Credit Metrics

1
Credit Metrics

• Lecture Notes for FIN 653
• Yea-Mow Chen
• Department of Finance
• San Francisco State University

2
I. Introduction

• Introduced in 1997 by J P Morgan and its
  co-sponsors (Bank of America, Union Bank of
  Switzerland, etc.) as a value at risk (VAR)
  framework to apply to the valuation and risk of
  nontradable assets such as loans and privately
  placed bonds.
• Credit-Metrics asks If next year is a bad year,
  how much will I lost on my loans and loan
  portfolio?

3
I. Introduction

• With RiskMetrics we look at the market value or
  price of an asset and the volatility of that
  asset's price or return in order to calculate a
  probability (e.g., 5 percent) that the value of
  that asset will fall below some given value
  tomorrow.
• In the case of RiskMetrics, this involves
  multiplying the estimated standard deviation of
  returns ? on that asset by 1.65 and then
  revaluing the current market value of the
  position (P) downward by 1.65?. That is, VAR for
  one day (or DEAR) is
• VAR P 1.65 ?

4
I. Introduction

• For loans, we observe neither P (the loan's
  market value) nor ? (the volatility of loan value
  over the horizon of interestassumed to be 1 year
  for loans and bonds under CreditMetrics).

5
I. Introduction

• However, using
• (l) available data on a borrower's credit rating,
  
  
• (2) the probability of that rating changing over
  the next year (the rating transition matrix),
  
• (3) recovery rates on defaulted loans, and
• (4) yield spreads in the bond market
• It is possible to calculate a hypothetical P and
  ? for any nontraded loan or bond and thus a VAR
  figure for individual loans and the loan
  portfolio.

6
I. Introduction

• Rather than defining comparable firms using an
  equity-driven distance to default, CreditMetrics
  utilizes external credit rating. That is, the
  CreditMetrics model is built around a credit
  migration or transition matrix that measures the
  probability that the credit rating of any given
  debt security will change over the course of the
  credit horizon.

7
Three Steps in CreditMetrics Modeling of Credit
Risk

• Step 1 estimate the credit exposure amount of
  each obligator in the portfolio
  
• Step 2 calculate the volatility of value due to
  credit quality changes
  
• Step 3 Calculate credit quality correlations and
  portfolio risk

8
Three Steps in CreditMetrics Modeling of Credit
Risk
9
Three Steps in CreditMetrics Modeling of Credit
Risk
10
Step 1 Estimating Credit Risk Exposure Amounts

• Future cash flows at risk beyond the time horizon
  for products such as
  
• Bonds ? face value
• Loans ? face value
• Receivables ?face amount
• Letters of credit ?full nominal amount

11
Step 1 Estimating Credit Risk Exposure Amounts

• Market-driven instruments
• Swaps
• Forwards

12
Step 2 Compute the Volatility in Value Caused by
Credit Quality Changes

• There are three steps
• A/ Estimate the credit quality migrations
• B/ Calculate the changes in value upon credit
  quality migration
  
• C/ Construct the distribution of bond value

13
Step 2 Compute the Volatility in Value Caused by
Credit Quality Changes

• Estimate credit quality migrations
• Risk is due to default but also to changes in
  value
  
• Upgrades
• Downgrades
• Transition matrices
• Chance of default
• Chance of migrating to other credit quality state

14
Step 2 Compute the Volatility in Value Caused by
Credit Quality Changes

• Estimate credit quality migrations

15
Step 2 Compute the Volatility in Value Caused by
Credit Quality Changes

• Calculate the changes in value upon credit
  quality migration
  
• Two different types of revaluation
• Revaluation in default
• ? based on recovery rates
• Revaluation upon upgrade / downgrade
• ? driven by credit spread changes

16
Step 2 Compute the Volatility in Value Caused by
Credit Quality Changes

• Based on recovery rates which depend on the
  seniority class of debt
  

17
Step 2 Compute the Volatility in Value Caused by
Credit Quality Changes

• Straightforward present value revaluation

One year forward zero curves for each credit
rating ()
18
Step 2 Compute the Volatility in Value Caused by
Credit Quality Changes

• Construct the distribution of bond value

19
Step 2 Compute the Volatility in Value Caused by
Credit Quality Changes

• Construct the distribution of bond value

20
Step 3 Correlations

• Estimating Credit Quality Correlations
• Calculating Portfolio Risk

21
Step 3 Correlations
22
Step 3 Correlations

• Rating outcomes of various obligors are
• not independent
• affected by same economic factors
• Thus, measure of correlation is needed in
  addition to individual likelihoods

23
Step 3 Correlations

• Approaches to Estimating Credit Quality
  Correlations
  
• Actual Rating and Default Correlations
• Bond Spread Correlations
• Uniform Constant Correlation
• Equity Price Correlation

24
Step 3 Correlations

• Actual Rating and Default Correlations
• Pro Derived from rating agencies data
• Pro Provides an objective measure of actual
  experience
  
• Con Suffers from sparse sample sets
• Con Requires that identical treatment for
  obligors with given credit ratings

