Economic Models of Credit Risk Lectures 10 & 11. Credit Risk Modeling. Based on Risk Management, Crouhy, Galai, Mark, McGrawHill, 2000. (Kealhofer / McQuown / Vasicek). The Contingent Claim Approach  Structural Approach: KMV . The Option Pricing Approach: KMV.
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Lectures 10 & 11
Based on Risk Management, Crouhy, Galai, Mark,
McGrawHill, 2000
KMV challenges CreditMetrics on several fronts:
1. Firms within the same rating class have the same default rate
2. The actual default rate (migration probabilities) are equal to the historical default rate (migration frequencies)
KMV challenges CreditMetrics on several fronts:
3. Default is only defined in a statistical sense without explicit reference to the process which leads to default.
KMV’s model is based on the option pricing approach to credit risk as originated by Merton (1974)
1. The firm’s asset value follows a standard geometric Brownian motion, i.e.:
dV
=
m
+
s
t
dt
dZ
t
V
t
ì
ü
s
2
=
m

+
s
Z
V
V
exp
(
)
t
t
í
ý
0
t
t
2
î
þ
Liabilities / Equity
Debt: Bt
(F)
Risky Assets: Vt
Equity: St
V
V
Total:
t
t
The Option Pricing Approach: KMV2. Balance sheet of Merton’s firm
Equity value at maturity of debt obligation:
(
)
=

S
max
V
F
,
0
T
T
Firm defaults if
<
V
F
T
with probability of default (“real world” probability measure)
æ
ö
æ
ö
s
2
V
ç
÷
+
ç
m

÷
Ln
T
0
ç
÷
F
2
ç
÷
è
ø
(
)
(
)
<
=
Z
<

=

0
P
V
F
P
N
d
ç
÷
T
T
2
s
T
ç
÷
ç
÷
è
ø
í
î
The Option Pricing Approach: KMV3. Probability of default (“real world” probability measure)
Assets Value
ü
æ
ö
s
2
=
m

+
s
Z
ç
÷
ý
V
V
exp
T
T
T
O
è
ø
T
2
þ
m
=
T
E
(
)
V
e
V
O
T
V
T
V
0
F
Probability of default
Time
T
rT
B
+ P
= Fe
0
0
The Option Pricing Approach: KMVBank’s payoff matrix at times 0 and T for making a loan to Firm ABC and buying a put on the value of ABC
Time
0
T
£
Value of Assets
V
V
F
V
F
>
0
T
T
Bank’s Position:
·
B
F
V
make a loan
0
T
·
P
F  V
O
buy a put
0
T
Total
B
P
F
F
0
0
Corporate loan = Treasury bond + short a put
Po = f ( Vo, F, sv, r, T ) (BlackScholes option price)
6Conditional recovery when default = VT
KMV: Merton’s ModelFirm ABC is structured as follows:
Vt = Value of Assets (at time t)
St = Value of Equity
Bt = Value of Debt (zerocoupon)
F = Face Value of Debt
Problem:
Vo ( say =100 ), F ( say = 77 ), sv ( say = 40% ),
r ( say =10% ) and T ( say = 1 year)
Solve for Bo,So,YT and Probability of Default
`
Solution:
P0( = 3.37) ® Bo( = 66.63) ® So( = 33.37) ® YT ( =15.6%) ®PT ( = 5.6%)
æ
ö
F
=
®
P
=

ç
÷
=

=

Y
L
Y
r
=

rT
P
f
(
V
,
T
)
S
V
B
B
Fe
P
K
T
N
T
T
o
o
o
o
o
o
o
è
ø
B
o
Note: In solving for P0 we get Probability of Default ( = 24.4% )
P
=

