Chp25CreditRisk

1、 k. cuthbertson, d. nitzschefinancial engineering:derivatives and risk management(j. wiley, 2001)k. cuthbertson and d. nitzschelecturecredit risk version 1/9/2001 k. cuthbertson, d. nitzschecreditmetrics (j.p. morgan 1997) transition probabilitiesvaluationjoint migration probabilitiesmany obligors:

2、mapping and mcsother modelskmv credit monitorcsfb credit risk plusmckinsey credit portfolio viewtopics k. cuthbertson, d. nitzschecreditmetrics (j.p. morgan 1997) k. cuthbertson, d. nitzschekey issues. creditmetrics (j.p. morgan 1997) calculating the probability of migration between different credit

3、 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. nit

4、zschefig 25.1:distribution (+1yr.), 5-year bbb-bond50607080901001100.0000.0250.0500.0750.1000.900defaultcccbbbbbbaaaaaarevaluation at risk horizonfrequency k. cuthbertson, d. nitzschefigure 25.2: calculation of c-var credit ratingsenioritycreditspreadsmigrationlikelihoodsrecovery rate indefaultvalue

5、 of bond innew ratingstandard deviation or percentilelevel for c-var k. cuthbertson, d. nitzschesingle bondmean and standard deviation of end-year valuecalculation end-yr value (3 states, a,b d)31iiimvpv3122312imiiimiivvvpvvp67, 132,)1 (106$.)049. 1 (6$)043. 1 (6$)037. 1 (6$6$fvaa67, 122,)1 (106$.)0

6、8. 1 (6$)07. 1 (6$)06. 1 (6$6$fvba k. cuthbertson, d. nitzschetable 25.1 : transition matrix (single bond) initialprobability : end-year rating (%)sumratingabdapaa = 92pab = 7pad = 1100 k. cuthbertson, d. nitzschetable 25.2 : recovery rates after default (% of par value) seniority classmean (%)stand

7、ard deviation (%)senior secured5327senior unsecured5125senior subordinated3824subordinated3320junior subordinated1711 k. cuthbertson, d. nitzschetable 25.3 : one year forward zero curves credit ratingf12f13f14ab6.07.08.0notes : f12 = one-year forward rate applicable from the end of year-1 t

8、o the end of year-2 etc. k. cuthbertson, d. nitzschetable 25.4 : probabilities and bond value (initial a-rated bond) year end ratingprobability %$valueapaa = 92vaa = 109bpab = 7vab = 107dpad = 1vad = 51notes : the mean and standard deviation for initial-a rated bond are vm,a = 108.28, v,a = 5.78.mea

9、n 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 - $108.2821/2 = $5.78 k. cuthbertson, d. nitzschetable 25.5 : probability and value (initial b-rated bond) year end ratingprobability$value1. apba = 3vba = 1082. bpbb = 90vbb =

10、983. dpbd = 7vbc = 51notes : the mean and standard deviation for initial-b rated bond are vm,b = 95.0, v,b = 12.19. k. cuthbertson, d. nitzschetable 25.6 : possible year end value (2-bonds) obligor-1 (initial-a rated)obligor-2 (initial-b rated)1. a2. b3. dvba = 108vbb = 98vbd = 511. avaa = 109217207

11、1602. bvab = 1072152051583. dvad = 51159149102notes : 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. nitzschetable 25.7 : transition matrix ( ij (percent) i

12、nitial ratingend year ratingrow sum1. a2. b3. d1. a92711002. b39071003. d00100100note: if you start in default you have zero probability of any rating change and 100% probabilityof staying in default. k. cuthbertson, d. nitzschetwo bondsrequires probabilities of all 3 x 3 joint end-year creditrating

13、s and for each state joint probability (see below) value of the 2 bonds in each state (t25.6 above) k. cuthbertson, d. nitzschetable 25.8 : joint migration probabilities : ij (percent) ( = 0) obligor-1 (initial-a rated)obligor-2 (initial-b rated)1. a2. b3. dp21 = pab = 3p22 = pbb = 90p23 = pbd = 71.

