Topic 6' Measuring credit risk Loan portfolio

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Topic 6. Measuring credit risk (Loan portfolio)

• 6.1 Credit correlation
• 6.2 Credit VaR
• 6.3 CreditMetrics

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6.1 Credit correlation

• Credit correlation measures the degree of
  dependence between the change of the credit
  quality of two assets/obligors.
• If Obligor As credit quality (credit rating)
  deteriorates, 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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6.1 Credit correlation

• Example 6.1
• (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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6.1 Credit correlation
The 99.9th percentile increases as the default
correlation among the loans increase. From
Exhibit 3, both the left and right tail of the
portfolio loss distribution are increasing with
the default correlation ? More likely for the
extreme events (no loss or large loss).
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6.1 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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6.1 Credit correlation
J.P. Morgan, CreditMetrics Technical
Document, Apirl, 1997.
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6.1 Credit correlation
J.P. Morgan, CreditMetrics Technical
Document, Apirl, 1997.
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6.1 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 bond
  spreads to estimate the credit correlation.
• Pros Objective measure of actual credit
  correlation and consistent with other models of
  risky assets.
• Limitation Limited data especially for low
  credit quality bonds.

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6.1 Credit correlation

• Asset value
• It is evident that the value of a firms 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.
• It is used in CreditMetrics for the estimation of
  credit correlation in loan/bond portfolio.

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6.2 Credit VaR

• Credit value at risk (Credit VaR) is defined in
  the same way as the VaR in eq. (4.1) of topic 4.
  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 4).

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6.2 Credit VaR
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6.3 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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6.3 CreditMetrics

• Single bond
• Procedures
• Credit rating migration
• Valuation
• 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.

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6.3 CreditMetrics
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6.3 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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6.3 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
  ones assessment of current environment.

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6.3 CreditMetrics
One-year transition matrix ()
Source Standard Poors CreditWeek (April 15,
1996)
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6.3 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 (6.1)
• 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.
• The uncertainly in recovery rate is not
  considered in this course.

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6.3 CreditMetrics
Recovery rate by seniority class ( of face value
(par))
Source Carty Lierberman (1996)
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6.3 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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6.3 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
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6.3 CreditMetrics
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6.3 CreditMetrics

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

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6.3 CreditMetrics
Table 6-1
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6.3 CreditMetrics
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6.3 CreditMetrics

• If L1 follows normal distribution, then

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6.3 CreditMetrics

• For a general distribution (discrete or discrete
  mixed with continuous) of L1, the 1-year X VaR
  (X percentile) is defined as
• (Compare with eq. (4.1) in topic 4.)
• Under the actual distribution of L1 (from table
  6-1 in p.25), using eq. (6.4),
• 1-year 99 credit VaR
• 9.45 (gt7.43 in normal dist.)

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6.3 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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6.3 CreditMetrics

• Assume X1, X2, , XN follow multivariate normal
  distribution and
• Since Xi relates to the asset return of firm i,
  the assumed value Xi, say 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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6.3 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
• Using the assumption (6.5), 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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6.3 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.

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6.3 CreditMetrics
Standardized asset return of firm i (BB rated)
xi(AAA) 3.43
xi(BB) -1.23
xi(A) 2.39
xi(AA) 2.93
xi(CCC) -2.30
xi(B) -2.04
xi(BBB) 1.37
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6.3 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 (6.5).
• Determine the new rating of the bonds at the end
  of one year by comparing xi with the credit
  quality thresholds in step 1.

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6.3 CreditMetrics

• Procedures (cont.)
• Revalue each bond at the end of one year in the
  portfolio by following the step 2 in the single
  bond case.
• Calculate the portfolio loss.
• Repeat step 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.

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6.3 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.

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