Measuring Name Concentration With the Herfindahl Index

Why Measure Name Concentration?

A portfolio may have many borrowers and still depend heavily on a few of them. For example, one name may account for a large share of total exposure while numerous small names make the portfolio appear diversified by count. The Herfindahl-Hirschman Index (HHI) summarizes how unevenly a chosen exposure measure is distributed across names.

The index is a concentration statistic, not a credit-loss model. It does not by itself estimate default probability, loss given default, or portfolio capital. Its result depends on which exposure measure is used and on how names are defined.

Definition: For a portfolio of $n$ names, let $E_i$ be the nonnegative exposure assigned to name $i$, and let $E=\sum_{i=1}^{n}E_i>0$ be the total exposure on the same basis. The exposure share of name $i$ is $s_i=E_i/E$, so $s_i\geq 0$ and $\sum_{i=1}^{n}s_i=1$. The Herfindahl index is $$ \operatorname{HHI}=\sum_{i=1}^{n}s_i^2. $$

The exposure basis might be drawn from a portfolio’s exposure-at-default measure or another defined exposure amount. The important requirements are consistency and a clear unit of account: calculate every name’s share using the same exposure definition, date, and currency basis, and decide which obligors count as the same name. A name with zero exposure can be omitted without changing the index.

The HHI can be reported on a fractional scale from 0 to 1, or on a 0-to-10,000 scale using percentage-point shares. These are two reporting conventions for the same calculation. If fractional shares are used, an HHI of 0.25 corresponds to 2,500 points. Do not square percentages such as 40% as though they were 40 percentage points and then interpret the result on the fractional scale.

Interpretation and Basic Properties

Squaring shares gives larger names more influence than smaller ones. A name with twice another name’s share contributes four times as much to the HHI. Thus the index responds to the distribution of exposure, not merely the number of names.

For $n$ positive-share names, the HHI is at least $1/n$ and at most 1. Equal shares produce the minimum, $1/n$; a portfolio with all exposure in one name has HHI 1. The bounds follow directly from the share conditions. By the Cauchy-Schwarz inequality,

\left(\sum_{i=1}^{n}s_i\right)^2\leq n\sum_{i=1}^{n}s_i^2 \quad\Longrightarrow\quad \operatorname{HHI}\geq \frac{1}{n},

because the shares sum to 1. Equality holds when every share is $1/n$. For the upper bound, $s_i^2\leq s_i$ for each share in $[0,1]$, so $\operatorname{HHI}\leq\sum_i s_i=1$. Equality requires one share to be 1 and all others to be 0.

One interpretation comes from a random-selection experiment. Imagine drawing two exposure units independently from the portfolio, with each unit’s name determined by the exposure shares. The probability that both draws belong to the same name is $\sum_i s_i^2$: the probability both draws are from name $i$ is $s_i^2$, and these events are mutually exclusive across names. The HHI is therefore the probability of a same-name match in this particular exposure-weighted experiment.

A related summary is the effective number of equally sized names, defined as $N_{\mathrm{eff}}=1/\operatorname{HHI}$. It is the number of equal-share names that would produce the observed HHI. This is a useful comparison, not a literal count of borrowers and not a measure of loss capital. If shares are unequal, the effective number can be fractional and is no greater than the actual number of positive-share names.

Worked Example: Calculating HHI From Exposure Amounts

A hypothetical insurer has credit exposures of $40 million, $30 million, $20 million, and $10 million to four separate obligors. Calculate the fractional HHI, its 0-to-10,000 representation, and the effective number of equally sized names.

Method and exposure shares. First find the total exposure: $E=40+30+20+10=\$100$ million. Divide each exposure by this total. The shares are $0.40$, $0.30$, $0.20$, and $0.10$, which sum to 1.

Square and add. The contributions are $0.40^2=0.16$, $0.30^2=0.09$, $0.20^2=0.04$, and $0.10^2=0.01$. Thus $\operatorname{HHI}=0.16+0.09+0.04+0.01=0.30$.