Default spread ( ) for corporate debt
( For V0 = 100, T = 1, and r = 10% )
Y
r
T
T
s
0.05
0.10
0.20
0.40
LR
0.5
0
0
0
1.0
0.6
0
0
0.1%
2.5%
0.7
0
0
0.4%
5.6%
0.8
0
0.1%
1.5%
8.4%
0.9
0.1%
0.8%
4.1%
12.5%
1.0
2.1%
3.1%
8.3%
17.3%

rT
Fe
Leverage ratio:
=
LR
V
0
4. Default point and distance to default
Observation:
Firms more likely to default when their asset values reach a certain level of total liabilities and value of shortterm debt.
Default point is defined as
DPT=STD+0.5LTD
STDshortterm debt
LTD longterm debt
Default point (DPT)
Probability distribution of V
Asset Value
Expected growth of
assets, net
E(V)
1
V0
DD
DPT = STD + ½ LTD
Time
1 year
0
Distancetodefault (DD)
DD is the distance between the expected asset value in T years, E(VT) , and the default point, DPT, expressed in standard deviation of future asset returns:
5. Derivation of the probabilities of default from the distance to default
EDF
40 bp
4
6
1
2
5
3
DD
KMV also uses historical data to compute EDFs
1
,
200
800
=
=
DD
4
100
KMV: EDFs (Expected Default Frequencies)Example:
V0 = 1,000
Current market value of assets:
Net expected growth of assets per annum:
Expected asset value in one year:
Annualized asset volatility,
Default point
20%
:
V1 = V0(1.20) = 1,200
sA
100
800
Assume that among the population of all the firms with DD of 4 at one point in time, e.g. 5,000, 20 defaulted one year later, then:
20
=
=
=
EDF
0
.
004
0
.
4%
or 40 bp
1
year
5
,
000
The implied rating for this probability of default is BB+
Example:Federal Express ($ figures are in billions of US$)
November 1997
February 1998
Market capitalization (S0 ) (price* shares outstanding)
Book liabilities
Market value of assets (V0 )
Asset volatility
Default point
Distance to default (DD)
EDF
$ 7.8
$ 4.8
$ 12.6
15%
$ 3.4
12.63.4
0.15·12.6
0.06%(6bp)
$ 7.3
$4.9
$ 12.2
17%
$ 3.5
12.23.5
0.17·12.2
0.11%(11bp)
= 4.9
= 4.2
º A
º AA
4. EDF as a predictor of default
EDF of a firm which actually defaulted versus EDFs of firms in various quartiles and the lower decile.
The quartiles and decile represent a range of EDFs for a specific credit class.
4. EDF as a predictor of default
EDF of a firm which actually defaulted versus Standard & Poor’s rating.
4. EDF as a predictor of default
Assets value, equity value, short term debt and long term debt of a firm which actually defaulted.
In CreditRisk+ no assumption is made about the causes of default: an obligor A is either in default with probability PA, or it is not in default with probability 1PA. It is assumed that:
Under those circumstances, the probability distribution for the number of defaults, during a given period of time (say one year) is well represented by a Poisson distribution:
where
m
= average number of defaults per year
æ
ö
å
m
m
=
P
It is shown that can be approximated as
ç
÷
A
è
ø
A
One year default rate
Credit Rating
Average (%)
Standard deviation (%)
Aaa
0.00
0.0
Aa
0.03
0.1
A
0.01
0.0
Baa
0.13
0.3
Ba
1.42
1.3
B
7.62
5.1
m
Note, that standard deviation of a Poisson distribution is .
For instance, for rating B: .
m
=
=
7
.
62
2
.
76
versus
5
.
1
CreditRisk+ assumes that default rate is random and has Gamma
distribution with given mean and standard deviation.
Source: Carty and Lieberman (1996)
Probability
Excluding default rate volatility
Including default rate volatility
Number of defaults
Source: CreditRisk+
Distribution of default events
1. Losses (exposures, net of recovery) are divided into bands, with the level of exposure in each band being approximated by a single number.
Notation
A
Obligor
Exposure (net of recovery)
LA
PA
Probability of default
lA=LAxPA
Expected loss
Example: 500 obligors with exposures between $50,000 and $1M (6 obligors are shown in the table)
Roundoffexposure(in $100,000)
Exposure ($)(loss given default)
Exposure(in $100,000)
Obligor
Band
L
n
n
A
j
j
j
A
1
150,000
1.5
2
2
2
460,000
4.6
5
5
3
435,000
4.35
5
5
4
370,000
3.7
4
4
5
190,000
1.9
2
2
6
480,000
4.8
5
5
The unit of exposure is assumed to be L=$100,000. Each band j, j=1, …, m, with m=10, has an average common exposure: vj=$100,000j
In Credit Risk+ each band is viewed as an independent
portfolio of loans/bonds, for which we introduce the
following notation:
Notation
Common exposure in band j in units of L nj
nj = $100,000, $200,000, …, $1M
Expected loss in band j in units of L ej
(for all obligors in band j)
Expected number of defaults in band j mj
ej = nj x mj
mj can be expressed in terms of the individual loan characteristics
Number
Band:
of
e
m
j
j
j
obligors
1
30
1.5 (1.5x1)
1.5
2
40
8 (4x2)
4
3
50
6 (2x3)
2
4
70
25.2
6.3
5
100
35
7
6
60
14.4
2.4
7
50
38.5
5.5
8
40
19.2
2.4
9
40
25.2
2.8
10
20
4 (0.4x10)
0.4
µ
å
å
n
n
=
=
=
n
G
(
z
)
P
(
lossj
nL
)
z
P
(
n
defaults
)
z
j
j
=
=
n
0
n
0
m

