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 hold that markup pricing-based profits are superior to “return on assets” on these two respects.

**Prior information**: 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):

*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. Again, to simplify exposition, we exclude an intercept from behavioral equation (2).

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:

**Dirty accounting 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.

**Walmart example**: 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 (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), which implies:

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

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

**Historical notes**: 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) was a prominent economists of the 20^{th} century, and he was the most complete economist that we studied.

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. See Richard Farebother, *Fitting Linear Relationships* (A history of the calculus of observations, 1750-1900), Springer, 1999, Chapter 5 (The method of least squares) and Prakash Gorroochurn, *Classic Topics in the History of Modern Mathematical Statistics*, Wiley, 2016, section 1.5 (Least squares and the normal distribution), pp. 149-184. The purpose of these notes is to show that least squares has an enduring academic legacy.

**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, we 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]. In this numerical mathematics treatise (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. But read the acerbic commentary by E. Jaynes, *Probability Theory* (*The Logic of Science*), Cambridge University Press, 2004, p. 703: “Notable because the fake ‘variable star’ data on p. 349 were used by Bloomfield (1976), who proceeded to make the author’s analysis, with absurd conclusions, the centerpiece of his textbook on spectrum.” Re Peter Bloomfield, *Fourier Analysis of Time Series*, John Wiley & Sons, 1976, Jaynes on p. 685 is even more scathing, and concludes his vituperation with this sectarian folly: “A Bayesian would not be able to make such errors [charge of using "fake data"], because he would be obliged to think about his prior information concerning the phenomenon and the data-taking procedure.”

**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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