Credit Risk
Chapter 22
02:58
Credit Ratings



02:58
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”
Historical Data
Historical data provided by rating agencies
are also used to estimate the probability of
default
02:58
Cumulative Ave Default Rates (%)
(1970-2006, Moody’s, Table 22.1, page 490)
1
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
19.476 30.494 39.717
46.904
52.622 59.938
Caa-C
02:58
2
69.178
Interpretation


02:58
The table shows the probability of
default for companies starting with a
particular credit rating
A company with an initial credit rating of
Baa has a probability of 0.181% of
defaulting by the end of the first year,
0.506% by the end of the second year,
and so on
Do Default Probabilities Increase
with Time?


02:58
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
Default Intensities vs Unconditional
Default Probabilities



02:58
The default intensity (also called hazard rate) is
the probability of default for a certain time period
conditional on no earlier default
The unconditional default probability is the
probability of default for a certain time period as
seen at time zero
What are the default intensities and
unconditional default probabilities for a Caa or
below rate company in the third year?
[（39.717%-30.494%)/(1-30.494%)]

Define V(t) as cumulative probability of the company
surviving to time t.
( 1- V( t +t ) ) - ( 1- V( t ) )
 t
V( t )
02:58
Recovery Rate
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
02:58
Recovery Rates
(Moody’s: 1982 to 2006, Table 22.2, page 491)
Class
02:58
Mean(%)
Senior Secured
54.44
Senior Unsecured
38.39
Senior Subordinated
32.85
Subordinated
31.61
Junior Subordinated
24.47
Estimating Default Probabilities

Alternatives:




02:58
Use Bond Prices
Use CDS spreads
Use Historical Data
Use Merton’s Model
Using Bond Prices
Average default intensity(h) over life of bond
is approximately
s
h
1 R
where s is the spread of the bond’s yield
over the risk-free rate and R is the recovery
rate
显然这样估计出来的是风险中性违约概率。
02:58
More Exact Calculation



02:58
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 riskfree bond is 5% (with continuous compounding)
Price of risk-free bond is 104.09; price of corporate bond
is 95.34; expected loss from defaults is 8.75
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).
Calculations
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
02:58
288.48Q
Calculations continued



02:58
We set 288.48Q = 8.75 to get Q = 3.03%
This analysis can be extended to allow
defaults to take place more frequently
With several bonds we can use more
parameters to describe the default
probability distribution
02:58
The Risk-Free Rate


02:58
The risk-free rate when default
probabilities are estimated is usually
assumed to be the LIBOR/swap zero rate
(or sometimes 10 bps below the
LIBOR/swap rate)
To get direct estimates of the spread of
bond yields over swap rates we can look
at asset swaps
18
19
Real World vs Risk-Neutral
Default Probabilities


02:58
The default probabilities backed out of
bond prices or credit default swap spreads
are risk-neutral default probabilities
The default probabilities backed out of
historical data are real-world default
probabilities
A Comparison



02:58
Calculate 7-year default intensities from
the Moody’s data (These are real world
default probabilities)
Use Merrill Lynch data to estimate
average 7-year default intensities from
bond prices (these are risk-neutral
default intensities)
Assume a risk-free rate equal to the 7year swap rate minus 10 basis point
Real World vs Risk Neutral Default
Probabilities, 7 year averages (Table
22.4, page 495)
Rating
Aaa
Aa
A
Baa
Ba
B
Caa-C
02:58
Real-world default
Risk-neutral default
Ratio
probability per yr (% per probability per yr (% per
annum)
year)
0.04
0.60
16.7
0.05
0.74
14.6
0.11
1.16
10.5
0.43
2.13
5.0
2.16
4.67
2.2
6.10
7.97
1.3
13.07
18.16
1.4
Difference
0.56
0.68
1.04
1.71
2.54
1.98
5.50

The calculation of default intensities using
historical data are based on equation(22.1) and
table 22.1. From equation (22.1),we have
 (7)  1/ 7ln[1  Q(7)]

