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EQ 2 – Discounted Dividend Valuation

Published: 10 August 2026
Updated: 3 November 2026

Worked examples are fully visible. Check-yourself items are study aids you can reveal one at a time.

A common share is a claim on a business, and a business generates cash over time. That single observation is the whole foundation of discounted cash flow valuation. If you own the claim, you own the stream, so the intrinsic value of the share must be the present value of the cash the claim is expected to deliver. The idea traces back to John Burr Williams in 1938 and remains the most widely used framework in professional equity research.

Applying the idea is harder than stating it. Four decisions have to be made in sequence: which class of model to use, which is the same thing as deciding what counts as cash flow; how to forecast that cash flow; which methodology to use for the discount rate; and what numerical value that discount rate should take. This reading fixes the first decision on dividends, that is, on distributions authorised by the board of directors. Models built on that definition are dividend discount models, or DDMs. The discount rate questions are handled identically for every discounted cash flow model and are treated separately elsewhere in the curriculum.

Why present value, and why a risk adjustment

Money arriving later is worth less than money arriving now, because money in hand can be spent or reinvested immediately. So every expected cash flow has to be pulled back to today before the pieces are added up. Where the timing and the amount are certain, the arithmetic is mechanical: a single certain receipt of $100 in two years, discounted at a risk-free rate of 5% a year, is worth $100/(1.05)² = $90.70.

Equity cash flows are not certain. Two adjustments handle that. First, treat each cash flow as a random variable and discount its expected value, the probability-weighted mean of the possible outcomes. Second, raise the discount rate above the risk-free rate to compensate for bearing the risk. The general expression is then:

V₀ is value today, CF_t is the expected cash flow at time t, r is the required rate of return, and n runs without end for equities.

The expression above uses one discount rate for every period, which assumes a flat term structure. Nothing prevents an analyst from applying different rates to different cash flows to reflect differences in business risk, operating risk or financial leverage, or simply different risk-free rates at different horizons. The single-rate version is enough for what follows.

One property of the expression matters more than it first appears. Value at any future date equals the present value, as of that date, of everything that comes after it. So a stream can be collapsed part way through and the answer does not change.

Example 1 · Worked

An asset is expected to produce $100 in one year, $150 in two years and $200 in three years. The discount rate is 10%.

1. What is the value of the asset today?

Solution. Discount each cash flow and add:
100 ÷ (1.10)¹ = 90.909
150 ÷ (1.10)² = 123.967
200 ÷ (1.10)³ = 150.263
Summing gives V₀ = $365.14.

2. What is the value of the asset at t = 1, immediately after the first cash flow?

Solution. Only the two remaining cash flows matter, discounted back one and two years respectively:
150 ÷ (1.10)¹ = 136.364
200 ÷ (1.10)² = 165.289
V₁ = $301.65.

3. Confirm that valuing the first cash flow plus V₁ gives the same answer as part 1.

Solution. Treat the asset as one dividend plus a resale value:
100 ÷ (1.10)¹ = 90.909
301.653 ÷ (1.10)¹ = 274.23
Total = $365.14, exactly as before. Collapsing the tail of the stream into a single terminal value changes nothing, provided the terminal value is itself the present value of what follows. This equivalence is what licenses every terminal value calculation later in the reading.

Three definitions of return

Three definitions of the cash flow stream dominate practice: dividends, free cash flow and residual income. Each one implies a different model, and each one suits a different valuation situation.

Figure 1: From operating cash to the shareholder

Each step down the cascade removes a claim. A minority shareholder can only reach the bottom box, which is why the DDM takes dividend policy as given rather than as something to be redirected.

Dividends. An investor who buys a share and holds it receives cash only as dividends. The obvious objection is that this ignores earnings that are retained rather than distributed. It does not: retained earnings fund the growth that produces larger dividends later, and the DDM captures all future dividends, so the reinvestment is already inside the model. Because dividends are far smoother than earnings, DDM values are less sensitive to short-run noise than values from the other two definitions, and analysts often read a DDM value as an estimate of long-run intrinsic value.

A company may pay no dividend for opposite reasons. It may be unprofitable with no cash to distribute, or it may be highly profitable and reinvesting everything into growth opportunities that beat the cost of capital. Firms of the second kind typically initiate a dividend as they mature and the supply of attractive projects thins out. Mature, profitable companies tend to pay dividends and are reluctant to cut them.

Dividend practice varies across markets and over time. Research has found that a lower share of companies listed in US markets pay dividends than in most other markets, although the US sample may contain a disproportionate share of smaller and younger companies, which pay dividends less often. The fraction of companies paying cash dividends has declined across most developed markets, including the United States, Canada, the European Union, the United Kingdom and Japan. The explanations offered are a growth in the number of small listed companies with low profitability and high growth potential, a reduced willingness to pay dividends even after controlling for profitability and growth, and the rise of share repurchases as a substitute channel for distributing cash.

The DDM can in theory be applied to a company that pays no dividend, and later in this reading it is. In practice it usually is not, because forecasting when a dividend will begin, and how large it will be, without any dividend record or stated policy to anchor the estimate, is guesswork. There is also a question of perspective. A small shareholder cannot influence the timing or size of distributions, so taking dividend policy as given is exactly the right stance for that investor. If dividends bear no sensible relation to the value the business creates, a DDM applied to that share will be unreliable.

Dividends and the DDM suit a valuation when the company pays dividends, so a record exists to analyse; when the board has a dividend policy that is consistent and comprehensible in relation to profitability; and when the investor takes a non-control perspective. Companies that fit are often seasoned and profitable but outside the fastest-growing parts of the economy.

Free cash flow. Cash flow from operations is the cash a company generates by selling goods and services, and it is the cash flow that speaks most directly to the economics of the business. But for a going concern, not all of it is available: assets wear out, become obsolete, or have to be added to support growth. Free cash flow to the firm (FCFF) is cash flow from operations less capital expenditure, including investment in working capital. It is the amount that bondholders and shareholders together could withdraw without damaging the business. Equity value on this basis is the present value of expected FCFF, the value of the whole company, less the market value of debt.

Free cash flow to equity (FCFE) goes one step further and subtracts all payments to debtholders, interest and principal repayment net of new issuance. Debt has a prior claim, so the money paid on it is not available to common shareholders. FCFF is a pre-debt concept and FCFE is a post-debt concept. FCFE is the baseline free cash flow model for equity, but FCFF is often easier to apply when leverage is expected to change substantially over the forecast period.

Accounting definitions do not always line up with these concepts. Under US generally accepted accounting principles, net interest payments, a financing item, sit inside cash flow from operating activities, so careful analysts add after-tax net interest back when computing cash flow from operations. Under International Accounting Standards, companies may or may not classify interest expense as operating.

Free cash flow can be computed for any company, dividend-paying or not, which is a large practical advantage. FCFE also measures what a company could afford to distribute, so it is informative even when actual dividends differ sharply from it. Because FCFE is cash that a controlling owner could redeploy outside the company, free cash flow valuation is the natural choice for an investor taking a control perspective. Even a minority holder may want that perspective if an acquisition is plausible, since the share price should then reflect what an acquirer would pay.

Free cash flow has its own failure mode. A company with intense capital demands, such as a retailer building outlets far from saturating its market, can have negative expected free cash flow for many years even while it is highly profitable. The present value of a run of negative numbers is negative, and capturing the point where free cash flow turns positive may require a forecast horizon so long that the estimates carry little confidence. Free cash flow models fit best when the company pays no dividend; when it does pay but dividends differ substantially from FCFE; when free cash flows line up with profitability inside a horizon the analyst is comfortable forecasting; and when the investor takes a control perspective.

Residual income. Residual income for a period is the earnings of that period in excess of the required return on the equity capital invested at the start of the period. Suppose shareholders have $200 million invested and the required return on the stock is 8%. The required return is an opportunity cost: the highest expected return available on equally risky alternatives, and therefore what investors give up by holding this stock. If the company earns $18 million during the year, the first 0.08 × $200 million = $16 million merely matches what could have been earned elsewhere. Only the remaining $18 million − $16 million = $2 million is value added.

The residual income model states that value per share is book value per share plus the present value of expected future residual earnings. Unlike the dividend and free cash flow models, it brings a stock variable, book value, into a present value expression. It is nevertheless a restatement of the DDM in company-level terms, because dividends, earnings and book value are tied together:

Book value of equity at t = book value of equity at (t − 1) + earnings for the period + minus dividends paid at t. This holds provided every item that reaches the balance sheet has first passed through the income statement, apart from transactions with owners.

Residual income can always be computed, so the model works for dividend payers and non-payers alike. It is often chosen for companies whose expected free cash flows are negative inside a comfortable forecast horizon, because it recognises value earlier than a free cash flow valuation does and therefore tends to produce higher estimates in those cases. Its focus on profitability net of opportunity cost is attractive, and executive compensation schemes are sometimes built on the same idea. The cost is that applying it well demands detailed knowledge of accrual accounting, and where management has used accounting discretion to obscure economic performance, the model becomes error-prone. Where a DDM is suitable, analysts often prefer it simply because it is simpler. Residual income suits a company that pays no dividend, as an alternative to free cash flow, or one whose expected free cash flows are negative within the forecast horizon.

Claims that one definition is inherently superior tend to track fashion rather than evidence. The workable position is that a given valuation problem suits one model better than the others, that an analyst tends to build more skill in one of them, and that the quality of the forecasts fed into any of them usually decides whether the work is useful.

Example 2 · Worked

You are director of equity research at a brokerage firm and have the final word on model choice. An analyst covering consumer non-cyclicals proposes a dividend discount model for two companies: Anheuser-Busch InBev SA/NV, referred to here as AB InBev (Euronext: ABI, NYSE: BUD), and Diageo plc (LSE: DGE, NYSE: DEO). Fifteen years of history follow. Earnings and dividends are both stated per share, and the payout column divides the second figure by the first.

BUD and DEO: the earnings and dividends record

EQ 2 – Discounted Dividend Valuation — table 1
Year	BUD EPS ($)	BUD DPS ($)	BUD payout (%)	DEO EPS (pence)	DEO DPS (pence)	DEO payout (%)
2018	2.17	2.05	94	121.1	65.3	54
2017	3.98	4.33	109	105.5	62.2	59
2016	0.71	3.85	542	89.1	59.2	66
2015	4.96	3.95	80	94.6	56.4	60
2014	5.54	3.52	64	89.3	51.7	58
2013	8.72	2.83	32	97.4	47.4	49
2012	4.40	2.24	51	75.8	43.5	57
2011	3.58	1.55	43	74.1	40.4	55
2010	2.50	1.07	43	64.3	38.1	59
2009	2.90	0.55	19	65.0	36.1	56
2008	1.93	0.35	18	58.9	34.4	58
2007	3.06	3.67	120	55.0	32.7	59
2006	1.81	0.95	52	66.9	31.1	46
2005	1.17	0.57	49	45.2	29.6	65
2004	NA	NA	–	48.2	27.6	57

Source: company websites and filings.

