Transcript Standard presentation September 1996

Credit Risk Modelling
Economic Models of Credit Risk
Based on Risk Management, Crouhy, Galai, Mark,
McGraw-Hill, 2000
The Contingent Claim Approach -
Structural Approach:
KMV
(Kealhofer / McQuown / Vasicek)
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.
3
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 firm’s
assets falls below some critical level
4
The Option Pricing Approach: KMV
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.:
dVt
= mdt +s dZt
Vt


s2
Vt = V0 exp (m )t + s t Zt 
2


5
The Option Pricing Approach: KMV
2. Balance sheet of Merton’s firm
Assets
Risky Assets: Vt
Total:
6
V
t
Liabilities / Equity
(F)
Debt: Bt
Equity: St
V
t
The Option Pricing Approach: KMV
Equity value at maturity of debt obligation:
ST = max (VT - F , 0 )
Firm defaults if
VT
< F
with probability of default (“real world” probability
measure)
2


 
s
V
0

+  m T 
Ln
F0 
2  

= N (- d2
P (VT < F ) = P  ZT < 
s T




7
)
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
2



VT =VOexp m-s2 T +s T ZT 


mT
E(V ) =V e
T
O
VT
V0
F
Probability of default
T
8
Time
The Option Pricing Approach: KMV
Bank’s 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
Value of Assets
V0
VT  F
VT > F
Bank’s Position:
make a loan
-B0
VT
F
-P0
F - VT
O
-B0 -P0
F
F
·
· buy a put
Total
T
B 0 + P0= Fe-rT
Corporate loan = Treasury bond + short a put
9
KMV: Merton’s 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
10
1
Po = f ( Vo, F, sv, r, T ) (Black-Scholes option price)
2
Bo = Fe-rT - Po
3
So = Vo - Bo (assuming markets are frictionless)
4
Bo = Fe-YTT where YT is yield to maturity
5
Probability of Default = g (Vo, F, sv, r, T) = N ( - d2 )
(“risk neutral” probability measure)
6
Conditional recovery when default = VT
KMV: Merton’s 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
11
KMV: Merton’s Model
Solution:
P0( = 3.37)  Bo( = 66.63)  So( = 33.37)  YT ( =15.6%)  PT ( = 5.6%)
`
Po = f ( Vo , KT )
- rT
Bo = Fe - Po
So = Vo - Bo
F 
YT = LN  
 Bo 
 P T = YT - r
Note: In solving for P0 we get Probability of Default ( = 24.4% )
12
The Option Pricing Approach: KMV
Default spread ( P T = YT - r ) for corporate debt
( For V0 = 100, T = 1, and r = 10% )
s
LR
0.5
0.6
0.7
0.8
0.9
1.0
0.05
0.10
0.20
0.40
0
0
0
0
0.1%
2.1%
0
0
0
0.1%
0.8%
3.1%
0
0.1%
0.4%
1.5%
4.1%
8.3%
1.0
2.5%
5.6%
8.4%
12.5%
17.3%
-
Leverage ratio:
13
Fe rT
LR =
V0
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
DPT=STD+0.5LTD
STD - short-term debt
LTD - long-term debt
14
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
0
15
1 year
Time
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:
DD =
16
E (VT ) - DPT
s A,T
KMV: EDFs
(Expected Default Frequencies)
5. Derivation of the probabilities of default from
the distance to default
EDF
EDF = N (- d2 )
40 bp
1
2
3
4
5
6
DD
KMV also uses historical data to compute EDFs
17
KMV: EDFs
(Expected Default Frequencies)
Example:
Current market value of assets:
Net expected growth of assets per annum:
Expected asset value in: one year:
Annualized asset volatility,
sA
Default point
DD =
V0 = 1,000
20%
V1 = V0(1.20) = 1,200
100
800
1, 200 - 800
=4
100
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:
EDF1 year =
20
= 0.004 = 0.4% or 40 bp
5,000
The implied rating for this probability of default is BB+
18
KMV: EDFs
(Expected Default Frequencies)
Example:
Federal Express ($ figures are in billions of US$)
November 1997
February 1998
Market capitalization (S0 )
(price* shares outstanding)
$ 7.8
$ 7.3
Book liabilities
$ 4.8
$4.9
$ 12.6
$ 12.2
Asset volatility
15%
17%
Default point
$ 3.4
$ 3.5
12.6-3.4 = 4.9
0.15·12.6
12.2-3.5 = 4.2
0.17·12.2
Market value of assets (V0 )
Distance to default (DD)
EDF
19
0.06%(6bp) AA
0.11%(11bp) A
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.
20
KMV: EDFs (Expected Default
Frequencies)
4.
EDF as a predictor of default
EDF
EDF of a firm which
actually defaulted
versus Standard &
Poor’s rating.
21
S&P
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.
22
IV
The Actuarial Approach:
CreditRisk+
Credit Suisse Financial Products
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
24
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:
P (n defaults ) =
where
m ne- m
n!
for n = 1,2,...
m = average number of defaults per year

