CA2326925A1 - Credit risk estimation system and method - Google Patents

Abstract

A method creates an adjusted credit risk transition matrix and related matrices. The method uses the steps of obtaining a real probability measure (P) transition matrix, and adjusting the (P) transition matrix to be consistent with default probabilities under martingale measure (Q) by making the last column (default probabilities) in the (P) transition matrix consistent with default probabilities calculated under (Q). The method further scales the other entries in the transition matrix (P) to compensate for the adjustment while retaining the relative weights among non-default classes from the (P) transition matrix in the adjusted transition matrix. A credit risk model for the analysis of complex credit instrument implements the method. The adjusted transition matrix may be for use in analysing complex credit instruments. The model may be implemented in computer software on a computer readable medium. The adjusted transition matrix may be used to price credit swaps or value credit instruments. The model may have a simulation engine to generate valuation scenarios based on the adjusted transition matrix. The simulation engine may generate scenarios based on a Monte Carlo method. The model may be extended for the risk analysis of complex portfolios of credit instruments with stochastic interest and foreign exchange rates. The model may have a Monte Carlo simulation engine to generate valuation scenarios based on the joint distribution of interest rates, spreads, foreign exchange rates and obligors' credit migrations consistent with obtained or adjusted (P) transition matrix.

Description

CREDIT RISK ESTIMATION SYSTEM AND METHOD
FIELD OF THE INVENTION
The field of the invention relates generally to systems and methods for estimating credit risk associated with credit instruments and portfolios of credit instruments.
BACKGROUND OF THE INVENTION
Credit risk models are used to estimate the risk associated with credit instruments and portfolios of credit instruments, such as bonds and loans. Various models have evolved overtime. Many run simulations to generate possible valuation scenarios. Simulations are typically generated using a Monte Carlo method or variant.
Credit risk models typically generate scenarios using probability distributions based on historical data. These probability distributions may be obtained in the form of a transition matrix from various sources, for example Standard & Poors~ and Moody'sTM. Such probability distributions are often referred to as (P), a real measure probability distribution.
In credit risk models, transition matrices under (P) probability measure, or (P) transition matrices, are used to predict transition, or migration, from one credit class to another, "credit migration", over a given time period. A set of probabilities of migration, or transition, for different credit classes forms a transition matrix.

Models are typically implemented in computer software as part of a credit risk system using compatible hardware.
Alternative matrices, models, systems and methods for estimating credit risk and credit migration associated with credit instruments are desirable.
SUMMARY OF THE INVENTION
In a first aspect the invention provides a method of creating an adjusted (P) transition matrix of probabilities of credit migration for different credit classes. The method uses the steps of ~ obtaining a transition matrix under probability measure (P) ~ adjusting the (P) transition matrix to be consistent with default probabilities under martingale measure (Q) computed, for instance, from bond market prices by making a column of default probabilities in the (P) transition matrix consistent with default probabilities calculated under (Q) ~ scaling the other entries in the (P) transition matrix to compensate for the adjustment while retaining the relative weights among non-default classes from the (P) transition matrix in the adjusted transition matrix.
The method may adjust the (P) transition matrix in accordance with the following:
R = f R, ,. . ., Rx } - are all credit classes RK - corresponds to default state T p = ~ ~ p;, ~ (to, t, ) ~ ~ - adjusted (P) transition matrix for the time interval ~to, t, p;,~(to,t, ) - transition probability from credit class R; to credit class R,~

N( z" + Co ), j = K
Pa.i(to~t~ ) = V pLn~, j < K (i ~ K) Px,i (to ~ ti ) = 0 if j < K and px,x (to , t~ ) =1 Co =Co(G) = 1 ~,(h'; -Z;) G I LEG
~'~'t = N ~ (Pr,x ) Zt=N ~(qr) _ 1-N(z; +Co) v1 1- Pr,x P~,x - obtained (P) probability of default for the credit class R; (from the obtained (P) transition matrix) qt - (Q) probability of default for the credit class R;
G - a set of credit classes (can be the whole set R = {R,,...,Rx} or some subset I G I - denotes the number of elements in G ; if G is empty, put Co = 0 N~ ~ - standard normal cumulative density function.
Alternatively, the method may adjust the (P) transition matrix in accordance with the following:
T p = II P; (to ~ t~ ) II - adjusted (P) transition matrix for the time interval ~to, t, ~ for obligor n p"i(to,t, ) - obligor's n transition probability to migrate from credit class R;
to credit class R~
" N(zn + Co ), j = K
Pr,i (to ~ t~ ) _ _n (i ~ K) urPr,.i ~ j < K

PK.i (t0 ~ tr ) = 0 if j < K and pK.x (to,t~ ) =1 r jrs-as s Cn = Cn t t _ ro 0 0~ 0' 1~ n var yr z~ = N_nR'n 1-N(z; +Co) = 1 n - Pr,x p;"x - obtained (P) probability of default for the obligor n over interval ~to,t, ~ if it is in the credit class R; at time to qt - (Q) probability of default for the obligor n over interval ~to,t,~ if it is in the credit class R; at time to r - instantaneous risk-free interest rate at time t y, - obligor n stock log-return over interval ~to,t,~
of - obligor n drift function of time t In a second aspect the invention provides a credit risk model for the risk analysis of complex portfolios of credit instruments. The adjusted (P) transition matrix may be used to value credit swaps and other complex credit instruments. The model may have a simulation engine to generate valuation scenarios based on the adjusted (P) transition matrix. The simulation engine may generate scenarios based on a Monte Carlo method. The model may be implemented in computer software on a computer readable medium.

S
In a third aspect the invention provides a method of pricing a credit swap.
The method utilizes a number of steps:
~ (P) transition matrix over a risk horizon ~to,t,~ and the (Q) default probabilities over the same period ~to, t, ~ are obtained ~ an adjusted (P) transition matrix and (Q) transition matrix over ~to,t, ~
are created ~ a first series of forward (Q) transition matrices for standard observation times t = t,,tZ,t3,...
is built from the (Q) transition matrix and an initial term structure of default probabilities under Q over ~to,t,~
~ the first series is translated to produce a second series, having premium payment dates as the observation times by working with 'fractional' powers of transition matrices for subintervals ~ a third series of transition matrices is derived from the second series for co-migration on credit classes for pairs of obligors ~ a risk-free discount factor and probabilities contained in the third series are used to discount cash flow for each payment date over the period, and the discounted cash flows are summed for all payment dates. The sum is subtracted from a payoff function for fixed payments to give the credit swap price.
In a fourth aspect the invention provides an extended credit risk model for the risk analysis of complex portfolios of credit instruments with stochastic interest and foreign exchange rates. The model may have a Monte Carlo simulation engine to generate valuation scenarios based on the obtained or adjusted (P) transition matrix. The simulation engine may generate scenarios from a joint lognormal distribution of the following risk factors:
~ country/industry market indices ~ obligors' idiosyncratic components ~ base forward rates f(to,tl),f(tl,t2),...,f(tb_I,tb) for time intervals ~to,t,~, ~t,,t,~,...,~tb-,,tb~
corresponding to given maturities to,tl,...,tb ~ credit spread s, (tl ~ for the highest credit class R, and for maturity equal to the first bucket point tl on the corporate zero curve for each currency ~ incremental spreads Ds; (t, ) = s; (tl ) - s;-, (tl ~ , i = 2, . . ., K -1, for maturity tl (by credit class and currency) ~ the difference ~sK_, (tb ) = sK_1 (tn J - sK-I ~tl ~ between the spreads for the lowest credit rating for the longest and shortest maturity (by currency) ~ foreign exchange rates.
From risk factors logreturns for each simulation run the model may compute:
~ obligors' credit ratings at risk horizon based on country/industry market index logreturns and idiosyncratic components ~ base zero interest rates for all currencies based on forward rates f (to, t, ~, f (tl , tz ),. . . , f (tb_, , tb ) ~ credit spreads s; (t, ) = s;_, (t~ ) + ~sl (tl ) for maturity t, , i =
2,...K -1 (by currency) ~ credit spread sK_I (tb ) = sK_, (tl ) + ~.sx-, (tb ) for the lowest credit rating and longest maturity tb (by currency) ~ the rest of the spreads for maturity tb by partitioning the interval ~0, sK-, (tb )~ proportionally to s; (tl ) by taking s; (tb ) = s; (tl ) sK-' (tb ) (by currency) SK-I (t1 ) ~ spreads for other maturities tZ,t3,...,tb-, by linear interpolation between s; (tl ) and s; (tb ) (by currency) ~ corporate zero curves at risk horizon by adding the appropriate spread to the base curves (by currency) ~ foreign exchange rates.
In other aspects the invention provides various models, matrices, software, systems and methods incorporating combinations and subsets of the aspects described above, or utilizing the aspects described above.
BRIEF DESCRIPTION OF THE DRAWINGS
For a better understanding of the present invention and to show more clearly how it may be carried into effect, reference will now be made, by way of example, to the accompanying drawings which show the preferred embodiment of the present invention and in which:
Fig. 1 is a block diagram of the structure of a credit risk model according to the preferred embodiment of the invention, Fig. 2 is an example of Portfolio Summary (analytic engine was not used) output from the credit risk model of Figure 1, Fig. 3 is an example of Obligor Summary output from the credit risk model of Figure l, Fig. 4 is an example of Distribution of the Portfolio Forward Value output from the credit risk model of Figure 1, and Fig. 5 is an example of Marginal Risk vs. Exposure output from the credit risk model of Figure l, and g Fig. 6 is a block diagram of the structure of a credit risk model with stochastic interest and foreign exchange rates according to the preferred embodiment of the invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
Probability distribution (probability measure) (P) is a probability distribution of risk factors based on historical data. In credit risk models, probability distribution (P) is used to predict transition, or migration, from one credit class to another, "credit migration". A group of probabilities of migration, or transition, for different credit classes forms a transition matrix. A
series of transition matrices can be built for different time intervals and for different obligors.
In the preferred embodiment, a given, or obtained, transition matrix under the probability measure (P) is adjusted to be consistent with default probabilities under martingale probability measure (Q). In the preferred embodiment, the (P) transition matrix is adjusted by making a column of default probabilities consistent with default probabilities calculated under (Q). The other entries in the transition matrix are scaled to compensate for the adjustment while retaining the relative weights among the non-default classes in the transition matrix.
The adjusted (P) transition matrix may be incorporated into a credit risk model to analyze complex portfolios of credit instruments, including credit derivatives. For example, the model may use the adjusted (P) transition matrix in place of the given (P) transition matrix in a simulation engine to generate scenarios. The simulation engine may generate scenarios based on any Monte Carlo method.

The model may be implemented in computer software as part of a credit risk system using compatible hardware.
The adjusted (P) transition matrix may be used in the pricing of credit swaps.
Credit swap pricing may be incorporated into the credit risk model as it is consistent with the methodology used in the model.
This description has four Parts.
Part 1 describes assignee's basic credit risk model, CreditVaRTM, which utilizes deterministic interest and foreign exchange rates. Each obligor's standardized equity returns are decomposed into weighted average of market indices returns (mufti-beta model) where the weights are specified to appropriately reflect the obligor's participation in the corresponding markets and to model obligor's idiosyncratic returns. Part 1 has Appendix l, 2 and 3.
Part 2 describes the derivation of an adjusted (P) transition matrix and related matrices. In particular the relationship between P and Q probabilities for obligors to migrate from one credit class to another including probabilities of default is derived. An adjusted transition matrix and related matrices are created based on the derived relationship. The adjusted (P) transition matrix and related matrices allow portfolios of different instruments, including credit derivatives, to be analyzed. In the preferred embodiment, the adjusted (P) transition matrix and related matrices have been incorporated into an extended version of the credit risk model of Part 1. Part 2 also describes a method of pricing credit swaps using the adjusted (P) transition matrix and related matrices. In the preferred embodiment, the method has been incorporated into the credit risk I~
model of Part 1. The adjusted transition (P) matrix, related matrices and correlations between equity returns are used to compute joint probability distribution of obligors' credit migrations.
Part 3 describes an extended version of the credit risk model that allows for stochastic interest and foreign exchange rates. The model generates correlated interest rates, foreign exchange rates and obligors' credit migrations according to their joint probability distribution. As stochastic rates are permitted valuation must be performed for each instrument in a generated scenario.
It will be recognized that the adjusted (P) transition matrix and related matrices, pricing method and stochastic features may also be incorporated into other compatible credit risk models.

