Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.1
Chapter 26
Credit Risk
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.2
Credit Ratings
? In the S&P rating system,AAA is the
best rating,After that comes AA,A,
BBB,BB,B,and CCC
? The corresponding Moody’s ratings are
Aaa,Aa,A,Baa,Ba,B,and Caa
? Bonds with ratings of BBB (or Baa) and
above are considered to be,investment
grade”
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.3
Information from Bond
Prices
? Traders regularly estimate the zero
curves for bonds with different credit
ratings
? This allows them to estimate
probabilities of default in a risk-neutral
world
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.4
Typical Pattern
(See Figure 26.1,page 611)
Spread
over
Treasuries
Maturity
Baa/BBB
A/A
Aa/AA
Aaa/AAA
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.5
The Risk-Free Rate
? Most analysts use the LIBOR rate as
the risk-free rate
? The excess of the value of a risk-free
bond over a similar corporate bond
equals the present value of the cost of
defaults
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.6Example (Zero coupon rates;
continuously compounded)
M a tu ri ty
(y e a rs )
Ri sk- f ree
y ield
Corp o rat e
b o n d y ield
1 5% 5,2 5 %
2 5% 5,5 0 %
3 5% 5,7 0 %
4 5% 5,8 5 %
5 5% 5,9 5 %
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.7
Example continued
One-year risk-free bond (principal=$1) sells for
One-year corporate bond (principal=$1) sells for
or at a 0.2497% discount
This indicates that the holder of the corporate bond
expects to lose 0.2497% from defaults in the first year
e ? ? ?0 05 1 0 951229.,
e ? ? ?0 0525 1 0 948854.,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.8
Example continued
? Similarly the holder of the corporate
bond expects to lose
or 0.9950% in the first two years
? Between years one and two the
expected loss is 0.7453%
e e
e
? ? ? ?
? ?
? ?0 05 2 0 0550 2
0 05 2 0 009950
.,
.,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.9
Example continued
? Similarly the bond holder expects to
lose 2.0781% in the first three years;
3.3428% in the first four years; 4.6390%
in the first five years
? The expected losses per year in
successive years are 0.2497%,
0.7453%,1.0831%,1.2647%,and
1.2962%
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.10
Summary of Results
(Table 26.1,page 612)
Maturity
(yea rs)
Cumu l,Los s,
%
Los s
During Yr (%)
1 0.2 49 7 0.2 49 7
2 0.9 95 0 0.7 45 3
3 2.0 78 1 1.0 83 1
4 3.3 42 8 1.2 64 7
5 4.6 39 0 1.2 96 2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.11Recovery Rates
(Table 26.3,page 614,Source,Moody’s Investor’s Service,2000)
Clas s Mea n(%) SD (%)
Sen ior Secured 52.31 25.15
Sen ior Unsecured 48.84 25.01
Sen ior Subordi nated 39.46 24.59
Sub ordinated 33.71 20.7 8
Junio r Sub ordinated 19.69 13.85
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.12
Probability of Default
0, 0 2 5 9 2 4 a n d 0, 0 2 5 2 9 4,0, 0 2 1 6 6 2,0, 0 1 4 9 0 6,
0, 0 0 4 9 9 4,a r e 5 a n d 4,,3 2,1,y e a r sin d e f a u l t of
i e sp r o b a b i l i t e x a m p l e,o u r in 0, 5R a t eR e c If
R a t e R e c.-1
L o ss % E x p,
D e f of P r o b
L o ss % E x p, R a t e ) R e c.-(1 D e f, of P r o b,
?
?
??
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.13
Reason Why This Analysis is
Simplistic
? Bonds are assumed to be zero-coupon
? The equation:
Prob,of Def.× (1-Rec,Rate)=Exp
Loss%
assumes that the claim in the event of
default equals the no-default value of
the bond
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.14A More Complete Analysis,
Definitions
B
j
,Pri ce t od ay of b ond m at urin g at t
j
G
j
,Pri ce t od ay of b ond m at urin g at t
j
i f ther e wer e no
pro babil it y of defa ult
F
j
( t ),For war d pric e at t im e t of G
j
( t < t
j
)
v ( t ),PV of $1 r ec ei ve d at t im e t with c er ta inty
C
j
( t ),Cla im ma de i f ther e is a def ault a t tim e t < t
j
R
j
( t ),Rec ov er y ra te i n the eve nt of a defa ult at t ime t < t
j
?
ij
,PV of l oss f rom a def ault a t tim e t i r elat ive to G j
p
i
,The r is k-neu tr al pro babil it y of def ault a t ti m e t
i
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.15Risk-Neutral Probability of Default
Page 616,equations 26.3 to 26.5
? PV of loss from default
? Reduction in bond price due to default
? Computing p’s inductively1
j
j j i ij
i
G B p ?
?
?? ?
1
1
j
j j i iji
j
jj
G B p
p
?
?
?
???? ?
? ? ? ? ? ? ? ?i j i j i j i j iv t F t R t C t? ??????
