Chris Leaney

The Herfindahl-Hirschman Index (HHI) is a measure of diversification, commonly used as an indicator to calculate banks’ credit concentration risk capital requirements (where credit concentration risk is potential losses from undiversified portfolios). According to BCBS (2019) HHI is employed by c. 50% of regulators, including the Prudential Regulation Authority (PRA) since 2016. However, despite some evidence that the data-light, easy-to-implement HHI produces broadly comparable outcomes with formal models (eg Bundesbank (2006)), such evidence is limited to large banks or theoretical datasets. In this post I examine the relationship between HHI and a formal model of sector and geographical concentration risk. I show that, for a wide sample of bank sizes, HHI is poorly correlated with the model outputs for both risk types.

How does the PRA assess credit concentration risk?

Credit concentration risk is one of the many risks considered under Pillar 2A, which is an assessment of additional capital to cover risks not adequately captured under Pillar 1. Pillar 1 capital requirements for credit risk are calibrated on a perfectly diversified book, and hence concentration risk must be added under Pillar 2. The PRA defines it as ‘the risk of losses arising as a result of concentrations of exposures due to imperfect diversifications’.

There are three types of concentration risk: single name (large individual exposures as a proportion of a portfolio), sector (imperfect diversifications across industries) and geographical (imperfect diversifications across regions).

As of 1 January 2016, the PRA requires banks to calculate the HHI for all three types of concentration risk. The HHI is defined as the sum of the squares of the relative portfolio shares of all borrowers (these portfolio shares are calculated using risk-weighted assets (RWAs)). Well-diversified portfolios have an HHI close to zero, whilst the most concentrated portfolios have a number close to one. Mapping models translate a bank’s HHI into one of five proposed capital add-on ranges or ‘buckets’, and then a bank’s exact add-on within the appropriate range is determined using supervisory judgement. The ability to provide judgement based on information no statistical model can incorporate is important. One type of mapping model is used for single name concentration risk, and another for both sector and geographical concentration risk. I focus on sector and geographical concentration risk in this post.

A model of sector and geographical concentration risk

To calculate banks’ capital add-ons for sector and geographical concentration risk, I use a multi-factor capital model which is the same as the mapping model used by the PRA and similar to that of Düllmann and Masschelein (2007). In theory, the Basel II formula should be a good approximation for the ‘true’ risk in a well-diversified portfolio. Hence, for such a portfolio the capital requirements under a multi-factor model will be roughly equal to the capital requirements under a single-factor setting. For a less diversified portfolio with concentrations in one or more sectors or geographical regions, the capital requirements under the multi-factor model will be greater than the single-factor setting and the difference should represent the amount by which the Basel II single-factor model underestimates the ‘true’ capital requirement, ie the amount of capital required under Pillar 2A to mitigate for sector or geographical concentration risk. The PRA ran a series of simulations when it developed its current mapping model which demonstrated that formal models like the one I also use are generally effective at capitalising actual concentration risk.

I calculate the capital add-ons using the following components for sector/geographical respectively:

  1. inter-sector/inter-region correlations (the correlations between each pair of sectors/regions);
  2. intra-sector/intra-region correlations (the ratio of the volatility of each individual sector/region to the volatility of the industry/world); and
  3. banks’ RWA profiles from regulatory reporting.

The individual sectors and regions are the same as currently defined by the PRA. The differences between my ‘challenger model’ and the PRA’s current mapping model are related to data inputs, rather than the model itself:

  1. the data inputs to calculate the correlations. I also use aggregated probability of default data for global companies, but from an alternative recent data set; and
  2. the number of banks (I use 38 of various sizes versus 23 predominantly large banks).

To test the effectiveness of RWA HHI as a mapping tool against the challenger model (which the PRA’s simulations suggested was a good proxy for actual concentration risk), I plot the add-ons for each bank against the respective RWA HHI values which I calculated from the banks’ RWA profiles. I then regress RWA HHI against the suggested capital add-on and analyse the resultant R-squared values. An R-squared value of 1 would suggest the entirety of the variance of the dependent variable (the capital add-on) is explained by the only independent variable (RWA HHI), while 0 would suggest none of the variance is explained.

What I found

The relationship between RWA HHI and suggested capital add-ons from my challenger model completely broke down for both sector and geographical concentration risk. This can be seen for sector concentration risk in Chart 1 and geographical in Chart 2, with both R-squared values very close to 0 and no clearly visible trends. It is important to note that all of the banks’ actual add-ons are within the relevant PRA add-on range for their RWA HHI value, whereas my modelling suggests only a handful should be for either risk type. This means that even the application of supervisory judgement can only partially, but not completely, mitigate the output.

Chart 1: RWA HHI mapping versus suggested capital add-on from challenger modelling for sector concentration

Chart 2: RWA HHI mapping versus suggested capital add-on from challenger modelling for geographical concentration

The discrepancies arise because the suggested capital add-on for each bank depends on the particular sectors and regions that they are concentrated in (information not captured by RWA HHI), rather than the overall level of concentration in a portfolio (captured by RWA HHI). Portfolios with greater exposures in the particular sectors and regions that are more volatile and/or more correlated with other sectors and regions are inherently more risky, and so require a greater capital add-on to compensate for that. Hence, we can end up in a situation where a bank with a high RWA HHI value can have a lower suggested add-on than a bank with a low RWA HHI value (as is often the case, shown in the charts above).

RWA HHI provides a broad measure of banks’ overall sector and geographic concentration, is simple to implement and regulate, and can easily be used for peer comparison. However, it is agnostic as to which sectors and regions these concentrations are in, and so it is therefore insufficiently granular to identify the true riskiness of a whole portfolio when compared to a formal model.

Implications and further questions

While the post provides new evidence about the effectiveness (or not) of RWA HHI as an indicative tool for sector and geographical concentration risk capital add-ons as calculated by a formal model, these findings hold true for the specific modelling technique and data inputs that I used and assume that formal models estimate concentration risk well. There will need to be further research conducted with different models, data inputs, and to examine the effectiveness of the formal models in capturing actual concentration risk before firmer conclusions can be drawn, although this has produced some thought-provoking outcomes. Also, I have not considered single name concentration risk in this post; it is not yet known whether there could potentially be similar problems with using RWA HHI as an indicator for single name risk.

If regulators ultimately determine that using RWA HHI as an indicator is not appropriate for some or all types of concentration risk, there is also the open question about a better policy alternative:

  • Should regulators and banks ramp up the judgement and/or qualitative part of the assessment? That could make consistency a challenge.
  • Should regulators and banks use a different indicator? I also ran my methodology for several others, including exposure at default HHI, and encountered the same concern as for RWA HHI.
  • Should all banks be made to model concentration risk rather than rely on mapping? That would be data heavy and resource intensive and likely more challenging for smaller banks.
  • Should regulators require banks to utilise simulation exercises and/or stress testing? Could anyone somehow build on an existing large exposures regime? Could there be different methodologies for small banks versus large banks?

You get my point. There are numerous options to consider, but all with the same fundamental goal in mind; ensuring that banks are adequately capitalised for the true risks in their portfolios.


Chris Leaney wrote this post when he worked in the Bank’s UK Global Banks Division.

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