1. Is a dividend discount model an appropriate choice for AB InBev?

Solution. On this record, no. Dividends per share have run from $0.35 to $4.33, and the annual payout ratio from 18% to 542%. The swings reflect a history of growth by major merger and acquisition: AB InBev was created when the Belgian company InBev acquired Anheuser-Busch in 2008, InBev itself came from a merger of Interbrew of Belgium with AmBev of Brazil, and in 2016 the group bought SABMiller. Nothing in that record suggests a stable link between dividends and earnings, and if dividends do not adjust to profitability, a DDM cannot be trusted. A company-level definition of cash flow is the better route.

Valuation is forward-looking, so an analyst would also check disclosures about future policy. Carrying the extra debt from the SABMiller purchase, AB InBev cut its dividend in 2018 and told shareholders in its annual report that paying down debt is the priority and could restrict what it can distribute.

2. Is a dividend discount model an appropriate choice for Diageo?

Solution. Yes. Diageo earnings show a fairly steady long-term upward trend and dividends have tracked that growth. Across the full period, earnings per share and dividends per share compounded at similar rates of 6.8% and 6.3% a year. In most years the payout ratio sat between 50% and 60%. Dividends are paid and they bear a comprehensible, consistent relation to earnings, which is precisely the condition a DDM needs.

On the forward-looking check, Diageo disclosed in its 2018 annual report that it continues to target dividend cover, defined as EPS divided by DPS, of between 1.8 times and 2.2 times, which corresponds to a payout ratio between 45% and 56%.

The DDM is the oldest and simplest present value approach to common stock. In a survey of CFA Institute members with equity analysis responsibilities, close to 80% of respondents reported using a discounted cash flow approach, and about 35.1% of those using discounted cash flow employed a dividend discount model. Its place in academic and practitioner research is as secure as its place on a desk.

Throughout what follows, a period dividend is assumed to be paid in a single amount at the end of the period.

One holding period

An investor who buys a share and sells it after one year receives two cash flows: the dividend and the sale proceeds. Value today is the present value of both.

D₁ is the expected dividend for Year 1, P₁ the expected price at t = 1, and r the required return on the stock.

Example 3 · Worked

You expect Carrefour SA (EN Paris: CA) to pay a dividend of €0.46 next year and expect the share price in one year to be €23.00. The required return on Carrefour equity is 8%.

Estimate the value of Carrefour stock.

Solution. Both cash flows arrive at t = 1, so discount their sum by one year:
(0.46 + 23.00) ÷ (1 + 0.08)¹ = 23.46 ÷ 1.08 = €21.72.

Several holding periods

Extending to two years simply adds a second dividend term, and the price now arrives at t = 2. For any finite horizon of n years, value is the present value of the n expected dividends plus the present value of the expected price at the end.

The finite-horizon DDM. P_n is the expected terminal share price.

This is not a theoretical curiosity. It is one of the two practical routes to a DDM valuation: forecast dividends individually over a horizon of, typically, two to five years, then estimate a terminal price by some separate method.

Example 4 · Worked

Over the next five years a stock is expected to pay annual dividends of $2.00, $2.10, $2.20, $3.50 and $3.75. The share price in five years is expected to be $40.00. The required return on equity is 10%.

What is the value of the stock?

Solution. Discount each dividend at 10% and add the discounted terminal price:
2.00 ÷ (1.10)¹ = 1.818
2.10 ÷ (1.10)² = 1.736
2.20 ÷ (1.10)³ = 1.653
3.50 ÷ (1.10)⁴ = 2.391
3.75 ÷ (1.10)⁵ = 2.328
40.00 ÷ (1.10)⁵ = 24.837

The five dividends have a combined present value of $9.926 and the terminal price contributes $24.837, for a total of $34.76. Note the split: more than two-thirds of the estimate comes from the terminal price. That dominance is a permanent feature of finite-horizon DDMs and it is the reason terminal value estimation deserves the care given to it later in this reading.

The general form

Lengthen the horizon by one year and one more dividend term appears while the price term moves out. Push the horizon out without limit and the price term vanishes into the distance, leaving only dividends:

The general dividend discount model, first written down by Williams in 1938. The series continues without end.

Even an investor with a short horizon is exposed to the whole stream. Value today depends directly on the dividends received before the sale and indirectly on every dividend after it, because those later dividends are what the buyer will pay for.

The general form also presents an impossible forecasting task, since no one can forecast an unbounded number of individual dividends. Two families of simplification exist. The first assigns the whole future stream to a stylised growth pattern: constant growth forever, which gives the Gordon growth model; two distinct stages of growth, which gives the two-stage model and the H-model; or three distinct stages, which gives the three-stage model. The second forecasts a finite number of dividends individually, typically three to ten years out, using pro forma financial statement analysis, and then either assigns the remaining dividends to a stylised pattern or forecasts the share price at the terminal point directly, for instance by applying a multiple to forecast book value or earnings per share. The horizon chosen often depends on how predictable the company earnings are, sometimes described as their visibility.

Spreadsheets make the second family practical and are convenient in every case. Whichever definition of cash flow an analyst uses, one of these two approaches is almost always behind the forecast. The real difficulty is choosing a pattern that matches the company and then supplying good inputs to it.

The simplest possible pattern is constant growth: every dividend exceeds the one before it by the same percentage, so that D_t = D_t−1(1 + g). If the most recent dividend D₀ was €10 and growth is forecast at 5%, then D₁ = €10 × 1.05 = €10.5, and the dividend five years out is D₅ = €10 × (1.05)⁵ = €10 × 1.276282 = €12.76. In general D_t = D₀(1 + g)^t.

Substituting that into the general DDM produces a geometric series in which each term equals the previous one multiplied by (1 + g)/(1 + r). Summing an infinite geometric progression whose first term is a and whose ratio is m, with the absolute value of m below one, gives a/(1 − m). Setting a = D₁/(1 + r) and m = (1 + g)/(1 + r) collapses the whole series into two symbols over a difference:

The Gordon growth model, developed by Gordon and Shapiro (1956) and Gordon (1962). The two forms are identical because D₁ = D₀(1 + g).

The formula requires r greater than g. If r = g, dividends grow exactly as fast as they are discounted and the undiscounted sum is unbounded. If r is below g, dividends outrun the discounting and the sum is again unbounded. Infinite values are not economically meaningful, so constant growth with r at or below g is not a coherent assumption.

A numerical illustration: an annual dividend of €5 has just been paid, so D₀ = €5. Long-term growth is 5% and the required return on equity is 8%. Then V₀ = (€5 × 1.05) ÷ (0.08 − 0.05) = €5.25 ÷ 0.03 = €175. The single most common error on this calculation is putting D₀ in the numerator instead of D₁.

What the model assumes, and when that is defensible

Because the model extends dividends indefinitely, both r and g must be long-term figures rather than next-year figures. And because the denominator is a small difference between two large-ish numbers, values are extremely sensitive to both inputs. Sensitivity analysis is not optional decoration here.

The conditions for any DDM apply: the company pays dividends and dividends relate comprehensibly to profitability. The Gordon growth model adds one more. It fits companies whose earnings are expected to grow at a rate comparable to, or below, the nominal growth rate of the economy. A business growing far faster than the economy will not do so forever, and the model horizon is forever.

To judge that, an estimate of nominal economic growth is needed, usually built from gross domestic product data published by national agencies and by the World Bank. Real GDP growth for a group of developed markets is shown below.

Average annual real GDP growth rates, 1988 to 2017

EQ 2 – Discounted Dividend Valuation — table 2
Country	1988–1997	1998–2007	2008–2017
Australia	3.2%	3.5%	2.6%
Canada	2.1	3.2	1.6
Denmark	2.0	2.0	0.8
France	2.2	2.4	0.8
Germany	2.6	1.7	1.3
Italy	1.9	1.5	−0.5
Japan	2.8	1.0	0.5
Netherlands	3.1	2.8	0.9
Sweden	1.4	3.5	1.6
Switzerland	1.5	2.4	1.4
United Kingdom	2.4	2.9	1.1
United States	3.1	3.1	1.5

Source: OECD. Twelve countries across three ten-year periods.

Nominal GDP growth is then the estimated real rate plus expected long-run inflation. Using ten years of history through 2018, one estimate of underlying real growth for Canada is 1.6%; adding the Bank of Canada inflation target of 2% gives a nominal rate of 1.6% + 2% = 3.6%. Two qualifications matter. Listed companies are always less than the whole corporate sector, so the growth of the listed universe can drift away from nominal GDP growth over long periods, and particular subsectors can sustain differentials for a long time. Even so, an earnings growth rate far above nominal GDP growth cannot continue in perpetuity. When an analyst forecasts one, the correct response is to move to a multistage model whose final stage carries a plausible mature rate, not to force the Gordon growth model to accept the high number.

Example 5 · Worked

Joel Williams follows Sonoco Products Company (NYSE: SON), which makes paper and plastic packaging for consumer and industrial customers. Sonoco appears to raise its dividend when the level of earnings rises in a way it regards as sustainable, and typically keeps the payout ratio between 40% and 60%. Williams notes the following.

The most recent quarterly dividend, declared 13 February 2019, was $0.41, which is consistent with an annual dividend of 4 × $0.41 = $1.64.
His forecast dividend growth rate is 4.5% a year.
Beta is 0.95. With an equity risk premium of 4.5% and a risk-free rate of 3%, the capital asset pricing model gives a required return on equity of 3.0 + 0.95(4.5) = 7.3%.

1. Calculate the Gordon growth model value for Sonoco stock.

Solution. Grow the current dividend one year and divide by the spread between r and g:
V₀ = ($1.64 × 1.045) ÷ (0.073 − 0.045) = $1.7138 ÷ 0.028 = $61.21.

2. Sonoco trades at $59.55. Is the stock fairly valued, undervalued or overvalued on this estimate?

Solution. The market price is $1.66 below the model value, which is approximately 2.7% less than the estimate of $61.21. Sonoco looks slightly undervalued. A gap of that size is well inside the range of ordinary input error, so the honest reading is that the model does not identify a material mispricing.

Example 6 · Worked

Robert Kim, an analyst for a US domestic equity income fund, is assessing Middlesex Water Company (NASDAQ: MSEX), a listed water utility, in early 2019. Most of the US population is supplied by government entities, but a group of investor-owned utilities also serves the public, and with a market capitalisation of about $880 million MSEX is among the ten largest of them. Its original franchise, the Middlesex System, supplies homes, factories and businesses across a built-up stretch of central New Jersey, while subsidiaries add water supply together with wastewater collection and treatment in southern New Jersey and in Delaware.

Return on equity has averaged 8.5% over the past ten years with little variation, and profit margins are above industry averages. When the credit rating was upgraded in 2015, Standard & Poor’s cited improving management of regulatory risk expected to produce less volatile profitability, moderately better cash flow measures, and the ability to earn consistently closer to authorised returns. Because most revenue comes from the regulated supply of an essential staple to a stable population, Kim is confident forecasting earnings and dividend growth. The payout ratio policy appears to average between 60% and 70%. Other facts:

Per-share dividends for 2018, D₀, were $0.911.
Kim forecasts long-term earnings growth of 4.5% a year.
Raw beta is 0.70 and adjusted beta is 0.80, based on 60 monthly returns, but the R² associated with beta is under 20%.
Pretax cost of debt is estimated at 4.8%, based on a Standard & Poor’s issuer rating of A and the corporate yield curve.
Kim estimates the required return on equity at 6.8%.
The current market price is $43.20.

