Two-stage (multistage) growth models

When do you need more than one growth rate?

A multistage growth model fits when expected cash flow growth changes over time, for example a company expanding fast into new markets before slowing to a stable rate. The fast stage can use one higher rate for $n$ periods or a different rate for each year; once growth is stable, a terminal value captures everything after. The first stage can also be below trend, a weak spell before a recovery. Present value models built on past cash flows are not useful for startups with little revenue and losses; they work from the growth stage on.

The simplest case is the two-stage growth model . It gives the company two growth rates: a short-term growth rate g_s for a finite number of periods $n$, and a long-term growth rate g_l forever after. It works for dividends, and the same method works for free cash flow to equity (FCFE), which the formulas below use.

Revenue growth is not cash flow growth, and the difference is the caution of this section (revenue growth versus cash flow growth ). A budget gym chain can grow membership fees 15% a year while its FCFE falls, because every new gym it opens eats cash. Grow the cash flow you discount, not the sales.

Choose by the growth pathstages you can defend

• Constant growth Use it already stable; mature
• Two-stage Use it a spell of above- or below-trend growth, then stable
  • Typical case advantage that fades as rivals arrive
  • Also fits a temporary slump before recovery
• A rate for each year Know the idea path forecast year by year
• No growth model Know the idea startups without revenue; transformational change

The method has two parts. Forecast the cash flows of the fast stage one at a time. Then collapse everything after that into one number at period $n$, the terminal value at the end of the fast stage , using the constant growth formula.

$$IV_t = \sum_{i=1}^{n} \frac{\text{FCFE}_t(1 + g_s)^{i}}{(1 + r_e)^{i}} + \frac{IV_{t+n}}{(1 + r_e)^{n}}$$

$$IV_{t+n} = \frac{\text{FCFE}_{t+n+1}}{r_e - g_l}$$

$$\text{FCFE}_{t+n+1} = \text{FCFE}_t(1 + g_s)^{n}(1 + g_l)$$

Here $IV_t$ is the intrinsic value today, $\text{FCFE}_t$ is the free cash flow to equity just earned (for a share paying dividends, use $D_t$), $r_e$ is the required return on equity and $IV_{t+n}$ is the value at period $n$.

The constant growth formula values cash flows starting one period after the date it is applied. So $IV_{t+n}$, applied at period $n$, values cash flows from period $n+1$ on, and it needs the first cash flow of the stable stage . The terminal value is a value at period $n$, so it is discounted $n$ periods, the same as the period-$n$ cash flow.

The four classic slips

Each of these changes the answer to the dividend example above, and each appears as an option in exam questions.

Slip	What it does	Result
Terminal value from $D_{t+n}$ instead of $D_{t+n+1}$	$IV_{t+3}$ = 1.8251 ÷ 0.05 = 36.50	$IV_t$ = 31.36
Discount $IV_{t+n}$ for $n + 1$ periods	38.33 ÷ 1.10⁴	$IV_t$ = 30.11
Grow the first stable year at $g_s$ instead of $g_l$	$D_{t+4}$ = 1.8251 × 1.15 = 2.0988	$IV_t$ = 35.47
Leave $IV_{t+n}$ undiscounted	3.937 + 38.326	$IV_t$ = 42.26

The correct answer is 32.73. Using $g_s$ in the terminal formula’s denominator breaks it: 15% is above $r_e$ = 10%.

How sensitive is it?

Because the terminal value dominates, small changes to $n$, $g_l$ or $r_e$ move the result a lot. This is the sensitivity to n, g_l and r_e . In the dividend example, with everything else unchanged:

Long-term growth $g_l$	$IV_t$
4%	27.70
5%	32.73
6%	40.27

Required return $r_e$	$IV_t$
9%	41.00
10%	32.73
11%	27.22

For the FCFE example: lengthening the fast stage from four to six years lifts the per-share value from 13.23 to 15.75; raising $r_e$ from 10% to 11% cuts it to 11.52. Analysts therefore test the terminal value, for example by comparing the multiple it implies with the multiples of mature peers.

Which companies need a multistage model?

Use the stages that match how growth will really behave. Constant growth fits a mature company already at a stable rate (lesson 3). Two stages fit a finite spell of above- or below-trend growth followed by a stable rate: a new product, an advantage that fades as rivals arrive, a temporary slump. A different rate for each year fits when the path can be forecast year by year. None of these fits a startup without revenue, or a company being transformed (lesson 7).

Age is not the only test. An old company can find fresh growth, for example by launching a new product line or buying a rival, and that burst is expected to fade back toward a normal rate. Two stages fit. So does a company going through a weak spell that is expected to climb back to its normal growth rate. The test is the shape of the growth path, not the company’s age. Any use of a multistage model must be backed by company and industry analysis (see the industry analysis module).

Exam tip Match the stages to growth you can defend with company and industry analysis. One stable rate: constant growth. A spell, then stability: two stages. Nothing you can defend: use detailed forecasts or multiples.

Go deeper Why the forecast horizon matters

The shorter the fast stage, the larger the share of value left in the terminal value: with 10 growing 8% a year, then 3%, at 9%, the terminal value is 85% of the value over three years, 77% over five and 62% over ten. This is the terminal value share of intrinsic value . Analysts keep the horizon as long as they can forecast with confidence. Lesson 5 picks the long-run growth rate itself.

Go deeper Checking the terminal value

A terminal value is a forecast of what the company will be worth at period $n$. To check it, convert it to a multiple. In the dividend example, $IV_{t+3}$ of 38.33 against an expected earnings per share can be turned into a price-to-earnings ratio and compared with the ratios of mature companies. If the implied multiple is far above what mature peers trade at, the long-term growth rate or the required return is probably too generous. Learning Module 7 shows how multiples can set or check a terminal value.