Justified Leading P/E

Builds onGrowing Perpetuity (Gordon Growth) · Sustainable Growth Rate — if this page feels steep, start there.

\frac{P_0}{E_1} = \frac{p}{r - g}

$P_0 / E_1$: the leading P/E: today's price over NEXT year's expected earnings
$p$: the dividend payout ratio: the fraction of earnings paid out (D₁ = p·E₁)
$r$: the required return on the stock (from CAPM or DDM)
$g$: the perpetual growth rate — itself tied to retention: g = (1−p)·ROE
$r - g$: the shrinking gap that powers all Gordon math: small gap, big multiple

Why it must be true

"The stock trades at 18 times earnings" — but should it? The justified P/E answers from first principles instead of peer gossip. Divide the Gordon model by next year's earnings and the fair multiple falls out of three fundamentals: **the payout ratio $p$ (how much of earnings become dividends), the required return $r$ , and growth $g$ .**

Read it as a machine: higher growth inflates the fair multiple (the classic intuition), higher required return deflates it (why multiples compress when rates rise), and payout cuts both ways — paying out more lifts the numerator but starves the growth that comes from retention. The formula turns "is 18× expensive?" into a checkable claim about $p$ , $r$ and $g$ .

The derivation

Start from Gordon with next year's dividend written as a slice of next year's earnings, $D_1 = p \cdot E_1$ :

P_0 = \frac{D_1}{r - g} = \frac{p\,E_1}{r - g}

Divide both sides by $E_1$ :

\frac{P_0}{E_1} = \frac{p}{r - g}

The subtle coupling: $g = (1-p) \times ROE$ , so payout appears twice — once openly in the numerator, once hidden inside $g$ . Raising the payout of a high-ROE firm can LOWER its justified multiple by killing better-invested growth.

When to reach for it

Deriving what P/E a stock's fundamentals deserve — to compare against its actual multiple, or to explain why multiples move with rates and growth.

Listen for

“justified (leading) P/E”“P/E based on fundamentals”“payout ratio … required return … growth”“what multiple should the stock command”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Denominator first: r − g is a small number, so tiny changes swing the multiple hugely. 40% payout over (10% − 6%) = 0.4/0.04 = 10× in one glance.
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Rate sensitivity check: each point OFF the r−g gap adds multiple non-linearly (gap 4%→3% takes 10× to 13.3×). If a question moves r or g, expect a big multiple move.
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Trailing vs leading: the trailing version (P₀/E₀) is the same fraction times (1+g). If an answer choice is exactly (1+g) bigger, it's the trailing twin.

Traps in applying it

✗Using the earnings retention ratio instead of the payout ratio in the numerator.
✗Confusing leading (E₁) and trailing (E₀) versions — the trailing multiple is (1+g) larger.
✗Treating p and g as independent: raising payout lowers sustainable growth via g = (1−p)·ROE.

Limits & criticisms

All of Gordon's fragility, squared: the multiple explodes as g approaches r, so aggressive growth assumptions manufacture any multiple you like — the formula justifies, it doesn't verify. It also assumes the payout policy and growth run forever, ignores buybacks (the modern payout channel), and quietly assumes earnings are honestly measured — the E in P/E is accounting's most massaged number.

Where it came from

The justified multiple comes straight out of Gordon and Shapiro's 1956 growth model, but its working life began when analysts needed to discipline the P/E — the market's favorite shorthand since the 1930s. It anchors equity research "target multiple" arguments, explains mechanically why low rates justified high multiples in the 2010s (smaller $r$ , bigger fraction), and is the standard exam bridge between dividend discounting and relative valuation.

One identity, 1 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

The multiple fundamentals deserve

\frac{P_0}{E_1} = \frac{p}{r-g}

The relative-valuation face: turns 'is 18× expensive?' into a testable claim about payout, required return and growth.

Drill this face →

On the BA II Plus

Worked example: With a payout ratio of 55%, required return 8.5%, and sustainable growth 4.5%, what is the justified leading P/E?

1.0.085 [−] 0.045 [=]the r − g gap
2.[STO] 1 0.55 [÷] [RCL] 1 [=]payout over the gap = P/E

→ 13.75

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