Quick answer: a bank can hold thousands of loans and still be poorly diversified if too much exposure depends on the same borrower, corporate group, industry, geography, collateral market or economic factor. Banks therefore measure concentration at several layers. The Herfindahl-Hirschman Index (HHI) measures how exposure shares are distributed. Large-exposure rules constrain single counterparties and connected groups. Credit-portfolio models add default correlation to show that individually small loans can still fail together. Granularity analysis asks whether a few names are large enough that idiosyncratic risk has not diversified away. Stress testing then asks what happens when the shared factor actually breaks.

Ten loans to ten names are not necessarily ten independent risks. They can all be one bet on the same future.

Why this belongs in mathematics

Credit concentration combines portfolio weights, quadratic indices, latent-factor models, correlated Bernoulli defaults, tail distributions, conditional probability and scenario analysis. It also teaches a critical anti-naturalisation habit: the number of borrowers is not the same as the number of independent risk sources.

The OCC describes concentration-risk management as a core part of loan-portfolio safety and soundness, covering direct, indirect and contingent exposures and concentrations by borrower, related group, industry or other common factor. See the OCC Concentrations of Credit handbook.

1. Exposure share is the simplest starting point

Suppose a loan portfolio has total exposure E and borrower i has exposure E_i. Define borrower share:

s_i = E_i / E.

A S$100 million borrower in a S$1 billion portfolio has a 10% share. That is already meaningful, but one share alone does not describe the rest of the distribution.

2. HHI compresses the exposure distribution

The Herfindahl-Hirschman Index can be written:

HHI = Σ s_i².

If ten borrowers each hold exactly 10% of the portfolio, HHI = 10×0.1² = 0.10. If one borrower holds 50% and five borrowers hold 10% each, HHI = 0.5² + 5×0.1² = 0.30.

Squaring gives disproportionate weight to large positions. That is precisely what concentration measurement needs.

3. The effective number of equal exposures

A useful interpretation is:

Effective equal-sized borrower count ≈ 1 / HHI.

An HHI of 0.10 corresponds to about ten equally weighted exposures. An HHI of 0.25 corresponds to about four. This does not mean the portfolio literally contains that many borrowers. It translates uneven concentration into an intuitive equivalent.

4. HHI sees weight concentration—but not default correlation

Imagine two portfolios, each with 100 equal-sized loans. They have the same borrower HHI. In Portfolio A the borrowers operate across unrelated sectors and countries. In Portfolio B every borrower is a property developer in the same city financed by the same refinancing market.

Borrower HHI says the two portfolios are equally granular. Economically, they are not. Portfolio B has a common-factor concentration that HHI by name cannot see.

This is why HHI must be calculated across several meaningful dimensions—single name, connected group, sector, geography, collateral type—and then complemented by correlation and stress analysis.

5. Connected counterparties prevent legal names from hiding one risk

Suppose a bank lends S$60 million each to three separate companies. If the companies are economically interdependent—perhaps one is the holding company and the others depend on the same cash source—the bank may not really have three independent S$60 million risks.

The Basel large-exposures framework therefore aggregates exposures to groups of connected counterparties where control relationships or economic interdependence mean distress can transmit across the group.

Under the Basel standard, an exposure at or above 10% of Tier 1 capital meets the definition of a large exposure for reporting purposes, and total exposure to a single counterparty or connected group generally must not exceed 25% of Tier 1 capital. G-SIB exposures to other G-SIBs face a tighter 15% limit. See Basel LEX20.

6. Why the large-exposure limit is not a full concentration model

A bank can remain below every single-name regulatory limit and still be heavily concentrated in commercial real estate, oil producers, one country, one employer ecosystem or one collateral market.

Large-exposure rules constrain catastrophic single-name loss. Portfolio concentration management asks a broader question: which exposures can become weak together?

7. Expected loss does not reveal diversification by itself

Suppose two portfolios both have simplified expected loss of S$10 million. One contains thousands of small, weakly related risks. The other contains four large borrowers exposed to the same commodity cycle.

