How do you derive the sustainable growth rate from the DuPont identity, and how does it connect to valuation?

Question

AcadiFi · Accepted Answer

The sustainable growth rate is the maximum rate a company can grow earnings without issuing new equity or changing its capital structure. It's foundational for any dividend discount model. Let me build the full derivation.

**Starting Point**

g = b x ROE

Where b = retention ratio = 1 - dividend payout ratio.

**DuPont Decomposition of ROE**

The 5-factor DuPont breaks ROE into:

ROE = (Net Income / EBT) x (EBT / EBIT) x (EBIT / Revenue) x (Revenue / Assets) x (Assets / Equity)

Or more commonly the 3-factor version:

ROE = Net Profit Margin x Asset Turnover x Equity Multiplier

ROE = (NI/Sales) x (Sales/Assets) x (Assets/Equity)

**Therefore:**

g = b x (NI/Sales) x (Sales/Assets) x (Assets/Equity)

```mermaid
flowchart TD
    SGR["Sustainable Growth Rate
g = b x ROE"] --> RR["Retention Ratio (b)
1 - Payout Ratio"]
    SGR --> ROE["ROE
= NI/Equity"]
    ROE --> NPM["Net Profit Margin
NI / Sales"]
    ROE --> AT["Asset Turnover
Sales / Assets"]
    ROE --> EM["Equity Multiplier
Assets / Equity"]
    style SGR fill:#1a1a2e,stroke:#c9a84c,color:#fff
    style ROE fill:#1a1a2e,stroke:#c9a84c,color:#fff
```

**Worked Example: Castleton Industries (fictional)**

| Metric | Value |
|---|---|
| Net income | $48 million |
| Sales | $320 million |
| Total assets | $400 million |
| Total equity | $200 million |
| Dividends paid | $12 million |

Step 1: Retention ratio = 1 - (12/48) = 1 - 0.25 = **0.75**

Step 2: Net profit margin = 48/320 = **15.0%**

Step 3: Asset turnover = 320/400 = **0.80x**

Step 4: Equity multiplier = 400/200 = **2.00x**

Step 5: ROE = 15.0% x 0.80 x 2.00 = **24.0%**

Step 6: g = 0.75 x 24.0% = **18.0%**

**Connection to Valuation**

In the Gordon Growth Model: P0 = D1 / (r - g). If Castleton pays D0 = $0.50 per share and the required return is 22%, then:

D1 = $0.50 x 1.18 = $0.59

P0 = $0.59 / (0.22 - 0.18) = $0.59 / 0.04 = **$14.75**

**Critical Insight for Exams:** If a company increases leverage (higher equity multiplier), ROE rises and so does the sustainable growth rate -- but this also increases financial risk, which should increase the required return r. Students often forget that changing one DuPont component has ripple effects on valuation through both numerator and denominator.

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How do you derive the sustainable growth rate from the DuPont identity, and how does it connect to valuation?

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Metric	Value
Net income	$48 million
Sales	$320 million
Total assets	$400 million
Total equity	$200 million
Dividends paid	$12 million