B. Exponential Functions · Financial Mathematics: Compound Interest and Annuities

The Compound Interest Formula: Finding the Amount (Future Value)

B3.2About 70 minutes

Learning goals

I can identify P, i, n and A in a compound interest problem, for any compounding period.
I can use A = P(1 + i)ⁿ with a scientific calculator to find the amount (future value).
I can find the total interest earned or paid with I = A − P.
I can solve multi-stage problems where the rate or compounding changes partway through.

Words to know

future value (FV)
Another name for the amount, A: what an investment or debt is worth at a future date.

present value (PV)
Another name for the principal, P: what money is worth now.

compounding period
The length of time after which interest is calculated and added to the balance (a year, six months, three months, a month, a day).

i (interest rate per compounding period)
The annual rate divided by the number of compounding periods per year: i = r ÷ m.

n (number of compounding periods)
The number of times interest is added: n = m × t, where t is in years.

GIC (Guaranteed Investment Certificate)
A Canadian savings product that pays a set interest rate for a set term, such as 1 to 5 years.

How often is "compounded"?

Open any Canadian bank's website and you will see offers such as "3.2% compounded semi-annually" or "7.5% compounded monthly". The rate tells you how much per year. The words after it tell you how often the interest is calculated and added. To work out what you will end up with, you need one formula and a careful reading of those words.

The compound interest formula

A = P(1 + i)ⁿ (also written FV = PV(1 + i)ⁿ)

A (or FV) = amount, or future value
P (or PV) = principal, or present value
i = interest rate per compounding period, as a decimal
n = number of compounding periods

To get i and n from the words in a problem:

i = r ÷ m and n = m × t

where r is the annual rate, m is the number of compounding periods per year, and t is the number of years.

The total interest is I = A − P.

Compounded…	Periods per year (m)	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
weekly	52	0.06 ÷ 52 ≈ 0.001154	260
daily	365	0.06 ÷ 365 ≈ 0.000164	1825

How the compounding words change i and n. In the calculator, type 0.06 ÷ 52 (not a rounded decimal) to avoid rounding errors.

Each compounding period multiplies the balance by (1 + i). Four periods means multiplying by 1.02 four times, which is (1.02)⁴. That is all the exponent in A = P(1 + i)ⁿ is doing.

Worked example

Example 1: A GIC

Ravi puts $3 500 in a 5-year GIC that pays 3.2% per year, compounded semi-annually. Find the amount at maturity and the interest earned.

Identify: P = 3500, r = 0.032, m = 2, t = 5.
i = 0.032 ÷ 2 = 0.016 and n = 2 × 5 = 10.
A = 3500(1 + 0.016)¹⁰ = 3500(1.016)¹⁰.
Calculator: 1.016 ^ 10 ≈ 1.1720256, so A ≈ 3500 × 1.1720256 ≈ 4102.09.
I = A − P = 4102.09 − 3500 = 602.09.

The GIC is worth $4 102.09 at maturity. Ravi earns $602.09 in interest.

Worked example

Example 2: Compound interest on a debt

Mei-Lin uses a line of credit to borrow $4 000 at 7.5% per year, compounded monthly. She makes no payments for 3 years. How much does she owe, and how much of that is interest?

P = 4000, i = 0.075 ÷ 12 = 0.00625, n = 12 × 3 = 36.
A = 4000(1.00625)³⁶.
Type it in one line: 4000 × (1 + 0.075 ÷ 12) ^ 36 ≈ 5005.78.
I = 5005.78 − 4000 = 1005.78.

She owes $5 005.78, and $1 005.78 of it is interest: more than 25% of what she borrowed.

Does compounding more often help?

Yes, but less than you might think. Keep the annual rate the same and compound more often: each period's interest starts earning interest sooner, so the amount goes up a little. But the gains get smaller and smaller. Going from annual to semi-annual adds about $15 below; going from weekly to daily adds about 54 cents.

Compounding more often always gives a bigger amount, but the increases shrink. The annual rate usually matters much more than the compounding frequency.

Worked example

Example 3: A two-stage investment

Tessa invests $2 000 at 4% compounded quarterly for 3 years. Then she moves the whole amount to an account paying 5% compounded monthly for 2 more years. What does she have at the end?

Stage 1: i = 0.04 ÷ 4 = 0.01, n = 4 × 3 = 12. A₁ = 2000(1.01)¹² ≈ 2253.65.
The amount from stage 1 is the principal for stage 2.
Stage 2: i = 0.05 ÷ 12, n = 12 × 2 = 24. A₂ = 2253.65(1 + 0.05 ÷ 12)²⁴ ≈ 2253.65 × 1.1049413 ≈ 2490.15.
Total interest over 5 years: 2490.15 − 2000 = 490.15.

