CreditRisk+ is a method for quantifying the probability of loss distributions and risk measures like Value at Risk for loan portfolios. It generates loss distributions based on probability generating functions and models defaults as independent Poisson processes. The document outlines the theoretical framework and assumptions of CreditRisk+, including how to model exposure bands, default correlations, and aggregate loans from multiple borrowers.

1. Credit Risk Management through
CreditRisk+
Dr Howard Haughton
Holistic Risk Solutions Limited

2. What is CreditRisk+?
CreditRisk+ is a method for quantifying:
The probability loss distribution for a portfolio of
loans. For example, it will be able to answer the question as
to what the probability would be corresponding to a certain
level of credit loss in the portfolio
A summary risk measure such as Value At Risk (VAR)
or Expected Shortfall (ES). Hence it can provide
information indicating the minimum loss in the portfolio that
will coincide with a given confidence level as well as the
average of those losses above that confidence level.
Quantify the contribution to either the VAR and/or ES
on a borrower, sector or portfolio basis. Hence it will be
able to provide insight as to which borrowers/loans, sectors
and portfolios are the most/least riskiest.

3. CreditRisk+ is not
A method for determining credit ratings
A method for determining the risk-adjusted price of
loans
A method for determining default probabilities/rating
transitions

4. Theoretical framework
some mathematics
CreditRisk+ generates probability loss distributions based
on the theory of probability generating functions (PGF).
In the above the powers of s denote the various states
that the random variable X might attain. The values of p
(for each state) denote the probability of the random
variable being in that state.
( ) ( )
( ) ( ) 11,0 0
1
1
0
0
0
==
++=== ∑
∞
=
XX
n
n
X
k
k
kX
FpF
where
spspspsEspsF L

5. Default Events
Assume that the event of default for a single
borrower can be viewed as a discrete event (i.e. it
either occurs or does not occur over a finite time
frame such as 1-year).
Assume that the probability of default is p and no
default is 1-p=q. There are therefore 2 states of
the world for the random variable denoting the
event of default: either default occurs (call this
state 1) or it does not occur (call this state 0).

6. Default Event continued
The probability generating function for this type of
default event would be:
In CreditRisk+ notation:
In the above, A denotes that the random defaults are
associated with a borrower A.
( ) ( ) ( ) psqpsppsspsEsF X
X +=+−=+−== 11 10
( ) ( )111 −+=+−= spsppsF AAAA

7. Some more mathematics
Independence of default events implies that:
For convenience:
( ) ( ) ( )( )∏∏ −+==
A
A
A
A spsFsF 11
( )( ) ( )( )
( )
( )
( )
( )
∑
∑
∑
=
=
∑
=
⇒
−
≈
−+=
−
−
A
A
s
sp
A
A
A
A
p
where
eesF
sp
spsF
A
A
μ
μ 1
1
1
11loglog

8. Taylor series representation
This implies that the probability distribution for the
number of defaults is:
This follows from observing the coefficient of the z
term in the expression.
( ) ( ) n
n
n
uss
z
n
eeeesF ∑
∞
=
−−−
===
0
1
!
μμμμ
( )
!n
edefaultsnp
n
μμ−
=

9. Common size losses
In CreditRisk+ losses are calculated with respect to
integer multiples of a common size loss.
AAAA
A
A
A
LvLL
LlossCommon
LossExpected
pyprobabilitDefault
LExposure
AObligor
ελ
λ
*,* ==
=

10. Common size continued
All exposures are scaled so that they can be expressed
as an integer multiple of the common loss L. The
common loss can be determined as follows:
( )
A
L
L
Intv
RELLthenLRELif
Biggest
RoundL
EL
RoundREL
EL
A
A
pp
p
p
A
AP
∀⎟
⎠
⎞
⎜
⎝
⎛
−−=
=>
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
= ∑
,
0,
100
0,
1000
λ

