We have well-specified “return on assets” showing that we must estimate reduced-forms (instead of structural) equations and run away from using this scrappy financial ratio to determine arm’s length profits subject to corporate income taxes. However, criticism is valid if we can provide a better substitute that can satisfy two conditions: First, the new alternative theory (markup pricing) resolves certain knotty issues of the old theory (such as avoid the cloudy base of “return on assets”); and second, the new theory provides *more reliable measures* of arm’s length profits. We submit that markup pricing-based profits are superior to “return on assets” on these two respects.

We posit an accounting equation, which is not controversial and is deemed to be true that the revenue (net sales) of any comparable company, and the “tested party” (controlled entity), is the sum of total costs and operating profits. For this purpose, we define Total Costs (Stricto) of each company (enterprise) to be equal to COGS plus XSGA (operating expenses before a depreciation deduction, which accountants call selling, general and administrative expenses):

(1) S(*t*) = C(

*t*) + P(

*t*)

for *t* = 1 to T fiscal periods.

To simplify exposition, we don’t show the comparable *i*-th subscript.

We introduce a behavioral equation that the operating *profits *(EBITDA) are proportional to the selected comparable company’s net sales during the same period:

*t*) = μ S(

*t*) + U(

*t*)

where the slope coefficient μ is the operating *profit margin* before depreciation, and U(*t*) is a random error.

We substitute (2) into (1) and obtain a reduced-form equation amenable to regression analysis:

(3) S(*t*) = λ C(

*t*) + V(

*t*)

where λ = 1 / (1 – μ) > 1 is the operating *profit markup* and V(*t*) is the transformed random variable.

Equation (3) shows that one unit increase in C(*t*) induces λ = 1 / (1 – μ) > 1 increase in S(*t*) (see equation A3 below), a feature reminiscent of the widow’s cruse of Sarepta. More the management of an enterprise spends greater are revenues, and thus greater are profits.

We can obtain the operating profit margin by *indirect least squares* using the equation of λ, which is the slope coefficient of (3). This means that the operating profit margin is calculated after we determine the slope coefficient of (3), that is, after we determine λ applying regression analysis to the selected comparable company time-series data:

Before we estimate the regression equation (3), we need to consider certain accounting measures of total costs because they differ from idealized economic concepts. First, COGS may include direct labor and materials, which are expected. However, COGS may include “other costs,” which is an anomalous category making cross-company comparisons difficult. Second, COGS include one-period change in inventories, which economists treat as inventory profit (or loss) and adjust profits to be free of this accounting anomaly. Thus, an empirical measure of arm’s length profits can consider these extraneous charges from an economic perspective; however, we recognize that because of accounting disclosure limitations reliable adjustments to COGS can be made only to one-period change in inventories.

Operating expenses (XSGA) are also a mixed bag of accounting intricacies, and we must ascertain that they don’t include extraordinary (non-recurring expenses) recorded under various guises. Some companies include depreciation in both COGS and XSGA, and thus diligence is required to review accounting footnotes to expunge these anomalies from total costs. In short, the quality of company financials is important, and we are sorry for gullible analysts who trust private company data (without accompanying accounting footnotes) under a myth that data can be used no-matter the source.

We illustrate the markup equation (3) using historical financials from Walmart (GVKEY 11259). We can consider any listed company in the Capital IQ (Compustat) database and our choice is made because Walmart is well-known. To keep this exercise simple to verify, we take COGS as normalized by Standard & Poor’s (original vendor of Capital IQ (Compustat), of which RoyaltyStat is a licensed distributor). Also, for the purpose of this example, we have not removed one-period change in inventories. Total costs in period *t*, C(*t*) = COGS(*t*) + XSGA(*t*), excluding DP(*t*) (depreciation of property, plant & equipment and amortization of acquired intangibles).

