Dividend Discount Model

Dividend Discount Model

Dividend Discount Model: Valuing a Share From Its Dividends

What it does

The only cash a company ever pays to a shareholder who keeps the share is its dividends. A dividend discount model (DDM) values the share as the sum of all the dividends expected, each brought back to today's money. The page takes last year's dividend, grows it at the rates you enter, discounts each year at the return you require, and adds the results. It reports a model value: what that stream of dividends is worth under those inputs. It does not say what the market price should be.

The method

Discounting. With r the required return (the yearly return an investor requires for holding the share, also called the cost of equity), a dividend D_t paid in t years is worth D_t/(1 + r)^t today, because that amount invested at r would grow to D_t by then. The value is V = D₁/(1 + r) + D₂/(1 + r)² + … for ever. Dividends are taken at the end of each year. D₀ is last year's dividend.

Constant growth (Gordon). If the dividend grows at one rate g for ever, D_t = D₀(1 + g)^t, and the endless sum has a closed form:

V = D₁/(r − g) = D₀(1 + g)/(r − g), for r above g

Two-stage. The dividend grows at g_S (the growth now) for N years and at g_L (the long-run growth) afterwards. Stage 1 is added year by year. Stage 2, the terminal stage, is the constant-growth formula applied at year N, which gives the share price the model expects then, P_N, discounted N years:

D_t = D₀(1 + g_S)^t for t = 1 … N
P_N = D_N(1 + g_L)/(r − g_L)
V = D₁/(1 + r) + … + D_N/(1 + r)^N + P_N/(1 + r)^N

H-model. Growth rarely drops in one step. In the H-model the growth declines in a straight line from g_S to g_L over N = 2H years (H is the half-life of the extra growth), and the value is approximated by one formula:

V ≈ D₀[(1 + g_L) + H(g_S − g_L)]/(r − g_L)

The first term is the constant-growth value at g_L; the second is an allowance for the extra growth. The page prints the exact year-by-year sum beside it, with the growth of year t equal to g_S − (g_S − g_L)(t − 1)/N for t = 1 … N and g_L afterwards, and states the difference.

Share of value in the terminal stage. [P_N/(1 + r)^N]/V: the part of today's value that comes from dividends after year N. For the constant-growth model, which has no stages, the page shows the split at year 10.

What a price implies. Setting the constant-growth value equal to a price P and solving gives the implied growth, g = (P r − D₀)/(P + D₀), and, solved for the return instead, the implied required return r = D₁/P + g: the dividend yield plus the growth.

Advanced: sustainable growth. Return on equity ROE = earnings per share / book value per share. Payout ratio = dividend / earnings; retention b = 1 − payout. A company that grows only by retaining earnings at a steady ROE grows at g = ROE × b.

Advanced: residual income. Residual income of year t is RI_t = E_t − r B_t−1: earnings less a charge, at the required return, for the book value B in use at the start of the year. The value is V = B₀ + RI₁/(1 + r) + RI₂/(1 + r)² + … The page forecasts E_t = ROE_t B_t−1 with ROE moving in a straight line from today's level to r over M years (ROE_t = ROE₀ + (r − ROE₀)(t − 1)/M, and r from year M + 1, so residual income is then zero), a constant payout ratio, and the clean-surplus relation B_t = B_t−1 + E_t − D_t: book value changes only by earnings less dividends. Under clean surplus the residual income value equals the discounted dividends of the same forecast, whatever the forecast; the page computes both and shows that they agree.

Advanced: present value of growth opportunities. PVGO = P − E/r: the price (or, with no price entered, the model value) less the worth of today's earnings paid out unchanged for ever.

What it assumes

Dividends are the only cash the shareholder receives; share buybacks are left out unless you add them to the dividend. One required return for all years. Growth follows the simple pattern chosen and then stays at one rate for ever. Dividends arrive once a year, at year end. The residual income form adds: book value is a fair measure of the capital in use, a constant payout ratio, and no excess return after the fade.

When it misleads

The value is very sensitive to r − g. The formula divides by the difference between two estimates, each uncertain by a point or more. In the default case one point more long-run growth raises the value by 16 percent and one point less required return raises it by 20 percent. As g approaches r the value grows without limit, and at g equal to or above r the model has no answer. A long-run growth above the growth of the whole economy cannot last for ever.

Most of the value is usually in the terminal stage. With 77 percent of the value coming from dividends more than five years away, as in the default case, the result is mostly a statement about the long-run inputs.

It needs a dividend that reflects the business. A company that pays nothing has no DDM value, though it is not worthless; one that pays out little of its earnings, or more than it earns, gives a value that says more about its dividend policy than about its earning power. Use the residual income form, Reverse DCF or DCF Valuation for those.

The required return is not observed. It is an estimate (the Cost of Capital tool works one out). The "implied" figures are true only inside the constant-growth model: the implied growth is the single rate held for ever that fits the price, not a forecast of any year.

The H-model is an approximation. It is close when the decline is short and the two growth rates are not far apart (1.9 percent low in the default case) and drifts as either grows.

Sustainable growth uses one year's ROE. An unusually good or bad year, or book value shrunk by buybacks, distorts it.

Worked example

The default case is an illustrative company, not a real one: D₀ = 2.00, r = 9 percent, two-stage with g_S = 8 percent for N = 5 years and g_L = 3 percent, price 45.00.

D₁ = 2.00 × 1.08 = 2.1600, worth 2.1600/1.09 = 1.9817 today; D₂ = 2.3328, worth 1.9635; D₃ = 2.5194, worth 1.9455; D₄ = 2.7210, worth 1.9276; D₅ = 2.9387, worth 1.9099. Stage 1 sums to 9.73.

P₅ = 2.9387 × 1.03/(0.09 − 0.03) = 50.45, worth 50.45/1.09⁵ = 32.79 today. V = 9.73 + 32.79 = 42.52, of which 32.79/42.52 = 77.1 percent is the terminal stage.

The same inputs give 34.33 in the constant-growth model (2.06/0.06) and 38.50 in the H-model (2.00 × [1.03 + 2.5 × 0.05]/0.06, against an exact sum of 39.23).

At the price of 45.00 the implied growth is (45 × 0.09 − 2)/(45 + 2) = 4.36 percent a year for ever, and the implied required return at 3 percent growth is 2.06/45 + 0.03 = 7.58 percent.

Advanced, with earnings of 4.00 and book value of 25.00 a share: ROE = 16 percent, payout 50 percent, sustainable growth 16 × 0.5 = 8 percent. Residual income of year 1 is 4.00 − 0.09 × 25 = 1.75; with ROE fading to 9 percent over ten years the residual income is worth 8.43, so V = 25.00 + 8.43 = 33.43; the dividends of that same forecast are worth 13.76 + 19.68 = 33.43 as well. PVGO = 45.00 − 4.00/0.09 = 0.56.

A calculator for education and analysis, not investment advice.