14、 ap11 = paa = 922.7682.86.442. bp12 = pab = 93. dp13 = pad = 10.030.90.07notes : 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 i

15、ndependence so forexample the bottom right entry 33 = p13 p23 = 0.07% = 0.07x0.01x100%). 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 rowentries (eg. 92= 2.76+82.8+6.44) and the figures at the top (

16、eg. p22 = 90%) equal the sum of the columnentries.assumes independent probabilities of migrationp(a at a, and b at b) = p(a at a) x p(b at b) k. cuthbertson, d. nitzschetwo bondsmean and standard deviation 29.203$31,jiijijpmvv49.13$)(2/131,22,jimijijpvvv k. cuthbertson, d. nitzschetable 25.9 : m arg

17、inal r isk b ond typestandard d eviationa5.87b12.19a + b13.49m arginal r isk of bond-b7.62n otes : the m arginal risk of adding bond-b to bond-a is $7.62 ( = a+b - a = 13.49- 5.87), w hich is m uch sm aller thanthe “stand-alone” risk of bond-b of b =12.19, because of portfolio diversification effect

18、s.marginal risk of adding bond-b k. cuthbertson, d. nitzschefig 25.3: marginal risk and credit exposurecredit exposure ($m)7.551015asset 18 (bbb)asset 15 (b)asset 9asset 16asset 7 (cc)00.0%2.5%5.0%7.5%10%marginal standard deviation (p+i - p)/misource : j.p. morgan (1997) creditmetricstm technical do

19、cument chart 1.2. k. cuthbertson, d. nitzschepercentile level of c-varorder va+b in table 25.6 from lowest to highestthen 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, , 2.76critical value closest t

20、o the 1% level gives $149hence: c-var = $54.29 (= vmp - $149 = $203.29 - $149) k. cuthbertson, d. nitzschecredit varthe c-var of a portfolio of corporate bonds depends onthe credit rating migration likelihoodsthe value of the obligor (bond) in default (based on the seniority class of the bond)the va

21、lue 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. nitzschemany obligors: mapp

22、ing and mcs k. cuthbertson, d. nitzschemany obligors: mapping and mcsasset returns are normally distributed and is knowninvert the normal distribution to obtain credit rating cut-off pointsprobability bbb-rated firm moving to default is 1.06%. then from figure 25.4 :25.12 pr(default) = pr(rzdef) = (

23、zdef/) = 1.06%hence:25.13 zdef = f-1 (1.06%) = -2.30suppose 1.00% is the observed transition probability of a move from bbb to ccc (table 25.10) then: 25.14 pr(ccc) = pr(zdefrzccc) = (zccc/) - (zdef/) = 1.00hence: (zccc/) = 1.0 + (zdef/) = 2.06and zccc = -1(2.06) = -2.04 k. cuthbertson, d. nitzschef

24、igure 25.4: transition probabilities: initial bb-rated probabilitytransition probability:defcccbbbbbbaaaaaa-2.301.06-2.041.00-1.238.8480.531.377.732.39 0.672.930.143.430.03standard deviation:we assume (for simplicity) that the mean return for the stock of an initial bb-rated firm is zeroprobability

25、of a downgrade to b-ratedprobability of defaultz k. cuthbertson, d. nitzschemany obligors: mapping and mcscalculating the joint likelihoods i,jasset returns are jointly normally distributed and covariance matrix is known, as is the joint density function ffor any given zs we can calculate the integr

26、al below and assume this is given by y 25.15 pr(zb rzbb, zbb rzbbb) = dr dr = y%y is then the joint migration probabilitywe can repeat the above for all 8x8 possible joint migration probabilities),(rrfbbbbbbbbzzzz k. cuthbertson, d. nitzschemcsfind the cut-off points for different rated bondsnow sim

27、ulate 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 vpread off the 1% left tail cut-off pointassumes asset return corre

28、lations reflect changing economic conditions, that influence credit migration k. cuthbertson, d. nitzschetable 25.10 : threshold asset returns and transition probabilities (initial bb rated obligor) final rating transition prob threshold asset return(cut off)aaaaaabbbbbbcccdefault0.030.140.677.7380.

