Credit Metrics By: A V Vedpuriswar November 11, 2010

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®

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 of a given rating being upgraded or downgraded within a given time horizon. Transition matrices are published by ratings agencies such as Standard and Poor's and Moody's.

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

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

Average cumulative default rates (%), 1970-2003 (Source: Moody’s) Rating Term (Years) 1 2 3 4 5 7 10 15 20 Aaa 0.00 0.04 0.12 0.29 0.62 1.21 1.55 Aa 0.02 0.03 0.06 0.15 0.24 0.43 0.68 1.51 2.70 A 0.09 0.23 0.38 0.54 0.91 1.59 2.94 5.24 Baa 0.20 0.57 1.03 1.62 2.16 3.24 5.10 9.12 12.59 Ba 1.26 3.48 6.00 8.59 11.17 15.44 21.01 30.88 38.56 B 6.21 13.76 20.65 26.66 31.99 40.79 50.02 59.21 60.73 Caa 23.65 37.20 48.02 55.56 60.83 69.36 77.91 80.23 5 5

Average recovery rate (%) Recovery rates on corporate bonds as a percent of face value, 1982-2004. (Source: Moody’s). Class Average recovery rate (%) Senior secured 57.4 Senior unsecured 44.9 Senior subordinated 39.1 Subordinated 32.0 Junior subordinated 28.9 6 6

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

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.

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.

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

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.

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.

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.

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

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 to calculate credit risk for the portfolio. Alternatively, we can use percentile levels.

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

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? 17

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.

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. 19

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.

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. 21

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. 22

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.

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. 24

Transforming distributions with Gaussian copula v1 p1 u1 v2 p2 u2 0.25 -0.67449 0.50 0.75 +0.67449 Bivariate distribution with 0.3 correlation 0.25 0.50 0.75 .0951 .1633 .2158 .2985 .4133 .5951

Transforming distributions with Gaussian copula v1 percentile u1 v2 u2 0.1 5.00 -1.64 2.00 -2.05 0.2 20.00 -0.84 8.00 -1.41 0.3 38.75 -0.29 18.00 -0.92 0.4 55.00 0.13 32.00 -0.47 0.5 68.75 0.49 50.00 0.00 0.6 80.00 0.84 68.00 0.47 0.7 88.75 1.21 82.00 0.92 0.8 95.00 1.64 92.00 1.41 0.9 98.75 2.24 98.00 2.05

Joint probability distribution using Gaussian Copula, ρ=0.5 V 1 V2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 .006 .017 .028 .037 .044 .048 .049 .050 .013 .043 .081 .120 .156 .181 .193 .198 .200 .061 .124 .197 .273 .331 .364 .381 .387 .019 .071 .149 .248 .358 .449 .505 .535 .548 .076 .164 .281 .417 .537 .616 .663 .683 .020 .078 .173 .301 .456 .600 .701 .763 .793 .079 .177 .312 .481 .642 .760 .837 .877 .080 .179 .318 .494 .667 .798 .887 .936 020 .180 .320 .499 .678 .816 .913 .970

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. 28

Credit Metrics Illustration

The Credit migration of a BBB Bond Gupton, Finger, Bhatia, Credit Metrics technical document

Using the forward curve to compute bond values Consider an A Bond. The present value of cash flows can be calculated as follows: V = 6 + 6/1.0372 + 6/1.04322 + 6/1.04933 + 106/1.05324 = 108.66 Consider a BBB Bond. The present value of cash flows can be calculated as follows: V = 6 + 6/1.041 + 6/1.04672 + 6/1.05253 + 106/1.05634 = 107.55

The credit migration of an A Bond Gupton, Finger, Bhatia, Credit Metrics technical document

Gupton, Finger, Bhatia, Credit Metrics technical document

Gupton, Finger, Bhatia, Credit Metrics technical document

Gupton, Finger, Bhatia, Credit Metrics technical document

Distribution of portfolio returns Gupton, Finger, Bhatia, Credit Metrics technical document