Comparing investments and effective interest rates (QCE General Mathematics Unit 4)

The nominal rates cannot be compared directly because they compound at different frequencies, so convert each one to an effective annual rate with $i_{\text{effective}} = (1 + i)^k - 1$, where $i$ is the rate per compounding period and $k$ is the number of compounding periods per year.

Option A (1 mark): $i = \frac{0.056}{12}$, $k = 12$:

Option B (1 mark): $i = \frac{0.0562}{4} = 0.014\,05$, $k = 4$:

Compare (1 mark): $5.746\% > 5.740\%$, so option A gives the (slightly) higher effective return.

Evaluate (1 mark): Ngarra's decision is reasonable. Option B has the higher nominal rate, but option A's more frequent (monthly) compounding lifts its effective rate just above option B's. The difference is very small (about $6 per year on each $100 000 invested), so either choice would give a similar return.

Modelled on the QCAA 2023 General Mathematics external assessment (not checked word for word against the paper).

(a) $i = \frac{0.06}{4} = 0.015$, $k = 4$: $i_{\text{effective}} = 1.015^4 - 1 \approx 0.061\,364$, so $6.136\%$. (1 mark)

(b) $i = \frac{0.06}{12} = 0.005$, $k = 12$: $i_{\text{effective}} = 1.005^{12} - 1 \approx 0.061\,678$, so $6.168\%$. (1 mark)

(a) $i = 0.004$, $k = 12$: $1.004^{12} - 1 \approx 0.049\,070$, so about $4.907\%$ p.a. (1 mark)

(b) One year at the effective rate: $20\,000 \times 1.049\,070 \approx 20\,981.40$, which matches $20\,000 \times 1.004^{12}$. (1 mark)

Daily 365, weekly 52, fortnightly 26, monthly 12, quarterly 4, six-monthly 2. (1 mark for all six.)

Monthly: $\left(1 + \frac{0.048}{12}\right)^{12} - 1 \approx 0.049\,07$, about $4.907\%$. (1 mark)

Quarterly: $\left(1 + \frac{0.0485}{4}\right)^{4} - 1 \approx 0.049\,39$, about $4.939\%$. (1 mark)

The quarterly option has the higher effective rate, so it is the better investment. (1 mark) Here the higher nominal rate wins despite less frequent compounding.

• P: $\left(1 + \frac{0.089}{12}\right)^{12} - 1 \approx 9.272\%$
• Q: $\left(1 + \frac{0.09}{4}\right)^{4} - 1 \approx 9.308\%$
• R: $\left(1 + \frac{0.0895}{26}\right)^{26} - 1 \approx 9.346\%$

(2 marks) For a loan, the lowest effective rate is cheapest: Loan P. (1 mark)

(a) $i = 0.015$: $1.015^{12} - 1 \approx 0.195\,62$, so about $19.56\%$ p.a. (1 mark)

(b) Each month's interest is added to the balance, and the next month's interest is charged on that larger balance: interest is charged on interest. Over a year this compounding makes the true cost higher than the nominal 18%. (1 mark)

(a) Quarterly: $\left(1 + \frac{0.065}{4}\right)^4 - 1 \approx 6.660\%$. Monthly: $\left(1 + \frac{0.064}{12}\right)^{12} - 1 \approx 6.591\%$. (2 marks)

(b) The advice is not reasonable. More frequent compounding only helps when the nominal rates are equal or very close; here the quarterly option's higher nominal rate more than makes up for its less frequent compounding, giving the higher effective rate (6.660% compared with 6.591%). Jess should choose 6.5% compounding quarterly. (1 mark)

(a) $(1 + i)^{12} - 1 = 0.05$, so $(1 + i)^{12} = 1.05$ and $1 + i = 1.05^{1/12} \approx 1.004\,074$. So $i \approx 0.004\,074$ ($0.4074\%$ per month). (1 mark)

(b) Nominal $= 12 \times 0.4074\% \approx 4.889\%$ p.a. (1 mark)

(c) The nominal rate simply multiplies the monthly rate by 12, ignoring interest on interest. The effective rate includes the extra interest earned when each month's interest itself earns interest, so it is higher. (1 mark)