The Distribution of Growing Perpetuities and the Gordon Growth Formula

A cash flow growing at expected rate $g$ with volatility $\sigma$ and discounted at $r > g$ for ever is worth $S_0/(r-g)$ by the Gordon growth formula. That number is the mean of the present value, not its typical size. The present value is an exponential functional of a drifted Brownian motion, and Dufresne’s identity gives its law exactly: $2S_0/(\sigma^2\gamma_\nu)$, with $\gamma_\nu$ a Gamma variable of shape $\nu = 1 + 2(r-g)/\sigma^2$. This one number fixes the shape. The mean is the Gordon value at every volatility; the median, as a share of it, is $0.689$ at $r – g = 3$ per cent and $\sigma = 20$ per cent, and $0.495$ at $\sigma = 30$ per cent. The variance is finite only while $\sigma^2 < 2(r-g)$. At those two settings $70.0$ and $77.4$ per cent of outcomes fall below the Gordon value, a share that exceeds one half for every $\nu$ and tends to one as volatility grows. A ten-year model's terminal value has a median of $0.584$ of its mean, and a simulation free of the identity reproduces its quantiles within $0.43$ per cent.