The flat perpetuity. Let the annuity run forever. As n grows without limit, (1+r)^(−n) shrinks to 0, and the annuity formula collapses to the most elegant result in finance:
PV = C / r
Perpetuity formula example, and real perpetuities
A promise of ₹8.50 a year forever, when comparable money earns 9.5%, is worth 8.50/0.095 = ₹89.47. That price is finite even though the payments never stop, because distant flows shrink geometrically faster than they accumulate. Here's a way to see it. With ₹89.47 in hand at 9.5%, you could manufacture ₹8.50 a year forever without touching the principal. A perpetual stream is worth the principal that would replicate it.
Real perpetuities exist. The UK's consols paid coupons from the 18th century until redeemed in 2015. A 2.5%-coupon consol at a 4% market yield priced at 2.5/0.04 = £62.5 per £100 face.
India's live example is the AT-1 perpetual bond issued by banks, and it comes with a caveat taught in blood. They're callable, their coupons can be skipped, and their principal can be written off entirely. In March 2020, holders of ~₹8,400 crore of Yes Bank AT-1s were wiped out while equity survived. You'll use that fact when you work out where instruments sit in the capital stack. It's also a permanent reminder that the formula prices promised cash flows. Whether they'll be paid is the analyst's job, and the algebra can't tell you.
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How do you value preferred stock as a perpetuity?
Preferred stock fits the flat perpetuity most cleanly, and it's worth a paragraph because you'll meet it again as the third leg of a cost of capital. A straight preferred share promises a fixed dividend, has no maturity, and ranks ahead of the ordinary shares. So its value is the level perpetuity, and its required return is the same equation solved the other way:
P = D / r, and therefore r = D / P
Take a US preferred with a $25 par value and a 6.5% rate. Its dividend is 25 × 0.065 = $1.625 a year, and an investor demanding 7.4% pays 1.625/0.074 = $21.96. Now run the inversion. It's the version you'll actually use, because you can observe the price and you can't observe the required return. If this share trades at $19.50, the market's required return is 1.625/19.50 = 8.33%. That figure is a real reading of where this issuer's fixed-claim capital prices, somewhere between its bond yield and its cost of equity. It tells you more than any conversation about whether the preferred "looks cheap." An Indian non-convertible redeemable preference share with a ₹100 face value and an 8.5% rate works the same way: ₹8.50 a year, ₹85.00 at a required 10%, and an implied 8.50/92 = 9.24% at a market price of ₹92.
What breaks the simple perpetuity formula?
Two features break the simple form, and both are common. A callable preferred gives the issuer the right to retire it at a set price. So the stream stops being infinite in the states of the world where that matters most. Rates fall, the issuer calls, and the investor's upside is capped near the call price. Price it to the call and forget the perpetuity. On the ₹92 share above, with a par call three years out, the yield to call is 11.82% against the 9.24% perpetual reading, and the gap is the option the investor wrote. A cumulative preferred accrues any skipped dividend as arrears, and those arrears must clear before the ordinary shares are paid. A missed year becomes a deferred claim. Value it as a stream with a gap in it plus a lump when the arrears clear, a shape no level perpetuity can show.