Transcript Topic 1. Introduction to financial derivatives

Topic 5. Measuring Credit Risk
(Loan portfolio)
5.1 Credit correlation
5.2 Credit VaR
5.3 CreditMetrics
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5.1 Credit correlation


Credit correlation measures the degree of dependence
between the change of the credit quality of two
assets/obligors.
“If Obligor A’s credit quality (credit rating) changes,
how well does the credit quality of Obligor B
correlate to A?”
The portfolio loss is highly sensitive to the credit
correlation.
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5.1 Credit correlation

Example 5.1
(Adelson, M. H. (2003), “CDO and ABS underperformance: A
correlation story”, Journal of Fixed Income, 13(3), December,
53 – 63.)
Consider a portfolio consisting 100 loans. Each loan
has 90% chance of paying $1 and 10% chance of
paying nothing. Simulation is used to examine the
performance of the portfolio.
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5.1 Credit correlation
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5.1 Credit correlation
Exhibit 3
Both the left and right tail of
the portfolio loss distribution
are increasing with the default
correlation.
Exhibit 4
The 99.9th percentile
increases as the default
correlation among the loans
increase.
The default correlation increases, more
likely for the extreme events (no loss or
large loss).
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5.1 Credit correlation
Ways to measure credit correlation
 Direct estimation of joint credit moves:
• Using the historical data of credit rating transition.
• Pro: No assumptions on the distribution of the underlying
processes governing the change of credit quality.
• Limitation: limited data and treat all firms within a given
credit rating class to be identical.
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5.1 Credit correlation
J.P. Morgan, “CreditMetrics – Technical Document”, Apirl, 1997.
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5.1 Credit correlation
J.P. Morgan, “CreditMetrics – Technical Document”, Apirl, 1997.
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5.1 Credit correlation

Bond spread:
• Bond (Yield) spread = Yield of risky bond – Risk free yield.
• The change of credit quality induces the change of bond
spread. It is reasonable to use the correlation between the
bond spreads to estimate the credit correlation.
• Pros: Objective measure of actual credit correlation and
consistent with other models for risky assets.
• Limitation: Limited data especially for low credit quality
bonds.
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5.1 Credit correlation

Asset value:
• It is evident that the value of a firm’s assets determines its
ability to pay its debts. So, it is reasonable to link up the
credit quality of a firm with its asset level.
• The credit correlation can be estimated from the correlation
between the asset values.
• It is used in CreditMetrics for the estimation of credit
correlation in loan/bond portfolio.
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5.2 Credit VaR



Credit value at risk (Credit VaR) is defined in the
same way as the VaR in Eq. (3.16) of Topic 3. The
credit VaR is to measure the portfolio loss due to
credit events.
The time horizon for credit risk is usually much
longer (often 1 year) than the time horizon for market
risk (1 day or 1 month).
As compared to the distribution of the portfolio loss
due to market risk, the distribution due to credit
events is highly skewed and fat-tailed. This creates a
challenge in determining credit VaR (not as simple as
the normal distribution in Topic 3).
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5.2 Credit VaR
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5.3 CreditMetrics



CreditMetrics was introduced in 1997 by J.P. Morgan
and its co-sponsors (Bank of America, Union Bank of
Switzerland, et al.). (See J.P. Morgan, “CreditMetrics
– Technical Document”, Apirl, 1997.)
It is based on credit migration analysis, i.e. the
probability of moving from one credit rating class to
another within a given time horizon.
Credit VaR of a portfolio is derived as the percentile
of the portfolio loss distribution corresponding to the
desired confidence level.
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5.3 CreditMetrics
Single bond
 Procedures:
1. Credit rating migration
2. Valuation
3. Credit risk estimation

We illustrate the above procedures with following
case:
Portfolio:
BBB rated 5-year senior unsecured bond has face
value $100 and pays an annual coupon at the rate of
6% p.a..
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5.3 CreditMetrics
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5.3 CreditMetrics
Step 1. Credit rating migration
 Rating categories, combined with the probabilities of
migrating from one credit rating class to another over
the credit risk horizon (1 year) are specified.
 Actual transition and default probabilities vary quite
substantially over the years, depending whether the
economy is in recession, or in expansion.
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5.3 CreditMetrics

Many banks prefer to rely on their own statistics
which relate more closely to the composition of their
loan and bond portfolios. They may have to adjust
historical values to be consistent with one’s
assessment of current environment.
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5.3 CreditMetrics
One-year transition matrix (%)
Initial
Rating
AAA
AA
A
BBB
BB
B
CCC
AAA
90.81
0.70
0.09
0.02
0.03
0
0.22
AA
8.33
90.65
2.27
0.33
0.14
0.11
0
Rating at year-end (%)
A
BBB
BB
B
0.68
0.06
0.12
0
7.79
0.64
0.06
0.14
91.05
5.52
0.74
0.26
5.95
86.93
5.30
1.17
0.67
7.73
80.53
8.84
0.24
0.43
6.48
83.46
0.22
1.30
2.38
11.24
CCC
0
0.02
0.01
1.12
1.00
4.07
64.86
Default
0
0
0.06
0.18
1.06
5.20
19.79
Source: Standard & Poor’s CreditWeek (April 15, 1996)
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5.3 CreditMetrics
Step 2. Valuation
 In this step, the value of the bond will be revalued at the
end of the risk horizon (1 year) for all possible credit
states. It is assumed all credit rating movements are
occurred at the end of the risk horizon (1 year).
 At the state of default:
• Specify the recovery rate (% of the face value can recover when
the bond defaults) for different seniority level.
• Value of the bond at default
= Face value  Mean recovery rate
(5.1)
• In our case, the mean recovery is 51.13%, the value of the bond
when default occurs at the end of one year is $51.13.
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5.3 CreditMetrics
Recovery rate by seniority class (% of face value (par))
Seniority Class
Senior Secured
Senior Unsecured
Senior subordinated
Subordinated
Junior subordinated
Mean (%)
53.80
51.13
38.52
32.74
17.09
Standard Deviation (%)
26.86
25.45
23.81
20.18
10.90
Source: Carty & Lierberman (1996)
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5.3 CreditMetrics

