CreditRisk+ Model Tutorial 3.0

Tutorial

CreditRisk+ Model

Melchiori, Mario R.

CreditRisk+ Model

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Tutorial CreditRisk+ Model

Example Spreadsheet-Based Implementation

The purpose of this tutorial is to illustrate the application of the CREDITRISK+ Model to an example portfolio.

For illustrative purposes, we have used as example portfolio such as that used in CREDITRISK+ Technical Document.

However, there is no limit, in principle, to the number of obligors that can be handled by the CREDITRISK+ Model. Increasing

the number of obligors has only a limited impact on the processing time.

Example Portfolio and Static Data

Like in CREDITRISK+ Technical Document, the examples are based on a portfolio consisting of 25 obligors of varying credit

quality and size of exposure.

The examples are given, each based on the same portfolio, as follows:

• All obligors are allocated to a single specific sector. (Example 1.xlsm)

• All obligors are allocated to a single systemic sector. (Example 2.xlsm)

• Each obligor is allocated to only one sector. This example assumes that each obligor is subject to only one

systematic factor, which is responsible for all of the uncertainty of the obligor’s default rate. (Example 3.xlsm)

• Each obligor is apportioned to a number of sectors. This example reflects the situation in which the fortunes of an

obligor are affected by a number of systematic factors. The sectors are non-correlated. (Example 4.xlsm)

• Hold To Maturity Analysis. (Example 7.xlsm)

The examples correspond to original version of CREDITRISK+ (Wilde, 1997). The following enhance are covered:

• Each obligor is apportioned to a number of sectors. This example reflects the situation in which the fortunes of an

obligor are affected by a number of systematic factors. The sectors are correlated (Giese, 2003). (Example 5.xlsm)

• Severity Variation from Specific Factors and Systematic one are modeled (Bürgisser, Kurth, & Wagner, 2001).

(Example 6.xlsm)

• Equalization Severity Variation. (Example 8.xlsm)

• Combining Profit and Loss. (Example 9.xlsm)

• Summary risk measures by sub portfolio.

Excel file names in parentheses.

The examples are installed on the spreadsheet, together with the results generated by the model. For each example, the

inputs to the model have been set to generate the following:

• Percentiles of loss.

• Full loss distribution.

• Expected Loss.

• Unexpected Loss.

and, where appropriate:

• Risk contributions by CREDITRISK+ Model. (Giese, 2003)

• Risk contributions by Haaf and Tasche. (Haaf & Tasche, 2002)

• Risk contributions by Götz Giese. (Giese, 2003)

• Expected Shortfall by Haaf and Tasche. (Haaf & Tasche, 2002)

• Expected Shortfall by Götz Giese. (Giese, 2003)

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These measures are generated for each obligor

and sub portfolio using Recurrence – Panjer - or

fast Fourier Transform – FFT – model (Melchiori,

2004), when required. The steps to reproduce

the results are described in all cases. Each

worksheet is equipped with a sheet named

Control Panel that looks like the table below.

Clear All button erases both input and output

data. Set names button sets the worksheet

ranges of data to be read into the model. To

activate of model implementation, press the

button Activate Model.

The examples use a credit rating scale, which can

be entered in the Excel sheet labeled IN_Default

Rate, to assign default rates and default rate

volatilities to each obligor. This is showed below.

However, they can be assigned without to

employ the credit rating scale. The credit rating

scale and other data in the table are designed for

the purposes of the example only.

NOTE: To activate the model implementation, prior to press the

button Activate Model, you must provide data input in the sheets

labeled IN_Obligors and IN_Default Rate.

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Example 1: All obligors are allocated to a single specific sector

Assumption Input Set up

All obligors are allocated to

a single sector

Sector 1 equals 100% for all

obligors in sheet IN_Obligors.

Zero in other sectors.

The Sector is a Specific one.

Switch on this facility via the

Execute Process Screen.

The exposure amounts are

net of recovery. There is not

Specific Severity Variation.

Each obligor belongs to the

same sub portfolio.

To each obligor corresponds to

one and only one Exposure.

Input 1 to each obligor in

column labeled Portfolio.

There is not Severity

Variation.

Switch off this facility via the


Compute Risk Contributions

to Unexpected Loss No input is required.


to Quantile Loss



Add up by Sub portfolio Switch on this facility via the


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Panjer Mode

The data are entered on Excel sheets

labeled IN_Obligors .

The example computes Risk Contributions

and Expected Shortfall using the model

proposed by Haaf & Tasche. (Haaf & Tasche,

2002)

On activation, the model will show the

Execute Process Screen. This screen is used

to specify the calculation mode, the output

data required. Worksheet ranges of data to

be read into the model are automatically

set.

