# Version 1/9/2001 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT Version 1/9/2001 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture Credit Risk K. Cuthbertson, D. Nitzsche Topics 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 K. Cuthbertson, D. Nitzsche CreditMetrics (J.P. Morgan 1997) )

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. K. Cuthbertson, D. Nitzsche Fig 25.1:Distribution (+1yr.), 5-Year BBB-Bond BBB Frequency 0.900 0.100 0.075

BB 0.050 0.025 Default 0.000 50 60 70 CCC 80 90 Revaluation at Risk Horizon K. Cuthbertson, D. Nitzsche B 100

A AA 110 AAA Figure 25.2: Calculation of C-VaR Credit Rating Seniority CreditSpreads Migration Likelihoods Recovery Rate in Default Value of Bond in new Rating

Standard Deviation or Percentile Level for C-VaR K. Cuthbertson, D. Nitzsche Single Bond Mean and Standard Deviation of end-year Value 3 Vm piVi i 1 3 v p V V i i 1 i m

2 3 2 p V V i i 2 m i 1 Calculation end-yr value (3 states, A,B D) VA, A \$6 \$6 \$6 \$6 \$106

... (1.037) (1.043) 2 (1.049) 3 (1 f1, 7 ) 6 VA, B \$6 \$6 \$6 \$6 \$106 ... (1.06) (1.07) 2 (1.08) 2 (1 f1, 7 ) 6 K. Cuthbertson, D. Nitzsche Table 25.1 : Transition Matrix (Single Bond) Initial

Rating A Probability : End-Year Rating (%) A B D PAA = 92 pAB = 7) pAD = 1 K. Cuthbertson, D. Nitzsche Sum 100 Table 25.2 : Recovery Rates After Default (% of par value) Seniority Class Senior Secured Senior Unsecured Senior Subordinated Subordinated Junior Subordinated Mean (%)

53 51 38 33 17) K. Cuthbertson, D. Nitzsche Standard Deviation (%) 27) 25 24 20 11 Table 25.3 : One Year Forward Zero Curves Credit Rating f12 f13 f14 A 3.7) 4.3 4.9 B

6.0 7) .0 8.0 Notes : f12 = one-year forward rate applicable from the end of year-1 to the end of year-2 etc. K. Cuthbertson, D. Nitzsche Table 25.4 : Probabilities and Bond Value (Initial A-Rated Bond) Year End Rating A B D Probability % pAA = 92 pAB = 7) pAD = 1 \$Value VAA = 109 VAB = 107) VAD = 51 Notes : The mean and standard deviation for initial-A rated bond are

Vm,A = 108.28, V,A = 5.7) 8. Mean and Standard Deviation Vm,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 - \$108.282]1/2 = \$5.7) 8 K. Cuthbertson, D. Nitzsche Table 25.5 : Probability and Value (Initial B-Rated Bond) Year End Rating Probability \$Value 1. A pBA = 3 VBA = 108 2. B pBB = 90 VBB = 98 3. D pBD = 7) VBC = 51 Notes : The mean and standard deviation for initial-B rated bond are

Vm,B = 95.0, V,B = 12.19. K. Cuthbertson, D. Nitzsche Table 25.6 : Possible Year End Value (2-Bonds) Obligor-1 (initial-A rated) Obligor-2 (initial-B rated) 1. 1. 2. 3. Notes A VBA = 108 2. B VBB = 98 3.

D VBD = 51 A VAA = 109 217) 207) 160 B VAB = 107) 215 205 158 D VAD = 51 159 149 102 : The values in the ith row and jth column of the central 3x3 matrix are simply the sum of the values in the appropriate row and column (eg. entry for D,D is 102 = 51 + 51). K. Cuthbertson, D. Nitzsche Table 25.7 : Transition Matrix (ij (percent)) Initial Rating

End Year Rating Row Sum 1. A 2. B 3. D 1. A 92 7) 1 100 2. B 3 90 7) 100 3. D 0 0 100 100 Note: If you start in default you have zero probability of any rating change and 100% probability of staying in default. K. Cuthbertson, D. Nitzsche

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) K. Cuthbertson, D. Nitzsche Table 25.8 : Joint Migration Probabilities : ij(percent) ( = 0) Obligor-1 (initial-A rated) 1. 2. 3. A B D p11 = pAA = 92 p12 = pAB = 7) p13 = pAD = 1

