Discounted Dividend Valuation: formula sheet

Key formulas

General DDM (infinite horizon)

V₀ = Σ [Dₜ ÷ (1 + r)ᵗ], t = 1 to ∞

V₀ is intrinsic value today, Dₜ is the expected dividend in year t, r is the required return on equity.

Finite holding period

V₀ = Σ [Dₜ ÷ (1 + r)ᵗ] for t = 1 to n + Vₙ ÷ (1 + r)ⁿ

Vₙ is the expected share price at the end of year n. It must be the value of dividends after year n.

Required return from CAPM

r = Rf + β × (Rm − Rf)

Use when the vignette gives the risk-free rate, beta and equity risk premium.

Dividend growth rate

g = (Dₜ ÷ Dₜ₋₁) − 1

Dividend for next year: D₁ = D₀ × (1 + g).

Implied value comparison

If V₀ > market price, undervalued; if V₀ < price, overvalued

Compare intrinsic value with the current market price.

Gordon growth value

V0 = D1 ÷ (r − g) = D0 × (1 + g) ÷ (r − g)

Requires g < r and constant growth forever. D1 is next year's dividend.

Implied required return

r = D1 ÷ P0 + g

Use the market price as P0. Dividend yield on D1 plus growth.

Implied growth rate

g = (P0 × r − D0) ÷ (P0 + D0)

Derived from P0 = D0(1 + g) ÷ (r − g). If D1 is given, use g = r − D1 ÷ P0.

Sustainable growth

g = b × ROE, where b = 1 − payout ratio

Retention rate times return on equity. Gives g when the vignette supplies payout and ROE.

Justified leading P/E

P0 ÷ E1 = (D1 ÷ E1) ÷ (r − g)

Payout ratio on next year's earnings divided by r − g.

Justified trailing P/E

P0 ÷ E0 = (D0 ÷ E0) × (1 + g) ÷ (r − g)

Equals leading P/E times (1 + g).

Value decomposition

V₀ = E₁ ÷ r + PVGO

E₁ is next year's expected EPS. E₁/r is the no-growth value per share.

PVGO

PVGO = V₀ − E₁ ÷ r

Use the market price if asked what the market implies; use the model value if asked for the intrinsic PVGO.

Justified leading P/E

P₀ ÷ E₁ = 1 ÷ r + PVGO ÷ E₁

1/r is the no-growth component. The rest is the growth component.

Gordon leading P/E

P₀ ÷ E₁ = (1 − b) ÷ (r − g)

b is the retention rate; g = b × ROE in the sustainable growth setting. Needs r > g.

Trailing P/E

P₀ ÷ E₀ = (1 + g) × (P₀ ÷ E₁)

Since E₁ = E₀ × (1 + g). Valid for constant growth.

Growth share of value

PVGO ÷ V₀

Fraction of price that depends on growth opportunities.

Retention rate

b = 1 − dividend payout ratio = 1 − (D ÷ EPS)

Payout and retention add to 1. Retention can also be (NI − dividends) ÷ NI.

Sustainable growth rate

g = b × ROE

Assumes stable capital structure, no new equity issued, and ROE constant.

DuPont ROE (three-step)

ROE = (NI ÷ Sales) × (Sales ÷ Assets) × (Assets ÷ Equity)

Margin × turnover × leverage. Use inside g = b × ROE.

Five-step DuPont ROE

ROE = (NI ÷ EBT) × (EBT ÷ EBIT) × (EBIT ÷ Sales) × (Sales ÷ Assets) × (Assets ÷ Equity)

Tax burden × interest burden × EBIT margin × turnover × leverage.

Gordon growth value

V0 = D1 ÷ (r − g) = D0 × (1 + g) ÷ (r − g)

Valid when g < r and growth is constant.

Implied payout and growth link

Payout = 1 − g ÷ ROE

Rearranged g = (1 − payout) × ROE. Use it to find the payout needed for a target g.

Two-stage DDM

V0 = Σ [D0 × (1 + gS)^t ÷ (1 + r)^t] for t = 1 to n + [Vn ÷ (1 + r)^n]

gS is the short-term (high) growth rate. The sum covers the n years of stage one.

Terminal value (Gordon growth)

Vn = D(n+1) ÷ (r − gL)

D(n+1) = Dn × (1 + gL). Requires r > gL. Value is at time n, so discount it n years.

H-model

V0 ≈ [D0 × (1 + gL) ÷ (r − gL)] + [D0 × H × (gS − gL) ÷ (r − gL)]

H = half the length of the decline period in years. gS is the starting high growth rate, gL the long-run rate. This is an approximation.

Three-stage DDM

V0 = PV(stage 1 dividends) + PV(stage 2 dividends) + PV(terminal value)

Terminal value at the end of stage 2 = D(next year) ÷ (r − gL), discounted by the total years of stages 1 and 2.

Dividend payout and growth link

g = b × ROE, where b = 1 − payout ratio

Useful when the vignette gives retention and ROE instead of growth rates.

CAPM required return

r = Rf + β × (E(Rm) − Rf)

(E(Rm) − Rf) is the equity risk premium. Use the beta and premium given in the vignette.

Build-up method

r = Rf + equity risk premium + size premium + specific-company premium

Premiums are added. Some versions use a beta on the equity risk premium. Follow the vignette's stated components.

Multifactor (Fama-French style)

r = Rf + β(mkt) × λ(mkt) + β(size) × λ(size) + β(value) × λ(value)

Each λ is a factor risk premium. Multiply each factor beta by its premium and add.

Implied return, Gordon growth

r = D1 ÷ P0 + g

Valid for a constant growth rate g below r. D1 is next year's dividend, not D0.