Execute Process Screen

Press Percentiles button to

change the percentiles values.

The following are the percentiles

set up by default:

Press OK, then Execute button on the Execute Process

Screen to proceed to the next step.

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Summary of Input Data Check

The model implementation has been preset to identify errors in the data read in before the calculation begins. The model

implementation ensures that the data satisfies the following three criteria:

The sector allocation table contains only numeric data.

The decomposition of each obligor to the various sectors adds up to 100%.

A sector must contain at least one allocation entry.

Other errors are identified during the process.

Press the OK button on the Summary

of Input Data Check Screen to proceed

with the calculation.

Output of the process

Loss Distribution

The model displays Loss Distribution and

its graph on the sheet named OUT_Loss

Distribution, using the results generated

from the steps above.

Mean, Unexpected and Percentiles Loss

Like the original CreditRisk+ model, if an

exact percentile does not exit, it is

compute by lineal interpolation. The

model shows on the sheet labeled

OUT_Percentiles, summary statistics of

the portfolio loss distribution.

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Risk Contributions

The model has been preset to output risk contributions for each obligor. The risk contributions calculated by the model are

defined as risk contributions to standard deviation on a chosen percentile of the loss distribution. This is the approach

employed by the original version of CreditRisk+. This risk measure will be present whatever model is chosen, unlike the

following risk measure, which shall be selected when required. The sum of the Risk Contributions computed via this

methodology equals to percentile chosen.

The model has been preset to calculate risk contributions by reference to the 99th percentile loss. This setting can be

altered to a different percentile via the Execute Process Screen.

The model permits to compute the Risk Contributions and Expected Shortfall per obligor using the methodology due to

Haaf and Tasche (Haaf & Tasche, 2002). It can always be chosen, except when the variable Sector covariance is greater than

zero, in this case we will use the Giese´s approach (Giese, 2003). The sum of the Risk Contributions computed via this

methodology equals to cumulated loss distribution greater than the percentile chosen, if it does not exist exactly. The

model computes the Risk Contributions and Expected Shortfall by sector.

Results are showed on OUT_Risk Contributions sheet.

Summary Information by sub portfolio

The information by sub portfolio is

displayed on OUT_RC Portfolio sheet.

FFT Mode

Execute Process screen changes when we choose FFT Mode. FFT 2N parameter is highlights. 2N governs the number of points

of the Loss distributions. A longer vector is generally required for a discrete representation of the loss distribution, since it

will take on large values with non-

zero probability. If there is not

enough room in the discrete vector,

then the tail probabilities will wrap

around and reappear at the

beginning. Therefore, it is crucial to

select a correct value for N. If N

equals to zero, the code calculates

the fit value. Sector Covariance

equals to zero in this example. Other

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inputs remain unchanged with respect to Panjer mode.

Summary of Input Data screen and other outputs are identical to the previous mode.

Example 2: All obligors are allocated to a single systematic sector



a single




The Sector is a Systemic one.

Switch off Sector 1 for specific

risk facility via the Execute

Process Screen.


net of recovery. There is not

Specific Severity Variation.

Each obligor belongs to the

same sub portfolio.






Variation.






to Quantile Loss





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Panjer Mode

Execute Process screen would look like below:

FFT Mode

The explanation of the Example 1, FFT Mode, applies. Execute Process screen would look like below:

Summary of Input Data screen and other outputs are similar to the previous example.

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Example 3: Each obligor is allocated to only one sector of several sectors. The sectors are non-

correlated


Each obligor is allocated to

only one sector of several

sectors

In sheet IN_Obligors, sector

where each obligor is allocated

equals to 100%. Zero in other

sectors.

There is not Specific Sector. Switch on this facility via the



net of recovery. Each obligor

belongs to the same sub

portfolio.






Variation.




to Unexpected Loss. No input is required.


to Quantile Loss.



Add up by Sub portfolio. Switch on this facility via the


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Panjer Mode


Summary of Input Data screen looks like below:

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FFT Mode


Summary of Input Data screen should look like in previous mode.

Example 4: Each obligor is apportioned to a number of sectors. The sectors are non-correlated



only one sector of several

sectors

In sheet IN_Obligors, each

obligor is apportioned to a

number of sectors. The

decomposition of each obligor to

the various sectors must add up

to 100%.

The Sector 1 is a Specific

one.




net of recovery. All obligors

belong to the same sub

portfolio.