Tessa has $2 490.15 after 5 years, having earned $490.15 in interest.

Calculator habits that prevent mistakes

Type the whole calculation in one line with brackets: 3500 × (1 + 0.032 ÷ 2) ^ (2 × 5).
Do not round i (for example 0.075 ÷ 12 = 0.00625 is exact, but 0.05 ÷ 12 = 0.0041666… is not). Rounding it to 0.0042 changes the answer.
Round money to the nearest cent only at the end.
Check that n counts periods, not years. Monthly for 3 years is n = 36, not 3.

Just for fun

Why did the GIC refuse to go to the party?

It was locked in for five years and didn't want to pay the early-withdrawal penalty.

Hands-on activity

Read a real rate sheet

You need: a phone or computer, and paper.

With a parent or guardian, look at the savings and GIC rates on the website of a Canadian bank or credit union (you do not need to log in). Write down two offers, including the rate, the term and the compounding period (it may be in the fine print or the product details).
For each offer, calculate what $1 000 would grow to by the end of the term using A = P(1 + i)ⁿ.
Compare: which offer would give more money? Is the difference big or small?
Write one question you would ask the bank before choosing (for example: can I take the money out early, and is there a penalty?).

What to look for: i and n correctly found from the words (e.g., "compounded annually" for a 3-year GIC gives n = 3). Rates change often, so write down the date you looked them up.

Your turn

For 4.8% compounded quarterly for 3 years, find i and n.

Find the amount: $1 500 at 4% compounded quarterly for 2 years.

Find the amount and the interest: $2 500 at 5.5% compounded semi-annually for 3 years.

A credit card balance of $900 is charged 18% compounded monthly. If no payments are made for 1 year, what is owed?

$10 000 is invested at 3.5% compounded semi-annually for 10 years. Find the amount.

Explain why "6% compounded monthly" gives a little more than "6% compounded annually".

More practice: warm-up

How many compounding periods are in 4 years, compounded semi-annually?

Find i for 9% compounded monthly.

What does PV stand for, and which letter in A = P(1 + i)ⁿ is it?

Find the amount: $1 000 at 2.5% compounded annually for 4 years.

An investment grew to $1 340.00 from $1 000. How much interest was earned?

More practice: on your own

Find the amount owed on a loan of $650 at 6% compounded monthly after 18 months with no payments.

Find the amount: $7 500 at 4.8% compounded quarterly for 3 years. How much interest is earned?

Which is better for a saver over 2 years on $5 000: (A) 2.5% compounded annually, or (B) 2.4% compounded monthly? Show the amounts.

$800 is invested at 3% compounded annually for 3 years, then the amount is reinvested at 3% for 3 more years. Show that this gives the same result as 6 years at 3%.

Sam says, "$300 owed at 21% compounded monthly for 6 months is 300 × 1.21⁶." Find and fix the mistake.

Challenge

$1 200 is invested at 8% compounded quarterly. Find the amount after 5 years, and the interest earned in year 5 alone.

Kenji borrows $2 000 at 6% compounded monthly. After 1 year he pays back $1 000. What does he owe 1 year after that payment, if he pays nothing more?

Show that $1 000 at 12% compounded monthly grows to more than $1 000 at 12.6% compounded annually after one year. By how much?

Check your understanding

Quiz

For 6% compounded quarterly for 4 years, what are i and n?

Which expression gives the amount of $2 000 at 3% compounded monthly for 5 years?

An investment of $5 000 grows to $6 083.26. How much interest was earned?

Keeping the annual rate the same, which compounding gives the largest amount after 10 years?

$1 000 is invested at 4% compounded semi-annually for 1 year. What is the amount?

In a two-stage investment, what is the principal for the second stage?

For parents and teachers

Big idea (B3.2): Students use the compound interest formula A = P(1 + i)ⁿ (FV = PV(1 + i)ⁿ) with a scientific calculator to find the amount (future value). The key skill is turning words like "compounded quarterly for 3 years" into i = r ÷ 4 and n = 12. Grade 9 introduced the formula; Grade 11 uses it more flexibly, including multi-stage problems and the FV/PV notation used by financial calculators.

How to help: Read a bank rate sheet together and ask, "What would i and n be?" Ask your teen to estimate before calculating ("about 6% of $1 000 per year, for 5 years, so a bit over $300 in interest").

Common difficulties:

Using the annual rate as i, or years as n, when compounding is not annual.
Rounding i too early (0.05 ÷ 12 should stay in the calculator).
Typing 3500 × 1 + 0.016 ^ 10 without brackets, which the calculator reads differently.
Forgetting that the interest is A − P, not A.

All rates in this lesson are practice examples, not current offers.