RoyaltyStat Blog

Return on Assets Using Adaptive Expectations

Posted by Ednaldo Silva

In transfer pricing, certain analysts prefer using “return on assets” even for businesses such as wholesale or retail trade in which assets are not expected to have a significant impact on operating profits. These analysts postulate a simple linear relationship between operating profits and accounting assets (variously defined) and calculate quartiles without respite. The econometric model underlying the single-variable computation of the quartiles of “return on assets” can be written as:

(1)     P(t) = β K(t) + U(t)

for t = 1 to T years of each selected comparable.

The annual variable P(t) denotes operating profits after depreciation (excluding impairment charges or the amortization of acquired intangibles), K(t) denotes net assets (aka “capital stock”) in period t, and U(t) denotes a residual error. See OECD (2017), ¶¶ 2.103-2.104 (Cases where the net profit is weighted to assets).

First, the definition of end-of-year assets K(t) must be consistent with the annual flow of investments generating such assets. See

Second, some analysts prefer using average assets, a decision that makes the algebra of equation (1) less palatable because of inconsistent manipulation on the RHS (right-hand side) but not on the LHS (left-hand side) of equation (1).

We have indicated on prior blogs that calculating the slope coefficient of equation (1) produces biased results because K(t) is not exogenous (“An endogenous variable is a variable that is correlated with the regression error term”). Hamilton (1994), p. 225. To correct this problem, we must introduce a structural equation for K(t) and estimate the partial slope coefficients of the reduced-form model of the selected profit indicator. See

Among others, we consider an economic model in which net investment is proportional to the difference between actual and desired net capital stock (aka net assets):

     (2)     K(t) − K(t − 1) = γ [(Z(t) − K(t − 1)],

              K(t) = γ [(Z(t) − K(t − 1)] + K(t − 1),

              K(t) = γ Z(t) + (1 – γ) K(t − 1)

where Z(t) denotes the desired capital stock.

In economics, equation (2) is called the “partial adjustment” (aka “adaptive expectations” or “habit persistence”) model of net investment. See Maddala (1977), pp. 142-143. This partial adjustment model (2) is equivalent to the distributed-lag model that we expounded on prior blogs. See

Since the desired capital stock is unknown, we can assume that it’s proportional to the net sales of the comparable entity whose profit indicator we want to estimate:

     (3)     Z(t) = α S(t),

where S(t) denotes actual net sales.

We substitute equation (3) into (2), and obtain:

     (4)     K(t) = γ α S(t) + (1 – γ) K(t − 1)

See Maddala (1977), pp. 142-143 and Kmenta (1986), pp. 528-531, 535, 590. As described on prior blogs, the term (γ α S(t)) is equivalent to gross investments in the perpetual inventory version of equation (4)

We substitute equation (4) into (1) and obtain a reduced-form regression equation to estimate “return on assets” correcting the missing S(t) variable bias:

     (5)     P(t) = β [γ α S(t) + (1 – γ) K(t − 1)] + U(t)

     (6)     P(t) = λ1 S(t) + λ2 K(t − 1) + U(t) ↔ Operating profits of each comparable.

where λ1 = α β γ and λ2 = β (1 – γ).

We posit that equation (6) is a proper method to test for the significance of asset intensity while the prevailing ad hoc accounting equation used ad nauseum (often accompanied by a “best practice” oxymoron) is indefensible. The OECD is a major source of misguidance proposing a contrived method in “Comparability Adjustments” (July 2010):

If we divide equation (6) by S(t), we obtain the profit margin as a linear function of the comparable company’s measure of asset intensity [K(t − 1) / S(t)]:

     (7)     M(t) = λ1 + λ2 [K(t − 1) / S(t)] + V(t) ↔ Operating margin.

It’s apparent that the intercept of equation (7) is the adjusted profit margin after this asset intensity consideration, because:

     (8)     M(t) − λ2 [K(t − 1) / S(t)] = λ1 + V(t) ↔ Adjusted operating margin.

Likewise, if we divide equation (6) by K(t − 1), we obtain “return on assets” (note that the denominator of “return on assets” is lagged by one period) as a linear function of the reciprocal of asset intensity:

     (9)     R(t) = λ1 [S(t) / K(t − 1)] + λ2 + V(t) ↔ Return on assets.

The same reduced-form equation (6) is adaptable and can be interpreted as profit margin or return on assets. However, we must test if the partial regression coefficients λ1 and λ2 of the reduced-form regression model (6) are significant, and if the adjusted R2 is acceptable.

We must test also the residuals U(t) of regression equation (6) to determine if they are well-behaved; if they are ill-behaved and show (e.g.) first-order serial correlation, we can estimate regression equation (6) using the method of Cochrane-Orcutt (CORC) or Prais-Winsten (PW). See Kmenta (1986), pp. 313-320.

The CORC and PW algorithms are incorporated in RoyaltyStat’s transfer pricing SaaS application, together with the Newey-West [1987] standard errors of the regression coefficients. See Hamilton (1994), pp. 281-282 and Hayashi (2000), pp. 408-410.

The Newey-West estimates correct for unknown time-series correlation among the residuals and for their heterogeneous variances in cross-section samples of comparables. The Newey-West estimates produce the same regression coefficients as OLS (ordinary least squares) and the same residual graphs. Therefore, they produce the same Durbin-Watson statistics as OLS. However, they provide different (more robust) standard errors of the regression coefficients, and thus they produce different t-statistics used to test hypotheses about the statistical significance of the regression coefficients.

We may obtain a better statistical fit using a power function version of equation (6), which means that determining a reliable profit indicator from selected comparables requires knowledge of data analysis beyond computation of quartiles. This blog shows that calculating quartiles of a univariate profit indicator is not likely to produce the most reliable estimate of the selected profit indicator—violating the parity principle that we must determine reliable measures of arm’s length taxable profits.

As noted on prior blogs, our major objection to using equation (6) is that the transfer pricing definitions of operating assets K(t) are flawed, making related-party compliance based on “return on assets” more vulnerable to attack. We can get resilience computing “return on assets” by restricting K(t) to be property, plant and equipment (PPE) because this more homogeneous mixture of assets relates to CAPX as its flow variable. To avoid heterogenous assets (which are difficult to establish comparability), we can consider the change in profits as a function of investments. Please revisit our discussion of the perpetual inventory model of net assets, and see IAS 16 (PPE):


James Hamilton, Time Series Analysis, Princeton University Press, 1994.

Fumio Hayashi, Econometrics, Princeton University Press, 2000.

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

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

OECD, Transfer Pricing Guidelines, 2017.


Published on Sep 27, 2019 12:03:39 PM

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:

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Topics: Net Profit Indicator, OECD adjustment is spurious