Part 1 - Basic CreditVaR Methodology 1. Introduction CreditVaRTM is assignee's proprietary model for measuring and analyzing credit risk in a portfolio context. This description is made in reference to CreditVaR as the preferred embodiment of the credit risk model. Those skilled in the art will recognise that many of the principles described herein are equally applicable to other credit risk models, such as J.P.
Morgan's implementation of CreditMetricsTM.
The methodology used in CreditVaR is based on credit migration analysis and on the assumption that an obligor's credit migration is driven by its asset value. For additional discussion of credit migration analysis, please see CreditMetricsT"' - Technical Document, April 2, 1997, J.P.
Morgan & Co. Incorporated that is hereby incorporated by reference and available at www.creditmetrics.com.
The methodology structure and implementation of CreditVaR are shown in Figure 1. Information required for credit risk analysis includes:
Obligor data:
Information about obligors is organized into a database containing details of their credit ratings, industries, and countries.
Portfolio data:

Information about financial positions is organized into portfolios of exposures. It allows to cover different types of instruments such as fixed income instruments, loans, commitments, letters of credit, etc.
Market data, transition probabilities and correlations:
These include yield curves, spread curves, foreign exchange rates, transition probabilities from one credit rating to another for different credit rating systems (Moody 8 states, Moody 18 states, S&P 8 states), correlations between market indices.
The CreditVaR methodology may be implemented both analytically and as a Monte Carlo simulation. It calculates two risk measures: standard deviation (in the analytic and simulation engines) and percentile level (in the simulation engine) of the portfolio value distribution for a given time horizon. The regulatory capital calculation is based on the percentile level, therefore, only simulation engine is used for the capital calculation. Accordingly, in the preferred embodiment the analytic engine was removed. Following sections describe the methodology and a possible implementation in more details.
2. Risk Statistics and Measurement The CreditVaR methodology can assess the impact of changes in debt value for a given portfolio due to credit quality movements of issuers (obligors) - including downgrades, upgrades and possible defaults - which can occur within the time horizon, typically a period of one year.
There are two computational engines developed in CreditVaR: Analytic Engine (AE) and Simulation Engine (SE). Both are capable of producing mean and standard deviation for the portfolio's forward value (risk horizon later). The two sets of numbers should be quite comparable as the same modelling framework is embodied in both engines. Though distribution of value for a portfolio of bonds is typically very much skewed, these statistics contribute to some understanding of credit risk present in the portfolio.
The Simulation Engine generates a distribution of forward portfolio values, and therefore has the advantage of estimating any statistic, including ones deemed useful in risk measurement. In particular, Yalue-at-Risk for any percentile level, together with a confidence band, can be computed. (Another risk statistic that can be produced from SE is average shortfall, which is the expected loss given that losses exceed a given level. The expected excession of a percentile level, which is the expected loss given that the loss is more extreme than the given percentile level, is another very useful risk measurement that can be obtained using SE
engine.) Results become more accurate as number of simulations increases, but will exhibit cost in terms of time and memory as with any Monte Carlo method.
With additional computation, both engines AE and SE produce marginal standard deviation by each obligor, i.e., the difference between the standard deviation of the entire portfolio and that of the portfolio ex the obligor. Plotting marginal standard deviation expressed as percentage of mean value for the given asset against the exposure size for each obligor, it can be used to identify that part of the portfolio which has concentration of risk, large exposure size and high percentage of marginal standard deviation. All of these measurements are useful in risk monitoring, as well as for capital allocation purpose.

The current CreditVaR method can be extended to produce measurement of risk which are due to obligor-specific returns. Risk due to market indices, in fact to any subset of market indices, can also be obtained.
3. Pre-processing The methodology requires that the three types of data mentioned in Introduction be given as inputs. They are then used to prepare the portfolio in an input format applicable to the CreditVaR Engine(s). We now indicate the required pre-processing.
For each instrument in the portfolio, there should be sufficient information to calculate its forward value for any credit migration with respect to a credit rating system chosen for CreditVaR. Specifications such as coupon rate, term to maturity, yield curve, spreads, etc. are needed. Recovery rate and its standard deviation, i.e. the seniority of the debt, are also required in case default occurs. We assume that obligor's exposures can be aggregated or netted across instruments having same seniority.
The credit quality movements can be tabulated in a transition matrix (for the given risk horizon) of probabilities for an obligor in one rating to end up in other ratings, including default, or remain unchanged in its rating. An additional ingredient essential in the pre-processing is a module which quantitatively measures the portfolio effect of credit by accounting for correlations of asset returns for the obligors. Once equipped with these asset correlations (which are on average typically between 20% and 35%), joint rating changes can be modelled across these obligors.

Ultimately, it turns out that in this model the only parameters which affect the risk of a portfolio are the two already mentioned, namely, the transition probabilities for each obligor and the correlations between asset returns. In this framework, the formulation of risk estimation is then reduced to considering the "standardized" (mean 0, standard deviation 1 ) asset returns, where the only parameters to be estimated are these correlations.
It should be noted that interest rates and spreads are assumed to be deterministic in this Part 1. In Parts 2 and 3 interest rates and foreign exchange rates are allowed to be stochastic. As stochastic rates are permitted valuation must be performed for each instrument in a generated scenario.
Part 3 describes a method and means for valuing such scenarios.
As asset values are in general difficult to observe, we use instead the correlation between equity returns as its proxy. The obvious advantage of this is that equity data are more readily available than data of credit spreads or of actual joint rating changes. In order to keep the size of correlation matrix reasonably small to be useful in computation as well, the methodology relies on some form of component analysis. Basically, each obligor's standardised asset returns are decomposed into a weighted average of some 'benchmark' country-industry indices where the weights are specified to appropriately reflect the obligor's participation in the corresponding markets, plus a residual component that is associated to obligor's idiosyncratic returns. These weights are specified for each obligor as part of input stored in Obligor database. (An example of such specification would be: Company X participates 70% in German Chemicals and 30% in German Electronics, and 15% of the movements in X's equity returns are company-specific.) In this way, the methodology incorporates the mix in obligor's market capitalization as well as obligor-specific movements in returns when computing credit risk of the portfolio. Given these weights and a correlation matrix for the indices, one can easily obtain a matrix of correlations between all of the obligors present in the portfolio. Appendix 2 presents in detail this methodology.
We remark that in applying CreditVaR to a number of portfolios there are several computational modules which can be made external to the pre-processing of individual portfolios. For example, the asset correlations between obligors can be stored in a database, and each time a portfolio is encountered for credit risk computation selecting appropriate data can be considered as part of the pre-processing.
4. Analytic Engine (AE) Mean for the portfolio's forward value can be calculated in a straightforward way, using transition probabilities and forward bond prices which have already been valued in the pre-process. Computing standard deviation requires more work, but it reduces to computing standard deviations of various obligor-pair subportfolios. The latter is straightforward as well since joint probabilities of credit quality co-movements are made available in the pre-process.
Marginal standard deviations can be computed in a similar fashion.

5. Simulation Engine (SE) First, threshold levels of standardised asset returns representing credit rating changes are determined for each obligor in the portfolio using the transition probabilities; this part depends only on the obligor's credit class and not on the obligor itself. Using the asset correlations as the covariance matrix, samples of standardised multivariate normal random variables are generated.

These are then bucketed using the threshold levels, thereby generating scenarios of credit rating co-movements for the obligors.
For each of these scenarios, portfolio is revalued quickly using pre-processed prices of exposures. Each time a default occurs in scenario, i.e., when the sampled asset return value of an obligor is below the default threshold level, a random recovery rate is generated according to a beta-distribution whose defining parameters are governed by seniority associated to the obligor's bonds. Finally, we obtain a distribution of portfolio values and from it the relevant risk statistics.

6. Implementation A CreditVaR software program utilizes the methodology described above. The core computational engine is implemented in C++TM. The interface for WindowsTM
could be a combination of Microsoft AccessTM/Microsoft ExcelTM. Windows version will consist of a Microsoft Access/Excel application and a DLL with the compiled computational engine. The selection of hardware is at the discretion of the user provided it has sufficient processing capability and is compatible with the programming environment chosen. Other programming environments could be used as would be evident to a person skilled in the art.
CreditVaR program uses a country-industry covariance matrix to compute asset correlations between the obligors in the portfolio. Matrices may be contained on a computer readable medium, for example on a hard disk, floppy disk, optical disk, or random access memory, as appropriate for the particular computing environment and stage of use.
Matrices may be contained in an appropriate signal for transmission, such as across the Internet or another computer network Computation engine consists of 2 parts - analytic and simulation. Analytic solution produces total mean and total and marginal standard deviation of every obligor.
Simulation produces total mean and standard deviation of every obligor, as well as Value-at-Risk numbers for the whole portfolio. For initial implementation time horizon is taken to be 1 year. The model can be expanded to handle other credit instruments like loans, loan commitments, etc.
Inputs The current implementation of the Global Analytics CreditVaR model evaluates the credit risk of a bond portfolio based on the following information:
1. Obligor data ~ Current credit ratings of the issuers (obligors), company-specific risk of the obligors, capitalization data 2. Market Data ~ Current zero-rate curves and current spread curves corresponding to different credit ratings and currencies ~ Forward (1 year from now) zero-rate curves and forward spread curves corresponding to different credit ratings and currencies ~ Transition probabilities matrix for credit rating migrations ~ Covariances between the market indices underlying obligors' capitalization 3. Spot and forward ( I year) exchange rates 4. Market Data exposure data (PortEfolio) ~ For bonds: principal amount of the bond, coupon rate or spread, seniority (defined by recovery rate and its standard deviation), maturity of the bond, currency, etc.

' ~ 19 ~ For loans and loan commitments: total authorized amount, loan amount, letters of credit amount, coupon or spread, fees on letters of credit and unused portions, seniority, etc.
Outputs The program produces the following outputs:
1. Aggregated by obligor:
~ Total present value of the bonds by a given obligor ~ Total expected 1 year forward value of the bonds (obtained analytically and by simulation) ~ Total standard deviation of 1 year forward value of the bonds (obtained analytically and by simulation) ~ Stand-alone VaR at different confidence levels ~ Marginal standard deviation and marginal risk of a given obligor ~ Delta standard deviation and delta VaR
2. Totals for the portfolio:
Total present value of the portfolio Total expected portfolio value in 1 year from now (obtained analytically and by simulation) Total standard deviation of the portfolio value in 1 year from now Skewness and kurtosis of the portfolio distribution ~ Percentiles of the portfolio value (VaR numbers of the portfolio) and expected excession at 0.1%, 0.5%, 1%, 5% and 10% levels ~ 95% confidence intervals for the VaR values Marginal standard deviation and marginal VaR for a given obligor i are defined as STDtotal 'STDtotal-obligor; ~
VCIRtotal -VCIRtotal_obligor;
where STDtotal, VaRtotal - are portfolio standard deviation and VaR, and STDtatal-obligor; , VQRtotal-obligor; - are standard deviation and VaR of the portfolio without positions corresponding to this obligor.
Obligor's Delta standard deviation (Delta VaR) is defined as the sensitivity of the portfolio's standard deviation (VaR number) with respect to the aggregated obligor's i position value Pabrgar; ~ multiplied by pabrgar;
DeltaSTD __ aSTDtotal obligor; a Pobligor; ~
Pobligor;
DeltaVaR __ aVaRtotal obligor; ~, Pobligor;
t' obligor;
DeltaSTDobligor, and DeltaVaRobl;gor; satisfy following equations [3]:
STDtatal = ~ DeltaSTDobligor , i VaRtatal = ~DeltaVaRobl,.gar;
i In other words, DeltaSTDobligor; (DeltaVaRobligor; ) is an obligor i contribution into the portfolio's total standard deviation (VaR).

Appendix 1 to Part 1 Sample Input and Output Data Obligor data table (example) - contains the following information:
~ Short and Full name of the obligor ~ Current credit rating of the obligor ~ Company specific % - the percent of the company's risk that is not explained by company's sensitivity to a particular country or sector of industry ~ Country 1 (2, 3) and Industry 1 (2, 3) for each country and appropriate percentages -capitalization of the company Short NameLong NameRatingCompany-CountryCapitalizIndustryCapitalizIndustryCapitalizIndustryCapitalizCountryCapit aliz Specific1 ation1 ation2 ation3 ation2 ation % in in in in in Country Industry Industry Industry Country 1, 1. 2. 3. 2, % % % %

AAAAKOT General Aaa 20 US 100 GNRL100 Electric AAAABPD Hydra-QuebecAa 61 CA 100 GNRL100 AAAAAUD Ontario, Aa 72 CA 100 GNRL100 Province Of 28664 Dharmala Ba 51 ID 100 GNRL50 AUTO 20 FOOD30 Sakti Sejahtera, PT

29715 Citicorp A 53 US 100 GNRL20 BFIN 80 29841 Anwar B 51 ID 100 GNRL100 Sierad, PT

27508 BOC GroupA 52 GB 70 GNRL50 ELCS 50 US 30 Plc, The 29842 Dharmala Ba 51 ID 100 GNRL100 Inti Utama Current zero-rate curves for computing present values of the bonds (fragment):

Currency.,.IVlaturityv...... _Aa _ _ ......... ~ __ ~~ .. .__,_ ~. w Aaa3 A .._... Ba;.. ~_ Caag m Baa; . Bl..

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.. __5 -.. ~ _ 7 ~ 0 078535'0.109867 _ ...
A.. 3 , D
U

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..-.~.5 BEF - 20 0.0686480.0651040.0662090.0694460.0827010.0940530.187946 ~ ~ t T ~

BEF 30 0.062548;0.0692350.0707660.0746960.094125E0.109520.187946 CAD 1 0.046084:0.0519080.0526080.0543080 0588080 0649080.075408 I = t CAD ~ m 0.0480690 0517970 0526000.0541970.0602420.065933'0.079768 ~ ! ~~ ~ ~.
~ 9 ~..._. _ ___..__ _ ~._ _ ._. _._ 0 0518180 0534090.0540000 055812;0 06291 0.0692720 091916 __. 3 ~ ~m CAD

CAD 5 0 055428'0 0553840.0560910 0575740 0667850.073954I0.105090 _ a , .. , ~ _ _ CAD . _ 0.0578110.0586380.060727:0.0701820.0792510.120275 .._._0.058771 ?
7 =

CAD ~ ~ 0.0624370.0601020.0609290.0637880.0739430.083994qJ 0.138775 ~ i ~ ~ k 10 , ._.__ ,_ ,.._ , "p,.._.,_.", . ,. . . . .
CAD _._.. 0 0671680.0646060.0655610.0685820 0800940.090640'0.156078 __ 1 _ _ . _ CAD 20 0 069212:0.0665220.067619;0.070830:0 0839750.0952060.187307 l ~

. __ ___...___ ~ _ CAD ._..30 0.0704990.0676550.0691030.0728140.0908770.1048480.187307 i ~ ? ; ~

CH 1 0.0257000.0181000.0188000.0205000.0250000.0311000.041600 F ; ; ~ ~

... , , 2 ~ 0.0292510.0213340.022137'0.023738, 0.0354760.049285 , , ~ - : = 0.029770~
CH ~
F

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CHF . __ 0 0296470 0245030 0251010 0269130.033995. 0 062777 .._. 0.040351 e._ ___ _ CHF .__._~ 0 029943~~~~0 0302500 0309600 032465E0.0416410.048827,0.079720 ~ _._ i~
~,5 ...A........_~...._...... .._._...... . _. W..... _.. .. ..~.. .._......
. ...__.w._. _. . ..._..~ . .._..__ _. _. ...._ CH 7 0 0304720 0353450 0361740 0382620 047756..... 0.097483 F ~ ? ~ 0 056809 _.~_ W__.m.~_..~.~ . _ _ .
0 041027~. . ._ ~..w __... 0 118510 CH ~~10 0 0310120 040189 ~.. _. ...
F ; 0 0438660 0541160.064209 ~ ~ =

_._. _.__ _~_._. . _ ~~,CHF.~._. 0 031123:0 0446270.045579;0.0485700 060025]0..0705550.134308 .15 ~ ~ "

_.. _ __.~ .~ . .....~._. .. ...... _._ _ 20 _ ._. .... ......... 0 0645930.0758040 162180 CHF 0 031259.. . 0 0516399 ~
0 047376, .

