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Module 7 · L3 of 20 ~40 min ⚡ +95 XP available

Effective Annual Rate of Interest

A bank advertises 12% p.a., but compounds monthly. A credit card charges 20% p.a., but compounds daily. The number on the poster is almost never the number you actually pay or earn. In this lesson you will strip away the marketing language and calculate the true rate: the Effective Annual Rate (EAR).

Today's hook, Two savings accounts both advertise "6% p.a.", but one compounds annually and one compounds monthly. They cannot both give you the same final amount. Which one wins, and by exactly how much?
0/5QUESTS

Get oriented

Recall what you already know, meet the key ideas and settle the terms.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Recall, your gut answer first
+5 XP warm-up

Two savings accounts both advertise "6% p.a.":

Account A: 6% p.a. compounded annually.

Account B: 6% p.a. compounded monthly.

Which account gives you more money after one year? Make a prediction and explain your reasoning before reading on.

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Beyond the syllabus. Effective annual rate conversion is supporting depth: it helps compare products but is not a stated core requirement of the 2024 Financial Mathematics focus. The exam's core content is reducing-balance loans and annuities modelled by recurrence — treat this lesson as useful support.
02
Nominal vs Effective Annual Rate
+5 XP to read

The nominal rate is the advertised annual rate before adjusting for compounding. The Effective Annual Rate (EAR) is what you actually earn or pay once compounding is included. They are the same only when compounding is annual, in every other case, the EAR is higher.

The EAR formula is derived directly from the compound interest formula with $P = 1$ over one year. Lock it in: divide the nominal rate by the number of compounding periods, add 1, raise to that power, then subtract 1.

The EAR converts a nominal rate compounded n times a year into a true annual rate.
$$r_{\text{eff}} = \left(1 + \dfrac{r}{n}\right)^n - 1$$

Consider $10,000 at 12% p.a. with different compounding frequencies:

Annual ($n=1$)
$A = 10{,}000(1.12)^1 = \$11{,}200$
EAR = 12.00%
Monthly ($n=12$)
$A = 10{,}000(1.01)^{12} = \$11{,}268.25$
EAR = 12.68%
Daily ($n=365$)
$A = 10{,}000\bigl(1+\tfrac{0.12}{365}\bigr)^{365} = \$11{,}274.96$
EAR = 12.75%
Financial Trap Alert, Credit Cards. A typical Australian credit card advertises "19.99% p.a." but compounds daily. The EAR is $(1 + 0.1999/365)^{365} - 1 = 22.13\%$. On a $\$5,000$ balance carried for one year, that is an extra $\$107$ in interest that the headline rate conceals.
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What you'll master
Know

Key facts

  • The EAR formula and when to apply it
  • The difference between nominal and effective rates
  • Common compounding frequencies and their period counts
Understand

Concepts

  • Why more frequent compounding always produces a higher EAR
  • How marketing uses nominal rates to make products appear better
  • The mathematical limit as compounding approaches continuous
Can do

Skills

  • Calculate EAR for any nominal rate and compounding frequency
  • Compare two or more products using EAR
  • Identify misleading financial advertising using mathematics
  • Convert between nominal and effective rates in both directions
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Key terms
Nominal rateThe advertised annual rate before compounding adjustments. Also called the "headline rate."
Effective Annual Rate (EAR)The true annual interest rate after accounting for compounding. Always $\geq$ nominal rate.
Compounding frequency ($n$)How many times interest is added per year. Monthly = 12, quarterly = 4, daily = 365, fortnightly = 26.
Continuous compoundingThe theoretical limit as $n \to \infty$; gives EAR $= e^r - 1$.
Compounding periodThe time between each interest calculation. The rate per period is $r/n$.
Salvage rate conversionFinding the nominal rate that matches a given EAR: $r = n\!\left[(1+r_{\text{eff}})^{1/n} - 1\right]$.

Tell nominal from effective

See what the same headline rate becomes at different compounding frequencies.

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Nominal vs effective, visual comparison
core concept

At 12% nominal, the effective rate grows with compounding frequency. The bar chart below shows $10,000 growing under three regimes.

Nominal vs Effective Rate Comparison 12.00% 12.68% 12.75% Annual Monthly Daily $10,000 at 12% p.a., Effective Rate by Compounding Frequency nominal = EAR

More frequent compounding always pushes the EAR above the nominal rate.

The difference between 12% and 12.75% may look trivial, but on a $500,000 mortgage over 25 years, that extra 0.75% costs approximately $65,000 in additional interest.

EAR formula: $r_{\text{eff}} = \left(1 + \dfrac{r}{n}\right)^n - 1$; $r$ = nominal annual rate as a decimal; $n$ = compounding periods per year

Pause, copy the effective annual rate (EAR) formula $r_{\text{eff}} = \left(1 + \dfrac{r}{n}\right)^n - 1$ where $r$ is the nominal annual rate and $n$ is the number of compounding periods per year, into your book.

Did you get this? True or false: the effective annual rate is always greater than or equal to the nominal rate.

Work the examples

Follow three worked conversions between nominal and effective rates.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · CALCULATE EAR

A term deposit advertises 5.4% p.a. compounded quarterly. What is the EAR?

1
$r_{\text{eff}} = \left(1 + \dfrac{0.054}{4}\right)^4 - 1$
Identify $r = 0.054$, $n = 4$ (quarterly). Substitute into the EAR formula.
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PROBLEM 2 · COMPARE PRODUCTS

Product X: 5.6% p.a. compounded semi-annually. Product Y: 5.5% p.a. compounded monthly. Which is better for the investor?