Obligor-2 (initial-B rated) 1. A 2. B 3. D p21 = pAB = 3 p22 = pBB = 90 p23 = pBD = 7) 2.7) 6 0.21 0.03 82.8 6.3 0.9 6.44 0.49 0.07) Notes : The sum of all the joint likelihoods in the central 3x3 matrix is unity (100). The joint migration probability i,j = p1,i p2,j (where 1 = initial A rated and 2 = initial B rated). We are assuming statistical independence so for example the bottom right entry 33 = p13 p23 = 0.07) % = 0.07) x0.01x100%). The transition probabilities (eg. p12 = 7) %) are included as an aide memoire. The figures on the left (eg. p12 = 7) %) equal the sum of the likelihood row entries (eg. 92= 2.7) 6+82.8+6.44) and the figures at the top (eg. p22 = 90%) equal the sum of the column entries. Assumes independent probabilities of migration

p(A at A, and B at B) = p(A at A) x p(B at B) K. Cuthbertson, D. Nitzsche Two Bonds Mean and Standard Deviation 3 Vm , p ijVij \$203.29 i , j 1 3 v, p 2 2 ( ijVij ) Vm i , j 1 1/ 2 \$13.49

K. Cuthbertson, D. Nitzsche Marginal Risk of adding Bond-B Table 25.9 : Marginal Risk Bond Type A B A+B Marginal Risk of Bond-B Standard Deviation 5.87) 12.19 13.49 7) .62 Notes : The marginal risk of adding bond-B to bond-A is \$7) .62 ( = A+B - A = 13.49- 5.87) ), which is much smaller than the stand-alone risk of bond-B of B =12.19, because of portfolio diversification effects. K. Cuthbertson, D. Nitzsche Fig 25.3: Marginal Risk and Credit Exposure

Marginal Standard Deviation (p+i - p)/i 10% Asset 15 (B) 7.5% Asset 7 (CC) 5.0% Asset 16 Asset 9 2.5% Asset 18 (BBB) 0.0% 0 5 7.5

10 Credit Exposure (\$m) Source : J.P. Morgan (1997) CreditMetricsTM Technical Document Chart 1.2. K. Cuthbertson, D. Nitzsche 15 Percentile Level of C-VaR Order VA+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] VA+B = {\$102, \$149, \$158, \$159, , \$217) } i,j = {0.07) , 0.9, 0.49, 0.43, , 2.7) 6} Critical value closest to the 1% level gives \$149 Hence: C-VaR = \$54.29 (= Vmp - \$149 = \$203.29 - \$149) 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 K. Cuthbertson, D. Nitzsche Many Obligors: Mapping and MCS 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] Hence: [25.13]

Pr(default) = Pr(R

BBB BB B Probability of default A Def Standard Deviation: CCC -2.30 Transition probability: 1.06 AA AAA -2.04 -1.23 1.00 8.84 80.53 1.37) 7) .7) 3 2.39 2.93 3.43

0.67) 0.14 0.03 We assume (for simplicity) that the mean return for the stock of an initial BB-rated firm is zero K. Cuthbertson, D. Nitzsche Z 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 Zs we can calculate the integral below and assume this is given by Y [25.15] Pr(ZB

f ( R, R ' , )dR dR = Y% Y is then the joint migration probability We can repeat the above for all 8x8 possible joint migration probabilities 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, Vp(1) Repeat above many times and plot a histogram of Vp Read off the 1% left tail cut-off point Assumes asset return correlations reflect changing economic conditions, that influence credit migration K. Cuthbertson, D. Nitzsche Table 25.10 : Threshold Asset Returns and Transition

Probabilities (Initial BB Rated Obligor) Final Rating AAA AA A BBB BB B CCC Default Transition Prob Threshold 0.03 0.14 0.67) 7) .7) 3 80.53 8.84 1.00 1.06 Source: J.P. Morgan (1997) ) Table 8.4 (amended)

K. Cuthbertson, D. Nitzsche ZAA ZA ZBBB ZBB ZB ZCCC ZDef Asset Return(cut off) 3.43 2.93 2.39 1.37) -1.23 -2.04 -2.30 Table 25.11 : Individual Firms Transition Probabilitie End-year Individual Transition Probabilities % Rating Firm 1(BBB)

