Description
measure credit risk using unconditional and conditional default probabilities. It also includes Morton model and calculation of VAR
Credit Risk
1
Credit Risk and Credit Ratings
• Different credit rating agencies have adopted
different expressions representing the credit level
of the clients.
• In the S&P rating system, AAA is the best rating.
After that comes AA, A, BBB, BB, B, CCC, CC, and C.
• The corresponding Moody’s ratings are Aaa, Aa, A,
Baa, Ba, B, Caa, Ca, and C.
• Bonds with ratings of BBB (or Baa) and above are
considered to be “investment grade”.
2
Credit Rating and Cumulative Average Default Rates (%)
(1970-2006, Moody’s, Table 22.1)
3
1 2 3 4 5 7 10
Aaa 0.000 0.000 0.000 0.026 0.099 0.251 0.521
Aa 0.008 0.019 0.042 0.106 0.177 0.343 0.522
A 0.021 0.095 0.220 0.344 0.472 0.759 1.287
Baa 0.181 0.506 0.930 1.434 1.938 2.959 4.637
Ba 1.205 3.219 5.568 7.958 10.215 14.005 19.118
B 5.236 11.296 17.043 22.054 26.794 34.771 43.343
Caa-C 19.476 30.494 39.717 46.904 52.622 59.938 69.178
For the companies that start with a good credit rating, the default
probabilities tend to increase with time, relative to poor credit companies.
For the companies that start with a poor credit rating, the default
probabilities tend to decrease with time, relative to good credit companies.
Measuring Credit Risk:
Unconditional vs. Conditional Default Probabilities
• The unconditional default probability is the probability
of default for a certain time period as seen at time zero.
• The conditional default probability (also called the
default intensity or hazard rate) is the probability of
default for a certain time period conditional on no
earlier default.
• What are the default intensities and unconditional
default probabilities for a Caa rate company in the third
year?
4
Recovery Rates following the Default
Moody’s: 1982 to 2006, Table (22.2)
5
Class Mean(%)
Senior Secured 54.44
Senior Unsecured 38.39
Senior Subordinated 32.85
Subordinated 31.61
Junior Subordinated 24.47
The recovery rate for a bond is usually defined as the price of the bond immediately
after default as a percent of its face value.
Estimating Default Probabilities
Alternative Bases for Estimation:
– Use Bond Prices
– Use CDS spreads
– Use Historical Data
– Use Merton’s Model
6
Using Bond Prices (Equation
22.2)
Average default intensity over life of bond is approximately
where s is the spread of the bond’s yield over the risk-free
rate and R is the recovery rate.
R
s
÷ 1
7
Step 1: Calculating Expected Loss:
Assume that a five year corporate bond pays a coupon of 6% per annum
(semiannually). The yield is 7% with continuous compounding and the yield on a
similar risk-free bond is 5% (with continuous compounding).
Given this, price of risk-free bond is 104.09; price of corporate bond is 95.34; the
expected loss from defaults is 8.75.
Step 2: Calculation of Probability of Default
(Table 22.3)
Time
(yrs)
Def.
Prob.
Recovery
Amount
Risk-free
Value
LGD Discount
Factor
PV of Exp
Loss
0.5 Q 40 106.73 66.73 0.9753 65.08Q
1.5 Q 40 105.97 65.97 0.9277 61.20Q
2.5 Q 40 105.17 65.17 0.8825 57.52Q
3.5 Q 40 104.34 64.34 0.8395 54.01Q
4.5 Q 40 103.46 63.46 0.7985 50.67Q
Total 288.48Q
8
We set 288.48Q = 8.75 which gives Q = 3.03%. Thus, the probability of default is 3.03% p.a.
This analysis can be extended to allow defaults to take place more frequently.
Suppose that the probability of default is Q per year, and that defaults always
happen half way through a year (immediately before a coupon payment).
Further, assume 40% recovery rate.
Using CDS Spreads
Instead of calculating the spread between the risk-free rate and the
yield on the bond as done in the previous approach, we can directly
observe the spread from the CDS, if they exist.
Rest of the procedure for calculating the probability of default will be
the same.
9
The probabilities calculated under the two approaches based on
the Bond Value and based on the CDS spreads is called as
‘risk-neutral default probability’.
Using Historical Data
Instead of calculating the spreads based on the bond prices or the CDS, we can use
historical data as viewed earlier, and infer about the probability of default for the
specific credit rating of our interest.
10
This is known as ‘real world default probability’.
