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Valuing a share: the Gordon-Shapiro model
What is a share worth? The discounted sum of all its future dividends.
P = D₁ / (r − g)Problem / motivation
A share promises neither a fixed coupon nor repayment (course “Share or bond?”): dividends voted year after year, uncertain, with no maturity. How can it be given a price today?
The course “Bond prices and interest rates” supplied the move: the value of a security is the sum of its future flows DISCOUNTED — each euro of tomorrow divided by (1 + r) as many times as there are years to wait. For a bond, the flows stop at maturity; you add a few terms and you are done. For a share, the dividends NEVER stop: you would have to add an infinity of them. An endless sum that would yield a finite price?
Yes — that is the whole point of this course. Gordon and Shapiro (1956) showed that if the dividend grows at a constant rate g and the required rate r stays above it, the infinite sum closes onto a one-line fraction: P = D₁ / (r − g). The formalisation makes it appear before your eyes, algebra included; then we use it as analysts do — to judge whether a market price is expensive or cheap.
Assumptions
In your view, which assumptions are needed for this model to hold? Jot down your ideas — no lead is wrong, this is your worksheet.
Your worksheet is still empty. Go for it: propose at least one idea.
0 idea(s) proposedFormalization
It all starts from the course “Bond prices and interest rates”: receiving €X in n years is worth X ÷ (1 + r)ⁿ today — investing means multiplying; discounting means dividing. The price of a share is therefore the sum of all its discounted dividends: P = D₁/(1 + r) + D₂/(1 + r)² + D₃/(1 + r)³ + … without end. D for Dividend, the subscript for the year: D₁ is next year's (D₀, the last one paid, is already banked — it no longer counts in the price, but it will serve to estimate D₁).
Enter assumption 1: each dividend is the previous one × (1 + g), so D₂ = D₁(1 + g), D₃ = D₁(1 + g)², and so on. Each term of the sum is then the previous one × (1 + g)/(1 + r) — a constant ratio, what mathematicians call the COMMON RATIO of the sequence. If r > g (assumption 2), that ratio is smaller than 1: the terms SHRINK, like 1 + ½ + ¼ + ⅛ + … which totals 2, not infinity. Mathematicians call such a sequence a GEOMETRIC SERIES — and its sum is known.
For a geometric series that shrinks, total = first term ÷ (1 − ratio). Here: P = [D₁/(1 + r)] ÷ [1 − (1 + g)/(1 + r)]. The bracket simplifies: 1 − (1 + g)/(1 + r) = (r − g)/(1 + r), and the (1 + r) cancel out. What remains is P = D₁ / (r − g): three lines of algebra, and the fraction of the header is born — with the condition r > g handed over by the same computation. Financiers call this result a GROWING PERPETUITY: payments without end, growing steadily.
Turn the fraction over: r = D₁/P + g. The required rate decomposes into the dividend yield (D₁/P) plus growth (g) — because under these assumptions the PRICE itself rises by g each year (next year: P′ = D₂/(r − g) = P × (1 + g)). Your R from the course “Return and risk” finds exactly its two pieces again: the income paid, and the capital gain. Click each term:
Solving / calculation
A mature company has just paid D₀ = €5.00; g = 2% is expected; shareholders require r = 8%. (In the poll, the €5 was D₁ — price €83.33. Here the €5.00 is D₀, one notch earlier: watch what that changes.) Let's compute, check it both ways, then it is your turn.
- Next year's dividend
D₁ = 5.00 × 1.02D₁ = €5.10 - The gap in the denominator
r − g = 0.08 − 0.02= 0.06 (6 points of gap: 8 − 2) - Gordon-Shapiro
P = 5.10 / 0.06P = €85.00 - Check, reading it in reverse
5.10/85 + 0.02= 6% + 2% = 8% = r ✓
The share is worth €85 — that is 16.7 times its dividend (financiers say: a MULTIPLE of 16.7, which is 1/(r − g)). And the reverse reading lands back on r: 6% dividend yield + 2% growth. If this share is quoted at €70, Gordon says it is undervalued; at €110, overvalued — unless the market expects a different g from yours (interpretation).
P = D₁ / (r − g)Set D₁, r and g: the price, its multiple, the reverse reading and the cost of one point of rate all follow. Reproduce limit 1: bring r − g down to 0.1 point (r = 3%, g = 2.9%) and watch the price take off.
Economic interpretation
A one-line formula, three uses: valuing, understanding multiples, understanding the nervousness of markets.
The formula manufactures a FUNDAMENTAL value, to be set against the quoted price: quoted below it, the share looks undervalued — ACCORDING TO YOUR assumptions (your g, your r). Analysts also make the return journey: start from the quoted price and deduce the g that “the market is pricing in” (the IMPLIED g) — the same gymnastics as the yield to maturity of the course “Bond prices and interest rates”. The gap between your g and the market's is precisely the debate that makes buyers and sellers.
The thinner the gap r − g, the smaller the denominator and the higher P climbs — the multiple 1/(r − g) goes from 16.7 (a 6-point gap) to 50 (a 2-point gap). This is the logic of GROWTH STOCKS: a high expected g justifies paying many times the dividend… as long as the expectation holds.
r = risk-free rate + risk premium (assumption 4). When the central bank raises its policy rate, the risk-free rate rises, hence r, hence ALL the r − g denominators in the market: prices fall mechanically — and all the more sharply where the gap was thin (point 2). The simulator puts a number on it: +1 point of r in our example, and the price goes from €85.00 to €72.86 (−14.3%). Symmetrically for g: a revision to growth moves the price just as much.
Limits / critiques
Exercises
D₁ = €4, r = 9%, g = 3%. What price (in €)?
True or false: if g becomes greater than r, the formula gives a negative price, to be read as “worthless share”.
A share is quoted at €50; D₁ = €2; you require r = 7%. What g is “the market pricing in” implicitly (in %)?
Rates rise by one point everywhere. Which shares fall THE MOST, in proportion?
A company HAS JUST PAID D₀ = €6.00; g = 5%, r = 10%. What price (in €)? (Mind which dividend to use.)
P = €60, D₁ = €3, g = 1.5%. What rate r are investors implicitly requiring (in %)?
D₁ = €5.10, g = 2%; r goes from 8% to 9%. What is the NEW price (in €)?
True or false: if the Gordon-Shapiro price exceeds the quoted price, the share is certainly a good deal.