Convert and interpret. On the 0-to-10,000 scale, $0.30\times10{,}000=3{,}000$ points. The effective number is $1/0.30=3.3333$, or about 3.33 equally sized names. As a check, four equal-share names would have HHI $1/4=0.25$, below 0.30; the unequal portfolio is more concentrated than that equal-share portfolio.

Common mistake avoided: using the number of obligors alone would suggest four names but would miss the uneven shares. The HHI incorporates both the number of names and the size distribution of their exposures.

How Changes in Shares Affect the Index

The HHI responds to a redistribution of exposure even when total exposure does not change. In particular, splitting one name into two separate names with the same combined exposure lowers the index, provided those exposures genuinely represent separate obligors under the chosen name definition. Conversely, combining two names raises the index when their shares are treated as one name. The algebra makes the effect clear: replacing a share $a+b$ with two shares $a$ and $b$ changes its contribution from $(a+b)^2$ to $a^2+b^2$, a decrease of $2ab$.

Worked Example: Effect of Splitting One Exposure

A hypothetical portfolio has five distinct obligors with exposure shares of 0.30, 0.20, 0.20, 0.15, and 0.15. The exposure to one of the 0.20-share obligors is replaced by equal exposures to two new, unrelated obligors, each with share 0.10. Find the HHI before and after the change, and explain what the comparison means.

Before the split. Square each original share and add: $0.30^2=0.09$, $0.20^2=0.04$ for each of two names, and $0.15^2=0.0225$ for each of two names. Therefore, $\operatorname{HHI}_{\mathrm{before}}=0.09+0.04+0.04+0.0225+0.0225=0.215$.

After the split. The shares are 0.30, 0.20, 0.15, 0.15, 0.10, and 0.10. Their squared contributions sum to $0.09+0.04+0.0225+0.0225+0.01+0.01=0.195$.

Compare and check. The index falls by $0.215-0.195=0.020$, or 200 points on the 0-to-10,000 scale. The algebraic check gives the same decrease: splitting 0.20 into 0.10 and 0.10 reduces the contribution by $2(0.10)(0.10)=0.020$. Total exposure is unchanged because $0.10+0.10=0.20$.

Method and common mistake: recalculate the shares by name before squaring them. Counting the two new exposures as one name would leave the index unchanged and miss the stated change in the obligor structure. In practice, a legal or administrative split does not necessarily create two economically independent names; name aggregation must reflect the purpose of the analysis.

Exposure Concentration Is Not Loss Concentration

An exposure-share HHI assigns each name a weight based on exposure alone. It does not account for differences in probability of default or loss given default. Two portfolios with the same exposure shares therefore have the same exposure HHI, even if the names have different expected loss rates.

To illustrate this distinction, an analyst may separately calculate a concentration statistic using expected-loss shares. That is a different, explicitly labelled weighting basis; it should not be confused with the exposure HHI. Expected loss for a name depends on the relevant default probability, exposure, and loss given default assumptions, and a share based on expected loss answers a different question.

Worked Example: Comparing Exposure and Expected-Loss Shares

Consider a hypothetical one-period portfolio with exposures of $50 million, $30 million, and $20 million to three obligors. For this example, assume each exposure remains fixed over the period, the default probabilities are 0.002, 0.010, and 0.005, respectively, and the loss given default is 100% for every name. Calculate the exposure HHI and, separately, the HHI using expected-loss shares.

Exposure HHI. Total exposure is $50+30+20=\$100$ million, so exposure shares are 0.50, 0.30, and 0.20. The exposure HHI is $0.50^2+0.30^2+0.20^2=0.25+0.09+0.04=0.38$, or 3,800 points.

Expected losses. With the stated assumptions, expected loss for each name is exposure multiplied by default probability and loss given default. In millions of dollars, the amounts are $50(0.002)(1)=0.10$, $30(0.010)(1)=0.30$, and $20(0.005)(1)=0.10$. Total expected loss is $0.10+0.30+0.10=\$0.50$ million.

