F. Financial Literacy · Financial Decisions: Interest, Borrowing, Appreciation, Depreciation and Budgets

Compound Interest and Compounding Periods

F1.3 A1About 70 minutes

Learning goals

I can explain the difference between simple and compound interest.
I can find the interest rate per period (i) and the number of periods (n) for different compounding periods.
I can use A = P(1 + i)ⁿ to find the amount for annual, semi-annual, quarterly and monthly compounding.
I can compare rates with different compounding periods using the effective annual rate.

Words to know

principal (P)
The amount of money first saved, invested or borrowed.

amount (A)
The principal plus all the interest, at the end of the time.

compounding period
How often interest is calculated and added to the balance: yearly (annually), twice a year (semi-annually), 4 times a year (quarterly), monthly or daily.

interest rate per period (i)
The annual rate divided by the number of compounding periods per year. 6% compounded monthly gives i = 0.06 ÷ 12 = 0.005.

number of periods (n)
The number of years × the number of compounding periods per year.

effective annual rate
The rate that, compounded once a year, would give the same growth as the stated rate with its compounding period. It lets you compare rates fairly.

annual percentage rate
The yearly interest rate that a lender or bank states, before you account for how often it compounds.

Remember from Grade 8

Simple interest is paid only on the principal: I = Prt. $1 000 at 6% for 5 years earns 1000 × 0.06 × 5 = $300, so A = $1 300.
Compound interest is paid on the principal and on interest already earned. Compounded once a year: A = P(1 + r)ⁿ, so $1 000 at 6% for 5 years grows to 1000 × 1.06⁵ ≈ $1 338.23.

In Grade 9 you will learn what happens when interest is compounded more often than once a year. This matters for savings, loans, mortgages and credit cards.

Hook: same rate, different results

Two banks both advertise 6% per year on savings. Bank A adds interest once a year. Bank B adds it every month. After 5 years, will you have the same amount? No: Bank B pays a bit more, because each month's interest starts earning interest right away. The compounding period is one of the "ways interest is calculated" that the curriculum asks you to compare.

The same year split into different compounding periods. Each dot is a moment when interest is calculated and added to the balance. More dots means interest starts earning interest sooner.

The compound interest formula with compounding periods

A = P(1 + i)ⁿ

P = principal
i = interest rate per compounding period = annual rate ÷ periods per year
n = number of compounding periods = years × periods per year
A = final amount. The interest earned is A − P.

Notice that in Grade 8, n meant the number of years. Now n means the number of periods. When interest is compounded annually, the two are the same.

Compounding	Periods per year	i for 6% per year	n for 5 years
Annually	1	0.06	5
Semi-annually	2	0.03	10
Quarterly	4	0.015	20
Monthly	12	0.005	60
Daily	365	0.06 ÷ 365	1825

How to find i and n. Always divide the annual rate and multiply the years by the same number.

Worked example

Example 1: Quarterly compounding

Find the amount after 10 years if $2 000 is invested at 4% per year, compounded quarterly. How much interest is earned?

Quarterly means 4 periods per year, so i = 0.04 ÷ 4 = 0.01.
n = 10 × 4 = 40 periods.
A = 2000(1 + 0.01)⁴⁰ = 2000 × 1.01⁴⁰ ≈ 2000 × 1.488864 ≈ 2977.73.
Interest = 2977.73 − 2000 = 977.73.

A ≈ $2 977.73, so the interest earned is about $977.73.

How interest is calculated	Amount after 5 years	Interest earned
Simple interest	$1 300.00	$300.00
Annually (n = 5)	$1 338.23	$338.23
Semi-annually (n = 10)	$1 343.92	$343.92
Quarterly (n = 20)	$1 346.86	$346.86
Monthly (n = 60)	$1 348.85	$348.85
Daily (n = 1825)	$1 349.83	$349.83

$1 000 at 6% per year for 5 years. More frequent compounding gives more interest, but the extra amount gets smaller and smaller.

After 30 years, simple interest gives $2 800 but compound interest gives about $5 743.49. Simple interest grows by the same $60 every year (a straight line). Compound interest grows by more every year (a curve).

Comparing rates fairly: the effective annual rate

How do you compare 5.00% compounded annually with 4.90% compounded monthly? Find what each one does to $1 in one year.

Effective annual rate = (1 + i)ᵖ − 1, where p is the number of periods in one year.

This matters for borrowing too. A credit card might state 19.99% per year. If the interest is compounded monthly and you never pay it off, the effective rate is (1 + 0.1999 ÷ 12)¹² − 1 ≈ 21.93% per year.

Worked example

Example 2: Which savings rate is better?

Bank A pays 5.00% compounded annually. Bank B pays 4.90% compounded monthly. Which is better for a saver?

Bank A: effective rate = 5.00% (compounded once a year, so nothing changes).
Bank B: i = 0.049 ÷ 12. In one year: (1 + 0.049 ÷ 12)¹² ≈ 1.050116.
Effective rate for Bank B ≈ 1.050116 − 1 = 0.050116, or about 5.01%.

