State what credit spread risk is and weigh its effect on the computation of credit VaR.

FRM® Part II · Credit Risk Measurement and Management · Credit Value at Risk · FRM2.CR.10.f

The idea

Credit spread risk is the risk that credit spreads change and so move the market value of a credit-risky position, even though no default or rating change occurs. For a small change in spreads, the approximation is Loss = market value × spread duration × change in spread. Migration-based models such as CreditMetrics revalue each rating state with fixed forward spreads. They therefore leave this risk out and understate credit VaR when spreads are volatile. Including it, by simulating spreads or adding a spread VaR, widens the distribution. Spreads tend to widen when defaults rise, so the two components are positively correlated and add by more than independence suggests. Spreads also include a risk premium, so a PD implied from spreads overstates the real-world PD. Ignoring that premium, the implied figure is PD = spread ÷ LGD.

Formulas

Mark-to-market loss from a spread widening

Loss = MV × spread duration × Δspread

MV: the market value of the position
spread duration: the percentage price sensitivity to a change in spread
Δspread: the change in credit spread, as a decimal

A first-order approximation for small spread changes. Write 40 bp as 0.004, not 0.04.

Default probability implied from a credit spread

PD = spread ÷ LGD

spread: the credit spread over the risk-free rate
LGD: loss given default, one minus the recovery rate

Ignores the risk premium in the spread, so the result is risk-neutral and above the real-world PD. Dividing by the recovery rate instead of LGD is the usual slip.

A practice question

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

Credit spread risk is the risk that:

a borrower fails to pay, causing a loss of principal and interest.Why not: This is default risk; spread risk can produce mark-to-market losses with no default.
the exposure to a counterparty at a future date is larger than expected.Why not: Describes counterparty exposure risk, not movement in credit spreads.
changes in credit spreads alter the market value of a credit-risky position even when there is no default or rating change. - correct
the recovery rate on a defaulted claim is lower than assumed.Why not: Describes recovery risk, which is separate from the pricing of credit risk through spreads.

Spreads move with market perception of credit quality and risk appetite; a widening reduces the value of a bond whose issuer has not defaulted or been downgraded.

A harder one

A bond portfolio has a market value of USD 50 million and a spread duration of 4.5. Credit spreads widen by 40 basis points with no default or downgrade. The mark-to-market loss is closest to:

USD 0.20 millionWhy not: Omits spread duration: 50 x 0.004.
USD 0.90 million - correct
USD 9.00 millionWhy not: Treats 40 basis points as 4%: 50 x 4.5 x 0.04.
USD 1.80 millionWhy not: Reports duration x spread change (4.5 x 0.4% = 1.8%) as if it were a currency amount, without multiplying by market value.

Loss = market value x spread duration x change in spread = 50 x 4.5 x 0.004 = 0.90 million.

The hardest tier

A bank's credit VaR from default and migration is USD 12 million and its credit spread VaR is USD 5 million. Assume the two are jointly normal with a correlation of 0.4, reflecting that spreads widen when defaults rise. The combined credit VaR is closest to:

USD 14.7 million - correct
USD 13.0 millionWhy not: Assumes independence: the square root of (12² + 5²) = 13.
USD 17.0 millionWhy not: Adds the two figures, assuming perfect correlation of 1.0.
USD 12.0 millionWhy not: Ignores spread risk altogether, leaving the default and migration figure.

Combined = sqrt(12² + 5² + 2 x 0.4 x 12 x 5) = sqrt(144 + 25 + 48) = sqrt(217) = 14.7 million. Positive correlation raises the total above the independent figure.

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