~

m_ _.. _._ _ __~
'~.~ CH _ . 0.0315820.0553770.0569000.0609040.08028 0.0962760.162180 F . ._ ~ ~ ~ 3 ] ~ ~ ~
~ __ ~
.

D EM 1 0 0436000 0402000 0409000 0426000 0471000.0532000.063700 j ~ i _._ A..~. .. _ _.~ _. ~~ .
~DEM W _.. _..w 0.0417310.042534;. 0.050175,0.0558730.069703 ._~2 0.045953; 0.044133:

__. _ D EM _ _.__ 0.0501430.0452790.0458740.0476890.0548 0.0611660.083789 . 3 E ~ , , ~ 01 !
. ~

_ _ DEM ~~~~~~~5 0.054534~~0.0499860.05069790.0521.910.0614350.0686390.099872 ~ [ .

__,.. _..~ ~... _ .
D EM , . 0 0582490 0546070.055441.. 0 0670920.0762400.11.7637 ~ Yp ~ . ' ~
7 0.057548 ~

_... _ .. _ D EM __._ 0 0616600 0584660.0593050 0621970 0724940.0826890.138390 ,~1.0 E !

_ _ ~ .. _ ~. _ _. _. .
D EM . 0 065009062601 0.063563=0 066605'0 078204. _._._._...
15 ~ ~ 0 r 0.088840I..
0.154962 D EM 20 , , 0.065974, 0.0824830.0938490.187930 0.0668470.064869. .. t 0.069215 DEM ~v~ 0.070431;0.074380,0.076066=0.080398:0.1025310.120894s0.187930 30 ~

Forward (1 year) zero-rate curves for computing forward values of the bonds (fragment):
_. _._._..__._. _,.. ._ . .._. . ___ .._ _ __ _.. _ ...,_Aaa ._ A. . .._ . _._.__.,__.._ , ~ _ _ . Ba B. ., Currency Matunty Aa _...__ Caa' Baa AUD 0 0.055559'0.053202'0.054107:0.055601:0.0631920.068470;0.085654 .. _ ...._... . _ z " . ..
AUD .....,....Ø055559:0.053202"0.0541070.0556010.063192-0.068470:0.085654 1 - ~ ~

__ ., _., ~ ,..._ _ , . 177 P-..~..-'57528-~--",0.058066.-:0 -0.103750 AUD , . - ~ 0 0 -0 0 0683710.074863 060 5993 = rv ~ 2 , _ . _ ~.__ _ AUD _.._ 0.065036:0.061292'0.0620030.063435'0.073895:0.081370.-0.-118040 AUD 6 0.067908=0.064212rt0.0650660.067236'0.0775910.087239:0.133997 ~

__ .. __... .~_ . . _ ..
AUD - 9 0.070155;0 0671620 0680100.071044,0.081944]0.0925750.153663 F

.",.,.,.,." ..,.,. _ ,. , , , .
-AUD.".. 0.071248'0 0698940 070867.. , ,.. 0.0968610.166664 ~ 14 .,. 0.086007 0.073988;

.__ _ ~_ _ 0071148 _., _ " ". . _.. q ."..- "."
AUD 19 0 078271_. ~.. _ _ 0 1104911 ~ ~ ~ 0 097859 0 65043 0.0794780 083029~

. _ ~ . - _ __ ~. . m....._ _ ry'AU_. ~~ 0.069150:0.076488~ 0.0780530.0820170.1020201~ ..~ _ . D . . ~ ~ a , 0.117422'0.163480 . .
i 29 ~

BEF 0 0 051264;0 0449940 045901'0 047400E0 054998'0.0602930 077496 ~ ~~~BEF~m~- rv-0.051264~0.044994:0.0459010.047400~rv~ ~ 0.060293-0.077496 ~ -- ~ E 0.0549981~

_ _ _;_.. ~. .. _ BEF 2 0.0544900 0481450 0486860 050557~ 0 0 0654770.094266 ~ ~ 058983 ~ T
~

_ . ~ ~_.__ _ 3 ~_ .
BEF 4 0 0529450 053657(0.055099,0 065547, 0.109610 __.__.. _ .
i BEF _..._ 0 0619420 0580840 0589420 0611220 0715421-0.0812300.1-28113 - ~ ~ ~

s.. .. _" __.~ _ _ ( BEF ,,., 0 0650710 0613970 0622520.0652850 0762520.086928-0.148053 T 9 ' ~ ~ ~~

.~d~a ~ __., .. ~_.. _ _ BEF ~ X14 . ~- 0.0649590 0680790.080118~~0.091007~0.160574 0.067782;_..M>_~_ ~ V ~
0.063984 w _.. ~ _ ..
_ BEF ._,,:~. 0 066404 0 070851 . ~_.. _ -~ 19 0 069923 0 067530, 0 0845810.~096219~0.194843 ~-~~~~~~ ~~~~~rv.-.-a-o63169~~0.070233 -~
0.1115060.192460 BEF ~.._ m~ ~~
29 0.071793 ~ ~-0.075803 0.095765 M~-CAD 0 0 050057'0 0516870.0525920 0540870 0616790.0669590 084146 ~ ~ ' ' _ . _. _..._~_~_ . _ _ ~... ._, E CAD 1 .. 0.0516870.0525920.054087-0.061679. 0.084146 0.050057 ~ 0.066959 . _ _. , .. , _ __.
. ~~CAD, ._. 0 0546960054160 0 0565660 064981 .._.
~._. 0 054697, ~ ~~0.071461 f ~rv 0.100264 i CAD ~' ~- ~-rv0.057776'-~m0.056255m~0.0569630.0583920.06878~0.076227~0.112638 ~~ ~~m -~
~-4 '9 _ ~~ ~ 0 0609010 0587990 0596470.0618010.0720909P0.0816600.127933 . CAD ~~~ ~ ~~~ ~mv ~ x ~ 6 ~

___ .. _ ._. ,.
. CAD ~9 0 0642700 0610170 0618580.0648470.075638;--~0.086136~0.146042 ~

_._ CAD _.._ 0~.068690~_ . 0.069609~~__ -0 092501-0.162067 __ 0.0655190.066492; 0.081631 1-4 , ;
( CAD _. . ~ 0.070443:-0 0672970.0684150.0717070.0853160.0968240.193509 . _ i :

_ _ ___ _ _ _.
-CAD. _ 0.0713510.0682020.0696770.0734580.092000-~~0.106251V0.191367 . 3- ry . - ~ ! ; ~

CH 0 0.0328150.0245790.0254840.0269860.0345620.0398710.057028 F _ , 3 ? ?
l . __ 4 _.

CH 1 0 0328150 0245790 02548 0 0269860 0345620 039871'0 057028 F ~

3-. . _.. _...~_ ._.~,__. ~_.. .
___ - __~ __ 0.027719;0.028266'0.0301350.038522~ 0.0450070.073527 CH ~2 0.031626: ~ ~~
F

_., _ __..,._... _ __., ____,._ _ .
M--CHF_ . 0 03100630 0333100 0340230 035479,~__, ..__. _ 4 0 045844..,...._..._.._ 0.053306;___ 0.089466 CH ~ ~ 0-.0312700.038247'~~~0.0390990.041253m 0.051598~0.0611550-.107084 F 6 , m ~ i - j ( ~

CH 9 0 0316040 0426730 0435270.0464950 0574010.0679530 127399 F ~ ~ - ~

~_ _ _.. _ _ _ ____CH ~4 14 0.0315110 0465480 0475180 050605'0.0625720.0734300.141238 F ~ ~ ~~ 3 ~ ~

_._.____ __ .~ "... ,_ _._. , -CH '~ ~ ~~ 0.031553;0.0489390.050041~_ 0.066718-rvrv~ 0.078210~ 0.168899 F 19 ~ - rv J 0.053304 s ~~m E~ ~~~ ~rv-0.031785;-0.056686:-0.05823930.062326-0.082248]--0.098595?
q~~~~CHF -m~m29 0.166578 DEM 0 0 0483110.0432650 0441710 0456690 05326010.058552:0.075739 . . . _ , _ __~ ~ _ __.,...rvDEM___._ 0.048311-i0.0432650.0441-710.045669~ 0.053260-0.0585520.075739 . . M - ~ ~ j _ _ _._ ., _._ ~ m~0.053430'_ ~0.048370~0.050243-~0.058673, ._ .
DEM _._ -Mrv 0.047828~~ Emm -0.0651710.093976 2 - ~

DEM 4 0.0572850 0524470 053160'0 054603;0.0650500.07253570.109106 ( DEM 6 0.060711'-0.0570270.0578850.060060;0.0704610.0801290.126889 ~ ~

___. .... _ ~ _.
DEM 636 ~6 0.06051.Ø061369x.0643. - .. . 0 147006 9 ~-, SwW' ~ ,. 8 ~.. 0 0860 0.0 8 ~ 0 0 6 f_ _ ._. _ _ ~~DEM. . ~0-.066555e~_, ,_ ~... "".,3.... 0.091432Ø161773 _. _. 0.064220'-0.0652010.0683410.080461 ; , : -. . .06618 .Ø o ?
DEM 9 0 068085~. 6731p?...6 0 08 0 0960310 194856 0 3 0 . 3 9 .0 _. ,..._ , DEM . :.71 .07.. .y77299 . 172 -1_...441 .- 0 192463 0 0 368 78,._, m0 - 0 0 0 23304 __ 29 ....... 0 55 0 08 93 ......
. .. _. __ ,. . ... 6 ... _ . ~.___ _, ..._...,_, .... . ._.
.. t., ..,., .. _..,.
__.
.

Note: forward curves were obtained from the current curves by bootstrapping method.

Table of foreign exchange rates (spot and forward):
~~rvCurrency USD per USD per Ccy, ' s of Cc , fwd 1 r' ATS 0.0812843 0.0811934' ~. . 0.7000000 0.6672129' _ -AU
......_.D

_ . .-.. . _ -__~
~_..__. _ . . .__~
..-__-., . - 0.0276916;
. _ __ _.
BEF 0.0277393 _.....CADØ7058159 0.6984693=

CHF 0 7041758 0 7163023;

i ~D 0.5720824 ~
~ E 0.5717753 W M

.____.,.pKK ~~ 0.1502765_ ~M0.1495353 ESP 0.0067732 0 0066965 , 1 FRF ~ 0.1708701, 0.170683 8''s ~ GBP 1.6898000 _ .TMrv~ 1.6363808 H 0.1293494 0.1221799, KD

I_ . ~ 0.00028050.0000571;
-___.-_-Ip-R

IEP rv 1.4936000_ ~ 1.41078781 IT ~ 0.00058380.0005726;
L

_._..- _ _ . _ ..__J 0.0079494 _.__.._;
PY 0.0083655:

~MXN0.1222046 ~~0.1055812 MYR 0 2928258 0.2 ~ 0896 19=

~ _ 0 5076400 _ N _ LG 0.507025 NOK ~ 0.1407856_ 0.137 ~~~~~NZD 0.6266000 _ 0.5710057;

~rnPTE ~~0.0056032~0.0055445i E ~
K

m~ SGD 0.6327912 0.5657095 ~~ T 0 0252048 0 0166732 ~ H
B

w S 1 0000000 1 0000000 U D

XEU 11342000 1.1190554 _. _.... __... _~ ~__ __..- _ ,_ . _w_ ~ ...._ ...

Covariance matrix of country-industry indices (fragment, actual size 211 by 211 ):
...__ __..,....._.__,-Covariance.... ... _.....__... . _ .._.__.... _.
....__..
matrix .. . .... ..... " ..... _ _ . .. ._ . ,...._ . ~
.
..
. _.. .._. . ... ......... ....

