Credit Risk Modeling

1
Credit Risk Modeling

• Economic Models of Credit Risk
• Lectures 10 11

Based on Risk Management, Crouhy, Galai,
Mark, McGraw-Hill, 2000
2
The Contingent Claim Approach - Structural
Approach KMV

• (Kealhofer / McQuown / Vasicek)

3
The Option Pricing Approach KMV
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)

• Default rates change continuously while ratings
  are adjusted in a discrete fashion.
• Default rates vary with current economic and
  financial conditions of the firm.

4
The Option Pricing Approach KMV
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 proposes a structural model which relates
  default to balance sheet dynamics
• Microeconomic approach to default a firm is in
  default when it cannot meet its financial
  obligations
• This happens when the value of the firms assets
  falls below some critical level

5
The Option Pricing Approach KMV
KMVs model is based on the option pricing
approach to credit risk as originated by Merton
(1974) 1. The firms asset value follows a
standard geometric Brownian motion, i.e.
dV
s
m


t
dZ
dt
t
V
t
ü
ì
s
2
Z

-

s
m
t
t
V
V
)
(
exp
ý
í
t
t
0
2
þ
î
6
The Option Pricing Approach KMV
2. Balance sheet of Mertons firm
7
The Option Pricing Approach KMV
Equity value at maturity of debt obligation
(
)
-

0
,
max
F
V
S
T
T
Firm defaults if
lt
F
V
T
with probability of default (real world
probability measure)
ö
æ
ö
æ
s
2
V

ç

ç
-

m
0
T
Ln

ç
2
F

ç
ø
è
(
)
(
)
-

-
lt
Z

lt
0
d
N
P
F
V
P

ç
2
T
T
s
T

ç

ç
ø
è
8
The Option Pricing Approach KMV
3. Probability of default (real world
probability measure)

• Distribution of asset values at maturity of the
  debt obligation

Assets Value
ü
æ
ö
s
2
s

-
ç


m
Z
V
V
T
T
ý
exp
T
O
T
è
ø
2
þ
m
T

E
V
e
(
)
V
O
T
V
T
V
0
F
Probability of default
Time
T
9
The Option Pricing Approach KMV
Banks pay-off 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
gt
0
T
T
Banks Position

-B
F
make a loan
V

0
T

buy a put
-P
F - V
O

0
T
Total
-B
-P
F
F
0
0
Corporate loan Treasury bond short a put
10
KMV Mertons Model
Firm ABC is structured as follows Vt Value
of Assets (at time t) St Value of Equity Bt
Value of Debt (zero-coupon) F Face Value
of Debt
11
KMV Mertons Model
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
12
KMV Mertons Model

Solution
P0( 3.37) Bo( 66.63) So( 33.37) YT (
15.6) PT ( 5.6)
æ
ö
F



-
P
-

-
Y
L
Y
r
ç


rT

-
B
Fe
P
S
V
B
P
f
V
T
(
,
)
K
T
N
T
T
o
o
o
o
o
è
ø
o
o
B
o
Note In solving for P0 we get Probability of
Default ( 24.4 )
13
The Option Pricing Approach KMV
-

P
Default spread ( ) for
corporate debt ( For V0 100, T 1, and r 10
)
r
Y
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
14
KMV EDFs (Expected Default Frequencies)
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
short-term debt. Default point is defined
as DPTSTD0.5LTD STD - short-term debt LTD -
long-term debt
15
KMV EDFs (Expected Default Frequencies)
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
16
KMV EDFs (Expected Default Frequencies)
Distance-to-default (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

17
KMV EDFs (Expected Default Frequencies)
5. Derivation of the probabilities of default
from the distance to default
EDF
40 bp
5
6
4
2
1
3
DD
KMV also uses historical data to compute EDFs
18
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
year
1
5
000
,
The implied rating for this probability of
default is BB
19
KMV EDFs (Expected Default Frequencies)
ExampleFederal 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 12.6-3.4 0.1512.6 0
.06(6bp)
7.3 4.9 12.2 17 3.5 12.2-3.5 0.1712.2
0.11(11bp)

