B. Exponential Functions · Financial Mathematics: Compound Interest and AnnuitiesWhat Is an Annuity? Regular Payments in Tables and Graphs
B3.5About 70 minutes
Learning goals- I can explain what an annuity is and give everyday Canadian examples.
- I can tell whether an annuity is an ordinary simple annuity (payments at the end of each period, payment and compounding periods the same).
- I can build a table, by hand or with a spreadsheet, that shows how an annuity grows period by period.
- I can read a graph of an annuity's future value and explain the gap between the balance and the total deposits.
Words to know
annuity
A series of equal payments (or deposits) made at regular intervals, such as every month or every year.
payment interval
The time between payments.
term
The total time from the start of the first payment interval to the end of the last one.
ordinary annuity
An annuity where each payment is made at the END of its payment interval.
simple annuity
An annuity where the payment interval equals the compounding period (e.g., monthly payments, interest compounded monthly).
ordinary simple annuity
Both: payments at the end of each period, and payment interval = compounding period. This is the kind studied in MCF3M.
future value of an annuity (FV)
The total of all the payments plus all the interest they earn, at the end of the term.
Little and often
So far every problem had one deposit (a lump sum) that grew. Real saving rarely works that way. Zoë puts $50 from every paycheque from her part-time job at a garden centre into savings. Her parents pay the same car loan payment every month. A retired neighbour receives a pension cheque every month. Each of these is a stream of equal payments at regular intervals. In math and finance, that is called an annuity.
Naming annuities
Two questions sort annuities:
- When in each interval is the payment made? At the end → an ordinary annuity. At the beginning → an annuity due.
- Does the payment interval match the compounding period? Yes → a simple annuity. No (for example, monthly payments but interest compounded semi-annually) → a general annuity.
In MCF3M you work with ordinary simple annuities: payments at the end of each period, and payment interval = compounding period. Annuities due and general annuities are studied in later courses.
Worked exampleExample 1: Is it an ordinary simple annuity?
Classify each situation.
(a) $75 deposited at the end of every month into an account paying 3% compounded monthly. (b) $1 500 deposited once into a 5-year GIC. (c) Rent of $900 paid at the start of every month. (d) $200 deposited at the end of every month into an account paying 4% compounded quarterly.
- (a) Equal regular payments at the end of each month, monthly compounding: ordinary simple annuity.
- (b) A single deposit: not an annuity (it is a lump sum, so use A = P(1 + i)ⁿ).
- (c) Payments at the start of each period: an annuity due, not ordinary.
- (d) Monthly payments but quarterly compounding: a general annuity, not simple.
Only (a) is an ordinary simple annuity.
Watching an annuity grow: a table
Suppose you deposit $1 000 at the end of each year for 5 years into an account paying 5% compounded annually. Each year, two things happen: the balance earns interest, and then a new $1 000 deposit arrives.
New balance = old balance × 1.05 + 1000
| Year | Balance at start | Interest (5%) | Deposit at end | Balance at end |
|---|
| 1 | $0.00 | $0.00 | $1 000.00 | $1 000.00 |
| 2 | $1 000.00 | $50.00 | $1 000.00 | $2 050.00 |
| 3 | $2 050.00 | $102.50 | $1 000.00 | $3 152.50 |
| 4 | $3 152.50 | $157.63 | $1 000.00 | $4 310.13 |
| 5 | $4 310.13 | $215.51 | $1 000.00 | $5 525.63 |
Another view: each payment grows separately
You can also follow each deposit on its own. The deposit at the end of year 1 grows for 4 years, so at the end it is worth 1000(1.05)⁴. The deposit at the end of year 2 grows for 3 years, and so on. The last deposit arrives on the final day and earns no interest.
FV = 1000(1.05)⁴ + 1000(1.05)³ + 1000(1.05)² + 1000(1.05) + 1000
Read from right to left, each term is the previous one multiplied by 1.05. A sum like this, where each term is a constant multiple of the one before, is called a geometric series. (Geometric series are studied in Grade 11 Functions, MCR3U, and in later courses; in the next lesson you will use the formula that adds one up quickly.)
