1. © K. Cuthbertson, D. Nitzsche FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture Credit Risk

2. © K. Cuthbertson, D. Nitzsche CreditMetrics  (J.P. Morgan 1997) Transition probabilities Valuation Joint migration probabilities Many Obligors: Mapping and MCS Other Models KMV Credit Monitor CSFB Credit Risk Plus McKinsey Credit Portfolio View Topics

3. © K. Cuthbertson, D. Nitzsche CreditMetrics  (J.P. Morgan 1997)

4. © K. Cuthbertson, D. Nitzsche Key Issues. CreditMetrics  (J.P. Morgan 1997)  calculating the probability of migration between different credit ratings and the calculation of the value of bonds in different potential credit ratings.  using the standard deviation as a measure of C-VaR for a single bond and for a portfolio of bonds.  how to calculate the probabilities (likelihood) of joint migration between credit ratings.

5. © K. Cuthbertson, D. Nitzsche Fig 25.1:Distribution (+1yr.), 5-Year BBB-Bond 50 60708090100110 0.000 0.025 0.050 0.075 0.100 0.900 Default CCC B BB BBB A AA AAA Revaluation at Risk Horizon Frequency

6. © K. Cuthbertson, D. Nitzsche

7. Single Bond Mean and Standard Deviation of end-year Value Calculation end-yr value (3 states, A,B D)

8. © K. Cuthbertson, D. Nitzsche

11. Mean and Standard Deviation V m,A = 0.92($109) + 0.07($107) + 0.01($51) = $108.28  v,A = [0.92($109) 2 + 0.07($107) 2 + 0.01($51) 2 - $108.28 2 ] 1/2 = $5.78

12. © K. Cuthbertson, D. Nitzsche

15. Two Bonds Requires probabilities of all 3 x 3 joint end-year credit ratings and for each state ~ joint probability (see below) ~ value of the 2 bonds in each state (T25.6 above)

16. © K. Cuthbertson, D. Nitzsche Assumes independent probabilities of migration p(A at A, and B at B) = p(A at A) x p(B at B)

17. © K. Cuthbertson, D. Nitzsche Two Bonds Mean and Standard Deviation

18. © K. Cuthbertson, D. Nitzsche Marginal Risk of adding Bond-B

19. © K. Cuthbertson, D. Nitzsche Fig 25.3: Marginal Risk and Credit Exposure Credit Exposure ($m) 7.55 1015 Asset 18 (BBB) Asset 15 (B) Asset 9 Asset 16 Asset 7 (CC) 0 0.0% 2.5% 5.0% 7.5% 10% Marginal Standard Deviation (  p+i -  p )/  i Source : J.P. Morgan (1997) CreditMetrics TM Technical Document Chart 1.2.

20. © K. Cuthbertson, D. Nitzsche Percentile Level of C-VaR Order V A+B in table 25.6 from lowest to highest then add up their joint likelihoods (table 25.8) until these reach the 1% value. [25.10] V A+B = {$102, $149, $158, $159, …, $217}  i,j = {0.07, 0.9, 0.49, 0.43, …, 2.76} Critical value closest to the 1% level gives $149 Hence: C-VaR = $54.29 (= V mp - $149 = $203.29 - $149)

21. © K. Cuthbertson, D. Nitzsche Credit VaR The C-VaR of a portfolio of corporate bonds depends on  the credit rating migration likelihoods  the value of the obligor (bond) in default (based on the seniority class of the bond)  the value of the bond in any new credit rating (where the coupons are revalued using the one-year forward rate curve applicable to that bonds new credit rating)  either use the end-year portfolio standard deviation or more usefully a particular percentile level

22. © K. Cuthbertson, D. Nitzsche Many Obligors: Mapping and MCS

23. © K. Cuthbertson, D. Nitzsche Many Obligors: Mapping and MCS Asset returns are normally distributed and  is known ‘Invert’ the normal distribution to obtain ‘credit rating’ cut-off points Probability BBB-rated firm moving to default is 1.06%. Then from figure 25.4 : [25.12] Pr(default) = Pr(R<Z Def ) =  (Z Def /  ) = 1.06% Hence: [25.13] Z Def = F -1 (1.06%)  = -2.30  Suppose 1.00% is the ‘observed’ transition probability of a move from BBB to CCC (table 25.10) then: [25.14]Pr(CCC) = Pr(Z Def <R<Z ccc ) =  (Z CCC /  ) -  (Z Def /  ) = 1.00 Hence:  (Z CCC /  ) = 1.0 +  (Z Def /  ) = 2.06 and Z CCC =  -1 (2.06) = -2.04 

24. © K. Cuthbertson, D. Nitzsche Figure 25.4: Transition Probabilities: Initial BB-Rated Probability Transition probability : Def CCC B BB BBB A AA AAA -2.30 1.06 -2.04 1.00 -1.23 8.8480.53 1.37 7.73 2.39 0.67 2.93 0.14 3.43 0.03 Standard Deviation : We assume (for simplicity) that the mean return for the stock of an initial BB-rated firm is zero Probability of a downgrade to B-rated Probability of default Z

