Perpetuity
A perpetuity is an annuity in which the periodic payments 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 value of the perpetuity is finite because receipts that are anticipated far in the future have extremely low present value (today's value of the future cash flows). Additionally, because the principal is never repaid, there is no present value for the principal. The price of a perpetuity is simply the coupon amount over the appropriate discount rate or yield, that is
To give a numerical example, a 3% UK government War Loan will trade at 50 pence per pound in a yield environment of 6%, whilst at 3% yield it is trading at par.
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.