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Outline why estimating default correlation for credit portfolios matters and tell reduced form default correlation models apart from structural ones.

FRM® Part II · Credit Risk Measurement and Management · Credit Risk · FRM2.CR.12.e

The idea

Default correlation determines how likely defaults are to cluster. Expected loss on a portfolio depends only on the individual default probabilities. The tail of the loss distribution, and so economic capital and the value of senior tranches, rises sharply with correlation, because many simultaneous defaults become more likely. For two names with default probabilities PDA and PDB and default correlation ρ, the joint default probability is: Joint = PDA × PDB + ρ × √(PDA(1 − PDA) × PDB(1 − PDB)). Reduced form models treat default as a random event with a hazard rate and create correlation by linking intensities or default times. They calibrate to market spreads but give low default correlation unless intensities are very volatile. Structural models of the Merton type say default occurs when asset value falls below debt, with correlation coming from correlated asset values. They have economic intuition but need balance sheet data. Exam point: correlation changes the tail, not the mean.

Formulas

Joint default probability of two names

JDP = PDA × PDB + ρ × √(PDA × (1 − PDA) × PDB × (1 − PDB))

PDA
the default probability of name A
PDB
the default probability of name B
ρ
the default correlation between them

With ρ = 0 it is the independent product. Leaving out the product term or the square root scaling are the usual errors.

A practice question

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

Why is default correlation important for a portfolio of loans?

  1. It leaves expected loss unchanged but fattens the tail of the loss distribution, raising unexpected loss and capital. - correct
  2. It raises expected loss because correlated loans default more often individually.Why not: Correlation changes the joint behaviour, not each loan's default probability, so expected loss is unchanged.
  3. It lowers portfolio risk because correlated exposures offset one another.Why not: Positive correlation reduces diversification, so risk rises, not falls.
  4. It matters only for equity tranches of securitisations, because senior claims are protected by subordination and are unaffected by it.Why not: Senior tranches are exposed to extreme clustering of defaults, so they are affected too.

Expected portfolio loss is the sum of individual expected losses. Correlation affects how often many defaults happen together, which drives the extreme percentiles used for credit VaR and economic capital.

A harder one

A credit desk wants to model default correlation across 300 corporate names. It has CDS spreads for all names but no usable balance sheet or asset value data. Which approach fits best, and why?

  1. A structural model, because asset value correlations can be inferred from CDS spreads alone.Why not: Structural models need asset values and debt levels, which this desk does not have.
  2. A reduced form model, because hazard rates can be calibrated to market spreads and linked through correlated intensities or default times. - correct
  3. A structural model, because it is always more accurate than reduced form models.Why not: Accuracy is not guaranteed and the data requirement rules it out here.
  4. Neither; default correlation cannot be modelled without a history of observed defaults.Why not: Both approaches use market-implied inputs, so a history of defaults is not required.

Reduced form models take default intensity from market prices and require no capital structure data. Structural models explain default economically but need asset value and liability data.

The hardest tier

A portfolio has two loans, each USD 100 million with zero recovery and a one-year default probability of 5%. The default correlation between the two loans is 0.30. The standard deviation of the portfolio loss is closest to:

  1. USD 35.1 million. - correct
  2. USD 30.8 million.Why not: Uses zero correlation, which understates the variance of the portfolio loss.
  3. USD 33.1 million.Why not: Adds the correlation term once (rho x s²) instead of twice (2 x rho x s²) in the variance.
  4. USD 43.6 million.Why not: Adds the two standard deviations, which assumes perfect correlation.

Each loan has loss variance 100² x 0.05 x 0.95 = 475. Portfolio variance = 475 x (1 + 1 + 2 x 0.30) = 1,235, so the standard deviation = 35.1. Correlation 0 would give 30.8 and perfect correlation 43.6.

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