Describe how strongly loan defaults in a bank's loan portfolio tend to depend on one another, and explain what this means for the portfolio's default rate.

FRM® Part I · Valuation and Risk Models · Measuring Credit Risk · FRM1.VR.6.b

The idea

Loan defaults in a portfolio are positively correlated, because borrowers share exposure to the same economy. In a recession many firms weaken together, and in a boom few fail. The correlation between any two borrowers' default events is small in absolute terms, a few percent or less, but it matters. If defaults were independent, a large portfolio would show almost the same default rate every year. For n loans with default probability p, Standard deviation = √(p(1 − p) ÷ n). Observed default rates instead swing from well below average to several times average. With pairwise default correlation ρ, Variance = p(1 − p) × [1 ÷ n + (n − 1) ÷ n × ρ]. This does not fall to zero as n grows. It approaches ρ × p(1 − p). Correlation leaves expected loss unchanged but fattens the tail, so capital cannot be set assuming independence.

Formulas

Standard deviation of the default rate with independent defaults

σ = √(p(1 − p) ÷ n)

p: the default probability of each loan
n: the number of loans

Holds only if defaults are independent, so it falls towards zero as n grows. Real default rates vary far more because defaults are correlated.

Variance of the portfolio default rate with correlated defaults

Variance = p(1 − p) × [1 ÷ n + (n − 1) ÷ n × ρ]

p: the default probability of each loan
n: the number of loans
ρ: the pairwise default correlation

As n grows it approaches ρ × p(1 − p), not zero. Take the square root for the standard deviation of the default rate.

A practice question

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

Which statement best describes how loan defaults in a bank's portfolio depend on one another?

Defaults are essentially independent, so the portfolio default rate is almost the same every year.Why not: Independence would give a very stable default rate; observed default rates swing widely, which shows positive dependence.
Defaults are almost perfectly correlated, so all loans in the portfolio tend to default at the same time.Why not: Overstates dependence: pairwise default correlation is small, otherwise portfolio default rates would be all-or-nothing.
Defaults are negatively correlated, because one default removes a competitor and helps the remaining borrowers.Why not: Gets the sign wrong: in downturns defaults cluster, which is positive correlation.
Defaults are positively correlated through common exposure to the economy, though pairwise correlation is modest. - correct

Common economic conditions drive defaults, so they cluster in recessions. The correlation between two borrowers is small, but it is positive and it makes the portfolio default rate vary materially from year to year.

A harder one

A bank has 400 equally sized loans, each with a one-year default probability of 3%. If defaults were independent, the standard deviation of the portfolio's default rate would be closest to:

17.06%Why not: Forgets to divide by the number of loans; this is the standard deviation of a single loan's default indicator.
0.85% - correct
0.0073%Why not: Reports the variance of the default rate rather than its standard deviation.
0.043%Why not: Divides the single-loan standard deviation by n instead of by the square root of n.

Standard deviation of default rate = sqrt(p(1-p)/n) = sqrt(0.03 x 0.97 / 400) = 0.85%. Real portfolios show much larger variation because defaults are positively correlated.

The hardest tier

A bank holds 100 equal loans, each with default probability 2% and a pairwise default correlation of 0.05. Using variance of default rate = p(1-p)[1/n + (n-1)/n x rho], the standard deviation of the portfolio default rate is closest to:

1.40%Why not: Assumes independence and drops the correlation term, which dominates for a portfolio of 100 loans.
3.13%Why not: Uses the infinite-portfolio limit sqrt(rho x p(1-p)) and omits the 1/n sampling term that still matters at n = 100.
3.41% - correct
0.117%Why not: Reports the variance of the default rate as if it were the standard deviation.

Variance = 0.02 x 0.98 x (0.01 + 0.99 x 0.05) = 0.0196 x 0.0595 = 0.001166; square root = 3.41%. Independence would give only 1.40%.

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