The Harmonic Mean Bias of Ratio Profit Indicators

Abstract

Let Y(i)=\alpha+\beta X(i) for i=1,\dots,N and let m(i)=Y(i)/X(i) denote the ratio indicator. We show that the arithmetic mean of m(i) overstates the slope \beta by exactly \alpha/X_{H}, where X_{H} is the harmonic mean of X(i); that the p-th quantile of m(i) overstates \beta by exactly \alpha/X_{H,p}, where X_{H,p} is the corresponding harmonic quantile; and that the interquartile range of m(i) is proportional to |\alpha| and to the dispersion of 1/X(i), and is independent of \beta. The bias is structural and does not vanish as N increases. The ratio indicator is unbiased if and only if \alpha=0.

1. Setup and Exact Decomposition

Let there be N observations on paired variables \{Y(i),X(i)\}, i=1,\dots,N, with X(i) > 0 for every i, related by:

\quad
(1)\qquad
Y(i)=\alpha+\beta X(i)

Here Y(i) denotes operating profit and X(i) denotes the profit denominator (revenue, total cost, or operating assets), \alpha is the intercept, and \beta is the slope parameter of interest. Define the ratio indicator:

\quad
(2)\qquad
m(i)=\frac{Y(i)}{X(i)}

Substituting (1) into (2) and dividing term by term:

\quad
(3)\qquad
m(i)=\frac{\alpha+\beta X(i)}{X(i)}=\beta+\frac{\alpha}{X(i)}

Write u(i)\equiv 1/X(i), so that:

\quad
(4)\qquad
m(i)=\beta+\alpha\,u(i)

Equation (4) is the whole argument. The ratio indicator is an affine function of the reciprocal of the scale variable. The slope \beta enters only as a location shift; the intercept \alpha enters only as a scale factor on u(i). Every result below is a direct consequence.

2. The Mean of Ratios

Proposition 1 (Harmonic mean bias). Let X_{H} denote the harmonic mean of X(i):

\begin{aligned}
&(5)\quad
X_{H}=\left[\frac{1}{N}\sum_{i=1}^{N}\frac{1}{X(i)}\right]^{-1}
\qquad
\Longleftrightarrow
\qquad
\frac{1}{N}\sum_{i=1}^{N}\frac{1}{X(i)}=\frac{1}{X_{H}}
\end{aligned}

Then:

\quad
(6)\qquad
\bar{m}-\beta=\frac{\alpha}{X_{H}}

Proof. Average (3) over i:

\begin{aligned}
&(7)\quad
\bar{m}=\frac{1}{N}\sum_{i=1}^{N}m(i)
=\frac{1}{N}\sum_{i=1}^{N}\left[\beta+\frac{\alpha}{X(i)}\right]\\
&\quad
=\beta+\frac{\alpha}{N}\sum_{i=1}^{N}\frac{1}{X(i)}
=\beta+\frac{\alpha}{X_{H}}
\end{aligned}

where the last equality applies definition (5).

The average ratio does not estimate the slope. It estimates the slope plus \alpha/X_{H}. The bias term contains no N: enlarging the comparable set sharpens the estimate of a biased quantity. This is the sense in which the defect is structural rather than a small-sample artifact.

3. The Quantiles of Ratios

3.1 Invariance of quantiles under affine maps

Let Q_p(\cdot) denote any sample quantile estimator that returns a convex combination of two adjacent order statistics:

\quad
(8)\qquad
Q_p(z)=(1-\lambda)\,z_{(k)}+\lambda\,z_{(k+1)},
\qquad
\lambda\in[0,1]

with k and \lambda determined by p and N alone. This class includes the order-statistic definition (\lambda=0), the Tukey hinges, and the linear-interpolation rule (type 7) used by default in R and numpy.

Lemma 2. For constants c and d \neq 0, and w(i)=c+d\,z(i):

\quad
(9)\qquad
Q_p(w)=
\begin{cases}
c+d\,Q_p(z), & d > 0\\[2pt]
c+d\,Q_{1-p}(z), & d < 0
\end{cases}

Proof. For d > 0 the map is strictly increasing, so ranks are preserved, w_{(k)}=c+d\,z_{(k)}, and (8) gives Q_p(w)=(1-\lambda)(c+d z_{(k)})+\lambda(c+d z_{(k+1)})=c+d\,Q_p(z); the convex combination passes through the affine map because affine maps commute with convex combinations. For d < 0 the map is strictly decreasing, so ranks reverse, w_{(k)}=c+d\,z_{(N+1-k)}, and the same argument applies at the reflected level 1-p.