It is shown that m can be approximated as  m =

25

PA 
A
CreditRisk+: Frequency of default events
One year default rate
Credit Rating
Aaa
Aa
A
Baa
Ba
B
Average (%)
0.00
0.03
0.01
0.13
1.42
7.62
Standard deviation (%)
0.0
0.1
0.0
0.3
1.3
5.1
Note, that standard deviation of a Poisson distribution is m .
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)
26
CreditRisk+: Frequency of default events
Probability
Excluding default rate volatility
Including default rate volatility
Source: CreditRisk+
Number of defaults
Distribution of default events
27
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)
28
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
Obligor
Exposure (net of recovery)
Probability of default
Expected loss
29
A
LA
PA
lA=LAxPA
CreditRisk+: Loss distribution
Example: 500 obligors with exposures between
$50,000 and $1M
(6 obligors are shown in the table)
(in $100,000)
A
Exposure ($)
(loss given
default)
LA
nj
Round-off
exposure
(in $100,000)
nj
1
2
3
4
5
6
150,000
460,000
435,000
370,000
190,000
480,000
1.5
4.6
4.35
3.7
1.9
4.8
2
5
5
4
2
5
Obligor
Exposure
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
30
Band
j
2
5
5
4
2
5
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 = $100,000, $200,000, …, $1M
Expected loss in band j in units of L
(for all obligors in band j)
Expected number of defaults in band j
nj
ej
mj
ej = nj x mj
mj can be expressed in terms of the individual loan characteristics
31
CreditRisk+: Loss distribution
Band:
j
32
Number
of
obligors
ej
mj
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
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:
G j (z ) =

P(lossj = nL) zn =
n =0

P(n defaults) z
nn j
n =0
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:
Gj (z) =
33


n =0
-mj
e
mnj
n!
nn j
z
nj
- mj + mj z
=e
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
n
-m +m z
j
j
G ( z) =  e
j
=e
m
m
j =1
j =1
-  m +  m zn j
j
j
j =1
m
where m =  m j denotes the expected number of defaults for the
j =1 entire portfolio.
34
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
1 dn G ( z)
=
P ( loss of nL )
|
z=0
n!
dz n
for n = 1, 2 ,...
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)
P (0 loss
) = G (0 ) =
P (loss of nL ) =

j : v j n
35
e
m
ej
n
= e
-
v
j
j
e
j
P (loss of (n - v j ) L )
V
Reduced Form Approach
Duffie-Singleton - Jarrow-Turnbull
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”.
37
Reduced Form Approach
Example: a two-year defaultable zero-coupon bond that pays 100 if no
default, probability of default l = 0.06, LGD=L=60%. The annual (riskneutral) risk-free rate process is :
p1 = 0.5
r = 12%
V11 =
0. 94  100 + 0. 06  0.4  100
= 86. 08
1. 12
V12 =
0.94 100 + 0.06  0.4 100
= 87.64
1.1
r = 8%
p2 = 0.5
r = 10%
(
 +
  )+  (
 +
  )
V0 = 0.5 0.94 V11 0.06 0.4 V11 0.5 0.94 V12 0.06 0.4 V12 = 77.52
1.08
38
Reduced Form Approach
“Default-adjusted” interest at the tree nodes is:
R11 =
100
-1 = 16.2%
86.08
R0 =
R12 =
100
-1 = 14.1%
87.64
0.5  86.08 + 0.5  87.64
-1 = 12%
77.52
In all three cases R is solution of the equation ( Dt = 1 ):
1
1
[(1 - l Dt ) + l Dt (1 - L ) ]
=
1 + R Dt 1 + r Dt
r Dt + l D tL
D
=
R t
1 - l Dt + l Dt ( 1 - L )
If Dt  0 , then R  r + lL , where lL 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.
39
Reduced Form Approach
General case: l (t ) is hazard rate, so that if t denotes
the time to default, the survival probability at horizon t is
t

Prob( t > t ) = E exp( -  l( s ) ds ) 
0


E is expectation under risk-neutral measure. For the
constant
we have:


t
>
=
l
Prob (
t ) E exp( t ) 


The probability of default over the interval (t , t + Dt )
provided no default has happened until time t is:


