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Under an assumed binomial distribution, derive the variance of the probability of default.

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

The idea

Treat one obligor's default over the horizon as a binomial, or Bernoulli, trial. An indicator D equals 1 with probability PD and 0 otherwise. Its mean is PD. Because D is either 0 or 1, D² equals D, so the expected value of D² is also equal to PD. The variance is the expected value of D² minus the squared mean. Variance = PD − PD² = PD × (1 − PD). The standard deviation is the square root of that variance, and it is largest at a PD of 50 percent. This is the building block of UL. The default rate of N independent identical obligors is the number of defaults divided by N. Writing Vrate for the variance of the default rate across those N obligors, Vrate = PD × (1 − PD) ÷ N. Now consider the number of defaults. Writing Vcount for its variance, Vcount = N × PD × (1 − PD). Diversification shrinks the variance of the rate, while the variance of the count grows with N. Do not confuse the variance with the PD itself or with its square root.

Formulas

Variance of a binomial default indicator

Var(D) = PD × (1 − PD)

PD
the probability of default
D
the default indicator, 1 on default and 0 otherwise

From E[D²] − (E[D])², since D² = D. It is the variance, not the standard deviation; take the root for that.

Variance of the default rate of N independent obligors

Vrate = PD × (1 − PD) ÷ N

PD
each obligor's probability of default
N
the number of obligors

Independent, identical obligors. The variance of the number of defaults is N × PD × (1 − PD) instead, which grows with N.

A practice question

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

The one-year PD of an obligor is 3 percent. Under the binomial assumption the variance of its default indicator is:

  1. 0.0300.Why not: Takes the variance to equal the PD, omitting the (1 - PD) correction.
  2. 0.0291. - correct
  3. 0.0009.Why not: Uses PD² alone, which is only one term of the variance.
  4. 0.1706.Why not: Reports the standard deviation rather than the variance.

Variance = PD x (1 - PD) = 0.03 x 0.97 = 0.0291.

A harder one

In deriving the variance of a default indicator D, equal to 1 with probability PD and 0 otherwise, which value does E[D²] take?

  1. PD², because the expectation of a square is the square of the expectation.Why not: Applies E[D²] = (E[D])², which holds only for a constant, not a random variable.
  2. PD x (1 - PD), because the second moment is the variance.Why not: Mixes up the second moment with the variance.
  3. PD, because D² equals D when D is 0 or 1. - correct
  4. 1, because D² is always one.Why not: Ignores that D is zero in the no-default case.

Because 0² is 0 and 1² is 1, D² = D, so E[D²] = E[D] = PD; subtracting PD² gives the variance PD(1-PD).

The hardest tier

A portfolio holds 400 equal, independent loans, each with a PD of 2 percent. Under the binomial assumption, the standard deviation of the portfolio default rate is closest to:

  1. 0.70 percent. - correct
  2. 14.00 percent.Why not: Uses the single-obligor standard deviation, with no diversification.
  3. 2.80 percent.Why not: Reports the standard deviation of the number of defaults as if it were a rate.
  4. 0.035 percent.Why not: Divides the standard deviation by N instead of by the square root of N.

Default-rate variance = PD(1-PD)/N = 0.0196/400; sd = 0.00700 = 0.70%.

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