Expected loss can be identical while tail risk is radically different. Concentration changes the distribution around the mean.

This is why expected-credit-loss accounting and concentration-risk analysis are related but different jobs.

8. A one-factor default model makes correlation visible

A useful teaching model represents borrower i with a latent credit variable:

Y_i = √ρ_iZ + √(1−ρ_i)ε_i

where:

Z is a common systematic factor such as the broad economy;
ε_i is borrower-specific risk;
ρ_i controls how strongly borrower i depends on the common factor.

The borrower defaults when Y_i falls below a threshold chosen to match its PD. If many borrowers share a large ρ, a bad common factor can push many below their thresholds at once.

This family of asymptotic single-risk-factor logic sits beneath the Basel IRB risk-weight framework. Basel’s current formulas use PD, LGD, EAD, maturity and supervisory correlation relationships to calculate unexpected-loss capital. See Basel CRE31.

9. Correlation changes tail loss far more than average loss

Imagine 1,000 equally sized borrowers each with a 1% unconditional PD and the same LGD. If defaults were independent, the law of large numbers would make total defaults relatively stable around the expected count.

Introduce a strong common economic factor and bad years become much worse: the portfolio can experience many defaults together. Good years can be even quieter. The average PD remains 1%, but the distribution becomes more clustered.

This is why a portfolio can look diversified borrower by borrower while remaining concentrated in one macroeconomic driver.

10. Granularity: the assumption that no single name matters too much

The Basel IRB model is built around an asymptotic idea: in a perfectly fine-grained portfolio, idiosyncratic borrower risk diversifies away, leaving systematic risk as the main portfolio uncertainty.

Real portfolios can violate this. If one borrower is 15% of the book, that borrower’s idiosyncratic outcome has not diversified away.

Basel research calls this residual single-name concentration granularity risk. A classic granularity adjustment estimates the extra portfolio capital associated with finite, uneven exposures beyond the infinitely fine-grained benchmark. See Studies on credit risk concentration.

11. Ten separate loans can still be one exposure if identity aggregation fails

Suppose the bank lends to a parent company, three subsidiaries and two special-purpose vehicles but its systems store them in different business lines. Each exposure looks modest locally. At group level they may form the bank’s largest risk.

Basel concentration research notes that effective granularity measurement requires aggregation by borrower identity across systems. Without identity resolution, the mathematics can be perfectly calculated on the wrong unit of risk.

This is the credit-risk version of the lesson in transaction reconciliation: before modelling, establish what the records refer to.

12. Sector concentration needs a second HHI

Aggregate exposures by sector and calculate sector shares. A borrower-level HHI can be low while sector HHI is high.

For example, 100 equal borrowers all in commercial real estate produce borrower HHI = 0.01, an apparently granular book. But if commercial real estate is 100% of the portfolio, the sector HHI across sectors is 1.00—the maximum possible concentration.

Neither index is “the right HHI.” They answer different questions.

13. Collateral can create hidden common-factor concentration

Borrowers can appear diversified by business while all loans are secured by the same asset class. A decline in one property market can simultaneously:

weaken borrower cash flow;
reduce collateral value;
raise LGD;
make refinancing harder;
increase default correlation.

That is a wrong-way concentration between borrower quality and recovery value. Measuring only borrower industry can miss it.

14. Stress testing turns concentration from index into consequence

An HHI tells the bank that exposure is concentrated. A stress test asks what the concentration does under a scenario.

Useful concentration stresses include:

default of the largest borrower;
simultaneous failure of connected counterparties;
severe recession in the dominant industry;
collapse in the common collateral market;
refinancing closure for one borrower segment;
higher PD and LGD occurring together;
withdrawal of a common guarantor or protection provider.

The point is not to assign one probability to each disaster. It is to reveal whether one common factor can consume an unacceptable share of capital.

