Measuring Name and Sector Credit Concentration

Why Measure Credit Concentration?

A credit portfolio can have many obligors and still depend heavily on a few of them. A portfolio with a large share of exposure to one borrower may be vulnerable to that borrower's deterioration, while a portfolio whose exposures cluster in one industry may be vulnerable to a common sector shock. Measuring concentration makes these patterns visible and supports comparison across portfolios or over time.

Concentration measures describe how exposure is distributed; they do not, by themselves, estimate default probabilities, loss given default, dependence, or tail loss. Those require additional assumptions and models. This tutorial focuses on exposure shares, name and sector concentration indices, and contributions to those indices, aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: Let $E_i$ be the nonnegative exposure assigned to obligor $i$, measured at a common valuation date and on a consistent basis. Let $E=\sum_{i=1}^{n}E_i$ be total portfolio exposure. When $E>0$, the exposure share of obligor $i$ is $w_i=E_i/E$. The shares satisfy $0\leq w_i\leq1$ and $\sum_i w_i=1$.

The exposure basis must be stated. For example, an analysis might use exposure at default, current drawn balances, or another defined measure. A comparison is meaningful only when the portfolios use compatible definitions, units, valuation dates, and treatment of guarantees or collateral. If the total exposure is zero, shares and the indices below are undefined.

Name Concentration and the HHI

The Herfindahl-Hirschman index (HHI) summarizes the distribution of exposure across obligors by squaring each obligor's share and adding the squares. Squaring gives disproportionate weight to large shares: a single large name has more influence than several small names with the same combined exposure.

Definition: The name HHI is $H_N=\sum_{i=1}^{n}w_i^2$. The contribution of obligor $i$ to the HHI is $w_i^2$. Its fraction of the total HHI is $w_i^2/H_N$, when $H_N>0$.

With shares expressed as decimals, the HHI is between $1/n$ and 1 for $n$ positive-exposure obligors: equal shares give $1/n$, while one obligor holding all exposure gives 1. A frequently used related summary is the effective number of equal-sized obligors, $1/H_N$. It is an interpretation of the index, not necessarily the actual number of obligors. If shares are instead entered as percentages, the sum of squared percentage shares is $10{,}000H_N$; do not compare that scale directly with an HHI calculated from decimal shares.

Worked Example: Name HHI and the Largest Obligor's Contribution

A hypothetical insurer has credit exposures of $50 million, $25 million, $15 million, and $10 million to four different obligors. Calculate the name HHI, the largest obligor's fraction of the HHI, and the effective number of equal-sized obligors. All exposures are on the same basis.

Method. Divide each exposure by the total, square each share, and sum. The total exposure is $50+25+15+10=100$ million, so the shares are $0.50,\ 0.25,\ 0.15,\ 0.10$.

$$ H_N=(0.50)^2+(0.25)^2+(0.15)^2+(0.10)^2 =0.2500+0.0625+0.0225+0.0100=0.3450. $$

The largest obligor's HHI contribution is $0.50^2=0.2500$. Its fraction of the total HHI is $0.2500/0.3450=0.7246$, to four decimals. The effective number is $1/0.3450=2.8986$, to four decimals. The check is that the four contributions sum to $0.3450$, and their fractions of the total sum to 1 (up to rounding).

Common mistake. Reporting the largest obligor's exposure share, $0.50$, as its fraction of HHI would be incorrect. HHI attribution uses the squared share divided by the total HHI: $0.2500/0.3450$, not $0.50$.

Sector Concentration

Name concentration and sector concentration answer different questions. Name concentration treats each obligor separately. For sector concentration, first assign each obligor to exactly one sector, sum exposures within each sector, and then calculate shares and the HHI using those sector totals. The classification should be mutually exclusive and exhaustive for the portfolio being measured.