_.__.. _ ...._... _.. Country AU AU __AU AU
._.. ~
... , , ._ .. .. .
~ Y

Index name CountryIndustryAU GNRL AU BFI AU AU CSTR
N BM ED

MSCI Australia Index (.CIAU) AU AU GNRL 0.0002008 __ ASX Banks & Finance Index (.ABII)AU AU BFI 0.00016580.0003709 N

ASX Media Index (.AMEI) AU AU BM 0.00023840.00014440.0005618 ~ ED

ASX Building Materials Index AU AU CSTR 0.0001718.437E-050.0001920.0004216 (.ABMI) ASX Chemicals Index (.ACII) AU AU CHEM
0.00025670.00016240.00035240.00023 V

ASX Energy Index (.AXEY) AU AU ENRG
0.00015180.00010790.00018710.0001241 ASX Food & Household Goods IndexAU _ AU FOOD
0.00020460.00019230.00024080.0001694 (.AFHI) ~ T~ rn~ ~

ASX Insurance Index (.Alll) AU AU I
0.00020080.0001950.0002560.0001246 NSU

ASX Paper & Packaging Index (.APPI)AU AU PAPR 0.00018117.415E-050.00027560.0002049 ~ ~ ~~ v ASX Transport Index (.ATII) AU AU TRAN 0.00017540.00013890.00025 0.0001775 M_SCI Austna Index (.CIAT) _ AT AT GNRL 5.911 9.668E-060.00010027.506E-05 MSCI Behgwm Index (.GIBE) BE BE GNRL 8.795E-055.433E-050.0001459.354E-05 ~

MSCI Canada Index (.CICA) CA CA GNRL 0.00011520.0001250.0001277.762E-~
~

Toronto SE Automobiles CA CA AUTO 7.868E-058.753E-058.144E-054.668E-05 & Parts Index (.TIPA) Toronto SE Financial Services CA CA BFI
0.00010360.00018050.00017390.0001015 Index (.TFS) N
~

Toronto SE Broadcasting Index CA CA BM 4.429E-051.239E-054.918E-056.247E-05 (.TCMB) ED

Toronto SE Cement & Concrete CA CA CSTR 0.00011217.239E-050.00017138.249E-05 Index (.TIPC) Toronto SE Chemicals Index (.TIPZ)CA CA CHEM
0.0001190.00014220.00013868.82E-05 20739940.3 Transition probabilities of credit rating migrations (Moody-8 rating system):
Rating Aaa Aa A Baa Ba B Caa Default Aaa 0.9338 0.0594 0.0064 0.0000 0.0002 0.0000 0.0000 0.0002 Aa 0.0161 0.9055 0.0746 0.0026 0.0009 0.0001 0.0000 0.0002 A 0.0007 0.0228 0.9242 0.0463 0.0045 0.0012 0.0001 0.0002 Baa 0.0005 0.0026 0.0551 0.8848 0.0476 0.0071 0.0008 0.0015 Ba 0.0002 0.0005 0.0042 0.0516 0.8691 0.0591 0.0024 0.0129 B 0.0000 0.0004 0.0013 0.0054 0.0635 0.8422 0.0191 0.0681 Caa 0.0000 0.0000 0.0000 0.0062 0.0205 0.0408 0.6919 0.2406 Default0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 Note: Any transition probability matrix may be used (not necessarily 8-state).

Example 1 Sample bond portfolio:
Exposure Obligor RatingCurrencyRecoveryRecovRateAmount Maturity Rate Frequency Name Name Rate StdDev 912810EN4-11AAAAKOT Aaa USD 40 25 5000000.00 15/11/20227.6252 021654878440001AAAAAUD Aa CAD 50 25 15151000.0011/03/20038 2 19971027-00539AAAAKOT Aaa USD 40 25 14000000.0015/08/20276.3752 027054867160001AAAABPD Aa CAD 60 30 4905000.00 15/08/202011 2 027052668030001AAAABPD Aa CAD 38 23 1300000.00 15/10/200411 2 021655365450002AAAAAUD Aa CAD 32 20 25000000.0002/06/20277.6 2 021655355640001AAAAAUD Aa CAD 17 11 15420000.0001/12/20058.25 2 "Rate" is the fixed coupon rate of the bond, "Frequency" is coupon payment frequency per year.
Obligors:
AAAAKOT - General Electric AAAABPD - Hydro-Quebec AAAAAUD - Ontario, Province Of Sample output (obtained by running CreditVaR program on the above portfolio):
Portfolio: Exp7 Number of simulation runs: 100000 Number of exposures: 7 Number of obligors: 3 Evaluation date: 14/04/98 Started: 11:14:37 Finished: 11:16:24 Simulation time: 00:01:47 Analytic solution time: 00:00:00 Portfolio Present Value = 71786803.00 Portfolio Mean (analytic) - 70144493.80 Portfolio StDev (analytic) - 523028.46 Portfolio Mean (simulation) - 70143025.60 Portfolio StDev (simulation) - 530826.76 PortfolioPercentile (10%) -105932.78 -PortfolioPercentile (5%) - -297306.87 PortfolioPercentile (lo) - -812778.79 PortfolioPercentile (0.5%) -970228.53 -PortfolioPercentile (0.1%) -4912661.85 -Obligor RatingPresent Mean Mean St.Dev. St.Dev. MarginalMarginal Value (analytic)(simulation)(analytic)(simulation)St.Dev. Risk AAAAKOT Aaa 19190259.3019093143.1919093014.2573207.8973760.893636.56 0.019 AAAAAUD Aa 45880465.7244557820.8744556843.65507005.01515008.37422137.890.920 AAAABPD Aa 6716077.996493529.746493167.7071014.5578555.9212079.970.180 Example 2 (actual program output) Exposure data (fragment of input, actual size is 1863 exposures, 207 obligors) 15'1 31312 'Aa ' 40 25 1000000; 3120108!6.4=

16:131312 A a r 40~ 25i 1000000 3120113 _ __ ~AUD 65751 2 __,. . E.. ~ , , 2_._, ._.....
17 131312 Aa AUD ... 25 1000000 12/1 ! . . __ .. 40= 3 0 8 7 2 _.... ..........._. _ .. .... . ...... ...
. '. --. ' . ....
. .

~8~ 31312 Aa BAUD 40 25' 1000000 3120_/28__..'.
7.05~~~...__._._.........,2 _. ___ . -__ ' . _ _... -. .__ - ~ ~ ..._,.__.._.- .-.. ._.._ - -___ _. _._ _ ~.
~

~9 A ~AUD 40 25 1000000 3120199;_.. ___2 y 31312 - 541 = _ .

31312 . ' AUD _ ' . ' .
,1 ..
_..... -.

2 A AUD 4 0! 25 1000000' 3120/01 5 1 31312 1 ~ ~ ~T._. ._ s4 .
_.___.._ _..__._~ ~~.y . ... '~~ .
. ~~~ .._ ._...__ - ~
-22N A AUD 40~ 25: X000000. 3120/_03'.~
3131 - ...._ ' 1 . ..... ..._.. 6 2 _ _ _ __ --~ ~ J ~
_._.... _.__. V. ~- --- .
' _ ._. _ __.

23 "p, ~AUD 0 25 10000003 3120l05~~V
_.___.._..2 y :31312 ~ a f ~~m~ 6.27 ~

24 1 31312 A AUD 40' 25 1000000 3120/08 -E ; 6.53 ~

25 1 31312 A AUD 40 25= 1000000 3120/13 6.71 ' '~
. __ _. ~ ~ 2 s _ ~

__.__. ._ .._.._..~ _. __ , __ .' ._ _.. _..._ " __,_ _ _... . . ~ AU '__, . 40 . ._..._. _. . .__. __ 26 1 31312 ., _'__ D 25 __.. _.. _.__.._.....
___.. ----- A ~~ 4 _ _. E 1000000 .. ~_....'_7 14 ~ 2 - . ._....._ _._ .. _._ 3/20/18 . ~ .
. ~ . _ . _. .. ._. ~...___.._..
_. .__ --~

2~ W p, f AUD 0 25 1000000 3/20128 7.2 .
~ 31312 - ... .. f 0 2 - .. ~..

2$ Baa BAUD 25f 100000 3120199 5 ~ 0 61 _... 29-~31312. .Baa_._.~.AUD........40~ 25~ __ 1000000:..._, .
. _... ....._ - _.___3120/0... 5.s._...
__..__..

Example of the program output - Portfolio Summary (analytic engine was not used) (see Figure 2) Example of the program output - Obligor Summary (see Figure 3) Example of the program output - Distribution of the portfolio forward value (See Figure 4) Example of the program output - Marginal Risk vs. Exposure (See Figure 5) Appendix 2 to Part 1 Correlation model 2.1 Market and specific components and country/industry weights Let us assume that company's standardized equity return r of the firm A can be expressed in the form:
r = NMYM +~srs ~ (1) where rM is standardized return of the market component of r and rs is standardized return of the idiosyncratic component, rM and rs are independent. From this follows that coefficients /3M
and ~3s satisfy condition ,13M + ~s =1. (2) We also assume that return of the market component rM is linear combination of standardized country/industry index returns r , which are independent of rs rM = ~3,r, + . . . + ~3"r" . (3) cor(r,rs)=0, i=1,...,n.
Equations (1), (3) can be obtained by regressing A equity returns against index returns in a usual way.
There is another way to represent r in terms of index returns r and idiosyncratic return rs .
This representation is consistent with (1)-(3) but based on country/industry weighting coefficients. We show below how this representation can be obtained from (1)-(3).

Denote by ~ , a; - standard deviations of the firm A equity return and index i return respectively, p« correlation between index returns r,. and r ( i, j =1,. .., n ). Then from (3) one has YM = ~I r + . . . + ~n rn --' ~Ir +...+-n 6nrn ~I ~n +'...+-n ' 6r +...+ n 6 r /~ (~ 1 1 n n 61 6n /~
6' ~1 ~- . . . + n ~'n ~' + . . . + ~n 61 ~n 61 ~n or r,~ _ ~' +...+ ~n ~w,~,r +...+WnQnYn), (4) 61 6n where wl, wz,. .., wn are given by w~ _ ~' , i =1,...,n . (5) ~, ~' + . . . + ~n ~n Note, that wl +wz +...+wn =1.
Coefficients wl,wz,...,wn are called country/industry weights and can be expressed in percents.
From equation (4) follows that z n n var~rM~= ~' +...+~n ~~w;w~6;Q~p;~ =1. (6) 61 6n i-1l-1 Denote ~~n--.. n ~~WiWj~i6.IpJ ' i-1 J-1 Then from (6) we have ~' +...+'~n 1 . (8) 61 6n a.
Using (1), (4), (8) one can express standardized equity return in terms of country/industry weights as r=~MrM +~srs Wl~'1 Wn6n M r + . . . + Yn + SYS .

Hence, to compute this representation one needs to have following parameters:
country/industry weights, index standard deviations and correlations, specific coefficient ~3s (coefficient ,Q~, for the market component can be computed from (2) as ,13~, = 1- ~3s ). In the current implementation CreditVaR I derives parameters ,~3s and ~3M from "specific risk percent" s defined as a fraction of total equity return changes explained by firm-specific movements:

~3s =s. (10) 2.2 Correlation between two obligors Equations (1), (3) as well as equation (9) allow to compute correlations between equity returns of any two firms.
Let rB, r~ are standardized equity returns of two different firms B and C
respectively and B B _ _ rB = ,B WI ~1 Y +...+ wn~n r +~BYB =~BY -E-...+~BY -~'~BYB
~I 1 n S S 1 1 n n S S
d- 6 YB = ~C w1 61 Y + . . . -~ wn 6n Y + ~CYC = ~CY -+- . . . + ~CY ~- ~CYC
M 1 n S S 1 1 n n S S

where n n 2 n n 2 _ ~~ B B _ ~ C C
~B GrGrwi wj ~i6jpij ~ ~C ~~Wi wj 6i6jp~ (11) i=_1 j=1 i=_1 j=1 _ B _ C
~B _ /~B w( ai /~C _ ~C w( 6i !"i - l"M ~ l"i - l''M ~ 1 = 1,...,h . (12) ~B ~C
Then correlation between rB and r~ is cor(rB,r~~=~~,~3iB~~ p~ . (13) i=1 j=1 Example.
Suppose we wish to compute correlation between two obligors B and C in the obligor table in Appendix 1 (Sample Input and Output Data). Obligors short names are "28664"
and "29715"
respectively. From the table we have:
Obligor B ("28664") Specific percent sB = 0.87 Country/industry weights wo,cNR~ = 0.5 , w,D,AUro = 0.2 > >'''p,FOOD = 0.3 .
Obligor C ("29715") Specific percent s~ = 0.88 Country/industry weights n'us,GNRL = 0.2 , WUS,BFrN = 0.8 .
Let us assume following standard deviations and correlations between market indices:
Index StandardCorrelations DeviationID ID ID US US BFIN
GNRL AUTO FOOD GNRL

Indonesia, General 0.03 1 0.4 0.2 0.01 0.03 Indonesia, Automobiles0.02 0.4 1 0.1 0.02 0.04 Indonesia, Food 0.04 0.2 0.1 1 0.01 0.02 US, General 0.01 0.01 0.02 0.01 1 0.3 US, Banking and 0.02 0.03 0.04 0.02 0.3 1 Finance According to (7):
QB=0.032x0.52+0.022x0.2z+0.042x0.32+2x0.03x0.02x0.5x0.2x0.4 +2x0.03x0.04x0.5x0.3x0.2+2x0.02x0.04x0.2x0.3x0.1 = 0.000471 6B = 0.000471 = 0.022 6~ =O.OIz x0.22 +0.022 x0.82 +2x0.O1x0.02x0.2x0.8x0.3 = 0.000279 &c = 0.000279 = 0.017 From (10), (11), (12) /3M = 1- 0.872 = 0.49 , ~3s =0.87, _ B
/~)B _ /~B wID,GNRL61D,GNRL = 0.49 0'S x 0.03 - 0.33 , l' lD,GNRL ~M ~.B O.O22 B
D,AUTO - /-'M wID.AUT061D,ALrTO = 0.49 0'2 x O.O2 - 0~O9, a~B 0.022 _ B
r ID, FOOD - /'M wID.FOOD~ID,FOOD = 0,49 0'3 x 0.04 - 0~27 , d~B 0.022 /j,= 1- 0.882 = 0.47 , ~3s =0.88, _ c (/]Jc __ /~c n'us,GNnL~us,GNRL = 0,47 0.2 x 0.01 = 0.06 i- US, GNRG i" M ~-C O. O 17 C
(/J]c _ /~c ~'US,BFIN6US,BFIN = 0.4.7 0'8 x 0.02 - 0~44 i US,BFIN - NM ~C 0.017 Finally, compute correlation between rB and rc using (13):
cor(rB,rc)=0.33x0.06x0.O1+0.33x0.44x0.03+0.09x0.06x0.02+0.09x0.44x0.04 +2x0.27x0.06x0.01+2x0.27x0.44x0.02 = 0.017 2.3 Simulation of correlated equity returns Obligors correlation matrix is used in the model to generate correlated samples of normally distributed random vectors of equity returns. In this section we describe the procedure based on Cholesky decomposition of the correlation matrix of index returns.
Suppose there are K obligors and their standardized equity returns are represented as (see (12) above):
Rk =,t3,kr +...+,(3n n +~3srs , k=1,...,K. (14) Denote by _ r _ R = (R~,...,RK)T , Y = _M , Y,N = (Y,...,Yn)T , YS = (YS,...YS
YS
~n+K~x1 Pl 1 PI 2 . . . P( n _ Pz~ Pzz . . . Pzn _ CM 0 C~' . : ~ . . CM O I
K ~n+K~~n+K~
Pnl Pn2 ... Pnn nxn ~1 /~2 K ~~ Q ... O
/''I/J1 ... ~S
~1 1 2 K O 2 ... O
~z ~z . . ~s . ,13z n W _ = ' ' ' ~. : . . . : n (n+K~xK

~1 /~ K Q Q /~K
' 2 ... ... ~
l'' ~

l n n nxK S KxK
n where CM is correlation matrix between indices and IK is K x K identity matrix.
Then correlation matrix C of equity returns equals to:
C=E(RRT)=E(WTYrTW)=WTE~rrT )W =WTCMW . (15) Let us assume that matrix HM is obtained by Cholesky decomposition of the matrix CM
CM = HMH,; . (16) Denote by _ HM 0 HM O I ~n+Khc~n+K~
H -WTHM (17) ( H is K x (n + K) matrix). Then _ T
CM HM HM
C=WTHMHMW =HHT (18) are Cholesky decompositions of CM and C respectively.