Real World vs. Risk Neutral Default Probabilities
7 year averages (Table 22.4)
Rating Real-world default
probability per year
(% per annum)
Risk-neutral default
probability per year
(% per year)
Ratio Difference
Aaa 0.04 0.60 16.7 0.56
Aa 0.05 0.74 14.6 0.68
A 0.11 1.16 10.5 1.04
Baa 0.43 2.13 5.0 1.71
Ba 2.16 4.67 2.2 2.54
B 6.10 7.97 1.3 1.98
Caa-C 13.07 18.16 1.4 5.50
11
Extra Risk Premiums Earned By Bond Traders
(Table 22.5)
Rating Bond Yield
Spread over
Treasuries
(bps)
Spread of risk-free
rate used by market
over Treasuries
(bps)
Spread to
compensate for
default rate in the
real world (bps)
Extra Risk
Premium
(bps)
Aaa 78 42 2 34
Aa 87 42 4 42
A 112 42 7 63
Baa 170 42 26 102
Ba 323 42 129 151
B 521 42 366 112
Caa 1132 42 784 305
12
Possible Reasons for Excess Spread
• Corporate bonds are relatively illiquid.
• The subjective default probabilities of bond traders may be
much higher than the estimates from Moody’s historical
data.
• Bonds do not default independently of each other. This leads
to systematic risk that cannot be diversified away.
• There is also a component of unsystematic risk. Since bond
returns are highly skewed with limited upside, the non-
systematic risk is difficult to diversify away and may be
priced by the market.
13
Which World Should We Use?
• We should use risk-neutral estimates for
valuing credit derivatives and estimating the
present value of the cost of default.
• We should use real world estimates for
calculating credit VaR and conducting
scenario analysis.
14
Merton’s Model: A Possible Reconciliation
• Merton’s model regards the equity as an
option on the assets of the firm.
• In a simple situation the equity value is
max(V
T
?D, 0)
where V
T
is the total value of the firm and
D is the debt repayment required.
• This gives us a new perspective that the
value of equity can be viewed as an
option value given the strike price equal
to D and current value of the underlying
equal to V
T.
15
Equity as an Option
An option pricing model enables the value of
the firm’s equity today (E
0
) to be related to the
value of its assets today (V
0
) and the volatility
of its assets (o
V
)
.
16
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 ( ) ( )
;
where
o
o
o
0 1 0 0
) ( V d N V
V
E
E
V V E
o o
c
c
o = =
The above equation has be solved by plugging in the values for Vo and
based on the following volatility equation.
V
o
Equity as an Option: 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 V
0
=12.40 and
o
v
=21.23%.
• The probability of default is N(-d
2
) or 12.7%
• The market value of the debt is 12.40 – 3.00 =
9.40
• The present value of the promised payment is
9.51
• The expected loss to debt financiers is 9.51 –
9.40 = 0.11 i.e. about 1.2% of debt’s no default
value of 9.51
• The recovery rate is (12.7% - 1.2%)/12.7% = 91%
17
Credit Risk Mitigation
• Netting
• Collateralization
• Downgrade triggers
18
Credit VaR
• Can be defined analogously to Market
Risk VaR
• A T-year credit VaR with an X% confidence
is the loss level that we are X% confident
will not be exceeded over T years
19
Calculation of Credit VaR:
Factor-Based Gaussian Copula Model
• Consider a large portfolio of loans, each of which has a
probability of Q(T) of defaulting by time T. Suppose that
all pairwise copula correlations are µ so that all a
i
’s are
• We are X% certain that F is less than
N
?1
(1?X) = ?N
?1
(X)
• It follows that the VaR is
µ
20
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¦
)
¦
`
¹
¦
¹
¦
´
¦
µ ÷
µ +
=
÷ ÷
1
) ( ) (
) , (
1 1
X N T Q N
N T X V
Calculation of Credit VaR:
Example of Factor-Based Gaussian Copula Model
• A bank has $100 million of retail exposures
• 1-year probability of default averages 2% and the
recovery rate averages 60%
• The copula correlation parameter is 0.1
• 99.9% worst case default rate is
• The one-year 99.9% credit VaR is therefore
100×0.128×(1-0.6) or $5.13 million
21
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1 0 1
999 0 1 0 02 0
1 999 0
1 1
.
.
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) , . ( =
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.
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|
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=
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N V
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¦
¹
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¦
µ ÷
µ +
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÷ ÷
1
) ( ) (
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1 1
X N T Q N
N T X V
doc_961067450.pptx
measure credit risk using unconditional and conditional default probabilities. It also includes Morton model and calculation of VAR
Credit Risk
1
Credit Risk and Credit Ratings
• Different credit rating agencies have adopted
different expressions representing the credit level
of the clients.