15. Creative-work lens: The Big Short and the illusion of many independent mortgages

The Big Short is not a credit-portfolio model, but it supplies a useful intuition: thousands of individual mortgages can look diversified while remaining tied to the same housing-price, underwriting and refinancing regime. The narrative helps a learner notice the common factor behind many separate contracts.

The quantitative task is then stricter: measure exposure shares, correlation, collateral dependence and scenario loss rather than using a dramatic story as proof.

16. The concentration-risk pipeline

Resolve borrower and connected-group identities.
Aggregate direct, indirect and contingent exposure.
Calculate single-name shares and HHI.
Repeat concentration measures by sector, geography and collateral type.
Identify large exposures relative to Tier 1 capital.
Map common economic factors and guarantors.
Estimate PD, LGD and EAD consistently.
Model default dependence or correlated stress.
Measure granularity and largest-name contribution.
Run single-name and sector stress scenarios.
Calculate marginal contribution to portfolio tail loss.
Compare concentration with limits and risk appetite.
Update after large new facilities, mergers or sector repricing.
Reconcile portfolio reports across systems and legal entities.

17. Failure modes

Borrower-count illusion. Thousands of loans are assumed diversified without identifying shared factors.
HHI absolutism. One index is treated as a complete concentration model.
Legal-name fragmentation. Connected counterparties appear as separate independent risks.
Correlation stationarity. Normal-period default dependence is assumed stable in recession.
Collateral independence. Recovery values are assumed independent of borrower distress.
Expected-loss comfort. The mean looks manageable while tail concentration is extreme.
Limit-only management. Staying below a regulatory single-name limit is treated as proof the portfolio is diversified.
Sector taxonomy blindness. Broad industry labels hide one common revenue or funding source.

18. Diagnostics and falsifiers

What is the borrower HHI? Sector HHI? Geographic HHI?
How many equal exposures would produce the same borrower HHI?
Which connected group becomes largest after entity aggregation?
What share of total unexpected loss comes from the top ten names?
Does portfolio tail loss change sharply when asset correlation rises?
Which collateral market supports the greatest share of exposures?
What happens if the largest borrower and dominant sector weaken together?
Does the portfolio remain acceptable after removing assumed diversification benefits?

Suppose someone claims, “We have 500 borrowers, so the portfolio is diversified.” A falsifier is evidence that 400 borrowers depend on the same commercial property market and that a common stress causes their PD and LGD to rise together. Borrower count measures records; diversification measures independent risk.

19. Verification and update triggers

reconcile group exposure across lending, derivatives and contingent facilities;
validate connected-counterparty mappings;
compare several concentration dimensions instead of one HHI;
backtest default-correlation assumptions through stressed periods;
stress collateral and borrower quality jointly;
update sector mappings after business-model changes;
re-run concentration metrics after mergers, acquisitions or large originations;
challenge any portfolio described as diversified without naming the factors that are supposed to be independent.

Connections across the finance-and-banking algorithms lane

Bank capital models — regulatory risk weights use PD, LGD, EAD and correlation but do not replace concentration management.
Credit-rating migration — deterioration can cluster across names exposed to one factor.
Expected credit loss — expected value can remain moderate while concentration increases tail severity.
Interbank network contagion — another form of concentration created by common exposures and connections.

Research anchors

The deeper lesson

Diversification is not the appearance of many lines in a loan system. It is the presence of genuinely different failure routes. HHI reveals weight concentration. Connected-counterparty analysis reveals hidden identity concentration. Correlation reveals common-factor dependence. Granularity reveals whether individual names still matter too much. Stress testing turns all of those abstractions back into the question that matters: how much of the bank can fail together?

Educational note: This article explains banking mathematics and public concentration-risk concepts. It is not credit advice, lending advice, investment advice or institution-specific regulatory guidance.

Small Group Tutorials

How Banks Measure Credit-Portfolio Concentration: HHI, Default Correlation, Granularity, Large Exposures and Stress Testing