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Variation.






to Quantile Loss





Panjer Mode



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FFT Mode



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Example 5: Each obligor is apportioned to a number of sectors. The sectors are correlated



several sectors.




decomposition of each obligor to

the various sectors must add up

to 100%.


one.



The Sector are correlated Input Sector Covariance equals

to 0.15.


net of recovery. Obligors

belong to different sub

portfolios.



Input a value for each obligor

in column labeled Portfolio, to

identify the portfolio where

obligor belongs.


Variation.






to Quantile Loss





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FFT Mode

In this example, only the FFT Mode is applied. Sector Covariance value must be lesser than a determined value, in this

example, that determined value equals to 0.25. Check out the Giese´s paper for more details. During the process, such

errors are identified.


Summary of Input Data screen should look like prior example.

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Example 6: Severity Variation from Specific Factors and Systematic one


All obligors must be

allocated to a single




The Sector must be a

Systemic one.



Process Screen.


net of recovery. Specific

Severity Variation is

modeled. All obligors belong

to the same sub portfolio.



Input a value for each obligor

in column labeled Portfolio, to

identify the portfolio where

obligor belongs.

Systematic Severity

Variation is modeled.


Execute Process Screen. Click

on Options to set the

parameters of Severity

Variation Process.

Options for Incorporating

Severity Variation. Put Systemic volatility = 0.20.


Severity Variation. Set Manual Input option.


Severity Variation.

Set Data Expand Mode to

“Normal” and Specific

Volatility = 0.15. This model

the severity density function

by discretizing a normal

distribution with mean equals

to, and standard deviation

equals to 15%.

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Panjer Mode


Click on Options to setup the parameters of Incorporating Severity Variation process:

For this implementation,

we have chosen to model

the obligor-specific

severity density function

by discretizing a normal

distribution with mean

equals to, and standard

deviation equals to 15% of,

non-stochastic exposure

used in previous example.

Systemic severity

variations are assumed

lognormal distributed

with mean parameter

equals to one and

standard deviation equals

to 0.20.

The implementation supports two modes of data expansion, “Normal” and “Lognormal”, other distributions to model the

severity variations can be implemented without any problem setting into expansion mode "None" and manually input data

into the worksheet “IN_Obligors”.

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FFT Mode

The explanation of the Example 6, Panjer Mode, applies. Execute Process screen would look like below:


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Example 7: Hold to Maturity Analysis



a single.




The Sector must be a

Systemic one.



Process Screen.


net of recovery. Obligors

belong to different sub

portfolios.

To each obligor has several

probably exposure with its

correspond Default

Probability. Input a value for

each obligor in column labeled

Portfolio, to identify the

portfolio where obligor

belongs.

There is not Systematic

Severity Variation.




to Unexpected Loss. No input is required.


to Quantile Loss.



Modes of Execution

Execute Process screen and Summary of Input Data screen are the same as in Example 2, for the modes of Panjer, FFT, and

Giese.

This Example illustrates the use of the model for analyzing the portfolio over its hold to maturity time horizon. To illustrate

a multi - year time horizon, the data used in this example has been extended as follows:

The obligor details used in the other examples have been extended to show the exposures rolling off over a

period of up to three years. Before use, the data is rearranged in the IN_Obligors.

The static data (default rates and default rate standard deviations) used in the other examples have been

extended over three years. The one-year default rates are the same as in the other examples, but this example

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also introduces a term structure of default rates by specifying marginal probabilities of default in years 2 and 3 of

the portfolio.

The model outputs are the same as the other example, but in this example, the model calculates a risk contribution for each

obligor for each year in which the obligor has an exposure outstanding.

Example 8: Equalization Severity Variation



several sectors.




decomposition of each obligor

to the various sectors must

add up to 100%.


several sectors of collateral.



number of collaterals. The

decomposition of each obligor

to the various collaterals must

add up to 100%.


one.




net of recovery. Specific


modeled. All obligors belong

to the same sub portfolio.

To each obligor has several

probably exposure with its

correspond Default

Probability. Input a value for

each obligor in column labeled

Portfolio, to identify the

portfolio where obligor

belongs.


modeled.


Execute Process Screen. Click

on Options to set the

parameters of Severity

Variation Process.

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Severity Variation. Put Systemic volatility = 0.20.


Severity Variation. Set Equalization Input option.


Severity Variation.

Set Calculate Mode to “Sys.

Default” and Specific Volatility

= 0.15.


Severity Variation.

