If you are interested in learning about the basic concepts of annuities and perpetuities, you might have come across some statements that seem confusing or contradictory. In this article, we will explain the differences between these two types of financial instruments and help you identify which one of the following statements related to annuities and perpetuities is correct.
Contents
What are Annuities and Perpetuities?
An annuity is a series of fixed payments that are made or received at regular intervals for a certain period of time. For example, if you buy a life insurance policy that pays you $10,000 every year for 20 years, you have purchased an annuity. An annuity can be either an ordinary annuity or an annuity due, depending on whether the payments occur at the end or the beginning of each period.
A perpetuity is a special type of annuity that lasts forever, or until the end of time. For example, if you buy a bond that pays you $100 every year indefinitely, you have purchased a perpetuity. A perpetuity can be either a constant perpetuity or a growing perpetuity, depending on whether the payments remain the same or increase at a constant rate over time.
How to Calculate the Present Value of Annuities and Perpetuities?
The present value of an annuity or a perpetuity is the amount of money that you would need to invest today to receive the same stream of cash flows in the future. To calculate the present value of an annuity or a perpetuity, you need to know the amount of each payment ©, the number of periods (n), and the interest rate ® that applies to each period.
The formula for calculating the present value of an ordinary annuity is:
PV = C * [ {1 – (1 + r)-n} / r ]
The formula for calculating the present value of an annuity due is:
PV = C * [ {1 – (1 + r)-n} / r ] * (1 + r)
The formula for calculating the present value of a constant perpetuity is:
PV = C / r
The formula for calculating the present value of a growing perpetuity is:
PV = C / (r – g)
where g is the growth rate of the payments.
Which One of the Following Statements Related to Annuities and Perpetuities is Correct?
Now that you have learned the definitions and formulas of annuities and perpetuities, let’s look at some statements that are commonly made about them and see which one is correct.
- Statement A: The present value of a perpetuity is always higher than the present value of an annuity.
- Statement B: The present value of an annuity due is always higher than the present value of an ordinary annuity.
- Statement C: The present value of a growing perpetuity is always higher than the present value of a constant perpetuity.
- Statement D: The present value of an annuity or a perpetuity depends only on the amount and frequency of the payments.
The correct statement is Statement B. The present value of an annuity due is always higher than the present value of an ordinary annuity, because the payments start earlier and therefore have less discounting. You can verify this by comparing the formulas for both types of annuities and noticing that the only difference is a factor of (1 + r), which is always greater than 1.
Statement A is false, because the present value of a perpetuity can be lower than the present value of an annuity if the interest rate is high enough. For example, if C = $100, r = 10%, and n = 10, then the present value of an ordinary annuity is $614.46, while the present value of a constant perpetuity is $1,000. However, if r = 20%, then the present value of an ordinary annuity is $379.08, while the present value of a constant perpetuity is $500.
Statement C is false, because the present value of a growing perpetuity can be lower than the present value of a constant perpetuity if the growth rate is higher than the interest rate. For example, if C = $100, r = 10%, and g = 5%, then the present value of a constant perpetuity is $1,000, while the present value of a growing perpetuity is $2,000. However, if g = 15%, then the present value of a constant perpetuity is still $1,000, while the present value of a growing perpetuity is -$666.67.
Statement D is false, because the present value of an annuity or a perpetuity also depends on the interest rate and the growth rate (if applicable). A higher interest rate or a lower growth rate will reduce the present value, while a lower interest rate or a higher growth rate will increase the present value.
Conclusion
We hope that this article has helped you understand the basic concepts of annuities and perpetuities and how to calculate their present values. We also hope that you have learned how to identify which one of the following statements related to annuities and perpetuities is correct. Remember, the correct statement is Statement B: The present value of an annuity due is always higher than the present value of an ordinary annuity. If you have any questions or comments, please feel free to share them with us. Thank you for reading!