<
t

+
D
Prob t
t
t  = l(t ) Dt
(similar to the example above).
40
l(t ) = l
Reduced Form Approach
R (t )
R, r
Corporate curve
Yield spread = l L
R
Treasury curve
r
t
Term structure of interest rates
41
Maturity
Reduced Form Approach
By modelling the default adjusted rate we can incorporate
other factors which affect spreads such as liquidity:
R = r + lL + l
where l denotes the “liquidity” adjustment premium.
l > 0 if there is a shortage of bonds and one can
benefit from holding the bond in inventory,
l < 0 if it becomes difficult to sell the bond.
Identification problem : how to separate l and L in l L.
Usually L is assumed to be given. Implementations differ with
respect to assumptions made regarding default intensity l.
42
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 L=50%).
(%)
Company X
one-year
forward rates
(%)
One-year
forward credit
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
Maturity
t (years)
43
Treasury
curve
Reduced Form Approach
Forward
Cumulative Conditional
probabilities
defauilt
default
Maturity of default probabilities probabilities
t (years)
pt (%)
l t (%)
Pt (%)
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
7
2.80
3.30
9.54
12.52
2.61
2.99
For example, for year 4:
Cumulative probability:
Conditional probability:
44
FS4 = l4  L = 1.08% , then
P4 = P3 + (1 - P3 ) l4 = 4.74
p4 = (1 - P3 )  l4 = 2.10
l4 = 2.16
Reduced Form Approach
Generalizations:
()
l t is modeled as a Cox process (CIR model),
• Intensity of the default
conditional on vector of state variables X t , such as default free
interest rates, stock market indices, etc.
()
d l (t ) = k (q - l (t ))dt + s l (t)dB ()
t ,
where
is a standard Brownian motion, q is the long-run mean of l
(
)
B
t
k is mean rate of reversion to the long-run mean, s is a volatility
coefficient.
Properties:
•
l (t )  0,
• Conditional survival probability
p (t , s ) = e
a (t -s )+ b (t -s )l (t ) , where
a and b are known time-dependent functions of time,
• The volatility of
45
p (t , s )is
p (t , s ) = b (s - t )s
l (t )
Reduced Form Approach
Generalizations:
• Intensity of the default
l (t ) can be modeled as a jump process:
d l (t ) = k (q - l (t ))dt + dZ (t ),
where Z (t ) = N (t ) - gJt , N (t )- cumulative jumps by t at
Poisson arrival times, g is mean arrival rate, J is mean jump
size.
Take jumps sizes
to be, say,
independent
and
exponentially
distributed.
l (b.p.)
0
46
Years
Reduced Form Approach
Generalizations:
• Risk free spot rate r (t )is modeled as one factor
extended Vasicek process.
dr (t ) = k1 (q1 (t ) - r (t ))dt + s1 dB1 (t ),
where B1 (t ), k1 , s1 are similar to parameters in CIR
model,
q1 (t )is a function defined from current term structure of
interest rates. B1 (t ) is correlated with Brownian motion
of the default intensity process.
Closed form solutions for the bond prices.
47
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
48
IX. Conclusion
Conclusion
Extensions of the basic framework:
Second Generation Credit VaR models
• Contingent exposures: swaps, loan
commitments
–
Stochastic interest rates
• Credit risk mitigation techniques: credit
derivatives
–
50
Problem: Pricing under Q measure while measure
risk under P measure
Example: Default Swap
Example 1: One year forward value of the default swap
Default swap:
Bond:
Maturity=3 years,
Maturity=7 years,
Premium=1%,
Coupon=7.9%
Recovery rate=40%.
Notional=$10,000,
Recovery rate=40%.
Correlation between asset returns = 0.465
Credit rating of Counterparty
Credit
rating of
underlying
bond
51
Aaa
Aa
A
Baa
Ba
B
Caa
Aaa
-273.07
-262.89
-193.83
70.62
219.51
294.74
676.01
Aa
-273.14
-263.11
-194.60
68.84
217.43
292.58
673.50
A
-273.54
-264.06
-198.40
59.73
206.19
280.65
659.19
Baa
-272.27
-264.22
-205.61
32.67
170.91
242.12
605.70
Ba
-271.46
-264.02
-209.07
19.08
152.40
221.51
576.32
B
-270.82
-263.73
-210.97
10.86
141.02
208.71
557.64
Caa
-266.15
-260.30
-214.92
-18.51
98.81
160.52
480.93
Example: Default Swap
Example 2: VaR calculation.
Bond:
Default swap:
Credit rating=BB,
Credit rating= AAA,
Maturity=7 years,
Maturity=3 years,
Coupon=7.9%
Premium=1%,
Notional=$10,000,
Recovery rate=40%.
Recovery rate=40%.
Correlation between asset returns = 0.465
1 year VaR at 99% confidence level:
Portfolio 1 (Bond):
4,177
Portfolio 2 (Bond and Default swap) : 727
52