Firm 2(A) Firm 3(CCC) AAA 0.02 0.09 0.22 AA 0.33 2.27) 0.00 A 5.95 91.05 0.22 BBB 86.93 5.52 1.30 BB B CCC Default 0.18 0.06 19.7) 9

Sum 100 100 100 Source: J.P. Morgan (1997) ) Table 9.1 K. Cuthbertson, D. Nitzsche Table 25.12 : Asset Return Thresholds Threshold ZAA ZA ZBBB ZBB ZB ZCCC ZDef Firm-1 (BBB) 3.54 2.7) 8 1.53 -1.49 -2.18 -2.7) 5

-2.91 Firm-2 (A) 3.12 1.98 -1.51 -3.19 -3.24 Firm-3 (CCC) 2.86 2.86 2.63 1.02 -0.85 Notes: The Zs are standard normal variates. For example, if the standardised asset return for firm-1 is 2.0 then this corresponds to a credit rating of BB. Hence if Z B R ZBB then the new credit rating is BB. If from run-1 of the MCS we obtain (standardised) returns of -2.0, -3.2 and +2.9 then the new credit ratings of firms 1, 2 and 3 respectively would be BB, CCC and AAA respectively. Source: J.P. Morgan (1997) ) Table 10.2 K. Cuthbertson, D. Nitzsche Other Models

K. Cuthbertson, D. Nitzsche KMV Credit Monitor Default model~ uses Mertons , equity as a call option Et = f(Vt, FB, 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 FB, r, T-t and E we can solve the above two equations to obtain v. V (1 v ) FB 100 80 2 Distance from default = std devns v 10 If V is normally distributed, the theoretical probability of default (i.e. of V < FB) is 2.5% (since 2 is the 95% confidence limit) and this is the required default frequency for this firm. K. Cuthbertson, D. Nitzsche CSFP Credit Risk Plus 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 e n p( n, defaults ) n! p(0) = = 0.049, p(1) = 0.049, p(2) = 0.149, p(3) = 0.224p(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 99th percentile Average loss given default LGD = \$10,000 then: K. Cuthbertson, D. Nitzsche CSFP Credit Risk Plus Average loss given default LGD = \$10,000 then: Expected loss = (3 defaults) x \$10,000 = \$30,000 Unexpected loss (99th 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) K. Cuthbertson, D. Nitzsche Figure 25.5: Probability Distribution of Losses Probability Expected Loss 0.224 Unexpected Loss 99th percentile 0.049 Economic Capital

0.008 \$30,000 \$80,000 K. Cuthbertson, D. Nitzsche Loss in \$s McKinseys Credit Portfolio View, CPV Explicitly model the link between the transition probability (e.g. p(C to D)) and an index of macroeconomic activity, y. pit = f(yt) where i = C to D etc. y is assumed to depend on a set of macroeconomic variables X it (e.g. GDP, unemployment etc.) Yt = g (Xit, vt) i = 1, 2, n Xit depend on their own past values plus other random errors it. It follows that: pit = k (Xi,t-1, vt, it) Each transition probability depends on past values of the macrovariables Xit and the error terms vt, it. Clearly the pit are correlated. K. Cuthbertson, D. Nitzsche

McKinseys Credit Portfolio View, CPV 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 Xit above). This may give rise to a simulated probability pis = 0.25 of whereas the historic (unconditional) transition probability might be p ih = 0.20 . This implies a ratio of ri = pis / pih = 1.25 Repeat the above for all initial credit rating states (i.e. i = AAA, AA, etc.) and obtain a set of rs. K. Cuthbertson, D. Nitzsche McKinseys Credit Portfolio View, CPV 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 1st percentile. Mark-to-market model with direct link to macro variables K. Cuthbertson, D. Nitzsche TABLE 25.13 : A COMPARISON OF CREDIT MODELS Characteristics J.P.Morgan CreditMetrics KMV Credit Monitor CSFP Credit Risk Plus Mark-to-Market (MTM) or Default Mode (DM) Source of Risk MTM

MTM or DM DM McKinsey Credit Portfolio View MTM or DM Multivariate normal stock returns Multivariate normal stock returns Stochastic default rate (Poisson) Macroeconomic Variables Stock prices Transition probabilities

Option prices Stock price volatility Correlation between mean default rates Correlation between macro factors Analytic or MCS Analytic Analytic MCS Correlations Solution Method K. Cuthbertson, D. Nitzsche End of Slides

K. Cuthbertson, D. Nitzsche