Explaining ROE Changes with DuPont Factors

Why Decompose a Change in ROE?

Return on equity (ROE) can change because a company earns a different amount per dollar of sales, generates a different amount of sales per dollar of assets, or uses a different amount of assets per dollar of equity. The three-factor DuPont identity puts these channels into one equation. Comparing the factors across periods helps explain how the reported ROE changed, rather than merely noting that it rose or fell.

A factor decomposition is an accounting analysis, not a causal experiment. It assigns the arithmetic change in ROE to changes in specified ratios. It does not, by itself, establish why management decisions, market conditions, or other events changed those ratios.

Definition: For a period, let $m$ be net profit margin, $t$ be total asset turnover, and $e$ be the equity multiplier. With net income, sales, average total assets, and average equity measured consistently for that period, $$ m=\frac{\text{net income}}{\text{sales}},\qquad t=\frac{\text{sales}}{\text{average total assets}},\qquad e=\frac{\text{average total assets}}{\text{average equity}}. $$ The three-factor DuPont identity is $$ \text{ROE}=\frac{\text{net income}}{\text{average equity}}=mte. $$

The sales and average-asset terms cancel when the factors are multiplied, leaving net income divided by average equity. For a two-period comparison, use subscript 0 for the earlier period and 1 for the later period. The total ROE change is $m_1t_1e_1-m_0t_0e_0$. Report this difference as a percentage-point change when the ROEs are expressed as percentages; it is not the same as a percentage change relative to the earlier ROE.

Set Up a Comparable Two-Period Analysis

Before assigning contributions, check that the ratios are constructed consistently. In each period, use net income and sales from the same income statement period, and use average total assets and average equity that correspond to that period. The same accounting definitions and units should be used in both periods. Otherwise, a computed difference can reflect inconsistent measurement rather than a meaningful change in the DuPont factors.

Once the factors are calculated, first recompute ROE as their product in each period. This is a useful check: if the product does not match net income divided by average equity, at least one input or calculation is inconsistent. Then calculate the total difference in ROE. The remaining question is how to divide that difference among the factors.

Worked Example: Reconcile ROE to Three Factor Changes

A hypothetical manufacturer reports the following ratios, calculated on a consistent basis. In the earlier period, its net profit margin is 6%, asset turnover is 1.5, and equity multiplier is 2.0. In the later period, the corresponding figures are 5%, 1.8, and 2.2. Find ROE in each period and the total change. Use an order-neutral attribution to allocate the change to the three factors.

Method. Calculate each period’s ROE from $mte$, then use the Shapley attribution described below. This method averages the factor effects over all six orders in which three factors can change.

Earlier-period ROE. Convert the margin to a decimal and multiply: $$ 0.06(1.5)(2.0)=0.18=18.00\%. $$

Later-period ROE. $$ 0.05(1.8)(2.2)=0.198=19.80\%. $$

The total change is $0.198-0.180=0.018$, or an increase of 1.80 percentage points.

Margin contribution. The margin change is $0.05-0.06=-0.01$. Weight the combinations of old and new turnover and multiplier: $$ C_m=(-0.01)\left[\frac{1}{3}(1.5)(2.0)+\frac{1}{6}(1.5)(2.2)+\frac{1}{6}(1.8)(2.0)+\frac{1}{3}(1.8)(2.2)\right]. $$ The bracket equals $1.0000+0.5500+0.6000+1.3200=3.4700$, so $C_m=-0.0347$, or $-3.47$ percentage points.

Turnover contribution. The turnover change is $1.8-1.5=0.3$: $$ C_t=(0.3)\left[\frac{1}{3}(0.06)(2.0)+\frac{1}{6}(0.06)(2.2)+\frac{1}{6}(0.05)(2.0)+\frac{1}{3}(0.05)(2.2)\right]. $$ The bracket equals $0.0400+0.0220+0.0167+0.0367=0.1154$ using displayed rounding. Using the unrounded terms, $C_t=0.0346$, or 3.46 percentage points.

