What a payment that never ends is worth today
Imagine a contract that pays you $1,000 every year, forever. Not for 30 years. Forever. Your instinct says that should be worth a fortune, because the payments never stop. Then you run the math and the present value at a 5% discount rate is $20,000. An infinite stream of payments has a finite, often surprisingly modest, value today. That single counterintuitive fact is the heart of how preferred stock, endowments, and certain real estate income get priced.
The reason is the time value of money. A dollar arriving in year 50 is worth almost nothing today once you discount it back, and a dollar in year 200 is essentially zero. Add up an infinite tail of payments that each shrink toward nothing, and the sum converges. The formula is elegant: present value equals the annual payment divided by the discount rate, or PV = PMT / r. For $1,000 a year at 5%, that is $1,000 / 0.05 = $20,000. Drop the rate to 4% and the value jumps to $25,000, because each future dollar is discounted less harshly.
This is exactly how preferred stock is valued. A preferred share paying a fixed $5 annual dividend, with investors demanding a 6% return, is worth $5 / 0.06, or about $83.33. No maturity date, no principal repayment, just a fixed perpetual dividend, which is the textbook definition of a perpetuity. The same logic prices a perpetual bond (a consol), a charitable endowment that must fund a scholarship forever, or a rental property treated as a perpetual income stream.
The growing perpetuity adds one more lever: payments that rise at a steady rate. If your $1,000 payment grows 2% a year and you discount at 5%, the formula becomes PV = PMT / (r - g), or 1,000 / (0.05 - 0.02) =33,333. The growth lifts the value substantially, which is why the spread between the discount rate and the growth rate matters so much in valuation.
One critical rule: the discount rate must exceed the growth rate. If payments grow as fast as or faster than the rate at which you discount them, the math breaks and the value runs to infinity, which is the model warning you that no real asset grows forever faster than money is discounted.