three-year term of the loan. Each short tick is one compounding period the option counts, and the bracket marks one year.
Choose an option to see the $i$ and $k$ it puts into $i_{\text{effective}} = (1 + i)^k - 1$.
Option (A) uses $i = 0.062$ and $k = 1$: $(1 + 0.062)^{1} - 1 = 0.062$, which is 6.20%.
Option (B) uses $i = 0.0155$ and $k = 4$: $(1.0155)^{4} - 1 = 0.0634565 \approx 6.35\%$.
Option (C) uses $i = 0.00516667$ and $k = 12$: $(1.00516667)^{12} - 1 = 0.0637925 \approx 6.38\%$.
Option (D) uses $i = 0.00516667$ and $k = 36$: $(1.00516667)^{36} - 1 = 0.203846 \approx 20.38\%$.
(A) is the nominal rate. With one period a year there is no compounding within the year, but this loan compounds monthly.
(B) compounds 4 times a year, but this loan compounds monthly.
(C) uses the monthly rate and the 12 months in one year. This is the effective annual rate.
(D) counts all 36 months of the loan, so it gives the interest over three years, not one year.
Try the question first. Choose a hint above when you want one.
$i = \dfrac{0.062}{12} = 0.00516667$ per month, and $k = 12$ compounding periods per year.
The effective rate is the interest over one year, so $k$ counts the periods in a year, not in the three-year term.
$i_{\text{effective}} = (1 + i)^k - 1 = (1.00516667)^{12} - 1 = 0.0637925 \approx 6.38\%$
The monthly rate is $i = \dfrac{0.062}{12} = 0.00516667$ with $k = 12$ periods in a year, so $i_{\text{effective}} = (1.00516667)^{12} - 1 = 0.0637925 \approx 6.38\%$.
Only mark this done when you could do it without help. Reading the solution does not count.
A Tangent Tuition mock exam question. It is not a past QCAA question, and Tangent Tuition is not affiliated with the QCAA.
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