How is credit unexpected loss calculated, and what is its relationship to economic capital?

Unexpected loss (UL) measures the volatility of credit losses around the expected loss. While expected loss is a cost of doing business (priced into loan spreads and reserved for), unexpected loss is the uncertainty that requires capital.

Single Exposure Formula

UL = EAD x sqrt(PD x sigma_LGD^2 + LGD^2 x PD x (1 - PD))

where:

• PD = probability of default
• LGD = loss given default (mean)
• sigma_LGD = standard deviation of LGD
• EAD = exposure at default

The formula captures two sources of uncertainty: variability in LGD (first term) and variability in default itself (second term, a Bernoulli variance).

Worked Example

Ridgemont Bank has a $10 million commercial loan to a mid-market manufacturer.

Expected Loss:

EL = 0.025 x 0.45 x $10M = $112,500

Unexpected Loss:

UL = $10M x sqrt(0.025 x 0.25^2 + 0.45^2 x 0.025 x 0.975)

UL = $10M x sqrt(0.025 x 0.0625 + 0.2025 x 0.024375)

UL = $10M x sqrt(0.001563 + 0.004936)

UL = $10M x sqrt(0.006499)

UL = $10M x 0.08062

UL = $806,200

Relationship to Economic Capital

Economic capital covers losses beyond expected loss up to a chosen confidence level (typically 99.9% for banks). The relationship is:

Economic Capital = Loss at 99.9% confidence - Expected Loss

For a portfolio, UL is not simply additive because default correlations matter. Correlated defaults increase portfolio-level unexpected loss. The capital multiplier (often 3-5x single-exposure UL for a diversified portfolio at 99.9%) bridges single-name UL to the capital requirement.

Key Exam Points:

• EL is a cost (reserved/provisioned); UL is a risk (capitalized)
• Higher LGD variability increases UL even if mean LGD stays the same
• Portfolio UL depends critically on default correlation — higher correlation = higher UL
• Basel IRB models use a specific correlation function that varies by PD

For more on credit risk capital, explore our FRM Part II question bank.