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Weigh up how granularity influences Credit VaR.

FRM® Part II · Credit Risk Measurement and Management · Portfolio Credit Risk · FRM2.CR.11.h

The idea

Granularity describes how evenly exposure is spread across names. A granular portfolio has many small exposures, while a concentrated one has a few large ones. A common measure is the Herfindahl index, the sum of squared exposure shares. One divided by the HHI is the effective number of equal-sized exposures. For given PDs, LGDs and correlation, expected loss does not depend on granularity, but Credit VaR does. In a granular portfolio idiosyncratic risk diversifies away and only systematic factor risk remains. The limiting loss rate per unit of exposure, before LGD, at confidence q is given next. Loss rate = N((N⁻¹(PD) + √ρ × N⁻¹(q)) ÷ √(1 − ρ)). A concentrated portfolio keeps extra idiosyncratic risk and has a higher Credit VaR. Adding many names never removes systematic risk, so Credit VaR has a floor above zero.

Formulas

Herfindahl index of a credit portfolio

HHI = Σ wᵢ²

Σ
the sum over every exposure in the portfolio
wᵢ
each exposure's share of total exposure

Equals 1 ÷ N for N equal exposures. 1 ÷ HHI is the effective number of equal-sized names, which can be far below the count of names.

Limiting loss rate of an infinitely granular portfolio

loss rate = N((N⁻¹(PD) + √ρ × N⁻¹(q)) ÷ √(1 − ρ))

N
the standard normal cumulative distribution function, with N⁻¹ its inverse
PD
the probability of default
ρ
the asset correlation
q
the confidence level

Per unit of exposure before LGD; only systematic risk remains. A concentrated portfolio sits above this, and Credit VaR subtracts the expected loss.

A practice question

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

All else equal (average PD, LGD, total exposure and correlation), how does greater granularity of a credit portfolio affect Credit VaR?

  1. It raises Credit VaR, because more names mean more chances of default.Why not: Expected loss does not change, and more names diversify idiosyncratic risk rather than add to it.
  2. It has no effect, because expected loss is the same.Why not: Expected loss is the same but the tail is lower with more, smaller exposures.
  3. It eliminates credit risk, so Credit VaR falls to zero as the number of names grows.Why not: Systematic factor risk remains, so Credit VaR stays above zero.
  4. It lowers Credit VaR, because idiosyncratic risk diversifies away and mainly systematic risk remains. - correct

More, smaller exposures diversify away name-specific risk. Only factor risk remains, which sets a floor for Credit VaR.

A harder one

A portfolio holds 81 loans: one of USD 20 million and 80 of USD 1 million each (total USD 100 million). The effective number of equal-sized exposures, 1/HHI, is closest to:

  1. 81Why not: Counts the names and ignores the size differences.
  2. 5.0Why not: Takes 1 divided by the largest share (20%).
  3. 20.8 - correct
  4. 0.048Why not: Reports the Herfindahl index itself rather than its reciprocal.

Shares: 0.20 and 80 x 0.01. HHI = 0.04 + 80 x 0.0001 = 0.048. Effective number = 1 / 0.048 = 20.8, far fewer than 81 names.

The hardest tier

A very granular portfolio of identical loans has PD of 2%, LGD of 100% and asset correlation 0.20. The 99.9% loss rate is N((N^-1(PD) + sqrt(rho) x N^-1(99.9%)) / sqrt(1 - rho)), with N^-1(2%) = -2.054 and N^-1(99.9%) = 3.090. Table values: N(-0.67) = 0.2514, N(-0.75) = 0.2266, N(-0.84) = 0.2005, N(-1.61) = 0.0537. The 99.9% Credit VaR (loss rate less expected loss) is closest to:

  1. 21% - correct
  2. 23%Why not: Reports the 99.9% loss rate and does not subtract the 2% expected loss.
  3. 18%Why not: Uses rho = 0.20 instead of sqrt(rho) = 0.447 as the weight on the tail quantile, giving z = -0.84.
  4. 3.4%Why not: Uses -3.090 for the tail quantile (the 0.1% quantile), giving z = -1.61.

z = (-2.054 + 0.4472 x 3.090) / 0.8944 = -0.751. N(-0.75) = 0.2266, a 99.9% loss rate of about 22.7%. Less the 2% expected loss, Credit VaR is about 21%.

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