For simulation of a sample of K dimensional normal vectors R with zero vector of means and covariance matrix C (standard deviations of coordinates of R equal 1 ) we first simulate n + K
dimensional vector of standard independent normal variables x = xM ~ NO~ In+x xs xM = x~ , . . . , x" , xs = xs , . . . xs , and then compute R as R =Hx=WTHMx H 0 x H x . (19) _ (Br A ,v nr = (BT n n~ M = BrHMxM + Macs 0 I xs xs From (18) follows that R has distribution N~O,C):
E(R)=0 , cov(R)=E(HxxTHT~=HE(xxT)FIT =HI"+KHT =HI"+rcHT =C.
Equation ( 19) can be used for simulating correlated sample of equity returns in the model.

Appendix 3 to Part 1 Pricing Procedures 3.1 Pricing Procedure for Floating Rate Notes Floating rate notes are designed so that the coupon payments depend on some current interest rate index (usually LIBOR or some other known rate), which is called reference rate. This causes the value of a Floating Rate Note (FRN) to be close to par at all times. Credit risky FRNs pay coupons combined of reference rate plus some spread over it, which depends on the credit rating of the issuer.
The notation is as follows:
P - price of note, r, - expected future reference rate yt - spot long-term yields, used to discount the cash-flows at time t;
s - stated credit spread, fixed under terms of the FRN
m - maturity of note in years n - coupon payment frequency Assuming flat structure of credit spreads, the price of the note per 100 of face value, is P-~100(rt+s)ln+ 100 i-1 (1 +' yi )tr (1 + ymxn )~~n,n Here, r; is forward reference rate for the period between i-th and (i+1)-st coupon payments.
Assuming the reference rate is annually compounded, the way to compute forward reference rate r~ is the following:

r. - (1 + R;+~ )';.~ /c';.~-', ~ -1 ' (1+R;)'' where R; is the rate on the reference curve for time point t; .
Within the current framework of CreditVaR I model, we will take existing current and forward corporate curves as a proxy for y;, since they incorporate market required spread. The reference rate R and credit spread s should be supplied as parameters of the FRN.
Note that for computing present value of an FRN we use the supplied reference curve, but to compute forward 1-year value of an FRN, a forward reference curve has to be obtained. The rates r; in the computation of forward value of an FRN are therefore "forward forward" rates.
3.2 Pricing Procedure for Interest Rate Swaps Swaps can be characterized as the difference between two bonds, which can be either fixed or floating rate. In the swap agreement, principal is not exchanged, which makes swap different from a bond from the point of view of credit risk. Another difference is that in the event of default by the counterparty, the sign of the value of a swap determines weather the other party pays or receives some recovery amount (this depends on netting rules).
Assumptions:
1 ) There is no more than one credit event per year 2) The credit event happens one year from evaluation date 3) Netting is done for positions with the same obligor and seniority 4) One recovery rate is applied to a netted position 5) Implementation allows different recovery models depending on the sign of exposure to a bank, including recovery rate 0 if the value of the swap to the bank is negative 6) A bank's credit events are not modelled (i.e., the bank's credit rating is constant).
The notation is as follows:
V - value of the swap to the bank, B, - value of the bond modeling the cash flow received by the bank, BZ - value of the bond modeling the cash flow paid by the bank, rl - recovery rate applied if bank receives in the event of default by counterparty, rz - recovery rate applied if bank pays in the event of default by counterparty, Q - notional pincipal in swap agreement.
The value of the swap to the bank then is:
V=B~-B2 B~ and Bz are present (or forward 1 year) values of the underlying bonds, obtained by standard procedure for fixed or floating rate bonds. Ideally, swap bears credit risk only if the value of the swap to the financial institution is positive. In practice, however, if counterparty defaults, it might not free the bank from liability to that counterparty (due to netting rules). Therefore, in case of default, the recovery amount is calculated as follows:
If V > 0, the recovered amount is r, *V, If V < 0, the amount paid is r2*V

Here, we don't take into consideration the event of default or credit rating migration by the bank.
Recovery rates rl and r2 are determined from legal considerations at the time of default. We can use Beta distribution again to simulate recovery rates r~ and r2 (in current implementation, we take r,=r2). Alternatively, rz can be put to 0 if the swaps with negative exposure are excluded from the computation.
3.3 Pricing Procedure for Loan Commitments A loan commitment is composed of a used portion and an unused portion. Used portion has a Loan (drawn funds) and a Letter of Credit (undrawn funds) parts. Interest (usually LIBOR plus spread) is paid on the Loan portion of a loan commitment, LIBOR plus LC fee is paid on the Letter of Credit part (throughout its lifetime), and an Unused Fee is paid on the unused portion of the loan commitment. There is also a Facility Fee applied to the total loan commitment amount throughout its lifetime. A 1-time Upfront Fee paid on the total loan commitment amount at the initiation can be ignored in the CreditVaR I framework since it bears no credit risk.
The amount drawn at the risk horizon is closely related to the credit rating of the obligor. If the obligor deteriorates, it is more likely to withdraw additional funds. On the other hand, if its prospects improve, it is unlikely to need the extra borrowings.
Let T be the total (authorized) loan commitment amount; L - the amount drawn (in loans); LC -the amount in Letters of Credit; U- unused portion of the loan commitment.
We have that T =L+LC+U.

Let l be LIBOR and s be spread over LIBOR, so (l+s) is the interest paid on the loan; lc - the fee paid on the letter of credit, of - the fee paid on the unused portion; ff -facility fee applied to the total authorized amount of loan commitment. Then present value (in the CreditVaR framework) is computed as follows: we treat the loan and letter of credit parts as floating rate notes (because of the way interest is paid); facility fee and unused portion fee are valued as bonds with no principal repayment.
PV - m" L x (l~ + s)/n + L +
~1 + yr ~~' (1 + ymn )',n.n ~LCxlcln + ~Uxufln + ~Tx ff/n (1 + yr ~f ' t ~l + Y. ~~' ~ ~1 + Yr )';
where y is the annualized rate taken from the corresponding current corporate curve used to discount cash flows (depends on credit rating of the obligor in the CreditVaR
framework); n is the frequency of coupon payments, and m is the number of years to maturity.
Forward value (FV ) is computed similarly, with y being the forward (1 year from now) rates taken from the corresponding corporate curve.
If downgrade or default happen to a particular obligor, it is likely to withdraw additional funds from the loan commitment line. To model this we assume that there is an "expected" drawdown (ED), or "average commitment usage" if the obligor changes credit rating (see Credit Metrics Technical Document, p.45 for appropriate information):
Credit rating ~ Average commitment Aaa 0.1 Aa 1.6%

A 4.6%

Baa 20.0%

Ba 46.8%

B 63.7%

Caa 75.0%

' 43 Then in the case of any credit event (downgrade, upgrade, default) the used portion of a loan commitment is adjusted in the following way:
Downgrade Let X % ( Y% ) is average commitment usage given in the Table above for the current (new) credit rating respectively, X < Y . Then new unused, loan and letter of credit portions are computed as new = ~ 1 - X ~
L"ew = L + U - U"ew LC"ew -LC, - new used amount T - U"ew is proportionally distributed between loan amount L"ew and letters of credit amount LC"e~. .
Upgrade In this case X > Y . New unused, loan and letter of credit portions are computed as U"e,y = T - (L + LC) ~ , L"ew = max(L - (U"ew - U~~~
L ~"ew = T - L"ew U"ew Default In the case of default it is assumed that used portion is the maximum of current used portion and 90% of the authorized amount T
Lnew = max(L + LC,0.9 x T) , LCnew ~ ~
Unew - T Lnew Forward value of the loan commitment in the case of upgrade or downgrade is computed as FV,o~pr = FV + (L - Lnew) .
In the case of default the loan commitment value is Ln~", x R , where R is the recovery rate sampled from the Beta distribution (consistent with CreditVaR I framework), and total forward value is FV,o«, = Lnew x R + ~L - Lnew l Part 2 - CreditVaR Methodology: Credit Migration Process Under Two Probability Measures As described in Part l, the CreditVaR is a model for measuring and analyzing credit risk in a portfolio context. The methodology used is based on credit migration analysis and on the assumption that an obligor's credit migration is driven by its asset value.
Correlations between equity returns are used to compute joint probability distribution of obligors' credit migrations.
Each obligor's standardized equity returns are decomposed into weighted average of market indices returns (multi-beta model) where the weights are specified to appropriately reflect the obligor's participation in the corresponding markets and to model obligor's idiosyncratic returns.
In the Monte Carlo simulation version the model calculates distribution of the portfolio values, statistical characteristics and risk measures defined as percentiles at given confidence levels (95%, 99%, etc). Simulation engine generates scenarios based on the "real world" probability distribution (P) of the risk factors. The distribution is assumed to be lognormal with parameters computed from the historical time series data. For each generated scenario the portfolio value is computed under the martingale probability measure (Q). The relationship between these two probability measures is derived from the assumptions about stochastic processes for index and stock returns. In particular, we derive the relationship between P and Q
probabilities for obligors to migrate from one credit class to another including probabilities of default (transition matrices). This allows one to analyze portfolios of different instruments including credit derivatives.

' 46 In the CreditVaR framework it is assumed that credit migration process forms a discrete Markov chain with fixed moments of time T = ~Uo < U, < ... < U, < . ..~, Uo >_ 0 . In sections 1-5 of this Part we present mathematical models of the migration process and define probabilities under P
and Q measures to migrate from one credit class at time U, to another credit class at time UI+, .
Theoretical results of sections 1-5 are applied in sections 6-8 to compute P
and Q transition probabilities from market credit spreads and credit migration historical data.
1. Stochastic processes for market index and stock returns under P measure Suppose that B = (B',BZ,...,B°) is a standard D-dimensional Brownian motion on a probability space ~S2,F,P) (B',Bz,...,8° are independent), where F =
~F,t >_ 0} is a standard filtration of B .
Let I = {I',I2,...,1"' ~ be a set of all market indices. Denote by V,"" the index I"' value at time t >_ 0 . It is assumed that Y'~ satisfies stochastic differential equation (SDE) of the form:
dV'" =Vrm(,umdt+Br";dBl +9,zdB; +...+e~°dB°) , t>_0 (1.1) Vo =v"', m=1,2,...,M, where ,u;" , B,'; are piecewise continuous deterministic functions of t, i =1,2,...,D .
Let us fix some time interval ~to, t, ~ corresponding to two consecutive time steps from T
to = U, , t, = U,+, , I > 0 . Denote x;" = ln(V,"' )- ln(Y,o ~ be the index I
"~ log-return for the time interval ~to,t~, to <_ t <_ t, . Then from Ito's lemma and (1.1) follows that ' ' 47 xm t~~~m 2 ~~~s 1 ~ + ~~ 2 ~ + . . . + (85"D ~ )~S
t (1.2) + f BS",dBs + 9s"2dBs + . . . + 8 DdB°
to for to <- t <- t, with initial condition x o = 0 . In a matrix form ( 1.2) can be written as t xt = a, + ~es dBf , to <- t <- t, , xto = 0 , ( 1.3 ) to where ( 1 M
xt = xt,..., xt , a=~al,...,aM~
z t~~s 21\vs.l~+~~s.2~+...+(85,D~)~5~..'~t~~M 2~~~M~+~~~~+...+(~ p~) S , ~0 Bt I Bm MxD' dBS = ~dBs , . . ., dB° ~ .
Let us consider N obligors (issuers) 5 = {S',S2,...,S"'}, and let Ht be an obligor S" stock price at time t . We assume that Ht satisfies SDE:
dHf =Ht (aldt+~~,dB; +a-t2dB; +...+o-;DdB°~, t>-0, (1.4) Ho = h" , n = l, 2, . . . , N , where a; , o-~; are piecewise continuous deterministic functions of time, i =1,2,...,D . Then for the stock log-return y~ = ln(Hf ~-ln(H~o ) one obtains yr - o' Las 2 ll6s 1 ~ ~' ~6s 2 ~ + . . . + (mss D ~ )Jds t (1.5) + f 6°,dBs + ~5.2dBs + . .. + 6s.DdB° , to ~t~tl~ Y~o =0.
Similar to (1.3) this can be written as:
r Y7 = br + J6SdB5 , to <_ t <_ t, ~ Yra = 0 > (1.6) to where T
Y7 (Yr~...,yN) , T
b = ~b; ,...,b;') T
o-Las 2\las,l~+~6s,z~+...+(~S,D~)Jds,..-'o-Las 2~~~ ~~-I'.(6 z~+...+(6 D~)~ds , ~~~~ 1 NxD
2. Multi-beta model In the CreditVaR framework correlations between stock returns are derived from the index return correlations using multi-beta model. This is linear regression model that represents stock returns as linear combinations of index returns and stock specific (idiosyncratic) components:
n _ n,0 ~n,l 1 n,2 2 n,M M n Yr -~r +~r xr +~r xr +...+~r xr +~7 , to <t<tl, (2.1) n=1,2,...,N, where ~; and x; ' are uncorrelated: cov(s; , x~" ) = 0 , and E(s° ) = 0 , m =1,2, . . . , M , n =1,2,...,N . In the matrix form:
Y7 = ~~ + ~rxr + Er~ (2.2) where ~3° _ ~~r'~o,. . , ~ ~7N,o ~ ~ ~r = prn,m NxM is a matrix of stock "betas", s, _ (s; ,..., ~N ~ is a vector of idiosyncratic components which are uncorrelated with index returns:

cove, , x, ~ = cov(E~ , x;" ~ = 0 (2.3) NxM
(here and below expected value E(.) and covariance cov(.) are conditional to 6 -algebra F ) It is assumed that rank(8r ) = M < D for all t E ~to, t, ~ . (Note, that if rank(Br ) < M <_ 1~ for some t = i then it can be shown that there exists time interval ~i , t ~ c ~to , t, ~, i E ~t , t ~ and index subset I c I , I ~ QS , such that for all indices in I their log-returns can be represented as linear functions of index I I I log-returns with time dependent deterministic coefficients. Then index set I can be reduced to subset I I I on ~t, t ~ .) Then from (1.8), (1.9) it follows that the (M x M)-matrix 9r9rT is nonsingular on ~to,t, ~ and _, r r /31 = J~S95 ds j6s9s ds . (2.4) ra to Indeed, from ( 1.3, ( 1.6) follows that r cov~xr,xr)= J9S9s ds, (2.5) ro r cov(yr.xl = J6ses ds, (2.6) ro Then (2.2) and (2.5) give r COV (.Yt ~ xr ~ = cov(~r~ + ~r xt + Et ~ xr ~ _ ~r COV (xr , xr ~ _ l-'r ~ea es ds .
to Therefore, r r J~ses ds = ~3r f 9S9s ds ro y and (2.4) holds.
Equations (2.2) and (2.4) also give the covariance matrix between equity specific components:

COV (E; El ) = COV (.Y - N o - ~ .Y - N o -r r r~ r r r ' (2.7) = JQs~s ds - ~~SBS ds JBSBS ds 8565 ds ru y y ro In the special case of constant volatility matrices B and ~ one has COV(Er . Er ~ - \t t0 ~~ (JD BT (B eT ) BST , where ~D is ~D x D~ identity matrix. In this case equity specific components EI are uncorrelated if and only if Q(~D -9T (B BT ~ ' B~ _ ~. , where ~, is a diagonal matrix.
This means that orthogonal projections of the rows of stock volatility matrix Q on the linear subspace, which is an orthogonal compliment in R° of the linear space generated by rows of B , are orthogonal.
Note, that in this case covariance matrix of stock returns equals CoV~yt, yt~
_ ~t - to~(~39BT,~T + ~,~.
3. Credit migration model under probability measure P
It is assumed that at any time U E T each obligor S" E S belongs to one of credit classes R = {R~,...,Rx ~, where Rx corresponds to default. Suppose that credit migration process forms a Markov chain on R with discrete times T and absorbing state Rx . Denote by R~
a credit class of obligor Sn at time U E T . Similar to section 1, we fix a time interval ~to,t~~ corresponding to two consecutive time steps from T : to = U, , t, = U,+, , l >- 0 . Suppose that R~o = R; and p"~ (to -~ t, ~ is a probability of S" to migrate to credit class R~ at time t, p"~ ~to ~ t, ~ = P(R;~ = R~ IR;o = R; ~. (3.1 ) Denote by TnP (to ~ tl ~ = P~ ,; ~to ~ t~ ~ (3.2) xXx a transition matrix of S" for the time interval ~to, t, ~ . Then K
p~ j (to -~ t, ) = l, ~ = l, . . . , K
j=1 PK.Oto-~t,)=0, j=1,...,K-1.
PK.K \t0 ~ t1 ~ - 1 For each nondefault state R; and probabilities p"j (to ~ t, ) , j =1, . . ., K
define thresholds h;"j (to, t, ~, j =1,. . ., K -1, for the standard normal distribution as follows:
P~ i ~to ~ ti ~ =1- N~ht , ~to ~ t~ ~~
Pi z ~to '-~ t~ ) = N(hr ~ (to, t, O - N~hrnz ~to ~ t~ ~~
... , (3.3) P~ x-~ ~to ~ t~ ~ = N~htnx-z ~to ~ t~ ~~- N~hrnK-~ ~to ~ t~ ~~
P~K ~to ~ ti ~ = N~htnK-i ~to ~ t~ O
where N(.) is standard normal cumulative density function.
Let wt = Zy.,'t (yt - E(yt )) be a vector of standardized stock returns with correlation matrix given by t cor(w,,wt)=Zy~cov(yt,y,)Zy~; =Zy~ J~S~Sds ZY~, (3.4) to where ZY,t is a (N x N) diagonal matrix with stock returns standard deviations var y~ 0 ~ ~ ~ 0 Z _ 0 var yl ... p (3.5) Y, t 0 0 ... var yN
( var(.) is a variance under P measure conditional to o--algebra F ) Similar to the CreditMetrics framework, the CreditVaR credit migration model is based on the assumption that credit class of the obligor S" at time t, is defined by the value of its normalized stock return w; and thresholds h;",~ (to, t~ ) Assumption: Given R;o = R; , R," = R~ if and only if wl E (h;°~ (to, t~
), h;"~_~ (to, t~ )~ , where i, j =1,2,...,K, h;"o(to,t,)=co, h;"x(to,t,)=-oo.
From the Assumption it follows that joint migration process for obligors can be defined using a joint probability distribution of their normalized stock returns. Indeed, suppose that Rio = R;" , S" E S , n =1, . . ., N . Then normalized obligors' stock returns have N -dimensional normal distribution with zero means, standard deviations equal to 1 and correlation matrix (4.4). Denote by fN (x, , xz , . . . , xN ) their joint density function. Then the joint probability that at time t, obligors will be in credit classes R~ , R~2 , . . ., R~.y, respectively is given by P~Rt = R~ ,. . ., Rt"' = RAN R;o = R; ,.. ., Rrp = R;,ry, ) h',.~,-, c~o.f, >h 2.;z-~ c~o.t, > h N.~,v-, c~o.~, > (3.6) - . . . J fN (x, , xz, . . . , xN ~dx~ dxz . . . dxN .
hi,.into~tn hi.iz~tov~ hlv.lv~~o~u~
4. Stochastic processes for index and stock returns under Q measure In this section we derive stochastic differential equations for the index and stock returns under the martingale probability measure Q.

Let r, be an instantaneous risk-free interest rate at time t . We assume that rt is a deterministic positive function of t . Then index values V,"' , m =1,2,...,M , and stock prices H; , n =1,2,...,N, for non-dividend-paying stocks (this assumption can be relaxed) satisfy following equations:
dV,"~ =V,"'(rr dt+Br';dB~ +B;zdB; +...+9,odBr°) , t>_0, (4.1) Vo =v"', m=1,2,...,M, dHf =H~ (rdt+o-F,dB~ +~lzdB; +...+~;DdB,°), t>_0, (4.2) Ho =hn, n=1,2,...,N, where r Bd =Bd + jzsds, d =1,...,D, (4.3) are Brownian motions under the probability measure Q defined by the Radon-Nikodym derivative:
p r r ~P = exp - ~ Jzd dB~ - ~ ~z5 zds , (4.4) d=10 0 Izt 2 - (z y -IyZr ~ -E . . . + (T D ~ , where vector-function zt = (z~,z; ,...,z°~ solves equations:
Br';z; +B;zzl +...+e; pz° =,um -rr , m=1,2,...,M, (4.5) n I n 2 n D n ~r.nt + ~t.zzr + . .. + 6r. pzr = at - rt , n = 1,2,. . . , N . (4.6) In the matrix form:
erzt = W
(4.7) 6rzr = ar where ;u =(,u' -r,..."u"' -r~ a =~a' -r,...,a'v -r)T . We assume that solution z of (4.7) t r r r ~ ~ r ~ < <
exists for all t E ~to, t, ~ .
Similar to (1.5), (1.8) one computes index and stock log-returns from (4.1), (4.2) for the time interval ~to, t, ~ _ ~U,, U,+, t x~ = al + J9SdBs , to 5 t S t, , x~o = 0 , (4.8) ro t Yr = bt + Ja'SdBs , to <- t <_ t, ~ yto = 0 ~ (4.9) where ' M
a, = ar,. .,at r trJCrs 2 (\Bs.l ~ + les.1 )Z + . . . -I- (es.D ~ )~'s~ . . . , t JCY - 2 ~~8~
11 + \Bs,1 ~ + . . . ~- (B fl ~ )~S
b, _ (b; , . . . , b,'~
r CoJlr 2(~~s.n+~6f.z~+...+(~f,D~)~ds,..., J~r -~((~sn'~~~ z~+...+(QsD~)~ds~ .
dBs = (dBs , . . . , dB° ~ .
Let EQ(.) covQ(.,.) be an expected value and covariance matrix under the probability measure Q, conditional to ~ -algebra F,o . Then from (4.8), (4.9) follows that EQ (x~ ) = a , (4.10) E~(y,)=b, (4.11) covQ(x,,xt)=cov(xt,x,)= 6585 ds, (4.12) ro $$
r covQ ~Yr ~ xr ~ = cov(yr ~ xr ~ = f 6s85 ds ~ (4.13) ro r covQ(yr,yr)=cov(yr,yr)= fa~s6s~ds. (4.14) a In particular, varQ (y; ) = vary; ) (4.15) for all n =1,2,..., N . Moreover, covQ (sr , x, ) = cov~s, , xr ) = 0 , (4.16) covQ (el , sr ) = cov(~r , Er ) , (4.17) but E~ ~~r ~ = br - ~rar - ~r~ ~ 0 . (4.18) 5. Credit migration model under Q measure In section 3 we defined the relationship between credit migration events and movements of the normalized stock returns. This allows to compute probabilities of credit migrations for different credit classes under the Q measure.
Let obligor S" E S be in the credit class R;o = R; at time to . According to the assumption in section 3 its credit rating at time t, will be R; = R~ if and only if w; E
(h;°~ (to, t, ), h;",~_, (to, t, )~ , j =1,2,. . ., K , where thresholds h;"~ (to, t, ) are derived from the standard normal distribution.
Then probability under Q measure for obligors S',SZ,...,5"' to migrate from the credit classes R;o = Ri ,...,Rto = Ri,, at time to to the credit classes R; = R~ ,...,RtN =
R~,y at time t, respectively is Q(R; = R~ ,...,Rt" = RAN R;o = Ri ,...,Rto = RiN ) - , > > ~ - ' t7 ( N ( N ' (5~1) wti E ,cii,Ji t0 t~ "ti,Ji ~ t~~t~ ~~'wt~ E ~~.v,.l.v t~'t~ ~hl:v,%;v t t0 tl As follows from the definition of wt = (wl ,. . ., wN ~ and (4.15) wr = 1 n lyr - E(Yr ~~
var yt var y~ ~y' E~ 'Y' ))+ var yr ~E~ 'yr )- E~Yr ~) (5.2) = w° + Cp (to,t) where Q n (y~ - E~ ~Yr ~) (5.3) t var yt and Co (to,t) is a constant given by Co(to~t)= 1 " ~E~(Ya~-E(.Y~ )) var yt _ 1 J (rs _ as ~s var y~ to rs _ as s ~~Os,l~+(~s,2~+...+(6sD~~S 2 to Then for the vector wt = (w; ,. . ., w;' ~ using (3.4), (4.14), (4.15) one has COVQ (Wt , Wt ) = COVQ ~Wr , wt = E~ ~Zv,~r (Yt - E(Yt )XYr - E(Y, ))T Z>..', = ZY'r covQ(Y,,Y,)z,,,'t (5.4) = Zv,l r Cov(Y, ~Yr )Zy,l, = cov(wt , wt ~.
From equations (4.9), (5.3), (5.4) it follows that the random vector wt =~w;,...,w;'~ has a normal distribution under the probability measure Q with zero vector of means, covariance matrix cove ~w, , w, ~ = cov(wt , w, ~ and probability density function fN (xl , x, , . . . , xN ~ . Then wl is a standard normal variable for each n =1,. .., N and qt ; ~to ~ tl ~ = Q~~'~'r, E ~ht~; ~to ~ tl ~~ hinj-I ~to ~ tl ~~) = Q(w" + C" (t t ~ E ~h" (t t ), h." _ (t t ~D
t, D 0 ~ 1 z j 0' 1 :, j 1 0' 1 / / '',l (5.5) Q \wti E lhinj \t0 ~ tl ~ CO \t0 ~ t1 ~~ hinj-1 lt0 ~ tl ~ CO \t0 ~ tl )~
= N~ht j -I ~to ~ tl J - Co ~to ~ tl ~) - N lhr j ~to ~ t~ ~ - Co ~to ~ tl )~
Therefore, similar to transition probabilities p,."j (to --~ t, ~, transition probabilities q"j (to ~ t, under martingale measure Q are defined by shifted thresholds r Ys -G~nI'S
l r _ o gf j lt0 ~ tl ~ - hin j lt0 ~ tl ~ ~
tJ \1as I ~ + \as . 2 ~ + . . . + (d's .. D ~ ~S 2 to Note that transition probabilities q°j (to -~ t, ) depend on n and can be different far different obligors. Equations (5.5), (5.6) show the relationship that exists between two transition matrices TnP~t° -~t,~= Ptj~to -~ty~ and TnQ(t° ~t,)=~~q'~j~t°
~t~~~' Similar to (3.6) for the Q probability of joint migrations we have Q(R;1 =R, ,...,R,"' =R%~ Rro =Ril,...,R,o =Ri,V
Wrl E\"fl..ll~t~'tlh"ti.JWl~tO~tl~l~..,Wr E(lliv,J,v~t~'tl~~~i~,.IN-~~t~'t~~~~
Q(Wa', E(bii..%nt~'t~~'gil,lWl~tO~tly~..,W~1 E(giN,.l.v(tp'tlOgi~,.IN-1~t0~tly (S. I) g!LJI-IOO~tl~ giN.lN-1\rp,lll - . . . f fN (x~ , x, , . . . , xN ~dx, dx2 . . . dxN .
gi~.ll ~r~'~I J OIN.JN ~t~'rl I
6. Construction of transition matrix under Q-measure over risk horizon In this section we suggest, as an application of the theory developed in earlier sections, a practical way to obtain a Q transition matrix for the period [to, t, ] , t, =
to + H , where H
denotes a risk horizon, typically 1 year. We suppose that we are provided with a P transition matrix and a set of Q default probabilities, one for each initial credit rating, over the period. The matrix obtained this way will be used in subsequent sections.
The main point here is that of implementation issue. In practice, the differences between default thresholds of the given P transition matrix and those determined by the given Q default probabilities may not be independent of the credit ratings, contrary to the theory (see (5.6)). This is the case usually because the P transition matrix is obtained empirically from past historical data while the Q default probabilities are obtained from current price data.
Below, we make a modification to the given P transition matrix and then apply the theory to obtain a Q transition matrix over the risk horizon. Note that the last column of such Q transition matrix will coincide with the given Q default probabilities.