3.2 The harmonic quantile

Define the harmonic quantile of X at level p as the reciprocal of the corresponding quantile of the reciprocals:

\quad
(10)\qquad
X_{H,p}\equiv\left[Q_{p}\!\left(1/X\right)\right]^{-1}

Definition (10) stands to Q_p exactly as the harmonic mean (5) stands to the arithmetic mean: in both cases one averages the reciprocals and inverts the result.

Proposition 3 (Quantile bias). For \alpha > 0 and any p \in (0,1):

\quad
(11)\qquad
Q_{p}\!\left(m\right)-\beta=\frac{\alpha}{X_{H,p}}

and for \alpha < 0:

\quad
(12)\qquad
Q_{p}\!\left(m\right)-\beta=\frac{\alpha}{X_{H,1-p}}

For \alpha=0, m(i)=\beta for every i, and every quantile of m equals \beta.

Proof. Apply Lemma 2 to (4) with c=\beta, d=\alpha, z=u=1/X. For \alpha > 0, Q_{p}\!\left(m\right)=\beta+\alpha\,Q_{p}\!\left(1/X\right)=\beta+\alpha/X_{H,p} by (10). For \alpha < 0 the rank reversal of Lemma 2 replaces p by 1-p. For \alpha=0, (3) collapses to m(i)=\beta.

Proposition 3 states the exact result. Its practical content is easier to read after the following corollary, which translates the harmonic quantile back into a quantile of X itself.

Corollary 4 (Rank inversion). Because t \mapsto 1/t is strictly decreasing on (0,\infty), the order statistics of u and X are related by u_{(k)}=1/X_{(N+1-k)}. Consequently, for a continuous population distribution of X:

\quad
(13)\qquad
Q_{p}\!\left(1/X\right)=\frac{1}{Q_{1-p}\!\left(X\right)},
\qquad
\text{so}
\qquad
X_{H,p}=Q_{1-p}\!\left(X\right)

and, for \alpha > 0:

\begin{aligned}
&(14)\quad
Q_{.25}\!\left(m\right)=\beta+\frac{\alpha}{Q_{.75}\!\left(X\right)}\\
&\quad
Q_{.50}\!\left(m\right)=\beta+\frac{\alpha}{Q_{.50}\!\left(X\right)}\\
&\quad
Q_{.75}\!\left(m\right)=\beta+\frac{\alpha}{Q_{.25}\!\left(X\right)}
\end{aligned}

Remark 5 (Finite-sample caution). Identity (13) is exact in the population and exact in finite samples under the order-statistic definition with the reflected index convention k \mapsto N+1-k. It is not exact under interpolating quantile rules, because reciprocation is nonlinear: interpolating X and then inverting differs from interpolating 1/X. Proposition 3, stated in terms of X_{H,p}, is exact under every rule in class (8). In applied work, compute Q_{p}\!\left(1/X\right) directly rather than inverting Q_{1-p}\!\left(X\right).

The economic content of Corollary 4 is that the rank order is reversed. When \alpha > 0, the lower quartile of the profit indicator is generated by the upper quartile of the scale variable. The interquartile range of a profit level indicator is populated by the smallest companies at the top and the largest companies at the bottom. Nothing in the ranking of m(i) conveys information about relative profitability; it conveys the inverse ranking of size.