m
n
e
j
µ
n
å
j
n

m
+
m
n
z
j
=
=
z
e
G
(
z
)
j
j
j
j
n
!
=
n
0
CreditRisk+: Loss distributionTo derive the distribution of losses for the entire portfolio we proceed as follows:
Step 1: Probability generating function for each band.
Each band is viewed as a portfolio of exposures by itself. The probability generating function for any band, say band j, is by definition:
where the losses are expressed in the unit L of exposure.
Since we have assumed that the number of defaults follows a Poisson distribution (see expression 30) then:
m
n

m
+
m
å
å
z
j
m
n
j
j

m
+
m
z
=
Õ
=
G
(
z
)
e
e
j
=
=
j
j
j
1
j
1
=
1
j
CreditRisk+: Loss distributionStep 2: Probability generating function for the entire portfolio.
Since we have assumed that each band is a portfolio of exposures, independent from the other bands, the probability generating function for the entire portfolio is just the product of the probability generating functions for all bands.
m
å
m
denotes the expected number of defaults for the entire portfolio.
m
where
=
j
1
j
=
1
d
G
(
z
)
=
=
P
(
loss
of
nL
)

for
n
1
,
2
,...
=
z
0
n
n
!
dz
j

(
)
(
)
v
m
=
=
=
P
0
loss
G
0
e
e
j
j
CreditRisk+: Loss distributionStep 3: loss distribution for the entire portfolio
Given the probability generating function (33) it is straightforward to derive the loss distribution, since
these probabilities can be expressed in closed form, and depend only on 2 sets of parameters: ej and nj . (See Credit Suisse 1997 p.26)
e
å
e
(
(
)
)
(
)
å
=

j
P
loss
of
nL
P
loss
of
n
v
L
j
n
£
j
:
v
n
j
Example: a twoyear defaultable zerocoupon bond that pays 100 if no default, probability of default , LGD=L=60%. The annual (riskneutral) riskfree rate process is :
=
r
12
%
=
p
0
.
5
´
+
´
´
1
0
.
94
100
0
.
06
0
.
4
100
=
=
V
86
.
08
11
1
.
12
=
r
8
%
´
+
´
´
0
.
94
100
0
.
06
0
.
4
100
=
=
V
87
.
64
12
1
.
1
=
p
0
.
5
=
r
10
%
2
(
)
(
)
´
´
+
´
´
+
´
´
+
´
´
0
.
5
0
.
94
V
0
.
06
0
.
4
V
0
.
5
0
.
94
V
0
.
06
0
.
4
V
=
V
=
11
11
12
12
77
.
52
0
1
.
08
“Defaultadjusted” interest at the tree nodes is:
100
100
=

