Measuring and interpreting credit risk measures – analogies to market risk measures
x = random variable profit/loss. If x is drawn from a normal distribution, N(,)
the standardized variable, z, can be constructed from the mean and standard deviation of
x.
x
z
Selecting a confidence level for the value at risk estimate determines the appropriate
quantile of the standard normal, N(0,1) distribution, i.e. the value of z.
For instance a confidence level, c = 0.99 results in the value of z (1-c =0.01) = -2.33.
Solving for the value of x (Value at Risk) corresponding to this confidence level.
-2.33 + = x. The 99% Value at Risk = -[-2.33 + ], is 2.33 below the mean. In
many market risk measurement settings (short (daily) horizons) it was argued that
estimates of the mean were so small and imprecise that it was better to ignore the mean.
When measuring credit risk it is not appropriate to ignore the mean credit loss. Notice
that the mean credit loss will be a negative value. The bulk of the probability mass of the
credit loss distribution is contained below zero.
If the definition of credit risk is expanded to include both the event of default and the
change of value resulting from credit rating changes, there is always a possibility that
credit ratings could improve and increase value.
Evaluating credit loss due to default event:
CL = b*CE*LGD = b*CE*(1-f)
b = 1 if default
b = 0 if not default
E(b) = Probability of default.
For a single credit risk exposure: E(CL) = PD * CE * (1-f)
n
For a portfolio of credit risk exposures: E (CL p ) PDi CEi (1 f i )
i 1
The value at risk due to credit risk lies below the expected credit loss by an amount that
must be inferred from the distribution of credit losses.
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The dispersion in credit losses for a portfolio of credit risks depends critically on the
correlation between default events across credit risks. This is analogous to the
dependence of a portfolio’s return variance on the correlation between the returns of the
portfolio’s assets.
For correlated default events (A,B) the expected probability of joint default:
E prob( A B Eb A bB PDA PDB ( A, B) A B
PDA PDB ( A, B) PDA (1 PDA ) PDB (1 PDB )
For a portfolio consisting of two credit risks (A,B) there are four possible outcomes.
2x2 Contingency table: PDA = 0.05, PDB = 0.07, (A,B) = 0.50
B
Default (B) Not (~B)
Default (A) 0.0313 0.0187 0.05
A
Not (~A) 0.0387 0.9113 0.95
0.07 0.93
E prob( A B PD A PD B ( A, B) PD A (1 PD A ) PD B (1 PD B )
0.05 0.07 0.50 0.05 (1 0.05 ) 0.07 (1 0.07 ) 0.03130
If CEA = $1,000 , CEB = $2,000 and the recovery rate fi = 0 for both A,B then:
Event CLp Likelihood Cumulative 1-Cum
~A ~B $0 0.9113 0.9113 0.0887
A ~B $1,000 0.0187 0.930 0.07
~A B $2,000 0.0387 0.9687 0.0313
AB $3,000 0.0313 1.000
E[CLp] = 0.05*$1,000 +0.07*$2,000 = $190
2 (CL p ) 0.9113 (0 190) 2 0.0187 (1,000 190) 2
0.0387 (2,000 190) 2 0.0313 (3,000 190) 2
419,100
(CL p ) $647.38
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The 95% confidence level quantile of the credit loss distribution:
Minimum CLi such that the cumulative probability, Prob(CL CLi) 95%
CLi(95%) = $2,000.
The 95% credit value at risk, VaRC(95%): distance between CLi(95%) and E(CL).
2,000 = VARC(95%) + 190
VaRc(95%) = $1,810.
It is useful to point out that for this portfolio of two credit risks the distribution of credit
losses is skewed left.
The iid normal methodology value at risk at the 95% confidence level
VaRC = -[-1.645 * $647.38 + 190] = -[-1,065 + 190] = $875.
Diversification of credit risks:
As identified earlier the correlation between default events for component credit risks
greatly influences the skew of the credit loss distribution.
Consider the following example calculating VARC(99%) for a portfolio of two credit
risks. In the fist pass the correlation between default events is zero. In this example
recovery rates for each credit risk are 50%.
PD(A) 5.00% B ~B
PD(B) 5.00% A 0.0025 0.0475 0.05
CORR(A,B) 0.00 ~A 0.0475 0.9025 0.95
CE(A) 100 0.05 0.95
CE(B) 200 Cumultaive
f(i) 50.00% ~A,~B 0 0.9025 0.9025
A,~B 50 0.0475 0.95
~A,B 100 0.0475 0.9975
A,B 150 0.0025 1
E(CL) 7.5
99% quantile 100
VARC(99%) 92.5
In the second pass the all settings are identical to the first pass except for the default
correlation which is now assumed to be, = 0.50.
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PD(A) 5.00% B ~B
PD(B) 5.00% A 0.02625 0.02375 0.05
CORR(A,B) 0.50 ~A 0.02375 0.92625 0.95
CE(A) 100 0.05 0.95
CE(B) 200 Cumultaive
f(i) 50.00% ~A,~B 0 0.92625 0.92625
A,~B 50 0.02375 0.95
~A,B 100 0.02375 0.97375
A,B 150 0.02625 1
E(CL) 7.5
99% quantile 150
VARC(99%) 142.5
The impact of the increased correlation in default events is an increase in VARC(99%).
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