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Math101learn.math101.caAnnuities
An annuity is a sequence of equal payments at regular intervals whose value depends on compounding and payment timing.
Annuity formulas add many deposits or payments after moving each one to the same point in time.
What an annuity is
An annuity consists of equal payments made at regular intervals. Examples include monthly savings deposits, loan payments, pensions, and some leases.
The payment interval and compounding period must be aligned or converted carefully.
Ordinary annuity and annuity due
In an ordinary annuity, payments occur at the end of each period. In an annuity due, payments occur at the beginning.
Each annuity-due payment earns or avoids one extra period of interest, so its value is an ordinary-annuity value multiplied by $(1+i)$ at the same comparison date.
Variables
Common notation includes:
- $R$: regular payment;
- $i$: interest rate per payment period;
- $n$: number of payments;
- $FV$: future value after the sequence;
- $PV$: present value equivalent.
Convert an annual nominal rate and time span into matching periodic $i$ and $n$.
Future value of an ordinary annuity
For end-of-period deposits,
This is a geometric-series sum. Earlier payments compound longer than later ones.
Worked example: savings plan
Total deposits are $200(60)=\$12\,000$; the difference is interest earned.
Present value of an ordinary annuity
The present value of end-of-period payments is
This answers: what single amount now is financially equivalent to the future payment stream at rate $i$?
Loan principal is often the present value of repayments.
Finding a payment
Rearrange the appropriate formula. To reach a target future amount,
For a loan,
Keep full precision until the final currency rounding.
Timeline method
Draw a timeline showing payment dates, present date, future date, and compounding periods. A timeline prevents beginning/end timing errors and clarifies whether an extra payment occurs at time $0$.
Count intervals, not only calendar labels.
Loans and amortization
Each loan payment covers interest on the outstanding balance plus some principal. Early payments often contain more interest; later payments contain more principal.
An amortization schedule tracks balance, interest, principal repaid, and remaining amount period by period.
Real-world cautions
Actual accounts may use fees, changing rates, daily compounding, missed payments, taxes, or contribution limits. Formula results assume the stated constant rate and exact payment schedule.
Compare effective costs and read contract terms, not only the advertised payment amount.
Common mistakes
Using the annual rate as $i$ for monthly payments. Convert to the period rate.
Using years as $n$. Count payments.
Confusing beginning and end payments. Identify ordinary versus due.
Treating total deposits as account value. Interest changes the total.
Rounding the periodic rate early. It affects many compounded periods.
Quick self-check
- Are payments equal and equally spaced?
- Do they occur at the beginning or end of periods?
- Do $i$ and $n$ use the same period as payments?
- Is the question asking for present value, future value, payment, or time?
- Does a timeline confirm the first and last payment dates?
- Are fees, rate changes, and currency rounding distinguished from the ideal model?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Deposit 200 dollars monthly for 5 years at 6% nominal interest compounded monthly. What is the approximate future value?
- i = 0.005 and n = 60.
- FV = 200((1.005)⁶⁰ − 1)/0.005.
- FV is approximately 13,954 dollars.
End of lesson
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