Transfer pricing practice usually determines the arm’s length taxable income using the interquartile range of the chosen profit indicator, defined as a ratio m(i) = Y(i)/X(i) across comparable companies. This post shows that when the data include a nonzero intercept—a common trait of corporate financial data—those quartiles are biased. The bias equals a/H, where the numerator is the intercept and H is the harmonic mean of X(i). We then describe a three-step regression process that properly accounts for the intercept and produces an unbiased (defensible) benchmark.

1. The Linear Model and the Profit Level Indicator

Suppose the relationship between a profit measure Y (e.g., operating income) and a base X (e.g., net sales, total cost, or total assets) across N comparable firms (or enterprises) is described by the linear regression model:

\quad Y_i = a + b \cdot X_i + \varepsilon_i, \quad i = 1, \dots, N

where a is the intercept (capturing fixed costs or scale effects), b is the slope (the proportional return), and ε(i) is a mean-zero disturbance. The profit level indicator (PLI) is the marginal (first derivative) return on the base:

\quad \mathrm{PLI} \equiv \frac{dY}{dX} = b

This is scale-invariant, an economically meaningful measure of arm’s length profitability. It is the same for a large firm and a small firm that share the same underlying economics. Standard OLS yields the best linear unbiased estimator b̂ of this quantity by the Gauss–Markov theorem.

2. Why the Ratio m(i) = Y(i)/X(i) Is Contaminated

Substituting the model into the ratio gives:

\quad m_i = \frac{Y_i}{X_i} = b + \frac{a}{X_i} + \frac{\varepsilon_i}{X_i}

The ratio does not equal the PLI b. It is inflated (or deflated) by the term a/X(i), which varies inversely with firm size. A comparable with a small base X(i) will show a high ratio even if its true proportional profitability is identical to a large comparable. This is the contamination (unreliable measure of the PLI).

2.1 The Bias of the Sample Mean

Taking the sample mean across comparables:

\quad \bar{m} = b + \frac{a}{N} \sum_{i=1}^{N} \frac{1}{X_i}

The harmonic mean of X(1), …, X(N) is defined by:

Substituting:

\quad \bar{m} = b + \frac{a}{H}

The mean of the ratio exceeds (or falls below) the true PLI by a/H

2.2 The Bias of the Quartiles

The same logic applies to every quantile of the distribution of m(i). Because each observation is shifted by a/X(i), the p-th quantile of m(i) satisfies

\quad Q_p[m] = b + a \cdot Q_p\!\left[\frac{1}{X}\right]

When averaged across the interquartile range, the shift reduces to a/H. The conventional arm’s length range [Q₁, Q₃] is therefore shifted entirely away from b by this amount. A tested party whose true PLI equals b could be found outside the range simply because the comparables happen to be smaller (or larger) firms. This is not an economic finding—it is a statistical artefact.

3. The Correct Three Steps Procedure

The following procedure eliminates the contamination and produces a benchmark that is both statistically rigorous and economically interpretable.

Step 1 — Test Whether the Intercept is Statistically Nonzero

Before concluding that the bias is material, test the null hypothesis that a = 0 using the standard OLS t-statistic:

\quad t = \frac{\hat{a}}{\mathrm{SE}(\hat{a})} \sim t(N - 2)

If H₀: a = 0 cannot be rejected at a chosen significance level, the proportional model Y(i) = b·X(i) is consistent with the data. In that special case the contamination term vanishes and ratio-based analysis is approximately unbiased. Proceed to the ratio range only in this case.

If H₀ is rejected—as it typically is with corporate financial data—the intercept is statistically significant and the contamination is real. Move to Step 2.

Step 2 — Use the Regression Benchmark, Not the Ratio Quartile

For a tested party with base K* (e.g., its net sales), the arm’s length benchmark for its profit is the OLS fitted value:

\quad \hat{Y}^{*} = \hat{a} + \hat{b} \cdot K^{*}

This single point estimate uses both the intercept and the slope and is, by the Gauss–Markov theorem, the minimum-variance linear unbiased estimate of the expected arm’s length profit at K*. It replaces the median of the ratio distribution.

Step 3 — Construct the Arm’s Length Range as a Regression Interval

There are two standard intervals, each appropriate for a different question:

Confidence Interval (for the expected arm’s length profit coefficient):

\quad \hat{Y}^{*} \pm t_{\alpha/2}(N-2) \cdot \mathrm{SE}(\hat{Y}^{*})

where SE(Ŷ*)² = s² · xᵀ(XᵀX)⁻¹x, with x = [1, K*]ᵀ and s² the OLS residual variance.

Prediction Interval (for the profit level of a single new entity):

\quad \hat{Y}^{*} \pm t_{\alpha/2}(N-2) \cdot s \cdot \sqrt{1 + x^{\top}(X^{\top}X)^{-1}x}

Wider than the confidence interval; accounts for both estimation error and residual variance.

In practice, the confidence interval for the regression coefficients is the more appropriate choice for transfer pricing, since the question is whether a specific tested party’s profit indicator falls within the arm’s length range—not whether it equals the expected value for a firm of its size. Both intervals correctly propagate the nonzero intercept and require no modification when a ≠ 0.

4. Summary: Ratio Quartiles vs. Regression Intervals

5. Practical Implications

The framework above has three direct implications for transfer pricing practice:

These recommendations do not require more data—the same comparables used to compute the ratio interquartile range are used to estimate the regression coefficients. They do require that practitioners move from a purely percentile-based framework to a model-based approach, which is already the standard in every other domain of applied econometrics.

I appreciate the comments of Jon Breslaw and Florian Semani — both are free of any errors.