0. Credit Risk Plus and Credit Metrics
By: A V Vedpuriswar
October 4, 2009

1. Credit Risk Plus

2. Introduction
CreditRisk+ is a statistical credit risk model launched by Credit Suisse First Boston (CSFB) in 1997.
CreditRisk+ can be applied to any type of credit product, including loans, bonds, financial letters of credit and derivatives.

3. Credit Risk Plus
Credit Risk + allows only two outcomes – default and no default.
In case of default, the loss is of a fixed size.
The probability of default depends on credit rating, risk factors and the sensitivity of the obligor to the risk factors. 3

4. Analytical techniques
CreditRisk+ uses analytical techniques, as opposed to simulations, to estimate credit risk.
The techniques used are similar to those applied in the insurance industry.
CreditRisk+ makes no assumptions about the cause of default.
It models credit risk based on sudden events by treating default rates as continuous random variables.

5. Data requirements Exposure Default rates Default rate volatilities
Recovery rates

6. Methodology Model the frequency of default events
Model the severity of default losses
Model the distribution of default losses
Sector analysis
Stress testing

7. Factors for Estimating Credit Risk
When estimating credit risk, CreditRisk+ considers :
credit quality and systematic risk of the debtor
size and maturity of each exposure
concentrations of exposures within a portfolio
CreditRisk+ accounts for the correlation between different default events by analyzing default volatilities across different sectors, such as different industries or countries.
This method works because defaults are often related to the same background factors, such as an economic downturn.
To estimate credit risk due to extreme events such as earthquakes, CreditRisk+ uses stress testing.
For low probability events that can't be covered under the statistical model, it uses a scenario-based approach.

8. Frequency of default events
The timing of default events cannot be predicted.
The probability of default by any debtor is relatively small.
CreditRisk+ concerns itself with sudden default – as opposed to continuous change – when estimating credit risk.

9. Poisson Distribution
CreditRisk+ uses the Poisson distribution to model the frequency of default events.
The Poisson distribution is used to calculate the probability that a given number of events will take place during a specific period of time.
The Poisson distribution is useful when the probability of an event occurring is low and there are a large number of debtors.
For this reason, it is more appropriate than the normal distribution for estimating the frequency of default events.

10. Using the Poisson distribution
Suppose there are N counterparties of a type and the probability of default by each counterparty is p.
The expected number of defaults, , for the whole portfolio is Np.
If p is small, the probability of n defaults is given by the Poisson distribution, i.e, the following equation:
p (n) = 10

11. Modeling the Severity of Default Losses
After calculating the frequency of default events, we need to look at the exposures in the portfolio and model the recovery rate for each exposure.
From this, we can conclude the severity of default losses.

12. Modeling the Distribution of Default Losses
After estimating the number of default events and the severity of losses, CreditRisk+ calculates the distribution of losses for the items in a portfolio.
In order to calculate the distributed losses, CreditRisk+ first groups the loss given default into bands of exposures.
The exposure level for each band is approximated by a common average. .

13. Sector analysis
Each sector is driven by a single underlying factor, which explains the volatility of the mean default rate over time.
Through sector analysis, CreditRisk+ can measure the impact of concentration risk and the benefits of portfolio diversification.
As the number of sectors is increased, the level of concentration risk is reduced.

14. Stress Testing
Stress tests can be carried out in CreditRisk+ and outside CreditRisk+.
CreditRisk+ can be stress tested by increasing default rates and the default rate volatilities and by stressing different sectors to different degrees.
Some stress tests, such as those that model the effect of political risk, can be difficult to carry out in CreditRisk+.
In this case, the effect should be measured without reference to the outputs of the model.

15. Applications of CreditRisk+
Calculating credit risk provisions
Enforcing credit limits
Managing credit portfolios

16. Calculating Credit Risk Provisions
When credit losses are modelled, the most frequent loss tends to be much lower than the estimated average loss.
This is because the estimated average takes into account the risk of occasional extreme losses.
Credit provisions, also known as economic capital, need to be set aside to protect against such losses.
CreditRisk+ can be used to set provisions for credit losses in a portfolio.

17. Enforcing Credit Limits
High concentrations of a small number of exposures can significantly increase portfolio risk.
Credit limits are an effective way of avoiding concentrations.
They provide a means of limiting exposure to different debtors, maturities, credit ratings and sectors.
An individual credit limit should be set at a level that is inversely proportional to the default rating associated with a particular debtor's credit rating.