At the state of up(down)grade:
• Obtain the one year forward zero curves (the expected
discount rate at the end of one year over different terms) for
each credit rating class.
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5.3 CreditMetrics
Let VR(t) be the value of the BBB rated bond at time t
(in year) with rating class changing to “R”.
Suppose the BBB rated bond is upgraded to “A”. The
value of the bond at the end of one year is given by
It should be noted that the maturity of the bond will
become four years at the end of year 1.
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5.3 CreditMetrics
4 years
0
1
2
3
4
5
6
6
6
6
106
Time (Year)
Cash flows
VA (1)
6
6
6
106
VA (1)  6 



 108.66
2
3
4
1.0372 (1.0432) (1.0493) (1.0532)
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5.3 CreditMetrics
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5.3 CreditMetrics
Step 3. Credit risk estimation
 The portfolio loss over one year L1 is given by
L1  VBBB (1)  VR (1)
(5.2)
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5.3 CreditMetrics
Table 5-1
Year-end
rating
AAA
AA
A
BBB
BB
B
CCC
Default
Probability
of state:
p (%)
0.02
0.33
5.95
86.93
5.30
1.17
0.12
0.18
VR(1) ($)
109.37
109.19
108.66
107.55
102.02
98.10
83.64
51.13
L1 ($)
(Eq. (5.2))
1.82
1.64
1.11
0
5.53
9.45
23.91
56.42
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5.3 CreditMetrics
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5.3 CreditMetrics
 E L1   m   pL1  0.46
All rating
varL1    
2
 pL  E L 
2
1
1
 8.95
(5.3)
All rating

If L1 follows normal distribution, then
1 - year99% credit VaR  m  2.33
 0.46  2.33 8.95
 $7.43
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5.3 CreditMetrics

For a general distribution (discrete or discrete mixed
with continuous) of L1, the 1-year X% VaR (X
percentile) is given by
1 - year X % VaR  minl : PrL1  l   X %

(5.4)
Under the actual distribution of L1 (from Table 5-1 in
p.25), using Eq. (5.4),
1-year 99% credit VaR
= $9.45 (>$7.43 under normal dist.)
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5.3 CreditMetrics
Portfolio of bonds
 The correlation among the bonds in the portfolio is
modeled through their asset correlations.
 Suppose the portfolio contains N bonds and all the
bonds are issued by different firms.
Let Xi be the standardized asset return of firm i in the
portfolio, for i =1, …, N.
The standardized asset return is defined as the asset
return (percentage change in asset value) adjusted to
have mean 0 and standard derivation 1.
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5.3 CreditMetrics

Assume X1, X2, …, XN follow multivariate normal
distribution and
X i ~ 0,1
for i  1,, N
corrX i , X j   ij
(5.5)
Since Xi relates to the asset return of firm i, the realized
value of Xi which is denoted by xi, will determine the
credit rating class of firm i.
 The range of xi in which firm i falls in the specified
rating class can be determined from the one-year rating
transition matrix in P.18. We illustrate this methodology
by an example.

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5.3 CreditMetrics

Suppose the current rating of firm i is BB.
We link up xi with the transition probabilities in P.18
as follows:
The probability of firm i defaults = 1.06%
Set
Pr X i xi CCC  1.06%
(5.6)
Using the assumption (5.5), we have xi(CCC) = 2.30.
If the realized value of the asset return is less than
2.30, then firm i defaults.
The prob. of firm i transiting from BB to CCC = 1%.
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5.3 CreditMetrics

Set
Prxi CCC   X i xi B  1%
Pr X i xi B  Pr X i xi CCC   1%
Pr X i xi B  2.06%
xi B  2.04
(fromEq. (5.6))
Similarly, we get xi(BB), xi(BBB), xi(A), xi(AA) and
xi(AAA).
The credit quality thresholds for other credit ratings
can also be derived by following the above procedure.
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Standardized asset return of firm i (BB rated)
6.3 CreditMetrics
xi(CCC) xi(B) xi(BB)
-2.30
-2.04 -1.23
xi(BBB) xi(A) xi(AA) xi(AAA)
1.37
2.39 2.93
3.43
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5.3 CreditMetrics
The Monte-Carlo simulation is employed to determine
the credit VaR of the portfolio.
 Procedures:

1. Determine the credit quality thresholds for each credit
rating class.
2. Simulate the standardized asset return xi of firm i, for i =1,
…, N, from the multivariate normal distribution in (5.5).
3. Determine the new rating of the bonds at the end of one
year by comparing xi with the credit quality thresholds in
Step 1.
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5.3 CreditMetrics

Procedures (cont.):
4. Revalue each bond at the end of one year in the portfolio
by following Step 2 in the single bond case.
5. Calculate the portfolio loss.
6. Repeat Steps 2 to 5 M times to create the distribution of the
portfolio losses.
7. The 1-year X % credit VaR can be calculated as the X
percentile of the portfolio loss distribution in Step 6.
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5.3 CreditMetrics

Weakness:
1. Firms within the same rating class are assumed to have the
same default (migration) probabilities.
2. The actual default (migration) probabilities are derived
from the historical default (migration) frequencies.
3. Default is only defined in a statistical sense (non-firm
specific) without explicit reference to the process which
leads to default or migrate.
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