Your weak spots

Insights load after your first practice round.

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

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

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.

auto-saved

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.

$$r_{\text{eff}} = \left(1 + \dfrac{r}{n}\right)^n - 1$$

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.

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

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: $r = n\!\left[(1+r_{\text{eff}})^{1/n} - 1\right]$.

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.

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.

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?

$r_{\text{eff}} = \left(1 + \dfrac{0.054}{4}\right)^4 - 1$

Identify $r = 0.054$, $n = 4$ (quarterly). Substitute into the EAR formula.

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?

Product X: $(1 + 0.056/2)^2 - 1$

$n = 2$ for semi-annual. Set up the EAR formula for each product separately.

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?

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.

Quick check: A savings account offers 4.8% p.a. compounded monthly. Which expression gives the EAR?

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%?

auto-saved

Quick-fire practice · 4 EAR calculations

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

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

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

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 [blank] rather than relying on the [blank].

Match each term with its meaning:

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.

auto-saved

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.

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)

auto-saved

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)

auto-saved

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)

auto-saved

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 despite having the lower nominal rate, because its EAR (5.88%) exceeds Product B's EAR (5.85%) [1 mark].

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].

Boss battle · The Loan Shark

earn bronze · silver · gold

Five timed questions on EAR and product comparisons. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.

Enter the arena

Science Jump · platform challenge

Climb platforms by answering EAR, nominal rates, and compounding frequency questions. Pool: lessons 1–3.

Mark lesson as complete

Tick when you've finished the practice and review.

← Compound Interest in Depth Lesson 4 · Depreciation →

Module overview · Maths Advanced · Checkpoint 1