25
Step 3 Correlations

• Bond Spread Correlations
• Pro Provides most objective measure of actual
  correlation
  
• Con Data quality problems, specifically for low
  credit quality users
  
• Con Impossible in practice

26
Step 3 Correlations

• Equity Price Correlations
• Uses fluctuations in value of underlying firms
  assets as a prediction of firms ability to meet
  its obligations
  
• Pro Provides forward-looking, efficient market
  information
  
• Pro Offers advantage of time series
• Con Require heavy processing to yield info
  about likely credit quality correlation

27
Step 3 Correlations

• Equity Price Correlations

28
Step 3 Correlations

• Equity Price Correlations
• Volatility of asset values should directly
  predict chance of default by a firm
  
• Positive correlation between asset returns of two
  firms directly implies positive correlation in
  default expectations
  

29
Step 3 Correlations

• Equity Price Correlations
• Creates a link between underlying firm value and
  firms credit rating
  

30
Step 3 Correlations

• Firms can be related to one another via there
  common sensitivity to industry/country sectors
  

31
Step 3 Correlations

• Obtaining a distribution of values for a
  portfolio of many bonds
  
• A random sample of possible portfolio states is
  used to obtain the value distribution for a
  portfolio of many bonds
  

32
Step 3 Correlations

• Obtaining a distribution of values for a
  portfolio of many bonds
  

33
Step 3 Correlations

• Credit Risk Measures Output at the Portfolio
  Level
  
• Standard Deviation
• Percentile Levels
• Marginal Risk Statistics

34
Example

• Consider the example of a five-year fixed-rate
  loan of 100 million made at 6 percent annual
  interest. The borrower is rated BBB. What is
  the credit risk of this loan?

35
The Distribution of an Individual Loans Value

• CreditMetrics evaluates each loans cash flows
  under eight possible credit migration
  assumptions, corresponding to each of eight
  credit ratings AAA, AA, . CCC, and default.
• The loans value over the upcoming year is
  calculated under different possible scenarios
  over the succeeding year, e.g., the rating
  improves to AAA, AA, etc.
• Historical data on publicly traded bonds are used
  to estimate the probability of each of these
  credit migration scenarios.
  
• Putting together the loan valuations under each
  possible credit migration and their likelihood of
  occurrence, we obtain the distribution of the
  loans value. At this point, standard VaR
  technology may be utilized.

36
The Distribution of an Individual Loans Value

• Based on historical data, it is estimated that
  the probability of a BBB borrower staying at BBB
  over the next year is 96.93 percent. There is
  also some probability that the borrower of the
  loan will be upgraded, and some probability that
  it will be downgraded or even default.

37
The Distribution of an Individual Loans Value

• Table One-Year Transition Probability for
  BBB-Rated Borrower
  
• Rating Transition Rating Transition
• Probability Probability
• ________________________________________
• AAA 0.02 BB 5.30
• AA 0.33 B 1.17
• A 5.95 CCC 0.12
• BBB 86.93 Default 0.18
• ________________________________________

38
The Distribution of an Individual Loans Value

• The migration process is modeled as a finite
  Markov chain, which assumes that the credit
  rating changes from one rating to another with a
  certain constant probability at each time
  interval.
• The credit migration matrix can be estimated from
  historical experience as tabulated by rating
  agencies, from Merton options-theoretical default
  probabilities, from bank internal rating systems,
  or even from intensity-based models.

39
Valuation

• Valuation
• The effect of rating upgrades and downgrades is
  to impact the required credit risk spreads or
  premiums on loans and thus the implied market
  value (or present value) of the loan. If a loan
  is downgraded, the required credit spread premium
  should rise so that the present value of the loan
  to the FI should fall the reverse is true for a
  credit rating upgrade.

40
Valuation

• Since we are revaluing the five-year 100
  million, 6 percent loan at the end of the first
  year after a credit event has occurred during
  that year, then
• 6 6 6
  106
  
• P 6 -------------- -----------------
  --------------- --------------
  
• (1r1 s1) (1r2 s2)2 (1r3
  s3)3 (1r4 s4)4
  
•   
• where
• ri the risk-free rates on T-bonds expected to
  exist one year, two years. and so on, into the
  future (i.e., they reflect forward rates from the
  current Treasury yield curve) and
• si annual credit spreads for loans of a
  particular rating class of one year, two years,
  three years, and four years maturity (the latter
  are derived from observed spreads in the
  corporate bond market over Treasuries).

41
Valuation

• In CreditMetrics, interest rates are assumed to
  be deterministic. Thus, the risk-free rates, ri,
  are obtained by decomposing the current spot
  yield curve in order to obtain the one-year
  forward zero coupon Treasury yield curve.
• For example, if todays risk free spot rate were
  3.01 p.a. for 1 year maturity pure discount US
  Treasury securities, and 3.25 for 2 year
  maturities, then the forward risk-free rate
  expected one yr from now on 1-year US Treasury
  is
• (1 0.0325)2 (1 0.0301) (1 r1)
• which gives r1 3.5

42
Valuation

• CreditMetrics obtains fixed credit spread si for
  different credit ratings from commercial firms
  such as Bridge Information Systems.
  
