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− Pablo Neruda (1904−1973)

U.S. 26 CFR § 1.482-5(b)(4)(i−ii) claims that the “return on capital employed” (return on assets, ROA) is less sensitive to “functional differences” than the operating profit margin or the operating profit markup. This claim is based on the unrealistic assumption that “capital flows” equalize profit rates (return on assets) among companies in the same (or different) industries by some “invisible hand.”

This idealism does not reflect corporate reality. The ideal conditions for the equalization of return on assets across companies are implausible. To test this unsupported claim, define the variables M (profit margin), R (return on assets), and T (asset turnover), and let the *coefficient of variation* (CV) be the selected measure of reliability (or efficiency).

We show that the coefficient of variation of M cannot exceed that of R = M T; that is, in the Dupont formula, M is more reliable than R. The cited postulate of the U.S. transfer pricing regulations is wrong.

The coefficient of variation is defined as the ratio of the standard deviation to the mean of the data. The coefficient of variation is a relative measure of dispersion (spread, concentration) of the selected data points around the mean and shows the extent of the data variability around the mean (typical value, central point of attraction). As a result, smaller values of CV are preferred to larger values.

Define the coefficient of variation as:

(1) K^{2} = ( σ^{2} / μ^{2} ) or K = √ ( σ^{2} / μ^{2} )

If M and T are independent variables, the mean and variance are defined as:

(2) μ_{R} = μ_{M} μ_{T}

(3) σ^{2}_{R} = σ^{2}_{M} σ^{2}_{T} + σ^{2}_{M} μ^{2}_{T} + σ^{2}_{T} μ^{2}_{M}

Substituting (2) and (3) into (1), we obtain:

(4) K^{2}_{R} = σ^{2}_{R }/ μ^{2}_{R}

(5) K^{2}_{R} = [ σ^{2}_{M} σ^{2}_{T} + σ^{2}_{M} μ^{2}_{T} + σ^{2}_{T} μ^{2}_{M }] / μ^{2}_{M }μ^{2}_{T}

Using (5) and the coefficient of variation (1), we obtain:

(6) K^{2}_{R} = K^{2}_{M} K^{2}_{T }+ K^{2}_{M} + K^{2}_{T}

(7) K^{2}_{R} = K^{2}_{M} ( K^{2}_{T} + 1 ) + K^{2}_{T}

Thus, K_{R} > K_{M} because K_{T} > 0.

In the Dupont formula R = M T, the return on assets is decomposed into the profit margin times asset turnover. We showed in equations (1) to (7) that the reliability (efficiency) of the return on assets is lower than the reliability of the profit margin. In practice, M must be preferred to R to determine an arm’s length profit indicator because 26 CFR § 1.482-1(e)(1) provides that the “most reliable” profit indicator should be used to determine arm’s length taxable income (operating profits).

We regard M and T as independent variables because M (operating profit / sales ratio) is a dimensionless profit indicator and T (sales / asset ratio) is a technology measure dependent on scale. The U.S. transfer pricing regulations claim that R is less sensitive to “functional differences” than M is false, and such an invalid claim should be abandoned.

In statistics, reliability is often called efficiency (see Thomas Wonnacott and Ronald Wonnacott, *Introductory Statistics*, John Wiley, 1969, pp. 136-137 (“We describe [parameter estimate] as more efficient because it has a smaller variance.”). See also Thad Mirer, *Economic Statistics and Econometrics *(2nd edition), Macmillan, 1988 [1983], p. 254 (“When we compare all unbiased estimators, the one with the smallest variance is said to be *efficient*.”). (Original emphasis).

James Stock and Mark Watson, *Introduction to Econometrics* (4th edition), Pearson, 2019, state (pp. 32, 59) the mathematical expectation of the product E(X Y) = μ_{x} μ_{y} + σ_{xy}. Thus, in equation (2), we assume the covariance σ_{xy} = 0 (variables X and Y are independent). Stock and Watson do not provide the properties of the variance of the product V(X Y).

Equation (2) is stated in Calyampudi Rao, *Linear Statistical Inference and its Applications*, John Wiley, 1965, p. 75, but not the variance (3). UK physics professors, Ifan Hughes & Thomas Hase, *Measurements and their Uncertainties* (Modern Error Analysis), Oxford University Press, 2010, include a comprehensive formulation of best estimates and the respective standard errors (uncertainties).

The synonyms we use for the arithmetic mean (typical value, central point) and for dispersion (spread or measure of data concentration) are found in Harald Cramér, *Mathematical Methods of Statistics*, Princeton University Press, 1946, pp. 178−179. He defines the coefficient of variation (1) on p. 357.

Joseph Gastwirth, *Statistical Reasoning in Law and Public Policy*, Vol. 2, Academic Press, 1988, p. 494, writes: “the *coefficient of variation* (CV) ... reflects the *relative accuracy* of the sample mean.... Of course, the precision required varies with the application. For example, a CV of 0.15 is typically sought by the U.S. Bureau of the Census, while the Consumer Product Safety Commission requires a CV of 0.33 or less for its estimates of the number of accidents. In the present application involving a payment of funds, a CV of 0.05 or 0.10 might be desirable.” (Original emphases).

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