Growing Perpetuities: Dividend Timing and Why r Must Exceed g

A growing perpetuity is a sequence of payments that increases at the same rate each period and continues indefinitely. Its present value depends on both growth and discounting; growth alone cannot determine value.

Put the first payment one period ahead

Let $D_1$ be the first future payment, $g$ its constant growth rate, and $r$ the discount rate. Assuming positive payments and ordinary rates $r,g>-1$,

$$P_0=\frac{D_1}{1+r}+\frac{D_1(1+g)}{(1+r)^2}+\cdots.$$

The ratio between successive discounted terms is $q=(1+g)/(1+r)$. The series converges when $q<1$, equivalent here to $r>g$. Summing it gives

$$P_0=\frac{D_1}{1+r}\frac{1}{1-q}=\frac{D_1}{r-g}.$$

This derivation explains the restriction: if growth matches or exceeds discounting, the positive terms do not produce a finite perpetuity value. A negative quotient from substituting $g>r$ is outside the formula’s valid domain.

A new dividend-timing example

A hypothetical firm has just paid $D_0=1.80$ USD per share. Assume annual dividend growth of 3% and a required annual equity return of 9%. The next dividend is

$$D_1=1.80(1.03)=\$1.854, \qquad P_0=\frac{1.854}{0.09-0.03}=\$30.90.$$

Using $D_0$ directly would give 30.00 USD and would omit a period of growth. Keep precision through the calculation and round the final currency result.

Test the spread, not just the growth forecast

With $D_1$ held at 1.854 USD, changing $r-g$ from 6% to 5% raises the estimate from 30.90 USD to 37.08 USD. This is a sensitivity calculation with the next dividend fixed, not a second forecast for the firm. A small positive spread can make the estimate highly sensitive to modest assumption changes.

A stable terminal phase needs consistent assumptions

Perpetual growth must be economically sustainable. A temporary expansion rate cannot simply be extended forever. In a multi-stage dividend model, a stable-growth terminal value at time $T$ uses $D_{T+1}$ and must itself be discounted back to today. Dividend policy, growth, and the required return should describe the same phase and time period.