• Using different credit spreads si for each loan
  payment date and the forward rates, we can solve
  for the end of year value of the loan that is
  upgraded from a BBB to an A rating within the
  next year
• 6 6 6
  106
  
• P 6 ----------- ---------------
  -------------- -------------
  
• (1.0372) (1.0432)2 (1.0493)3
  (1.0532)4
  
• 108.66

43
Valuation

• Table 2 Value of the Loan at the End of 1 Year
  under Different Ratings
  
• __________________________________________________
  ______Year-End Loan Value () Year-End Loan
  Value ()
  
• Rating Including first Rating Including
  first
  
• year coupon year coupon
• __________________________________________________
  ______
  
• AAA 109.37 BB 102.22
• AA 109.19 B 98.10
• A 108.66 CCC 83.64
• BBB 107.55 Default 51.13
• __________________________________________________
  ______
  

44
Valuation

• Table 2 shows the value of the loan if other
  credit events occur. Note that the loan has a
  maximum market value of 109.37 (if the borrower
  is upgraded to AAA) and a minimum value of 51.13
  if the borrower defaults.
• The distribution of loan values on the one-year
  credit horizon data can be drawn using the
  transition probabilities and the loan valuations.

45
Valuation

• It is clear that the value of the loan is not
  symmetrically (or normally) distributed.
  
• Thus CreditMetrics produces two VAR measures
• 1. Based on the normal distribution of loan
  values
  
• 2. Based on the actual distribution of loan
  values
  

46
Calculation of VaR

• Assumed scenarios the 5 percent worst-case and
  the 1 percent worst-case scenarios.
  
• The first step in calculating VAR is to calculate
  the mean of the loan's value, or its expected
  value, at year 1, which is 107.09. If next year
  is a bad year, how much can it expect to lose?
  We could define a bad year as occurring once
  every 20 years (the 5 percent VAR) or once every
  100 years (the 1 percent VAR).
• Assuming that loan values are normally
  distributed, the variance of loan value around
  its mean is 8.9477 (squared) and its standard
  deviation or volatility is the square root of the
  variance equal to 2.99. Thus the 5 percent VAR
  for the loan is 1.65 2.99 4.93 million,
  while the 1 percent VAR is 2.33 2.99 6.97
  million.

47
Calculation of VaR

• However, this is likely to underestimate the
  actual or true VAR of the loan because the
  distribution of the loan's value is clearly
  nonnormal. In particular, it demonstrates a
  negative skew or a long-tail downside risk.
  Using the actual distribution of loan values and
  probabilities, we can see from Table 11A3 that
  there is a 6.77 percent probability that the loan
  value will fall below 102.02, implying an
  "approximate" 5 percent actual VAR of over
  107.09 - 102.02 5.07 million, and that there
  is a 1.47 percent probability that the loan value
  will fall below 98.10, implying an "approximate"
  1 percent actual VAR of over 107.09 - 98.10
  8.99.

48
Calculation of VaR

• Table 3 VaR Calculations for the BBB Loan

49
Calculation of VaR

• Assuming Normal Distribution
• 5 VAR 1.65 ? 4.93
• 1 VAR 2.33 ? 6.97
• Assuming Actual Distribution
• 5 VAR 95 of actual distribution 107.09-
  102.02 5.07
  
• 1 VAR 99 of actual distribution 107.09-
  98.10 8.99

50
Capital Requirements

• For the example of a 10 million face value BBB
  loan, the capital requirements by the Federal
  Reserve and the BIS would be 8 million.
  
• Using the 1 percent VAR based on the normal
  distribution, a capital requirement of 6.97
  million would be required, while using the 1
  percent VAR based on the iterated value from the
  actual distribution, a 14.80 million capital
  requirement would be required.

51
Capital Requirements

• It should be noted that under the CreditMetrics
  approach every loan is likely to have a different
  VAR and thus a different implied capital
  requirement. This contrasts to the current BIS
  regulations, where all private sector loans of
  different ratings and different maturities are
  subject to the same 8 percent capital
  requirement. Thus, an important objective of the
  CrecitMetrics sponsors is to get regulators to
  move toward accepting "internal model" - based
  measures of capital requirements for credit risk
  similar to the way in which they have accepted
  internal model - based measures for market risk
  capital requirements.

52
The Value Distribution for a Portfolio of Loans

• The major distinction between the single loan
  case and the portfolio case is the introduction
  of correlations across loans.
  
• CreditMetrics solves for correlations by first
  regressing equity returns on industry indices.
  
• The correlation between any pair of equity
  returns is calculated using the correlations
  across the industry indices.
  
• Once we obtain equity correlations, we can solve
  for joint migration probabilities to estimate the
  likelihood that the joint credit quality of the
  loans in the portfolio will be wither upgraded or
  downgraded.
• Finally, each loans value is obtained for each
  credit migration possibility.
  
• The first two moments (mean and standard
  deviation) of the portfolio value distribution
  are derived from the probability-weighted loan
  values to obtain the normally distributed
  portfolio value distribution.