6.3 Discounting: perpetuities, Gordon growth, and the two-stage DCF

6.3.1Growing annuities and perpetuities#

Proof

Let $q = (1+g)/(1+r)$. The sum is $\frac{c_1}{1+r}\sum_{k=0}^{n-1} q^k$. For $q \neq 1$, the finite geometric series gives $\frac{c_1}{1+r}\cdot\frac{1-q^n}{1-q}$. Now $1 - q = \frac{(1+r)-(1+g)}{1+r} = \frac{r-g}{1+r}$, so the factor $(1+r)$ cancels and the sum is $c_1(1-q^n)/(r-g)$. If $q = 1$ each term is $c_1/(1+r)$.∎

Proof

With $r > g$ we have $0 < q < 1$, so $q^n \to 0$ in (6.7). If $g \ge r$ then $q \ge 1$, every term is at least $c_1/(1+r) > 0$, and the partial sums are unbounded.∎

Setting $g = 0$ gives the level perpetuity $c/r$. Applied to dividends, (6.8) is the Gordon growth model: $P_0 = D_1/(k_e - g)$. Rearranged, $k_e = D_1/P_0 + g$: the market's implied cost of equity is the forward dividend yield plus growth. This is how Damodaran estimates the implied equity risk premium of an index each month.

6.3.2The two-stage DCF convention#

Two-stage FCFF DCF The convention used in this course's code (m06_dcf.dcf_value):

• $F_0$ is the free cash flow of the year just ended; it is not discounted.
• Explicit stage: $F_t = F_0(1+g)^t$ for $t = 1, \dots, N$.
• Terminal value at $N$: $TV_N = F_N(1+g_T)/(W - g_T)$, a growing perpetuity whose first flow is $F_{N+1}$. Requires $W > g_T$.

By (6.7) with $c_1 = F_0(1+g)$, the first term equals $\frac{F_0(1+g)}{W-g}\big[1-\big(\tfrac{1+g}{1+W}\big)^N\big]$ when $W \neq g$.

6.3.3Terminal value and sensitivity#