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.