4. The Interquartile Range Is Independent of \beta

Corollary 6 (Range distortion).

\begin{aligned}
&(15)\quad
\mathrm{IQR}(m)
=|\alpha|\left[Q_{.75}\!\left(1/X\right)-Q_{.25}\!\left(1/X\right)\right]
\\[8pt]
&\phantom{(15)\quad \mathrm{IQR}(m)}
=|\alpha|\cdot\mathrm{IQR}(1/X)
\end{aligned}

and, using (13):

\begin{aligned}
&(16)\quad
\mathrm{IQR}(m)
=|\alpha|\left[\frac{1}{Q_{.25}\!\left(X\right)}-\frac{1}{Q_{.75}\!\left(X\right)}\right]
\\[12pt]
&\phantom{(16)\quad \mathrm{IQR}(m)}
=\frac{|\alpha|\cdot\mathrm{IQR}(X)}
{Q_{.25}\!\left(X\right)\,Q_{.75}\!\left(X\right)}
\end{aligned}

Proof. Differencing Proposition 3 at p=.75 and p=.25 eliminates \beta, which enters both quantiles as the same additive constant. The final equality follows from A^{-1}-B^{-1}=(B-A)/(AB) with A=Q_{.25}\!\left(X\right) and B=Q_{.75}\!\left(X\right).

The slope has cancelled identically. Under the maintained linear hypothesis (1), the interquartile range of the ratio indicator carries no information whatever about the economic relation between Y and X. It is a function of the intercept and of the size dispersion of the selected companies. Extend the screen to smaller companies and the range widens mechanically; restrict it to companies of comparable scale and the range narrows toward zero, without any change in the underlying economics. An arm’s length range constructed in this way is a statement about the sampling of scale, not about profitability.

5. Comparison with the Aggregate Ratio

Practitioners sometimes compute the ratio of sums rather than the mean of ratios. This is a different, and smaller, error. From (1):

\quad
(17)\qquad
\frac{\sum_{i}Y(i)}{\sum_{i}X(i)}
=\frac{N\alpha+\beta\sum_{i}X(i)}{\sum_{i}X(i)}
=\beta+\frac{\alpha}{\bar{X}}

The bias is \alpha/\bar{X} rather than \alpha/X_{H}. By the arithmetic-harmonic mean inequality, X_{H} \leq \bar{X}, with equality only if X(i) is constant. Hence for \alpha > 0:

\quad
(18)\qquad
\frac{\alpha}{X_{H}}\geq\frac{\alpha}{\bar{X}} > 0

The mean of ratios is more biased than the ratio of means. The gap is the Jensen discrepancy \alpha\left[\mathbb{E}_N(1/X)-1/\mathbb{E}_N(X)\right] \geq 0. Expanding 1/X about \mu_X to second order, \mathbb{E}(1/X) \approx \mu_X^{-1}(1+\mathrm{CV}^2) with \mathrm{CV}=\sigma_X/\mu_X, so that:

\quad
(19)\qquad
\frac{\alpha}{X_{H}}\approx
\frac{\alpha}{\bar{X}}\left(1+\mathrm{CV}_{X}^{2}\right)

To this order, the excess bias of the mean of ratios over the ratio of means equals the squared coefficient of variation of the denominator. Neither quantity estimates \beta.

6. Numerical Illustration

The table below takes eight companies with revenue X in millions of dollars, \alpha=5.0, and \beta=0.06. The data satisfy (1) exactly: there is no sampling error of any kind, and the true operating margin parameter is 6.00 percent.

The relevant statistics are:

\begin{aligned}
&(20)\quad
\bar{X}=113.75,
\qquad
X_{H}=34.5324,
\qquad
\mathrm{CV}_{X}=1.0820
\\[8pt]

&\quad
\bar{m}=0.204792
=0.06+\frac{5.0}{34.5324}
=\beta+\frac{\alpha}{X_{H}}
\\[8pt]

&\quad
\frac{\sum Y}{\sum X}=0.103956
=0.06+\frac{5.0}{113.75}
=\beta+\frac{\alpha}{\bar{X}}
\\[12pt]

&\quad
Q_{.25}\!\left(m\right)=0.0975,
\qquad
Q_{.50}\!\left(m\right)=0.14125,
\qquad
Q_{.75}\!\left(m\right)=0.2475
\\[8pt]

&\quad
\mathrm{IQR}(m)=0.1500
=5.0\times\left(0.0375-0.0075\right)
=|\alpha|\cdot\mathrm{IQR}(1/X)
\end{aligned}

The true slope is 6.00 percent. The mean of ratios reports 20.48 percent. The aggregate ratio reports 10.40 percent. The interquartile range runs from 9.75 percent to 24.75 percent and therefore does not contain the true parameter at all: the arm’s length range brackets nothing. Note also that \mathrm{IQR}(m)=0.15 is two and a half times \beta itself, generated entirely by \alpha and by the spread of 1/X. The tested party could be assessed as low-margin for no reason other than being large.