1. Calculate the Gordon growth model value using Kim’s required return estimate.

Solution. V₀ = $0.911(1.045) ÷ (0.068 − 0.045) = $41.39.

2. Does MSEX appear overvalued, fairly valued or undervalued on that estimate?

Solution. The estimate of $41.39 differs from the market price of $43.20 by less than 5%, so MSEX appears fairly valued.

3. Justify the selection of the Gordon growth model for MSEX.

Solution. The assumption of stable perpetual growth is realistic here for four reasons. Profitability is stable, as the return on equity record shows, because demand is predictable and prices are regulated. Dividends bear a comprehensible and consistent relationship to earnings, evidenced by the payout ratio policy. The forecast growth rate of 4.5% a year is both attainable and reasonable against long-term nominal annual GDP growth for the United States of approximately 4.3% over the twenty years from 1998 to 2018, even though recent earnings growth has been faster. And the forecast contains no period of very high or very low growth that would demand a multistage treatment.

4. Calculate the CAPM required return assuming beta reverts to the mean, with an equity risk premium of 4.5% and a risk-free rate of 3%.

Solution. Reversion to a mean of 1.0 is the characteristic of adjusted historical beta, so use 0.80:
3% + 0.80(4.5%) = 6.6%.

5. Value the shares twice more: (A) with the required return obtained in question 4, and (B) with a required return built by adding a 2.5% premium to the cost of debt.

Solution.
A. At r = 6.6%: V₀ = $0.911(1.045) ÷ (0.066 − 0.045) = $45.33.
B. Bond yield plus risk premium gives r = 4.8% + 2.5% = 7.3%. Then V₀ = $0.911(1.045) ÷ (0.073 − 0.045) = $34.00.

6. Evaluate how uncertainty in the required return affects the conclusion in question 2.

Solution. With the adjusted-beta CAPM estimate, the value of $45.33 exceeds the market price but by only around 5%, so MSEX still looks fairly valued, and in any case beta explains less than 20% of the variation in MSEX returns. With the bond-yield-plus-risk-premium estimate, $34.00 is more than 20% below the price of $43.20, which would make the stock significantly overvalued. But no specific evidence supports the particular 2.5% premium chosen. Given that the two better-supported estimates both point to fair value, the reasonable conclusion is that MSEX is relatively fairly valued, and that the required return estimate, not the growth estimate, is where the uncertainty sits.

How sensitive is the answer

Because the denominator is a difference, small movements in either input move the value a long way. The disciplined response is to lay out a grid rather than quote a point estimate.

Example 7 · Worked

The MSEX value of $41.39 rested on D₀ of $0.911, growth of 4.5% and a required return of 6.8%. Suppose each of r and g could be wrong by 25 basis points in either direction.

Estimated price given uncertain inputs

EQ 2 – Discounted Dividend Valuation — table 3
Required return	g = 4.25%	g = 4.50%	g = 4.75%
r = 6.55%	$41.29	$46.44	$53.02
r = 6.80%	$37.24	$41.39	$46.55
r = 7.05%	$33.92	$37.33	$41.49

What does the grid say about confidence in the $41.39 estimate?

Solution. Read the corners first. The widest spread between r and g, r = 7.05% against g = 4.25%, produces the smallest value at $33.92. The narrowest spread, r = 6.55% against g = 4.75%, produces the largest at $53.02. As the spread closes toward zero the value rises without limit. The largest figure in the grid is more than 55% above the smallest, and the inputs never moved by more than a quarter of a percentage point.

Six of the nine cells sit below the market price of $43.20, so two-thirds of the grid points to overvaluation. Five of the nine sit within 10% of the market price, the four exceptions being $53.02, $37.33, $37.24 and $33.92. Taken together, the grid supports a conclusion of relatively fair value or slight overvaluation. The best single estimate remains $41.39, but the point estimate carries far less information than the grid does.

Utilities are a traditional showcase for this model because supplying an essential service under regulation produces stable numbers. Industry membership alone is not enough, though. A utility pursuing growth by acquisition can have earnings and dividend growth quite unlike its peers, and many US utility holding companies own large unregulated subsidiaries, so the picture of steady, slow growth does not always hold.

The model is also applied to broad equity indexes, especially in developed markets, where listed issues represent a large fraction of the corporate sector, index growth therefore approximates average economic growth, and a sustainable trend growth rate can plausibly be identified. Analysts have used it both to judge whether a market is fairly valued and to back out the equity risk premium implied by the current market level.

The constant growth formula is more flexible than its usual presentation suggests, because g is not required to be positive.

Fixed-rate perpetual preferred stock

Fixed-rate perpetual preferred stock pays a specified dividend, ranks ahead of common stock in its claim on earnings, and has no maturity date. Financial institutions, banks in particular, have used it to raise permanent equity capital without diluting common shareholders. Most issues are callable by the issuer after some period, so a full valuation has to price the call option. The non-callable form is straightforward: the dividend is level, so g = 0 and the Gordon growth model reduces to a perpetuity.

Value of non-callable fixed-rate perpetual preferred stock. In this context r is often called a capitalisation rate.

Example 8 · Worked

Kansas City Southern Preferred 4% (KSU-P), issued 2 January 1963, has a par value of $25 per share. The required return on the security is estimated at 5.5%.

Estimate the value of the issue.

Solution. The annual dividend is 0.04($25) = $1.00. Capitalising it at the required return:
V₀ = D ÷ r = 1.00 ÷ 0.055 = $18.18.
The share trades below par because the required return of 5.5% exceeds the 4% coupon rate on par value.

Negative growth

A level dividend is the boundary case of zero growth. Dividends can also be expected to shrink, and the same formula handles that once the negative sign is carried through the subtraction correctly.

Example 9 · Worked

Afton Mines is profitable and is expected to pay a $4.25 dividend next year. Because its mining properties are being depleted, the best estimate is that dividends decline forever at 4% a year. The required return on Afton stock is 9%.

What is the value of Afton shares?

Solution. The $4.25 is already the next-year dividend, so use the D₁ form directly. The growth rate is −4%, so the denominator becomes 0.09 − (−0.04) = 0.13:
V₀ = 4.25 ÷ 0.13 = $32.69.
Negative growth widens the denominator rather than narrowing it, which is why a declining dividend stream still carries a finite and quite ordinary-looking value.

Dividend growth, earnings growth and value appreciation

Constant growth ties several quantities together tightly. Start from V₀ = D₁/(r − g) and multiply both sides by (1 + g). The left side becomes V₀(1 + g) and the right side becomes D₁(1 + g)/(r − g), which is D₂/(r − g), which is V₁. So value grows at exactly g, holding r constant. With a constant payout ratio, earnings grow at g as well, because dividends are a fixed proportion of them.

So g in the Gordon growth model is the rate of value appreciation, sometimes called the capital gains yield. Some texts describe it as the rate of price appreciation. That is true only if prices are efficient, so that price equals value. Where a stock is mispriced, the realised rate of capital appreciation depends on the nature of the mispricing and on how quickly, if at all, it is corrected.

A second consequence is that the components of total return stay constant through time, again assuming price tracks value. The dividend yield D₁/P₀ does not move, because the numerator and the denominator grow at the same rate. Take a share quoted at €50.00 whose expected payment over the coming twelve months is €1, giving a forward dividend yield of 2%, with g put at 5.50% a year. Every component then stands still: 2% from the dividend, 5.50% from appreciation, and 7.50% in total, today and at any date you care to pick in the future.

Cash dividends are not the only way to hand cash to shareholders. Share repurchases, also called buybacks, have grown steadily in importance in developed markets, and in the United States more than half of dividend-paying companies also repurchase regularly. Anyone using a DDM has to know what that does to the model.

Repurchases differ from cash dividends in four ways that matter for forecasting.

A repurchase reduces the number of shares outstanding, all else equal. Shareholders who sell see their relative ownership fall against those who do not.
Companies with established cash dividends are reluctant to cut or omit them, but companies generally do not regard themselves as committed to any particular level of repurchase.
Cash dividends are more predictable both in amount and in timing. US evidence suggests that for companies with active repurchase programmes the amount repurchased over two-year intervals does bear a relationship to earnings, but companies appear opportunistic about exactly when they buy. Repurchases are therefore harder to forecast than the dividends of a company with an identifiable policy.
As a baseline, repurchases carried out at market prices are neutral for the wealth of continuing shareholders.

An analyst could model repurchases directly by forecasting total earnings, total distributions through both channels, and the share count. Far less practical experience exists with such models than with DDMs. The workable alternative is to keep the DDM focused on cash dividends but to allow expected repurchases to feed through into the per-share growth rate of dividends. Applied that way, the DDM remains a valid approach to valuing a company that repurchases its own shares.

Reading a growth rate out of the market price

Because value is so sensitive to g, a gap between a model value and the market price can often be recast as a disagreement about growth. Given the price, the expected next dividend and an estimate of the required return, the Gordon growth model can be inverted to give the growth rate the market appears to be assuming. The analyst then judges whether that implied rate is reasonable, demanding or unambitious given what is known about the business. It is a second, and often more intuitive, perspective on whether a stock is fairly valued. The same inversion can be performed on other DDMs, though rarely in closed form.

Example 10 · Worked

Consider a share whose beta is 1.1. Government paper in this market yields 5.6%, and the premium demanded for equity risk is taken as 6%. Dividends stand at $2.00 per share today and are forecast to compound at 5% without end. The share changes hands at $40.

1. Estimate the value of the stock.

Solution. The required rate of return is 5.6% + 1.1(6%) = 12.2%. Then:
V₀ = 2.00(1.05) ÷ (0.122 − 0.05) = 2.10 ÷ 0.072 = $29.17.

2. What constant dividend growth rate would justify the market price of $40?

Solution. The model value of $29.17 is well below the market price of $40.00, so the market must be assuming growth above 5%. Substitute everything except g and solve:
40 = 2.00(1 + g) ÷ (0.122 − g)
4.88 − 40g = 2 + 2g
42g = 2.88
g = 0.0686
An expected dividend growth rate of 6.86% is needed to justify the $40 price. The analytical question then becomes concrete: is 6.86% a plausible perpetual growth rate for this business, given the economy it operates in? That is a far easier question to argue about than whether $40 is the right price.

A share price can be split into two economically distinct pieces: what the business is worth if it never reinvests another currency unit, and what its future investment opportunities add. The second piece is the present value of growth opportunities, or PVGO, also called the value of growth. More precisely, PVGO is the forecast total net present value of the projects the company will undertake.

Growth is not automatically good

Whether earnings growth helps shareholders depends entirely on the return earned on the reinvested money relative to the opportunity cost of funds. Take a company with a required return on equity of 10% that has earned €1 per share and is deciding whether to pay it out or reinvest it for a year. If it reinvests at 10%, the present value of the investment is €1.10/1.10 = €1.00, exactly equal to its cost, so the net present value of reinvesting is zero. If instead it can reinvest at 25% for the year, the per-share net present value of the opportunity is €1.25/1.10 − €1, approximately €0.14.

The trap is reinvestment at a positive return below 10%. That raises earnings per share, which looks like progress, while destroying value. Shareholder wealth rises only when reinvested earnings earn more than the opportunity cost of funds, which is the condition that return on equity exceeds r, with return on equity here computed on the market value of equity rather than book value. So investors have a direct incentive to assess how many profitable projects a company will find, and how profitable they will be.