11. Common loss example
A Pa La lambda a Va Exp band
1 30.00% 358,475 107,542.50 2 2
2 30.00% 1,089,819 326,945.70 6 6
3 10.00% 1,799,710 179,971.00 9 9
4 15.00% 1,933,116 289,967.40 10 10
5 15.00% 2,317,327 347,599.05 12 12
6 15.00% 2,410,929 361,639.35 12 14
7 30.00% 2,652,184 795,655.20 14 15
8 15.00% 2,957,685 443,652.75 15 16
9 5.00% 3,137,989 156,899.45 16 24
10 5.00% 3,204,044 160,202.20 16 25
11 1.50% 4,727,724 70,915.86 24 27
12 5.00% 4,830,517 241,525.85 24 28
13 5.00% 4,912,097 245,604.85 25 29
14 30.00% 4,928,989 1,478,696.70 25 32
15 10.00% 5,042,312 504,231.20 25 33
16 7.50% 5,320,364 399,027.30 27 39
17 5.00% 5,435,457 271,772.85 27 77
18 3.00% 5,517,586 165,527.58 28 100
19 7.50% 5,764,596 432,344.70 29
20 3.00% 5,847,845 175,435.35 29
21 30.00% 6,466,533 1,939,959.90 32
22 30.00% 6,480,322 1,944,096.60 33
23 1.60% 7,727,651 123,642.42 39
24 10.00% 15,410,906 1,541,090.60 77
25 7.50% 20,238,895 1,517,917.13 100

12. Example continued
389,2020,
100
222,140,
1000
48.863,221,14
=⎟
⎠
⎞
⎜
⎝
⎛
=
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
== ∑
Biggest
RoundL
EL
RoundREL
EL
p
p
A
AP λ

13. Exposure bands
∑
∑
=
=
==
==
⇒
=
m
j
j
vvA A
A
j
j
j
jjj
j
j
j
jA
vv
v
defaultsnumberExpected
jbandlossExpected
vjbandlossCommon
1
:
134.3
*
μμ
εε
μ
με
μ
ε

14. Exposure bands continued
band number band eta j mu j
1 2 0.531 0.266
2 6 1.615 0.269
3 9 0.889 0.099
4 10 1.433 0.143
5 12 3.504 0.292
6 14 3.931 0.281
7 15 2.192 0.146
8 16 1.567 0.098
9 24 1.544 0.064
10 25 11.011 0.440
11 27 3.314 0.123
12 28 0.818 0.029
13 29 3.003 0.104
14 32 9.585 0.300
15 33 9.606 0.291
16 39 0.611 0.016
17 77 7.614 0.099
18 100 7.500 0.075

15. Default loss distribution
( ) ( )∑
∞
=
==
0
*
n
n
sLnlossesaggregatepsG
( ) ( )
( ) ( )
( ) ( )( )
( )( )
( )
∑
∑∑
∏
∑∑
∏
=
==
−
+−
=
+−
+−
∞
=
−∞
=
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
==
==
∑∑
==
⇒
===
=
==
m
j j
j
v
m
j j
j
m
j
v
j
sp
sm
j
s
s
n
nv
n
j
n
nv
j
m
j
j
v
s
v
s
sP
where
sPFeeesG
es
n
e
sdefaultsnpsG
sGsG
jj
m
j
jv
j
m
j
jjv
jj
jv
jjj
j
j
1
11
1
1
00
1
11
!
ε
ε
μ
μ
μ
μ
μμ
μμ
μμ
μ

16. Loss distribution expression
Differentiate the Taylor series expansion of G to
determine probability expression, see Wilson [W]:
( ) ( )
( ) ( )( )00
!
1
*
1
0
:
0
PFeGA
A
n
A
A
ds
sGd
n
Lnoflossp
m
j j
j
j
j
v
nvj
vn
j
n
nsn
n
=
∑
==
=
⇒
==
=
−
≤
−
=
∑
ε
ε

17. Alternative representation
( )
( ) ( )[ ] ( )
( )
( ) ( ) ( ) 2
1
2
,min
1
22
2
10deg
1
1
1
σμσμ
μσσ
μσ
−
=
=
+===
−−+
+
=
∑
∑
S
jv
Aj
vm
j
jSS
S
AandpandPmwhere
jvAjv
v
vA
A
Note the above representation (based on Panjer [P]) is useful as it
explicitly shows the relationship between loss probabilities and the
mean and variance of portfolio losses. Other means of
implementing the PGF are due to Melchiori [M] using FFT methods.
It is assumed in the above that there is only one sector. However
as shown by Kurth [K], Tashe [T] and others the formalism can be
used for a multi-sector case with the corresponding portfolio
expected loss and volatility of losses being calculated and
substituted.

