A bank's loan book has expected loss of $40 million per year. Capital is sized to absorb a once-per-thousand-years loss. The question that drives the capital number is not "what's the average loss". It's "how do defaults cluster when the economy turns?" Same average, different correlation, very different capital.
Banks hold two flavors of capital. Regulatory capital is the amount Basel rules require for a given risk-weighted-asset calculation. Economic capital is the bank's own internal estimate of capital needed to survive at a chosen confidence level. The amount that limits insolvency probability to a target rate (often 0.03% or 0.05%).
The two numbers can differ. Regulatory capital uses standardized formulas with conservative parameters; economic capital uses bank-specific models with empirical parameters. Banks track both because they pay regulatory capital costs but make business decisions on economic-capital-driven risk-adjusted return on capital (RAROC).
KEY: Economic capital sized to a 99.97% confidence level corresponds to a one-in-3,333 year survival probability, roughly the historical default rate of an AA-rated firm.
Common mistakes
- Treating EL as the capital number. EL is priced into spreads and reserves; capital absorbs UL beyond EL. A choice that recommends "hold capital equal to expected loss" is wrong.
- Using equity correlation as default correlation. Equity returns of the same firms might correlate at 0.5; default events typically correlate at 0.05 to 0.15. A question that supplies an equity correlation and asks for portfolio credit risk is testing whether you swap to default correlation.
- Adding unexpected losses linearly. Portfolio UL is NOT the sum of single-loan ULs unless correlation equals 1. With ρ < 1, portfolio UL is less than the linear sum (diversification). With ρ = 0, portfolio UL scales with √N.
Bottom line
- Expected loss (EL) = PD × LGD × EAD, priced into spreads and absorbed by reserves. Unexpected loss (UL) is the volatility of losses around EL, absorbed by economic capital (never set capital equal to EL).
- Single-asset UL under binomial default with deterministic LGD: . Portfolio UL rises with default correlation, not the linear sum of single-loan ULs.
- Default correlations typically run 0.05 to 0.15 and Basel IRB asset correlations 0.12 to 0.24; never substitute the far higher equity correlations, which overstate UL.
- Gaussian copula drives credit portfolio models: each loan defaults when its standardized asset return falls below a threshold, with a shared common factor creating correlation.
Exam shortcut
When a credit-portfolio question gives you PD, LGD, EAD, and asset correlation ρ, reach for Vasicek immediately. Most FRM credit VaR problems are Vasicek with rounded inputs. The conditional default rate at 99.9% is always larger than PD; the difference grows with ρ. For ρ = 0, conditional rate equals PD (no concentration); for ρ = 1, every loan defaults together at the tail.
The full lesson (about 3,428 words, 23 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
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