Multiplier contribution. The multiplier change is $2.2-2.0=0.2$: $$ C_e=(0.2)\left[\frac{1}{3}(0.06)(1.5)+\frac{1}{6}(0.06)(1.8)+\frac{1}{6}(0.05)(1.5)+\frac{1}{3}(0.05)(1.8)\right]. $$ The bracket is $0.0300+0.0180+0.0125+0.0300=0.0905$, so $C_e=0.0181$, or 1.81 percentage points.

Check. The contributions sum to $-0.0347+0.0346+0.0181=0.0180$, matching the total ROE change of 0.0180. Displayed contributions are rounded to four decimal places in decimal form; unrounded contributions reconcile exactly.

Sequential Bridges and Their Order Dependence

A sequential bridge changes one factor at a time, holding the remaining factors at their current values. For example, one could change margin first, then turnover, then the multiplier. The resulting contributions telescope to the total ROE difference, but the allocation can change if the order changes. This happens because the effect of changing one factor depends on the values of the other factors with which it is multiplied.

Worked Example: A Sequential Bridge Depends on Order

A hypothetical service firm has earlier-period factors $m_0=0.04$, $t_0=2.5$, and $e_0=1.6$. Its later-period factors are $m_1=0.05$, $t_1=2.4$, and $e_1=1.5$. Calculate the total ROE change and attribute it first by changing margin, then turnover, then the multiplier.

Earlier and later ROE. $$ \text{ROE}_0=(0.04)(2.5)(1.6)=0.160=16.0\%, \qquad \text{ROE}_1=(0.05)(2.4)(1.5)=0.180=18.0\%. $$ The total change is $0.180-0.160=0.020$, or 2.0 percentage points.

Change margin first. Holding the other factors at their earlier values gives a contribution of $$ (0.05-0.04)(2.5)(1.6)=0.040. $$

Change turnover second. Margin is now at its later value, while the multiplier remains at its earlier value: $$ (2.4-2.5)(0.05)(1.6)=-0.008. $$

Change the multiplier last. Both margin and turnover are now at their later values: $$ (1.5-1.6)(0.05)(2.4)=-0.012. $$

Check. The sequential contributions sum to $0.040-0.008-0.012=0.020$, the total ROE change. If the multiplier changes first, its contribution is $(-0.1)(0.04)(2.5)=-0.010$; the later contributions then differ, although their sum remains 0.020. The bridge reconciles exactly but does not give a unique allocation without a specified order.

Order-Neutral Attribution by Averaging All Orders

When no order is naturally preferred, an order-neutral method is to calculate the effect of each factor in every possible sequence and average that factor’s contribution. For three factors, there are six sequences. The resulting Shapley contributions assign interaction effects symmetrically and sum exactly to the change in the product.

Theorem: Let $m_0,t_0,e_0$ and $m_1,t_1,e_1$ be the factors in two periods, with $\Delta m=m_1-m_0$, $\Delta t=t_1-t_0$, and $\Delta e=e_1-e_0$. The Shapley contributions are $$ C_m=\Delta m\left[\frac{1}{3}t_0e_0+\frac{1}{6}t_0e_1+\frac{1}{6}t_1e_0+\frac{1}{3}t_1e_1\right], $$ $$ C_t=\Delta t\left[\frac{1}{3}m_0e_0+\frac{1}{6}m_0e_1+\frac{1}{6}m_1e_0+\frac{1}{3}m_1e_1\right], $$ $$ C_e=\Delta e\left[\frac{1}{3}m_0t_0+\frac{1}{6}m_0t_1+\frac{1}{6}m_1t_0+\frac{1}{3}m_1t_1\right]. $$ They satisfy $C_m+C_t+C_e=m_1t_1e_1-m_0t_0e_0$.

The weights follow from averaging across the six sequences. For the margin effect, consider the positions of the other two factors when margin changes. In two sequences, both turnover and the multiplier are still at their old values; in two, both have already changed; and in one each, exactly one has changed. Thus the old-old and new-new combinations each receive weight $2/6=1/3$, while each mixed combination receives weight $1/6$. The same counting applies to turnover and the multiplier.