18. Assumptions revisited
The exposures are expressed net of collateral value (i.e. recovery)
The recovery values are constant
The exposures can be approximated as integer multiples of a fixed unit
of loss. This is a necessary assumption for the discrete probability
model used
The distribution of the number of defaults can be approximated via the
Poisson distribution (valid for small default probabilities)
In the case of multi-sectors it is assumed that sectors are independent.
It is shown by Kurth et al [K] how the above assumptions can be
relaxed to incorporate correlation between sectors and non-constant
severity of loss.

19. Average correlation
Default correlations can be modeled in a number of
ways:
Equity
Asset
Credit spreads
Implied based on model
The use of an average correlation (i.e. all pair-wise
credits have the same default correlation) makes
calibration/modeling easier.

20. Correlation continued
Kurth et al., [K] show how an average correlation can
be derived in their extension of CreditRisk+
incorporating default correlations and severity
variations. Whilst useful these estimates suffer from
the assumption that correlations are constant.
Giese [G] also shows how dependent risk factors can
be modeled in the CreditRisk+ framework.

21. Some research finding about
correlation
1. Default correlations are an increasing function of time
(see Standard & Poor’s [S&P], Zhou [Z])
2. Default correlations increase in recessionary periods
and decrease in boom times (see Gersbach and
Lipponer [GL])
3. Default correlations are, on the whole, positive (see
Standard & Poor’s [S&P])
4. The higher the default probabilities the higher the
default correlations (above references and others)
5. Default correlations less relevant for higher grade
credits than lower ones

22. Conditional correlation
Average default correlation should be conditional on the state of the
economy:
( )
( ) 01
1
=
+=
−
−
tt
tt
t
Ave
E
where
XF
ε
εβρ
In the above, F is any suitable function for mapping the product of the
sequence of macro-economic variables (along with the estimated
regression coefficients) into a forecasted correlation value.

23. Conditional correlation
continued
Note that if a Probit model is used (e.g. the Normal
distribution) then:
Note that, more generally, rather than just assuming an
expected value any percentile for the forecasted correlation
could be derived via sampling of the distribution associated
with the error term.
Given the above it would be easy to incorporate scenario
analysis/stress testing into the credit modeling framework.
For example, a 99% “worst case” correlation could be
obtained on the basis of the generated values.
( )( ) βρ ˆ
1
1
−
−
=Φ t
t
Ave XE

24. Estimation of default probabilities
1. Actuarial approach via default history for each credit
rating category (based on in-house data)
2. Inferred from delinquency probabilities for rating
classes in the absence of sufficient history of defaults.
Note some conjectured relationship between
delinquency and default (possibly for each rating
class) must be used here.
3. Implied based on in-house rating model e.g.
Logit/Probit modeling approaches.
4. Note default probabilities from (3) can be made
conditional on the state of the economy.

25. Loan aggregation
The original CreditRisk+ formalism does not specify
how to deal with cases where a borrower has more
than 1 loan in a portfolio. A simple aggregation rule
can however be applied to determine a single
aggregate loan from a collection of loans (possibly
multi-currency).

26. Aggregation example
Sectors Split Total Collateral Split Total
Exposure Sector 1 Sector 2 Sector 3 Sector N Sector Sector 1 Sector 2 Sector 3 Sector N Collateral
358,475 50.0% 30.0% 10.0% 10.0% 100.00% 10.0% 20.0% 30.0% 40.0% 100.0%
358,475 50.0% 30.0% 10.0% 10.0% 100.00% 20.0% 10.0% 5.0% 65.0% 100.0%
The above shows 2 partial loan details for a borrower. To aggregate these loans we
assume that:
1. Sector contributions will remain the same for the aggregate loan
2. The total exposure is the sum of the exposure for both loans
3. PD and STD estimates are the same for both loans and aggregate loan
4. Multiply collateral splits by respective exposure amount, sum the result for each
sector and divide individual sector amount by total aggregate exposure.