A company with no positive net present value projects available is a no-growth company. It should distribute all its earnings, because retained earnings cannot be deployed profitably. Its earnings then stay flat in perpetuity, assuming a constant return on equity: earnings equal return on equity multiplied by equity, and equity does not grow because nothing is retained. The right earnings measure for this calculation is E₁, the t = 1 figure, since the assets needed to produce it are already in place; E₁ is the constant level of earnings, or the average level if return on equity varies around its mean. The no-growth value per share is therefore E₁/r, a perpetuity capitalised at the required return. It can equally be read as the per-share value of assets already in place. Total value is that plus the value of growth:

Any share splits into the value of assets in place and the value of future investment.

If price equals value, then P₀ less E₁/r is the market estimate of the value of growth. Return to MSEX and suppose it would have average earnings per share of $1.52 if it distributed everything. With a required return of 6.8% and a price of $43.20:

$43.20 = ($1.52 ÷ 0.068) + PVGO = $22.42 + PVGO, so PVGO = $43.20 − $22.42 = $20.78. On that basis 48% of the market value of MSEX ($20.78/$43.20 = 0.48) is attributable to growth. Two points on the arithmetic: dividing $1.52 by exactly 0.068 gives $22.35, so the source figure of $22.42 corresponds to a required return carried to more decimal places than the rounded 6.8%; and the subtraction and the percentage are internally consistent with $22.42.

Figure 2: Splitting a market price into assets in place and growth

Each bar is scaled to the whole of that company’s own share price, so the two are comparable in proportion and not in level. A high PVGO share says the market is paying mainly for projects not yet undertaken, which carries different risk from paying for assets already generating cash.

Applying the same decomposition across three very different businesses shows how much the split can vary.

Estimated PVGO as a percentage of price, early 2019

EQ 2 – Discounted Dividend Valuation — table 4
Company	Beta	r	E₁	Price	E₁/r	PVGO	PVGO/Price
Alphabet, Inc.	1.16	8.2%	$47.49	$1,236.34	$579.14	$657.20	53.16%
McDonald’s Corporation	0.52	5.3%	$8.23	$194.12	$155.28	$38.84	20.01%
Macy’s, Inc.	0.45	5.0%	$3.09	$25.11	$61.80	($36.69)	n.m.

Earnings estimates from NASDAQ, betas from S&P equity research. Required returns use the capital asset pricing model with a risk-free rate of 3.0% and an equity risk premium of 4.5%.

Growth accounts for about 53% of the market value of Alphabet and a much smaller share of the other two. The negative PVGO for the retailer admits several readings. It may reflect the pressure traditional retailers face from online competition. It may equally mean that the no-growth value has been overstated, either because the earnings estimate is too high or because the required return estimate is too low. A negative PVGO is a signal to re-examine inputs before drawing a conclusion about the business.

What determines PVGO? First, the set of investment options a company holds, which is the substance behind the word opportunities. Second, the flexibility to change course as circumstances change: the option to delay a start, to scale a project up or down, or to abandon it. That second element is the value of the real options the company holds. Businesses with good opportunities, high managerial flexibility, or both, should carry higher PVGO than businesses without those advantages.

Restating the split as a price-to-earnings ratio

Dividing the decomposition through by forecast earnings turns it into the most familiar valuation ratio there is.

The leading P/E splits into a no-growth component and a growth component.

The first term, 1/r, is the P/E a no-growth company should carry. The second is the part of the multiple attributable to growth opportunities. For MSEX the actual P/E is $43.20/$1.52 = 28.4. The no-growth P/E is 1/0.068 = 14.7, the multiple the company should trade at with no growth prospects at all. The growth component is $20.78/$1.52 = 13.67. The distinction is worth making because the value of growth and the value of assets in place do not carry the same risk, which is exactly what reading PVGO as a bundle of real options implies.

Justified leading and trailing P/Es

The Gordon growth model can also be turned into an expression for P/E in terms of fundamentals. This has two uses. Fed with forecast inputs it produces a justified, or fundamental, P/E: the multiple that is warranted given the fundamentals, assuming the model applies. Because P/E is so widely understood, stating a view that way often communicates better than quoting a value per share. Used in reverse, it tests whether the earnings growth implied by the current multiple is plausible.

Two versions exist because there are two earnings figures in common use. The leading, or forward, P/E divides the current price by a forecast of the next twelve months of earnings, or sometimes by the next fiscal year figure. The trailing, or current, P/E divides the current price by the last twelve months of earnings. Define b as the retention rate, the fraction of earnings kept in the business, so the payout ratio is (1 − b) = D_t/E_t. Dividing the Gordon growth model by E₁ and by E₀ in turn gives:

Justified leading P/E.

Justified trailing P/E. The two differ by exactly the factor (1 + g).

Example 11 · Worked

Harry Trice wants a justified P/E for the French cosmetics manufacturer L’Oréal SA (EN Paris: OR) using the Gordon growth model. He has assembled the following.

L’Oréal inputs

EQ 2 – Discounted Dividend Valuation — table 5
Item	Value
Market price per share	€242.70
Earnings per share, trailing twelve months	€7.08
Annual dividend at its present level	€3.85
Forecast growth in dividends	4.25%
Yield on risk-free paper	2.0%
Premium demanded for equity risk	5.0%
Beta measured against the CAC index	0.72

1. Calculate the justified trailing and leading P/Es.

Solution. First the required return, from the capital asset pricing model:
r = 2.0% + 0.72(5.0%) = 5.6%.

Then the payout ratio:
(1 − b) = D₀/E₀ = 3.85 ÷ 7.08 = 0.54, carried as 0.5438 in the calculations.

Justified leading P/E = 0.5438 ÷ (0.056 − 0.0425) = 40.28.
Justified trailing P/E = 0.5438(1.0425) ÷ (0.056 − 0.0425) = 42.00.

2. On the trailing comparison, is L’Oréal fairly valued, overvalued or undervalued?

Solution. The actual trailing P/E is €242.70 ÷ €7.08 = 34.3. That is below the justified trailing P/E of 42.0, so L’Oréal appears undervalued.

The same conclusion can be stated as a price. On Trice’s assumptions the Gordon growth model gives 3.85(1.0425) ÷ (0.056 − 0.0425) = €297.31, comfortably above the market price of €242.70. The two statements are the same statement: multiplying the justified trailing P/E of 42.00 by trailing earnings of €7.08 reproduces the same value.

Multistage models can also be turned into P/E expressions, but the complexity is not repaid by the insight. For a multistage model the practical route to a justified leading P/E is simply to divide the model value by first-year expected earnings. In every case the multiple resolves into the same three ingredients: the required return on equity, the expected dividend growth rate or rates, and the payout ratio or ratios. All else equal, a higher expected dividend growth rate supports a higher price.

Assuming one stable growth rate from today into the indefinite future is unrealistic for most companies. Practitioners commonly assume instead that growth passes through three phases.

Growth phase. Markets open up quickly, margins are wide, and earnings per share climb at a pace no business can hold forever, which is what the term supernormal growth points at. Free cash flow to equity is frequently negative, because the company is pouring money into expansion. Given high prospective returns on equity, payout ratios are typically low or zero. As markets mature, or as the unusual returns pull in competitors, earnings growth rates eventually fall.
Transition phase. Growth slows as competition presses on prices and margins, or as sales growth decelerates because the market is saturating. Earnings growth may still be above average but is declining toward the growth rate of the wider economy. Capital requirements typically fall, which often turns free cash flow positive and lifts payout ratios or triggers the initiation of a dividend.
Mature phase. Equilibrium arrives. New investment, taken on average, earns no more than the cost of the capital funding it. The return the business earns converges toward the return investors demand, and the earnings growth rate, the payout ratio and profitability all settle at levels the company can hold indefinitely. Growth at that settled level is the mature growth rate, and this is precisely the phase the Gordon growth model was built to value.

The picture is an approximation, not a law. A company can restart its growth phase by changing strategic focus or business mix, and technological change can transform growth prospects in either direction with startling speed. But the lifecycle picture supplies the intuition behind every multistage discounted cash flow model, and multistage models are a staple of investment firms that value on a discounted cash flow basis.

Figure 3: The three phases of dividend growth

Supernormal growth is competed away, not sustained. The mature rate must be at or below the nominal growth rate of the economy, which is why the solid line finishes below the dashed reference.

Survey evidence backs the practical weight of this picture. Among CFA Institute members who reach for a dividend discount model, the single-stage version is the least popular choice. The reported figures are 55% for a two-stage model, 11% for an H-model, which is itself a two-stage form, and 50% for something with more than two stages. Since analysts frequently run several models side by side, those shares add to well above 100%.

The general two-stage model

Two versions of the two-stage DDM are in common use, and both assume constant growth at a mature rate in Stage 2. In the general version the whole of Stage 1 is a period of abnormal growth, say 15%, and the transition to mature growth, say 7%, is abrupt. In the H-model, treated in the next section, growth declines steadily through Stage 1 until it reaches the mature rate at the end.

The general version rests on the finite-horizon model, with V_n standing in for P_n. The first n dividends grow at the short-term rate g_S, so D_t = D₀(1 + g_S)^t. After time n the rate switches to the long-term rate g_L, and the dividend at time n + 1 is D_n(1 + g_L) = D₀(1 + g_S)ⁿ(1 + g_L). Because the stream from n + 1 onward grows at a constant rate, the Gordon growth model prices it as of time n:

Terminal value at the end of Stage 1, computed with the first Stage 2 dividend in the numerator.

The general two-stage dividend discount model.

Terminal value

The terminal value V_n, also called the continuing value, is the value at the end of the explicit forecast of everything that comes after it. Estimating it accurately is the largest single determinant of whether a multistage DDM is any good, because it usually carries most of the answer. Two alternative approaches to determining it are standard. The first applies the Gordon growth model to the first dividend of the mature phase, as above. The second applies a multiple to a projected fundamental at the terminal date, most often a price-to-earnings multiple applied to forecast earnings per share, or sometimes a multiple of forecast book value per share. Both appear in the worked examples below.

A single discount rate is used across all phases in the examples here. That reflects both a preference for simplicity and the absence of any clear objective basis for varying it by phase. Some analysts do use different discount rates for different growth phases.

Example 12 · Worked

Carl Zeiss Meditec AG (AFX: GR), 65% owned by the Carl Zeiss Group, supplies screening, diagnostic and therapeutic systems for treating vision problems. Reviewing the shares in early 2019 at €80.55, buy-side analyst Hans Mattern forecasts that the current dividend of €0.55 will grow by 9% a year for the next ten years, after which growth falls to 5% and stays there indefinitely. Mattern estimates the required return on equity at 5.88%, from a beta of 0.90 against the DAX, a risk-free rate of 1.2% and an equity risk premium of 5.2%.

1. Calculate the terminal value at t = 10.

Solution. Grow the current dividend for ten years at 9%, then once more at 5% to obtain the first mature-phase dividend, and capitalise:
V₁₀ = 0.55(1.09)¹⁰(1.05) ÷ (0.0588 − 0.05) = 155.358.
Equivalently, D₁₀ = €1.302 and the Period 11 dividend is €1.3671 = €1.302 × 1.05, so V₁₀ = €1.3671 ÷ 0.0088 = €155.36.