Definition: If sector $s$ contains obligors in set $I_s$, its exposure is $E_s=\sum_{i\in I_s}E_i$ and its share is $W_s=E_s/E$. The sector HHI is $H_S=\sum_s W_s^2$. The contribution of sector $s$ is $W_s^2$, and its fraction of the sector HHI is $W_s^2/H_S$, when $H_S>0$.

Worked Example: Comparing Name and Sector Concentration

A hypothetical lending portfolio has six obligors. Sector A contains exposures of $30 million and $20 million; Sector B contains $20 million and $10 million; Sector C contains $10 million and $10 million. Calculate both HHIs and identify what the comparison shows.

Name HHI. Total exposure is $30+20+20+10+10+10=100$ million. The six name shares are $0.30,\ 0.20,\ 0.20,\ 0.10,\ 0.10,\ 0.10$. Therefore,

$$ H_N=0.30^2+0.20^2+0.20^2+3(0.10^2) =0.09+0.04+0.04+0.03=0.20. $$

Sector HHI. Sector A has exposure $30+20=50$ million and share $0.50$. Sector B has exposure $20+10=30$ million and share $0.30$. Sector C has exposure $10+10=20$ million and share $0.20$. Thus,

$$ H_S=0.50^2+0.30^2+0.20^2=0.25+0.09+0.04=0.38. $$

Sector A's contribution is $0.25$, or $0.25/0.38=0.6579$ of the sector HHI, to four decimals. The sector HHI exceeds the name HHI because aggregating obligors into sectors combines their exposures before squaring. It indicates that exposure is more concentrated across the three sector categories than across the six individual names. Neither result establishes how much loss a sector event would cause; that would require further risk assumptions.

Common mistake. Do not calculate the sector HHI by adding the name contributions within a sector. The correct sector contribution squares the sector's combined share. For Sector A, that is $(0.30+0.20)^2=0.25$, not $0.30^2+0.20^2=0.13$.

Why the HHI Reacts to Exposure Changes

The effect of changing one obligor's exposure is not determined by whether that obligor's balance rises or falls alone. Shares are relative to total exposure, so changing one exposure changes both its numerator and the portfolio denominator. The following derivative gives a useful local interpretation.

Write $Q=\sum_i E_i^2$ and $E=\sum_i E_i$. Then $H_N=Q/E^2$. Holding all exposures other than $E_k$ fixed, $dQ/dE_k=2E_k$ and $dE/dE_k=1$. Applying the quotient rule gives

\frac{\partial H_N}{\partial E_k} =\frac{2E_kE^2-2QE}{E^4} =\frac{2(E_kE-Q)}{E^3} =\frac{2(w_k-H_N)}{E}.

Since $E>0$, a small increase in $E_k$ lowers the HHI when $w_k<H_N$, raises it when $w_k>H_N$, and has zero first-order effect when $w_k=H_N$. This is a local result; for a finite change, recalculate the index using the new exposures. It explains why adding exposure to a relatively small name can reduce measured concentration even though the portfolio's total credit exposure rises.

Worked Example: A Small Name Grows While Other Exposures Stay Fixed

A hypothetical portfolio has four obligors with exposures of $40 million, $30 million, $20 million, and $10 million. Find the largest obligor's contribution as a fraction of name HHI. Then increase the $10 million exposure by $5 million, holding the other absolute exposures fixed. Calculate the revised HHI and determine whether concentration rises or falls.

Initial index. Total exposure is $40+30+20+10=100$ million, with shares $0.40,\ 0.30,\ 0.20,\ 0.10$. The initial HHI is

$$ H_N=0.40^2+0.30^2+0.20^2+0.10^2 =0.16+0.09+0.04+0.01=0.30. $$

The largest obligor contributes $0.40^2=0.16$. Its fraction of the total HHI is $0.16/0.30=0.5333$, to four decimals.

Revised index. The exposures are now $40,\ 30,\ 20,\ 15$ million, for a total of $105$ million. Using shares, or equivalently the sum of squared exposures divided by the square of total exposure,

H_N^{\text{new}} =\frac{40^2+30^2+20^2+15^2}{105^2} =\frac{1600+900+400+225}{11025} =\frac{3125}{11025} \approx 0.2834.