• In the S&P rating system, AAA is the best rating.
After that comes AA, A, BBB, BB, B, CCC, CC, and C.
• The corresponding Moody’s ratings are Aaa, Aa, A,
Baa, Ba, B, Caa, Ca, and C.
• Bonds with ratings of BBB (or Baa) and above are
considered to be “investment grade”.
2
Credit Rating and Cumulative Average Default Rates (%)
(1970-2006, Moody’s, Table 22.1)
3
1 2 3 4 5 7 10
Aaa 0.000 0.000 0.000 0.026 0.099 0.251 0.521
Aa 0.008 0.019 0.042 0.106 0.177 0.343 0.522
A 0.021 0.095 0.220 0.344 0.472 0.759 1.287
Baa 0.181 0.506 0.930 1.434 1.938 2.959 4.637
Ba 1.205 3.219 5.568 7.958 10.215 14.005 19.118
B 5.236 11.296 17.043 22.054 26.794 34.771 43.343
Caa-C 19.476 30.494 39.717 46.904 52.622 59.938 69.178
For the companies that start with a good credit rating, the default
probabilities tend to increase with time, relative to poor credit companies.
For the companies that start with a poor credit rating, the default
probabilities tend to decrease with time, relative to good credit companies.
Measuring Credit Risk:
Unconditional vs. Conditional Default Probabilities
• The unconditional default probability is the probability
of default for a certain time period as seen at time zero.
• The conditional default probability (also called the
default intensity or hazard rate) is the probability of
default for a certain time period conditional on no
earlier default.
• What are the default intensities and unconditional
default probabilities for a Caa rate company in the third
year?
4
Recovery Rates following the Default
Moody’s: 1982 to 2006, Table (22.2)
5
Class Mean(%)
Senior Secured 54.44
Senior Unsecured 38.39
Senior Subordinated 32.85
Subordinated 31.61
Junior Subordinated 24.47
The recovery rate for a bond is usually defined as the price of the bond immediately
after default as a percent of its face value.
Estimating Default Probabilities
Alternative Bases for Estimation:
– Use Bond Prices
– Use CDS spreads
– Use Historical Data
– Use Merton’s Model
6
Using Bond Prices (Equation
22.2)
Average default intensity over life of bond is approximately
where s is the spread of the bond’s yield over the risk-free
rate and R is the recovery rate.
R
s
÷ 1
7
Step 1: Calculating Expected Loss:
Assume that a five year corporate bond pays a coupon of 6% per annum
(semiannually). The yield is 7% with continuous compounding and the yield on a
similar risk-free bond is 5% (with continuous compounding).
Given this, price of risk-free bond is 104.09; price of corporate bond is 95.34; the
expected loss from defaults is 8.75.
Step 2: Calculation of Probability of Default
(Table 22.3)
Time
(yrs)
Def.
Prob.
Recovery
Amount
Risk-free
Value
LGD Discount
Factor
PV of Exp
Loss
0.5 Q 40 106.73 66.73 0.9753 65.08Q
1.5 Q 40 105.97 65.97 0.9277 61.20Q
2.5 Q 40 105.17 65.17 0.8825 57.52Q
3.5 Q 40 104.34 64.34 0.8395 54.01Q
4.5 Q 40 103.46 63.46 0.7985 50.67Q
Total 288.48Q
8
We set 288.48Q = 8.75 which gives Q = 3.03%. Thus, the probability of default is 3.03% p.a.
This analysis can be extended to allow defaults to take place more frequently.
Suppose that the probability of default is Q per year, and that defaults always
happen half way through a year (immediately before a coupon payment).
Further, assume 40% recovery rate.
Using CDS Spreads
Instead of calculating the spread between the risk-free rate and the
yield on the bond as done in the previous approach, we can directly
observe the spread from the CDS, if they exist.
Rest of the procedure for calculating the probability of default will be
the same.
9
The probabilities calculated under the two approaches based on
the Bond Value and based on the CDS spreads is called as
‘risk-neutral default probability’.
Using Historical Data
Instead of calculating the spreads based on the bond prices or the CDS, we can use
historical data as viewed earlier, and infer about the probability of default for the
specific credit rating of our interest.
10
This is known as ‘real world default probability’.
