1. The issue: a ratio is a restricted model
Financial ratios are the lingua franca of transfer pricing and credit analysis. Profit indicators such as \mathrm{OIBDP}/\mathrm{REVT} and coverage indicators such as \mathrm{OIBDP}/\mathrm{XINT}, leverage covenants expressed as net debt/EBITDA, and distress scores associated with Altman (1968) and Ohlson (1980) have made ratio analysis a widespread practice. A ratio may be useful as a descriptive statistic, but its interpretation as a scale-independent benchmark imposes a proportionality assumption. The restriction should be examined, not presumed.
Let Y and X be two measured accounting variables, with X \neq 0. Define the observed ratio R=Y/X. Two distinct departures from proportionality merit separate treatment: an additive intercept in a linear specification and nonunit elasticity in a power specification. The identities below are deterministic; disturbances and statistical inference are introduced separately.
2. Linear specification: the intercept problem
Start with the affine relation:
\quad (1)\qquad Y=\alpha+\beta X
Dividing by X yields the exact identity:
\quad
(2)\qquad
R\equiv\frac{Y}{X}=\beta+\frac{\alpha}{X}.Equation (2) shows that the ratio equals the marginal slope \beta only when \alpha=0. If \alpha \neq 0, the ratio varies inversely with the denominator, despite a constant marginal slope. Indeed:
\quad
(3)\qquad
\frac{dR}{dX}=-\frac{\alpha}{X^{2}},\qquad\frac{dY}{dX}=\beta.For \alpha > 0, R decreases as positive X increases; for \alpha < 0, it increases. Two enterprises with identical \beta but different X may therefore have different ratios even though the model specifies the same marginal relation. Ratios can confound size with the relation they purport to summarize.
For a sample of n observations with nonzero X(i), the unweighted mean ratio obeys:
\quad
(4)\qquad
\bar{R}=\frac{1}{n}\sum_{i=1}^{n}\frac{Y(i)}{X(i)}=\beta+\alpha\,\frac{1}{n}\sum_{i=1}^{n}\frac{1}{X(i)}.The ratio of sums is a different quantity:
\quad
(5)\qquad
\frac{\sum_{i=1}^{n}Y(i)}{\sum_{i=1}^{n}X(i)}=\beta+\frac{n\alpha}{\sum_{i=1}^{n}X(i)},provided the sum of the denominators is nonzero. Neither equals \beta generally when \alpha \neq 0. Thus, averaging ratios, aggregating accounts before division, and estimating the regression slope are distinct operations.
2.1 Statistical version and the 68% convention
Introduce a regression disturbance:
\quad (6)\qquad Y(i)=\alpha+\beta X(i)+\varepsilon(i),\qquad E\left(\varepsilon(i)\mid X(i)\right)=0.
\quad
(7)\qquad
\frac{Y(i)}{X(i)}=\beta+\frac{\alpha}{X(i)}+\frac{\varepsilon(i)}{X(i)}.Conditional on nonzero X(i), E\left(Y(i)/X(i)\mid X(i)\right)=\beta+\alpha/X(i). Under \alpha=0 and the stated conditional-mean assumption, the conditional mean ratio equals \beta. This does not imply that sample quartiles equal \beta, nor that ratios are statistically efficient. If denominators approach zero, the dispersion of \varepsilon(i)/X(i) may become large; the existence of unconditional moments also requires regularity conditions.
For a parsimonious initial diagnostic, compare the fitted intercept with one of its estimated standard errors:
\quad
(8a)\qquad
\left|\hat{\alpha}\right|<\operatorname{SE}(\hat{\alpha})\;\to\;\text{zero lies inside }\hat{\alpha}\pm\operatorname{SE}(\hat{\alpha}).\quad
(8b)\qquad
\left|\hat{\alpha}\right|\ge\operatorname{SE}(\hat{\alpha})\;\to\;\text{zero lies outside or on that interval.}This is the author’s one-standard-error (approximately 68% under normal large-sample inference) screening convention, not an equivalence test. Failure to cross the threshold does not prove \alpha=0; crossing it does not amount to rejection at 5%. Use an appropriate standard error, such as HC3 for cross-sectional heteroskedasticity or HAC for serial dependence, when justified by the data structure. For a large t-statistic reference, a two-sided one-SE cutoff corresponds to roughly 32% significance, not 5%.
3. Power specification: the elasticity problem
A second source of scale dependence arises even without an additive intercept. Consider positive X and Y:
\quad
(9)\qquad
Y=aX^{b},\qquad a>0,\quad X>0.Dividing by X gives:
\quad
(10)\qquad
R\equiv\frac{Y}{X}=aX^{b-1}.The ratio is constant in X if and only if b=1 (for a > 0 and a nondegenerate range of X). The cases are immediate: b > 1 implies the ratio increases with X; b < 1 implies it decreases, including when b is negative. The economically common diminishing-but-positive elasticity case, 0 < b < 1, produces a decreasing ratio.
