Question 15
Which of the following investment options gives the best return?
Not quite. Daily compounding has the most periods, but it also has the lowest nominal rate. Its effective annual rate is 6.1088%, just under the 6.1150% for (B). One way to get this is to assume more compounding always wins and skip the calculation. In 2021, 35.4% of Queensland students chose (A).
That’s it. Step through the hints below to see why. In 2021, 34.0% of Queensland students chose (B).
Not quite. Quarterly compounding at 5.97% gives an effective annual rate of 6.1050%. That’s below both (B), 6.1150%, and (A), 6.1088%. In 2021, 9.9% of Queensland students chose (C).
Not quite. 5.99% is the highest nominal rate, but interest is only added twice a year. Its effective annual rate is 6.0797%, the lowest of the four. One way to get this is to compare the nominal rates and ignore the compounding. In 2021, 19.6% of Queensland students chose (D).
Calculate each option’s effective annual rate. The bars only show the narrow band from 6.05% to 6.12%, so small gaps look big.
The four options pull in opposite directions. The nominal rate goes up from (A) to (D), while the number of compounding periods goes down.
You can’t compare them by eye. Turn each one into an effective annual rate: the actual percentage growth over one year. Then calculate, compare, conclude.
Split the annual rate into the rate per period, $i$, and count the periods in a year, $k$. Then use the formula book:
$$i_{\text{effective}}=(1+i)^k-1$$
$$\text{(B): }\left(1+\frac{0.0595}{12}\right)^{12}-1=0.061\,149\ldots\approx6.1150\%$$
In 2021 the formula sheet wrote this as $\left(1+\frac{i}{n}\right)^n-1$ with $i$ the annual rate. It’s the same calculation: the old $\frac{i}{n}$ is the 2025 book’s $i$, and the old $n$ is $k$.
The rates are very close. (B) beats (A) by only 0.006 of a percentage point, so keep plenty of decimal places until you compare.
Switch the left panel to 1 dp and all four options read 6.1%. Try 2 dp as well. Rounding early would leave you guessing. Type each rate in one line and read off at least four decimal places of the percentage.
Monthly compounding at 5.95% has the highest effective annual rate, so it gives the best return. Using 360 or 366 days for (A) doesn’t change the order.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2021 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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