Regular payments
An annuity is a series of equal payments at regular intervals. Examples:
- Saving $200 every month
- Car or mortgage payments
- A pension paying $2,000 a month in retirement
In an ordinary annuity, payments come at the end of each payment interval. We’ll assume the compounding period matches the payment interval.
Future value of an annuity
How much will regular deposits grow to?
FV = PMT × [(1 + i)ⁿ − 1] / i
Example: Deposit $200 at the end of each month for 10 years at 5% per year compounded monthly.
- i = 0.05/12, n = 120
- FV ≈ $31,056.46
- Total deposited: $24,000; interest earned ≈ $7,056
Present value of an annuity
How much is needed now to provide regular payments in the future?
PV = PMT × [1 − (1 + i)⁻ⁿ] / i
Example: To withdraw $1,500 a month for 20 years from an account earning 4% compounded monthly:
- i = 0.04/12, n = 240
- PV ≈ $247,532.79
This is also how loan amounts relate to payments: a loan’s amount is the present value of its payments.
Saving for retirement
Example: Save $100 a month from age 20 to 60 (40 years) at 6% compounded monthly.
FV ≈ $199,149, from only $48,000 in deposits.
Starting earlier makes an enormous difference because of compounding.
Canadian registered plans
| Plan | Purpose | Key feature |
|---|---|---|
| RRSP | retirement | contributions reduce taxable income now; withdrawals are taxed later |
| TFSA | any goal | growth and withdrawals are tax-free |
| RESP | child’s education | government adds grants (the Canada Education Savings Grant) |
| FHSA | first home | tax-deductible contributions and tax-free withdrawals for a qualifying home |
Contribution limits and rules are set by the government and change over time.
Technology
Spreadsheets: =FV(rate, nper, pmt) and =PV(rate, nper, pmt). Financial calculators use TVM (time value of money) keys.
Worked examples
Example 1: $5,000 at the end of each year for 30 years at 6% compounded annually. FV? ≈ $395,291.
Example 2: Present value of $800 per month for 5 years at 6% compounded monthly? ≈ $41,380.
Example 3: Is a car loan an annuity? Yes: equal payments at regular intervals.
Common mistakes
- Using the annual rate instead of the rate per period.
- Mixing up FV and PV situations. Saving up → FV; paying out or borrowing → PV.
- Forgetting that payments must be equal and regular.
Try it yourself
- Is a weekly allowance of $20 an annuity?
- Which formula for "how much do I need now to receive $1,000/month for 10 years"?
- Which formula for "how much will $50/week grow to"?
- Why does starting to save early matter so much?
Answers: 1) yes 2) present value 3) future value 4) Money has more time to compound.
Quick recap
- An annuity is equal payments at regular intervals.
- Future value of an annuity: savings growth; present value of an annuity: what’s needed now.
- Plans like the RRSP and TFSA help Canadians save.