How do you construct the loss distribution for a credit portfolio, and what is the difference between expected and unexpected loss?

Building the credit portfolio loss distribution is the core challenge of credit risk management. Expected loss is straightforward to compute; the difficulty lies in capturing how losses cluster due to correlations.

Expected Loss (EL)

For each exposure i:

EL_i = PD_i x LGD_i x EAD_i

Portfolio expected loss is simply the sum:

EL_portfolio = SUM(EL_i)

This is the mean of the loss distribution — the amount you should provision for through loan pricing and reserves.

Example Portfolio

Greystone Bank has a small loan portfolio:

Portfolio EL = $226,000

Unexpected Loss (UL) and the Full Distribution

Unexpected loss is the volatility of losses around the expected value. For a single exposure:

UL_i = EAD_i x sqrt[PD_i x (1-PD_i) x LGD_i^2 + LGD_var x PD_i^2]

But the portfolio UL is NOT the sum of individual ULs — it depends on default correlations.

UL_portfolio = sqrt[SUM_i SUM_j UL_i x UL_j x rho_ij]

Higher default correlation means losses tend to cluster: when one borrower defaults, correlated borrowers are more likely to default too. This fattens the tail of the loss distribution.

Credit VaR

Credit VaR = Quantile Loss (e.g., 99.9%) - Expected Loss

This is the "unexpected loss" at a specific confidence level — the capital the bank needs to hold.

Approaches to Building the Full Distribution

1. CreditMetrics — Monte Carlo simulation of correlated rating migrations using asset correlations (Merton model)
2. CreditRisk+ — Actuarial approach treating defaults as Poisson-distributed events, grouped by sectors
3. Vasicek single-factor model — Analytic formula assuming one systematic risk factor; used in Basel IRB
4. Monte Carlo with copulas — Most flexible; can incorporate non-normal dependence

Key Distinction

Practice credit portfolio problems in our FRM Part II question bank.