In DuPont’s profit identity, “return on assets” (profit rate) is equal to profit margin multiplied by asset turnover. Using DuPont’s profit formula, combined with an assumption that profit margin and asset turnover (measures of economic performance and adopted technology) are independent, we showed on a prior blog that the coefficient of variation of return on assets is greater than of the profit margin. It follows that profit margin is more reliable (has a lower coefficient of variation) than return on assets. In science, coefficient of variation is an accepted method of determining the reliability of a selected variable.

Here, we expand the short formula of the coefficient of variation of return on assets to show in more detail the algebra involved. We start with several useful definitions and statistical results obtained on a prior blog published on October 22, 2018:

Define the relevant M, R, and T variables and assume that M and T are independent (assume they are not correlated):

(1) R = [M ∙ T] = *asset return *(OECD lingo: “return on assets”), where
(2) M = Profit / Revenue = *profit margin*

(3) T = Revenue / Assets = *asset turnover*

(4) E[M ∙ T] = µ_{R} = µ_{M} µ_{T}

(5) Var[M ∙ T] = σ_{R}^{2} = σ_{M}^{2} σ_{T}^{2} + σ_{M}^{2} µ_{T}^{2} + σ_{T}^{2} µ_{M}^{2}

(6) K_{R}^{2} = (σ_{R}^{2} / µ_{R}^{2}) = {σ_{M}^{2} σ_{T}^{2} + σ_{M}^{2} µ_{T}^{2} + σ_{T}^{2} µ_{M}^{2}} / µ_{M}^{2} µ_{T}^{2}

(7) K_{R}^{2} = {K_{M}^{2} (K_{T}^{2} + 1) + K_{T}^{2}} > K_{M}^{2}

Now, we expand the coefficient of variation formula (6) to obtain a short version (7):

(6) K_{R}^{2} = (σ_{R}^{2} / µ_{R}^{2}) = {σ_{M}^{2} σ_{T}^{2} + σ_{M}^{2} µ_{T}^{2} + σ_{T}^{2} µ_{M}^{2}} / µ_{M}^{2} µ_{T}^{2}

= (σ_{M}^{2} σ_{T}^{2} / µ_{M}^{2} µ_{T}^{2}) + (σ_{M}^{2} µ_{T}^{2} / µ_{M}^{2} µ_{T}^{2}) + (σ_{T}^{2} µ_{M}^{2} / µ_{M}^{2} µ_{T}^{2})

= K_{M}^{2} K_{T}^{2} + K_{M}^{2} + K_{T}^{2}

(7) K_{R}^{2} = {K_{M}^{2} (K_{T}^{2} + 1) + K_{T}^{2}},

which shows that K_{R}^{2 }is greater than K_{M}^{2} because K_{T}^{2} is positive.

Takeaway: Under the M and T independence assumption, equation (7) shows that the claimed *à priori* advantage of return on asset over profit margin can't hold, and we must rely on analysis of facts and circumstances of the audit case.