Transcript Chp25 - staff.city.ac.uk

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
B
Default
CCC
A
AA
0.000
50
60
70
80
90
Revaluation at Risk Horizon
© K. Cuthbertson, D. Nitzsche
100
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
Vm 
v 
3
 pV
i
i 1
i
3
 p V  V 
2
i 1
i
i
m

3
 p V  V 
2
2
i 1
i i
m
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 = 5.78.
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
= $5.78
© 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
VAA = 109
217
207
160
A
VAB = 107
215
205
158
B
VAD = 51
159
149
102
D
: 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
2.76
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% = 0.07x0.01x100%). 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 row
entries (eg. 92= 2.76+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
 v, p


2
2
   ( ijVij )  Vm 
i , j 1

3
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 $7.62 ( = A+B - A = 13.49- 5.87), which is much smaller than
the “stand-alone” risk of bond-B of B =12.19, because of portfolio diversification effects.
© K. Cuthbertson, D. Nitzsche
Fig 25.3: Marginal Risk and Credit Exposure
Marginal Standard Deviation
(p+i - p)/mi
10%
Asset 15 (B)
7.5%
Asset 7 (CC)
5.0%
Asset 16
Asset 9
Asset 18 (BBB)
2.5%
0.0%
0
5
7.5
10
Credit Exposure ($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, …, 2.76}
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<ZDef) = (ZDef/) = 1.06%
ZDef = 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(ZDef<R<Zccc) = (ZCCC/) - (ZDef/) = 1.00
Hence:
(ZCCC/) = 1.0 + (ZDef/) = 2.06
and
ZCCC
= -1(2.06)
© K. Cuthbertson, D. Nitzsche
= -2.04
Figure 25.4: Transition Probabilities: Initial BB-Rated
Probability
Probability of a
downgrade to B-rated
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
7.73
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 Z’s we can calculate the integral below and assume
this is given by ‘Y’
[25.15] Pr(ZB <R<ZBB, Z’BB <R’<Z’BBB) = Z
Z BB
B

'
Z BBB
'
Z BB
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
7.73
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 Firm’s 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
19.79
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
2.78
1.53
-1.49
-2.18
-2.75
-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 Z’s 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 ZB  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 firm’s 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 Merton’s , 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.
Distance from default = V (1  m v )  FB  100  80  2
v
10
std devn’s
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 m
can vary with the economic cycle.
Assume bank has 100 loans outstanding and
implying m = 3 defaults per year.
Probability of n-defaults
estimated 3% p.a.
em m n
p(n, defaults) 
n!
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 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 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 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
McKinsey’s 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 Xit (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
McKinsey’s 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 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 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 r’s.
© K. Cuthbertson, D. Nitzsche
McKinsey’s Credit Portfolio View, CPV
Then take the (CreditMetrics type) historic 8 x 8 transition matrix Tt
and multiply these historic probabilities by the appropriate ri 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