Constant growth models, implied growth and the sustainable growth rate

What is the constant growth dividend discount model?

Forecasting an endless stream of dividends is impractical, so analysts assume a pattern. If dividends grow at a constant rate $g$ forever and the required return on equity, $r_e$ (also called the cost of equity), is above $g$, the endless stream sums to one fraction, the intrinsic value $IV_t$. This is the constant growth dividend discount model .

$$IV_t = \frac{D_t(1 + g)}{r_e - g} = \frac{D_{t+1}}{r_e - g}$$

The numerator can be total dividends (giving the value of all the equity) or dividends per share (giving a value per share). It is next period’s dividend, $D_{t+1}$. The model is the present value of a growing perpetuity, and a perpetuity’s first payment is one period away. Using the dividend just paid, $D_t$, is the single most common error.

With $g = 0$ the formula becomes a no-growth perpetuity : 2.50 ÷ 0.09 = 27.78. A higher $r_e$ always lowers the value, all else equal.

Which companies suit constant growth?

Mature, well-established companies with stable cash flows. A stable-growth firm has roughly average market risk, a debt ratio near its industry’s, a return on capital close to its weighted average cost of capital (WACC), and fewer new investment opportunities, so lower capital expenditures and a higher payout than a fast-growing firm.

The company life cycle explains why. A startup has little revenue and losses, so present value models built on past cash flows are not useful. A growth company is expanding rapidly. A mature company has stable cash flows, and constant growth fits. A declining company’s growth can turn negative (lesson 5 values one). Every firm eventually approaches stable growth.

Is constant growth appropriate?stable cash flow growing below the required return, forever

• Good fit Use it mature, stable
  • Traits average risk; industry-average debt; return on capital near WACC
  • Cash lower capital spending, higher payout
• Poor fit Use it use multistage growth or detailed forecasts
  • Startup little revenue, losses
  • Fast or changing growth growth above the required return or set to change

Does constant growth work for free cash flow too?

Yes. The same formula applies to any cash flow measure, discounted at the matching rate. This is the constant growth model for any cash flow .

$$IV_t = \frac{E(CF)_{t+1}}{\text{Required rate of return} - g}$$

Here $E(CF)_{t+1} = CF_t \times (1 + g)$, or a direct forecast of next period’s cash flow.

With free cash flow to the firm (FCFF) the same formula at WACC gives the value of the whole firm, and the market value of debt is then subtracted (lesson 5). When constant growth starts from the first year, the numerator, next period’s cash flow, is also what sets the terminal value.

How sensitive is the value to g and the required return?

The value rises very quickly as $g$ approaches $r_e$. With $D_t$ = 2.50 and $r_e$ = 9%:

Growth $g$	$D_{t+1}$	Value $IV_t$
3%	2.575	42.92
4%	2.600	52.00
5%	2.625	65.63
6%	2.650	88.33

One point of extra growth adds between 9 and 23 to the value. A change in $r_e$ matters just as much. With $g$ = 4%, a required return of 8% gives 65.00, 9% gives 52.00 and 10% gives 43.33. This is the sensitivity to g : always test the inputs before trusting a constant growth value.

Three rules follow. First, $g$ must be below the required return for a positive, finite value. Second, it should not exceed the long-run nominal growth rate of the economy (growth that includes inflation), because a firm growing faster than the economy forever would eventually be the economy. A stable firm’s realistic $g$ is below the economy’s. Third, use a nominal $g$ in the same currency as the cash flows and the discount rate; the risk-free rate can stand in for the economy’s nominal growth. A declining firm can have a negative $g$.

What growth rate does the price imply?

Put the market price in place of $IV_t$ and solve for $g$. This is the implied constant growth rate .

$$g_{implied} = \text{Required rate of return} - \frac{E(CF)_{t+1}}{P_t}$$

Implied growth works with any measure and lets analysts compare the growth priced into companies in an industry, across industries or against the economy. Constant growth from a single year’s cash flow is a simplification, though; most firms’ growth changes, which lesson 4 handles.

Where do growth estimates come from?

From history or from fundamentals.

Historical growth and the base effect

Dividends are reported directly; FCFE and FCFF must be built from several lines and are often volatile. A historical growth rate is the compound rate between two past years. The base effect is the effect of the starting year chosen.

Take FCFE of 40, 95, 44, 47 and 50 over five years (the 95 was a one-off). From year 2: (50 ÷ 95)^(1/3) − 1 = −19.3% a year. From year 1: (50 ÷ 40)^(1/4) − 1 = 5.7%. The unusual year 2 biases any rate that starts there.

The sustainable growth rate

Growth a company can fund from its own earnings depends on how much it invests and how productive that investment is. The sustainable growth rate (SGR) measures it.

$$SGR_t = \left(1 - \frac{\text{Dividends}_t}{\text{Net income}_t}\right) \times \frac{\text{Net income}_t}{\text{BVE}_{t-1}}$$

The first term is the share of earnings retained; the second is return on equity (ROE) on the book value of equity (BVE) at the start of the year. Use it where both are expected to stay stable.

The same idea works with cash flows other than dividends. The FCFE sustainable growth rate replaces dividends with FCFE; the FCFF sustainable growth rate uses net operating profit after tax (NOPAT) , which is operating income × (1 − tax rate), and the return on capital (ROC) on book value of total capital (book debt plus book equity) at the start of the year.

$$\begin{aligned} \text{FCFE SGR}_t &= \left(1 - \frac{\text{FCFE}_t}{\text{Net income}_t}\right) \times \frac{\text{Net income}_t}{\text{BVE}_{t-1}} \\ \text{FCFF SGR}_t &= \left(1 - \frac{\text{FCFF}_t}{\text{NOPAT}_t}\right) \times \frac{\text{NOPAT}_t}{\text{Book value of total capital}_{t-1}} \end{aligned}$$

Using it: with dividends per share of CHF 1.60 just paid and $r_e$ of 12%, the constant growth value at $g$ = 8.75% is 1.60 × 1.0875 ÷ (0.12 − 0.0875) = 1.74 ÷ 0.0325 = CHF 53.54. Forgetting to grow the dividend gives 49.23; using the payout ratio in place of the retained share (0.30 × 12.5% = 3.75%) gives 20.12.

Long-run growth g

• Historical growth Know the idea watch the base effect
• Sustainable growth rate Memorize share retained × ROE; FCFE and FCFF versions
• The economy Know the idea nominal growth, about the risk-free rate

Go deeper Where the formula comes from

Write the dividend in period $t+i$ as $D_t(1+g)^i$. The present value of the stream is $D_t\left[\frac{1+g}{1+r_e} + \frac{(1+g)^2}{(1+r_e)^2} + \cdots\right]$. This is an infinite geometric series with ratio $(1+g)/(1+r_e)$, which is below 1 only if $r_e > g$. It sums to $\frac{(1+g)/(1+r_e)}{1 - (1+g)/(1+r_e)} = \frac{1+g}{r_e-g}$. Multiply by $D_t$ and you have $D_t(1+g)/(r_e-g)$. With $g = 0$, it reduces to the no-growth perpetuity $D \div r_e$, the same form lesson 6 uses for perpetual preferred stock.