Differentiation makes the distinction between the average and marginal relations explicit:
\quad
(11)\qquad
\frac{dY}{dX}=abX^{b-1}=b\,\frac{Y}{X}.\quad
(12)\qquad
\frac{dY}{dX}\cdot\frac{X}{Y}=b;\qquad\frac{Y}{X}=\frac{1}{b}\,\frac{dY}{dX},\quad b\neq 0.Thus b is the constant elasticity. The average ratio equals the marginal slope only at b=1 (apart from degenerate cases). Directly differentiating the ratio also gives:
\quad
(13)\qquad
\frac{dR}{dX}=a(b-1)X^{b-2}.A comparison of firms with different X therefore mixes scale with the dependent-variable relationship whenever b differs from unity.
3.1 Log-log estimation and inference
Taking natural logarithms of the positive deterministic variables in (9):
\quad (14)\qquad \ln Y=\ln a+b\ln X.
A log-log regression introduces u(i):
\quad (15)\qquad \ln Y(i)=c+b\ln X(i)+u(i),\qquad c=\ln a.
The analogous one-standard-error screen is centered on unit elasticity:
\quad
(16a)\qquad
\left|\hat{b}-1\right|<\operatorname{SE}(\hat{b})\;\to\;\text{unit elasticity remains within one SE.}\quad
(16b)\qquad
\left|\hat{b}-1\right|\ge\operatorname{SE}(\hat{b})\;\to\;\text{flag evidence of scale dependence.}Again, this is a screening convention. In the stochastic log-log model, Y(i)/X(i)=\exp(c)\,X(i)^{b-1}\exp(u(i)). If E\left(u(i)\mid X(i)\right)=0, the conditional geometric mean ratio is \exp(c)\,X(i)^{b-1} under suitable integrability. But the arithmetic conditional mean is:
\quad
(17)\qquad
E\left(\frac{Y(i)}{X(i)}\mid X(i)\right)=\exp(c)\,X(i)^{b-1}\,E\left[\exp(u(i))\mid X(i)\right].Even if b=1, the arithmetic mean ratio is scale-independent only if the retransformation factor E\left[\exp(u(i))\mid X(i)\right] is constant in X. It equals \exp(\sigma^{2}/2) under conditional normality with constant log-error variance \sigma^{2}, but is not one. Consequently, exponentiating fitted log values does not automatically recover conditional arithmetic means.
4. What is being tested?
The two models imply alternative restrictions, not two hurdles that every financial ratio must independently pass. For an affine conditional-mean model, proportionality entails \alpha=0. For a constant-elasticity specification, proportionality entails b=1, with separate qualifications for retransformation and heteroskedasticity. Model adequacy matters before either restriction is interpreted.
| Specification | Ratio identity | Proportionality restriction |
|---|---|---|
| Affine | Y/X=\beta+\alpha/X | \alpha=0 |
| Power | Y/X=aX^{b-1} | b=1 |
A practical sequence is to inspect the Y–X scatterplot and LOWESS, estimate plausible specifications, examine residual patterns and influential observations, and only then interpret the relevant coefficient restriction. The one-SE screens are deliberately transparent but must not substitute for specification checks or an explicit equivalence margin when the objective is to establish practically negligible departure from proportionality.
5. Application to financial distress indicators
Coverage, leverage, and profitability measures often place an accounting size in the denominator. Comparing their quartiles across enterprises implicitly treats scale differences as either negligible or substantively appropriate. A nonzero intercept can produce an inverse-denominator term; a nonunit elasticity can produce a systematic power of scale. Either may alter a ranking or a quartile without a corresponding change in the underlying slope or elasticity.
An April 2026 author study of eleven U.S. high-technology firms found four estimated intercepts within the chosen one-SE band around zero and seven outside it. These counts are retained here as an author-supplied illustration pending insertion of the underlying regression tables. They do not, by themselves, prove unbiased ratios for four firms or quantify distortion for the remaining seven.
Altman’s discriminant score and Ohlson’s probabilistic bankruptcy model show influential uses of financial ratios in distress prediction; they are not direct validations of scale-independent ratio benchmarking. A predictive score can be useful even when an individual ratio is not a structural or marginal coefficient. Prediction, descriptive comparison, and inference on economic relationships are separate tasks.
6. Conclusion
The two algebraic results are simple: an affine relation produces Y/X=\beta+\alpha/X; a power relation produces Y/X=aX^{b-1}. The first ratio is constant when the intercept vanishes; the second is constant when elasticity equals one. These are restrictions to be investigated, not default properties of accounting data. Financial ratios should remain in credit reports, but regression diagnostics should precede claims that their means or quartiles provide scale-neutral benchmarks.
References
Altman, Edward I. “Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy.” The Journal of Finance 23, no. 4 (September 1968): 589–609. https://doi.org/10.1111/j.1540-6261.1968.tb00843.x.
Ohlson, James A. “Financial Ratios and the Probabilistic Prediction of Bankruptcy.” Journal of Accounting Research 18, no. 1 (Spring 1980): 109–31. https://doi.org/10.2307/2490395.