The index falls from $0.3000$ to $0.2834$. The increased exposure belongs to a name whose initial share, $0.10$, was below the initial HHI, $0.30$, consistent with the derivative result. The revised total is $105$, not $100$; using the original denominator would be a denominator error.

Common mistake. A larger absolute exposure does not guarantee a larger HHI. Recalculate every share using the new total. Here the correct revised index is $0.2834$, not $0.2800$.

Interpretation and Practical Limits

Concentration measures are useful screening and comparison tools. A risk manager can track a name HHI alongside the largest-name share, or track a sector HHI alongside the largest-sector share. The largest share is easy to communicate, while HHI reflects the entire distribution. The effective number $1/H_N$ can make the index intuitive: a portfolio with HHI $0.25$ has the same HHI as four equally sized exposures, although its actual exposures need not be equal.

An HHI is not a credit capital requirement or a direct estimate of expected or unexpected loss. Two portfolios with the same exposure shares can have different credit quality, maturities, recoveries, or default dependence. Sector aggregation also depends on classification choices: a broad sector grouping can conceal concentration within a subsector, while a narrow classification can make small differences in labels appear important. Interpret results together with the chosen exposure definition and classification.

Define the measurement basis.
Specify the valuation date, exposure definition, currency and treatment of any adjustments. Confirm that total exposure is positive.

Calculate shares.
Divide each obligor exposure by total exposure. For sector analysis, sum obligor exposures within each consistently assigned sector first.

Calculate and attribute the index.
Square the shares and sum them. Report important shares and, where useful, each contribution as a fraction of the relevant HHI.

Interpret within scope.
Compare like with like and describe exposure distribution. Do not infer loss severity or dependence from concentration alone.

A frequent exam trap is to mix the decimal-share and percentage-share conventions, or to use an old total after an exposure changes. Another is to present a sector index as if it were the name index. State which index is being calculated, show the shares, and verify that the contributions add to the reported HHI.

Key takeaway: Name HHI squares obligor exposure shares; sector HHI squares aggregated sector shares. Contribution fractions show where each index comes from, but concentration indices describe portfolio composition rather than credit loss by themselves.

Check Your Understanding

For each hypothetical portfolio, use decimal shares and show enough intermediate work to check the result.

A portfolio has exposures of $45 million, $35 million, and $20 million. Calculate its name HHI and effective number of equal-sized obligors. (Answer: $H_N=0.3650$; effective number $=2.7397$, to four decimals.)
Four obligors have exposures of $24 million, $18 million, $18 million, and $12 million. Calculate the name HHI and the largest obligor's fraction of that HHI. (Answer: $H_N=38/144=0.2639$; largest-name fraction $=(1/9)/(38/144)=16/38=0.4211$, to four decimals.)
A hypothetical portfolio has name exposures of $25 million and $15 million in Sector A, $20 million in Sector B, and $10 million each in Sectors C and D. Calculate the name HHI and sector HHI. (Answer: $H_N=1450/6400=0.2266$; sector shares are $0.50, 0.25, 0.125, 0.125$, so $H_S=0.3438$, to four decimals.)
A portfolio has exposures of $50 million, $30 million, and $20 million. The smallest exposure increases by $10 million and the others stay fixed. Calculate both HHIs and state the direction of change. (Answer: initial $H_N=0.3800$; revised $H_N=4300/12100=0.3554$, to four decimals; concentration falls.)
Three sectors have exposures of $40 million, $35 million, and $25 million. Calculate the sector HHI and the fraction of that HHI contributed by the $35 million sector. Explain what the fraction attributes. (Answer: $H_S=0.40^2+0.35^2+0.25^2=0.3450$; the $35 million sector contributes $0.1225/0.3450=0.3551$, or $35.51\%$, to four decimals; it is that sector's fraction of the sector HHI, not its fraction of portfolio exposure.)

What You'll Learn