Expected Loss, Unexpected Loss and Credit VaR

A lender making thousands of loans knows, in aggregate, roughly how many will default — not which ones, but how many. That known, average loss rate is priced into every loan through the interest rate charged and is absorbed through ordinary provisions, like a routine cost of doing business. What a lender cannot price into the rate is the risk that losses in a bad year run far above that average — and that gap between the average and the bad-year outcome is what capital exists to absorb.

Defining the pieces

Expected loss on a loan or portfolio is:

$$EL = PD \times LGD \times EAD$$

where PD is the probability of default over the horizon, LGD is loss given default (the fraction not recovered), and EAD is exposure at default (the amount outstanding when default happens). This is a single number, an average, and a bank prices its lending margin to cover it plus a profit spread.

Unexpected loss (UL) is a measure of how much actual losses could deviate above that average in a bad scenario. Formally it is often expressed as the standard deviation of the loss distribution, but in practice risk managers care most about a specific high credit VaR: the loss level that will not be exceeded at some confidence level, say 99.9 percent over a year — the number regulatory capital is meant to cover, so that the bank survives all but the worst 1-in-1,000 years.

Worked example: EL on a loan book

A bank has a $500 million portfolio of commercial loans with an average PD of 2 percent over one year, an LGD of 45 percent, and an EAD equal to the full outstanding balance.

$$EL = 0.02 \times 0.45 \times 500{,}000{,}000 = 4{,}500{,}000$$

Expected loss is $4.5 million a year, or 0.9 percent of the book. If the portfolio earns an average interest margin of 3 percent over funding cost, the bank has $15 million of margin against $4.5 million of expected loss — $10.5 million left to cover operating costs and unexpected loss.

Worked example: why unexpected loss is a different number entirely

Suppose the loss distribution for this same portfolio, from a credit risk model, has a mean (expected) loss of $4.5 million but a 99.9th percentile loss — the credit VaR — of $38 million in a severe recession year, because defaults become correlated across borrowers exposed to the same economic downturn (the same correlation mechanic explored in Default Correlation and the Asset Threshold Model). Unexpected loss, defined as the gap between the tail outcome and the average, is:

$$UL = 38{,}000{,}000 - 4{,}500{,}000 = 33{,}500{,}000$$

That $33.5 million is what regulatory capital against this portfolio is meant to cover — not the $4.5 million average, which the bank should already be earning back through its margin. A bank that only reserves $4.5 million and holds no further capital cushion is solvent in every ordinary year and insolvent the moment a bad year actually arrives, because it never provisioned for the difference between average and tail.

Where you meet it in practice

Basel-style regulatory capital formulas, internal bank capital allocation, loan pricing models, and portfolio credit risk systems all separate EL from UL explicitly. A credit analyst pricing a new loan is really solving two problems at once: what margin covers the average loss, and what capital charge — priced implicitly through required return on equity — covers the tail. Missing either one shows up as either an uncompetitive price or an under-capitalized bank.