In several blogs, we postulated that an autoregressive (AR) model can produce more reliable measures of comparable company profit ratios (operating margin over revenue or profit rate over assets) than the naive profit model prescribed by the OECD transfer pricing guidelines. We prefer to work with profit margins because they are pure numbers, unlike profit rates over assets of different vintages. Here, we show the fixed-point equilibrium and the variance of an AR(1) model allowing the computation of a comparable profit ratio interval to benchmark related party transfers of goods and services.

**Fixed-Point Equilibrium**

Consider a simple AR(1) model of company profit ratios:

(1) Y(*t* + 1) = α + β Y(*t*) + U(*t*),

where Y(*t*) is stable when the absolute value of the slope coefficient is lower than one (|β| < 1), and the error term U(*t*) has zero mean and constant variance, σ^{2}.

The variable Y(*t*) is the profit ratio in year *t* = 0, 1, 2, 3, …, T of a selected comparable company, and U(*t*) is a random error because the relationship is stochastic. In transfer pricing, T ≤ 5 years. However, we get more reliable results if we consider more data points than the usual three to five years of company profit ratios. Three years of data are not sacrosanct.

Let μ be the mean (average) operating profit ratio, and E is the expected value operator. The expected value of equation (1) is the simple formula:

(2) E(Y(*t* + 1)) = α + β E(Y(*t*)), where E(U(*t*)) = 0.

(3) μ = α + β μ from which we get the *fixed-point equilibrium*:

(4) μ = α / (1 − β).

If α = 0, then μ = 0 and Y(*t*) fluctuates around zero. See Walter Enders, *Applied Econometric Time Series*, John Wiley & Sons, 1995, which is a good reference for AR models, including coverage of the Dickey-Fuller test for β = 1 (*i.e*., the random walk model). A *fixed point* “is also called a *rest point* of a flow, or an *equilibrium solution* of a differential equation. The rates of change vanish at these points and the variables of the differential equation remain constant.” Florin Diacu & Philip Holmes, *Celestial Encounters* (The Origins of Chaos and Stability), Princeton University Press, 1996, pp. 31, 153, showing the behavior of various flows near a rest point.

**Calculating the Equilibrium Profit Margin**

A chart of the operating profit margin of Target Corp. (CIK 27419; GVKEY 3813) from 1978 to 2016 appears stationary (without a discernible up or down trend). Using Compustat®’s acronym for the “operating margin after depreciation” (OMAD), we used OLS (ordinary least squares) regression analysis to obtain the AR(1) estimates:

(5) OMAD = 2.037 + 0.696 OMAD(-1), where OMAD(*t*) = Y(*t*),

from which we computed the equilibrium value (using equation (4)):

(4) μ = 2.037/(1 – 0.696) = 6.7%.

As stated, Target Corp.’s historical OMAD appears stationary; thus, its equilibrium OMAD is close to the mean = 6.773%, with a small standard deviation = 0.98%, and quartiles Q1 = 6.276%, Q2 (Median) = 6.911%, and Q3 = 7.432%. This is an unusually narrow interquartile range (IQR = 1.156%) that vitiates our need to compute a confidence interval for Target's equilibrium point. The action proposed here is that we would estimate the intercept (α) and slope (β) coefficients of the AR(1) model for every comparable company to the tested party, and then compute a range of their equilibrium values. FYI, we used 39 data points and the Ljung-Box Q statistics = 20.23. The *t*-statistic of the slope coefficient of AR(1) is significant (4.33), but not that of overfitting AR(2) (1.78).

**Variance and Profit Interval**

The variance of Y(*t*) is computed with the formula:

(6) V(Y(*t* + 1)) = β^{2} V(Y(*t*)) + σ^{2},

which is larger than the variance of the random error (σ^{2}) because the term β^{2} V(Y(*t*)) is greater than zero.

Using equation (4) and the square root of (6), we can compute the central value (fixed-point equilibrium) and its encompassing interval for the selected operating profit ratio. In RoyaltyStat®, we have a built-in AR(*p*) function (where the order *p* = 1 to 5) that clients can combine with our subscription-based Global Compustat company financials to select comparables to the tested party and determine a range of equilibrium profit ratios to benchmark the arm’s length transfer of traded goods and services.

**Ednaldo Silva**(Ph.D.) is founder and managing director at 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: [email protected]

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