Worked exampleExample 2: Adding up the contributions
Kiran deposits $200 at the end of every 3 months for 1 year into an account paying 4% compounded quarterly. Find the future value by adding each deposit's value at the end of the year.
- i = 0.04 ÷ 4 = 0.01 per quarter. There are 4 deposits.
- Deposit 1 (end of quarter 1) grows for 3 quarters: 200(1.01)³ ≈ 206.06.
- Deposit 2 grows for 2 quarters: 200(1.01)² = 204.02. Deposit 3 grows for 1 quarter: 202.00. Deposit 4 earns nothing: 200.00.
- FV = 206.06 + 204.02 + 202.00 + 200.00 = 812.08.
The future value is $812.08: $800 of deposits plus $12.08 of interest.
Watching an annuity grow: a graph
If the $1 000 deposits continue for 30 years, a spreadsheet produces the graph below. The total deposited grows in a straight line (another $1 000 each year). The balance curves upward, because the bigger the balance, the more interest it earns. The gap between them is the interest, and it grows faster and faster: after 10 years the balance is $12 577.89, after 20 years $33 065.95, and after 30 years $66 438.85, more than twice what was deposited.
Just for funFun fact: the word annuity comes from the Latin annus, meaning "year", because early annuities paid once a year. Today most are monthly, but nobody has renamed them "monthuities".
Hands-on activityBuild an annuity spreadsheet
You need: a computer or tablet with a free spreadsheet.
- Type the headings
Year, Start balance, Interest, Deposit, End balance in row 1. - Row 2: Year
1, Start balance 0, Interest =B2*0.05, Deposit 1000, End balance =B2+C2+D2. - Row 3: Year
=A2+1, Start balance =E2, then copy C2:E2 down into row 3. Fill rows 3 to 31 down. - Add a column
Total deposited with =1000*A2 and fill down. - Make a line chart of End balance and Total deposited against Year.
- Investigate: change the deposit to $500, then the rate to 3% and 7%. Describe what changes.
What to look for: the end balance after 5 years is $5 525.63 and after 30 years $66 438.85. Halving the deposit halves every balance. A higher rate makes the balance curve bend upward more, and the gap (interest) grows much faster.
Your turn
1In your own words, what is an annuity? Give two Canadian examples.
A series of equal payments made at regular intervals. Examples: monthly deposits into a TFSA or RESP savings plan; monthly car loan or student loan payments; monthly pension payments.
2$40 is deposited at the end of each week into an account that compounds interest weekly. Is it an ordinary simple annuity? Explain.
Yes. Payments are at the end of each interval, and the payment interval (weekly) equals the compounding period (weekly).
3$300 is deposited at the end of each month into an account paying interest compounded annually. Ordinary simple annuity?
No. It is ordinary (end of period), but the payment interval (monthly) is different from the compounding period (annual), so it is a general annuity.
4$500 is deposited at the end of each year for 3 years at 4% compounded annually. Make a table showing the balance at the end of each year.
Year 1: $500.00. Year 2: 500(1.04) + 500 = $1 020.00. Year 3: 1020(1.04) + 500 = $1 560.80.
5Find the future value in the last question by adding each deposit's value at the end.
500(1.04)² + 500(1.04) + 500 = 540.80 + 520.00 + 500.00 = $1 560.80 ✓
6Why does the last payment of an ordinary annuity earn no interest?
It is deposited at the end of the last period, the same moment the future value is measured, so it has no time to earn interest.
More practice: warm-up
1What does "ordinary" mean for an annuity?
Payments are made at the end of each payment interval.
2What does "simple" mean for an annuity?
The payment interval is the same as the compounding period.
3Monthly deposits of $25 for 3 years: how many payments?
4Total deposited: $150 per quarter for 2 years?
5Is a single $2 000 deposit an annuity?
No, an annuity needs a series of regular payments.
