DDM inputs: g from ROE, r from the premium, and the implied return

◈ 9 cards

Where D₁, g and r come from — EPS × payout, ROE × retention, risk-free plus premium — and the model rearranged to read the implied return and next year’s price off a market price.

Three inputs, three sources

Lesson 11.3 took $D_{1}$ , $g$ and $r$ as given. The paper often gives the ingredients instead:

$D_{1}$ = EPS × payout ratio. The dividend is the share of next year's earnings the board pays out.
$g$ = ROE × retention ratio — the sustainable growth rate. The earnings the firm keeps (retention = 1 − payout) are reinvested at its return on equity; that is how fast the equity, and so the dividend, can grow without new capital.
$r$ = risk-free rate + risk premium. The GoC yield plus what the market demands for this share's risk (Module 13 makes the premium beta × market premium).

Worked example — Ossington Brewing

Next year's EPS 3.20, payout 40 %, ROE 12 %, GoC yield 3.5 %, risk premium 6.5 %:

$D_{1}$ = =3.20*0.40 = 1.28
$g$ = =0.12*(1-0.40) = 7.2 %
$r$ = =0.035+0.065 = 10 %
$P_{0}$ = =1.28/(0.10-0.072) = 1.28 ÷ 0.028 = $45.71

The implied return

The market disagrees: Ossington trades at $40.00. Rearrange the model for $r$:

$r = \frac{D _{1}}{P _{0}} + g = \frac{1.28}{40} + 0.072 = 3.2 % + 7.2 % = 10.4 %$

=1.28/40+0.072. The return a share offers decomposes into a dividend yield ( $D_{1} / P_{0}$ ) and growth ( $g$ ) — the capital-gain yield. At 40 the share offers 10.4 % against the 10 % you required, so it is cheap to you; at 45.71 it offers exactly 10 %. Note it is $D_{1}$ over price, not $D_{0}$ — the same trap as Lesson 11.3.

Next year's price

If the model holds, every year the dividend is $(1 + g)$ larger and so is the price: $P_{1} = D_{2} / (r - g) = P_{0} (1 + g)$ — =45.71*(1+0.072) = $49.01; the long way, =1.28*1.072/(0.10-0.072) = 49.01 too. The DDM promises a capital gain of exactly $g$ a year, which is why the total return is dividend yield plus $g$.

Sensitivity — and the model's edge

Rebuild with ROE 15 %: $g$ = 0.15 × 0.60 = 9 %, and $P_{0}$ = 1.28 ÷ (0.10 − 0.09) = $128.00 — nearly triple, from three points of ROE. As $g $a pp r o a c h es$ r$ the price runs away, and a growth rate that close to the required return is not sustainable forever. When the arithmetic produces a number like that, the lesson is that the inputs, not the share, are wrong. Use ROE 10 % instead: $g $= 6$ P_0 $= 1.28 \div 0.04 = * *$ 32.00**.

Recompute

The decomposition, cold: required return = D₁ ÷ P₀ + g — dividend yield plus growth. And $g$ from accounting data: ROE × (1 − payout).

The value-prediction chain. Each input has an accounting or market source; the model turns them into today’s price and, one growth step later, next year’s.