Where  (t ) is the average default intensity(or
hazard rate)by time t and Q(t) is the cumulative
probability of default by time t. The value of Q(7)
are taken directly from Table 22.1.Consider, for
example, an A-rated company. The value of
Q(7) is 0.00759.The average 7-year default
intensity is therefore
 (7)  1/ 7ln(0.99241)  0.0011
02:58

The calculations using bond prices are based on
equation(22.2)and bond yields published by Merrill
Lynch. The results shown are averages between
December 1996 and Oct. 2007.The recovery rate is
assumed to be 40% and, the risk-free interest rate is
assumed to be the 7-year swap rate minus 10 basis
points. Foe example, for A-rated bonds the average
Merrill Lynch yield was 5.993%. The average swap
rate was 5.398%, so that the average risk-free rate
was 5.298%. This gives the average 7-year default
probability as
0.05993  0.05298
 0.0116
1  0.4
02:58
Risk Premiums Earned By Bond
Traders (Table 22.5, page 496)
Rating
Aaa
Aa
A
Baa
Ba
B
Caa
02:58
Bond Yield Spread of risk-free
Spread to
Spread over rate used by market compensate for
Treasuries
over Treasuries default rate in the
(bps)
(bps)
real world (bps)
78
42
2
87
42
4
112
42
7
170
42
26
323
42
129
521
42
366
1132
42
784
Extra Risk
Premium
(bps)
34
42
63
102
151
112
305
Possible Reasons for These Results




02:58
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.
Bond returns are highly skewed with limited
upside. The non-systematic risk is difficult to
diversify away and may be priced by the
market
Which World Should We Use?


02:58
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 scenario
analysis
Merton’s Model


02:58
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
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 0  V0 N ( d 1 )  De  rT N ( d 2 )
where
d1 
02:58
ln (V0 D )  ( r  s V2 2 ) T
sV
T
; d 2  d1  sV
T
Volatilities
E
s E E0 
s V V0  N (d 1 ) s V V0
V
This equation together with the option pricing
relationship enables V0 and sV to be
determined from E0 and sE
02:58
Example



02:58
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%
Example continued





02:58
The probability of default is N(-d2) or
12.7%
The market value of the debt is V0-E0=
9.40
The present value of the promised
payment is 9.51
The expected loss is about (9.519.4)/9.51=1.2%
The recovery rate is 91%
The Implementation of Merton’s
Model (e.g. Moody’s KMV)


02:58
Moody公司则利用股票可视为公司资产期权这一思想计算
出风险中性世界的违约距离（如图1所示），之后再利用
其拥有的海量历史违约数据库，建立起风险中性违约距离
与现实世界违约率之间的对应关系，从而得到预期违约频
率（Expected Default Frequency, EDF），作为违约概率
的预测指标。
图2就是Moody公司用这种方法计算出来的贝尔斯登预期
违约频率时间序列。从图上可以看出，在2008年3月14日
贝尔斯登被摩根大通接管前后，其预期违约频率最高飙升
到80％左右。可见，从股票价格中提炼出来的违约概率具
有很强的信息功能。
02:58
02:58
从期权价格中可以推导出风险中性违约概率
02:58
从期权价格中可以推导出
风险中性违约概率

02:58
运用上述方法，我们就可根据2008年3月14日贝
尔斯登将于2008年3月22日到期的期权价格，计
算出贝尔斯登的风险中性违约概率和公司价值的
概率分布（如图13所示）。贝尔斯登于2008年3
月14日被摩根大通接管。图13显示，市场对贝尔
斯登一周后的命运产生巨大分歧，公司价值大涨
大跌的概率远远大于小幅变动的概率，这样的分
布与正常情况的分布有天壤之别。可见期权价格
可以让我们清楚地看出市场在非常时期对未来的
特殊看法。
02:58
风险中性违约概率

