Measuring Competitiveness - The Herfindahl-Hirschman Index

While CFA candidates are aware of the Herfindahl-Hirschman Index (HHI), let's look at the fundamental math behind the formula to understand exactly what is going on (and why this metric is a mathematically good formula for competitiveness).

First, look at the different models available for describing a market place (in decreasing competitiveness): Perfectly competitive, monopolistic competition, oligopoly, and monopoly.

Looking at the extreme cases, we would expect a company in a perfectly competitive industry would have an insignificant market share (mathematically represented by an infinite number of firms with an infinitesimal market share). A monopoly would only have one firm will all the market share.

How can we use an index to describe the competitiveness of the intermediate competitive states? Number of firms is one option, however it needs to incorporate the relative market share for each.

Now let's return to the formula for HHI:

HHI = Σ Xi, i from 1 to n

• Xi is the percent market share of firm i x 100
  
• n is the number of firms (or 50 if more than that)

Upon further inspection, the HHI index is strikingly similar (identical with some modifications) of the idea of standard deviation with a mean of 0. Each companies' real percentage market share is described as a variance from perfect competition of zero. The more any individual firms have a disproportionate control (variation) from the expected average, the more weight it is assigned mathematically by squaring it's value.

Also, why choose a limit of 50 firms? Why place any limit at all? Well first of all, for every 50 firms, each additional firm contributes less than 2% (remember that firms are added from largest to smallest) and therefore affects the HHI less and less (less than 4 points out of a possible 10k). This was probably put in as a computational limit in order to simplify calculation. There is very little precision or accuracy lost by discounting remaining firms beyond 50.

An economically and mathematically perfectly competitive market will have an HHI of near 0 (in theory only, as a nearly perfect competitive market with 100 firms with 1% will still have a score of 100). A maximum HHI score (indicating a monopoly) occurs at 10,000.

The CFA text book proposes the following HHI metrics for the various competition levels:

• Perfect Competition less than 100
  
• Monopolistic Competition 101 to 999
• Oligopoly 1,000+
• Monopoly 10,000

However, they also mention that the Department of Justice using different metrics (in the measure of degree of competition in the case of approving merger decisions) as:

• Competitive less than 1000
  
• Moderately competitive 1000 to 1800
• Uncompetitive 1800+
  

In this respect, HHI is more descriptive of the actual competition rather than a four firm concentration ratio. For example, a monopoly is much less competitive than an industry with four firms equally sharing 25% of the market. This would be captured by the HHI score (monopoly 10,000 and four firm 2,500), but would be reflected in a four firm ratio as 100% as both cases being identical.