Transcript שקופית 1 - Bar-Ilan University

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 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
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
Measuring Credit Risk-Distribution of loss
The distribution of losses due to credit risk can be described
as:
N
~ ~
~
CL   bi  CE i  (1  f i )
i 1
Assuming the only random variable is bi:
N
E[CL]  pi  CE i  (1  f i )
i 1
Joint Events
The CL distribution depends on the correlation between the
default events.
When the defaults events are uncorrelated:
p(A&B) p(A) p(B)
When the defaults events are perfectly correlated
p(A&B) p(B| A)  p(A)  1 p(A)  p(A)
Joint Events
For Instance, p(A)=p(B)=1%
In the uncorrelated case:
p(A&B) p(A) p(B)  0.01 0.01  0.0001 0.01%
In the perfectly correlated case:
p(A&B) p(B| A)  p(A)  0.01  1%
Joint Events
When r <1:
p(A&B) r  A  B  p(A) p(B)
σA  p(A) 1  p(A)
p(A&B) r  p(A)[1 p(A)] p(B)[1 p(B)] p(A) p(B)
Joint Events
Consider the pervious example and assume the r=0.5:
σA  B  0.01 1  0.01  0.09949
p(A&B) r  p(A)[1  p(A)] p(B)[1  p(B)] p(A) p(B)
 0.5  0.099499  0.01 0.01  0.00505 0.5%
2
Credit VaR
Consider a portfolio of $100M composed of 3 bonds A, B and
C with the following default probabilities and CE:
Bond
CE ($M)
Default
Prob.
A
25
0.05
B
30
0.10
C
45
0.20
For simplicity, assume: 1. Exposures are constant;
2. The recovery rates are zero; 3. The default events are
independent
Credit Var
L ($M)
P(L)
None
0
0.684
0.6840
0
120.8
A
25
0.0360
0.7200
0.900
4.97
B
30
0.0760
0.7960
2.280
21.32
C
45
0.1710
0.9670
7.695
172.38
A&B
55
0.0040
0.9710
0.220
6.97
A&C
70
0.0090
0.9800
0.630
28.99
B&C
75
0.0190
0.9900
1.425
72.45
A&B&C
100
0.0010
1.0000
0.100
7.53
13.25
434.7
Sum
C. Prob. E(L)=Lp(L)
2=(L-(EL))2p(L)
Default
p(None) (1  0.05)  (1  0.1)  (1  0.2)  0.684
p(onlyA) 0.05 (1  0.1)  (1  0.2)  0.036
p(A& B)  0.05 0.1 (1  0.2)  0.004
Credit VaR
n
N
i 1
i 1
E(CL)   p(Li )  Li  p i  CE i 
0.05 25  0.1 30  0.2  45  13.25
n
2(CL)   pi  (Li  E(L))2 p(Li )  434.7
i 1
( CL )  434.7  20.9
Credit VaR
Expected Loss
0.8
0.7
Unexpected
Loss
Frequency
0.6
0.5
0.4
0.3
0.2
0.1
0
-100
-75
-70
-55
-45
-30
-25
0
Loss ($M)
With a confidence level of 95% the VaR is $45M
The unexpected loss is:
45  13.25  $31.75M
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
Credit Diversification
In the first case:
EL  0.01 100  $1M
σ  0.01(1  0.01) 100  $10M
100%
80%
60%
40%
20%
0%
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Credit Diversification
In the second case:
EL  $1M
σ  3M
100%
80%
60%
40%
20%
0%
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Credit Diversification
In the third case:
EL  $1M
σ  1M
100%
80%
60%
40%
20%
0%
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Credit Diversification
In the last case:
EL  $1M
σ  0.3M
100%
80%
60%
40%
20%
0%
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
This reflects the Central Limit Theory by which the distribution of
the sum of independent variables tends to normal distribution.
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:
N
E(CL)   pLi  Np 
i 1
100
 p 100
N
2
100
 100
2(CL)   p(1  p)L2i  Np(1  p)  

p
(
1

p
)


N
N


i 1
N
(CL)  p(1  p) 
100
N
2
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.
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.
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
Historical Default Rates
The marginal default rate during year T:
The survival rate:
MT
dT 
NT
T
ST   (1  d t )
t 1
The probability of defaulting in year 1:
p1  d1
In order to default in year 2, the firm must have survived the first
year and default in the second
p2  S1d 2
Cumulative Default Rates
Thus, the cumulative default rate at end of year 2:
C2  C1  p2  d1  S1d 2
In order to default in year 3, the firm must have survived the first
and the second years and default in year 3.
p3  S2d3
C3  C2  p3  d1  S1d 2  S2d3
Default Process
Default
d1
Default
1-d1
d2
No default
1-d2
Default
d3
No default
1-d3
C1  d1
C2  d1  S1d 2
C3  d1  S1d 2  S2d 3
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?
Historical Default Rates
Numerical Example
C1  p1  d1  4%
S1  1  d1  1  0.04  96%
p2  S1d 2  0.96 0.06  5.76%
C2  C1  p2  4%  5.76%  9.76%
S2  (1  d1 )(1  d 2 )  S1 (1  d 2 )  0.96 (1  0.06)  90.24%
p3  S2d3  0.9024 0.08  7.2%
C3  C2  p3  9.76%  7.2%  16.96%
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
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
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
Recovery Rates
S&P’s Historical RR for Corporate Debt
Seniority Ranking
Weighted Average
Senior secured
49.32
Senior unsecured
47.09
Subordinated
32.46
Junior subordinated
35.51
Total
40.23