Expected-loss shares and separate index. The shares of expected loss are $0.10/0.50=0.20$, $0.30/0.50=0.60$, and $0.10/0.50=0.20$. Their squared-share sum is $0.20^2+0.60^2+0.20^2=0.04+0.36+0.04=0.44$, or 4,400 points on the same reporting scale.

Check and interpretation. Both sets of shares sum to 1. The expected-loss-weighted statistic is higher here because the second name contributes 60% of expected loss, despite having only 30% of exposure. It does not replace the exposure HHI: the two statistics answer different questions, and neither alone describes the full distribution of portfolio losses.

Limitations and Common Traps

The HHI is deliberately compact, so it leaves out information. It is most useful when its inputs and interpretation are stated clearly and when comparisons use consistent definitions.

It depends on the exposure basis. Gross exposure, net exposure, and exposure at default can produce different shares. A comparison is difficult to interpret if the basis, valuation date, or treatment of offsets changes.
It does not measure credit quality. A high or low HHI alone says nothing about default probabilities, recoveries, or the timing of credit events. The example above shows how exposure shares and expected-loss shares can give different concentration readings.
It does not describe dependence between different names. The index summarizes the distribution of exposure across names. It does not model common drivers or specify how defaults occur jointly.
It is sensitive to name aggregation. Treating related borrowers as one name can increase concentration relative to treating them separately. The analyst must apply a defensible, consistent definition of a name.
It compresses a distribution to one number. Different portfolios can have the same HHI but different individual share patterns. The index is a summary, not a complete portfolio description.

A frequent calculation error is to square dollar exposures directly. For the first example, that mistaken calculation would be $40^2+30^2+20^2+10^2=3{,}000$, in units of millions of dollars squared. It is not an HHI and changes if the exposure unit is changed. The correction is to divide each exposure by total exposure first, then square the dimensionless shares: $0.40^2+0.30^2+0.20^2+0.10^2=0.30$. If reporting points, multiply that result by 10,000 to obtain 3,000 points.

Key takeaway: The HHI is the sum of squared shares on a clearly specified basis. Larger names have disproportionate influence, and the reciprocal gives an effective number of equally sized names. Interpret it as a name-concentration summary—not as a standalone measure of credit quality, dependence, or capital.

Check Your Understanding

For each problem, use shares that sum to 1, and distinguish the fractional HHI from its points representation where requested.

A hypothetical portfolio has exposure shares of 0.50, 0.25, 0.15, and 0.10. Calculate the fractional HHI, its points representation, and the effective number of equally sized names. (Answer: HHI $=0.345$, or 3,450 points; effective number $=1/0.345\approx2.90$.)
Five obligors have exposures of $24 million, $18 million, $18 million, $12 million, and $8 million. Calculate the exposure HHI. (Answer: Total exposure is $80 million; shares are 0.30, 0.225, 0.225, 0.15, and 0.10; HHI $=0.22375$, or 2,237.5 points.)
A portfolio has six equally exposed obligors. What is its HHI and effective number of equally sized names? (Answer: HHI $=1/6\approx0.1667$, or about 1,666.7 points; effective number $=6$.)
Exposure shares are initially 0.40, 0.30, 0.20, and 0.10. Ten percentage points of total exposure are transferred from the largest name to the smallest. Calculate the new HHI and the change from the original HHI. (Answer: New shares are 0.30, 0.30, 0.20, and 0.20; new HHI $=0.26$, down from 0.30 by 0.04, or 400 points.)
Three obligors have exposure shares 0.50, 0.30, and 0.20, and expected losses of $0.10 million, $0.30 million, and $0.10 million. Calculate the exposure HHI and the separate HHI based on expected-loss shares. (Answer: Exposure HHI $=0.38$, or 3,800 points; expected-loss shares are 0.20, 0.60, and 0.20, giving an expected-loss-weighted HHI of 0.44, or 4,400 points.)

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