Annuities Introduction
Understand how regular payments grow over time using the future value of an annuity, and distinguish between ordinary annuities and annuities due.
What Is an Annuity?
An annuity is a series of equal payments made at regular intervals over a fixed period of time. Examples include superannuation contributions, loan repayments, and regular savings deposits.
There are two main types of annuities depending on when each payment is made within the period:
Ordinary Annuity
Payments are made at the end of each period. This is the most common type (e.g. monthly loan repayments).
Annuity Due
Payments are made at the beginning of each period. Rent payments and insurance premiums often follow this pattern.
Visual: Payment Timeline (Ordinary Annuity)
Each green dot represents a payment of $R made at the end of the period. The total future value is the sum of all payments plus the compound interest earned on each.
Future Value of an Ordinary Annuity
The future value (FV) of an ordinary annuity tells us how much a series of regular payments will be worth at a future date, including compound interest.
Formula: Future Value of an Ordinary Annuity
FV = R × (1 + r)n − 1r
- R = regular payment amount per period
- r = interest rate per period (as a decimal)
- n = total number of payments
For an annuity due, each payment earns one extra period of interest, so we multiply by (1 + r):
Formula: Future Value of an Annuity Due
FVdue = R × (1 + r)n − 1r × (1 + r)
How Each Payment Grows
In an ordinary annuity with n payments, each payment earns a different amount of interest because it is invested for a different number of periods:
- 1st payment (made at end of period 1) earns interest for n − 1 periods, growing to R(1 + r)n−1
- 2nd payment earns interest for n − 2 periods, growing to R(1 + r)n−2
- ...
- Last payment (made at end of period n) earns no interest, remaining as R
The future value formula is the sum of this geometric series.
Key Vocabulary
Annuity
A series of equal payments made at regular time intervals over a specified period.
Ordinary Annuity
An annuity where payments are made at the end of each compounding period.
Annuity Due
An annuity where payments are made at the beginning of each compounding period.
Future Value (FV)
The total accumulated amount at the end of the annuity, including all payments and compound interest earned.
Worked Examples
Find the future value of $500 deposited at the end of each month for 3 years at 6% p.a. compounded monthly.
Step 1: Identify the values. R = $500, annual rate = 6%, so r = 0.06/12 = 0.005 per month, n = 3 × 12 = 36 payments.
Step 2: Substitute into the ordinary annuity formula: FV = 500 × [(1.005)36 − 1] / 0.005
Step 3: Calculate (1.005)36 = 1.19668... so (1.19668 − 1) / 0.005 = 39.3361...
Answer: FV = 500 × 39.3361 = $19,668.05 (to the nearest cent).
Maya deposits $200 at the beginning of each quarter into a savings account earning 8% p.a. compounded quarterly. What is the future value after 5 years?
Step 1: This is an annuity due (payments at the beginning). R = $200, r = 0.08/4 = 0.02, n = 5 × 4 = 20.
Step 2: FVdue = 200 × [(1.02)20 − 1] / 0.02 × (1.02)
Step 3: (1.02)20 = 1.48595... so [(1.48595 − 1) / 0.02] = 24.2974...
Answer: FVdue = 200 × 24.2974 × 1.02 = $4,956.67 (to the nearest cent).
Compare the future values of two annuities: both pay $1,000 annually for 10 years at 5% p.a. One is ordinary, the other is an annuity due.
Step 1: R = $1,000, r = 0.05, n = 10.
Step 2: Ordinary: FV = 1000 × [(1.05)10 − 1] / 0.05 = 1000 × [1.62889 − 1] / 0.05 = 1000 × 12.5779 = $12,577.89
Step 3: Annuity Due: FVdue = 12,577.89 × 1.05 = $13,206.79
Answer: The annuity due is worth $628.90 more because every payment earns one extra year of interest.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
In an ordinary annuity, when are payments made?
Question 2
$1,000 is deposited at the end of each year for 4 years at 10% p.a. compounded annually. What is the future value? (Round to the nearest dollar.)
Question 3
How does the future value of an annuity due compare to an ordinary annuity with the same R, r, and n?
Question 4
$250 is deposited at the end of each month for 2 years at 12% p.a. compounded monthly. How many payments (n) and what is the periodic interest rate (r)?
Question 5
Rent of $400 is paid at the beginning of each week for 52 weeks. The interest rate is 5.2% p.a. compounded weekly. Which formula correctly gives the future value?
Key Concepts Summary
- ● An annuity is a sequence of equal payments made at regular intervals over a fixed time.
- ● An ordinary annuity has payments at the end of each period; an annuity due has payments at the beginning.
- ● The future value formula for an ordinary annuity is FV = R × [(1+r)n − 1] / r.
- ● For an annuity due, multiply the ordinary FV by (1 + r) to account for the extra compounding period.
- ● Always convert the annual interest rate to the periodic rate (r = annual rate / periods per year) and calculate the total number of payments (n = years × periods per year).