A perpetuity is a stream of cash flow that supposedly goes on forever. The present value of a perpetuity is its price, the amount you would pay today to receive the cash flow stream. The present value calculation accounts for the time value of money by discounting the cash flows using an appropriate factor. There are a couple of ways to calculate the present value of a perpetuity.
Read More: Define "Time Value of Money"
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The Perpetuity Formula
The present value of a perpetuity in which the cash payments are fixed is:
PV = A/r
where A is the amount of the periodic payment and r is a suitable discount factor, such as an interest rate or yield.
Notice that the formula does not account for the return of the principal or purchase price of the perpetuity. That helps to differentiate a perpetuity from an ordinary bond, which pays cash flows for a given period and then returns your principal.
To take a numerical example, suppose a perpetual bond (sometimes called an irredeemable bond) is issued for at par value of $1,000 and pays out $50 per year in perpetuity. The coupon rate of the bond is thus $1,000 / $50, or 5 percent, which we can use as the discount rate. Thus, A = $50 and r = 0.05 on an annual basis.
The present value of the perpetuity is equal to $50 / 0.05, or $1,000. In other words, the price of this perpetuity is equal to the present value of its discounted cash flows. This makes intuitive sense, since the issue price should match the present value of the discounted cash flows according to the theory that money's value erodes the longer you have to wait to get it.
Read More: How to Calculate a Bond Price
Present Value of a Growing Perpetuity
Some perpetuities may pay a growing amount over time. Consider a common stock with a long history of increasing its dividend by 1 percent per year. If we consider the growth rate to be fixed, then the present value of this perpetuity is
PVgrowing = A1 / (r - g)
where A1 is the cash flow for the first period, r is the discount rate and g is a fixed growth rate.
To conjure an example, Stock XYZ has a current price of $50, a current dividend of $2 per year and an annual dividend growth rate of 1 percent that it has maintained for decades. The current dividend yield is $2/$50, or 4 percent, and the dividend a year from now should be 1.01 x $2, or $2.02. The present value of this perpetuity is $2 /(4 percent - 1 percent), or $66.67.
Choosing a Discount Factor
In the examples presented, the discount factor was set equal to a perpetual bond's coupon rate and a stock's current dividend yield. However, there is no rule that requires these choices. For example, suppose that 30-year bonds (not quite perpetual but close enough) of similar qualify and size were currently yielding 4.76 percent. Using this as the discount factor, the present value of the perpetual bond would be ($50 / 0.0476) or $1,050.42. This would be the fair market price based on prevailing interest rates. It's typical to value debt instruments using a discount factor matching the prevailing interest rates.
Shortfalls of the Perpetuity Formula
The first shortfall occurs when you apply the formula to stock dividends. Unlike bond interest, stock dividends are not guaranteed and can change over time. The perpetuity formula only works for constant or constantly growing cash flows.
Secondly, eternity doesn't really exist in the financial world. Many things can happen that can modify or terminate a perpetuity's cash flow. For example, the issuer may go bankrupt or may decide to buy back the perpetuity instrument. Thirdly, the equations can be criticized for not including important variables like inflation and tax rates.
A cautious investor might assign a probable holding period to a perpetuity and then treat it as an annuity when calculating its present value, assuming the sale of the instrument at the end of the holding period. This, too, is imperfect but may give a more realistic estimate of the instrument's value.
Read More: Annuity vs Perpetuity