02:58
风险中性违约概率虽然不同于现实概率，但其变
化可以反映现实世界违约概率的变化。在金融危
机时期，它可能比信用违约互换（CDS）的价差
能更敏感地反映出违约概率的变化（如图14所示
）。在贝尔斯登于2008年3月14日被接管前后，
根据上述方法计算出来的风险中性概率每天的变
化比CDS的价差更敏感。这是因为在金融危机期
间，金融机构自身的信用度大幅降低，造成在场
外（OTC）市场交易的CDS交易量急剧萎缩，价
差大幅扩大，信号失真。 CDS价差所隐含的信息
将在本文第五点讨论。
02:58
Credit Risk in Derivatives
Transactions (page 491-493)
Three cases
 Contract always an asset
 Contract always a liability
 Contract can be an asset or a liability
02:58
General Result




Assume that default probability is independent of the
value of the derivative
Consider times t1, t2,…tn and default probability is qi
at time ti. The value of the contract at time ti is fi and
the recovery rate is R
The loss from defaults at time ti is qi(1-R)E[max(fi,0)].
Defining ui=qi(1-R) and vi as the value of a derivative
that provides a payoff of max(fi,0) at time ti, the cost
of defaults is n
u v
i 1
02:58
i i
If Contract Is Always a Liability
(equation 22.9)
f  f 0e
*
0
 ( y *  y )T
where derivative provides a payoff at time T .
f 0* is the actual value of the derivative and
f 0 is the default - free value.
y * is the yield on zero coupon bonds issued
by the seller of the derivative and y is the
risk - free yield for this maturity
02:58
Credit Risk Mitigation



02:58
Netting
Collateralization
Downgrade triggers
Default Correlation



02:58
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, eg a first-to-default CDS and
CDO tranches.
Correlation


Correlation measures the degree to which the
probability of one event happening moves in sync
with the probability of another event happening.
In terms of default correlation,
 Zero correlation means that the default of one
company has no bearing on the default of another
company – the companies are completely
independent of each other.
 Perfect positive correlation means that if one
company defaults, the other will automatically
follow suit.
 Perfect negative correlation means that if one
company defaults the other one will certainly not.
Factors of Default Correlation



macroeconomic environment: good
economy = low number of defaults
same industry or geographic area:
companies can be similarly or inversely
affected by an external event
credit contagion: connections between
companies can cause a ripple effect
Binomial Correlation Measure
(page 508)

02:58
uses a set of rules to change
time-to-default values into either
the value 1 or 0, so that if a
company defaults within time T,
it is assigned the variable 1, and
otherwise it is assigned a 0.
Binomial Correlation continued
Denote Qi(T) as the probability that company i
will default between time zero and time T, and
Pij(T) as the probability that both i and j will
default. The default correlation measure is
 ij (T ) 
02:58
Pij (T )  Qi (T )Q j (T )
[Qi (T )  Qi (T ) 2 ][Q j (T )  Q j (T ) 2 ]
David Li
Statistician who moved over to business
 Worked at a credit derivative market in 1997
and knew about the need to measure default
correlation
 Colleagues in actuarial science working on
solution for death correlation, a function
called the copula
 “Default is like the death of a company, so we
should model this the same way as we model
human life” (Li)
(Whitehouse, Salmon)

Survival Time Correlation


02:58
Define ti as the time to default for company
i and Qi(ti) as the cumulative probability
distribution for ti
If the probability distributions of t1 and t2
were normal, we could assume the joint
probability distribution is bivariate normal.
But they are not.
Gaussian Copula Model



Define a one-to-one correspondence between the
time to default, ti, of company i and a variable xi
by
Qi(ti ) = N(xi ) or xi = N-1[Qi (ti)]
where N is the cumulative normal distribution
function.
This is a “percentile to percentile” transformation.
The p percentile point of the Qi distribution is
transformed to the p percentile point of the xi
distribution. xi has a standard normal distribution
Then we assume that xi are multivariate normal.
The default correlation between ti and tj is
measured as the correlation between xi and
xj .This is referred to as the copula correlation.
02:58
Marginal Probability
Distributions
Copula