18. Managing Portfolios
CreditRisk+ incorporates all the factors that determine credit risk into a single measure.
This is known as a portfolio-based approach.
The four factors that determine default risk are:
size
maturity
probability of default
concentration risk
CreditRisk+ provides a means of measuring diversification and concentration by sector.
More diverse portfolios with fewer concentrations require less economic capital.

19. Credit Metrics

20. Introduction CreditMetrics™ was launched by JP Morgan in 1997
It evaluates credit risk by predicting movements in the credit ratings of the individual investments in a portfolio.
CreditMetrics consists of three main components:
Historical data sets
A methodology for measuring portfolio Value at Risk (VAR)
A software package known as CreditManager®

21. Transition Matrices and Probability of Default
CreditMetrics uses transition matrices to generate a distribution of final values for a portfolio.
A transition matrix reflects the probability that a bond with a given rating will be upgraded or downgraded within a given time horizon.
Transition matrices are published by ratings agencies such as Standard and Poor's and Moody's.

22. Data requirements Credit ratings for the debtor
Default data for the debtor
Loss given default
Exposure
Information about credit correlations

23. Methodology
CreditMetrics™ measures changes in portfolio value by predicting movements in a debtor's credit ratings and accordingly the values of individual portfolio investments.
After the values of the individual portfolio investments have been determined, CreditMetrics™ can then calculate the credit risk.

24. CreditMetrics™ Software – CreditManager®
The software used by Credit Metrics is called CreditManager.
CreditManager® enables a financial institution to consolidate credit risk across its entire organization.
CreditManager® automatically maps each credit that the user loads into the system to its appropriate debtor and market data
It computes correlations and changes in asset value over the risk horizon due to upgrades, downgrades and defaults.
In this way, it arrives at a final figure for portfolio credit risk.
The software uses two types of data :
Position
Market

25. Steps for calculating credit risk for a single-credit portfolio
Determine the probability of credit rating migration.
Calculate the current value of the bond's remaining cashflows for each possible credit rating.
Calculate the range of possible bond values for each rating.
Calculate the credit risk.

26. Steps for calculating credit risk for a two-credit portfolio
Examine credit migration.
Calculate the range of possible bond values for each rating using independent or correlated credit migration probabilities.
Calculate the credit risk.

27. Steps for calculating credit risk for a multiple-credit portfolio
Calculate the distribution of values using a Monte Carlo simulation.
Use the standard deviation and percentile levels for this distribution to calculate credit risk for the portfolio.

28. Single credit portfolios
The steps to calculate distributed values for single-credit portfolios are:
Determine the probability of change in credit ratings.
Calculate the value of remaining cash flows for each possible credit rating.
Calculate the range of possible credit values for each rating.
The first step is to examine the probability of the bond moving from an one credit rating to another say within of one year.
The movement from one credit rating to another is known as credit migration.
Credit rating agencies publish credit migration probabilities based on historic data.

29. Bond values for different ratings
Having examined the different probabilities for credit rating migration, the next step is to calculate the range of possible bond values for each rating.
That means calculating the value of Bond X for a credit rating of Aaa, Aa, A, Baa, Ba, B, Caa, Ca, C.
To do this, we first need to calculate the value of the bond's remaining cash flows for each possible rating.

30. Discounting the cashflows
We use discount rates to calculate the current value of the bond's remaining cashflows for each credit rating.
These discount rates are taken from the forward zero coupon curve for each rating.
The forward zero coupon curve ranges from the end of the risk horizon – one year from now – to maturity.

31. Given a distribution of final values for Bond X, we can then calculate two risk measurements for the portfolio:
Standard deviation
Percentile

32. Multiple-Credit Portfolios
Because of the exponential growth in complexity as the number of bonds increases, a simulation-based approach is used to calculate the distribution of values for large portfolios.
Using Monte Carlo simulation, CreditMetrics simulates the quality of each debtor, which produces an overall value for the portfolio.
This procedure is then repeated many times in order to get the distributed portfolio values.
After we have the distributed portfolio values, we can then use the standard deviation and percentile levels for this distribution to calculate credit risk for the portfolio.

