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No deben admitirse más causas de las cosas naturales que aquellas que sean verdaderas y suficientes para explicar sus fenómenos.

− Isaac Newton, *Principios Matemáticos de la Filosofía Natural*, Alianza Editorial, 2022 (Regla primera), p. 615.

Determining an arm’s length profit indicator (univariate profit ratio) requires two equations, not one, as misstated in financial statement analysis textbooks (e.g., Bernstein, 1993 or Drake & Fabozzi, 2012. An accounting critique of univariate profit ratios is found in Whittington, 1986.

26 CFR § 1.482-5(b)(4)(i) (Rate of return on capital employed) and 26 CFR § 1.482-5(b)(4)(ii) (Financial ratios, including the ratio of operating profit to sales) suffer from the same univariate conceptual error. However, the head sentence of 26 CFR § 1.482-5(b)(4) states: “A variety of profit level indicators can be calculated in any given case,” provided that the “profit level indicator … produce[s] a reliable measure of the income [operating profit] that the tested party would have earned had it dealt with controlled taxpayers at arm’s length, taking into account all of the facts and circumstances.”

Moving to our economic conception of reliable profit indicators based on two equations: First, the *accounting equation* (1) is true by definition and is not subject to controversy. Second, the *behavioral* *equation *(2) posits that profits (dependent variable) are a function of specific X-factors. In reduced form, the behavioral equation is subject to empirical (statistical) testing. We illustrate the two-equations determination of “return on assets” (ROA):

(1) S(*t*) = C(*t*) + *d* K(*t* – 1) + P(*t*)

The variable **S**(*t*) is net sales of the selected enterprise in period *t* = 1 to T; **C**(*t*) is the total cost, consisting of COGS plus XSGA during the same period; **K**(*t* – 1) is the sum of PPENT and INTANO at the beginning of the period; and **P**(*t*) is OIADP (operating profit after depreciation and amortization). These variables are defined by using acronyms from Standard & Poor’s Compustat® database of publicly traded company financials.

Operating assets are a composite (PPENT + INTANO) variable because the “reasonable” depreciation rate includes the amortization rate. The stock (balance sheet) variable **K**(*t* – 1) must be lagged one period because of the “law of motion of capital.” (Klein, 1983, pp. 9, 19).

A behavioral profit equation can be postulated such that annual profits **P**(*t*) are proportional to **S**(*t*) and **K**(*t* – 1), which means that the profit margin is adjusted for asset-intensity:

(2) P(*t*) = *m* S(*t*) + *r* K(*t* – 1) + U(*t*)

The unknown residual variable **U**(*t*) is the uncertainty or random error.

Equation (2) may include an intercept. Unlike the accounting equation (1), the postulated behavioral equation (2) can be subject to statistical errors that may not satisfy standard assumptions, such as unequal variances or near-zero serial correlation among the residual errors.

Substitute equation (2) into (1) and we obtain the *reduced*-*form* equation that is amenable to regression analysis:

(3) S(*t*) = C(*t*) + *d* K(*t* – 1) + *m* S(*t*) + *r* K(*t* – 1) + U(*t*)

(4) S(*t*) = β_{1} C(*t*) + β_{2} K(*t* – 1) + V(*t*)

This reduced-form equation (4) has two X-factors (independent variables), **C**(*t*) and **K**(*t* – 1).

The partial regression coefficients are the profit markup, β_{1} = 1 / (1 – *m*) > 1, and the composite ROA, β_{2} = [(*d* + *r*) / (1 – *m*)]. The profit margin *m* is embedded in the profit markup.

If the absolute value of *m *is less than 1, like we expect from the operating profit margin, we obtain the convergent geometric series 1 / (1 – *m*) = 1 + *m* + *m*^{2} + *m*^{3 }+ ….

In equation (4), net sales **S**(*t*) is equivalent to a weighted sum of past total costs (or past investments) (see Lax & Terrell, 2014, Theorem 1.13 [Geometric series], p. 29 or Thomas, 1983, p. 564.

Because **K**(*t* – 1) is lagged one period (operating assets are measured at the beginning of the accounting period), the regression equation (4) must be run per individual company using its historical data.

If regression equation (4) produces β_{2} = 0, then model (4) reduces to a simple profit markup without asset intensity adjustment:

(5) S(*t*) = β_{1} C(*t*) + V(*t*)

Regression analysis based on comparable data to the tested party (taxpayer), using **S**(*t*), **C**(*t*), and **K**(*t* – 1) per company, can determine the statistical validity of model (4) versus model (5). Hence, the top reference to Newton’s (Occam’s razor) aphorism (Rule 1 [for the study of natural philosophy, ergo Physics], p. 794): “No more causes of natural things should be admitted than are both true and sufficient to explain their phenomena.”

Unlike the parsimonious equation (5), we do not get consistent regression results using the expanded equation (4). The choice of model (4) or model (5) is also affected by omitted variable bias (see Kmenta, 1986, pp. 443-446).

A prevailing vice in transfer pricing is to assert structural model (2), without proper statistical testing. Another vice is to assert that β_{2} < 0 in model (4), also without statistical testing.

The main takeaway here is that the calculation of univariate profit indicators (reducing perspective to quartiles of the profit margin, profit markup, or ROA) is misleading, because the univariate calculation of profit ratios does not benefit from economics or statistics learning. *Economics learning* forces us to posit causal modeling (such as equation (4) or (5)), and *statistics learning* forces us to seek reliable (smallest variance around the center value) estimates.

Leopold Bernstein, *Analysis of Financial Statements* (4^{th} edition), Irwin, 1993.

Pamela Drake & Frank Fabozzi, *Analysis of Financial Statements* (3rd edition), Wiley, 2012.

Lawrence Klein, *Lectures in Econometrics*, Elsevier, 1983. The “law of motion of capital” is equivalent to the “perpetual inventory” equation in which PPENT or INTANO is subject to a recurrent mechanism. Recurrence is an equation (algorithm) that describes a function in terms of its prior (typically smaller) value. The perpetual inventory method (PIM) of determining the present stock of PPENT or INTANO is equivalent to a weighted sum of past net investments to accumulate the respective asset stock.

Jan Kmenta, *Elements of Econometrics* (2nd edition), Macmillan, 1986.

Peter Lax & Maria Terrell, *Calculus with Applications* (2^{nd} edition), Springer, 2014.

Isaac Newton, *Mathematical Principles of Natural Philosophy* (*Principia*), University of California Press, 1999 [1st edition 1686], including a long guide written by Bernard Cohen. In English translation from the original Latin by Bernard Cohen and Anne Whitman, p. 794: “Nature is simple and does not indulge in the luxury of superfluous causes.” In Spanish translation from Latin with notes by Eloy Rada García, p. 616: “Pues la naturaleza es simple y no derrocha en superfluas causas de las cosas.”

George Thomas, *Calculus and Analytic Geometry* (Classical edition), Addison-Wesley, 1983.

Geoffrey Whittington, “On the Use of the Accounting Rate of Return in Empirical Research,” in Richard Brief (Editor), *Estimating the Economic Rate of Return from Accounting Data*, Routledge, 1986.

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