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.16
Relaxing Assumptions
? This analysis assumes constant interest
rates,and known recovery rates and claim
amounts
? If default events,risk-free rates,and recovery
rates are independent,results hold for
stochastic interest rates,and uncertain
recovery rates providing the recovery rate is
set equal to its expected value in a risk-
neutral world,?R
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.17
? The analysis can be extended to allow defaults at any
time
? It is important to distinguish between the default
probability density and the hazard rate
? The default probability density,q(t) is defined so that
q(t)dt as the probability of default between times t and
t+dt as seen at time zero
? The hazard rate is the probability of default between
times t and t+dt conditional on no earlier default
?
?
?
t
dh
ethtq 0
)(
)()(
tt
Extending the Analysis to Allow
Defaults at Any Time
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.18
What Should We Use as the
Claim Amount
The best assumption seems to be that
the claim amount for a bond equals the
face value plus accrued interest --- not
the no-default value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.19
Sample Data (Risk-free Rate=5%;
Expected Recovery Rate=30%)
Bond Life Coupon (%) Yield (%)
1 7.0 6.6
2 7.0 6.7
3 7.0 6.8
4 7.0 6.9
5 7.0 7.0
10 7.0 7.2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.20Implied Default Probabilities Assuming
That Default Can Happen on Bond
Maturity Dates (Table 26.5,page 617)
Time
(yrs)
Claim = No-
Def Value
Claim=Face
Val+Accr Int
1 0.0224 0.0224
2 0.0249 0.0247
3 0.0273 0.0269
4 0.0297 0.0291
5 0.0320 0.0312
10 0.1717 0.1657
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.21
Value Additivity
? If claim amount equals no-default value,
value of a coupon bond is sum of values
of constituent zero-coupon bonds
? The same is not true when claim
amount equals face value plus accrued
interest
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.22
Asset Swaps (page 618)
? An asset swap exchanges the return
on a bond for a spread above LIBOR
? Asset swaps are frequently used to
extract default probabilities from bond
prices,The assumption is that LIBOR
is the risk-free rate
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.23
Asset Swaps,Example 1
? An investor owns a 5-year corporate
bond worth par that pays a coupon of
6%,LIBOR is flat at 4.5%,An asset
swap would enable the coupon to be
exchanged for LIBOR plus 150bps
? In this case Bj=100 and Gj=106.65 (The
value of 150 bps per year for 5 years is
6.65.)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.24
Asset Swap,Example 2
? Investor owns a 5-year bond is worth $95 per
$100 of face value and pays a coupon of 5%,
LIBOR is flat at 4.5%,
? The asset swap would be structured so that
the investor pays $5 upfront and receives
LIBOR plus 162.79 bps,($5 is equivalent to
112.79 bps per year)
? In this case Bj=95 and Gj=102.22 (162.79 bps
per is worth $7.22)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.25
Theory and Practice
? In theory asset swap spreads should be
slightly dependent on the bond’s coupon
? In practice it is assumed to be the same for all
bonds with a particular maturity and the
quoted asset swap spread is assume to apply
to a bond selling for par
? This means that the spread would be quoted
as 162.79 bps in Example 2 and when
calculating default probabilities we would
assume Bj=100 and Gj=107.22
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.26
Historical Data
Historical data provided by rating
agencies are also used to estimate the
probability of default
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.27
Cumulative Average Default Rates (%)
(Table 26.7,page 619; S&P Report,January 2001)
1 2 3 4 5 7 10
AAA 0,00 0,00 0,04 0,07 0,12 0,32 0,67
AA 0,01 0,04 0,10 0,18 0,29 0,62 0,96
A 0,04 0,12 0,21 0,36 0,57 1,01 1,86
BBB 0,24 0,55 0,89 1,55 2,23 3,60 5,20
BB 1,08 3,48 6,65 9,71 1 2,57 1 8,09 2 3,86
B 5,94 1 3,49 2 0,12 2 5,36 2 9,58 3 6,34 43,41
CC C 2 5,26 3 4,79 4 2,16 4 8,18 5 4,65 5 8,64 6 2,58
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.28Interpretation
? The table shows the probability of
default for companies starting with a
particular credit rating
? A company with an initial credit rating of
BBB has a probability of 0.24% of
defaulting by the end of the first year,
0.55% by the end of the second year,
and so on
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.29
Do Default Probabilities
Increase with Time?
? For a company that starts with a good
credit rating default probabilities tend to
increase with time
? For a company that starts with a poor
credit rating default probabilities tend to
decrease with time
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.30
Bond Prices vs,Historical
Default Experience
? The estimates of the probability of
default calculated from bond prices are
much higher than those from historical
data
? Consider for example a 5 year A-rated
zero-coupon bond
? This typically yields at least 50 bps
more than the risk-free rate
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.31
Bond Prices vs,Historical
Default Experience
yp r o b a b i l i t h i st o r i cal
0, 5 7 % t h e t h a n g r e a t e r m u ch is T h i s 2, 4 7 %, is d e f a u l t
ofy p r o b a b i l i t yr-5 T h e r a t e,r e cov e r y z e r o A ss u m e
p e r i o d, y e a r-5 a o v e r v a l u e sb o n d ' t h e of 2, 4 7 % or
l o se to e x p e ct w et h a t m e a n s T h i s
0247.01
5005.0
??
