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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

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 4The 99.9th percentile increases as the default correlation among the loans increase.

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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

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5.1 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: All firms are treated equal 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 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 bond is different)

• Cons: Limited data especially for low credit quality bonds.

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5.1 Credit correlation Correlation of 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 lecture 4. The credit VaR is to measure the portfolio loss due to credit events.

The time horizon for credit VaR 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).

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5.2 Credit VaR

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5.3 CreditMetrics

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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 CreditMetricsSingle bond Procedures:

1. Credit rating migration

2. Valuation

3. Credit risk estimation We illustrate the above procedures with following

case:

A BBB rated 5-year senior unsecured bond has face value $100 and pays an annual coupon at the rate of 6%

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5.3 CreditMetricsStep 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.

Many banks prefer to rely on their own statistics which relate more closely to the composition of their loan and bond portfolios.

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5.3 CreditMetrics

Initial Rating at year-end (%) Rating AAA AA A BBB BB B CCC Default

AAA 90.81 8.33 0.68 0.06 0.12 0 0 0 AA 0.70 90.65 7.79 0.64 0.06 0.14 0.02 0 A 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06 BBB 0.02 0.33 5.95 86.93 5.30 1.17 1.12 0.18 BB 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06 B 0 0.11 0.24 0.43 6.48 83.46 4.07 5.20 CCC 0.22 0 0.22 1.30 2.38 11.24 64.86 19.79

Source: Standard & Poor’s CreditWeek (April 15, 1996)

One-year transition matrix (%)

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5.3 CreditMetricsStep 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

Seniority Class Mean (%) Standard Deviation (%)Senior Secured 53.80 26.86Senior Unsecured 51.13 25.45Senior subordinated 38.52 23.81Subordinated 32.74 20.18Junior subordinated 17.09 10.90

Source: Carty & Lierberman (1996)

Recovery rate by seniority class (% of face value (par))

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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 CreditMetricsLet 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”.

It should be noted that the maturity of the bond will become four years at the end of year 1.

The value of the bond at the end of one year is given by:

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5.3 CreditMetrics

66.108)0532.1(

106

)0493.1(

6

)0432.1(

6

0372.1

66 (1)

432A V

Time (Year)

Cash flows

0 1 2 3 4 5

6 6 6 6 106

)1(AV

4 years

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5.3 CreditMetrics

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

(5.2) )1()1( RBBB1 VVL

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5.3 CreditMetrics

Year-end rating

Probability of state: p (%)

VR(1) ($) L1 ($) (Eq. (5.2))

AAA 0.02 109.37 1.82 AA 0.33 109.19 1.64 A 5.95 108.66 1.11 BBB 86.93 107.55 0 BB 5.30 102.02 5.53 B 1.17 98.10 9.45 CCC 0.12 83.64 23.91 Default 0.18 51.13 56.42

Table 5-1

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5.3 CreditMetrics

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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

Assuming the actual distribution of L1 (from Table 5-1 in p.27), using Eq. (5.3),

1-year 99% credit VaR

= $9.45

(5.3) %Pr:min VaR %year -1 1 XlLlX

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5.3 CreditMetrics

Assuming L1 follows normal distribution, then

(<$9.45 under actual dist., heavy tail !)

(5.4) 95.8var

46.0

rating All

211

21

rating All11

LELpL

pLmLE

43.7$

95.833.246.0

33.2VaRcredit %99year -1

m

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5.3 CreditMetricsPortfolio 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.

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5.3 CreditMetrics Assume asset value changes are 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.

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5.3 CreditMetrics Assume X1, X2, …, XN follow multivariate normal distribution and

Xi is the asset return random variable of firm i,

we denoted its realized value by xi,

which will be 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.19. We illustrate this methodology by an example.

(5.5) ,corr

,,1for 1,0~

ijji

i

XX

NiX

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5.3 CreditMetrics

Initial Rating at year-end (%) Rating AAA AA A BBB BB B CCC Default

AAA 90.81 8.33 0.68 0.06 0.12 0 0 0 AA 0.70 90.65 7.79 0.64 0.06 0.14 0.02 0 A 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06 BBB 0.02 0.33 5.95 86.93 5.30 1.17 1.12 0.18 BB 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06 B 0 0.11 0.24 0.43 6.48 83.46 4.07 5.20 CCC 0.22 0 0.22 1.30 2.38 11.24 64.86 19.79

Source: Standard & Poor’s CreditWeek (April 15, 1996)

One-year transition matrix (%)

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5.3 CreditMetrics Suppose the current rating of firm i is BB.

We link up xi with the transition probabilities as follows:

1. The probability of firm i defaults = 1.06% (reading from the table), then set:

2.

3. we have xi(CCC) = 2.30, under the normal assumption from (5.5),

If actual asset return is < 2.30, firm i defaults.

(5.6) %06.1CCCPr ii xX

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5.3 CreditMetrics The prob. of firm i transiting from BB to CCC = 1%. 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.

)assumption (normal 04.2B

(5.6)) Eq.(from %06.2BPr

%1CCCPrBPr

%1BCCCPr

i

ii

iiii

iii

x

xX

xXxX

xXx

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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

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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. (e.g. BB)

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.