Why does the Basel IRB formula include a maturity adjustment, and how does longer loan maturity increase capital requirements?

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AcadiFi · Accepted Answer

The **maturity adjustment** in the Basel IRB formula accounts for the fact that longer-maturity loans face greater **migration risk** — the risk that the borrower's credit quality deteriorates before the loan matures, requiring additional capital or provisioning. **Economic Rationale** A 1-year loan to a BBB borrower that defaults in year 1 causes a loss. But a 5-year loan to the same borrower faces additional risk: even if the borrower doesn't default immediately, a downgrade to BB or B would reduce the loan's market value. The bank is locked into a riskier exposure for a longer period. **The Maturity Adjustment Formula** **MA = (1 + (M - 2.5) x b(PD)) / (1 - 1.5 x b(PD))** where **b(PD) = (0.11852 - 0.05478 x ln(PD))^2** is a smoothing function that increases with credit quality (lower PD = higher sensitivity to maturity). **Key Properties:** - At M = 2.5 years (F-IRB default), MA = 1.0 (no adjustment) - At M > 2.5 years, MA > 1.0 (capital increases) - At M < 2.5 years, MA < 1.0 (capital decreases) - The adjustment is more pronounced for better-rated borrowers (lower PD) **Worked Example** Fieldstone Commercial Bank evaluates two loans to different borrowers: | Parameter | Loan A (High Quality) | Loan B (Lower Quality) | |---|---|---| | PD | 0.50% | 3.00% | | Maturity | 4.0 years | 4.0 years | **Loan A: b(PD) = (0.11852 - 0.05478 x ln(0.005))^2** b = (0.11852 - 0.05478 x (-5.298))^2 = (0.11852 + 0.2902)^2 = (0.4087)^2 = 0.1670 **MA_A = (1 + (4.0 - 2.5) x 0.1670) / (1 - 1.5 x 0.1670)** MA_A = (1 + 0.2505) / (1 - 0.2505) = 1.2505 / 0.7495 = **1.668** **Loan B: b(PD) = (0.11852 - 0.05478 x ln(0.03))^2** b = (0.11852 - 0.05478 x (-3.507))^2 = (0.11852 + 0.1921)^2 = (0.3106)^2 = 0.0965 **MA_B = (1 + (4.0 - 2.5) x 0.0965) / (1 - 1.5 x 0.0965)** MA_B = (1 + 0.1448) / (1 - 0.1448) = 1.1448 / 0.8552 = **1.339** ```mermaid flowchart LR A["Maturity = 4 years"] --> B{"PD Level?"} B -->|"Low PD (0.5%)"| C["MA = 1.668 — large uplift"] B -->|"Higher PD (3.0%)"| D["MA = 1.339 — moderate uplift"] C --> E["Better credits: more room to migrate downward"] D --> F["Weaker credits: already near default, less migration risk"] ``` **Intuition: Why Is the Adjustment Larger for Low-PD Borrowers?** A high-quality (AA) borrower has many possible downgrades (AA to A, BBB, BB, B, CCC, default) — lots of "room to fall." A borrower already rated CCC is near default; the additional risk from extending maturity is proportionally smaller because the base PD is already high. **Practical Impact:** For the same 4-year $10M loan: - Loan A (PD=0.5%, MA=1.668): capital ~67% higher than at 2.5 years - Loan B (PD=3.0%, MA=1.339): capital ~34% higher than at 2.5 years This incentivizes banks to manage portfolio tenor and explains why short-term revolving facilities are more capital-efficient than long-term term loans. For more on the IRB formula, visit our FRM Part II course.

Why does the Basel IRB formula include a maturity adjustment, and how does longer loan maturity increase capital requirements?

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