Valuing a stock from its dividends

What is a share worth?

The dividend discount model says a share is worth the present value of all the dividends it will ever pay, discounted at the required return r. Dividends are uncertain, so every version of the model needs an assumption about how they grow. Three assumptions cover the exam.

Value of a sharepresent value of all future dividends

• No growth Memorize a fixed dividend forever: preferred stock, or a common share with a flat dividend
  • V₀ = D ÷ r a perpetuity
• Constant growth Memorize dividends grow at g forever
  • V₀ = D₁ ÷ (r − g) needs r above g; D₁ = D₀ × (1 + g)
• Changing growth Use it a fast stage, then a stable stage
  • Stage 1 discount each dividend one by one
  • Terminal value Gordon value at the end of stage 1, discounted n years

With no growth: a perpetuity

If the dividend never changes, the share is a perpetuity, exactly like the perpetual bond. The standard case is preferred stock : a fixed dividend, paid forever.

$$V_0 = \frac{D}{r}$$

The same formula fits a common share whose dividend is expected to stay flat (constant dividends ): a flat EUR 1.80 a year at a 9% required return is worth 1.80 ÷ 0.09 = EUR 20.00.

With constant growth: the Gordon model

When dividends grow at the same rate g forever, the infinite sum still has a closed form. This is the Gordon growth model , and the pattern is constant growth .

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

The numerator is the next dividend. If a question gives the dividend just paid, grow it once. The model only works when r is above g; otherwise the sum never settles. The required return r here is the return investors expect for the share’s risk; the Deeper below shows where it comes from.

Exam tip Some questions write dividends as Div and the terminal value as the expected share price at year n. Same quantities, same formulas.

Exam tip Value is very sensitive to r − g. With the same numbers and growth of 2.5% instead of 2%, the value jumps from 57.12 to 63.78. That is why analysts test their assumptions.

With changing growth: two stages

Many companies grow fast for a while and then settle. Value that in two parts. The two-stage dividend growth method is:

1. List the dividends in the high-growth years, D₀ × (1 + g_s)ᵗ, and discount each one.
2. At the end of that stage, value every later dividend as a growing perpetuity, using the long-term growth rate g_l. That is the terminal value , built from the first dividend of the stable stage.
3. Discount the terminal value by the number of years in stage 1, not one more, and add.

$$V_0 = \sum_{t=1}^{n} \frac{D_0 (1 + g_s)^{t}}{(1 + r)^{t}} + \frac{V_n}{(1 + r)^{n}}, \quad V_n = \frac{D_{n+1}}{r - g_l}$$

Almost 88% of this value (33.25 of 37.92) sits in the terminal value, far in the future. That is typical of growth stocks.

Go deeper Where r and g come from

The required return r comes from a risk model: the capital asset pricing model builds it from the risk-free rate plus a premium for market risk (Learning Module 8 and Equities). The growth rate g comes from what the company can earn and keep, which the Equities module on discounted cash flow and growth models develops. Both are estimates, which is why a small change in r − g moves the value so much.

Go deeper When the model fails, and preferred shares that are called

If g is equal to or above r, the formula gives a negative or infinite value. A company cannot grow faster than the required return forever, so the constant-growth stage must use a g below r. For a stage that does exceed r, use the two-stage method with a stable second stage.

Some preferred shares can be called (repaid) on a set date. Then the cash flows end, and the share is priced like a bond: discount the dividends and the call price.