How the retention ratio works

Retention asks what share of this year's profit was not sent out as a dividend:

$$b = 1 - \text{payout} = \frac{\text{Net income} - \text{Dividends}}{\text{Net income}}$$

On $45,000 of net income and $18,000 of dividends, payout is 40 percent, so $b$ is 60 percent. Return on equity on $300,000 of book equity is 15 percent.

The payout ratio calculator on this page prints payout, retention, ROE and sustainable growth from one sheet, so $b$ and ROE cannot be typed from different years. How the payout ratio works owns the 40 percent. This page owns the 60 percent.

Payout against retention is the split on this same residual. They add to 100.

ROE is the rate the book earned this period. Only the kept share of that residual is still on the book to fund next period's assets. The internally fundable growth rate is therefore the product, not the ROE alone:

$$g = \text{ROE} \times b$$

Fifteen percent times 60 percent is 9 percent. Pay the 40 percent out and that slice cannot buy assets. Keep the 60 percent and that slice can. That is why the identity multiplies by $b$, not by payout.

How sustainable growth works owns the 9 percent. This page is why the second factor is retention. How ROE works is the one division that produces the 15 percent.

Net income $60,000, dividends $0, equity $600,000. Payout is 0 percent. Retention is 100 percent. ROE is 10 percent. Then $g = 10 \times 1$, 10 percent.

With $b = 1$, every unit of ROE is kept, so sustainable growth equals ROE. That is the identity at the extreme, not a claim that the firm should pay nothing. Paying nothing is a retention of 100 percent, not a missing ratio.

Net income $80,000, dividends $40,000, equity $400,000. Payout is 50 percent. Retention is 50 percent. ROE is 20 percent. g is 10 percent.

ROE rose from 15 percent to 20 percent. Retention fell from 60 percent to 50 percent. The product only moved from 9 percent to 10 percent. Ranking names on $b$ and ranking them on $g$ are not the same order.

How DuPont analysis works would split that 20 percent ROE into margin, turnover and the equity multiplier. This calculator does not take sales or assets. Do not paste another sheet's sales line onto this $300,000 of equity.

The 60 percent is this year's split. Next year's $b$ matches it only if the payout stays put. Next year's ROE matches the 15 percent only if margin and the equity multiplier stay put. Write $g = \text{margin} \times \text{turnover} \times \text{equity multiplier} \times b$ and every piece has to hold for the product to repeat.

If the payout rises, $b$ falls and $g$ falls even if ROE does not move. If the margin compresses, ROE falls and $g$ falls. If the equity multiplier changes, ROE changes and $g$ follows. That is a teaching condition, not a forecast.

How the Gordon growth model works uses a $g$ in a price. That $g$ is an assumed perpetual dividend growth rate. This $b$ is an accounting split on one year's sheet.

It is not a payout target, not a sales forecast, and not a claim that 60 percent is the right split. The three sheets are 60 percent kept on $18,000 over $45,000, 100 percent when dividends are $0, and 50 percent on $40,000 over $80,000. This is educational material, not financial advice.