Introduction to Credit Risk. Credit Risk - Definitions Credit risk - the risk of an economic loss...
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Transcript of Introduction to Credit Risk. Credit Risk - Definitions Credit risk - the risk of an economic loss...
![Page 1: Introduction to Credit Risk. Credit Risk - Definitions Credit risk - the risk of an economic loss from the failure of a counterparty to fulfill its contractual.](https://reader035.fdocuments.net/reader035/viewer/2022062422/56649ec95503460f94bd6dac/html5/thumbnails/1.jpg)
Introduction to Credit Risk
![Page 2: Introduction to Credit Risk. Credit Risk - Definitions Credit risk - the risk of an economic loss from the failure of a counterparty to fulfill its contractual.](https://reader035.fdocuments.net/reader035/viewer/2022062422/56649ec95503460f94bd6dac/html5/thumbnails/2.jpg)
Credit Risk - Definitions
Credit risk - the risk of an economic loss from the failure of
a counterparty to fulfill its contractual obligations.
Credit Exposure (CE) or Exposure at Default (EAD) – the
economic value of the claim on the counter party at time of
default.
Recovery Rate (RR) – the payment ratio given default
Loss Given Default (LGD) – the fractional loss to default,
which is equal to 1 - RR
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Measuring Credit Risk-Distribution of loss
Definitions
bi - a “bernoulli” random variable that take the value of 1 if
default occurs and 0 otherwise, with probability of pi.
CEi - the credit exposure at the time of default.
fi - the recovery rate (RR)
(1-fi) – the loss given default (LDG)
N – number of instruments
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Measuring Credit Risk-Distribution of loss
The distribution of losses due to credit risk can be described
as:
Assuming the only random variable is bi:
)1(ECCLN
1ii ii f
~~b~
)1(CEE[CL]N
1ii ii fp
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Joint Events
The CL distribution depends on the correlation between the
default events.
When the defaults events are uncorrelated:
When the defaults events are perfectly correlated
p(B)p(A)p(A&B)
p(A)p(A)1p(A)A)p(Bp(A&B) |
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Joint Events
For Instance, p(A)=p(B)=1%
In the uncorrelated case:
In the perfectly correlated case:
%.... 01000010010010p(B)p(A)p(A&B)
%.| 1010p(A)A)p(Bp(A&B)
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Joint Events
When <1:
p(B)p(A)p(A&B) BA
p(A)1p(A)σA
p(B)p(A)p(B)]1p(B)[p(A)]1p(A)[p(A&B)
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Joint Events
Consider the pervious example and assume the =0.5:
09949.001.0101.0σ BA
%...... 50005050010010099499050
p(B)p(A)p(B)]1p(B)[p(A)]1p(A)[p(A&B)2
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Credit VaR
Consider a portfolio of $100M composed of 3 bonds A, B and
C with the following default probabilities and CE:
BondCE ($M)Default Prob.
A250.05
B300.10
C450.20
For simplicity, assume: 1. Exposures are constant;
2. The recovery rates are zero; 3. The default events are
independent
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Credit VarDefaultL ($M)P(L)C. Prob.E(L)=Lp(L)2=(L-(EL))2p(L)
None00.6840.68400120.8
A250.03600.72000.9004.97
B300.07600.79602.28021.32
C450.17100.96707.695172.38
A&B550.00400.97100.2206.97
A&C700.00900.98000.63028.99
B&C750.01900.99001.42572.45
A&B&C1000.00101.00000.1007.53
Sum13.25434.7
6840)2.01()1.01()05.01(p(None) .0360)2.01()1.01(05.0p(onlyA) .
0040)2.01(1.005.0B)p(A .&
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Credit VaR
25134520301025050
CEpL)p(LE(CL)N
1iii
n
1iii
....
9207434CL
7434)p(LE(L))(Lp(CL) i2
n
1iii
2
..)(
.