More practice: on your own
1$100 is deposited at the end of each month for 4 months at 6% compounded monthly. Find the future value with a table.
i = 0.005. Month 1: 100.00. Month 2: 100(1.005) + 100 = 200.50. Month 3: 200.50(1.005) + 100 ≈ 301.50. Month 4: 301.50(1.005) + 100 ≈ $403.01.
2For the last question, how much of the future value is interest?
3$500 is deposited at the end of every 6 months for 3 years at 4% compounded semi-annually. Write the future value as a sum of six terms (do not add yet), then add.
i = 0.02. FV = 500(1.02)⁵ + 500(1.02)⁴ + 500(1.02)³ + 500(1.02)² + 500(1.02) + 500 ≈ 552.04 + 541.22 + 530.60 + 520.20 + 510.00 + 500.00 = $3 154.06.
4On a graph of an annuity's balance and its total deposits, which one is a straight line? Why?
Total deposits is a straight line: the same amount is added every period (linear). The balance curves upward because interest is added too.
5Two friends each deposit $1 000 a year for 30 years at 5%. One deposits at the end of each year, the other at the start. Who ends up with more and why? (No calculation needed.)
The one who deposits at the start (annuity due): each deposit earns interest for one extra year.
Challenge
1Using the 30-year graph in this lesson, estimate the year in which the interest earned so far first exceeds the total deposited. Then check with a spreadsheet or the table rule (balance × 1.05 + 1000).
Interest exceeds deposits when the balance is more than twice the deposits. From the balance values: year 20 balance $33 065.95 vs 2 × $20 000 = $40 000 (not yet); year 25: balance ≈ $47 727.10 vs $50 000 (not yet); year 27: ≈ $54 669.13 vs $54 000 (yes); year 26: ≈ $51 113.45 vs $52 000 (no). So in year 27.
2Show that the end balances of an annuity follow B(k) = 1.05 B(k − 1) + 1000 with B(1) = 1000, and explain why this is not the same as exponential growth B(k) = 1000(1.05)ᵏ.
Each year the old balance grows by 5% and then a new $1 000 is added: B(k) = 1.05B(k − 1) + 1000. Pure exponential growth only multiplies; here a constant is added each period, so the ratio of consecutive balances is not constant (2050/1000 = 2.05, 3152.50/2050 ≈ 1.54, …).
3An account has $5 525.63 after the 5 deposits of $1 000 in this lesson. If no more deposits are made and the money stays at 5% compounded annually, what is it worth 5 years later? Compare with depositing $1 000 for 10 years.
5525.63(1.05)⁵ ≈ $7 052.26. Ten years of deposits gives $12 577.89 (from the lesson). The extra $5 000 of deposits in years 6 to 10 adds 12 577.89 − 7052.26 = $5 525.63, exactly the future value of a 5-year annuity, as it should be.
Quiz
1. Which is an ordinary simple annuity?
2. In an ordinary annuity of 10 annual payments, how many years does the FIRST payment earn interest?
3. $1 000 at the end of each year, 5% compounded annually. Balance at the end of year 2?
4. On a graph of an annuity's future value against time, the balance:
6. Each term in FV = R(1 + i)ⁿ⁻¹ + … + R(1 + i) + R is the previous term (reading right to left) multiplied by:
Big idea (B3.5): Students explain the meaning of the term annuity through numerical and graphical representations, using technology. In MCF3M they work only with ordinary simple annuities: payments at the end of each period, with the payment interval equal to the compounding period.
How to help: Name annuities in your family's life: a monthly car payment, automatic monthly savings, RESP contributions, a pension. Ask your teen whether each is "ordinary" (end of period) and whether payments and compounding match. If possible, let them build the spreadsheet in the activity and show you the graph.
Common difficulties:
- Giving the first payment interest for the full term. In an ordinary annuity, the first payment grows for n − 1 periods and the last earns nothing.
- Mixing up annuities with lump sums. A one-time deposit uses A = P(1 + i)ⁿ.
- Thinking the balance grows in a straight line. Only the total deposits do.