Mathematics Advanced • Year 12 • Module 7 • Lesson 3

Effective Annual Rate of Interest

Build fluency converting between nominal and effective annual rates and ranking financial products fairly.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Write the EAR formula:

r_eff = ________________________________

Q1.2 State, in one sentence, the difference between a nominal annual rate and an effective annual rate.

Q1.3 When comparing two financial products with different compounding frequencies, which rate must you use to make a fair comparison?

Stuck? Revisit lesson § Formula Reference and § Nominal vs Effective.

2. Worked example — comparing two car loans

Follow each line. Each step has a short reason.

Problem. Loan 1: 8.9% p.a. compounded fortnightly. Loan 2: 9.1% p.a. compounded monthly. Which has the lower true cost?

Step 1 — Identify n (periods per year) for each.

Loan 1: n = 26 (fortnightly = every 2 weeks)

Loan 2: n = 12 (monthly)

Step 2 — Apply r_eff = (1 + r/n)ⁿ − 1 for Loan 1.

r_eff,1 = (1 + 0.089/26)²⁶ − 1 = (1.003423)²⁶ − 1

r_eff,1 = 1.09280 − 1 = 0.09280 = 9.28%

Step 3 — Apply the formula for Loan 2.

r_eff,2 = (1 + 0.091/12)¹² − 1 = (1.007583)¹² − 1

r_eff,2 = 1.09479 − 1 = 0.09479 = 9.48%

Step 4 — Compare and conclude.

Reason: lower EAR = cheaper loan.

Conclusion. Loan 1 is cheaper (9.28% vs 9.48%), even though it has the higher nominal rate. Fortnightly compounding (less frequent than monthly) closes the nominal-vs-effective gap less, leaving the effective rate lower.

3. Faded example — EAR of a 4.2% p.a. quarterly account

Fill in each blank line. 4 marks

Step 1 — Identify n: Quarterly compounding means n = ____________ per year.

Step 2 — Substitute into the EAR formula:

r_eff = (1 + ________ / ________)^________ − 1

Step 3 — Evaluate the bracket:

r_eff = ( ____________ )⁴ − 1

Step 4 — Raise to the power and subtract 1:

r_eff = ____________ − 1 = ____________ = ____________ %

Conclusion. EAR ≈ ____________ %, which is ____________ percentage points above the nominal 4.2%.

Stuck? Revisit lesson § EAR Formula — Worked Example.

4. Graduated practice — compute EAR and compare

For each, state the nominal rate, the value of n, the formula substitution, and the EAR (to 2 dp where helpful).

Foundation — single EAR calculations (4 questions)

Q	Scenario	EAR
4.1 1	6% p.a. compounded annually
4.2 1	6% p.a. compounded quarterly
4.3 1	6% p.a. compounded monthly
4.4 1	6% p.a. compounded daily (n = 365)

Standard — typical HSC difficulty (6 questions)

Show working: write the substitution line for each.

4.5 A term deposit advertises 5.4% p.a. compounded quarterly. Find the EAR to 2 dp. 2 marks

4.6 A credit card advertises 19.99% p.a. compounded daily. Find the EAR. State the extra dollar interest per year on a $5,000 balance vs the nominal rate. 2 marks

4.7 Product X: 5.6% p.a. compounded semi-annually. Product Y: 5.5% p.a. compounded monthly. Compute the EAR of each and state which is better. 2 marks

4.8 A bonus savings account pays 4.8% p.a. compounded monthly for the first 6 months, then 3.2% p.a. compounded monthly. Compute the EAR for each phase separately. 2 marks

4.9 Express 8% nominal compounded monthly as an EAR; then express 7.85% nominal compounded daily as an EAR. Which earns more on the same deposit? 2 marks

4.10 A loan is advertised at 8.4% p.a. compounded monthly. (a) Find the EAR. (b) Explain in one line why the effective rate is higher than the nominal. 2 marks

Extension — combine concepts (2 questions)

4.11 A payday lender advertises "only 1% per week." Use r = 0.01, n = 52 to find the annual EAR. The ASIC cap on small-amount credit contracts is 48% EAR — is this loan legal? 3 marks

4.12 Show that as n → ∞ (continuous compounding) the formula gives r_eff = e^r − 1. Hence find the continuous-compounding EAR for a nominal rate of 10% p.a. and compare with the EAR for daily compounding at the same nominal rate. 3 marks

Stuck on 4.12? Use the limit definition: lim_n→∞ (1 + r/n)ⁿ = eʳ.