First, we introduce some general notation. For obligor S" E S and observation times t > s , s, t E T , let q~" (s ~ t) denote the Q default probability of S" over the period [s, t] given its initial rating of R; at time s , i.e., q,."(s-~t)=Q(Rt =D~Rs =R;), i=1,2,...,K.
Recall that the default state RK = D is absorbing so that qK (s ~ t) =1. Put K
= K -1, and package them in a x -dimensional vector q; (s ~ t) qn (s -~ t) = qi (S ~ t) qK (s ~ t) we have q" (s ~ t) >_ 0 , where 0 denotes the K -dimensional zero vector.
Now suppose we are given a P transition matrix Tp = ~ ~ p"~ ~ ~ for each individual obligor S" , respectively, and Q default probabilities q" (t° -~ t, ) over the risk interval [to, t, ] . Put q;' = qr (to ~ t, ) and z" = N-' (q n ) , 1 <_ i <_ K . Let r jrs-as s n _ n/ _ ro CO CO l t0~t1 t~
r(( 2 ( 2 2 ~~~6s.~ ~ + ~~s.z ~ + . . . + (O-s.D
ro as in Section 5. Define transition matrix TP = ~~ p"~ (to,t, ) ~~ by n N(zn + Co )~ j = K
Pr,i(to~ti) _ _n (i ~ K) (6.1) u~Pr..a ~ j < K
where u. = 1 1 (zn + C° ) . " t t = 0 if < K and "
n ~ Px.; ( o > > ) j Px,K (to ~ t~ ) =1. We have adjusted the - Pr.K
last column of the given matrix TP so that the new P default thresholds satisfy N-' (p; x (to,t' ))= z" + Co for i < K , i.e., they differ from the Q default thresholds z" by constant Co . The other entries corresponding to the probabilities of migration into non-default states are then adjusted while maintaining, for each row, the same relative weights with respect to the survival probability, i.e., Pi, j (t0 ~ tl ) _ Pint _ for all i, j < K . (6.2) 1-P~x(to~t~) 1-Pi~x Now, we apply the theory of change of measure to T P to obtain the desired Q
transition matrix over the risk horizon.
With T P = ~~ p,.""~ (to , t, ) (~ of (6.1 ), let h; , (to, t, ) be its thresholds. In particular, we have h nK (to ~ t~ ) = N ' (P nx (to ~ t~ )) = zi" '+~ Co . (6.3) Let TQ = ~~ q"~ (to, t, ) ~~ be the corresponding transition matrix under Q;
by (5.5) and (5.6), its (i, j) -th entries are defined by q n; (to ~ t~ ) = N(g> >-~ (to ~ t~ )) - N(g n; ~to ~ ti )) ~ i ~ .7 = 1~2~ .
. . , K (6.4) where g"~ (to, t' ) = h;"~ (to, t' ) - Co (to, t, ) . Using (6.3), (6.4) and N-' (-oo) = 0 , we have q ~x (to ~ ti ) = N(hinK (to ~ t' ) - Co ) - N(-°~) = N(Z n ) _ qt (to ~ t~ ) for each i =1,2,..., K . Thus, by construction, the last column of TQ fits exactly the given default probabilities under Q over the period [to,t' ]
qi x ~to ~ t~ ~ qi ~to ~ t~ ) qz.x ~to ~ t~ ) qa ~to ~ t' ) q" (to -~ t' ~ . (6.5) qx.x ~to ~ t' ) qK ~to ~ t~ ) 1 In summary, a suitable adjustment is made to the given P transition matrix such that its last column is compatible with the given Q default probabilities under the theory, i.e., their thresholds differ by a constant C independent of initial credit rating class.
For each row, the remaining entries in the P transition matrix are also adjusted by retaining same relative weights with respect to the survival probability. The constant C is then subtracted from other non-default thresholds of the modified P matrix to obtain the corresponding thresholds, and therefore the entries, of the desired Q matrix.
Note, that in most practical applications transition matrices for individual obligors are not available. Instead only one common for all obligors P transition matrix is given. It can be, for instance, transition matrix published by one of rating agencies like Moody's or S&P, or the internal bank's transition matrix. For such case following modification of the procedure above can be suggested. We assume that Q default probabilities q" (to ~ t, ) are also obligor independent so that q° (to ~ t, ) = q(to ~ t, ) . Let q; be the i -th component of q(to ~ t, ) , and put zr= N ~(qr)~
~'r = N ~ (Pr,rc ) K
Co= 1 ~~~'r-Zr x ;-, Here, for Co , we may want to average the differences w; - z; over only the i -th terms for which q; and p;.x are not too small (so that the thresholds are away from - oo );
also, instead of taking the average, one can choose Co which gives the least mean square error.
Similar to (6.1 ), define the new transition matrix T P = ~~ p~,~ (to, t, ) ~~ by N(z;+C°), j=K
P~.a (t° ~ t~ ) _ (i ~ K) u~p~,J, j <K
1-N(z. +C ) where u; = 1 _' ° ; pK,~ (to,t, ) = 0 if j < K and pK,x (to,t, ) =1.
Therefore, the last - Pr.rc column of the given matrix Tp is adjusted in such way that the new P default thresholds satisfy N-' (p~.x (t°, t, )) = z; + Co , i.e., they differ from the Q default thresholds z; by constant C°
independent of i . Similar to (6.2) we have p''' (to, t, ) _ p''' for all i, j < K .
1- Pt.K (to ~ t~ ) 1- Pr.K
7. Forward transition matrices under Q
In this section, we suggest an algorithm that builds a series of transition matrices matching the given term structure of default probabilities over various maturities.
Throughout this section, we assume that probabilities and transition matrices are under Q -measure. Such series will be useful, for example, when pricing credit derivatives via risk neutral valuation.
Recall that for each obligor S° E S we assume that its credit migration process observed at times in T forms a Markov chain on states R under Q-measure. We view T as 'universal' in that all observation times considered from now on are elements of T . In practice, we have a particular finite subset of observation times known from beginning.

Let T = ~ uo < u~ < ~ ~ ~ < uB ~ ( c T ) be a set of observation times. By a series of transition matrices on T , we mean a sequence of transition matrices over the periods [uo,u~], [u,,u,), ..., [ue_~,ue].
With such series, the transition matrix over any period [u~,ub] with a <b is given by the product of transition matrices over [u" , u"+~ ] , . . ., [ub_~ , ub ] , respecting the order.
Let To = ~ t, =to + lH ~ 1= 0,1,2,...,L ~ be a set of 'standard' observation times, where H is the risk horizon; we assume that To c T . For obligor S" E S , suppose we are given a term structure of default probabilities under Q -measure. By appropriately interpolating if necessary, this means that we are given the Q default probabilities satisfying p < q" (to ~ t~ ) < q" (to ~ t2 ) < . . . ~ q" (to ~ tL ) < a , (7.1 ) where a is the K -dimensional vector whose components are all equal to 1. For b , 0 < b < L , and an invertible transition matrix M over the interval [to, tb ] define vectors qM (tb ~ tr ) as follows:
q M (tb ~ tr ) = M _i q" (to ~ tr ) (7.2a) for l E ~b, b + 1, . . . , L} . Potentially, these vectors may have negative components because M-' can have negative entries. However, if these vectors satisfy O -- qM (tb ~ tb ) C qM (tb ~ tb+1 ~ ~ . .. < qM (tb ~ tL ) < a ~ (7.2b) then we refer to them as the time tb (forward) default probabilities obtained, from the initial term structure of default probabilities (7.1 ) using transition matrix M . Note that the equality in the first relation of (7.2b) is equivalent to the condition that the first K
components of the last column of M equals q°(to -~ tb), i.e., it matches the initially given default probabilities for the period [to > tb ] .
We wish to construct certain series of Q transition matrices T ~ (to ~ ti ) ~ T'"Q (t~ -~ tz ) ~ . . . , T Q (tL_~ -~ tL ) (7.3) on To such that the following proposition is satisfied with Zj = T Q (t~_, ~
tl ) .
Proposition: Given initially a term structure of time to default probabilities satisfying (7.1), there is a set of invertible matrices Z, , Zz , .. ., Z~ such that for each b , 0 < b < L , the time tb forward default probabilities obtained from (7.1 ) using M = Z, ZZ ~ ~ ~ Zb satisfy (7.2b), i.e., they are non-negative and increasing.
Proof of Proposition. We shall produce Z, 's recursively. At each stage, we aim to keep the entries corresponding to the probabilities of non-default migrations as 'close', in terms of maintaining their relative weights (This can be replaced by any other reasonable preference. For example, instead of seeking to retain same relative weights amongst the non-default probabilities, one can skew them in the direction of default probabilities: if default probability has increased then one can increase the relative weights towards lower credit classes, etc.), to those of the transition matrix TQ which was produced from the given data for [to,t~ ] in Section 6.

For the initial interval [t°,t,], let Y =TQ; its last column, qn(t° Wit') by (6.7), matches the given default probabilities for the interval. If Y is invertible and condition (7.2b) already holds for M = Y, , then we set TQ (t° ~ t, ) = Y, . More generally, consider the following 1-parameter family of matrices Zl(a)=aX,+(1-a)Y, 0<_a_<1, (7.4a) where r~ q~
- diag(r,',...~rK) 9n (t° ~tO ~. ~ (7.4b) 0 1 rK gK
0 ~~~ 0 1 with qi = q; (t° ~ t, ) < 1 and r,.' = I - g" (t° -~ t, ) for i =1,2,. . ., x . The matrices Z, (a) all have the same last column and in particular is independent of a and equal to that of Y .
Choose the "smallest" a , call it a, , in (7.4a) such that vectors qM (t, ~ t~
) defined by (7.2a) with invertible M = Z, (a) satisfy (7.2b) with b =1 there. (in practice, we fix a positive integer 1 > ~ from beginning, and restrict values of a in the set A(I) _ {o,l l 1.2 /
l,. --, (1-1) l 1,1} . Then a, is chosen to be the smallest a in A~l) for which condition (7.2b) holds.) That such a, exists is clear since for a =1, the condition is satisfied. Thus, we have the time t~
forward default probabilities p < q ~ (t~ ~ t2 ) < q~ (t~ ~ t3 ) < . .. < q ~ (t~ ~ tL ) < a (7.5) with respect to M = Z, (a, ) ; here, subscript M is omitted. We set TnQ
(t° ~ t, ) = Z, (a, ) .

Now, for subsequent periods [ti , tJ+~ ] , l >_ 1, we proceed recursively to construct invertible transition matrices. Using the time t, forward default probabilities (7.5) as given, we can obtain in a similar way an invertible transition matrix ZZ on [t, , t~ ] whose last column is compatible with the (forward) default probability q n (t, -~ t, ) and whose other entries have, in each row, the same relative weights as those of TQ (or, as before, have the weights distributed according to some preference). Note that the time t2 forward default probabilities obtained from the time t, default probabilities (7.5) using ZZ of [t, , t, ) will be the same as the ones obtained from the time to default probabilities (7.1) using Z,ZZ on [to,t,]. Repeating the procedure for intervals [tZ , t3 ] , [t3 , t4 ] and so on, we construct a desired series of transition matrices, (7.3), on To .
The key point about the construction is that the series of transition matrices built up this way fits the given term structure of default probabilities over each maturity tr , 0 <
l < L , in that the cumulative transition matrix Z, ZZ ~ ~ ~ ZI on the interval [to , t, ] has (the first x components ofJ
the last column matching the given q" (to ~ t~ ) .
Remark. There are many choices for such series. We have suggested simply one method that is practical and reasonable in incorporating much information about the P
transition matrix given for the initial period as well as preserving non-negativity and increasing of the forward default probabilities defined by the constructed transition matrices. Other alternate methods for producing such series can be made available. Actually, in practice, one may prefer to be given as exogenous input a series of forward Q transition matrices in place of the initial term structure of Q default probabilities.