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Credit VaR
With a confidence level of 95% the VaR is $45M
The unexpected loss is:
$31.75M13.2545
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0-25-30-45-55-70-75-100
Loss ($M)
Fre
qu
ency
Unexpected Loss
Expected Loss
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Credit Diversification
A portfolio of loan is less risky than single loans
Consider different alternatives for $100M loan portfolio:
One loan of $100M
10 loans each for $10M
100 loans each for $1M
1,000 loan each for $0.1M
Assume a fixed default probability of 1% for all loans and
are independence across loans
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Credit Diversification
In the first case:M1$10001.0EL
M10$100)01.01(01.0σ
0%
20%
40%
60%
80%
100%
0-10-20-30-40-50-60-70-80-90-100
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Credit Diversification
In the second case:
M1$EL M3σ
0%
20%
40%
60%
80%
100%
0-10-20-30-40-50-60-70-80-90
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Credit Diversification
In the third case:
M1$EL M1σ
0%
20%
40%
60%
80%
100%
0-10-20-30-40-50-60-70-80-90-100
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Credit Diversification
In the last case:
M1$EL M30σ .
0%
20%
40%
60%
80%
100%
0-1-2-3-4-5-6-7-8-9-10
This reflects the Central Limit Theory by which the distribution of
the sum of independent variables tends to normal distribution.
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Credit Diversification
The loans diversification does not effect the expected loss
but decreases the variance.
With N independent defaults events with the same
probability of p, we have:
100pN
100NppLE(CL)
N
1ii
22N
1i
2i
2
N
100)p1(p
N
100)p1(NpL)p1(p(CL)
N
100)p1(p(CL)
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Credit Diversification
In reality, there is some correlation between the defaults
events, which are all affected by the general state of the
economy:
many more defaults occur in a recession than in
expansion.
In this case the distribution will lose its asymmetry more
slowly.
The solution for this is to limit the exposure to a particular
sectors – defaults are more correlated among sectors than
across sectors.
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Historical Default Rates
Cumulative default rate measure the total frequency of
default at any time between the starting date and year T.
According to the S&P experience - from 10,000 BBB rated
firms, there where 36 defaults over one year, and 96 defaults
over 2 years.
Based on the Cumulative default rate one can derives the
marginal default rate, which is the frequency of default during
year T.
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Historical Default Rates
Definitions
MT – The number of issuers rated R that default in year T
NT – The number of issuers rated R that have no default by the
beginning in year T.
dT – The marginal default rate during year T – the proportion
of issuers, relative to the number at the beginning of year T.
ST – The survival rate - The number of issuers rated R that
will not have default by T.
PT – The probability of defaulting in year 2.
CT – The cumulative default rate at the end of year T
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Historical Default Rates
The marginal default rate during year T:
The survival rate:
The probability of defaulting in year 1:
In order to default in year 2, the firm must have survived the
first year and default in the second
T
TT N
Md
)d1(ST
1ttT
11 dp
212 dSp
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Cumulative Default Rates
Thus, the cumulative default rate at end of year 2:
In order to default in year 3, the firm must have survived the
first and the second years and default in year 3.
323 dSp
211212 dSdpCC
32211323 dSdSdpCC
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Default Process
Default
Default
Default
No default
No default
d1
1-d1
d2
d3
1-d3
1-d2
322113
2112
11
dSdSdC
dSdC
dC
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Historical Default Rates
Numerical Example
Consider a BBB rated firm that has default rates of d1=4%,
d2=6% and d3=8%
What are the survival rates at the end of years 1,2 and 3?
What is the probability of defaulting in years 1,2 and 3?
What is the cumulative default rates at the end of years 1,2 and
3?
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Historical Default Rates
Numerical Example
%4dpC 111
%9604.01d1S 11
%76.506.096.0dSp 212
%24.90)06.01(96.0)d1(S)d1)(d1(S 21212
%2.708.09024.0dSp 323
%76.9%76.5%4pCC 212
%96.16%2.7%76.9pCC 323
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Recovery Rates
Credit rating agencies measure recovery rates using the
historical observations of the value of the debt right after default.
The historical observations reveal that the RR depend on:
The state of the economy
The seniority of debtor – the proceeds from liquidation
should be divided according to the absolute priority rule
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Recovery Rates
Credit rating agencies measure recovery rates using the
historical observations of the value of the debt right after default.
The historical observations reveal that the RR depend on:
The state of the economy
The seniority of debtor – the proceeds from liquidation
should be divided according to the absolute priority rule
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Recovery Rates
Priority rule
Secured creditors – up to the extent of secured collateral
Priority creditors – post-bankruptcy creditors and taxes.
General creditors – unsecured creditors before bankruptcy
Shareholders
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Recovery Rates
S&P’s Historical RR for Corporate Debt
Seniority RankingWeighted Average
Senior secured49.32
Senior unsecured47.09
Subordinated32.46
Junior subordinated 35.51
Total40.23