To justify the reconciliation, take any one sequence. Changing the first factor creates an intermediate product; changing the second creates another; changing the last creates the final product. Adding the three successive differences cancels the intermediate products and leaves exactly $m_1t_1e_1-m_0t_0e_0$. Averaging this equality over all six sequences preserves the total difference. The average contributions are precisely $C_m$, $C_t$, and $C_e$, so they must sum to the full ROE change.

Worked Example: Correctly Average the Margin Effect

A hypothetical distributor has an earlier-period margin of 8%, turnover of 2.0, and equity multiplier of 2.0. In the later period, its margin is 7.5%, turnover remains 2.0, and its multiplier rises to 2.5. Find the ROE change and the Shapley contributions.

Calculate the two ROEs. $$ \text{ROE}_0=(0.08)(2.0)(2.0)=0.320=32.0\%, \qquad \text{ROE}_1=(0.075)(2.0)(2.5)=0.375=37.5\%. $$ The total increase is $0.375-0.320=0.055$, or 5.50 percentage points.

Margin contribution. Turnover is unchanged. Averaging the two orders in which margin and the multiplier can change gives equal weight to the old and new multipliers. The average multiplier is $(2.0+2.5)/2=2.25$. Therefore, $$ C_m=(0.075-0.08)(2.0)(2.25)=-0.0225. $$ The margin contribution is $-2.25$ percentage points.

Multiplier contribution. The average margin is $(0.08+0.075)/2=0.0775$. Thus, $$ C_e=(2.5-2.0)(2.0)(0.0775)=0.0775. $$ The multiplier contribution is 7.75 percentage points.

Check. Since turnover did not change, its contribution is zero. The two nonzero contributions sum to $-0.0225+0.0775=0.0550$, matching the 5.50-percentage-point ROE increase. Weighting the old and new multipliers unequally here would not average the two possible change orders and would give an incorrect margin allocation.

Reading and Reporting the Decomposition

A positive contribution means that the factor’s change, averaged under the chosen attribution method, raised ROE; a negative contribution means it lowered ROE. Contributions are measured in ROE units. For example, a contribution of $0.02$ is 2 percentage points, not a 2% relative increase. State the method used, especially for a sequential bridge, and verify that the displayed contributions reconcile to the total within rounding.

A factor’s direction alone does not determine its contribution. A margin decline can reduce ROE while higher turnover or a higher equity multiplier more than offsets the decline. Likewise, a higher multiplier can raise the DuPont ROE arithmetically without showing that the firm’s overall financial risk improved. The decomposition explains the identity; evaluating risk or the quality of earnings requires additional analysis.

Keep the ratio definitions and period measurements consistent.
Recompute each period’s ROE from the three factors before attributing the change.
Use percentage points for differences between ROEs expressed as percentages.
Do not interpret an arithmetic contribution as proof of a business cause.

Check Your Understanding

For each question, calculate the requested ROE change or contribution and check that the units and method are clear.

A hypothetical firm’s factors change from $m=0.07,t=1.2,e=2.0$ to $m=0.06,t=1.5,e=2.0$. Find both ROEs and the change in percentage points. (Answer: 16.80% and 18.00%; increase of 1.20 percentage points.)
A hypothetical retailer changes from $m=0.05,t=2.0,e=1.5$ to $m=0.055,t=1.8,e=1.5$. Find the total ROE change in percentage points. (Answer: 15.00% to 14.85%; decrease of 0.15 percentage points.)
Turnover is fixed at 2.4, while margin changes from 9% to 8% and the multiplier changes from 1.5 to 1.9. Find both ROEs and the Shapley contribution of each changing factor. (Answer: 32.40% to 36.48%; margin contribution $-4.08$ percentage points and multiplier contribution $+8.16$ percentage points.)
Factors change from $m=0.04,t=1.8,e=2.5$ to $m=0.05,t=2.0,e=2.0$. Using the sequential order margin, turnover, then multiplier, calculate the three contributions. (Answer: $+4.50$, $+2.50$, and $-5.00$ percentage points, respectively; total increase of 2.00 percentage points.)
Explain why a sequential bridge can give different individual contributions when the order changes, even though the total ROE change stays the same. (Answer: Each factor’s effect depends on the current values of the other multiplicative factors; changing the order changes those values, while the successive differences still telescope to the same total.)

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