2. Value the shares.

Solution. Discount the ten Stage 1 dividends and the terminal value at 5.88%.

Carl Zeiss Meditec AG: dividends and present values

EQ 2 – Discounted Dividend Valuation — table 6
Time	Calculation	D_t or V_t	Present value
1	0.55 × (1.09)¹	€0.600	€0.5662
2	0.55 × (1.09)²	0.653	0.5829
3	0.55 × (1.09)³	0.712	0.6001
4	0.55 × (1.09)⁴	0.776	0.6178
5	0.55 × (1.09)⁵	0.846	0.6360
6	0.55 × (1.09)⁶	0.922	0.6547
7	0.55 × (1.09)⁷	1.005	0.6740
8	0.55 × (1.09)⁸	1.096	0.6938
9	0.55 × (1.09)⁹	1.195	0.7143
10	0.55 × (1.09)¹⁰	1.302	0.7353
10	V₁₀ from the Gordon growth model	155.358	87.7395
Total			€94.2145

The estimated value is €94.21 per share, against a market price of €80.55.

3. How is that value distributed between the two stages?

Solution. The present value of the terminal value is €87.74, which is approximately 93% of the total. The ten explicitly forecast dividends contribute the balance, €94.21 − €87.74 = €6.47. Given how sensitive the Gordon growth model is to its inputs, and given that 93% of this answer runs through one such calculation, the sensible next step is to recompute the value across a range of plausible mature growth rates and quote an interval rather than a single number.

Why the terminal value nearly always dominates

Two forces work together. First, the explicit forecast covers a handful of years while the terminal value covers everything after that, an unbounded stretch of time. Second, the terminal value is itself a Gordon growth calculation, so it divides by a small number, r minus the mature growth rate. In the Meditec case that spread is only 0.88 percentage points, which multiplies the final dividend by more than 113. Discounting by ten years shrinks the result but nothing like enough to offset the multiplication. The practical consequence is that arguing about the Stage 1 dividends is usually a poor use of time, while arguing about the mature growth rate and the required return is where the valuation is actually decided.

The two-stage model earns its place because a temporary lead is a common commercial situation: a patent, a first-mover advantage or some other transient edge lets a company grow abnormally for a few years, after which competition and the growth of the wider economy pull it back. Its main weakness is the mirror image of its main simplification: the jump from abnormal growth to steady-state growth is abrupt, and few businesses change gear that suddenly.

Terminal value from a multiple instead

The second route to a terminal value applies a market multiple to a forecast fundamental. This mixes a dividend discount model for the explicit period with a relative valuation for the tail.

Example 13 · Worked

An analyst is reviewing the Procter & Gamble Company (NYSE: PG) at the start of 2019, when it was selling for $96.47. In the previous year P&G paid a dividend of $2.79, which the analyst expects to grow at 4% a year for four years. At the end of Year 4 the analyst expects the dividend to be 60% of earnings per share and the trailing P/E to be 22. The required return on P&G common stock is 6.5%.

Calculate the per-share value of P&G common stock.

Solution. Build the four dividends, then construct the terminal price from the Year 4 dividend.

The Year 4 dividend is $2.79(1.04)⁴ = $3.2639. Because dividends are then 60% of earnings, EPS₄ = $3.2639 ÷ 0.60 = $5.4398. Applying the trailing P/E of 22.0 gives a Year 4 value of 22.0($5.4398) = $119.6765. Discounted at 6.5% for four years, that is worth $93.0273 today.

Value of Procter & Gamble common stock

EQ 2 – Discounted Dividend Valuation — table 7
Time	Calculation	D_t or V_t	Present value at 6.5%
1	$2.79(1.04)¹	$2.9016	$2.7245
2	$2.79(1.04)²	3.0177	2.6606
3	$2.79(1.04)³	3.1384	2.5981
4	$2.79(1.04)⁴	3.2639	2.5371
4	22 × (3.2639 ÷ 0.60) = 22 × 5.4398	119.6765	93.0273
Total			$103.5476

The four dividends contribute $10.52 and the terminal value contributes $93.03, giving an estimated value of $103.55 per share against a market price of $96.47.

A company that pays no dividend

A share currently paying nothing is not outside the reach of the model. D₀ and D₁ may both be zero and the first payment may be years away, but the present value of the eventual stream can still capture the value of the business. The one genuinely worthless case is a company that pays nothing and will never be able to distribute cash at all. The practical technique is a multistage model whose first-stage dividend is zero.

Example 14 · Worked

A company currently pays no dividend and will not pay one for several years. It is expected to begin paying $1.00 five years from now, growing at 5% a year thereafter. The required rate of return is 11%.

Value the shares.

Solution. The Gordon growth model prices a stream as of one period before its first payment, so the t = 5 dividend gives the value at t = 4:
V₄ = D₅ ÷ (r − g) = 1.00 ÷ (0.11 − 0.05) = $16.67.

Then discount that back four years:
V₀ = 16.67 ÷ (1.11)⁴ = $10.98.
The share is worth $10.98 today even though it pays nothing until Year 5. The mechanical step that traps candidates is the timing: V₄, not V₅, is what D₅ produces.

If a company pays nothing but is highly profitable, an analyst may still be willing to forecast its eventual dividends. For an unprofitable non-payer that forecast is close to impossible, and even for the profitable case it is usually hard to pin down when dividends will start and what policy the board will then adopt. In both situations a free cash flow or residual income model is normally the better tool.

The general two-stage model holds growth constant at an extraordinary rate and then drops it in a single step. The gap can be large: in the Meditec valuation, growth ran at 9% for ten years and then fell to 5% in Year 11 and stayed there. Many businesses decelerate more gently than that. Fuller and Hsia (1984) built a variant in which growth starts high and falls in a straight line across the supernormal period, arriving at the normal rate exactly at the end.

Figure 4: Two ways of getting from high growth to mature growth

Both paths start and finish at the same growth rates over the same horizon. The H-model pays out less in the middle years, which is why it produces the lower value of the two.

The H-model. H is the half-life in years of the high-growth period, so that period lasts 2H years. g_S is the initial short-term growth rate and g_L the normal long-term rate after Year 2H.

The two terms have clean interpretations. The first is the value the dividend stream would have if it grew at g_L forever, which is just the Gordon growth model. The second approximates the extra value that accrues because growth is above g_L during Years 1 through 2H. It follows immediately that a longer supernormal period, meaning a larger H, and a wider gap between g_S and g_L both raise the share value, all else equal.

The intuition behind the second term is worth carrying. Over the supernormal period the average excess growth rate is (g_S − g_L)/2, since it starts at g_S − g_L and falls linearly to zero. Across 2H periods the cumulative excess dividend, measured against what a constant g_L would have paid, is 2H D₀(g_S − g_L)/2, which simplifies to D₀H(g_S − g_L). That is the upward adjustment applied to the first term. Note what is missing: the individual excess dividends are never discounted at their own dates. That omission is what makes the expression an approximation rather than an identity.

When the approximation stops being good enough

Because the excess dividends are lumped together rather than discounted individually, the error grows with the size of the lump and with how far out in time it sits. So the H-model is very close to the exact answer for a short supernormal period and a modest gap between the two growth rates, and drifts further from it as H grows large or as g_S minus g_L widens. When either condition bites, abandon the approximation and discount the dividends one by one. A spreadsheet makes the exact calculation cheap enough that there is little reason to accept a known approximation error.

Apply it to Meditec. Suppose Mattern had forecast growth declining in a straight line from 9% to 5% over the next ten years rather than holding at 9% and then stepping down. An H of 5 corresponds to that ten-year high-growth period. Then:

V₀ = [0.55(1.05) + 0.55(5)(0.09 − 0.05)] ÷ (0.0588 − 0.05) = €78.13, against €94.21 from the general two-stage model. The difference of about €16 is entirely down to the shape of the decline: the H-model gives up nine per cent growth immediately and never gets it back, while the two-stage model enjoys the full rate for a decade.

Example 15 · Worked

An analyst has decided to use the H-model and has gathered the following.

H-model inputs

EQ 2 – Discounted Dividend Valuation — table 8
Item	Value
Share price	€41.70
Current dividend	€1.77
Initial dividend growth rate	7%
Final and perpetual growth rate	4%
Length of the decline	10 years, linear
Required rate of return on equity	8.0%

1. Estimate the per-share value.

Solution. A ten-year decline means H = 5. Substituting:
V₀ = [1.77(1.04) + 1.77(5)(0.07 − 0.04)] ÷ (0.08 − 0.04)
= (1.84 + 0.27) ÷ 0.04 = €52.75.
The two numerator terms are carried at two decimals here, as in the source. Carrying them unrounded gives (1.8408 + 0.2655) ÷ 0.04 = €52.66, so the rounding costs about nine cents on a fifty-euro estimate.

2. What would the shares be worth if normal growth began immediately?

Solution. That is the first term of the H-model on its own:
1.84 ÷ 0.04 = €46.
The faster initial growth therefore adds 0.27 ÷ 0.04 = €6.75, and 46 + 6.75 = €52.75. Splitting the estimate this way is a useful discipline: it shows exactly how much of the valuation depends on the supernormal growth assumption, which is the assumption most likely to be wrong.

3. Do the shares appear fairly valued, overvalued or undervalued?

Solution. The estimate of €52.75 is approximately 26% above the market price of €41.70, so the shares appear undervalued.

Two versions of the three-stage DDM are in use, distinguished entirely by how the middle stage behaves. In the general three-stage model the company passes through three distinct stages and the second-stage growth rate is typically constant: for instance 20% for three years, then 10% for four years, then 5% thereafter. In the second version the middle stage declines linearly to the mature rate, so that Stages 2 and 3 together are exactly an H-model.

Version one: three distinct constant rates

Example 16 · Worked

An analyst covering a technology company estimates that the required return on equity is 9% and that dividends will grow at 14% for the next two years, 12% for the following five years, and 6.75% thereafter. The company pays a dividend of $3.30 per year and the stock trades at $194.98.

1. Estimate the value of the stock.

Solution. Grow the dividend through the two stages, discount each at 9%, and capitalise the first mature-phase dividend at the end of Year 7.

Estimated value using a three-stage DDM

EQ 2 – Discounted Dividend Valuation — table 9
Time	Calculation	D_t or V_t	Present value at 9%
1	3.30(1.14)	$3.7620	$3.4514
2	3.30(1.14)²	4.2887	3.6097
3	3.30(1.14)²(1.12)	4.8033	3.7090
4	3.30(1.14)²(1.12)²	5.3797	3.8111
5	3.30(1.14)²(1.12)³	6.0253	3.9160
6	3.30(1.14)²(1.12)⁴	6.7483	4.0238
7	3.30(1.14)²(1.12)⁵	7.5581	4.1346
7	3.30(1.14)²(1.12)⁵(1.0675) ÷ (0.09 − 0.0675)	$358.5908	196.161
Total			$222.8171

The three-stage model gives a fair price of $222.82, more than 14% above the market price of $194.98.

2. How much of the value comes from the terminal value?

Solution. The present value of the terminal value is $196.16 out of $222.82, or about 88% of the total. That concentration is characteristic rather than exceptional. Seven years of explicitly forecast dividends account for the remaining $26.66.

