Present Value of a Perpetuity

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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

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.

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