33. Portfolio Value Estimates at Risk Horizon
CreditMetrics requires three types of data to estimate portfolio value at risk horizon:
coupon rates and maturities for loans and bonds
drawn and undrawn amounts of a loan, including spreads or fees
market rates for market driven instruments, such as swaps and forwards

34. Correlations
One key issue in using Credit Metrics is handling correlations between bonds.
While determining credit losses, credit rating changes for different counterparties cannot be assumed to be independent.
How do we determine correlations? 34

35. Gausian Copula A Gaussian Copula Model comes in useful here.
Gaussian Copula allows us to construct a joint probability distribution of rating changes.
The Copula correlation between the ratings transitions for two companies is typically set equal to the correlation between their equity returns using a factor model.

36. Implementing Credit Metrics
The first step is to estimate the rating class for a debt claim.
The rating may remain the same, improve or deteriorate, depending on the firm’s performance.
Ratings transition matrix gives us the probability of the credit migrating from one rating to another during one year.
Next, we construct the distribution of the value of the debt claim.
We compute the value we expect the claim to have for each rating in one year. 36

37. Based on the term structure of bond yields for each rating category, we can get today’s price of a zero coupon bond for a forward contract to mature in one year.
If the migration probabilities are independent, we can compute the probabilities for transition of each bond independently and multiply them to obtain the joint probability.
By computing the value of the portfolio for each possible outcome and the probability of each outcome, we can construct the distribution for the portfolio value.
We can then find out the VAR at a given level of confidence.

38. The probabilities differ depending on the migration correlations.
If the different migration probabilities are not independent, we cannot multiply the individual probabilities.
Instead, we need to know the joint distribution of the bond migrations.
The values of the portfolios for each possible outcome are the same whether the bond migrations are independent or not.
The probabilities differ depending on the migration correlations.
But once we know the probabilities for each outcome, we can again construct a distribution for the bond portfolio and calculate the VAR. 38

39. However, this approach is often not enough.
The historical record of rating migration can be used to estimate the different joint probabilities.
However, this approach is often not enough.
Credit Metrics proposes an approach based on stock returns.
Say a firm has a given stock price and we want to estimate the credit risk.
Using the rating transition matrix, we know the probability of the firm migrating to different credit ratings.
We then use the distribution of stock returns to find out the ranges of returns that correspond to the various ratings. 39

40. We can produce stock returns corresponding to the various rating outcomes for each firm whose credit is in the portfolio.
The correlations between stock returns can be used to compute the probabilities of various rating outcomes for the credits.
For example, if we have two stocks we can work out the probability that one stock will be in the A rating category and other in AAA category.
When a large number of credits is involved, a factor model can be used.

41. Bivariate normal distributions
Let us now briefly discuss bivariate normal distributions.
Suppose there are two variables V1, V2. If V1 takes on a value v1, the value of V2 is normal with mean μ2 + ρ
and standard deviation, , where μ1, μ2 are defined as the respective unconditional means, 1, 2 are the unconditional standard deviations, ρ is the correlation coefficient between V1 and V2.
The expected value of V2 is linearly dependent on the value of V1.
But the standard deviation remains constant. 41

42. Factor models
Factor models also can be used to measure the correlation between normally distributed variables.
We can identify the component that is dependent on a common factor F and the remaining part which is uncorrelated with the other variables.
So if we have n distributions, U1, U2, … Un, we can write: Ui = aiF+1-a12Zi. F, Zi have normal distributions. ai is a constant between -1 and +1.
The Zi are uncorrelated with each other and the value F. The coefficient of correlation between Ui and Uj is aiaj. Without the factor model, we would have to estimate nc2 or correlations. With the single factor model, we have to estimate only n values, ai… an. 42

43. Problem
Consider a bank with a very large portfolio of similar loans. The probability of default is the same for each loan and the correlation between each pair of loans is the same. We are therefore X% certain that the percentage of loses over T years on a large portfolio will be less than V(X,T), where:
This result was first produced by Vasicek.
Suppose that a bank has a total of $100 million of retail exposures. The 1-year probability of default averages 2% and the recovery rate averages 60%. The copula correlation parameter is estimated as 0.1. What is the credit VAR at 99.9% confidence level ?

44. Solution
. In this case,
showing t hat the 99.9% worst case default rate is 12.8%. The 1-year 99.9% credit VaR therefore 100 x x (1 – 0.6) or $5.13 million.