We ran several regressions and can show the stability of the profit markup regression coefficients. We don’t report the intercept because they are insignificant; and the regression using first differences was run without an intercept. See Maddala (1977), pp. 91-92. We use 41 years of Walmart's data from 1978 to 2018:

(A1) Method: OLS (Ordinary Least Squares),

S(*t*) = 1.0722 C(*t*) + V(*t*), Newey-West *t*-stat = 345.4, R^{2} = 0.9999

(A2) Method: Prais-Winsten (Correction for first-order serial correlations among the residuals),

S(*t*) = 1.0673 C(*t*) + V(*t*), *t*-stat = 150.5, R^{2} = 0.9983, ρ = 0.9217

(A3) Method: OLS (using first differences of the dependent and independent variables),

∆S(*t*) = 1.078 ∆C(*t*) + ∆V(*t*), Newey-West *t*-stat = 63.5, R^{2} = 0.9933

Although these three measures of Walmart’s operating profit markup are stable (λ = 7.22%, 6.73% or 7.8%), we select the OLS (A1) results to compute Walmart’s long-term profit margin because they are the most reliable measured by the comparative *t*-statistics:

(A4) Walmart’s *profit margin* before depreciation is computed by using *indirect least squares* (see equation (4):

μ = (λ – 1) / λ = 0.0722 / 1.0722 = 0.0673; that is, μ = 6.73%

It’s difficult to produce such stable profit margins as we obtained and reliable (high *t*-statistics after correction for serial correlation using either the Prais-Winsten algorithm or using the Newey-West *t*-statistics) estimates using *structural models* of profit indicators. Among the profit indicators specified on the OECD Guidelines (2017), the *reduced-form* profit margin for controlled importers (or the profit markup for controlled exporters) can produce more reliable estimates of arm’s length profit indicators. One great advantage is the parsimony obtained by using the same equation (3) to produce a comparable profit markup or a comparable profit margin—depending on the facts and circumstances.

**Historical Note**: We first learned of the *markup price equation* (3) reading (as an undergraduate) the magnificent book by Michael Kalecki, *Theory of Economic Dynamics*, George Allen & Unwin, 1954, Chapter 1 (Cost and prices). Kalecki (1899-1970) is regarded as one of the most prominent economists of the 20^{th} century. Kalecki is the most complete economist that I have read.

According to Malinvaud (1970), p. 187, the method of *least squares* (such as OLS (ordinary least squares)) was first proposed by Legendre (1806) [*New methods for the determination of the orbits of comets* (in French)], and its optimal (minimal variance) properties were later extended by Laplace (1812) and Gauss (1821). Also, this Legendre reference, including derivation details, is provided in Whittaker and Robinson (1944), pp. 209-215. Today, the method of least squares is the workhorse of economics and other applied sciences.

**References**:

Michael Kalecki, *Theory of Economic Dynamics*, George Allen & Unwin, 1954. Reproduced in Kalecki’s *Selected Essays on the Dynamics of the Capitalist Economy*, Cambridge University Press, 1971, and in Kalecki’s *Collected Works*, Vol. II, edited by Jerzy Osiatyński, Oxford University Press, 1991.

G. Maddala, *Econometrics*, McGraw-Hill, 1977.

Edmond Malinvaud, *Statistical Methods of Econometrics* (2^{nd} revised edition), North-Holland, 1970, published first by Dunod in French, Paris, 1964, revised in 1969. Malinvaud was the textbook assigned at our Ph.D. econometrics course at the University of California at Berkeley. Later, I taught econometrics at the Graduate Faculty of the New School in New York using Maddala (1977) because Maddala's book was less terse and contained more applied economics illustrations.

Edmund Whittaker and G. Robinson, *The Calculus of Observations* (4th edition), Blackie & Son (Van Nostrand), 1944 [1924]. This is a fascinating treatise about numerical mathematics because (unlike many more recent textbooks) the authors write their derivations in historical context and provide original references in multiple languages, including Latin, French and German.

**Ednaldo Silva**(Ph.D.) is founder and managing director of RoyaltyStat. He helped draft the US transfer pricing regulations and developed the comparable profits method called TNNM by the OECD. He can be contacted at: esilva@royaltystat.com

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