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Compute UL for a portfolio of credit assets, along with each asset's contribution to it, across differing portfolio mixes, asset features and sizes.

FRM® Part II · Credit Risk Measurement and Management · Capital Structure in Banks · FRM2.CR.4.f

The idea

With the loss rate treated as fixed, each asset's standalone unexpected loss is ULi = EAi × LGDi × √(PDi × (1 − PDi)). Here EA is the exposure, LGD the loss given default and PD the probability of default. Portfolio UL combines the asset ULs through their pairwise correlations: ULP = √(Σi Σj ρij × ULi × ULj), summing over every pair. Portfolio UL is below the sum of the standalone ULs unless every correlation equals 1, and for two assets ULP = √(UL1² + UL2² + 2ρ × UL1 × UL2). Asset i's contribution, which equals ULi times its correlation with the portfolio, is ULCi = ULi × (Σj ρij × ULj) ÷ ULP. The contributions add up exactly to ULP. Larger exposures, higher LGDs, higher PDs (for any PD below 50%) and higher correlations raise UL and the contribution. Common errors: adding standalone ULs, using the wrong sign on the correlation term, and allocating UL in proportion to standalone UL, which ignores correlation.

Formulas

Unexpected loss of a two-asset portfolio

ULP = √(UL1² + UL2² + 2ρ × UL1 × UL2)

UL1
the standalone UL of asset 1
UL2
the standalone UL of asset 2
ρ
the correlation between their losses

Below UL1 + UL2 unless ρ is 1. The cross term keeps its plus sign for positive correlation.

Risk contribution of asset i to portfolio UL

ULCi = ULi × (Σj ρij × ULj) ÷ ULP

ULi
the standalone UL of asset i
ρij
the loss correlation between assets i and j, with ρii equal to 1
ULj
the standalone UL of each asset j
j
the index running over every asset, i included
ULP
the portfolio UL

The contributions add exactly to ULP. Allocating in proportion to standalone UL ignores correlation.

A practice question

From the question bank, with the reason each wrong option is wrong.

Two loans have stand-alone ULs of USD 300,000 and USD 400,000, and their losses are uncorrelated. The portfolio UL is:

  1. USD 700,000.Why not: Adds the ULs, which assumes perfect correlation.
  2. USD 500,000. - correct
  3. USD 350,000.Why not: Averages the two ULs, which has no basis in the formula.
  4. USD 400,000.Why not: Takes the larger stand-alone UL, as though the smaller loan added no risk at all.

With zero correlation, UL(P) = sqrt(300,000² + 400,000²) = 500,000.

A harder one

Two loans have stand-alone ULs of USD 2.0 million and USD 1.5 million with a loss correlation of 0.3. The portfolio UL is closest to:

  1. USD 3.50 million.Why not: Adds the stand-alone ULs, ignoring diversification.
  2. USD 2.50 million.Why not: Sets the correlation to zero.
  3. USD 2.84 million. - correct
  4. USD 2.11 million.Why not: Subtracts the correlation term instead of adding it.

UL(P) = sqrt(2.0² + 1.5² + 2 x 0.3 x 2.0 x 1.5) = sqrt(8.05) = 2.837.

The hardest tier

A portfolio has three assets with stand-alone ULs of USD 4, 3 and 2 million. Correlations: assets 1 and 2 are 0.5, assets 1 and 3 are 0.2, assets 2 and 3 are 0. The contribution of asset 3 to portfolio UL is closest to:

  1. USD 0.84 million. - correct
  2. USD 2.00 million.Why not: Takes the stand-alone UL, ignoring diversification.
  3. USD 0.60 million.Why not: Counts only the asset's own variance and drops its covariances with the other assets.
  4. USD 1.48 million.Why not: Allocates portfolio UL in proportion to stand-alone ULs.

UL(P) = sqrt(16+9+4+2x(0.5x12+0.2x8+0)) = sqrt(44.2) = 6.648. ULC(3) = 2 x (0.2x4 + 0x3 + 1x2)/6.648 = 0.842.

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