??
e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.32
Possible Reasons for These
Results
? The liquidity of corporate bonds is less
than that of Treasury bonds
? Bonds traders may be factoring into
their pricing depression scenarios much
worse than anything seen in the last 20
years
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.33
A Key Theoretical Reason
(page 621)
? The default probabilities estimated from
bond prices are risk-neutral default
probabilities
? The default probabilities estimated from
historical data are real-world default
probabilities
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.34
Risk-Neutral Probabilities
The analysis based on bond prices assumes
that
? The expected cash flow from the A-rated
bond is 2.47% less than that from the risk-
free bond
? The discount rates for the two bonds are the
same
This is correct only in a risk-neutral world
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.35
The Real-World Probability
of Default
? The expected cash flow from the A-rated
bond is 0.57% less than that from the risk-
free bond
? But we still get the same price if we discount
at about 38 bps per year more than the risk-
free rate
? If risk-free rate is 5%,it is consistent with the
beta of the A-rated bond being 0.076
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.36
Which World Should We
Use?
? We should use risk-neutral estimates for
valuing credit derivatives and estimating
the cost of default
? We should use real world estimates for
calculating credit VaR and scenario
analysis
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.37
Merton’s Model (page 621 to 623)
? Merton’s model regards the equity as an
option on the assets of the firm
? In a simple situation the equity value is
max(VT - D,0)
where VT is the value of the firm and D
is the debt repayment required
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.38
Equity vs,Assets
An option pricing model enables the
value of the firm’s equity today,E0,to be
related to the value of its assets today,
V0,and the volatility of its assets,sV
E V N d De N d
d
V D r T
T
d d T
rT
V
V
V
0 0 1 2
1
0
2
2 1
2
? ?
?
? ?
? ?
?
( ) ( )
ln ( ) ( );
w h e r e
s
s
s
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.39
Volatilities
s ?? s sE V VE EV V N d V0 0 1 0? ? ( )
This equation together with the option pricing
relationship enables V0 and sV to be
determined from E0 and sE
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.40
Example
? A company’s equity is $3 million and the
volatility of the equity is 80%
? The risk-free rate is 5%,the debt is $10
million and time to debt maturity is 1
year
? Solving the two equations yields
V0=12.40 and sV=21.23%
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.41
Example continued
? The probability of default is N(-d2) or
12.7%
? The market value of the debt is 9.40
? The present value of the promised
payment is 9.51
? The expected loss is about 1.2%
? The recovery rate is 91%
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.42The Implementation of Merton’s
Model (e.g,KMV Moody’s)
? Choose time horizon
? Calculate cumulative obligations to time
horizon,This is termed by KMV the,default
point”,We denote it by D
? Use Merton’s model to calculate a theoretical
probability of default
? Use historical data or bond data to develop a
one-to-one mapping of theoretical probability
into either real-world or risk-neutral probability
of default.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.43
The Loss Given Default
(LGD)
? For derivatives we need to distinguish
between a) those that are always
assets,b) those that are always
liabilities,and c) those that can be
assets or liabilities
? What is the loss in each case?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.44
Netting (page 624)
Netting clauses state that is a company
defaults on one contract it has with a
financial institution it must default on all
such contracts
?
?
?
?
?
?
?
?
?
?
?
?
N
i
i
N
i
i
VR
VR
1
1
0,m a x)1(
)0,m a x ()1( to
f r o m l o ss t h e ch a n g e s T h i s
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.45
Reducing Credit Exposure
(page 625)
? Collateralization
? Downgrade triggers
? Diversification
? Contract design
? Credit derivatives
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.46One-Year Transition Matrix
(S&P,January 2001,p626)
Year E nd
Ra ting
Init
Ra t
AAA AA A BBB BB B CCC De f
AAA
93,66 5.8 3 0.40 0.09 0.03 0.00 0.00 0,00
AA
0.66 91.7 2 6,9 4 0.49 0.06 0.09 0.02 0,01
A
0.07 2.25 91.7 6 5.18 0.49 0.20 0.01 0,04
BBB
0.03 0.26 4.83 89.2 4 4.44 0.81 0.16 0,24
BB
0.03 0.06 0.44 6.66 83.2 3 7.46 1.05 1,08
B
0.00 0.10 0.32 0.46 5.72 83.6 2 3.84 5,94
CCC
0.15 0.00 0.29 0.88 1.91 10.2 8 61.2 3 25,26
De f
0.00 0.00 0.00 0.00 0.00 0.00 0.00 100
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.47
Risk-Neutral Transition
Matrix
? A risk-neutral transition matrix is
necessary to value derivatives that have
payoffs dependent on credit rating
changes
? A risk-neutral transition matrix can (in
theory) be determined from bond prices
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.48
Example
Cumulative probability of default
1 2 3 4 5
A 0.67% 1.33% 1.99% 2.64% 3.29%
B 1.66% 3.29% 4.91% 6.50% 8.08%
C 3.29% 6.50% 9.63% 12.69% 15.67%
Suppose there are three rating categories and
risk-neutral default probabilities extracted from
bond prices are:
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.49
Matrix Implied
Default Probability
? Let M be the annual rating transition
matrix and di be the vector containing
probability of default within i years
? d1 is the rightmost column of M
? di = M di-1 = Mi-1 d1
? Number of free parameters in M is
number of ratings squared
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.50
Transition Matrix Consistent
With Default Probabilities
A B C Default
A 98.4% 0.9% 0.0% 0.7%
B 0.5% 97.1% 0.7% 1.7%
C 0.0% 0.0% 96.7% 3.3%
Default 0.0% 0.0% 0.0% 100%
This is chosen to minimize difference between all
elements of Mi-1 d1 and the corresponding cumulative
default probabilities implied by bond prices.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.51
Credit Default Correlation
? The credit default correlation between two
companies is a measure of their tendency to
default at about the same time
? Default correlation is important in risk
management when analyzing the benefits of
credit risk diversification
? It is also important in the valuation of some
credit derivatives
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.52
Measure 1
? One commonly used default correlation
measure is the correlation between
1,A variable that equals 1 if company A defaults
between time 0 and time T and zero otherwise
2,A variable that equals 1 if company B defaults
between time 0 and time T and zero otherwise
? The value of this measure depends on T,
Usually it increases at T increases.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.53
Measure 1 continued
Denote QA(T) as the probability that
company A will default between time
zero and time T,QB(T) as the probability
that company B will default between
time zero and time T,and PAB(T) as the
probability that both A and B will default,
The default correlation measure is
])()(][)()([
)()()()(
22 TQTQTQTQ
TQTQTPT
BBAA
BAAB
AB
??
