How does the Internal Loss Multiplier (ILM) adjust capital based on loss history, and why is the logarithmic function used to dampen extreme values?

The Internal Loss Multiplier (ILM) is the mechanism through which a bank's actual operational loss experience adjusts its SMA capital requirement. The formula uses a logarithmic function that increases with loss severity but at a decreasing rate, preventing astronomical capital charges for banks with extreme loss histories while still penalizing poor loss records.\n\nILM Formula:\n\nILM = ln(exp(1) - 1 + (LC/BIC)^0.8)\n\nWhere LC = Loss Component and BIC = Business Indicator Component.\n\nKey Properties:\n\n- When LC = BIC: ILM = ln(exp(1) - 1 + 1) = ln(e) = 1.0 (neutral -- capital equals BIC)\n- When LC < BIC: ILM < 1 (capital reduction for banks with better-than-average loss histories)\n- When LC > BIC: ILM > 1 (capital surcharge for banks with worse-than-average loss histories)\n- When LC >> BIC: ILM grows logarithmically, not linearly (dampened growth)\n\n`mermaid\ngraph LR\n A[\"LC/BIC Ratio\"] --> B{\"Ratio value?\"}\n B -->|\"< 1.0
(good loss record)\"| C[\"ILM < 1.0
Capital below BIC\"]\n B -->|\"= 1.0
(average)\"| D[\"ILM = 1.0
Capital equals BIC\"]\n B -->|\"2.0\"| E[\"ILM = 1.26\"]\n B -->|\"5.0\"| F[\"ILM = 1.69\"]\n B -->|\"10.0\"| G[\"ILM = 2.04\"]\n B -->|\"50.0\"| H[\"ILM = 3.14\"]\n E --> I[\"Logarithmic dampening:
10x loss ratio only
doubles the multiplier\"]\n G --> I\n`\n\nWorked Example -- Feldspar Regional Bank vs. Kingsbridge Global Bank:\n\n| Metric (EUR M) | Feldspar | Kingsbridge |\n|---|---|---|\n| BIC | 280 | 2,100 |\n| Average annual losses | 18 | 520 |\n| Loss Component (LC) | 126 | 3,640 |\n| LC / BIC ratio | 0.45 | 1.73 |\n| ILM | 0.80 | 1.21 |\n| OpRisk Capital | 280 x 0.80 = 224 | 2,100 x 1.21 = 2,541 |\n\nFeldspar's strong loss record (LC/BIC = 0.45) earns a 20% capital discount. Kingsbridge's elevated losses (LC/BIC = 1.73) incur a 21% capital surcharge.\n\nWhy Logarithmic Dampening:\n\nBasel considered linear and exponential alternatives. A linear function (ILM = LC/BIC) would impose punitive capital on banks with extreme historical events, potentially threatening solvency. The logarithm ensures:\n\n1. A bank with 10x average losses pays roughly 2x capital, not 10x\n2. Very large historical events (rogue trading, massive litigation) create significant but survivable capital charges\n3. The incentive to reduce losses remains (ILM is still increasing) but doesn't create cliff effects\n\nNational Discretion:\n\nCritically, supervisors can set ILM = 1 for all banks in their jurisdiction, effectively eliminating the internal loss data component. Several jurisdictions (including the EU in CRR3) have exercised this discretion, making capital purely BI-driven. This was controversial because it removes the loss-sensitivity that was the SMA's key innovation over the old BIA.\n\nDive deeper into ILM calibration in our FRM Part II course.