4.9
4.2
º A?
º AA?
20
KMV EDFs (Expected Default Frequencies)
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.
21
KMV EDFs (Expected Default Frequencies)
4. EDF as a predictor of default
EDF of a firm which actually defaulted versus
Standard Poors rating.
22
KMV EDFs (Expected Default Frequencies)
4. EDF as a predictor of default
Assets value, equity value, short term debt and
long term debt of a firm which actually
defaulted.
23
IV The Actuarial Approach CreditRisk

• Credit Suisse Financial Products

24
The Actuarial Approach CreditRisk
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 1-PA. It is assumed
that

• for a loan, the probability of default in a given
  period, say one month, is the same for any other
  month

• for a large number of obligors, the probability
  of default by any particular obligor is small and
  the number of defaults that occur in any given
  period is independent of the number, of defaults
  that occur in any other period

25
The Actuarial Approach CreditRisk
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
ö
æ
å

m
m

ç
P
It is shown that can be approximated as
A
ø
è
A
26
CreditRisk Frequency of default events
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
1
.
5


76
.
2
62
.
7
versus
CreditRisk assumes that default rate is random
and has Gamma distribution with given mean and
standard deviation.
Source Carty and Lieberman (1996)
27
CreditRisk Frequency of default events
Probability
Excluding default rate volatility
Including default rate volatility
Number of defaults
Source CreditRisk
Distribution of default events
28
CreditRisk Loss distribution

• In CreditRisk, the exposure for each obligor is
  adjusted by the anticipated recovery rate in
  order to produce a loss given default (exogenous
  to the model)

29
CreditRisk Loss distribution
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
lALAxPA
Expected loss
30
CreditRisk Loss distribution
Example 500 obligors with exposures between
50,000 and 1M (6 obligors are shown
in the table)
Exposure ()(loss given default)
Round-offexposure(in 100,000)
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 L100,000.
Each band j, j1, , m, with m10, has an
average common exposure vj100,000j
31
CreditRisk Loss distribution
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
32
CreditRisk Loss distribution
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
33
CreditRisk Loss distribution
To 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
34
CreditRisk Loss distribution
Step 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.
where
m

j
1

j
35
CreditRisk Loss distribution
Step 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
-

L
v
n
P
nL
P

of

loss

of

loss
j
n

n
v
j

j
36
V Reduced Form Approach

• Duffie-Singleton - Jarrow-Turnbull

37
Reduced Form Approach

• Reduced form approach uses a Poisson process like
  environment to describe default.
• Contrary to the structural approach the timing of
  default takes the bond-holders by surprise.
  Default is treated as a stopping time with a
  hazard rate process.
• Reduced form approach is less intuitive than the
  structural model from an economic standpoint, but
  its calibration is based on credit spreads that
  are observable.

38
Reduced Form Approach
Example a two-year defaultable zero-coupon bond
that pays 100 if no default, probability of
default , LGDL60. The annual
(risk-neutral) risk-free rate process is


12
r

5
.
0
p




1
100
4
.
0
06
.
0
100
94
.
0


08
.
86
V
11
12
.
1


8
r




100
4
.
0
06
.
0
100
94
.
0


64
.
87
V
12
1
.
1

5
.
0
p


10
r
2
(
)
(
)











4
.
0
06
.
0
94
.
0
5
.
0
4
.
0
06
.
0
94
.
0
5
.
0
V
V
V
V


V
52
.
77
12
12
11
11
0
08
.
1
39
Reduced Form Approach
Default-adjusted interest at the tree nodes is
100
100

-


-


1
.
14
1
R

2
.
16
1
R
12
11
64
.
87
08
.
86



64
.
87
5
.
0
08
.
86
5
.
0

-


12
1
R
0
52
.
77
In all three cases R is solution of the equation
( )