Version two: an H-model in the middle

In this version dividends grow at a high constant supernormal rate through Stage 1, decline linearly through Stage 2 exactly as in the H-model, and then grow at a sustainable constant rate in Stage 3. The procedure has four steps.

Gather the inputs: the current dividend; the lengths of the three stages and the growth rate expected in each; and an estimate of the required return on equity.
Compute the Stage 1 dividends and sum their present values.
Apply the H-model to Stages 2 and 3 to value them as of the beginning of Stage 2, then discount that value back to today.
Add the results of the second and third steps.

In the first step analysts often go deeper than applying a growth rate to the current dividend, and instead build explicit year-by-year earnings and dividend forecasts for the near term, commonly three, five or ten years.

Example 17 · Worked

Elsie Bouvier is evaluating Rhinestone Energy, a hypothetical diversified energy company active in oil and gas exploration and in natural gas distribution, for a small-cap growth portfolio. Given an aggressive programme of buying producing oil and gas properties, she expects above-average growth for five years. Her facts and forecasts:

The current market price is $56.18 and the current dividend is $0.56.
An initial five-year period of 11% a year earnings and dividend growth.
Mature growth of 6.5% a year, with ten years allowed for the transition.
For the capital asset pricing model, an adjusted beta of 1.2 from two years of weekly observations, an equity risk premium of 4.2%, and a risk-free rate of 3% from long bond yields.
Any security trading within plus or minus 20% of her intrinsic value estimate is treated as inside a fair value range.

1. Estimate the required return on equity, to one decimal place.

Solution. r = 3% + 1.2(4.2%) = 8%.

2. Value the common stock using a three-stage model with a linearly declining growth rate in Stage 2.

Solution. First compute the five Stage 1 dividends and discount them at 8%. Then value Stages 2 and 3 with the H-model as of t = 5, and discount that back five years.

The H-model inputs at t = 5 are D₅ = D₀(1 + g_S)⁵ = 0.56(1.11)⁵ = $0.9436, g_S = 11.0%, g_L = 6.5%, r = 8.0% and H = 5, since the second stage lasts 2H = 10 years. Then:
V₅ = 0.9436(1.065) ÷ (0.08 − 0.065) + 0.9436(5)(0.11 − 0.065) ÷ (0.08 − 0.065)
= 66.9979 + 14.1545 = $81.1524.
The present value of V₅ is $81.1524 ÷ (1.08)⁵ = $55.2310.

Rhinestone Energy: Stage 1 dividends and the H-model value

EQ 2 – Discounted Dividend Valuation — table 10
Time	Item	Calculation	Value	Present value at 8%
1	D₁	0.56(1.11)¹	$0.6216	$0.5756
2	D₂	0.56(1.11)²	0.6900	0.5915
3	D₃	0.56(1.11)³	0.7659	0.6080
4	D₄	0.56(1.11)⁴	0.8501	0.6249
5	D₅	0.56(1.11)⁵	0.9436	0.6422
5	V₅	H-model	$81.1524	55.2310
Total				$58.2731

The total value of Rhinestone Energy is $58.27.

3. What percentage of total value comes from Stage 1, and what percentage from Stages 2 and 3 together?

Solution. The five Stage 1 present values sum to $3.0422, which is $3.0422 ÷ $58.2731 = 5.2% of total value. Stages 2 and 3 together therefore account for 100% − 5.2% = 94.8%, which checks against $55.2310 ÷ $58.2731 = 94.8%.

4. Is the stock undervalued or overvalued from Bouvier’s perspective?

Solution. Her fair value band is $58.27 plus or minus 0.20($58.27), which runs from $58.27 − $11.65 = $46.62 up to $58.27 + $11.65 = $69.92. The market price of $56.18 falls inside that band, so Bouvier would call Rhinestone Energy fairly valued.

5. Some analysts forecast essentially flat earnings and dividends in the second year. Re-estimate the value assuming EPS is flat in Year 2 and 11% growth resumes in Year 3.

Solution. Holding D₂ at the Year 1 level pushes every later Stage 1 dividend down one step, so the Year 5 dividend becomes $0.8501 rather than $0.9436. Recompute the H-model value on that smaller base:
V₅ = 0.8501(1.065) ÷ (0.08 − 0.065) + 0.8501(5)(0.11 − 0.065) ÷ (0.08 − 0.065) = $73.1103.

Rhinestone Energy with no growth in Year 2

EQ 2 – Discounted Dividend Valuation — table 11
Time	Item	Explanation	Value	Present value at 8%
1	D₁	0.56(1.11)¹	$0.6216	$0.5756
2	D₂	No growth in Year 2	0.6216	0.5329
3	D₃	0.56(1.11)²	0.6900	0.5477
4	D₄	0.56(1.11)³	0.7659	0.5629
5	D₅	0.56(1.11)⁴	0.8501	0.5786
5	V₅	H-model	$73.1103	49.7576
Total				$52.5553

The estimated value falls to $52.56. A single flat year, six years before the terminal date, cuts the valuation by nearly $6 because it permanently lowers the base from which the entire terminal value is computed.

The three-stage model with a declining Stage 2 is widely used by firms that value on a DDM basis. One well-known implementation is the model built into the terminals supplied by Bloomberg L.P. to professional investors and analysts, which will produce an estimated value for any stock the user selects. It is a three-stage model with declining growth in Stage 2. It takes earnings estimates for Stage 1 and the cost of capital for the Stage 3 growth rate, and treats the Stage 2 rate as declining linearly between the two. It also estimates the required rate of return and the lengths of the three stages, giving faster-growing companies shorter growth stages and longer transitions, and slower-growing companies the reverse. Holding the combined length of the growth and transition phases fixed at 17 years, the durations for its four growth classifications are 3 years and 14 years for explosive growth, 5 and 12 for high growth, 7 and 10 for average growth, and 9 and 8 for slow or mature growth. Those are four classifications, and in each the two durations sum to 17. An analyst who tailors the stage lengths to the specific company should be able to beat a fixed specification of that kind.

What multistage models do well, and where they fail

Multistage DDMs can accommodate a very wide range of dividend patterns. They make stylised assumptions about growth grounded in a lifecycle view of business: the first stage often carries explicit analyst forecasts for the next two to five years or longer, and the final stage is usually a Gordon growth calculation on a long-run sustainable rate. The H-model smooths the transition inside the first stage; the standard two-stage model switches abruptly in the second period; three-stage models give the transition a stage of its own.

The limitations follow from the same structure. The present value of the terminal stage frequently exceeds three-quarters of total share value, and that terminal value is highly sensitive to the growth and required return assumptions behind it. And technological change can make the whole lifecycle picture a crude representation of what actually happens to a business.

Every model so far imposes a stylised pattern on dividends. Nothing requires that. With a spreadsheet an analyst can specify any dividend path at all, value it, and test how the answer responds to changes in growth and return assumptions. Spreadsheet modeling is what makes the second family of forecasting approaches practical, and it is the only route when the expected dividend path has kinks that no stylised pattern can represent.

Example 18 · Worked

Yang Co. is expected to pay a $21.00 dividend next year. An analyst estimates that the dividend will then decline by 10% a year for the following three years, so the growth rate over that stretch is −10%. In Year 5 Yang is expected to sell assets worth $100 per share, and the Year 5 dividend, which includes a distribution of part of the proceeds, is expected to be $60. In Year 6 the dividend is expected to fall to $40 and to stay at $40 for one further year. Thereafter it is expected to grow by 5% a year. The required rate of return is 12%.

What is the value of one Yang share?

Solution. Lay the dividends out year by year, discount each at 12%, and treat everything from t = 8 onward as a Gordon growth stream valued at t = 7.

Value of Yang Co. stock

EQ 2 – Discounted Dividend Valuation — table 12
Year	Item	Value	Present value at 12%	Explanation
1	D₁	$21.00	$18.75	Given as $21
2	D₂	18.90	15.07	Down a tenth on Year 1
3	D₃	17.01	12.11	Down a tenth on Year 2
4	D₄	15.31	9.73	Down a tenth on Year 3
5	D₅	60.00	34.05	Asset sale year, given as $60
6	D₆	40.00	20.27	Given as $40
7	D₇	40.00	18.09	Held at $40
7	V₇	600.00	271.41	(40.00 × 1.05) ÷ (0.12 − 0.05)
Total			$399.48

The value of one Yang share is $399.48. Note that this dividend path falls, jumps, falls again, plateaus and only then settles into steady growth. No two-stage or three-stage template would describe it, but a spreadsheet handles it without difficulty, and the final row is still an ordinary Gordon growth calculation.

Choosing the mature growth rate

In the example above the terminal value used a mature growth rate of 5%. Where should such a rate come from? Several routes are available.

Use g = (b in the mature phase) × (return on equity in the mature phase). Mature-phase return on equity can itself be estimated in several ways: by applying the DuPont decomposition to forecasts of its components; by setting return on equity equal to r, on the argument that in maturity a company can do no better than earn the opportunity cost of capital; or by setting it equal to the median return on equity for the industry.
Estimate g with other models by relating the mature growth rate to macroeconomic and industry growth projections.

Where the sustainable growth expression is used, the retention ratio b may be set empirically. The Bloomberg model, for instance, has assumed b = 0.55 in the mature phase, equivalent to a payout ratio of 45%, which is a long-run average payout ratio for mature dividend-paying companies in the United States. Analysts also sometimes project the payout ratio for the individual company.

Example 19 · Worked

An analyst is estimating the dividend growth rate to use in the final stage of a multistage dividend discount model. The company’s payout ratio is 25% and its return on equity equals its estimated required return on equity of 9%.

Estimate the sustainable growth rate.

Solution. A payout ratio of 25% means a retention rate of b = 0.75. Then:
g = (b in the mature phase) × (return on equity in the mature phase) = 0.75(9%) = 6.75%.
Setting return on equity equal to r is the middle option listed above, and it embodies the economic content of maturity: new projects earn their cost of capital and no more.

Turning a DDM around to give a required return

Everything so far has taken dividends, growth and a required return as inputs and produced a value. Given a market price and all the inputs except the required return, the same models can be solved for an internal rate of return instead. That figure has been used as an estimate of the required return, although feeding it straight back into a DDM is circular and inappropriate. It is better read as the expected return implied by the market price, an efficient-market expected return. If price does not in fact equal intrinsic value, the expected return has to be adjusted for the extra component of return that arrives when the mispricing corrects.

For some models the inversion is trivial. Rearranging the Gordon growth model gives:

The required return estimate is the forward dividend yield plus the expected dividend growth rate. Technically r here is an internal rate of return.

For a security priced at $10 with an expected dividend of $0.50 and expected growth of 8%, the estimate is 0.50/10 + 0.08 = 13%.

Example 20 · Worked

Bob Inguigiatto, CFA, is developing mean return estimates for a list of stocks ahead of a portfolio optimisation. One of them is NextEra Energy, Inc. (NYSE: NEE). He judges the Gordon growth model appropriate and treats prices as reflecting value. The company paid dividends of $4.44 in 2018 and in February 2019 raised the quarterly dividend from $1.11 to $1.25, implying an annual dividend of $5.00. The current stock price is $169.83. Dividends per share have grown at around 11.0% a year over the past five years, but recent earnings growth has been distorted by non-recurring items, and Inguigiatto settles on 5.50% as his best estimate of long-term earnings and dividend growth.