Bank B is very slightly better (about 5.01% compared with 5.00%), even though its stated rate is lower. The difference is small, so other things like fees may matter more.

Worked example

Example 3: Compound interest on a debt

Taylor borrows $3 000 at 9% per year, compounded monthly, and makes no payments for 2 years (a "deferred payment" plan). How much does Taylor owe?

i = 0.09 ÷ 12 = 0.0075 and n = 2 × 12 = 24.
A = 3000 × 1.0075²⁴ ≈ 3000 × 1.196414 ≈ 3589.24.
Interest = 3589.24 − 3000 = 589.24.

Taylor owes about $3 589.24. Compound interest works against you when you borrow: interest is charged on unpaid interest.

Calculator tip

Type the whole thing in one line: 2000 × (1 + 0.04 ÷ 4) ^ 40. Use the bracket keys, and do not round i before raising it to the power. Rounding too early can change the answer by several dollars.

Hands-on activity

The bean bank (simple vs. compound)

You need: about 200 dried beans or other small counters (grown-up note: keep small objects away from young children), two bowls, paper.

Put 50 beans in each bowl. Bowl S is a simple-interest account. Bowl C is a compound-interest account. The rate is 10% per round (each round is one "year").
Bowl S: each round add 10% of the original 50 beans, which is 5 beans.
Bowl C: each round add 10% of what is in the bowl now, rounded to the nearest whole bean.
Do 5 rounds and record a table.
Then do 5 rounds with a new rule for bowl C: add 5% of the balance twice per round (semi-annual compounding). Compare.

Model answer: Bowl S: 50, 55, 60, 65, 70, 75. Bowl C: 50, 55, 61 (60.5 rounded), 67 (67.1), 74 (73.7), 81 (81.4). The formula without rounding gives 50 × 1.1⁵ ≈ 80.5. Compound grows faster because the "interest" itself earns interest.

Just for fun

Why did the savings account love monthly compounding? Because it got a little interest in it twelve times a year instead of once.

Your turn

Find the amount after 3 years: $500 at 4% per year, compounded semi-annually.

Find i and n for 7.2% per year, compounded monthly, for 5 years.

Find the amount: $1 500 at 3% per year, compounded monthly, for 2 years.

Find the amount for the same $1 500 at 3% for 2 years using simple interest. Compare.

Which is better for a saver: 4.8% compounded annually or 4.6% compounded monthly?

Find the effective annual rate of 12% compounded monthly.

Pat owes $800 at 18% per year, compounded monthly, and makes no payments for 6 months. How much is owed?

More practice: warm-up

6% per year is compounded quarterly. What is the rate per quarter?

How many monthly periods are in 3 years?

Find the simple interest on $400 at 5% per year for 2 years.

Find the amount: $100 at 10% compounded annually for 2 years.

Same principal, same yearly rate, same time. Which gives more interest: compounding annually or monthly?

More practice: on your own

Find the amount: $2 500 at 5% per year compounded quarterly for 4 years.

Find the amount: $10 000 at 3.6% per year compounded monthly for 10 years.

A credit card charges 19.99% per year compounded monthly. Find the effective annual rate.

$1 000 is invested at 6% for 5 years. How much more does monthly compounding earn than annual compounding?

How much must you invest now at 4% compounded annually to have $5 000 in 5 years?

Challenge

$1 000 is invested at 6% compounded monthly. Use guess-and-check to find the number of months it takes to double. Compare with the Rule of 72.

For $10 000 over 10 years, compare Bank A (5.00% compounded annually) with Bank B (4.90% compounded monthly).

Find the effective annual rate of 6% compounded daily and hourly (8760 hours in a year). What do you notice compared with monthly (6.17%)?

Check your understanding

Quiz

A rate of 8% per year is compounded quarterly. What is i?

$1 000 is invested at 6% compounded monthly for 5 years. What is n?

$1 000 at 6% for 5 years. Which way of calculating interest gives the largest amount?

Which situation does A = 2000(1.01)⁴⁰ describe?

A credit card states 19.99% per year, compounded monthly. Its effective annual rate is about…

Why does compound interest grow faster than simple interest?

For parents and teachers

Big idea: F1.3 asks students to compare how different interest rates, lengths of time and ways interest is calculated affect the cost of buying things. This lesson extends Grade 8's annual compounding to semi-annual, quarterly, monthly and daily compounding, and introduces the effective annual rate for fair comparisons.

Common difficulties:

Forgetting to adjust both i and n. If the rate is divided by 12, the years must be multiplied by 12.
Rounding i too early (writing 0.049 ÷ 12 as 0.004). Keep full calculator precision until the end.
Confusing n with years. In this unit, n is the number of compounding periods.

How to help: Look at a real savings account or credit card statement together and find the stated rate and how often interest is calculated. Ask your teen to estimate the effective annual rate. All rates here are examples; real rates change often.