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

The Future Value of an Ordinary Simple Annuity

B3.7 B3.5About 75 minutes

Learning goals

I can use FV = R[(1 + i)ⁿ − 1] ÷ i to find the future value of an ordinary simple annuity with a scientific calculator.
I can find the future value with a TVM Solver or a spreadsheet FV function.
I can find the total deposits and the interest earned (FV − nR).
I can explain where the formula comes from as the sum of a geometric series.

Words to know

regular payment (R or PMT)
The equal amount paid or deposited each period.

future value of an annuity (FV)
The value at the end of the term of all the payments and the interest they earn.

total deposits
The number of payments times the payment: n × R.

interest earned (annuity)
Future value minus total deposits: FV − nR.

TFSA (Tax-Free Savings Account)
A Canadian registered account, open to residents 18 and older, in which investment growth and withdrawals are not taxed.

Adding 36 terms? There's a shortcut

Zoë deposits $50 at the end of every month for 3 years into a savings account paying 3% compounded monthly. In the last lesson you could have found her future value by adding 36 separate terms: 50(1.0025)³⁵ + 50(1.0025)³⁴ + … + 50. That is a lot of button pressing. In this lesson you will use one formula (and technology) to do it in a single step.

The future value formula for an ordinary simple annuity. Find i and n exactly as for compound interest: i = annual rate ÷ periods per year, and n = the number of payments.

The formula

For an ordinary simple annuity with regular payment R, interest rate i per period and n payments:

FV = R[(1 + i)ⁿ − 1] ÷ i

Then:

total deposited = nR
interest earned = FV − nR

Why it works (optional): The future value is the geometric series S = R + R(1 + i) + R(1 + i)² + … + R(1 + i)ⁿ⁻¹. Multiply every term by (1 + i): (1 + i)S = R(1 + i) + R(1 + i)² + … + R(1 + i)ⁿ. Subtract the first line from the second. Almost everything cancels, leaving (1 + i)S − S = R(1 + i)ⁿ − R, so iS = R[(1 + i)ⁿ − 1], and S = R[(1 + i)ⁿ − 1] ÷ i.

Worked example

Example 1: Zoë's savings

Find the future value of $50 deposited at the end of every month for 3 years at 3% compounded monthly. How much interest does Zoë earn?

R = 50, i = 0.03 ÷ 12 = 0.0025, n = 12 × 3 = 36.
FV = 50[(1.0025)³⁶ − 1] ÷ 0.0025.
Calculator: 1.0025 ^ 36 ≈ 1.0940514; subtract 1: 0.0940514; divide by 0.0025: 37.620560; multiply by 50: ≈ 1881.03.
Total deposited: 36 × 50 = $1 800. Interest: 1881.03 − 1800 = $81.03.

Zoë will have $1 881.03, of which $81.03 is interest.

Calculator and technology tips

One line: 50 × ((1 + 0.03 ÷ 12) ^ 36 − 1) ÷ (0.03 ÷ 12). Watch the brackets around the denominator.
TVM Solver: N = 36, I% = 3, PV = 0, PMT = −50, FV = ? (solve), P/Y = 12, C/Y = 12, PMT: END.
Spreadsheet: =FV(0.03/12, 36, -50, 0) returns 1881.03.

Example 1 in a TVM Solver. Each $50 deposit is money Zoë pays out, so PMT is negative; the future value comes out positive. PV = 0 because the account starts empty.

Worked example

Example 2: Filling a TFSA

The Canada Revenue Agency set the 2026 TFSA dollar limit at $7 000. Suppose Mateo, at age 18, starts depositing $7 000 at the end of each year for 10 years, earning 4% compounded annually. (A TFSA can hold savings accounts, GICs or other investments; 4% is an example rate, not a promise.) Find the future value and the interest.

R = 7000, i = 0.04, n = 10.
FV = 7000[(1.04)¹⁰ − 1] ÷ 0.04 ≈ 7000 × 12.006107 ≈ 84 042.75.
Deposits: 10 × 7000 = $70 000. Interest: 84 042.75 − 70 000 = $14 042.75.
In a TFSA, this growth is tax-free.

About $84 042.75, including $14 042.75 of tax-free interest. (Real limits are set each year; this example assumes $7 000 every year.)

Time is the secret ingredient

The graph below shows $100 a month at 6% compounded monthly. For the first 10 years, most of the balance is your own deposits. From about year 22, the interest earned is bigger than all the deposits combined (the 25-year bar shows this clearly). After 40 years, $48 000 of deposits has grown to almost $200 000.

The bottom part of each bar is what was deposited ($1 200 per year); the top part is interest. After 10 years the total is $16 388; after 40 years it is $199 149, of which more than $150 000 is interest. (Values rounded to the dollar.)

Worked example

Example 3: Comparing two savers

Saver A deposits $1 000 at the end of each year for 10 years at 6% compounded annually. Saver B deposits $2 000 at the end of each year for 10 years at 3% compounded annually. Who has more? Who earns more interest?