???
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.54
Measure 2
? Based on a Gaussian copula model for time to
default,
? Define tA and tB as the times to default of A and B
? The correlation measure,rAB,is the correlation
between
uA(tA)=N-1[QA(tA)]
and
uB(tB)=N-1[QB(tB)]
where N is the cumulative normal distribution function
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.55
Use of Gaussian Copula
? The Gaussian copula measure is often used
in practice because it focuses on the things
we are most interested in (Whether a default
happens and when it happens)
? Suppose that we wish to simulate the defaults
for n companies, For each company the
cumulative probabilities of default during the
next 1,2,3,4,and 5 years are 1%,3%,6%,
10%,and 15%,respectively
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.56
Use of Gaussian Copula continued
? We sample from a multivariate normal
distribution for each company
incorporating appropriate correlations
? N -1(0.01) = -2.33,N -1(0.03) = -1.88,
N -1(0.06) = -1.55,N -1(0.10) = -1.28,
N -1(0.15) = -1.04
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.57
Use of Gaussian Copula continued
? When sample for a company is less than
-2.33,the company defaults in the first year
? When sample is between -2.33 and -1.88,the
company defaults in the second year
? When sample is between -1.88 and -1.55,the
company defaults in the third year
? When sample is between -1,55 and -1.28,the
company defaults in the fourth year
? When sample is between -1.28 and -1.04,the
company defaults during the fifth year
? When sample is greater than -1.04,there is no
default during the first five years
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.58
Measure 1 vs Measure 2
n o r m a l tem u l t i v a r i a be to a ssu m e d be ca n t i m e s su r v i v a l dt r a n sf o r m e
b e ca u se co n si d e r e d a r e co m p a n i e sm a n y w h e nu se to e a si e r m u ch is It
1,M e a su r e t h a n h i g h e rt l y si g n i f i ca nu su a l l y is 2 M e a su r e
f u n ct i o n, ond i st r i b u t i
y p r o b a b i l i t n o r m a l b i v a r i a t e cu m u l a t i v e t h e is w h e r e
a n d
:v e r sa v i ce a n d 2 M e a su r e f r o m ca l cu l a t e d be ca n 1 M e a su r e
M
TQTQTQTQ
TQTQTuTuM
T
TuTuMTP
BBAA
BAABBA
AB
ABBAAB
])()(][)()([
)()(]);(),([
)(
]);(),([)(
22
??
?r
??
r?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.59
Modeling Default
Correlations
Two alternatives models of default
correlation are:
? Structural model approach
? Reduced form approach
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.60
Structural Model Approach
? Merton (1974),Black and Cox (1976),
Longstaff and Schwartz (1995),Zhou
(1997) etc
? Company defaults when the value of its
assets falls below some level,
? The default correlation between two
companies arises from a correlation
between their asset values
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.61
Reduced Form Approach
? Lando(1998),Duffie and Singleton
(1999),Jarrow and Turnbull (2000),etc
? Model the hazard rate as a stochastic
variable
? Default correlation between two
companies arises from a correlation
between their hazard rates
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.62
Pros and Cons
? Reduced form approach can be
calibrated to known default probabilities,
It leads to low default correlations,
? Structural model approach allows
correlations to be as high as desired,
but cannot be calibrated to known
default probabilities.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.63
Credit VaR (page 630)
Credit VaR asks a question such as:
What credit loss are we 99% certain will
not be exceeded in 1 year?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.64
Basing Credit VaR on Defaults Only
(CSFP Approach)
? When the expected number of defaults
is m,the probability of n defaults is
? This can be combined with a probability
distribution for the size of the losses on
a single default to obtain a probability
distribution for default losses
e
n
n?mm
!
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.65
Enhancements
? We can assume a probability
distribution for m,
? We can categorize counterparties by
industry or geographically and assign a
different probability distribution for
expected defaults to each category
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.66Model Based on Credit
Rating Changes
(Creditmetrics)
? A more elaborate model involves
simulating the credit rating changes in
each counterparty.