25. © K. Cuthbertson, D. Nitzsche Many Obligors: Mapping and MCS Calculating the Joint Likelihoods  i,j Asset returns are jointly normally distributed and covariance matrix  is known, as is the joint density function f For any given Z’s we can calculate the integral below and assume this is given by ‘Y’ [25.15] Pr(Z B <R<Z BB, Z’ BB <R’<Z’ BBB ) = dR dR’ = Y% ‘Y’ is then the joint migration probability We can repeat the above for all 8x8 possible joint migration probabilities

26. © K. Cuthbertson, D. Nitzsche MCS Find the cut-off points for different rated bonds Now simulate the joint returns (with a known correlation) and associate these outcomes with a JOINT credit position. Revalue the 2 bonds at these new ratings ~ this is the 1st MCS outcome, V p (1) Repeat above many times and plot a histogram of V p Read off the 1% left tail cut-off point Assumes asset return correlations reflect changing economic conditions, that influence credit migration

27. © K. Cuthbertson, D. Nitzsche

30. Other Models

31. © K. Cuthbertson, D. Nitzsche KMV Credit Monitor Default model~ uses Merton’s, equity as a call option E t = f(V t, F B,  v, r, T-t) KMV derive a theoretical relationship between the unobservable volatility of the firm  v and the observable stock return volatility  E :  E = g (  v ) Knowing F B, r, T-t and E we can solve the above two equations to obtain  v. Distance from default = std devn’s If V is normally distributed, the ‘theoretical’ probability of default (i.e. of V < F B ) is 2.5% (since 2 is the 95% confidence limit) and this is the required default frequency for this firm.

32. © K. Cuthbertson, D. Nitzsche Uses Poisson to give default probabilities and mean default rate  can vary with the economic cycle. Assume bank has 100 loans outstanding and estimated 3% p.a. implying  = 3 defaults per year. Probability of n-defaults p(0) = = 0.049, p(1) = 0.049, p(2) = 0.149, p(3) = 0.224…p(8) = 0.008 ~ humped shaped probability distribution (see figure 25.5). Cumulative probabilities: p(0) = 0.049, p(0-1) = 0.199, p(0-2) = 0.423, … p(0-8) = 0.996 “p(0-8)” indicates the probability of between zero and eight defaults in Take 8 defaults as an approximation to the 99 th percentile Average loss given default LGD = $10,000 then: CSFP Credit Risk Plus

33. © K. Cuthbertson, D. Nitzsche Average loss given default LGD = $10,000 then: Expected loss = (3 defaults) x $10,000 = $30,000 Unexpected loss (99 th percentile) = p(8) x 100 x 10,000 = $80,000 Capital Requirement = Unexpected loss-Expected Loss = 80,000 - 30,000 = $50,000 PORTFOLIO OF LOANS Bank also has another 100 loans in a ‘bucket’ with an average LGD = $20,000 and with  = 10% p.a. Repeat the above exercise for this $20,000 ‘bucket’ of loans and derive its (Poisson) probability distribution. Then ‘add’ the probability distributions of the two buckets (i.e. $10,000 and $20,000) to get the probability distribution for the portfolio of 200 loans (we ignore correlations across defaults here) CSFP Credit Risk Plus

34. © K. Cuthbertson, D. Nitzsche Figure 25.5: Probability Distribution of Losses Loss in $’s Probability Unexpected Loss Expected Loss Economic Capital $30,000$80,000 0.224 0.049 0.008 99th percentile

35. © K. Cuthbertson, D. Nitzsche Explicitly model the link between the transition probability (e.g. p(C to D)) and an index of macroeconomic activity, y. p it = f(y t )where i = “C to D” etc. y is assumed to depend on a set of macroeconomic variables X it (e.g. GDP, unemployment etc.) Y t = g (X it, v t )i = 1, 2, … n X it depend on their own past values plus other random errors  it. It follows that: p it = k (X i,t-1, v t,  it ) Each transition probability depends on past values of the macro- variables X it and the error terms v t,  it. Clearly the p it are correlated. McKinsey’s Credit Portfolio View, CPV

36. © K. Cuthbertson, D. Nitzsche Monte Carlo simulation to adjust the empirical (or average) transition probabilities estimated from a sample of firms (e.g. as in CreditMetrics). Consider one Monte Carlo ‘draw’ of the error terms v t,  it (which embody the correlations found in the estimated equations for y t and X it above). This may give rise to a simulated probability p i s = 0.25 of whereas the historic (unconditional) transition probability might be p i h = 0.20. This implies a ratio of r i = p i s / p i h = 1.25 Repeat the above for all initial credit rating states (i.e. i = AAA, AA, … etc.) and obtain a set of r’s. McKinsey’s Credit Portfolio View, CPV

37. © K. Cuthbertson, D. Nitzsche Then take the (CreditMetrics type) historic 8 x 8 transition matrix T t and multiply these historic probabilities by the appropriate r i so that we obtain a new ‘simulated ‘transition probability matrix, T. Then revalue our portfolio of bonds using new simulated probabilities which reflect one possible state of the economy. This would complete the first Monte Carlo ‘draw’ and give us one new value for the bond portfolio. Repeating this a large number of times (e.g. 10,000), provides the whole distribution of gains and losses on the portfolio, from which we can ‘read off’ the portfolio value at the 1 st percentile. Mark-to-market model with direct link to macro variables McKinsey’s Credit Portfolio View, CPV

38. © K. Cuthbertson, D. Nitzsche

39. End of Slides