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Product X: $(1 + 0.056/2)^2 - 1$
$n = 2$ for semi-annual. Set up the EAR formula for each product separately.
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PROBLEM 3 · LOAN COMPARISON

A car loan advertises 8.9% p.a. compounded fortnightly. A competitor advertises 9.1% p.a. compounded monthly. Which loan has the lower true interest cost?

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Loan 1: $r_{\text{eff}} = (1 + 0.089/26)^{26} - 1$
Loan 2: $r_{\text{eff}} = (1 + 0.091/12)^{12} - 1$
Fortnightly: $n = 26$. Monthly: $n = 12$. Set up both.
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Quick check: A savings account offers 4.8% p.a. compounded monthly. Which expression gives the EAR?

Avoid the traps

Stop comparing nominal rates directly across different frequencies.

Common errors · the 3 traps that cost marks

Trap 01
Confusing nominal with effective
Students compare two products by their advertised rates and pick the wrong one. Always compute EAR first, a higher nominal rate with annual compounding can lose to a lower nominal rate with daily compounding.
Trap 02
Forgetting to subtract 1
The formula is $(1 + r/n)^n - 1$. Writing just $(1 + r/n)^n$ gives 1 plus the EAR. The final $-1$ converts the growth multiplier into a rate. Never leave it out.
Trap 03
Using percentage instead of decimal
If the nominal rate is "12%", use $r = 0.12$, not $r = 12$. Using $r = 12$ gives an absurd answer. Convert to decimal before substituting.

Tell the logic: A credit card charges 20% p.a. compounded daily. Why is the true annual cost to the cardholder more than 20%?

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Drill it and revisit

Run the drill, then return to your opening answer and name what has changed.

Quick-fire practice · 4 EAR calculations

1

Find the EAR for 6% p.a. compounded quarterly. Give your answer to 2 decimal places.

2

Find the EAR for 18% p.a. compounded monthly.

3

A payday lender charges 1% per week. Use $r = 0.01$, $n = 52$ to find the EAR.

4

For $r = 10\%$ continuous compounding, use $e^{0.10} - 1$ to find the EAR limit.

Fill in the blank: To compare two financial products fairly, you must always compute the rather than relying on the .

Match each term with its meaning:

  • n = 12
  • n = 4
  • n = 26
  • n = 365
  • Fortnightly compounding
  • Daily compounding
  • Monthly compounding
  • Quarterly compounding
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Revisit your initial thinking

Account B (6% p.a. compounded monthly) wins. Account A: $A = 10{,}000(1.06) = \$10{,}600$. Account B: $A = 10{,}000(1.005)^{12} = \$10{,}616.78$. The difference is only $\$16.78$ in one year, but over 20 years at 6%, monthly compounding yields approximately $\$600$ more than annual compounding. The power of compounding frequency is subtle in the short term and substantial in the long term.

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Practise effective annual rates

Answer the question bank, then write full short-answer responses.

01
Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

02
Short answer
ApplyBand 43 marks

Q1. A home loan advertises 8.4% p.a. compounded monthly. (a) Calculate the EAR to 2 decimal places. (b) Explain in one sentence why the EAR is higher than the nominal rate. (3 marks)

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ApplyBand 43 marks

Q2. Product A: 5.8% p.a. compounded semi-annually. Product B: 5.7% p.a. compounded monthly. Which product offers a better return to an investor? Show all working. (3 marks)

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AnalyseBand 54 marks

Q3. A credit card charges 20% p.a. compounded daily. (a) Calculate the EAR (to 2 d.p.). (b) On a $5,000 balance carried for one year, how much extra interest does the daily compounding cause compared to annual compounding? (c) Explain why the EAR reveals the true cost that the nominal rate hides. (4 marks)

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Comprehensive answers (click to reveal)

Drill 1: $(1 + 0.06/4)^4 - 1 = 6.14\%$

Drill 2: $(1 + 0.18/12)^{12} - 1 = 19.56\%$

Drill 3: $(1 + 0.01)^{52} - 1 = 67.77\%$

Drill 4: $e^{0.10} - 1 = 10.52\%$

Q1 (3 marks): (a) $r_{\text{eff}} = (1 + 0.084/12)^{12} - 1 = (1.007)^{12} - 1 = 8.73\%$ [2 marks]. (b) Interest is added monthly and then earns further interest over the remaining months, so the total exceeds the simple annual rate [1 mark].

Q2 (3 marks): Product A: $(1 + 0.058/2)^2 - 1 = (1.029)^2 - 1 = 5.88\%$ [1 mark]. Product B: $(1 + 0.057/12)^{12} - 1 = 5.85\%$ [1 mark]. Product A is better because its EAR (5.88%) exceeds Product B's EAR (5.85%) [1 mark]. Note that A also has the higher NOMINAL rate (5.8% against 5.7%), so here the two orderings happen to agree. That is not guaranteed: more frequent compounding can let a lower nominal rate win, which is exactly why the EAR comparison is the one that settles it.

Q3 (4 marks): (a) $(1 + 0.20/365)^{365} - 1 = 22.13\%$ [1 mark]. (b) Extra interest = $5{,}000 \times (0.2213 - 0.20) = \$106.50$ [1 mark]. (c) The nominal rate assumes interest is only calculated once per year. Daily compounding means each day's interest earns interest itself; by year-end the total growth exceeds what the headline rate predicts [2 marks].