D
1
t
1
1


-
D

D
-

l
l
)
1
(
)
1
(
L
t
t
D

D

1
1
t
r
t
R
D

D
l
tL
t
r

D
t
R
-
D

D
-
l
l
)
1
(
1
L
t
t
If , then ,
where is the risk-neutral expected loss
rate, which can be interpreted as the spread over
the risk-free rate to compensate the investor for
the risk of default.
l



D
l
L
r
R
0
t
L
40
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
-

gt
l
t
ds
s
E
t
Prob
)
)
(
exp(
)
(
0
(
)
l
l

E is expectation under risk-neutral measure. For
the constant we have
t

l
t
-

gt
)
exp(
)
(
t
E
t
Prob
(
)
The probability of default over the interval
provided no default has happened until time t is
D

t
t
t
,
D

D


lt
l
t
t
t
t
t
t
Prob
)
(
(similar to the example above).
41
Reduced Form Approach
Corporate curve
(
)
r
R
,
t
R
Yield spread
l
L
R
Treasury curve
r
t
Maturity
Term structure of interest rates
42
Reduced Form Approach
By modelling the default adjusted rate we can
incorporate other factors which affect spreads
such as liquidity



l
l
L
r
R
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.
gt
0
l
lt
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
l
43
Reduced Form Approach
How to compute default probabilities and
l
Example. Derive the term structure of implied
default probabilities from the term structure of
credit spreads (assume L50).
Company X
One-year
Maturity
Treasurycurve
one-year
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
44
Reduced Form Approach
Forward
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
16
.
2

08
.
1
L
FS
For example, for year 4
, then
4
4
4
(
)
Cumulative probability


-


l
74
.
4
1
P
P
P
4
3
3
4
(
)
Conditional probability


-

l
10
.
2
1
P
p
4
3
4
45
Reduced Form Approach

• Generalizations
• Intensity of the default is modeled as
  a Cox process (CIR model), conditional on vector
  of state variables , such as default
  free interest rates, stock market indices, etc.
• where is a standard Brownian
  motion, is the long-run mean of
  is mean rate of reversion to
  the long-run mean, is a volatility
  coefficient.
• Properties
• ,
• Conditional survival probability
  , where
• and are known time-dependent
  functions of time,
• The volatility of is

(
)
l
t

(
)
t
X

(
)
(
)
(
)
(
)
(
)
l
l
q
l

-

s
,

t
dB
t
dt
t
k
t
d
(
)
q
l

t
B
s
k
(
)
³
l
0

t
(
)
(
)
(
)
(
)
l
b
a
-

-

t
s
t
s
t
e
s
t
p
,

b
a
(
)
(
)
(
)
(
)
l
s
b
-

t
t
s
s
t
p
,

s
t
p
,

46
Reduced Form Approach

• Generalizations
• Intensity of the default can be
  modeled as a jump process
• where ,
  - cumulative jumps by at Poisson
  arrival times, is mean arrival rate,
  is mean jump size.

(
)
l
t

(
)
(
)
(
)
(
)

-

l
q
l
,

t
dZ
dt
t
k
t
d
(
)
(
)
(
)
g
-

t
Jt
t
N
t
Z
t
N
g
J
Take jumps sizes to be, say, independent and
exponentially distributed.
47
Reduced Form Approach

• Generalizations
• Risk free spot rate is modeled as one
  factor extended Vasicek process.
• where , , are similar to
  parameters in CIR model,
• is a function defined from current
  term structure of interest rates. is
  correlated with Brownian motion of the default
  intensity process.
• Closed form solutions for the bond prices.

(
)
t
r

(
)
(
)
(
)
(
)
(
)
s
q

-

,

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

1
(
)
t
B
1
48
Reduced Form Approach

• Inputs
• the term structure of default-free rates
• the term structure of credit spreads for each
  credit category
• the loss rate for each credit category

• Model assumptions
• zero correlations between credit events and
  interest rates
• deterministic credit spreads as long as there are
  no credit events
• constant recovery rates