=
=

=
R
1
14
.
1
%
R
1
16
.
2
%
12
11
87
.
64
86
.
08
´
+
´
0
.
5
86
.
08
0
.
5
87
.
64
=

=
R
1
12
%
0
77
.
52
In all three cases R is solution of the equation ( ):
D
=
t
1
1
1
[
]
=

l
D
+
l
D

(
1
t
)
t
(
1
L
)
+
D
+
D
1
R
t
1
r
t
D
+
l
D
r
t
tL
D
=
R
t

l
D
+
l
D

1
t
t
(
1
L
)
®
+
l
If , then , where is the riskneutral expected loss rate, which can be interpreted as the spread over the riskfree rate to compensate the investor for the risk of default.
l
D
®
R
r
L
L
t
0
é
é
ù
ù
ù
ê
ê
ê
ú
ú
ú
ë
ë
ë
û
û
û
Reduced Form Approach(
)
l
t
General case:is hazard rate, so that if denotes the time to default, the survival probability at horizon t is
t
t
ò
t
>
=

l
Prob
(
t
)
E
exp(
(
s
)
ds
)
0
(
)
l
=
l
E is expectation under riskneutral measure. For the constant we have:
t
t
>
=

l
Prob
(
t
)
E
exp(
t
)
(
)
The probability of default over the interval provided no default has happened until time t is:
+
D
t
,
t
t
<
t
£
+
D
=
l
D
Prob
t
t
t
(
t
)
t
(similar to the example above).
Corporate curve
(
)
R
,
r
R
t
Yield spread =
l
L
R
Treasury curve
r
t
Maturity
Term structure of interest rates
By modelling the default adjusted rate we can incorporate other factors which affect spreads such as liquidity:
=
+
l
+
R
r
L
l
where l denotes the “liquidity” adjustment premium.
if there is a shortage of bonds and one can benefit from holding the bond in inventory,
if it becomes difficult to sell the bond.
>
l
0
<
l
0
l
l
Identification problem : how to separate and in . Usually is assumed to be given. Implementations differ with respect to assumptions made regarding default intensity .
L
L
L
l
How to compute default probabilities and
l
Example. Derive the term structure of implied default probabilities from the term structure of credit spreads (assume L=50%).
Company X
Oneyear
Maturity
Treasurycurve
oneyear
forward credit
t (years)
forward rates
spreads
FS t
(%)
(%)
(%)
1
5.52
5.76
0.24
2
6.30
6.74
0.44
3
6.40
7.05
0.65
4
6.56
7.64
1.08
5
6.56
7.71
1.15
6
6.81
8.21
1.40
7
6.81
8,47
1.65
t
Reduced Form ApproachForward
Cumulative
Conditional
probabilities
defauilt
default
Maturity
of default
probabilities
probabilities
(years)
t
p
(%)
(%)
(%)
l
t
t
1
0.48
0.48
0.48
2
0.88
1.36
0.88
3
1.30
2.64
1.28
4
2.16
4.74
2.10
5
2.30
6.93
2.19
6
2.80
9.54
2.61
3.30
12.52
2.99
7
l
=
=
l
´
=
2
.
16
FS
L
1
.
08
%
For example, for year 4: , then
4
4
4
(
)
Cumulative probability:
=
+

´
l
=
P
P
1
P
4
.
74
4
3
3
4
(
)
Conditional probability:
=

´
l
=
p
1
P
2
.
10
4
3
4
(
)
l
t
(
)
X
t
(
)
(
(
)
)
(
)
(
)
l
=
q

l
+
l
s
d
t
k
t
dt
t
dB
t
,
(
)
q
l
B
t
s
k
(
)
l
³
t
0
(
)
(
)
(
)
(
)
a

+
b

l
=
t
s
t
s
t
p
t
,
s
e
b
a
(
)
(
)
(
)
(
)
=
b

s
l
p
t
,
s
s
t
t
p
t
,
s
(b.p.)
0
Years
Reduced Form Approach(
)
l
t
(
)
(
(
)
)
(
)
l
=
q

l
+
d
t
k
t
dt
dZ
t
,
(
)
(
)
(
)
t
=

g
Z
t
N
t
Jt
N
t
g
J
Take jumps sizes to be, say, independent and exponentially distributed.
(
)
r
t
(
)
(
(
)
(
)
)
(
)
=
q

+
s
dr
t
k
t
r
t
dt
dB
t
,
1
1
1
1
(
)
s
B
t
k
1
1
1
(
)
q
t
1
(
)
B
t
1