Ordinary Annuity Formula
An ordinary annuity is a fixed amount of income that is given annually or at regular intervals. An annuity is an agreement with an insurance firm during which you create a payment (one-time big payment) or series of payments and, in return, receive a regular fixed income, beginning either immediately or after some predefined time within the future. The ordinary annuity formula is used to find the present and future value of an amount. Let us understand the ordinary annuity formula using solved examples.
What is the Formula for Ordinary Annuity?
Finding the future value of the annuity is important to accommodate inflation with time. The ordinary annuity formula is explained below along with solved examples. Annuity formulas for future and present value is also given.
- \(\begin{array}{l}\text { Ordinary Annuity }=P \times \frac{\left[1-(1+r)^{-n}\right]}{\left[(1+r)^{t} \times r\right]}
\end{array}\) - The future value of an ordinary annuity
FV = P×((1+r)n−1) / r
- The present value of an ordinary annuity
PV = P×(1−(1+r)-n) / r
where,
- P = Value of each payment
- r = Rate of interest per period in decimal
- n = Number of periods
Solved Examples Using Ordinary Annuity Formula
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Example 1: Alan was getting $100 for 5 years every year at an interest rate of 5%. Find the future value using the ordinary annuity formula at the end of 5 years?
Solution:
The future value
Given: r = 0.05, 5 years = 5 yearly payments, so n = 5, and P = $100
FV = P×((1+r)n−1) / r
FV = $100 × ((1+0.05)5−1) / 0.06
FV = 100 × 55.256
FV = $552.56
Answer: The longer-term value of annuity after the end of 5 years is $552.56. -
Example 2: If the present value of the annuity is $20,000. Assuming a monthly interest rate of 0.5%, find the value of each payment after every month for 10 years.
Solution
To find: The value of each payment
Given:
r = 0.5% = 0.005
n = 10 years x 12 months = 120, and PV = $20,000
Using formula for present value
PV = P×(1−(1+r)-n) / r
Or, P = PV × ( r / (1−(1+r)−n))
P = $20,000 × (0.005 / (1−(1.005)−120))
P = $20,000 × (0.005/ (1−0.54963))
P = $20,000 × 0.011...
P = $220
Answer: The value of each payment is $220.
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