1. Topic 5. Measuring Credit Risk (Loan portfolio)
5.1 Credit correlation
5.2 Credit VaR
5.3 CreditMetrics

2. 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.

3. 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. 3

4. 4

5. 5.1 Credit correlation 5

6. 5.1 Credit correlation
The default correlation increases, more likely for the extreme events (no loss or large loss).
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. 6

7. 5.1 Credit correlation Ways to measure(quantify) credit correlation
Direct estimation of joint credit rating moves
Correlation of Bond spread
Correlation of Asset Value 7

8. 5.1 Credit correlation Ways to measure credit correlation
Direct estimation of joint credit rating moves:
Using the historical data of credit rating transition.
Pros: No assumptions on the distribution of the underlying processes governing the change of credit quality. (distribution free, data driven)
Cons: limited data and treat all firms within a given credit rating class to be identical. 8

9. 5.1 Credit correlation
J.P. Morgan, “CreditMetrics – Technical Document”, Apirl, 1997. 9

10. 5.1 Credit correlation
J.P. Morgan, “CreditMetrics – Technical Document”, Apirl, 1997. 10

11. 5.1 Credit correlation Correlation of 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. (each name is different)
Cons: Limited data especially for low credit quality bonds. 11

12. 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. 12

13. 5.2 Credit VaR
Credit value at risk (Credit VaR) is defined in the same way as the VaR in Eq. (3.16) . 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). 13

14. 5.2 Credit VaR 14

15. 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. 15

16. 5.3 CreditMetrics Single bond Procedures:
Credit rating migration
Valuation conditional on rating
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.. 16

17. 5.3 CreditMetrics 17

18. 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.
-- historical information
Actual transition and default probabilities vary quite substantially over the years, depending whether the economy is in recession, or in expansion. 18

19. 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. 19

20. 5.3 CreditMetrics One-year transition matrix (%)
Source: Standard & Poor’s CreditWeek (April 15, 1996) 20

21. 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. 21

22. Recovery rate by seniority class (% of face value (par))
5.3 CreditMetrics
Recovery rate by seniority class (% of face value (par))
Source: Carty & Lierberman (1996) 22

23. 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. 23

24. 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. 24

25. 5.3 CreditMetrics 4 years Time (Year) 0 1 2 3 4 5 6 6 6 6 106
Cash flows
4 years 25

26. 5.3 CreditMetrics 26

27. 5.3 CreditMetrics Step 3. Credit risk estimation
The portfolio loss over one year L1 is given by 27

28. 5.3 CreditMetrics
Table 5-1 28

29. 5.3 CreditMetrics 29

30. 5.3 CreditMetrics
Assuming L1 follows normal distribution, then 30

31. 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
Assuming the actual distribution of L1 (from Table 5-1 in p.28), using Eq. (5.4),
1-year 99% credit VaR
= $9.45 (>$7.43 under normal dist.) 31

32. 5.3 CreditMetrics Portfolio of bonds
The correlation among the bonds in the portfolio is modeled through their asset correlations.
To construct correlation, we posit an “unseen” driver of credit migration  changes in asset value.
Intuition is that default occurs when the value of a firm’s asset drop below the mkt value of its liabilities.
Assets only used to build interactions b/w obligors. 32

33. 5.3 CreditMetrics Assume asset value changes is normally distributed
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. 33

34. 5.3 CreditMetrics
Assume X1, X2, …, XN follow multivariate normal distribution and
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.20. We illustrate this methodology by an example. 34

35. 5.3 CreditMetrics Suppose the current rating of firm i is BB.
We link up xi with the transition probabilities as follows:
The probability of firm i defaults = 1.06%
Set
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%. 35

36. 5.3 CreditMetrics
Set
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. 36

37. 6.3 CreditMetrics Standardized asset return of firm i (BB rated)
xi(CCC)
-2.30
xi(B)
-2.04
xi(BB)
-1.23
xi(BBB)
1.37
xi(A)
2.39
xi(AA)
2.93
xi(AAA)
3.43 37

38. 5.3 CreditMetrics One-year transition matrix (%)
Source: Standard & Poor’s CreditWeek (April 15, 1996) 38

39. 5.3 CreditMetrics
The Monte-Carlo simulation is employed to determine the credit VaR of the portfolio.
Procedures:
Determine the credit quality thresholds for each credit rating class.
Simulate the standardized asset return xi of firm i, for i =1, …, N, from the multivariate normal distribution in (5.5).
Determine the new rating of the bonds at the end of one year by comparing xi with the credit quality thresholds in Step 1. 39

40. 5.3 CreditMetrics Procedures (cont.):
Revalue each bond at the end of one year in the portfolio by following Step 2 in the single bond case.
Calculate the portfolio loss.
Repeat Steps 2 to 5 M times to create the distribution of the portfolio losses.
The 1-year X % credit VaR can be calculated as the X percentile of the portfolio loss distribution in Step 6. 40

41. 5.3 CreditMetrics Weakness:
Firms within the same rating class are assumed to have the same default (migration) probabilities.
The actual default (migration) probabilities are derived from the historical default (migration) frequencies.
Default is only defined in a statistical sense (non-firm specific) without explicit reference to the process which leads to default or migrate. 41