# Perpetuity

A **perpetuity** is a constant stream of cash flows that begin on a fixed date and continue indefinitely. It is sometimes referred to as a "perpetual annuity". Fixed coupon payments on permanently invested (irredeemable) sums of money are prime examples of perpetuities. Scholarships paid perpetually from an endowment fit the definition of perpetuity.

The price of a perpetuity is simply the present value of all coupons , received at the end of each period, forever:

where is the value of each identical coupons and r is the discount rate. Series like this formula are also know as geometric progression. Even if the series has an infinite number of terms, the whole series has a finite sum as each term is only a fraction of the preceding term. Additionally, because the principal is never repaid, there is no present value for the principal. The infinite sum can be simplified to:

## Due vs. Immediate Perpetuity

A perpetuity receivable starting at the present time is called a *perpetuity due*. If the frequency of payments equals the frequency of interest compounding, the present value of a perpetuity due with payments of 1, is given by *d*^{−1}, where *d* = 1 − (1 + *i*)^{−1}, and is called the *rate of discount*. In this case, *i* is the interest rate *per period*, not necessarily per year. If the first payment is 1 period in the future, the annuity is a *perpetuity immediate*, and the present value is *i*^{−1}.

## Real-life examples

For example, UK government bonds, called consols, that are undated and irredeemable (e.g. War Loan) pay fixed coupons (interest payments) and trade actively in the bond market. Very long dated bonds have financial characteristics that can appeal to some investors and in some circumstances, e.g. long-dated bonds have prices that change rapidly (either up or down) when yields change (fall or rise) in the financial markets.

A more current example is the convention used in real estate finance for valuing real estate with a cap rate. Using a cap rate, the value of a particular real estate asset is either the net income or the net cash flow of the property, divided by the cap rate. Effectively, the use of a cap rate to value a piece of real estate assumes that the current income from the property continues in perpetuity.

Another example is the constant growth Dividend discount model for the valuation of the common stock of a corporation. If the discount rate for stocks (shares) with this level of systematic risk is 12.50%, then a constant perpetuity of per dollar of dividend income is eight dollars. However if the future dividends represent a perpetuity increasing at 5.00% per year, then the Dividend Discount Model, in effect, subtracts 5.00% off the discount rate of 12.50% for 7.50% implying that the price per dollar of income is $13.33.

## References

- Ross S., Westerfield R., Jaffe J.(2005)
*Corporate Finance*, 6th Edition, Mc-Graw Hill