Introduction

Understanding how to calculate the price of an annuity and a perpetuity is fundamental in bond valuation and financial analysis. In this chapter, we will explore these concepts in depth, providing you with the knowledge and tools needed to determine the present value of these cash flows. Whether you’re assessing the value of a series of regular payments or evaluating a financial instrument with infinite periodic payments, mastering these calculations is essential. Let’s dive into the world of annuities and perpetuities.

Annuity

An annuity is a series of periodic payments that are made or received at equal intervals. To calculate the price of an annuity, you need to find the present value of these cash flows. The formula to compute the price of an annuity is:

$$PV=\frac{PMT}{r}\left(1-\frac{1}{(1+r)^n}\right)$$ Where:

• $PV$ is the present value of the annuity
• $PMT$ is the periodic payment
• $r$ is the interest rate per period
• $n$ is the total number of periods

Example: Let’s say you are considering an investment that promises to pay you USD 1,000 every year for the next 5 years, and the discount rate is 5%. Using the annuity formula:

$$PV=\frac{\text{USD 1,000}}{0.05}\left(1-\frac{1}{(1+0.05)^5}\right)=\text{USD 4,329.48} $$ So, the present value of this annuity is approximately USD 4,329.48.

Perpetuity

A perpetuity is a financial instrument that provides regular payments indefinitely. To calculate the price of a perpetuity, you can use the simplified formula:

$$PV=\frac{PMT}{r}$$ Where:

• $PV$ is the present value of the perpetuity.
• $PMT$ is the periodic payment.
• $r$ is the interest rate per period.

Example:Suppose you have an investment that pays you USD 500 every year indefinitely, and the discount rate is 6 %. Using the perpetuity formula: $$P V=\frac{\text{USD 500}}{0.06}= \text{USD 8,333.33}$$ The present value of this perpetuity is approximately USD 8,333.33.