Antitrust M&A practitioners are well familiar with the Herfindahl-Hirschman Index for measuring market concentration. HHI measurements allow parties and regulators to quickly identify mergers that might cause large increases in concentration and thus may raise questions under the antitrust laws. HHIs, however, can be cumbersome to calculate: Sometimes, just to get a single number — the “delta” between the premerger HHI and post-merger HHI — a practitioner is required to conduct dozens of underlying calculations. However, with just one weird trick, antitrust practitioners can cut to the chase and get deltas quickly, easily and accurately.

## The Herfindahl-Hirschman Index

For any given market (however defined), the HHI is based on the estimated market shares of all the individual competitors in that market. Each competitor’s market share, expressed as a percentage, is squared, and then the squared figures are all added together. Thus, in a market with three firms with respective market shares of 50 percent, 30 percent and 20 percent, the HHI would be (50² + 30² + 20²), or 3,800. Using this index, HHIs can theoretically range from zero (a market with infinitely many competitors, each having infinitesimal shares) to 10,000 (a market with a single monopolist, having a 100 percent share).

Under the horizontal merger guidelines, the U.S. Department of Justice and the Federal Trade Commission use post-merger HHIs and deltas as screens to help identify mergers that warrant closer review. Mergers that result in unconcentrated markets ( less than 1,500), or that result in small deltas (less than 100), are generally unproblematic. Mergers that result in moderately concentrated markets with deltas greater than 100, or mergers that result in highly concentrated markets with moderate deltas (100-200), are deemed to “potentially raise significant competitive concerns and often warrant scrutiny.” Mergers that result in highly concentrated markets and deltas greater than 200 are “presumed to be likely to enhance market power” and, in appropriate cases, will be challenged. In a chart, these presumptions look like this:

Under the merger guidelines, mergers that are in the “red” corner of the chart are presumptively unlawful. Such mergers can — and often do — overcome this presumption of illegality with “persuasive evidence showing that the merger is unlikely to enhance market power.” Indeed, in practice, the agencies tend to focus their energies on mergers that are well above the 2,500 HHI/200 delta thresholds. However, the HHI measurement is still a valuable initial screen to identify mergers that might become subject to a closer review.

## Calculating Deltas the Hard Way

Although the merger guidelines’ presumptions reference both the post-merger HHI and the delta, in some cases the delta can be the more interesting single piece of information. Additionally, it is common for parties to have good market share estimates for themselves, but poor insight into the market shares of their competitors; in these situations, HHIs cannot be calculated, but deltas can be. However, the delta can be a real pain in the neck to figure out.

Take, for example, a market with ten total firms: one firm with a 20 percent share, two firms with 15 percent shares, three firms with 10 percent shares, and four firms with 5 percent shares. And suppose that one of the 15 percent firms wants to merge with a 10 percent firm. A seasoned practitioner might be able to tell intuitively that this is a relatively unconcentrated market, but might be interested in knowing the delta to see where the merger presumptively falls. To calculate the delta in this case, one would calculate the pre-merger HHI, the post-merger HHI, and then subtract the difference. The math would look like this:

Post-merger HHI = (15+10)² + 20² + 15² + (2 × 10²) + (4 × 5²) = 1550

Premerger HHI = 20² + (2 × 15²) + (3 × 10²) + (4 × 5²) = 1250

Delta = 1550 – 1250 = 300

Altogether then, calculating the delta required eight steps of addition, five steps of multiplication, nine steps of squaring, and one step of subtraction, for a total of 23 separate calculations. And once finished, a responsible practitioner would likely do the math again, just to make sure it was done correctly.

This is not only cumbersome, it is also potentially inaccurate — since with each of the 23 calculations there are opportunities for mistakes in applying orders of operations, rounding, or significant figures. Some readers might scoff that they use an automatic Excel formula to calculate HHIs and therefore avoid these problems. Not so — some of these issues, particularly applying significant figures, can actually be made worse by using Excel. Fortunately, however, there is a way to do the math that is easier and quicker, and eliminates these opportunities for errors.

## Calculating Deltas The Easy Way

Tucked at the end of a paragraph in the horizontal merger guidelines is a weird trick for calculating HHI deltas (or, as the guidelines call it, the “increase in the HHI”) with ease: “The increase in the HHI is equal to twice the product of the market shares of the merging firms.” See Horizontal Merger Guidelines § 5.3.

Therefore, in the example we just considered, the delta is twice the product of 15 and 10. Sure enough, 2 × 15 × 10 equals 300 — the same number we reached before. Voila! Twenty-three complicated calculations have been reduced to two simple ones. With one weird trick, we have saved time, energy and opportunities for mistakes.

## Wait, Really?

Skeptical practitioners might want a little more information before entrusting their clients’ legal interests to an unexplained statement in the merger guidelines. For those readers, the algebra that underlies the shortcut is below.

Assume a merger with market participants “a,” “b,” “c,” and an undetermined number of other firms up to “z.” Participants “a” and “b” are merging.

Post-merger HHI = (a + b)² + c² + … + z²

Pre-merger HHI = a² + b² + c² + … + z²

Delta (∆) = Post-Merger HHI – Pre-merger HHI

∆ = ((a + b)² + c² + … + z²) – (a² + b² + c² + … + z²)

∆ = (a + b)² + c² + … + z² – a² – b² – c² – … – z²

∆ = (a + b)² – a² – b²

∆ = (a + b) × (a +b) – a² – b²

∆ = a² + (a × b) + (a × b) + b² – a² – b²

∆ = (a × b) + (a × b)

∆ = 2 × a × b

Q.E.D. Accordingly, even skeptical practitioners should take comfort in knowing that this one weird trick is, in fact, undeniably, demonstrably, mathematically true.

## Two More Uses for the Trick

The goal of this article was to give readers an easy solution for a common inconvenience. But if having a shortcut for calculating HHI deltas is not useful enough in its own right, there are at least two other good applications for the trick.

First, it can be used as an easy double-check on HHI calculations done the traditional way. In other words, if a practitioner uses the traditional approach of calculating the delta based on the difference between the post-merger HHI and premerger HHI, then the practitioner can quickly double-check his or her own work by using this shortcut. This double-check is not only convenient, but it also should give practitioners the comfort of reaching the same result two different ways.

Second, given that there is a shortcut for calculating deltas, it is no longer necessary to calculate premerger HHIs the hard way. Instead, practitioners who want to know a premerger HHI can get that number by calculating the post-merger HHI and then subtracting the delta. Alternatively, practitioners can calculate the premerger HHI first and then add the delta to determine the post-merger HHI.