A: FV = 1000[(1.06)¹⁰ − 1] ÷ 0.06 ≈ 13 180.79. Deposits $10 000. Interest ≈ $3 180.79.
B: FV = 2000[(1.03)¹⁰ − 1] ÷ 0.03 ≈ 22 927.76. Deposits $20 000. Interest ≈ $2 927.76.

B ends with more money ($22 927.76), because B deposited twice as much. But A earned more interest ($3 180.79), thanks to the higher rate.

Just for fun

Why did the annuity bring a calendar to every party?

It never misses a payment date, and it always shows up at the END.

Hands-on activity

Plan your own savings stream

You need: a scientific calculator (or a spreadsheet/TVM Solver) and paper.

Pick a realistic amount you could save at the end of each month (for example, part of a part-time paycheque or babysitting money).
Pick a time frame: 2 years (a car or a trip), or 5 years (post-secondary costs).
Use 3% compounded monthly, or a savings rate you find on a Canadian bank's website (write down the date).
Calculate the future value, the total deposited and the interest earned.
Repeat with the deposit doubled. Then repeat with the original deposit but double the time. Which change helped more?

What to look for: correct i and n (monthly: i = rate ÷ 12, n = months). Doubling the deposit exactly doubles the future value; doubling the time more than doubles it, because the money has longer to compound.

Your turn

Find the future value of $75 deposited at the end of each month for 2 years at 3.6% compounded monthly.

Find the future value of $1 200 at the end of each year for 8 years at 4.5% compounded annually. How much is interest?

$250 at the end of every quarter for 5 years at 5% compounded quarterly. Find FV.

$500 at the end of every 6 months for 5 years at 3.5% compounded semi-annually. Find FV and the interest.

Write the TVM Solver entries for the last question.

Write the spreadsheet formula for the future value of $150 a month for 5 years at 4.25% compounded monthly, and evaluate.

More practice: warm-up

For monthly payments for 4 years at 6% compounded monthly, find i and n.

Total deposited: $80 per month for 3 years?

An annuity's FV is $5 300 and $5 000 was deposited. How much interest?

Evaluate [(1.1)² − 1] ÷ 0.1 and explain what it means for $1 per year at 10%.

In a TVM Solver, what is PV for a savings plan that starts with an empty account?

More practice: on your own

$40 at the end of every week for 2 years at 2.6% compounded weekly. Find FV.

$300 per month for 3 years at 4.8% compounded monthly. Find FV and the interest.

Which gives more after 20 years: $1 000 per year at 3%, or $1 000 per year at 6% for only 10 years then left alone (no deposits) for 10 more years at 6%? (Annual compounding, end-of-year deposits.)

Kai says the future value of $100 per month for 12 months at 6% compounded monthly is 1200(1.005)¹². Explain the mistake and find the correct answer.

Show that doubling R doubles FV, but doubling n more than doubles FV, using R = $100, i = 0.005: compare n = 120 and n = 240.

Challenge

Leah deposits $200 a month at 6% compounded monthly from age 25 to age 65 (480 deposits). Sam deposits $200 a month at the same rate from age 35 to age 65 (360 deposits). Find each future value and the difference. What does this say about starting early?

Using the sum S = R + R(1 + i) + … + R(1 + i)ⁿ⁻¹, check the formula for R = 100, i = 0.1, n = 3 by adding the terms.

An account already has $2 000. Its owner adds $100 at the end of each month for 5 years at 4.2% compounded monthly. Find the total at the end. (Hint: the $2 000 is a lump sum; the deposits are an annuity.)

Check your understanding

Quiz

Which is the future value formula for an ordinary simple annuity?

For $60 at the end of every month for 5 years at 2.4% compounded monthly, n and i are:

FV of $100 at the end of each year for 2 years at 5% compounded annually is:

An annuity of $250 per quarter for 3 years has FV = $3 200. The interest earned is:

In a TVM Solver for a savings plan of $80 per month, PMT should be entered as:

Doubling the number of payments (same R and i) makes the future value:

The 2026 TFSA dollar limit set by the Canada Revenue Agency is:

For parents and teachers

Big idea (B3.7, with B3.5): Students solve problems involving the amount (future value) of an ordinary simple annuity, using technology such as a scientific calculator, spreadsheet or TVM Solver. The formula FV = R[(1 + i)ⁿ − 1] ÷ i is shown so every answer can be checked on a scientific calculator; the curriculum emphasizes using technology.

How to help: Make it personal: "If you saved $25 a week from your job for 2 years, what would you have?" Discuss the starting-early comparison in the Challenge; it is one of the most useful money lessons a teenager can learn.

Common difficulties:

Bracket errors in the formula. Encourage typing the whole expression in one line, or computing (1 + i)ⁿ first.
Using A = P(1 + i)ⁿ with the total of all deposits (as if they were all deposited on day one).
Sign errors in the TVM Solver.

Canadian fact used (checked October 2026): the 2026 TFSA dollar limit is $7 000 (Canada Revenue Agency, as reported by financial institutions such as Harvest Portfolios and Questrade). TFSAs are for Canadian residents 18 and older. Rates used are examples, not current offers.