7. Introducing a Disturbance

Nothing above depends on (1) holding exactly. Let

\begin{aligned}
&(21)\quad
Y(i)=\alpha+\beta X(i)+\varepsilon(i)\\
&\quad
\mathbb{E}[\varepsilon(i)\mid X(i)]=0,
\qquad
\operatorname{Var}[\varepsilon(i)\mid X(i)]=\sigma^{2}
\end{aligned}

Dividing by X(i):

\quad
(22)\qquad
m(i)=\beta+\frac{\alpha}{X(i)}+\frac{\varepsilon(i)}{X(i)}

so that:

\begin{aligned}
&(23)\quad
\mathbb{E}[m(i)\mid X(i)]=\beta+\frac{\alpha}{X(i)}\\
&\quad
\operatorname{Var}[m(i)\mid X(i)]=\frac{\sigma^{2}}{X(i)^{2}}
\end{aligned}

Two consequences follow. First, the bias results of Propositions 1 and 3 hold unchanged in conditional expectation; the disturbance adds variance, not correction. Since \mathrm{MSE} = \mathrm{Bias}^2 + \mathrm{Variance}, no amount of attention to the variance term discloses the bias term.

Second, the ratio transformation manufactures exact heteroskedasticity with known skedastic function \sigma/X(i). The smallest companies are the noisiest observations, and by Corollary 4 they are precisely the observations that the rank inversion drives into the upper quartile. The upper tail of a profit level indicator is where bias and variance are simultaneously largest.

7.1 The efficient estimator is the levels regression

Given (23), efficient estimation of (22) weights each observation by X(i)^2. The weighted criterion is:

\begin{aligned}
&(24)\quad
\sum_{i=1}^{N}X(i)^{2}
\left[m(i)-\frac{\alpha}{X(i)}-\beta\right]^{2}
\\[8pt]
&\quad
=\sum_{i=1}^{N}
\left[X(i)\,m(i)-\alpha-\beta X(i)\right]^{2}
\\[8pt]
&\quad
=\sum_{i=1}^{N}
\left[Y(i)-\alpha-\beta X(i)\right]^{2}
\end{aligned}

which is the ordinary least squares criterion applied to (21) in levels. Generalized least squares on the ratio is algebraically identical to ordinary least squares on the levels (Aitken, 1935). The regression is therefore not an alternative to the ratio analysis. It is the ratio analysis, carried out with the intercept estimated rather than assumed to be zero.

8. Implications

Note on Notation

The symbol H is reserved throughout EdgarStat documents for the projection (hat) matrix, with M = I - H the residual maker. The harmonic mean is therefore written X_{H} here, and the harmonic quantile X_{H,p}, to avoid collision. Readers who encounter the result stated elsewhere as “bias = \alpha/H” should read H as the harmonic mean of X.

References

Aitken, A. C. (1935). On least squares and linear combinations of observations. Proceedings of the Royal Society of Edinburgh, 55, 42–48. Canonical statement of generalized least squares; establishes the equivalence invoked in equation (24).

Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and Boyd, Edinburgh, p. 47. Origin of the convenience threshold underlying the reporting of estimates with their standard errors.

JCGM 100:2008. Evaluation of Measurement Data: Guide to the Expression of Uncertainty in Measurement (GUM). BIPM. Metrological authority for the k = 1 standard uncertainty; a scalar uncertainty-propagation framework that presumes the measurement model is known.

Weisberg, S. (2014). Applied Linear Regression, 4th ed. Wiley, Hoboken. Standard reference for the levels regression and for weighted least squares under known skedastic function.

Ednaldo Silva (Ph.D.) is founder and managing director of EdgarStat LLC. He was the first senior economic advisor to the IRS Office of the Chief Counsel and the first senior economist in the APA program, and participated in drafting the 1992–1994 section 482 regulations.