? This enables the credit losses arising
from both credit rating changes and
defaults to be quantified
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.67
Correlation between Credit
Rating Changes
? The correlation between credit rating
changes is assumed to be the same as
that between equity prices
? We sample from a multivariate normal
distribution and use the result to
determine the rating change (if any) for
each counterparty
26.1
Chapter 26
Credit Risk
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.2
Credit Ratings
? In the S&P rating system,AAA is the
best rating,After that comes AA,A,
BBB,BB,B,and CCC
? The corresponding Moody’s ratings are
Aaa,Aa,A,Baa,Ba,B,and Caa
? Bonds with ratings of BBB (or Baa) and
above are considered to be,investment
grade”
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.3
Information from Bond
Prices
? Traders regularly estimate the zero
curves for bonds with different credit
ratings
? This allows them to estimate
probabilities of default in a risk-neutral
world
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.4
Typical Pattern
(See Figure 26.1,page 611)
Spread
over
Treasuries
Maturity
Baa/BBB
A/A
Aa/AA
Aaa/AAA
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.5
The Risk-Free Rate
? Most analysts use the LIBOR rate as
the risk-free rate
? The excess of the value of a risk-free
bond over a similar corporate bond
equals the present value of the cost of
defaults
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.6Example (Zero coupon rates;
continuously compounded)
M a tu ri ty
(y e a rs )
Ri sk- f ree
y ield
Corp o rat e
b o n d y ield
1 5% 5,2 5 %
2 5% 5,5 0 %
3 5% 5,7 0 %
4 5% 5,8 5 %
5 5% 5,9 5 %
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.7
Example continued
One-year risk-free bond (principal=$1) sells for
One-year corporate bond (principal=$1) sells for
or at a 0.2497% discount
This indicates that the holder of the corporate bond
expects to lose 0.2497% from defaults in the first year
e ? ? ?0 05 1 0 951229.,
e ? ? ?0 0525 1 0 948854.,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.8
Example continued
? Similarly the holder of the corporate
bond expects to lose
or 0.9950% in the first two years
? Between years one and two the
expected loss is 0.7453%
e e
e
? ? ? ?
? ?
? ?0 05 2 0 0550 2
0 05 2 0 009950
.,
.,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.9
Example continued
? Similarly the bond holder expects to
lose 2.0781% in the first three years;
3.3428% in the first four years; 4.6390%
in the first five years
? The expected losses per year in
successive years are 0.2497%,
0.7453%,1.0831%,1.2647%,and
1.2962%
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.10
Summary of Results
(Table 26.1,page 612)
Maturity
(yea rs)
Cumu l,Los s,
%
Los s
During Yr (%)
1 0.2 49 7 0.2 49 7
2 0.9 95 0 0.7 45 3
3 2.0 78 1 1.0 83 1
4 3.3 42 8 1.2 64 7
5 4.6 39 0 1.2 96 2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.11Recovery Rates
(Table 26.3,page 614,Source,Moody’s Investor’s Service,2000)
Clas s Mea n(%) SD (%)
Sen ior Secured 52.31 25.15
Sen ior Unsecured 48.84 25.01
Sen ior Subordi nated 39.46 24.59
Sub ordinated 33.71 20.7 8
Junio r Sub ordinated 19.69 13.85
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.12
Probability of Default
0, 0 2 5 9 2 4 a n d 0, 0 2 5 2 9 4,0, 0 2 1 6 6 2,0, 0 1 4 9 0 6,
0, 0 0 4 9 9 4,a r e 5 a n d 4,,3 2,1,y e a r sin d e f a u l t of
i e sp r o b a b i l i t e x a m p l e,o u r in 0, 5R a t eR e c If
R a t e R e c.-1
L o ss % E x p,
D e f of P r o b
L o ss % E x p, R a t e ) R e c.-(1 D e f, of P r o b,
?
?
??
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.13
Reason Why This Analysis is
Simplistic
? Bonds are assumed to be zero-coupon
? The equation:
Prob,of Def.× (1-Rec,Rate)=Exp
Loss%
assumes that the claim in the event of
default equals the no-default value of
the bond
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.14A More Complete Analysis,
Definitions
B
j
,Pri ce t od ay of b ond m at urin g at t
j
G
j
,Pri ce t od ay of b ond m at urin g at t
j
i f ther e wer e no
pro babil it y of defa ult
F
j
( t ),For war d pric e at t im e t of G
j
( t < t
j
)
v ( t ),PV of $1 r ec ei ve d at t im e t with c er ta inty
C
j
( t ),Cla im ma de i f ther e is a def ault a t tim e t < t
j
R
j
( t ),Rec ov er y ra te i n the eve nt of a defa ult at t ime t < t
j
?
ij
,PV of l oss f rom a def ault a t tim e t i r elat ive to G j
p
i
,The r is k-neu tr al pro babil it y of def ault a t ti m e t
i
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.15Risk-Neutral Probability of Default
Page 616,equations 26.3 to 26.5
? PV of loss from default
? Reduction in bond price due to default
? Computing p’s inductively1
j
j j i ij
i
G B p ?
?
?? ?
1
1
j
j j i iji
j
jj
G B p
p
?
?
?
???? ?
? ? ? ? ? ? ? ?i j i j i j i j iv t F t R t C t? ??????
