2 Basel II and III’s Credit Value at Risk (CVaR) Approach
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We deﬁned credit risk in Chapter 10 as the risk of a ﬁnancial loss due to
an adverse change in the credit quality of a debtor, and mentioned the two
types of credit risk: (1) migration risk and (2) default risk.
To value CVaR, it is tempting to just take the market VaR equation (1.8)
pﬃﬃﬃ
VaRP = sP a x and transfer it to CVaR. However, there are two main
problems when using equation (1.8) for CVaR:
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1. The portfolio variance, deﬁned in equation (1.9), sP = bh Cbv , would
require input data as standard deviations of relative credit rating changes,
and the correlation coefﬁcient between the changes. However, these data
for credit risk are rare, since credit rating changes for most entities rarely
occur, often only once a year or even not at all.
2. The value for a in equation (1.8) would assume a normal distribution of
relative credit rating changes. However, credit rating changes are typically not normally distributed and depend on the current credit rating,
past credit rating changes, country, sector, seniority, coupon, yield, and
so on. See Meissner (2005) for a further discussion.
Since credit data are much scarcer than market data, in practice a much
more granular approach is used to derive CVaR. Basel II uses the one-factor
Gaussian copula (OFGC) model, which we discussed in detail in Chapter 6
for valuing CDO tranches. Let’s apply the OFGC to value CVaR.
We start with the core equation of the OFGC, which we discussed in
Chapter 6,
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
xi = rM + 1 − rZi
(12.1)
where
r: Default correlation parameter for the companies in the portfolio, 0 £
r £ 1. r is assumed identical and constant for all company pairs in the
portfolio.
M: Systematic market factor, which impacts all companies in the portfolio. M can be thought of as the general economic environment, for
example, the return of the S&P 500. M is a random drawing from a
standard normal distribution, formally M = n ∼ (0, 1). M is the same
as e in Chapter 4, section 4.1.
Zi: Idiosyncratic factor of asset i. Zi expresses ith company’s individual
strength, possibly measured by company i’s stock price return.
As M, Zi is also a random drawing from a standard normal
distribution.
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xi: The value for xi results from equation (12.1) and is interpreted as a
“Default indicator variable” for company i. The lower i, the earlier is
the default time T for company i. xi is by construction standard
normal.
Solving equation (12.1) for Zi, we derive
pﬃﬃﬃ
xi − rM
p
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ
Zi =
1−r
Taking cumulative values, we get
pﬃﬃﬃ
xi − rM
N(Zi ) = N pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1−r
(12.2)
where N(x) is the cumulative standard normal distribution at x. Since we use
the standard normal cumulative distribution N, this approach is called
Gaussian copula.
We now equate the individual default probability of entity i at time T,
PDi(T), which is given or estimated from the market data with the modelsimulated barrier N(xi), which includes the default correlation via the
xi: PDi(T) = N(xi). Solving for xi, we derive xi = N−1(PDi(T)), where
N−1 is the inverse of N. See Chapter 4, Figure 4.3 for details of the mapping
procedure N−1(PDi(T)). Inputting xi = N−1(PDi(T)) into equation (12.2),
we get
pﬃﬃﬃ !
N −1 PDi (T) − rM
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
N(Zi ) = N
(12.3)
1−r
Now the strong assumption is made that all entities i have the same
default probability PD at a certain time T. Hence we can drop the index i
and get
pﬃﬃﬃ !
N −1 PD(T) − rM
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
N(Z) = N
(12.4)
1−r
For a large homogeneous portfolio (LHP) with identical pairwise correlation r and identical default correlation PD(T), the right side of equation
(12.4) is approximately the percentage of entities in the portfolio defaulting
at T. For example, if there is no correlation between the entities (i.e., r = 0),
then equation (12.4) reduces to N[N−1(PD(T))] = PD(T). In this case, if the
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individual default probability is PD(T) = 10%, we can assume that approximately 10% of the entities will default by T.
