Perpetuities and growing perpetuities

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CF ÷ r for a level stream that never ends, then CF₁ ÷ (r − g) for one that grows — always next period’s cash flow, and only when g < r.

An annuity with no last payment

Let the number of payments grow without limit and the annuity factor $\frac{1 - ( 1 + r ) ^{- n}}{r}$ loses its second term, because $(1 + r)^{- n} \to 0$ . What is left is

$P V_{perpetuity} = \frac{C F}{r}$

A perpetuity is a level cash flow paid at the end of every period, forever; its value is one payment divided by the rate per period. Excel has no function for it — =CF/r is the whole formula. Like the ordinary annuity, the first payment is one period from now.

Worked example — the Lakehead scholarship

Lakehead Robotics endows a scholarship of $800 a year, paid at the end of each year, in perpetuity. At 5 %, the endowment needed is

$\frac{800}{0.05} = 16, 000$

Watch the annuity converge on it: the PV of $800 a year at 5 % is 6,177.39 for 10 years, 9,969.77 for 20, 14,604.74 for 50, 15,878.33 for 100 — and 16,000 in the limit. Payments a century away are worth almost nothing today, so "forever" adds little beyond the first hundred years.

A stream that grows

If the cash flow grows at a constant rate $g$ per period, the perpetuity formula gains a term:

$P V = \frac{C F _{1}}{r - g} (g < r)$

$C F_{1}$ is next period's cash flow — the first one you will receive — not this period's. If a question quotes the flow just paid ( $C F_{0}$ ), grow it once: $C F_{1} = C F_{0} (1 + g)$ . This is the exam's favourite trap, and it returns in Module 11 as $D_{0}$ versus $D_{1}$ in the dividend model.

A stream paying $3.00 next year, growing 4 % a year, valued at 9 %: $3.00 / (0.09 - 0.04) = \mathbf{60.00}$. =3/(0.09-0.04).
A stream that paid $2.50 this year, growing 3 %, at 8 %: first grow it — $2.50 \times 1.03 = 2.575$ — then $2.575 / 0.05 = \mathbf{51.50}$. =2.5*(1+0.03)/(0.08-0.03). Using 2.50 unadjusted gives 50.00, and it is marked wrong.
A shrinking stream: $1.00 just paid, declining 2 % a year, at 7 %: $CF_1 = 0.98$, and $r - g = 0.07 - (-0.02) = 0.09$, so $0.98 / 0.09 = \mathbf{10.89}$. A negative $g$ is fine; subtracting a negative adds.

When the formula breaks

The formula requires $g < r$ . If the growth rate reaches or exceeds the discount rate, each future flow is worth as much as or more than the last in present-value terms, the series does not converge, and the stream has no finite value. Excel will happily divide by a negative and print a negative price; the number means nothing. Check $g < r$ before you trust the cell.

The annuity’s PV rises toward CF ÷ r and never passes it. By 100 years it is within 1 % of the perpetuity value — distant payments are nearly worthless today.