Latin word that means “to fasten or fit”
Bridge between marginal distributions and
a joint distribution.
 In the case of death correlation, the
marginal distribution is made up of
probabilities of time until death for one
person, and joint distribution shows the
probability of two people dying in close
succession.
Joint distribution
Marginal distributions
(http://www.mathworks.com/access/helpdesk/help/toolbox/stats/copula_17.gif)
Skylar’s Theorem (1959)

If you have a joint distribution function
along with marginal distribution functions,
then there exists a copula function that
links them; if the marginal distributions are
continuous, then the copula is unique.
(http://www.mathworks.com/access/helpdesk/help/toolbox/stats/copula_14.gif)
Gaussian copula

Assumes that if the marginal probability
distributions are normal, then the joint probability
distribution will also be normal.
C (Qi (ti ), Q j (t j ))  M [ xi , x j ; ij ]  Pij (T )
where C is the copula function,
M is the cumulative bivariate normal
probability distribution function.
Copula Correlation Number




Specifies the shape of the multivariate distribution
 Zero correlation = circular

 Positive or negative correlation
= ellipse
The correlation number is always independent of the
marginals (Hull 507).
Assumptions
 one-to-one relationship between asset correlation
and default correlation based on the definition of
default as an asset falling below a certain value.
 Correlation number is always positive (Li 11-12).
The correlation number is an extremely important factor
in this model because it determines the information you
get out of the model.
(http://www.mathworks.com/access/helpdesk/help/toolbox/stats/gmdistribution_fit1.gif)
Marginal distributions
(default probability distributions from market data)
+
Correlation number
(estimated by asset correlation)
+
Choice of copula
(Gaussian / normal copula)
=
Fully defined multivariate distribution of the
probability of defaulting within time T
Binomial vs Gaussian Copula
Measures
The measures can be calculated from
each other
Pij (T )  M [ xi , x j ; ij ]
so that
ij (T ) 
M [ xi , x j ; ij ]  Qi (T )Q j (T )
[Qi (T )  Qi (T ) 2 ][Q j (T )  Q j (T ) 2 ]
where M is the cumulative bivariate normal
probability distributi on function
02:58
Comparison



02:58
The correlation number depends on the
way it is defined.
Suppose T = 1, Qi(T) = Qj(T) = 0.01, a
value of ij equal to 0.2 corresponds to a
value of ij(T) equal to 0.024.
In general ij(T) < ij and ij(T) is an
increasing function of T
Example of Use of Gaussian Copula
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
02:58
Use of Gaussian Copula continued


02:58
We sample from a multivariate normal
distribution to get the xi
Critical values of xi are
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
Use of Gaussian Copula continued






02:58
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
A One-Factor Model for the
Correlation Structure
xi  ai M  1 ai2 Zi

The correlation between xi and xj is aiaj

The ith company defaults by time T when xi < N-1[Qi(T)]
or
Zi 

N 1[Qi (T )]  ai M
1  ai2
Conditional on M the probability of this is
1

 N Qi (T )  ai M 

Qi (T M )  N 

2
1

a


i


02:58
A One-Factor Model for the
Correlation Structure

Suppose that Qi (T )  Q(T ) for all i and the
common correlation is  ,so that a   for
all i.Then
i
Q(T M )  N (
02:58
N 1 (Q(T ))   M
1 
)
Credit VaR


02:58
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
Calculation from a 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
 so that all ai’s are 
We are X% certain that M is greater than N-1(1−X)
= −N-1(X)
It follows that the VaR is
 N 1 Q(T )   N 1 ( X ) 
V ( X ,T )  N 


1 

02:58
CreditMetrics


02:58
Calculates credit VaR by considering
possible rating transitions
A Gaussian copula model is used to define
the correlation between the ratings
transitions of different companies
Future of the Model



Schönbucher and Schubert devised a
method that allows the default correlation
to be dynamic (2001) (Meneguzzo and Vecchiato 41)
Fit of different copulas, like Student-t
copula (Meneguzzo and Vecchiato 41, Jabbour et al. 32)
Implied correlation from new credit
derivative indices (Jabbour et al. 43)