We now replace the market factor M with a conﬁdence level X. M is
standard normal. Therefore, for a certain abscise value N−1(Y) of M, we have:
N −1 (Y )
Pr(M £ N −1 (Y)) =
∫
n(M)dM = N(N −1 (Y)) = Y
(12.5)
−∞
Equation (12.5) reads: The probability of M being smaller than or equal
to N−1(Y) is the surface of a normal distribution from −∞ to N−1(Y), where
n(M) is the normal distribution of M. This can be written as N(N−1(Y)), since
N(N−1(Y)) is the cumulative normal distribution from −∞ to N−1(Y). N is
the inverse of N−1; therefore N(N−1(Y)) = Y.
Graphically, we can express this as shown in Figure 12.1.
Replacing M in equation (12.4) with N−1(Y), we get
!
pﬃﬃﬃ
N −1 PD(T) − rN −1 (Y)
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
N
(12.6)
1−r
The term (12.6) tells us the probability
Y of the percentage
of defaults in
pﬃ
N −1 PD(T ) − rN −1 (Y )
pﬃﬃﬃﬃﬃﬃﬃﬃ
. We are interested in
the portfolio being bigger than N
1−r
probability of defaults smaller than Y. This is 1 − Y. Let’s set 1 − Y = X,
n(M)
0.5
0.45
0.4
0.35
0.3
0.25
0.2
Y
0.15
X=1–Y
0.1
0.05
0
–3.5
–2.5
–1.5
–0.5
0.5
1.5
2.5
N –1(Y)
FIGURE 12.1 Graphical Representation of a Normally Distributed Default
Distribution with Pr(N−1(Y)) £ M = Y, and the Conﬁdence Level X
3.5
M
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where X is a certain conﬁdence level. Replacing Y with 1 − X, and using
N−1(1 − X) = −N−1(X), we derive
!
pﬃﬃﬃ
N −1 PD(T) + rN −1 (X)
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
CVaR(X; T) = N
(12.7)
1−r
where
CVaR(X,T): credit value at risk for the conﬁdence level X for the time
horizon T
N: cumulative normal distribution
N−1: inverse of the cumulative normal distribution
PD(T): average probability of default of the assets in the portfolio for the
time horizon T
r: pairwise correlation coefﬁcient of the assets in the portfolio (r is
assumed constant for all asset pairs)
Equation (12.7) reads: We are X%
regarding
our loan
certain that
pﬃ
N −1 PD(T ) + rN −1 (X)
pﬃﬃﬃﬃﬃﬃﬃﬃ
due to (correportfolio we will not lose more than N
1−r
lated) default risk, for the time horizon T.
Equation (12.7) is an important result, which was ﬁrst published by
Vasicek in 1987.2 It is derived from the one-factor Gaussian copula (OFGC)
model of equation (12.1); see above. Equation (12.7) is currently used in the
Basel II accord as the basis to value credit risk in a portfolio. It takes into
consideration default risk, not migration risk (see Figure 10.1). CVaR is also
called credit at risk or worst-case default rate (WCDR).
Let’s look at an example of equation (12.7).
EXAMPLE 12.1: CALCULATING CREDIT VALUE
AT RISK (CVaR)
Suppose JPMorgan has given loans to several companies in the amount
of $100,000,000. The average 1-year default probability of the companies is 1%. The copula default correlation coefﬁcient between the
companies is 5%. What is the 1-year CVaR on a 99.9% conﬁdence level?
(continued)
2. O. Vasicek, “Probability of Loss on a Loan Portfolio,” KMV Working paper, 1987.
Results published in Risk magazine with the title “Loan Portfolio Value,” December
2002.
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It is
!
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
N −1 (0:01) + 0:05 N −1
0:999
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
CVaR(0:999; 1) = N
= 4:67%
1 − 0:05
We can derive N−1(x) via Excel’s =normsinv(x) or MATLAB’s
norminv(x) function. N(x) is =normsdist(x) in Excel and normdist(x)
in MATLAB.
Interpretation: JPMorgan is 99.9% sure that it will not lose more
than 4.67% of its loan exposure of $100,000,000 due to (correlated)
default risk of its debtors for a 1-year time horizon. In dollar amounts
and including a recovery rate of 40%, we derive that JPMorgan is
99.9% sure that it will not lose more than $100,000,000 ´ 0.0467 ´
(1 − 0.4) = $2,802,000.
12.2.1 Properties of Equation (12.7)
Equation (12.7) has some interesting properties.
