Gordon growth model and preferred stock valuation

How do you value preferred stock?

Preferred stock is a form of equity, generally without a vote, that has priority over common stock for dividends and for assets if the issuer is liquidated. The simplest case pays a fixed dividend forever and has no call or conversion feature. A fixed payment that lasts forever is a perpetuity , so the dividend discount model collapses into one fraction.

$$V_0 = \frac{D}{r}, \quad D = \text{dividend rate} \times \text{par}$$

Some preferred stock has a maturity date, when the par value $F$ is repaid. Value it like a bond: the present value of the dividends plus the present value of par.

$$V_0 = \sum_{t=1}^{n} \frac{D}{(1 + r)^t} + \frac{F}{(1 + r)^n}$$

Non-callable, non-convertible preferred stockthe base case: V = D ÷ r (perpetual) or bond-style (dated)

• Callable by the issuer Use it issuer holds the option, so value to the investor is lower
• Retractable (putable) by the holder Use it holder holds the option, so value is higher
• Convertible into common shares Know the idea conversion right adds value for the holder

Embedded options change the value. The issuer’s right to call the shares will be used when it suits the issuer, so a callable share is worth less to the investor. A retraction right, where the holder can sell the shares back to the issuer, will be used when it suits the holder, so it adds value. A conversion right also adds value for the holder. Precise valuation of those options is beyond this module.

What is the Gordon growth model?

Forecasting an endless stream of dividends is impractical, so analysts assume a pattern. The Gordon growth model , also called the constant growth DDM , assumes dividends grow at a constant rate $g$ forever. If the required return is above $g$, the endless series sums to a simple fraction.

$$V_0 = \frac{D_1}{r - g} = \frac{D_0\,(1 + g)}{r - g}$$

The numerator is next year’s dividend, $D_1$. The model is the present value of a growing perpetuity, and a perpetuity’s first payment is one year away. Using the dividend just paid, $D_0$, is the single most common error.

Working backward: required return and implied growth

Rearranged, the model says the required return is the dividend yield on next year’s dividend plus growth. If the share’s price equals its intrinsic value, this is the return the market is pricing in.

$$r = \frac{D_1}{P_0} + g$$

At a price of EUR 52.00 with D₁ = 2.60, the yield is 5% and g is 4%, so r = 9%, which recovers the starting point. Solving for growth instead gives the growth rate the price implies.

$$g = \frac{P_0\,r - D_0}{P_0 + D_0}$$

Where does g come from?

Analysts estimate the long-term growth rate from the history of dividends or earnings, the industry median, or from fundamentals. The fundamental route is the sustainable growth rate . Earnings that are not paid out are retained and reinvested at the return on equity, so growth equals the retention rate times ROE. The retention rate $b$ is one minus the payout ratio.

$$g = b \times ROE, \quad b = 1 - \text{payout ratio}$$

Be skeptical of any long-run g well above the economy’s long-run nominal growth rate. A mature company cannot keep growing dividends faster than nominal gross domestic product (GDP) forever.

How sensitive is the value?

The value rises very quickly as $g$ approaches $r$. With D₀ = 2.50 and r = 9%:

Growth g	D₁	Value V₀
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 $r$ matters just as much. With g = 4%, a required return of 8% gives 65.00, 9% gives 52.00 and 10% gives 43.33. Always test the inputs before trusting a Gordon value.

Which companies suit the constant growth model?

Is the Gordon growth model appropriate?it needs stable dividends growing below r, forever

• Good fit Use it mature, profitable, pays a dividend
  • Stable payout and growth near or below long-run nominal economic growth
  • Not sensitive to the business cycle regulated water utility, household goods maker
• Poor fit Use it use FCFE, a multistage DDM or multiples
  • No dividend, or erratic dividends nothing stable to extrapolate
  • Growth above r, or growth set to change formula fails, or growth fades

The model’s assumptions are four: dividends are the right cash flow, growth is constant forever, the required return is constant, and growth is strictly below the required return. It is most natural for mature, profitable dividend payers that do not swing with the business cycle, such as a regulated water utility or a long-established maker of household goods, particularly one that has raised its dividend at a steady rate for years.

When the assumptions fail, there are three alternatives: a model that allows varying growth (the next lesson), a different cash flow such as free cash flow to equity (FCFE), or a multiplier model. A company with no dividend today can still be valued if the first dividend and constant growth after it can be forecast, but that extrapolation is highly uncertain.

Exam tip The Gordon value always sits one year before the first dividend in the stream. Count that gap when you discount.

Go deeper Where the formula comes from

Write the dividend in year $t$ as $D_0(1+g)^t$. The present value of the stream is $D_0\left[\frac{1+g}{1+r} + \frac{(1+g)^2}{(1+r)^2} + \cdots\right]$. This is an infinite geometric series with ratio $(1+g)/(1+r)$, which is below 1 only if $r > g$. It sums to $\frac{(1+g)/(1+r)}{1 - (1+g)/(1+r)} = \frac{1+g}{r-g}$. Multiply by $D_0$ and you have $D_0(1+g)/(r-g)$. With $g = 0$, it reduces to the preferred stock perpetuity $D/r$.

Go deeper How much of the value is growth?

Subtract the no-growth value from the Gordon value. With D₀ = 2.50, r = 9% and g = 4%, a company that paid 2.50 forever would be worth 2.50 ÷ 0.09 = 27.78. The Gordon value is 52.00, so growth supplies 52.00 − 27.78 = 24.22, about 47% of the value. The larger that share, the more the value depends on a growth assumption that may not hold.