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Worked-example setupScope and assumptions
- Both opening and ending positions satisfy the accounting equation and use aligned definitions.
- Total equity is positive; preferred equity and noncontrolling interests are excluded.
- Liabilities are not relabeled debt; the packet does not supply financing cost, maturity, collateral, or covenant data.
- A separate teaching variant stipulates $80,000 of Beacon ending debt and $100,000 of Harbor ending debt solely to demonstrate a declared debt-to-total-assets definition.
- Period
- Year 5 ended December 31
- Units
- Whole USD; ratios in decimal form, percentages, or times as labeled
- Rounding
- Full precision internally; percentages and multipliers displayed to two decimals
Preserve the equal-ROA starting point
Beacon and Harbor each report 12% ROA. Any ROE difference must therefore enter through the average-equity relationship under the three-step identity.
Compute and cross-check leverage
| Measure | Beacon | Harbor |
|---|---|---|
| Average assets | $200,000 | $200,000 |
| Average liabilities | $100,000 | $120,000 |
| Average equity | $100,000 | $80,000 |
| Equity multiplier | 2.00 | 2.50 |
| Average liabilities ÷ equity | 1.00 | 1.50 |
The accounting equation supplies an independent check:
equity multiplier = 1 + average liabilities / average equity
Beacon: 2.00 = 1 + 1.00
Harbor: 2.50 = 1 + 1.50
The liabilities-to-equity term is deliberately not called debt-to-equity. The packet contains total liabilities, not a defined debt subtotal.
Recompose ROE
Beacon: 0.10 margin × 1.20 turnover × 2.00 multiplier = 0.24 = 24% ROE
Harbor: 0.08 margin × 1.50 turnover × 2.50 multiplier = 0.30 = 30% ROE
Writing the margin as a decimal makes the scale control visible; multiplying
the whole number 10 instead of 0.10 would overstate Beacon's result by 100.
Direct calculations agree: $24,000 ÷ $100,000 = 24% for Beacon and $24,000 ÷ $80,000 = 30% for Harbor.
Run a separately defined debt screen
The teaching variant supplies $80,000 of Beacon ending debt and $100,000 of Harbor ending debt. With same-date ending assets, the declared screens are:
Beacon: $80,000 debt / $220,000 assets = 36.36%
Harbor: $100,000 debt / $210,000 assets = 47.62%
Neither numerator is inferred from total liabilities. The ratio describes a date-specific financing share under the supplied debt definition; it does not show maturity, collateral, covenant headroom, cash generation, or repayment capacity. The equity multiplier continues to use average total assets and average total equity, so the two leverage measures must not be substituted for one another.
Interpretation
Harbor's six-percentage-point ROE advantage is leverage amplification in this bounded comparison, not higher ROA. A finance recommendation would require liability composition, interest cost, maturity, liquidity, covenant, collateral, and downside evidence. An accounting review should also reconcile both equity rollforwards and confirm entity scope.
Leverage can amplify positive asset return. It can also amplify losses. The packet contains no forecast or stress case, so “higher ROE” is not a risk- adjusted conclusion.
Common wrong paths
- Attribute 30% ROE to operations alone: ROA is equal; the multiplier creates the difference.
- Call every liability debt: use the label supported by the source data.
- Treat three components as independent controls: operational and financing choices can affect one another across periods.
Use the separate leverage question to test the interpretation with the same controlled source packet.
Verified calculation · performance return analysis
The curriculum loader recomputed this example before it entered the site build. Expand any structured input to inspect the stated facts.
- entities
- 2 fields
Inspect data
{
"beacon": {
"defined_debt_at_end": 80000,
"ending": {
"total_assets": 220000,
"total_equity": 110000,
"total_liabilities": 110000
},
"net_income": 24000,
"net_sales": 240000,
"opening": {
"total_assets": 180000,
"total_equity": 90000,
"total_liabilities": 90000
},
"reported_operating_cash_flow": 30000
},
"harbor": {
"defined_debt_at_end": 100000,
"ending": {
"total_assets": 210000,
"total_equity": 85000,
"total_liabilities": 125000
},
"net_income": 24000,
"net_sales": 300000,
"opening": {
"total_assets": 190000,
"total_equity": 75000,
"total_liabilities": 115000
},
"reported_operating_cash_flow": 18000
}
}Recomputed result
| Measure | Value |
|---|---|
| beacon average liabilities to equity | 1 |
| beacon debt to total assets | 0.3636 |
| beacon dupont return on equity | 0.24 |
| beacon equity multiplier | 2 |
| beacon leverage identity difference | 0 |
| beacon return on equity | 0.24 |
| beacon roe identity difference | 0 |
| harbor average liabilities to equity | 1.5 |
| harbor debt to total assets | 0.4762 |
| harbor dupont return on equity | 0.3 |
| harbor equity multiplier | 2.5 |
| harbor leverage identity difference | 0 |
| harbor return on equity | 0.3 |
| harbor roe identity difference | 0 |