Mathematics Advanced • Year 12 • Module 7 • Lesson 3
Effective Annual Rate of Interest
Practise HSC-style writing on EAR — including a structured extended response with marking criteria.
1. Short-answer questions
1.1 A savings account is advertised at 8.4% p.a. compounded monthly.
(a) Calculate the effective annual rate to two decimal places.
(b) Explain in one sentence why the effective rate is higher than the nominal. 3 marks Band 3
1.2 Product A: 5.8% p.a. compounded semi-annually. Product B: 5.7% p.a. compounded monthly. Compute the EAR of each and state which is the better investment. 3 marks Band 3-4
1.3 A credit card advertises 20% p.a. compounded daily.
(a) Find the EAR to two decimal places.
(b) Find the extra annual interest, in dollars, that the EAR implies on a carried balance of $5,000 compared with the nominal rate.
(c) Explain in 1-2 lines why a borrower paying only the minimum each month rarely reduces the principal. 4 marks Band 4
2. Extended response
2.1 A consumer must choose between three home-loan products to borrow $400,000. The lender advertises each at a different nominal rate and compounding frequency.
Loan P: 5.95% p.a. compounded annually.
Loan Q: 5.85% p.a. compounded monthly.
Loan R: 5.80% p.a. compounded daily.
(a) Compute the effective annual rate of each loan to two decimal places.
(b) Rank the loans from cheapest to most expensive based on EAR.
(c) On a $400,000 balance carried for one year, compute the EAR-based interest cost for each loan and the dollar saving of the cheapest over the most expensive.
(d) Write a short paragraph (3-4 sentences) advising the consumer why ranking by nominal rate would have given the wrong answer here, and how the rate-and-frequency interaction explains the swap. 8 marks Band 5-6
Explicit marking criteria
Part (a) — 3 marks
• 1 mark — EARP = 5.95% (annual compounding ⇒ EAR = nominal).
• 1 mark — EARQ = (1 + 0.0585/12)¹² − 1 ≈ 6.01%.
• 1 mark — EARR = (1 + 0.058/365)³⁶⁵ − 1 ≈ 5.97%.
Part (b) — 1 mark
• Ranking cheapest → most expensive: P (5.95%) < R (5.97%) < Q (6.01%).
Part (c) — 2 marks
• 1 mark — correct interest costs (≈ $23,800 / $23,880 / $24,040).
• 1 mark — correct saving figure ≈ $240 of P over Q.
Part (d) — 2 marks
• 1 mark — explicitly states ranking by nominal rate gives R < Q < P (the reverse of the true ranking).
• 1 mark — explains the frequency-rate interaction (monthly compounding of 5.85% beats the smaller nominal of 5.80% when daily/yearly compounding is compared because the (1 + r/n)ⁿ structure depends on both r and n).
Your response:
Stuck on (d)? Pair "ranking by nominal would have picked R" with one sentence on why frequency × rate matters.How did this worksheet feel?
What I'll revisit before next class:
1.1 — 8.4% p.a. monthly (3 marks)
Sample response. (a) reff = (1 + 0.084/12)¹² − 1 = (1.007)¹² − 1 = 1.08731 − 1 = 8.73%. (b) The interest added each month earns interest itself in subsequent months, so the actual annual return exceeds the nominal 8.4%.
Marking notes. 2 marks — formula substitution and final figure. 1 mark — explanation references compounding-on-interest within the year. A bald answer of 8.73% scores 1/2 for the calculation.
1.2 — Product A vs B (3 marks)
Sample response. A: reff = (1 + 0.058/2)² − 1 = (1.029)² − 1 = 5.88%. B: reff = (1 + 0.057/12)¹² − 1 = 5.85%. Product A is the better investment (higher EAR) despite the lower compounding frequency — its higher nominal rate dominates.
Marking notes. 1 mark each for the two EARs (1 dp acceptable). 1 mark for the comparison statement with the reason (the higher nominal beats the more frequent compounding here).
1.3 — Credit card 20% p.a. daily (4 marks)
(a) Sample. reff = (1 + 0.20/365)³⁶⁵ − 1 = 1.22134 − 1 = 22.13%.
(b) Sample. Extra interest on $5,000 = 5,000 × (0.2213 − 0.20) = 5,000 × 0.0213 ≈ $106.58 per year.
(c) Sample. On a $5,000 balance at 22.13% EAR the monthly interest is about $92. A typical "minimum payment" of 2-3% of the balance is around $100-$150 — barely above the interest cost — so the principal hardly drops while compounding keeps accelerating.
Marking notes. (a) 1 mark for substitution and final figure. (b) 1 mark for the calculation. (c) 2 marks for tying the monthly-interest figure to the minimum-payment figure (1 mark for one without the other).
2.1 — Extended response (8 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Effective annual rates.
Loan P (annual). EARP = 5.95% (annual compounding gives EAR = nominal). [1 mark]
Loan Q (monthly). EARQ = (1 + 0.0585/12)¹² − 1 = (1.004875)¹² − 1 = 1.06012 − 1 = 6.01%. [1 mark]
Loan R (daily). EARR = (1 + 0.058/365)³⁶⁵ − 1 = 1.05971 − 1 = 5.97%. [1 mark]
(b) Ranking cheapest to most expensive. P (5.95%) < R (5.97%) < Q (6.01%). [1 mark]
(c) Annual interest on $400,000. P: 400,000 × 0.0595 = $23,800.00. R: 400,000 × 0.0597 = $23,884.00. Q: 400,000 × 0.0601 = $24,048.00. [1 mark] Saving of P over Q = 24,048 − 23,800 = $248.00. [1 mark]
(d) Why nominal-rate ranking misleads. Ranking by nominal rate gives R (5.80%) < Q (5.85%) < P (5.95%), the reverse of the true EAR ranking. The reason is structural: the EAR depends on the product r · (1 + r/n)n−1-type effect of both r and n. Monthly compounding of 5.85% adds 12 cycles of interest-on-interest per year, which raises Q's true cost above P's annual 5.95%; daily compounding of 5.80% adds the most cycles but cannot overcome a 15-basis-point nominal deficit. The consumer should always compare on EAR to capture both effects. [2 marks]
Total: 8/8.
Band descriptors for marker.
Band 3: EAR formula applied to one or two loans correctly, ranking attempted but not justified. ≈ 3-4 marks.
Band 4: All three EARs correct, ranking correct, dollar interest correct but no critique of nominal-rate ranking. ≈ 5-6 marks.
Band 5: All calculations correct and ranking justified; critique present but lacks the explicit "reverse of nominal ranking" observation. ≈ 6-7 marks.
Band 6: Full calculations, ranking justified, dollar saving computed, and critique explicitly contrasts the EAR ranking with the nominal ranking and explains why both r and n matter via the (1 + r/n)ⁿ structure. 8/8.