Estimate the expected rate of return on NextEra Energy.

Solution. Next-year dividend: D₁ = $5.00(1.055) = $5.275. Then:
r = 5.275 ÷ 169.83 + 0.055 = 0.0311 + 0.055.
The two components are a dividend yield of 3.11% and a capital gains yield of 5.50%, and the source reports the expected rate of return as 8.60%. Adding the unrounded components gives 8.61%, so the reported figure is the lower of the two rounding conventions; either way the estimate is close to 8.6%.

Notice what has been assumed. The 11.0% historical dividend growth rate was rejected in favour of 5.50%, and that single choice moves the answer by more than five percentage points. In this inversion, as in the valuation, g does almost all the work.

The H-model can also be inverted in closed form:

Expected rate of return implied by the H-model. When the short-term and long-term growth rates are equal, this collapses to the Gordon growth model expression.

Take a security with a current dividend of $1, a current price of $20, and a short-term growth rate of 10% declining over ten years, so H = 5, to a long-term rate of 6%. Then r = ($1/$20)[(1 + 0.06) + 5(0.10 − 0.06)] + 0.06 = 12.3%.

For multistage and spreadsheet models no single equation exists. The procedure is the same as finding an internal rate of return on any irregular cash flow series: search, by computer or by trial and error, for the discount rate that sets the present value of the expected dividends equal to the current market price.

Example 21 · Worked

An analyst expects the 2019 dividend of Johnson & Johnson (NYSE: JNJ) of $3.60 to grow by 7.0% for six years and then by 5% in perpetuity. A recent price in early 2019 is $136.61.

1. Find a starting approximation for the internal rate of return.

Solution. Use the Gordon growth expression with the long-term growth rate:
r = ($3.60 × 1.07) ÷ $136.61 + 0.05 = 7.8%.
Because growth over the first six years is above the long-term rate of 5%, the true internal rate of return must exceed 7.8%.

2. Value the stream at 8% and at 8.5% and narrow the answer.

Solution. Grow the dividend at 7% for six years, then apply the Gordon growth model to D₇ = $5.6726 to obtain the value at t = 6, and discount everything.

Estimation of required return: Johnson & Johnson

EQ 2 – Discounted Dividend Valuation — table 13
Time	D_t	Present value at r = 8%	Present value at r = 8.5%
1	$3.8520	$3.5667	$3.5502
2	$4.1216	$3.5336	$3.5011
3	$4.4101	$3.5009	$3.4527
4	$4.7188	$3.4685	$3.4050
5	$5.0491	$3.4363	$3.3579
6	$5.4025	$3.4045	$3.3114
7	$5.6726
Subtotal 1 (t = 1 to 6)		$20.91	$20.58
Subtotal 2 (t = 7 onward)		$119.16	$99.34
Total		$140.07	$119.92
Market price		$136.61	$136.61

Subtotal 1 is the present value of the six explicitly forecast dividends. Subtotal 2 is the present value of the terminal value, which at 8% is [5.6726 ÷ (0.08 − 0.05)] ÷ (1.08)⁶ = $119.16.

At 8.0% the estimated value of $140.07 is about 2.5% above the market price, so the internal rate of return exceeds 8%. At 8.5% the value of $119.92 is well below the price, so the internal rate of return is only slightly above 8%. Solving precisely gives an internal rate of return of 8.08%. In a spreadsheet this is a Goal Seek operation: set the total present value cell to the current price by changing the discount rate cell.

The sharp fall in Subtotal 2 between the two discount rates, from $119.16 to $99.34, is worth pausing on. A 50 basis point move in r cuts the terminal value component by about 17%, because it widens the spread over the 5% mature growth rate from 3.0 to 3.5 percentage points.

Forecasting dividends from a full financial model

The most detailed version of spreadsheet modeling forecasts dividends out of a complete pro forma model of the operating and financial position of the company. Income statement and balance sheet are projected together, and the dividend falls out of the payout assumption applied to modelled net income. The example below shows what a highly profitable, fast-growing company can distribute as growth rates and margins are eroded by competition over time.

Example 22 · Worked

An analyst is forecasting dividends for Hoshino Distributors, a fictional company, over five years, on these assumptions.

Sales open at $100 million and then rise by 20%, 15%, 10% and 10% across Years 2 through 5.
Operating profit runs at a fifth of sales for the opening two years, slips to 18% in the third, and settles at 16% for the last two.
Interest is charged at a tenth of the debt balance carried during the year, and tax takes 40% of pre-tax profit.
The payout ratio climbs through the forecast: a fifth of earnings in each of the first two years, then 30%, 40% and 50%.
Whatever is held back joins the equity base one year later, and the asset base is pinned at 80% of that year’s sales throughout.
The opening year carries $40 million of borrowings against $40 million of shareholders’ equity. Borrowings act as the plug, being assets less equity, while equity in any year is the prior year’s figure lifted by that year’s retention.
Four million shares are in issue, and equity investors demand 15%.
At the close of the fifth year the business is expected to be worth ten times its earnings.

Estimate the current value per share.

Solution. Build the pro forma statements first. Each year the balance sheet drives the interest charge, which drives net income, which drives the dividend and the addition to equity, which drives next year balance sheet.

Hoshino Distributors pro forma financial statements (in millions)

EQ 2 – Discounted Dividend Valuation — table 14
Item	Year 1	Year 2	Year 3	Year 4	Year 5
Sales	$100.00	$120.00	$138.00	$151.80	$166.98
EBIT	20.00	24.00	24.84	24.29	26.72
Interest	4.00	4.83	5.35	5.64	6.18
EBT	16.00	19.17	19.49	18.65	20.54
Taxes	6.40	7.67	7.80	7.46	8.22
Net income	9.60	11.50	11.69	11.19	12.32
Dividends	1.92	2.30	3.51	4.48	6.16
Total assets	$80.00	$96.00	$110.40	$121.44	$133.58
Total debt	40.00	48.32	53.52	56.38	61.81
Equity	40.00	47.68	56.88	65.06	71.77

Dividing total dividends by the 4 million shares gives dividends per share, and discounting at 15% gives their present values.

Hoshino Distributors dividends per share and present values

EQ 2 – Discounted Dividend Valuation — table 15
Item	Year 1	Year 2	Year 3	Year 4	Year 5	Total
DPS	$0.480	$0.575	$0.877	$1.120	$1.540	$4.59
Present value	0.417	0.435	0.577	0.640	0.766	2.84

Earnings per share in Year 5 are $12.32 million ÷ 4 million shares = $3.08. At a P/E of 10, the Year 5 market price is $30.80, whose present value at 15% is $15.31. Adding the $2.84 present value of the five dividends gives a total value today of $18.15 per share.

The structure of the answer repeats the pattern seen throughout: the terminal price supplies $15.31 of $18.15, or about 84%, and the five years of explicitly modelled dividends supply the rest.

Several of the examples above have quietly used the relationship that the dividend growth rate equals the earnings retention ratio multiplied by the return on equity. This section explains why that holds and then combines it with the DuPont decomposition of return on equity to give a workable tool for forecasting dividend growth.

The sustainable growth rate is the rate of dividend and earnings growth that a given level of return on equity can support, assuming that the capital structure stays constant over time and that no additional common stock is issued. Its practical value is that it supplies either the stable growth rate for a Gordon growth model valuation or the mature growth rate in a multistage DDM whose terminal value is a Gordon growth calculation.

g is the dividend growth rate, b the earnings retention rate (one minus the payout ratio), and ROE the return on equity.

Strictly, b should be multiplied by the rate of return expected on new investment rather than on existing equity. Analysts commonly treat return on equity as a good enough proxy, but whether it is should be checked case by case.

Example 23 · Worked

A company earns a return on equity of 25% and retains 60% of earnings. In the year just ended it began with shareholders’ equity of $1,000,000, earned $250,000 of net income, and paid dividends of $100,000.

1. What growth rate does the sustainable growth expression predict?

Solution. g = b × ROE = 0.60 × 25% = 15%.

2. Trace the mechanism through one year to show why 15% is right.

Solution. Retaining 60% of $250,000 adds $150,000 to equity, so the next year opens with $1,000,000 + $150,000 = $1,150,000 of shareholders’ equity. No shares are issued, so all of the increase comes from retention.

Earning 25% on that base gives net income of 0.25 × $1,150,000 = $287,500. That is an increase of $287,500 − $250,000 = $37,500, and $37,500 ÷ $250,000 = 0.15, a 15% rise. (The source prints this ratio as 0.15%, which is a slip: the ratio 0.15 is 15 percent, as the dividend calculation that follows confirms.)

Retaining 60% of the new figure keeps 60% × $287,500 = $172,500 and distributes the other 40%, or 40% × $287,500 = $115,000. Dividends therefore rise from $100,000 to $115,000, which is exactly 15%. So long as the company keeps earning 25% on the 60% of earnings it reinvests, dividends keep growing at 15%.

Two implications follow. Higher return on equity means faster dividend growth, all else constant, and that relationship appears to be reliable. Lower retention means slower dividend growth, again all else constant, a relationship that has been called the dividend displacement of earnings. But all else is often not equal. The return on reinvested earnings may vary with the amount invested, and companies whose growth prospects change may change their dividend policy at the same time. Research has also found that dividend-paying companies had higher future growth rates over the period studied, which counsels caution about assuming that dividends displace earnings mechanically.

Why define sustainable in terms of internally generated funds? Because external equity is considerably more expensive than reinvested earnings, not least because of investment banker fees on secondary issues, and continuous new share issuance is not a practical funding route for most companies. Growth of capital through new debt issuance can be sustained for long periods, and indeed a company managing to a target ratio of debt to total capital must issue debt as equity grows through retention, simply to hold the ratio. Finally, actual retention ratios vary from year to year, for instance because earnings contain transitory components that management prefers not to reflect in the dividend. Actual dividend growth will therefore stray from the predicted rate even when the input estimates are unbiased. The expression is best used as a simple approximation of the average rate at which dividends can grow over a long horizon.

Decomposing return on equity

Return on equity is net income divided by shareholders’ equity: a company with a return on equity of 15% generates $15 of net income for every $100 of equity invested. It can be broken into components in stages. First into return on assets and the equity multiplier, so that a company can raise return on equity either by improving return on assets or by using more leverage, provided it can borrow at a rate below what it earns on assets. Then return on assets splits into profit margin and asset turnover:

The three-way DuPont decomposition: profit margin, asset turnover and the equity multiplier.

A higher profit margin raises return on equity. A higher asset turnover, which measures efficiency, does the same; a turnover of one means the company generates $1 of sales for every $1 of assets. The final term, the equity multiplier, measures leverage. A five-way decomposition exists, but the three-way version carries everything needed to understand the growth rate. Combining it with the sustainable growth expression gives the PRAT model:

Four drivers, in the order that spells the acronym: margin, retention, turnover and leverage.

Figure 5: The PRAT decomposition of sustainable growth

Profit margin and asset turnover together determine return on assets, which is an operating result. Retention and leverage are financial policy choices. Sustainable growth therefore rests on operating performance and on two decisions the board controls.