Implied return, multistage

P0 = Σ Dt ÷ (1 + r)^t + Vn ÷ (1 + r)^n, solve for r

Vn = Dn+1 ÷ (r − gL) if terminal value uses Gordon growth. Set P0 to market price and find r by IRR or trial and error.

Decision rule

Implied r > required r: undervalued. Implied r < required r: overvalued

Assumes the dividend forecasts are reasonable.

Finite horizon DDM

V₀ = Σ [Dₜ ÷ (1 + r)ᵗ] for t = 1 to n + Pₙ ÷ (1 + r)ⁿ

Dₜ may be zero in early years. Pₙ is the terminal value at the end of year n.

Terminal value by Gordon growth

Pₙ = Dₙ₊₁ ÷ (r − g)

Valid only if g < r and growth is stable after year n. Use the dividend for year n + 1.

Terminal value by terminal P/E

Pₙ = EPSₙ × (terminal P/E)

Use the forecast EPS at the horizon and a justified or comparable P/E.

Justified forward P/E from DDM

P₀ ÷ E₁ = (D₁ ÷ E₁) ÷ (r − g) = payout ratio ÷ (r − g)

Links the P/E multiple to payout, required return and growth. Assumes constant growth.

Two-stage DDM (Gordon terminal)

V₀ = Σ [D₀(1 + gₛ)ᵗ ÷ (1 + r)ᵗ] for t = 1 to n + [Dₙ₊₁ ÷ (r − gₗ)] ÷ (1 + r)ⁿ, where Dₙ₊₁ = D₀(1 + gₛ)ⁿ(1 + gₗ)

gₛ is short-term growth, gₗ is long-term growth, and gₗ < r. Dₙ₊₁ is the first dividend of the long-term stage: grow D₀ at gₛ for n years, then once at gₗ. The terminal value is Dₙ₊₁ ÷ (r − gₗ), measured at the end of year n.

Sustainable growth rate

g = b × ROE, where b = 1 − payout ratio

Use b as retention rate. Useful for estimating long-run growth.

Quick revision

• General DDM: value = present value of all expected future dividends discounted at the required return on equity.
• Gordon growth: V0 = D0 × (1 + g) ÷ (r − g) = D1 ÷ (r − g). It needs g < r and stable growth.
• Terminal value at time n = Dn+1 ÷ (r − g). Discount it back n periods.
• Implied return under Gordon growth: r = D1 ÷ P0 + g.
• Sustainable growth: g = b × ROE, where b = 1 − dividend payout ratio.
• No-growth value per share = E1 ÷ r; PVGO = V0 − E1 ÷ r.
• Justified leading P/E = (1 − b) ÷ (r − g); justified trailing P/E = D0 × (1 + g) ÷ (E0 × (r − g)), which equals (1 − b) × (1 + g) ÷ (r − g).
• Higher growth or a lower required return raises value, and value is very sensitive as r − g shrinks.
• Multistage models: explicit dividends in the high-growth stage, then a terminal value from a stable-growth stage.
• The H-model approximates a linear decline in growth from a high rate to a long-run rate; it is an approximation, not an exact value.
• For non-dividend payers, value using expected future dividends once payments begin, or use other methods when dividends are not a good guide to value.
• Always check the units and timing: annual versus quarterly dividends, and whether the vignette gives D0 or D1.

Common mistakes

• Discounting the terminal price at the wrong year or leaving it out. Fix: Add Vₙ to Dₙ in year n and discount both by (1 + r)ⁿ.
• Using D₀ as if it were next year's dividend. Fix: Check the timing. If a growth rate is given with D₀, compute D₁ = D₀ × (1 + g) first.
• Using D0 in the numerator instead of D1 Fix: Always ask which year the dividend belongs to. Multiply D0 by (1 + g) unless D1 is stated.
• Using the model when g ≥ r Fix: Check r > g first. If it fails, the constant-growth model is not valid and a multistage model is needed.
• Using E₀ instead of E₁ in E/r. Fix: The no-growth perpetuity starts next year. Use E₁. Convert E₀ × (1 + g) when needed.
• Assuming growth always means positive PVGO. Fix: PVGO is positive only when projects earn more than r. If ROE equals r, PVGO is zero despite growth.
• Using the payout ratio instead of the retention rate in g = b × ROE. Fix: Always write b = 1 − payout first. Growth comes from what is kept, not paid.
• Using ROA instead of ROE. Fix: Growth from retained earnings uses ROE. Multiply by leverage if you only have ROA.
• Discounting the terminal value n+1 years instead of n years. Fix: The Gordon formula values dividends one period ahead, so the result sits at time n. Discount it n years.
• Using the high growth rate to compute D(n+1). Fix: D(n+1) = Dn × (1 + gL). The new stage starts with the stable rate.

Exam tips

• Read the exhibit carefully for which dividend is D₀ and which is D₁. Timing errors are the most common trap.
• When a question asks which model suits a company, link the answer to the facts: stable, earnings-linked dividends favour DDM; low or irregular payout favours FCFE.
• If the required return is not given, look for CAPM inputs or a build-up and compute it before discounting.
• Use the calculator cash flow worksheet for multi-year forecasts to save time and avoid arithmetic slips.
• There is no penalty for wrong answers, so always pick an option.
• Look at the dividend label first. Whether the vignette gives D0 or D1 decides the first step and is a common trap.
• Expect the model to be paired with CAPM for r and with b × ROE for g. Practise chaining them in one pass.
• When asked about suitability, link the model to stable, mature, dividend-paying firms and note sensitivity to r − g.