5. Self-check the easy 3

Tick once you have checked your method.

For 4.1 I obtained 6.00% — the EAR of an annually-compounded product equals the nominal rate.

For 4.2–4.4 the EAR strictly increases with compounding frequency: 6.14% < 6.17% < 6.18%.

For 4.6 I noticed the EAR exceeds the nominal by about 2 percentage points — daily compounding compounds at a higher rate.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — EAR formula

r_eff = (1 + r/n)ⁿ − 1, where r is the nominal annual rate and n is the number of compounding periods per year.

Q1.2 — Nominal vs effective

The nominal rate is the advertised annual rate before adjusting for compounding; the effective rate is the actual annual cost or return after compounding is included.

Q1.3 — Fair comparison

Always compare using the effective annual rate (EAR), not the nominal rate, because the EAR puts all products on the same time scale (one year, ignoring compounding-frequency differences).

Q3 — Faded example: 4.2% p.a. quarterly

Step 1: n = 4. Step 2: r_eff = (1 + 0.042/4)⁴ − 1. Step 3: (1.0105)⁴ − 1. Step 4: 1.04267 − 1 = 0.04267 = 4.27%. EAR exceeds nominal by 0.07 percentage points.

Q4.1–4.4 — EAR at 6% p.a. across frequencies

Annual: 6.00%. Quarterly: (1.015)⁴ − 1 = 6.14%. Monthly: (1.005)¹² − 1 = 6.17%. Daily: (1 + 0.06/365)³⁶⁵ − 1 = 6.18%.

Q4.5 — 5.4% p.a. quarterly

r_eff = (1 + 0.054/4)⁴ − 1 = (1.0135)⁴ − 1 = 1.05509 − 1 = 5.51%.

Q4.6 — 19.99% p.a. credit card daily

r_eff = (1 + 0.1999/365)³⁶⁵ − 1 = 1.22125 − 1 = 22.13%. Extra interest on $5,000: 5,000 × (0.2213 − 0.1999) = 5,000 × 0.0214 ≈ $107 per year hidden by the nominal rate.

Q4.7 — Product X vs Y

X: (1 + 0.056/2)² − 1 = (1.028)² − 1 = 5.68%. Y: (1 + 0.055/12)¹² − 1 = 5.64%. X is better despite the lower-frequency compounding — its higher nominal rate dominates.

Q4.8 — Bonus-savings phases

Phase 1 (4.8% p.a. monthly): (1.004)¹² − 1 = 4.91%. Phase 2 (3.2% p.a. monthly): (1 + 0.032/12)¹² − 1 = 3.25%.

Q4.9 — 8% monthly vs 7.85% daily

8% monthly: (1 + 0.08/12)¹² − 1 = 8.30%. 7.85% daily: (1 + 0.0785/365)³⁶⁵ − 1 = 8.16%. The 8% monthly product earns more — daily compounding cannot overcome a 0.15-point nominal deficit at these rates.

Q4.10 — 8.4% p.a. monthly loan

(a) r_eff = (1 + 0.084/12)¹² − 1 = (1.007)¹² − 1 = 1.08731 − 1 = 8.73%. (b) Each month's interest is added to the balance and itself earns interest the following month — that "interest-on-interest" pushes the true rate above the nominal.

Q4.11 — Payday lender

r_eff = (1.01)⁵² − 1 = 1.67769 − 1 = 67.77% p.a. This exceeds the ASIC EAR cap of 48% on small-amount credit contracts, so the loan as advertised would be illegal in Australia (additional fees would also count toward the cap).

Q4.12 — Continuous compounding limit

lim_n→∞ (1 + r/n)ⁿ = e^r, so r_eff,∞ = e^r − 1. At r = 0.10: e^0.10 − 1 = 1.10517 − 1 = 10.52%. Daily compounding at the same nominal: (1 + 0.10/365)³⁶⁵ − 1 = 10.516% — almost identical, because daily compounding is already very close to the continuous limit.