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.16
Relaxing Assumptions
? This analysis assumes constant interest
rates,and known recovery rates and claim
amounts
? If default events,risk-free rates,and recovery
rates are independent,results hold for
stochastic interest rates,and uncertain
recovery rates providing the recovery rate is
set equal to its expected value in a risk-
neutral world,?R
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.17
? The analysis can be extended to allow defaults at any
time
? It is important to distinguish between the default
probability density and the hazard rate
? The default probability density,q(t) is defined so that
q(t)dt as the probability of default between times t and
t+dt as seen at time zero
? The hazard rate is the probability of default between
times t and t+dt conditional on no earlier default
?
?
?
t
dh
ethtq 0
)(
)()(
tt
Extending the Analysis to Allow
Defaults at Any Time
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.18
What Should We Use as the
Claim Amount
The best assumption seems to be that
the claim amount for a bond equals the
face value plus accrued interest --- not
the no-default value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.19
Sample Data (Risk-free Rate=5%;
Expected Recovery Rate=30%)
Bond Life Coupon (%) Yield (%)
1 7.0 6.6
2 7.0 6.7
3 7.0 6.8
4 7.0 6.9
5 7.0 7.0
10 7.0 7.2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.20Implied Default Probabilities Assuming
That Default Can Happen on Bond
Maturity Dates (Table 26.5,page 617)
Time
(yrs)
Claim = No-
Def Value
Claim=Face
Val+Accr Int
1 0.0224 0.0224
2 0.0249 0.0247
3 0.0273 0.0269
4 0.0297 0.0291
5 0.0320 0.0312
10 0.1717 0.1657
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.21
Value Additivity
? If claim amount equals no-default value,
value of a coupon bond is sum of values
of constituent zero-coupon bonds
? The same is not true when claim
amount equals face value plus accrued
interest
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.22
Asset Swaps (page 618)
? An asset swap exchanges the return
on a bond for a spread above LIBOR
? Asset swaps are frequently used to
extract default probabilities from bond
prices,The assumption is that LIBOR
is the risk-free rate
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.23
Asset Swaps,Example 1
? An investor owns a 5-year corporate
bond worth par that pays a coupon of
6%,LIBOR is flat at 4.5%,An asset
swap would enable the coupon to be
exchanged for LIBOR plus 150bps
? In this case Bj=100 and Gj=106.65 (The
value of 150 bps per year for 5 years is
6.65.)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.24
Asset Swap,Example 2
? Investor owns a 5-year bond is worth $95 per
$100 of face value and pays a coupon of 5%,
LIBOR is flat at 4.5%,
? The asset swap would be structured so that
the investor pays $5 upfront and receives
LIBOR plus 162.79 bps,($5 is equivalent to
112.79 bps per year)
? In this case Bj=95 and Gj=102.22 (162.79 bps
per is worth $7.22)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.25
Theory and Practice
? In theory asset swap spreads should be
slightly dependent on the bond’s coupon
? In practice it is assumed to be the same for all
bonds with a particular maturity and the
quoted asset swap spread is assume to apply
to a bond selling for par
? This means that the spread would be quoted
as 162.79 bps in Example 2 and when
calculating default probabilities we would
assume Bj=100 and Gj=107.22
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.26
Historical Data
Historical data provided by rating
agencies are also used to estimate the
probability of default
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.27
Cumulative Average Default Rates (%)
(Table 26.7,page 619; S&P Report,January 2001)
1 2 3 4 5 7 10
AAA 0,00 0,00 0,04 0,07 0,12 0,32 0,67
AA 0,01 0,04 0,10 0,18 0,29 0,62 0,96
A 0,04 0,12 0,21 0,36 0,57 1,01 1,86
BBB 0,24 0,55 0,89 1,55 2,23 3,60 5,20
BB 1,08 3,48 6,65 9,71 1 2,57 1 8,09 2 3,86
B 5,94 1 3,49 2 0,12 2 5,36 2 9,58 3 6,34 43,41
CC C 2 5,26 3 4,79 4 2,16 4 8,18 5 4,65 5 8,64 6 2,58
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.28Interpretation
? The table shows the probability of
default for companies starting with a
particular credit rating
? A company with an initial credit rating of
BBB has a probability of 0.24% of
defaulting by the end of the first year,
0.55% by the end of the second year,
and so on
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.29
Do Default Probabilities
Increase with Time?
? For a company that starts with a good
credit rating default probabilities tend to
increase with time
? For a company that starts with a poor
credit rating default probabilities tend to
decrease with time
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.30
Bond Prices vs,Historical
Default Experience
? The estimates of the probability of
default calculated from bond prices are
much higher than those from historical
data
? Consider for example a 5 year A-rated
zero-coupon bond
? This typically yields at least 50 bps
more than the risk-free rate
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.31
Bond Prices vs,Historical
Default Experience
yp r o b a b i l i t h i st o r i cal
0, 5 7 % t h e t h a n g r e a t e r m u ch is T h i s 2, 4 7 %, is d e f a u l t
ofy p r o b a b i l i t yr-5 T h e r a t e,r e cov e r y z e r o A ss u m e
p e r i o d, y e a r-5 a o v e r v a l u e sb o n d ' t h e of 2, 4 7 % or
l o se to e x p e ct w et h a t m e a n s T h i s
0247.01
5005.0
??
