A perpetual annuity, also called a perpetuity, promises to pay a certain amount of money to its owner forever. A classic example would be that of a perpetual bond, which promises to pay interest each year, for eternity (or for as long as the borrower can afford to pay). Historically issued by governments, companies like Volkswagen have issued perpetual bonds to raise money at low interest rates.
Though a perpetuity may promise to pay you forever, its value isn't infinite. The bulk of the value of a perpetuity comes from the payments that you receive in the near future, rather than those you might receive 100 or even 200 years from now.
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Calculating the present value of a perpetual annuity
We can use a simple formula to calculate the present value of a perpetuity annuity. This formula will tell us what a perpetuity is worth based on a discount rate, or a required rate of return.
Present Value of a Perpetuity = Annual Payment ÷ Discount Rate
Suppose that you own a perpetual bond that promises to pay you $500 each year. You believe the borrower is creditworthy, and thus think that a 6% return is suitable for this perpetual bond. Using this information, we can determine the present value for this perpetuity.
Present Value of a Perpetuity = Annual Payment ÷ Discount Rate
PV = $500 ÷ 0.06
PV = $8,333.33
This tells us that someone could pay you $8,333.33 for your bond and receive a 6% return on their money.
The present value of a perpetuity has an inverse relationship to the discount rate you use to value it. If we were to value this bond at a 4% discount rate, the present value would jump to $12,500 (PV = $500 ÷ 0.04). If we valued it with a 10% discount rate, the present value would fall to $5,000 (PV = $500 ÷ 0.10).
Calculating the present value of a growing perpetuity
Not all perpetuities pay the same amount each year forever. Some promise to pay a growing amount of money over time, perhaps to compensate for inflation, or because the earnings of a business are expected to grow.
The formula for growing perpetuities is only slightly more complicated than the formula for perpetuities that promise flat payments over time.
Present Value of a Growing Perpetuity = Next Annual Payment ÷ (Discount Rate – Payment Growth Rate)
Let's assume that you invested in a company that will pay you a dividend of $2 per share in one year. You expect the dividend to grow at 4% in perpetuity, and believe that the stock should be valued at a discount rate of 12%, given its risks. What is a share worth under these assumptions?
Present Value of a Growing Perpetuity = Next Annual Payment ÷ (Discount Rate – Payment Growth Rate)
PV = $2.00 ÷ (0.12-0.04)
PV =$2.00 ÷ 0.08
PV = $25.00
This formula thus reveals that if our assumptions are right -- the dividend will grow at 4% in perpetuity, and 12% is a sufficient return for the risk of owning the company -- shares should trade for $25 each. I used a stock in this example, because this formula is also known as the Dividend Discount Model, which is a basic formula for valuing a share of stock as if it were a perpetuity.
As with all financial formulas, the present value you calculate for any given perpetuity will only be as good as your inputs. Assuming ridiculously high growth rates, or using exceptionally low discount rates, will result in values that are much too high. Likewise, growth rates that are too low, or discount rates that are too high, will result in overly pessimistic present values. The formulas are simple; the hard part is getting the assumptions right.