Read the diagram from the bottom up. Profit margin and asset turnover fix return on assets, while retention and financial leverage reflect financial policy. So the dividend growth rate can be understood as determined by the return on assets of the business plus two policy choices. Analysts use the expanded expression to forecast the mature-phase growth rate.

One technical qualification. Both the sustainable growth expression and its DuPont expansion hold exactly only when return on equity is computed on beginning-of-period shareholders’ equity, as in the worked example above, which assumes retained earnings are not available for reinvestment until the period ends. Analysts and financial databases more often use average total assets, and DuPont analysis is commonly performed on that basis in practice.

Example 24 · Worked

Baggai Enterprises, a fictional company, earns 10% on its assets, keeps 30% of what it earns, and carries an equity multiplier of 1.25. Mondale Enterprises earns the same 10% on assets. It keeps two-thirds of earnings, however, and runs an equity multiplier of 2.00.

1. Calculate the sustainable dividend growth rate for each.

Solution. Multiply retention by return on assets by the equity multiplier, since return on assets times the equity multiplier is return on equity.
Baggai: g = 0.30 × 10% × 1.25 = 3.75%.
Mondale: g = (2/3) × 10% × 2.00 = 13.33%.

2. Identify what drives the difference.

Solution. Operating performance is identical, since both earn 10% on assets. The entire gap comes from financial policy: Mondale both retains a much larger share of earnings and carries much higher leverage, so its sustainable growth rate is far higher. Whether that higher growth is worth having depends on whether the extra leverage is prudent and whether the retained earnings can actually be deployed at the assumed return.

The horizon matters when applying this. If growth is being forecast for the next five years, use expectations of the four factors over those five years. If growth is being forecast into perpetuity, use very long-term forecasts of the same four factors.

As an illustration, suppose g = b × ROE = 0.60(15%) = 9%, where the 15% return on equity comes from a profit margin of 5%, an asset turnover of 2.0 and an equity multiplier of 1.5. With the sales-to-assets and assets-to-equity ratios fixed, sales, assets and debt all grow at 9% too. Because dividends are fixed at 40% of income, dividends grow at the same rate as income, 9%. Pushing dividend growth above 9% would not be sustainable on internally generated funds: retention would have to fall, and the company would then be unable to finance the assets that the sales growth requires without going outside for money.

Projecting historical ratios forward demands care. A company may have grown at 25% a year for five years, but that pace is unlikely to continue indefinitely, and the abnormally high returns on equity behind it are unlikely to persist either, because competitors arrive and because technology and demand can turn against the business. The example that follows shows a case where an above-average terminal growth rate is nevertheless defensible, and equally shows why the raw historical numbers had to be discarded.

Example 25 · Worked

An analyst is estimating a mature-phase growth rate for International Business Machines (NYSE: IBM) for a multistage dividend discount model. IBM’s return on equity for 2018 was around 52%, and over the past ten years its retention rate has averaged around 62%. Annual investment in property, plant and equipment has fallen from $3.7 billion in 2014 to $3.4 billion in 2018, and sales and earnings have declined over several years.

ROE decomposition for IBM

EQ 2 – Discounted Dividend Valuation — table 16
Year	ROE	Profit margin	Asset turnover	Financial leverage
2018	52.0%	11.0%	0.65	7.35
2017	32.7%	7.3%	0.63	7.12
2016	65.1%	14.9%	0.68	6.44
2015	92.5%	16.1%	0.74	7.75
2014	101.3%	13.0%	0.79	9.90
2013	72.3%	16.5%	0.79	5.54
2012	88.0%	15.9%	0.88	6.32
2011	78.7%	14.8%	0.92	5.78
2010	64.4%	14.9%	0.88	4.92
2009	59.3%	14.0%	0.88	4.82
Benchmark average	13.5%	10.5%	0.62	2.07

The benchmark row gives median values of the ROE components for a group of firms sharing the same two-digit SIC code as IBM. Ten years of company data are shown above it.

1. Why is the direct sustainable growth calculation unusable here?

Solution. Applying g = b × ROE with a retention rate of 62% and a return on equity of 52% would imply a perpetual growth rate above 32%, which is absurd for a mature company, and doubly so for one whose sales and earnings have been falling. The historical return on equity is not a forecast of the mature phase.

2. What does the decomposition reveal about the source of the high return on equity?

Solution. Compare each component with the benchmark. IBM’s profit margin of 11.0% in 2018 is close to the benchmark of 10.5%, and its asset turnover of 0.65 is close to the benchmark of 0.62. Financial leverage of 7.35, however, is more than three times the benchmark of 2.07. The high return on equity is therefore driven almost entirely by leverage rather than by superior operations.

3. Estimate the mature-phase growth rate using the PRAT approach.

Solution. The analyst assumes profit margin and asset turnover roughly match the benchmark average, and that as capital investment continues to fall, cash previously spent on investment will be used to retire debt and pay dividends, bringing financial leverage down to 2.0, close to the industry benchmark. The payout ratio is assumed to continue its recent rise and settle at 50%.

Forecast return on equity: (10.5%)(0.62)(2.0) = 13.0%.
Sustainable growth: g = (0.50)(13.0%) = 6.5%.

The estimate falls from an impossible 32% to a defensible 6.5% purely by replacing historical financial policy with forecast financial policy. That is the practical payoff of the DuPont decomposition: it tells you which component to override.

Check yourself

1. A company pays a large and erratic dividend, has negative expected free cash flow for the next eight years, and reports high-quality accruals. Which of the three cash flow definitions fits best, and why?

Residual income. Dividends are ruled out because they bear no consistent relationship to profitability. Free cash flow is ruled out because a long run of negative figures pushes the point at which value appears far into the future, where forecasts are least reliable. Residual income can always be computed, recognises value earlier than free cash flow, and its main drawback, sensitivity to accounting distortion, is not binding when disclosure quality is good.

2. A stock has just paid a dividend of $2.00. Growth is expected to be 4% forever and the required return is 9%. A colleague computes 2.00/(0.09 − 0.04) = $40.00. What is wrong, and what is the correct value?

The numerator must be the next dividend, not the last one. The correct calculation is 2.00(1.04)/(0.09 − 0.04) = 2.08/0.05 = $41.60. Using D₀ in place of D₁ understates the value by exactly the factor (1 + g), so the error is always in the same direction and grows with the growth rate.

3. Two companies have identical current dividends, identical required returns, and identical growth rates at the start and end of a ten-year supernormal period. One is valued with a general two-stage model and the other with an H-model. Which produces the higher value, and why?

The general two-stage model. It holds growth at the high rate for the entire ten years and drops to the mature rate only afterwards, whereas the H-model starts shedding growth immediately and reaches the mature rate at the end of the period. The Meditec figures make the size of the gap concrete: €94.21 against €78.13 on otherwise identical inputs.

4. A share trades at $50. Forecast next-year earnings are $2.50 and the required return is 8%. What is PVGO, and what does its sign tell you?

The no-growth value is E₁/r = 2.50/0.08 = $31.25, so PVGO = $50.00 − $31.25 = $18.75, which is 37.5% of the price. A positive figure means the market expects the company to find projects earning more than its cost of capital. Had the figure been negative, the correct first move would be to re-examine the earnings estimate and the required return before concluding anything about the business, since a negative PVGO is as often an input problem as a genuine market judgement.

5. Why does a 25 basis point error in the mature growth rate matter far more in a multistage DDM than a 25 basis point error in the first-stage growth rate?

Because the mature rate enters the terminal value, which typically carries three-quarters or more of the total value, and it enters through the denominator r minus g, which is a small difference. The first-stage rate only affects a handful of dividends whose combined present value is small. In the Meditec valuation the terminal value was about 93% of the answer, and in the three-stage technology example it was about 88%.

6. A company retains 40% of earnings and reports a return on equity of 30%, giving a sustainable growth rate of 12%. Its DuPont components are a profit margin of 8%, an asset turnover of 0.75 and an equity multiplier of 5.0. What should an analyst do before using 12% as a perpetual growth rate?

Test each component against a peer benchmark. An equity multiplier of 5.0 is doing most of the work here, and leverage of that size is unlikely to persist into a mature phase where cash is being returned rather than reinvested. Replacing it with a plausible mature multiplier, say 2.0, cuts the return on equity to 8% × 0.75 × 2.0 = 12%, and the growth rate to 0.40 × 12% = 4.8%. The value of the decomposition is that it identifies exactly which input to override.

7. An analyst inverts a DDM, obtains an internal rate of return of 9.4%, and proposes to use it as the required return in the same DDM to value the stock. What is the objection?

It is circular. The internal rate of return was derived by forcing the model value to equal the market price, so feeding it back in must reproduce the market price and can never identify a mispricing. The figure is better interpreted as the expected return the market price implies, given the dividend forecast. If price differs from intrinsic value, the expected return also needs adjusting for the return that arrives when the mispricing corrects.

End of lesson.

Back to 05 Equity Investments

EQ 2 – Discounted Dividend Valuation

Present value models and the choice of cash flow definition

Why present value, and why a risk adjustment

Three definitions of return

The dividend discount model over single and multiple holding periods

One holding period

Several holding periods

The general form

The Gordon growth model and its assumptions

What the model assumes, and when that is defensible

How sensitive is the answer

Perpetual preferred stock, negative growth, and what g actually measures

Fixed-rate perpetual preferred stock

Negative growth

Dividend growth, earnings growth and value appreciation

The present value of growth opportunities and justified price-to-earnings ratios

Growth is not automatically good

Restating the split as a price-to-earnings ratio

Justified leading and trailing P/Es

Growth phases and the two-stage dividend discount model

The general two-stage model

Terminal value

Terminal value from a multiple instead

A company that pays no dividend

The H-model

Three-stage dividend discount models

Version one: three distinct constant rates

Version two: an H-model in the middle

What multistage models do well, and where they fail

Spreadsheet modeling and estimating a required return from any DDM

Choosing the mature growth rate

Turning a DDM around to give a required return

Forecasting dividends from a full financial model

Sustainable growth and DuPont analysis

Decomposing return on equity

EQ 2 – Discounted Dividend Valuation

Present value models and the choice of cash flow definition

Why present value, and why a risk adjustment

Three definitions of return

The dividend discount model over single and multiple holding periods

One holding period

Several holding periods

The general form

The Gordon growth model and its assumptions

What the model assumes, and when that is defensible

How sensitive is the answer

Perpetual preferred stock, negative growth, and what g actually measures

Fixed-rate perpetual preferred stock

Negative growth

Dividend growth, earnings growth and value appreciation

Share repurchases and the implied dividend growth rate

Reading a growth rate out of the market price

The present value of growth opportunities and justified price-to-earnings ratios

Growth is not automatically good

Restating the split as a price-to-earnings ratio

Justified leading and trailing P/Es

Growth phases and the two-stage dividend discount model

The general two-stage model

Terminal value

Terminal value from a multiple instead

A company that pays no dividend

The H-model

Three-stage dividend discount models

Version one: three distinct constant rates

Version two: an H-model in the middle

What multistage models do well, and where they fail

Spreadsheet modeling and estimating a required return from any DDM

Choosing the mature growth rate

Turning a DDM around to give a required return

Forecasting dividends from a full financial model

Sustainable growth and DuPont analysis

Decomposing return on equity

Related lessons