# Decomposing Changes in ROA into Changes in Operating Margin and Asset Turnover via Total Factor Productivity (TFP)

I am currently reading a paper on the influence of hedge fund activism on plant-level productivity. In the appendix the author describes how he decomposes changes in ROA into changes in operating margin and asset turn-over. I am trying to redo the calculation explained by the author but am unable to do so.

The author is calculating the change in operating margin via a increase in TFP using the following formula:

Operating Margin = 1-1/(TFP x price change ratio)

The author calculates 26,6 percentage as the new operating margin but I cannot replicate the calculation. Any help is appreciated.

Here the part in question:

In Appendix B, we link formally the magnitude of the change in ROA to the change in raw TFP from years tt to t+3t+3 . In particular, we use a modified version of the decomposition in Bosch-Badia (2010), in which ROA is decomposed into TFP, input and output price changes, and asset turnover. Using the “DuPont decomposition” of ROA, we obtain the following relation:

(A2) ROA=Operating margin × Asset turnover,ROA=Operating margin × Asset turnover, where ROA is the ratio of earnings before interests and taxes (“operating profits”) to lagged total assets, operating margin is the ratio of operating profits to concurrent sales, and asset turnover is the ratio of sales to lagged assets and, as Bosch-Badia (2010) shows, operating margin = 1–1/(TFP × price change ratio). The price change ratio is the change in output price divided by the change in input price. All price changes are relative to the benchmark year (i.e., year tt ). In addition, we further make the following two assumptions: (i) The baseline operating margin is 24.7% (see Table 2, Column (1)), and (ii) the price change ratio is equal to one (i.e., input and output prices change by the same magnitude). With these assumptions we can link the change in ROA to the changes in TFP and in asset turnover. First, we estimate the change in TFP using the specification in Table 4, Column (4). Specifically, we narrow the estimation to only manufacturing firms based on Compustat SIC codes and find average productivity gains of 2.6% from years tt to t+3t+3 for plants owned by manufacturing target firms. Second, given the baseline operating margin of 24.7%, the increase in TFP of 2.6% translates into an expansion in operating margin by 1.9 percentage points to 26.6%. Third, the magnitude of the change in ROA also depends on the change in firm-level asset turnover, which is driven by reductions in capital at the plants that are not sold, and by divestitures and/or closures of plants. Using Compustat data, we find that for manufacturing target firms asset turnover increases from 1.07 to 1.20 on average from years tt to t+3t+3 . Taking the two changes together, the implied ROA increases by 5.5 percentage points from 26.4% in year tt (= 24.7% × 1.07) to 31.9% in year t+3t+3 (= 26.6% × 1.20).

And here a direct link to the paper.

Go to Appendix B.

The formula for OM is:

$$OM = 1 - \frac{1}{TFP\times PCR}$$

In the baseline year, you have that Operative Margin (OM) = 0.247. Further, by assumption, price change ratio (PCR) = 1. Then:

$$TFP = \frac{1}{1-OM} = \frac{1}{1-0.247}=1.328$$

Then, you are told that the TFP changed by 2.6%. Therefore, the new value for TFP is 1.3625 (equivalent to 1.328 $\times$ 1.026). With this new TFP, and assuming that PCR = 1, you can find the new value for OM:

$$OM = 1 - \frac{1}{1.3625\times 1}$$ $$OM = 1 - 0.734$$ $$OM = 0.266$$

There you get the 26.6% result.

• This number is towards the end of Column 4 in Table 4. In the row which starts with "d[t+3] – d[t]". Commented Mar 18, 2017 at 17:10
• I appreciate it! Commented Mar 19, 2017 at 0:38