Gini Coefficient and Credit Concentration Risk

Credit → Default/distribution risk
| 2025-11-13 17:30:16

Introduction Slide – Gini Coefficient and Credit Concentration Risk

Understanding Credit Portfolio Concentration Through the Gini Coefficient

Overview

Introduction to concentration risk and the role of the Gini Coefficient in measuring credit concentration across counterparties, sectors, or regions.
The importance of identifying credit concentration risk to prevent excessive exposure and mitigate potential large losses in credit portfolios.
Coverage includes key concepts, mathematical foundations, graphical analyses, and implications for risk management.
Summarizes how the Gini Coefficient quantifies inequality in exposures and supports informed decision-making to enhance portfolio diversification.

Key Discussion Points – Gini Coefficient and Credit Concentration Risk (Updated 2025)

Major Drivers and Implications of Credit Concentration Risk in 2025

Main Points

Credit concentration risk stems from uneven exposure distributions to counterparties, sectors, or geographies within loan portfolios.
The Gini Coefficient (ranging from 0 to 1) captures concentration and inequality in exposures.
Within loan portfolios, the coefficient typically increases in growth industries reflecting focused investment, but may spike in struggling industries requiring financing to survive during downturns.
Heightened Gini values signify increased vulnerability to default contagion and systemic shocks, mandating portfolio diversification and risk monitoring.
Continuous surveillance and dynamic stress testing are vital to effective concentration risk management in 2025.

Analytical Explanation & Formula – Gini Coefficient and Credit Concentration Risk

Analytical Foundations and Quantitative Specification

Concept Overview

The Gini Coefficient quantifies how unevenly credit exposures are distributed, by comparing the actual distribution to a perfectly equal one.
Formally, it is calculated with reference to the Lorenz curve $L(p)$, which shows the cumulative share of exposure as a function of cumulative share of counterparties (ranked by size).
The coefficient's value reflects portfolio concentration: near zero for uniform exposures, near one for highly concentrated portfolios.
Important Note: The Lorenz curve and line of equality lie within a 1x1 square, but the area under the equality line is a triangle of area 0.5, not 1. All area-based calculations must reference this triangular area to ensure correct scaling and interpretation of the Gini coefficient.

General Formula Representation

The Gini Coefficient $G$ is:

$$ G = 1 - 2 \int_0^1 L(p) \, dp $$

Where:

$ L(p) $: Lorenz curve, the cumulative proportion of exposure up to percentile $p$.
$ p $: cumulative proportion of counterparties ranked by exposure.

This formula assumes integration relative to the triangular area under the equality line (area = 0.5), ensuring Gini $G$ lies between 0 and 1, supporting meaningful risk-based diversification decisions.

Lorenz Curve and Gini Coefficient Understanding

Visualizing Portfolio Concentration Using the Lorenz Curve

Conceptual Overview

The Lorenz curve plots the cumulative share of credit exposure against the cumulative share of counterparties, depicting distribution inequality.
The area between the line of perfect equality (45° diagonal) and the Lorenz curve is used to calculate the Gini Coefficient.
This area lies within the triangle of total area 0.5 in the 1x1 plot.
The Gini coefficient is calculated as twice the gap area $A$, where $A = 0.5 - B$ and $B$ is the area under the Lorenz curve.
Key Insight: Proper scaling requires these areas be relative to the triangle, ensuring Gini values between 0 and 1.
A larger gap area $A$ signals greater concentration and risk, whereas a smaller $A$ indicates a more diversified portfolio.

Figure: Gini Coefficient Trend in Credit Portfolio (2020–2025)

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Derivation & Interpretation – Gini Coefficient in Credit Concentration

Geometric Derivation and Conceptual Insight

Geometric Derivation and Interpretation

The Gini Coefficient equals twice the area $A$ between the line of perfect equality and the Lorenz curve: $$ G = 2A $$.
The area under the equality line (45°) is $ \frac{1}{2} $; the area under the Lorenz curve is $ B = \int_0^1 L(p)\,dp $.
The gap between these is $$ A = \frac{1}{2} - B $$, so $$ G = 1 - 2B = 2A $$.
Crucial Note: Using the triangle with area 0.5 as the baseline ensures correct normalization of $G$ between 0 and 1.
Higher $G$ indicates higher concentration, lower $G$ means more diversified portfolios.

Visual & Mathematical Summary

Key relationships between area and index:

$$ A = \tfrac{1}{2} - \int_0^1 L(p)\,dp $$

$$ G = 2A = 1 - 2\int_0^1 L(p)\,dp $$

Where:

$ L(p) $: Lorenz curve (cumulative exposure share).
$ p $: cumulative share of counterparties (sorted by exposure).

As $ G $ rises, concentration and systemic vulnerability increase; as $ G $ approaches zero, the portfolio is more diversified.

Calculation Example – Lorenz Curve Area & Gini Coefficient

Analytical Summary & Table – Lorenz Curve and Gini Coefficient

Illustrative Data Table

Example portfolio of 10 loan exposures ordered ascendingly by size.

Loan ID	Exposure Amount	Percentile of Counterparties	Cumulative Exposure (Amount and %)	Incremental Area Below Lorenz Curve	Cumulative Area Below Lorenz Curve $B$
10	1,800	10%	1,800 (2.46%)	0.00123	0.00123
9	2,400	20%	4,200 (5.74%)	0.00347	0.0047
8	3,800	30%	8,000 (10.94%)	0.00753	0.01223
7	4,700	40%	12,700 (17.37%)	0.01220	0.02443
6	5,300	50%	18,000 (24.65%)	0.01841	0.04284
5	6,500	60%	24,500 (33.56%)	0.02596	0.0688
4	8,000	70%	32,500 (44.52%)	0.03598	0.10478
3	9,500	80%	42,000 (57.56%)	0.04785	0.15263
2	11,000	90%	53,000 (72.66%)	0.06075	0.21338
1	19,300	100%	72,300 (100.00%)	0.08631	0.29969
Area Below Lorenz Curve (B)					0.29969
Area Between Lorenz Curve and Line of Equality (A) = 0.5 - B					0.20031
Gini Coefficient (G) = 2A = 1 - 2B					0.40062

Key Discussion Points

Ensure Gini coefficient calculations anchor on the triangular area under the equality line (area = 0.5), avoiding values > 1.
Use cumulative exposures and trapezoidal increments to approximate the Lorenz curve area $B$.
Compute gap area $A = 0.5 - B$ before deriving Gini $G = 2A = 1 - 2B$.
Sorting exposures ascendingly affects Lorenz curve shape and concentration interpretation.

Graphical Analysis – Gini Coefficient and Credit Concentration Risk

Trend Analysis of Credit Concentration Over Time

Context and Interpretation

This line chart illustrates the trend in a portfolio’s Gini Coefficient from 2020 to 2025, showing concentration changes over time.
An increasing trend indicates growing concentration risk, highlighting exposure unevenness.
Such trends require management attention for intervention to enhance diversification.
Key insight: rising Gini values warn of escalating credit concentration risk that may impact portfolio stability.