??
e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.32
Possible Reasons for These
Results
? The liquidity of corporate bonds is less
than that of Treasury bonds
? Bonds traders may be factoring into
their pricing depression scenarios much
worse than anything seen in the last 20
years
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.33
A Key Theoretical Reason
(page 621)
? The default probabilities estimated from
bond prices are risk-neutral default
probabilities
? The default probabilities estimated from
historical data are real-world default
probabilities
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.34
Risk-Neutral Probabilities
The analysis based on bond prices assumes
that
? The expected cash flow from the A-rated
bond is 2.47% less than that from the risk-
free bond
? The discount rates for the two bonds are the
same
This is correct only in a risk-neutral world
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.35
The Real-World Probability
of Default
? The expected cash flow from the A-rated
bond is 0.57% less than that from the risk-
free bond
? But we still get the same price if we discount
at about 38 bps per year more than the risk-
free rate
? If risk-free rate is 5%,it is consistent with the
beta of the A-rated bond being 0.076
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.36
Which World Should We
Use?
? We should use risk-neutral estimates for
valuing credit derivatives and estimating
the cost of default
? We should use real world estimates for
calculating credit VaR and scenario
analysis
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.37
Merton’s Model (page 621 to 623)
? Merton’s model regards the equity as an
option on the assets of the firm
? In a simple situation the equity value is
max(VT - D,0)
where VT is the value of the firm and D
is the debt repayment required
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.38
Equity vs,Assets
An option pricing model enables the
value of the firm’s equity today,E0,to be
related to the value of its assets today,
V0,and the volatility of its assets,sV
E V N d De N d
d
V D r T
T
d d T
rT
V
V
V
0 0 1 2
1
0
2
2 1
2
? ?
?
? ?
? ?
?
( ) ( )
ln ( ) ( );
w h e r e
s
s
s
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.39
Volatilities
s ?? s sE V VE EV V N d V0 0 1 0? ? ( )
This equation together with the option pricing
relationship enables V0 and sV to be
determined from E0 and sE
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.40
Example
? A company’s equity is $3 million and the
volatility of the equity is 80%
? The risk-free rate is 5%,the debt is $10
million and time to debt maturity is 1
year
? Solving the two equations yields
V0=12.40 and sV=21.23%
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.41
Example continued
? The probability of default is N(-d2) or
12.7%
? The market value of the debt is 9.40
? The present value of the promised
payment is 9.51
? The expected loss is about 1.2%
? The recovery rate is 91%
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.42The Implementation of Merton’s
Model (e.g,KMV Moody’s)
? Choose time horizon
? Calculate cumulative obligations to time
horizon,This is termed by KMV the,default
point”,We denote it by D
? Use Merton’s model to calculate a theoretical
probability of default
? Use historical data or bond data to develop a
one-to-one mapping of theoretical probability
into either real-world or risk-neutral probability
of default.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.43
The Loss Given Default
(LGD)
? For derivatives we need to distinguish
between a) those that are always
assets,b) those that are always
liabilities,and c) those that can be
assets or liabilities
? What is the loss in each case?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.44
Netting (page 624)
Netting clauses state that is a company
defaults on one contract it has with a
financial institution it must default on all
such contracts
?
?
?
?
?
?
?
?
?
?
?
?
N
i
i
N
i
i
VR
VR
1
1
0,m a x)1(
)0,m a x ()1( to
f r o m l o ss t h e ch a n g e s T h i s
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.45
Reducing Credit Exposure
(page 625)
? Collateralization
? Downgrade triggers
? Diversification
? Contract design
? Credit derivatives
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.46One-Year Transition Matrix
(S&P,January 2001,p626)
Year E nd
Ra ting
Init
Ra t
AAA AA A BBB BB B CCC De f
AAA
93,66 5.8 3 0.40 0.09 0.03 0.00 0.00 0,00
AA
0.66 91.7 2 6,9 4 0.49 0.06 0.09 0.02 0,01
A
0.07 2.25 91.7 6 5.18 0.49 0.20 0.01 0,04
BBB
0.03 0.26 4.83 89.2 4 4.44 0.81 0.16 0,24
BB
0.03 0.06 0.44 6.66 83.2 3 7.46 1.05 1,08
B
0.00 0.10 0.32 0.46 5.72 83.6 2 3.84 5,94
CCC
0.15 0.00 0.29 0.88 1.91 10.2 8 61.2 3 25,26
De f
0.00 0.00 0.00 0.00 0.00 0.00 0.00 100
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.47
Risk-Neutral Transition
Matrix
? A risk-neutral transition matrix is
necessary to value derivatives that have
payoffs dependent on credit rating
changes
? A risk-neutral transition matrix can (in
theory) be determined from bond prices
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.48
Example
Cumulative probability of default
1 2 3 4 5
A 0.67% 1.33% 1.99% 2.64% 3.29%
B 1.66% 3.29% 4.91% 6.50% 8.08%
C 3.29% 6.50% 9.63% 12.69% 15.67%
Suppose there are three rating categories and
risk-neutral default probabilities extracted from
bond prices are:
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.49
Matrix Implied
Default Probability
? Let M be the annual rating transition
matrix and di be the vector containing
probability of default within i years
? d1 is the rightmost column of M
? di = M di-1 = Mi-1 d1
? Number of free parameters in M is
number of ratings squared
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.50
Transition Matrix Consistent
With Default Probabilities
A B C Default
A 98.4% 0.9% 0.0% 0.7%
B 0.5% 97.1% 0.7% 1.7%
C 0.0% 0.0% 96.7% 3.3%
Default 0.0% 0.0% 0.0% 100%
This is chosen to minimize difference between all
elements of Mi-1 d1 and the corresponding cumulative
default probabilities implied by bond prices.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.51
Credit Default Correlation
? The credit default correlation between two
companies is a measure of their tendency to
default at about the same time
? Default correlation is important in risk
management when analyzing the benefits of
credit risk diversification
? It is also important in the valuation of some
credit derivatives
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.52
Measure 1
? One commonly used default correlation
measure is the correlation between
1,A variable that equals 1 if company A defaults
between time 0 and time T and zero otherwise
2,A variable that equals 1 if company B defaults
between time 0 and time T and zero otherwise
? The value of this measure depends on T,
Usually it increases at T increases.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.53
Measure 1 continued
Denote QA(T) as the probability that
company A will default between time
zero and time T,QB(T) as the probability
that company B will default between
time zero and time T,and PAB(T) as the
probability that both A and B will default,
The default correlation measure is
])()(][)()([
)()()()(
22 TQTQTQTQ
TQTQTPT
BBAA
BAAB
AB
??
