The article explains the concept of perpetuities, where equal payments continue indefinitely, and how to calculate their present value using a specific formula. It also illustrates the calculation with an example of preferred share valuation, determining the present value of perpetual dividend payments.
Annuities are flows of equal payments that carry on for a set number of periods. If the annuity goes on forever, it’s known as a perpetuity. The word perpetuity derives from the word perpetual, which means “continual or everlasting.”
There are a few real examples of perpetuities. One example is a Consol bond (short for consolidated annuity) issued by the Bank of England. Consol bonds were perpetual and paid a fixed rate of interest. The first consol was issued in 1752 with an annual interest rate of 3.5%.
Formula
In order to calculate the present value of perpetuity, the following formula is used:
\[P{{V}_{perpetuity}}=\frac{PMT}{i}\]
Where
PMT=the periodic payment in the annuity
i=the interest rate
Present Value of a Perpetuity Example
Preferred Share Valuation Using the Present Value of a Perpetuity
Assume we would like to find out the present value of the dividends paid off by a share of preferred stock. We recognize that the stock assures to pay the holder an annual $100 dividend forever, and we presume a 10% interest rate. We also presume that the first dividend will be paid up in 1 year.
Solution
\[\begin{align}& PMT=100 \\& i=0.10 \\\end{align}\]
This question asks for the present value of a level perpetuity, so
\[P{{V}_{perpetuity}}=\frac{PMT}{i}=\frac{100}{0.1}=\$1,000\]
As investors, we would be willing to pay no more than $1,000 for a share of this stock.
Present Value of a Perpetuity Key Takeaways
The concept of perpetuities is vital for financial applications, particularly in valuing long-term investments such as preferred stocks or bonds that provide consistent, indefinite payments. By calculating the present value of these payments, investors can assess the fair price they should be willing to pay, ensuring informed investment decisions and effective financial planning. This method is crucial for valuing assets that generate continuous income, offering a reliable way to evaluate their worth in today’s terms.