The problem is how to estimate the "average". A standard practice, as adopted by Thomson Reuters, is to calculate a market cap-weighted average of the ratios in question. But taking an arithmetic average of ratios is fundamentally skewed.

Let's think about the PE ratio. Some investors elect to use PE while some others use its reciprocal, the Earnings Yield. This is only a personal choice on the flavor of numbers and shouldn't affect the final result of investment. Indeed, this isn't a problem for individual companies. Because EY = 1 / PE, the cheapest company always has the lowest PE and the highest EY.

But it will cause problems if an investor uses the arithmetic average of PE or EY to value an industry. In most cases, the reciprocal of average PE is not equal to the average EY. That is

Avg(EY) ≠ 1 / Avg(PE).

It's really an awkward situation when the cheapest industry measured by average PE is not the same one measured by average EY. For example, consider an industry with two companies whose PEs are 10 and 20, and another one with 14 and 15. The former has a higher average PE at 15, but the latter has a lower average EY at 6.9%.

A financial ratio R = Y / X is determined by two numbers, its dividend Y and its divisor X. Geometrically, we can represent the pair of numbers by a point (X, Y) on a two-dimensional coordinate system. We can draw a straight line from the origin point to the point (X, Y). An angle A is formed by the line and the X-axis. The ratio R is simply the tangent of the angle A. This relation is expressed by the following equation.

Tan(A) = R = Y / X

Unlike ratios where the reciprocal of average is not equal to the average of reciprocals, angles do preserve such equality. This suggests that we convert ratios into angles when estimating industry averages. We define the Industry Ratio Average function IRAvg to be

IRAvg(R) = Tan(Avg(Atan(R)))

Remember that 1 / Tan(x) = Tan(π/2 – x), and π/2 – Atan(x) = Atan(1/x). We have

1 / IRAvg(R) = 1 / Tan(Avg(Atan(R))) = Tan(π/2 – Avg(Atan(R)))

= Tan(Avg(π/2 – Atan(R))) = Tan(Avg(Atan(1/R))) = IRAvg(1/R)

Thus IRAvg(EY) = IRAvg(1 / PE) = 1 / IRAvg(PE). The definition guarantees that an industry has the lowest "average" PE also has the highest "average" EY.

**A Little Bit of Probability Theory**

Because a company can be affected by various one-time items or the accounting scheme it adopts, the number X and Y come with errors. Usually we can assume such errors following a two-dimensional normal distribution, which can be represented by a series of circles centered at (X, Y). Points on the same circle have the same distance to (X, Y) and, roughly speaking, come with the same probability.

Consider the rays shooting out of the origin point. Points on the same ray have the same ratio R and angle A and make no difference to our calculation. The rays project the error distribution of (X, Y) to an error distribution of ratios R and an error distribution of angles A. As shown on the chart by angles At and Ab, as well as ratios Rt and Rb, the projected distribution on angles is symmetric while that on ratios is skewed.

For arithmetic average to work, the error distribution has to be symmetric such that errors has the same probability can cancel out one another. That's why the average of angles works, but not the average of ratios.

Using a circle to represent a two-dimensional normal distribution requires that the error of X and the error of Y are of comparable magnitude. If not, the distribution is shaped in an oval and its projection on angles will also be skewed. In this case, even average of angles won't work.

One possibility with financial rations is that X and Y are not in the same unit. For example, ROE is usually expressed in percentage. If we take the number at face value, we are accounting the Return in cents and the Equity in dollars. Consequently an error of one dollar on the Return is expressed in cents as 100, while the same one dollar error on the Equity is expressed in dollar as 1, a 100 times difference. To avoid this problem, we need to divide percentage numbers by 100 before converting it to angles.

A theory is good only if it closely models the reality. The charts below shows the distribution of PB for the regional banks industry, as well as Atan(PB). It shows that PB is skewed compared to its estimated normal distribution. But Atan(PB) is symmetric and matches closely to a normal distribution.

**Other Techniques**

An analyst will adopt the following routine techniques to arrive at a better estimation.

Our approach is merely a step toward the real industry ratios. We welcome competitive ideas from the community thus together we can be closer to the finish line.

#### About the author:

*Author of ETF Ranking Newsletter.
*

An amateur trader / investor that tried many methods including buy and hold, technical analysis, quantitative analysis, day trading with pattern, and finally settled with fundamental analysis to discover undervalued quality stocks.

**Trustamind**

An amateur trader / investor that tried many methods including buy and hold, technical analysis, quantitative analysis, day trading with pattern, and finally settled with fundamental analysis to discover undervalued quality stocks.