The Collateral 1 is a Specific

one. Check this option.


to Quantile Loss





Panjer Mode


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Click on Options to setup the parameters of Incorporating Severity Variation process:

Equalization input:

First, we calculate the unexpected loss of the portfolio, taking into account the segment structure. Then we estimate single

systematic default and severity volatilities and such that the unexpected loss of the portfolio, computed with the single

segment formula matches the unexpected loss computed before. Finally, the loss distribution is calculated as in the single

segment situation, where the systematic default behavior is gamma and systematic severity variation is lognormally

distributed. There are three possibilities to estimate the implied overall systematic volatilities in the number of defaults

and in the severities :

Alternatives modes of calculating the Equalization of Incorporating Severity Variation:

Systematic default: We estimate (systematic default volatility) by equating the unexpected loss formulas of

the single and multisegment situation by setting (systematic severity volatility) and (specific severity

volatility) equals to zero. This mode permits to compute both Risk Contributions and Expected Shortfall.

Systematic severity: We focus on severity systematic risk and determine by equating the unexpected Loss

formulas of the single and multisegment situation by setting = 0 and = 0. This mode does not permit to

compute Risk Contributions and Expected Shortfall.

Specific severity: We focus on severity specific risk and determine by equating the unexpected loss formulas

of the single and multisegment situation by setting = 0 and = 0. This mode permits to compute Risk

Contributions and Expected Shortfall.

Note: The example permits to model both Correlation Sectors and Correlation Collaterals via the average correlation

approach. In this environment, the specific sector and specific collateral are independent.

Summary of Input Data screen looks like the

following:

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FFT Mode



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Example 9: Combining Profit and Loss

This example combines the rating migration concept of CreditMetrics with the approach of CreditRisk+, incorporating the

effect of ratings changes in CreditRisk+.

It allows integrating the rating migration concept into CreditRisk+, modeling the possible profits and losses due to rating

changes in the same way as the default events are modeled.

Migration rates must be assigned separately to a subportfolio of profits due to upgrades and a subportfolio of losses due to

downgrades.

In the example, twenty, equal and independent obligors of bonds BBB are chosen, for each it is necessary to calculate the

total value of each bond for different rating categories at the end of the period.

Determine the possible value changes caused by individual up/downgrades, and assign the migration rates (four rating of

profits, and four rating of losses). The chart below shows the first three steps of the approach:

The step four consists in evaluates the distributions of profits and losses separately. The absolute amounts of profits and

losses are used as net exposures and the default rate corresponds to the migration rate. It is not adequate to use the

CreditRisk+ concepts of default rate volatility and sector analysis because of the assumption of independent obligors.

Finally, the convolution process obtains the total loss distribution and the Risk contribution of each obligor.

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a single sector




Migration rates must be

assigned separately to a

subportfolio of profits and a

subportfolio of losses.

Each obligor has several

exposures with associated

default probability. Input a

value for each obligor in

column labeled Portfolio, to

identify the profits (upgrades)

and losses (downgrades).

Combining Profit and Loss

Migrations is modeled.



Panjer Mode



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FFT Mode



Note:

The Giese Model always applies.

Risk Contributions and Expected Shortfall always apply even though the severity specific variation is used. It permits to

model stochastically the Loss Given Default without to limit the output, i.e. it is possible to calculate Risk Contributions and

Expected Shortfall in all its dimensions and to allocate each obligor to one or more sectors and to take into account the

correlation between several sectors. To model stochastic Loss Given Default goes the following steps:

References

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Incorporating Severity Variations into Credit Risk: Bürgisser, P., Kurth, A., & Wagner, A. (2001). Downloded 17/05/2009, from

http://math-www.uni-paderborn.de/agpb/work/CRQ.pdf

Enhancing CreditRisk+: Giese, G. (April 2003). Downloded 17/05/2009, from: http://www.defaultrisk.com/pp_model162.htm

Calculating Value-at-Risk Contributions in CreditRisk+.: Haaf, H., & Tasche, D. (28/02/2002). Downloded 17/05/2009, from:

http://www.defaultrisk.com/pp_model_26.htm

CreditRisk+ by FFT: Melchiori, M. (July 2004).. Downloded 17/05/2009, from Social Science Research Network

http://ssrn.com/abstract=1122844

Good Migrations: Rolfes,Bernd, Broeker, Frank (November 1998). Downloded 17/05/2009, from

http://www.gloriamundi.org/ShowTracking.asp?ResourceID=453055008

CreditRisk+ Technical Document: Wilde, T. (October 1997). Downloded 17/05/2009, from

http://www.defaultrisk.com/pp_model_21.htm

CreditRisk+ Model Tutorial 3.0

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