Real World vs. Risk Neutral Default Probabilities
7 year averages (Table 22.4)
Rating Real-world default
probability per year
(% per annum)
Risk-neutral default
probability per year
(% per year)
Ratio Difference
Aaa 0.04 0.60 16.7 0.56
Aa 0.05 0.74 14.6 0.68
A 0.11 1.16 10.5 1.04
Baa 0.43 2.13 5.0 1.71
Ba 2.16 4.67 2.2 2.54
B 6.10 7.97 1.3 1.98
Caa-C 13.07 18.16 1.4 5.50
11
Extra Risk Premiums Earned By Bond Traders
(Table 22.5)
Rating Bond Yield
Spread over
Treasuries
(bps)
Spread of risk-free
rate used by market
over Treasuries
(bps)
Spread to
compensate for
default rate in the
real world (bps)
Extra Risk
Premium
(bps)
Aaa 78 42 2 34
Aa 87 42 4 42
A 112 42 7 63
Baa 170 42 26 102
Ba 323 42 129 151
B 521 42 366 112
Caa 1132 42 784 305
12
Possible Reasons for Excess Spread
• Corporate bonds are relatively illiquid.
• The subjective default probabilities of bond traders may be
much higher than the estimates from Moody’s historical
data.
• Bonds do not default independently of each other. This leads
to systematic risk that cannot be diversified away.
• There is also a component of unsystematic risk. Since bond
returns are highly skewed with limited upside, the non-
systematic risk is difficult to diversify away and may be
priced by the market.
13
Which World Should We Use?
• We should use risk-neutral estimates for
valuing credit derivatives and estimating the
present value of the cost of default.
• We should use real world estimates for
calculating credit VaR and conducting
scenario analysis.
14
Merton’s Model: A Possible Reconciliation
• Merton’s model regards the equity as an
option on the assets of the firm.
• In a simple situation the equity value is
max(V
T
?D, 0)
where V
T
is the total value of the firm and
D is the debt repayment required.
• This gives us a new perspective that the
value of equity can be viewed as an
option value given the strike price equal
to D and current value of the underlying
equal to V
T.
15
Equity as an Option
An option pricing model enables the value of
the firm’s equity today (E
0
) to be related to the
value of its assets today (V
0
) and the volatility
of its assets (o
V
)
.
16
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 ( ) ( )
;
where
o
o
o
0 1 0 0
) ( V d N V
V
E
E
V V E
o o
c
c
o = =
The above equation has be solved by plugging in the values for Vo and
based on the following volatility equation.
V
o
Equity as an Option: 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 V
0
=12.40 and
o
v
=21.23%.
• The probability of default is N(-d
2
) or 12.7%
• The market value of the debt is 12.40 – 3.00 =
9.40
• The present value of the promised payment is
9.51
• The expected loss to debt financiers is 9.51 –
9.40 = 0.11 i.e. about 1.2% of debt’s no default
value of 9.51
• The recovery rate is (12.7% - 1.2%)/12.7% = 91%
17
Credit Risk Mitigation
• Netting
• Collateralization
• Downgrade triggers
18
Credit VaR
• Can be defined analogously to Market
Risk VaR
• A T-year credit VaR with an X% confidence
is the loss level that we are X% confident
will not be exceeded over T years
19
Calculation of Credit VaR:
Factor-Based Gaussian Copula Model
• Consider a large portfolio of loans, each of which has a
probability of Q(T) of defaulting by time T. Suppose that
all pairwise copula correlations are µ so that all a
i
’s are
• We are X% certain that F is less than
N
?1
(1?X) = ?N
?1
(X)
• It follows that the VaR is
µ
20
| |
¦
)
¦
`
¹
¦
¹
¦
´
¦
µ ÷
µ +
=
÷ ÷
1
) ( ) (
) , (
1 1
X N T Q N
N T X V
Calculation of Credit VaR:
Example of Factor-Based Gaussian Copula Model
• A bank has $100 million of retail exposures
• 1-year probability of default averages 2% and the
recovery rate averages 60%
• The copula correlation parameter is 0.1
• 99.9% worst case default rate is
• The one-year 99.9% credit VaR is therefore
100×0.128×(1-0.6) or $5.13 million
21
128 0
1 0 1
999 0 1 0 02 0
1 999 0
1 1
.
.
) . ( . ) . (
) , . ( =
|
|
.
|
\
|
÷
· +
=
÷ ÷
N N
N V
| |
¦
)
¦
`
¹
¦
¹
¦
´
¦
µ ÷
µ +
=
÷ ÷
1
) ( ) (
) , (
1 1
X N T Q N
N T X V
doc_961067450.pptx