A bank's loan book has expected loss of $40 million per year. A 99.9% annual loss quantile allows a modeled 0.1% chance of exceedance in any year. How defaults cluster when the economy turns can change that quantile substantially even when average loss stays the same.
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: For a one-year horizon, economic capital sized to 99.97% confidence targets an annual failure probability of 0.03%, about 1 in 3,333 under the model. It does not promise survival for 3,333 years.
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. Identify the random variables the correlation measures. A loan-loss covariance calculation requires loss correlation, while the Vasicek formula uses latent asset correlation. An equity correlation cannot be substituted for either without a model connecting them.
- 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, asset and equity correlations measure different variables. Use the correlation required by the specified model; regulatory asset-correlation inputs depend on exposure class.
- 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
For a granular homogeneous loan pool, calculate the Vasicek stressed default fraction first, then multiply by LGD and EAD. Check whether the question asks for total-loss VaR or capital above expected loss. At zero asset correlation the limiting default fraction is PD because idiosyncratic risk diversifies away; at perfect correlation all defaults move together and the outcome is either no default or total default, with the selected quantile depending on...
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