27. 35,847.50 71,695.00 107,542.50 143,390.00
71,695.00 35,847.50 17,923.75 233,008.75
107,542.50 107,542.50 125,466.25 376,398.75
15.00% 15.00% 17.50% 52.50%
Sectors Split Total Collateral Split Total
Exposure Sector 1 Sector 2 Sector 3 Sector N Sector Sector 1 Sector 2 Sector 3 Sector N Collateral
716,950 50.0% 30.0% 10.0% 10.0% 100.00% 15.0% 15.0% 17.5% 52.5% 100.0%
Sectors Split Total Collateral Split Total
Exposure Sector 1 Sector 2 Sector 3 Sector N Sector Sector 1 Sector 2 Sector 3 Sector N Collateral
358,475 50.0% 30.0% 10.0% 10.0% 100.00% 10.0% 20.0% 30.0% 40.0% 100.0%
358,475 50.0% 30.0% 10.0% 10.0% 100.00% 20.0% 10.0% 5.0% 65.0% 100.0%
Note that for cases where a borrower has multiple currency loans then all loan
amounts must be converted to a chosen base currency.

28. Risk Measures
Value At Risk (VAR)
The VAR can be defined as the maximum loss credit loss, based
on a given confidence level, that a portfolio might incur over the
year. One can calculate the VAR by simply reading of the loss
probability distribution. For example once loss probabilities are
calculated a VAR at a 99% confidence level can be calculated by
observing the losses (ordered by size) that correspond to the
cumulative probability >=99%. Another way of stating this is to
say the VAR (given a confidence level of 99% say) is the
smallest loss such that the probability of exceeding this loss is
greater than or equal to 1-99%=1%.
Expected Shortfall (ES)
The ES can be viewed as the average of the losses conditional
on those losses being greater than or equal to the VAR.

29. Risk contributions
It would be advantageous to be able to ascertain the
contributions that each borrower makes to the VAR
and ES.
Knowing the contributions provides a measure as to
the relative riskiness of one borrower over another.

33. Conclusions
The actuarial approach as popularized via the CreditRisk+
method is widely used for the modeling of credit risk
This type of model is more practically applied than many
other portfolio methods to developing economies as
significantly less parameters need to be
calibrated/estimated much of which can’t be observed in
any event
The method easily lends itself to be combined with factor
models allowing for incorporation of credit-cycle factor
considerations into the risk process.
Stress testing is easily accommodated in the framework.
Values can provide checks on the adequacy of provisions
and capital and compared to regulatory standards.

34. References
[S&P] -Arnaud de Servigny & Olivier Renault (2002): Standard & Poor’s
presentation of Default Correlation: Empirical Evidence, November.
[K] - Burgisser, A. Kurth, and A. Wagner. Incorporating severity variations
into credit risk. Journal of Risk; 3(4):5-31, 2001
[C1] -Crouhy, M, D Galai and R Mark (2001): “Prototype Risk Rating
System”, Journal of Banking and Finance, January, pp 47-95
[C2] - Crouhy, M, D Galai and R Mark (2000): “A Comparative Analysis of
Current Credit Risk Models”, Journal of Banking and Finance, January, pp
57-117
[G]- Giese, G. 2004. Dependent Risk Factors. In: CreditRisk+ in the Banking
Industry. Berlin, Heidelberg: Springer Verlag. 153–165.
[GL]- Gersbach, H & Lipponer, A (2000): “The correlation effect”. University
of Heidelberg working paper, October
[M]- Mario Melchiori (2004): CreditRisk+ by Fast Fourier Transform,
Universidad Nacional del Litoral

35. References continued
[T]- Tasche, D. Expected shortfall and beyond. Journal of Banking
& Finance, 26(7):1519-1533, 2002.
[P]- Panjer, H. Recursive evaluation of a family of compound
distributions. ASTIN Bulletin, 12:22-26, 1981.
[W]- Wilson, T. CreditRisk+: A credit risk management framework.
London 1997, Credit Suisse Financial Products.
[Z]- Zhou, C (2001): “An Analysis of Default Correlations and
Multiple Defaults”, The Review of Financial Studies, Summer, pp
555-576.