■
■
We observe that for zero default correlation between the debtors in the
portfolio r = 0, it follows that CVaR = PD(T). So for a 99.9% conﬁdence
level, we are 99.9% sure that we will not lose more than the average default
rate PD(T). This is reasonable because there is no effect from correlation
and the maximum loss is just the average default probability of the debtors.
CVaR is a function of the default probability PD for the time horizon T,
the conﬁdence level X, and the pairwise default correlation r. T is
typically set to one year and the conﬁdence level used is typically
99.9%. In this case we get a relationship between CVaR and PD(T)
and r as displayed in Figure 12.2.
From Figure 12.2 we observe that CVaR is a positive function with
respect to PD(T) and r. The positive relationship between CVaR and PD(T),
∂CVaR > 0 is obvious: The higher the default probability PD(T), the higher is
∂PD(T )
the maximum loss CVaR. The positive relationship between CVaR and the
default correlation between the debtors r, ∂CVaR
∂r > 0, is also plausible: The
higher the default correlation, the higher is the probability that many debtors
default and the higher is the maximum loss CVaR. This is especially the case
for high default probability PD(T). For value of PD(T) higher than 20% and r
close to or higher than 80%, the maximum loss CVaR is close to 100% of the
total loan exposure; see Figure 12.2.
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CVaR as a Function of PD(T) and ρ
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
80%
40%
Default 5%
Correlation
1%
ρ
CVaR
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3%
5%
7%
9%
15% 17% 19%
11% 13%
Default Probability PD(T)
FIGURE 12.2 CVaR as a Function of the Default Probability PD(T) where T = 1 and
the default correlation between the debtor’s asset r. The conﬁdence level is X = 99.9%.
For a model that displays the CVaR, see “CVaR.xlsm” at www.wiley
.com/go/correlationriskmodeling, under “Chapter 12.”
12.3 BASEL II’s REQUIRED CAPITAL (RC)
FOR CREDIT RISK
Basel II uses equation (12.7) as a basis to calculate the capital charge for
credit risk. However, the capital charge is reduced by the expected loss,
which is measured by PD(T). The rationale is that banks cover the expected
loss with their own provisions as the interest rate that they charge. (Naturally, low-rated debtors have to pay a higher interest rate on their loans than
highly rated debtors.) Therefore the required capital RC for credit risk in the
Basel II accord is
RC = EAD ´ (1 − R) ´ CVaR − PD(T)
(12.8)
where
RC is the required capital by Basel II for credit risk in a portfolio
EAD is the exposure at default (for loans EAD is equal to the loan amount)
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R is the recovery rate (rate that is recovered from the defaulted loan)
CVaR is the credit value at risk derived by equation (12.7)
PD(T) is the average probability of default of the debtors in the portfolio
for time horizon T
Let’s look at equation (12.8) in an example. Let’s expand example 12.1.
EXAMPLE 12.2: CALCULATING REQUIRED CAPITAL
(RC) FOR CREDIT RISK
Suppose JPMorgan has given loans to several companies in the
amount of $100,000,000. The average 1-year default probability
PD of the companies is 1%. The copula default correlation coefﬁcient
between the companies is 5%. What is the 1-year capital charge of
Basel II on a 99.9% conﬁdence level assuming the recovery rate is
40%?
Answer:
We had already derived CVaR in example 12.1 as CVaR = 4.67%.
Following equation (12.8), the required capital charge of Basel II is
RC = $100;000;000 ´ (1 − 0:4) ´ (4:67% − 1%) = $2;202;000
Credit value at risk CVaR is typically calculated for a 1-year time
horizon. If a different time horizon is used, Basel II adds a maturity adjustment (MA). In this case the equation (12.8) changes to
RC = EAD ´ (1 − R) ´ CVaR − PD(T) ´ MA
where MA =
(12.8a)
1 + (M − 2:5) ´ b
and M is the maturity date. b is a constant set at b =
1 − 1:5 ´ b
2
[0.11852 − 0.05478 ´ ln(PD(T))] .
12.3.1 The Default Probability–Default
Correlation Relationship
Interestingly, the correlation coefﬁcient r is not an exogenous input in
equations (12.8) or (12.8a), but r is a function of the default probability