???
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.54
Measure 2
? Based on a Gaussian copula model for time to
default,
? Define tA and tB as the times to default of A and B
? The correlation measure,rAB,is the correlation
between
uA(tA)=N-1[QA(tA)]
and
uB(tB)=N-1[QB(tB)]
where N is the cumulative normal distribution function
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.55
Use of Gaussian Copula
? The Gaussian copula measure is often used
in practice because it focuses on the things
we are most interested in (Whether a default
happens and when it happens)
? Suppose that we wish to simulate the defaults
for n companies, For each company the
cumulative probabilities of default during the
next 1,2,3,4,and 5 years are 1%,3%,6%,
10%,and 15%,respectively
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.56
Use of Gaussian Copula continued
? We sample from a multivariate normal
distribution for each company
incorporating appropriate correlations
? N -1(0.01) = -2.33,N -1(0.03) = -1.88,
N -1(0.06) = -1.55,N -1(0.10) = -1.28,
N -1(0.15) = -1.04
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.57
Use of Gaussian Copula continued
? When sample for a company is less than
-2.33,the company defaults in the first year
? When sample is between -2.33 and -1.88,the
company defaults in the second year
? When sample is between -1.88 and -1.55,the
company defaults in the third year
? When sample is between -1,55 and -1.28,the
company defaults in the fourth year
? When sample is between -1.28 and -1.04,the
company defaults during the fifth year
? When sample is greater than -1.04,there is no
default during the first five years
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.58
Measure 1 vs Measure 2
n o r m a l tem u l t i v a r i a be to a ssu m e d be ca n t i m e s su r v i v a l dt r a n sf o r m e
b e ca u se co n si d e r e d a r e co m p a n i e sm a n y w h e nu se to e a si e r m u ch is It
1,M e a su r e t h a n h i g h e rt l y si g n i f i ca nu su a l l y is 2 M e a su r e
f u n ct i o n, ond i st r i b u t i
y p r o b a b i l i t n o r m a l b i v a r i a t e cu m u l a t i v e t h e is w h e r e
a n d
:v e r sa v i ce a n d 2 M e a su r e f r o m ca l cu l a t e d be ca n 1 M e a su r e
M
TQTQTQTQ
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r?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.59
Modeling Default
Correlations
Two alternatives models of default
correlation are:
? Structural model approach
? Reduced form approach
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.60
Structural Model Approach
? Merton (1974),Black and Cox (1976),
Longstaff and Schwartz (1995),Zhou
(1997) etc
? Company defaults when the value of its
assets falls below some level,
? The default correlation between two
companies arises from a correlation
between their asset values
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.61
Reduced Form Approach
? Lando(1998),Duffie and Singleton
(1999),Jarrow and Turnbull (2000),etc
? Model the hazard rate as a stochastic
variable
? Default correlation between two
companies arises from a correlation
between their hazard rates
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.62
Pros and Cons
? Reduced form approach can be
calibrated to known default probabilities,
It leads to low default correlations,
? Structural model approach allows
correlations to be as high as desired,
but cannot be calibrated to known
default probabilities.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.63
Credit VaR (page 630)
Credit VaR asks a question such as:
What credit loss are we 99% certain will
not be exceeded in 1 year?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.64
Basing Credit VaR on Defaults Only
(CSFP Approach)
? When the expected number of defaults
is m,the probability of n defaults is
? This can be combined with a probability
distribution for the size of the losses on
a single default to obtain a probability
distribution for default losses
e
n
n?mm
!
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.65
Enhancements
? We can assume a probability
distribution for m,
? We can categorize counterparties by
industry or geographically and assign a
different probability distribution for
expected defaults to each category
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.66Model Based on Credit
Rating Changes
(Creditmetrics)
? A more elaborate model involves
simulating the credit rating changes in
each counterparty.
? This enables the credit losses arising
from both credit rating changes and
defaults to be quantified
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
26.67
Correlation between Credit
Rating Changes
? The correlation between credit rating
changes is assumed to be the same as
that between equity prices
? We sample from a multivariate normal
distribution and use the result to
determine the rating change (if any) for
each counterparty