29、538.841.001.06-zaazazbbbzbbzbzccczdef-3.432.932.391.37-1.23-2.04-2.30source: j.p. morgan (1997) table 8.4 (amended) k. cuthbertson, d. nitzschetable 25.11 : individual firms transition probabilitie end-year individual transition probabilities % rating firm 1(bbb) firm 2(a) firm 3(ccc)aaaaaabbbbbbccc

30、default0.020.335.9586.93-0.180.092.2791.055.52-0.060.220.000.221.30-19.79sum100100100source: j.p. morgan (1997) table 9.1 k. cuthbertson, d. nitzschetable 25.12 : asset return thresholds threshold firm-1 (bbb) firm-2 (a) firm-3 (ccc)zaazazbbbzbbzbzccczdef3.542.781.53-1.49-2.18-2.75-2.913.121.98-1.51

31、-3.19-3.242.862.862.63-1.02-0.85notes: the zs are standard normal variates. for example, if the standardised asset return for firm-1 is 2.0 then thiscorresponds to a credit rating of bb. hence if zb r zbb then the new credit rating is bb. if from run-1 of the mcs weobtain (standardised) returns of -

32、2.0, -3.2 and +2.9 then the new credit ratings of firms 1, 2 and 3 respectively would bebb, ccc and aaa respectively.source: j.p. morgan (1997) table 10.2 k. cuthbertson, d. nitzscheother models k. cuthbertson, d. nitzschekmv credit monitor default model uses mertons , equity as a call option et = f

33、(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. distance from default = std devnsif v is normally distributed, t

34、he 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. 21080100)1 (vbvfvm k. cuthbertson, d. nitzscheuses poisson to give default probabilities and mean default rate m can vary with the economic cyc

35、le.assume bank has 100 loans outstanding and estimated 3% p.a. implying m = 3 defaults per year.probability of n-defaultsp(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) = 0.19

36、9, 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: !),(nedefaultsnpnmmcsfp credit risk plus k. cuthbertson, d. nitzscheaverage loss given d

37、efault lgd = $10,000 then:expected loss = (3 defaults) x $10,000 = $30,000unexpected loss (99th percentile) = p(8) x 100 x 10,000 = $80,000capital requirement = unexpected loss-expected loss = 80,000 - 30,000 = $50,000portfolio of loansbank also has another 100 loans in a bucket with an average lgd

38、= $20,000 and with m = 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 ign

39、ore correlations across defaults here) csfp credit risk plus k. cuthbertson, d. nitzschefigure 25.5: probability distribution of lossesloss in $sprobabilityunexpected lossexpected losseconomic capital$30,000$80,0000.2240.0490.00899th percentile k. cuthbertson, d. nitzscheexplicitly model the link be

40、tween 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 xit (e.g. gdp, unemployment etc.)yt = g (xit, vt)i = 1, 2, nxit depend on their own past values plus other random er

41、rors it. it follows that:pit = k (xi,t-1, vt, it) each transition probability depends on past values of the macro-variables xit and the error terms vt, it. clearly the pit are correlated. mckinseys credit portfolio view, cpv k. cuthbertson, d. nitzschemonte carlo simulation to adjust the empirical (

42、or average) transition probabilities estimated from a sample of firms (e.g. as in creditmetrics). consider one monte carlo draw of the error terms vt, it (which embody the correlations found in the estimated equations for yt and xit above). this may give rise to a simulated probability pis = 0.25 of whereas the historic (unconditional) transition probability might be pih = 0.20 . this implies a ratio ofri = pis / pih = 1.25repeat the above for all initial credit rating states (i.e.