8. Joint process under Q on T and application to pricing of credit swaps In the previous section, we have built series of Q transition matrices on standard observation times To for single processes R; . In application, for example in pricing of credit derivatives, we need to 'translate' this series to cover a different set of observation times T = ~ uo < u, < ~ ~ ~ < ue ( T includes 'nonstandard' time steps corresponding to payment dates, times to instrument's maturity, etc., which need not belong to To ) for a joint process. For this, it is sufficient to describe it for single processes, for then the joint migration Markov process will be determined by (5.7) on each period [uk_, , uk ] as the co-migration of the obligors are governed by the correlation among their asset returns. We make the assumption, in lack of any better information, that this correlation is constant in all periods, but this too can be given as input and made different for different periods. For the single process, it essentially requires that we provide a way to 'interpolate' a transition matrix for a given period to define a transition matrix for a subinterval of that period. We shall simply take a certain Taylor series expansion of the 'fractional' power of the transition matrix, the power being proportional to the subinterval's length, and when necessary make a minor adjustment in some cases.
Let Z be a transition matrix for the period [t~ , t~+, ] and suppose [u, v] c [t~ , to+, ] is a proper subinterval; put 8 = (v - u) l(tQ+, - t~ ) . Now consider the matrix Z B
defined by Ze =~((e))k(z-I)k ~ (s.l) k=0 where ((8)) k = e(e -1) ~ ~ ~ (8 - k + 1) / k! , as in the Taylor expansion around the identity matrix 1.
When series (8.1) converges, we may truncate it after a certain number of terms to evaluate ZB
with a small error. Note that a typical transition matrix, especially over a period that is not too long, is 'near' I . In actual implementation of (8.1), it may give rise to possibly negative entries, though small in size. In this case, one makes an adjustment in the resulting matrix, for example by replacing such entries by 0 and decreasing the adjusted amount from the diagonal probability.
In most cases, this procedure works well and is robust.
We summarize various constructions made thus far. Starting with a P transition matrix on the risk horizon and an initial term structure of Q default probabilities we construct series of transition matrices for single processes on standard observation times. These are then translated to produce series on a prescribed set of observation times T , first for the single processes and then for the joint process. Of course, for single processes, the Q series thus obtained can be used to imply the corresponding P series if differences in their thresholds are provided.
Application: We indicate how the constructed series of Q transition matrices can be used in evaluating credit swaps. A credit swap on a corporate bond or a reference facility provides protection to the holder of the reference bond from the loss of value when the bond issuer defaults. The buyer of the contract makes a series of premium payments, contingent upon the survival of either the seller or the bond issuer. Here, the scheduled dates for the premium payments are used as observation times T and the joint process will be that of the asset returns of two obligors, namely the counterparty C and the issuer B of the reference facility.

The price of a credit swap can be computed as the expected value, under Q-measure, of the difference between fixed rate and contingent payments. Also, when valuing the credit swap for risk measurement, it is important to consider whether the counterparty is a seller or a buyer, as the two situations are not exactly the opposite of each other. The payoff function for the contingent payments can be given generically in the form Xcp - ~.fEIF ~ (8.2) E
the sum is over an exhaustive set ~E~ of all possible, mutually exclusive events, IE is the indicator function of event E , and fE is the cashflow, usually of what was promised in the contract or some fraction of it due to seller's own default, when event E
occurs. Similarly, one has the payoff function XFp for the fixed payments side (buyer).
Denoting by T~ and iB , respectively, the random default times of the counterparty C (seller) and the reference party B , then typically E is an event with a specified range for values of i~
and iB . For example, E = ( event that u,_, < zB _< u, and r~ > u, ) where u,_, < u~ are two specified times, usually a pair of consecutive premium payment times.
Depending on E , fE is a function involving notional amount or forward values of the reference facility, a strike price, recovery rates of B and C, etc. and depends on whether premium payments are made in advance or in arrears. Both E and fE can be made explicit according to the credit swap's specification.
The price of the contingent claim is the expected discounted cashflows under the risk neutral pricing measure Q. It is equal to 7~
dEfE Q(E) ~ (8.3) E
Given a term structure of interest rates, the risk free discounting, d E fE , of cashflows fE can be approximated reasonably in a straightforward manner. Therefore, the remaining key ingredient in pricing credit swaps is to evaluate Q(E), where E usually are some joint events pertaining to the default times i~ and iB . In practice, all the probabilities Q(E) appearing in pricing formula (8.3) can be expressed in terms of the entries of the constructed transition matrices. We illustrate this, for example, for the following event:
E = ( event that uo < zB <_ uk-, and uo < z~ <_ uk ) .
Let to = (R;o , R jo ) be the initial ratings of (B, C) at time uo and Bk the rating of B at time uk (and similarly for Ck , etc). Then, we have Q(E) = Q(Bk-I = D~ Ck = D ~ I o ) x - ~ Q(Ck - D~ Bk-1 - D~ Ck-t - R j ~ 10 ) J=I
K
- ~ Q(Ck - D ~ Ck-1 - R j ) Q(Bk-1 - D~ Ck-I - R j ~ 10 ) j=1 where, for the last equality, we have used the Markov property for both the joint and single processes. Note that the probabilities appearing in the final stage are entries of the transition matrices, over periods [uk-I,uk ] and [uo,uk-I ] . Hence Q(E) are easily attained from the transition matrices. This completes the final piece of computation in pricing credit swaps.

~1 Part 3 - Stochastic Rates 1. Introduction CreditVaR model with stochastic interest and FX rates (CreditVaR II) is an extension of the CreditVaR model described in the Part I (referred further as CreditVaR I). The chart in Figure 6 describes the implementation of CreditVaR II model. CreditVaR II is a purely simulation engine and relies on generating correlated risk factors without pre-generation (like CreditVaR I), and full re-pricing of the portfolio for each scenario. Outputs are Value-at-Risk and various other statistics for portfolio as a whole as well as standalone and marginal statistics for each instrument in the portfolio.

2. Simulation Methodology 2.1. Simulation of Base Curves Let 0 be the current time and h be the risk horizon. Let y(t,T) be the continuously compounded zero coupon rate seen at time t for a bond maturing at t + T . Note that T is the term to maturity, not an absolute time point. Let f (t, T, , Tz ) be the forward rate seen at time t for a contract starting at t + T, and maturing at t + T, . We assume that the zero curve maturity buckets are the same for all currencies and all credit ratings, and we denote the maturity terms by B = ~t, , tz , . . . , tb ~ , where to = 0 < t, < tZ < ~ ~ ~ < tb . For convenience, we denote the zero curve as seen at time h by y = ( y, , y2 , . . . , yb ) _ ( y(h, t; ), y(h, t2 ), ~ ~ ~
, y(h, tb )) and the one seen at time zero by y° _ (y,°, y°,..., yb ) _ (y(0,t;),y(O,tZ),~ ~
~,y(0,tb)) . The zero rate for a term between two bucket points is determined by linear interpolation. (If t < t, or t > tb then the zero rate is set to y, or yb , respectively.) Our first objective is to generate samples of y, which are correlated within itself, spreads, FX
rates and equity returns. (Note that there is a vector y for each currency. We dropped the subscript for currency to avoid heavy notation.) Consider the forward rates seen at h , denoted by x = (x, , xZ ,. .., xb ) _ ( f (h, to, t, ), f (h, t, , tZ ), ~ ~ ~, f (h, tb_, , tb ) ) . x can be given by x, = y, and x; = y't' y'-'t'-' for i > 1. ( 1 ) tr - t~-i Conversely, r y. _ - ~ xk (tk - tk-~ ) tr x=~

Written in matrix forms, Y1 t' t z t' xl t, tz Yz xz t, tz - t, t3 - tz x (3) Y3 t3 t3 t3 3 Yb xb tt t2 - tl tb - tb_I
tb tb tb Denoting the above transition matrix by T, then x ' = T -' y ' where the inverse matrix is given explicitly as t~ tz tz - tt tz - t~
T-' (4) tb_1 tb tb - tb_t tb - tb_1 The following method is proposed to generate interest rate scenarios. Let x° _ (x°,x°,...,x°) _ i z h ( f (0, t°, t, ), f (0, t, , tz ), ~ ~ ~, f (0, tb_, , tb ) ) be the forward rates seen at time 0. We assume that the forward rates x are lognormally distributed and that the log relative changes ~ _ (~, , ~ z , . . ., ~b ) _ (ln xo , In xo , . . ., In xp ) follow a multivariate normal distribution N( ~,, S2) , x, xz xb where p.=(,u,"uz,...,,ub) are drifts and S2 is the covariance matrix.
Therefore, the forward rates are guaranteed to be positive.
To simulate the state of zero rates at time h in the future (risk horizon), we first generate a correlated normal sample ~ _ (~, , ~z ,. .., ~b ) from the distribution as defined above, obtain x _ (x,° exp ~~ , . . . , xb exp ~b ) , then apply equation ( 1 ) to get continuously compounded zero rates y.
2.2. Simulation of Spreads Given a credit rating R; , i =1, . . . , K -1, let s; be the spread curve for Ri over that of the base curve y . Thus, y + s; is the zero curve for R; rating. The restrictions on variability of the spreads are the following: a) the corporate curves should increase with decreasing the credit rating; b) the corporate curves should not overlap for any time point t. We propose the following scheme for generating stochastic credit spreads. In order to reduce the number of risk factors in the simulation and satisfy the restrictions above, we generate the spread curves by a combination of Monte Carlo simulation and linear interpolation. The risk factors involved in simulation of spreads are:
- Credit spread s, (t, ) for the highest credit rating for (typically Aaa) and for maturity equal to the first bucket point on the zero curve ( t~ ) - incremental spreads ~.s; (t, ) = s; (t, ) - s;_1 (t~ ) , i = 2, ..., K -1 for maturity t, - ~sx-, (tb ) = sx-, (tb ) - sK-, (t~ ), i.e., the difference between the spreads for the lowest credit rating for the longest and shortest maturity.
The simulation relies on the lognormal assumption for the above risk factors.
After they are generated as a part of a simulation scenario, we obtain spread curves by the following procedure:

1. Obtain s~ (t, ) = s;_, (t, ) + ~.s; (t, ) . Lognormal assumption on 4s; (t, ) ensures that spreads are increasing when credit rating is decreasing, i.e. s; (t, ) > s;_, (t, ) for i = 2,...R .
2. Obtain the sK_, (tb ) = sx_, (t, ) + ~sx_, (tb ) . We assume that Osx_, (tb ) has lognormal distribution. This way we make sure that the spread curve doesn't decrease with maturity for any scenario.
3. Obtain the rest of the spreads for maturity tb by partitioning the interval ~0, sx_, (tb )~
proportionally to s; (t, ) by taking s; (tb ) = s; (t, ) sx-' (tb ) . This way, we get spreads that Sx-~ (t~ ) increase when credit rating declines for maturity tb.
4. Obtain the interior of the spread curves by linear interpolation between s;
(t, ) and s; (tb ) .
The corporate zero curves at risk horizon are obtained by adding the appropriate spread to the base curves.
2.3. Simulation of FX rates The FX rates at risk horizon are assumed to be lognormaly distributed.
2.4. Simulation of credit migrations Credit migrations and defaults are simulated similarly to the methodology of CreditVaR I (see Part 1). The assumption is that credit migrations are generated by the change in asset value of the obligor, which is linked to industry participation and idiosyncratic component. In the framework of CreditVaR II, the migrations are based on the transition matrix, idiosyncratic obligor returns and the correlation of industry indices with the rest of the risk factors and within themselves.
2.5. Statistics Statistics contain means and standard deviations for some risk factors (forward rates, spreads, FX
rates) and the covariance matrix of all the risk factors including industry indices.
3. Input Data Obligor data:
Information about obligors is organized into a database containing details of their credit ratings, industries, and countries.
Portfolio data:
Information about financial positions is organized into portfolios of exposures. It covers different types of instruments such as fixed income instruments, loans, commitments, letters of credit, etc.
Market data and transition probabilities:

These include yield curves, spread curves, foreign exchange rates, transition probabilities from one credit rating to another for different credit rating systems (Moody 8 states, Moody 18 states, S&P 8 states) Statistics:
Statistics contain means and standard deviations for some risk factors (forward rates, spreads, FX
rates) and the covariance matrix of all the risk factors including industry indices.
4. Risk Statistics and Measurement CreditVaR II produces risk measurements that are similar to the ones produces by CreditVaR I.
On the portfolio level, the following quantities are calculated:
- Present value and Forward value of the portfolio at risk horizon - Distribution of the portfolio value at risk horizon and its mean, standard deviation, skewness and kurtosis - Value-at-Risk numbers at various percentile levels with 95% confidence bands - Expected Excession (or Expected Loss Given Default) of the percentile level On the instrument level, the following is calculated and reported:
- Present value and Forward value of the instrument at risk horizon - Mean and Standard Deviation of the instrument value at risk horizon - Marginal Standard Deviation and Delta-Standard Deviation (see below) of the instrument for a given portfolio - Capital amount (see below) - Optionally: Standalone Value-at-Risk numbers at various percentile levels - Marginal Standard Deviation of instrument - Delta-Standard Deviation of instrument 5. Credit Migration Model Credit quality migrations are modelled according to Credit VaR I methodology ( Part 1).
6. Simulation Engine First, the engine prepares the necessary data for simulation. Threshold levels of standardized asset returns representing credit rating changes are determined for each obligor in the portfolio using the transition probabilities; this part depends only on the obligor's credit class and not on the obligor itself. Relevant data is read from the database and is stored in the appropriate data structures in memory.
The scenarios are generated based on the covariance matrix and other statistics of the risk factors that are relevant for a particular portfolio. Each scenario is composed of the base curve part, spread curve part, FX part and credit migration part.
For each of these scenarios, portfolio has to be revalued. In CreditVaR I
model it was done quickly using pre-processed prices of instrument. CreditVaR II does full revaluation of the portfolio, without cash flow bucketing or any portfolio compression, based on the set of generated risk factors. Each time a default occurs in scenario, i.e., when the sampled asset return value of an obligor is below the default threshold level, a random recovery rate is generated according to a beta-distribution whose defining parameters are governed by seniority associated to the obligor's instruments. Finally, we obtain a distribution of portfolio values and from it the relevant risk statistics.
7. Implementation The Global Analytics CreditVaR II program utilizes the methodology described above. The core computational engine is implemented in C++. Currently, interface for Windows is being developed, using the ADO database technology. Windows version currently utilizes a Microsoft Access database. ADO technology allows CreditVaR II to be used with the majority of available databases.
It is noted that those skilled in the art will appreciate that various modifications of detail may be made to the preferred embodiments described throughout this specification, which modifications would come within the spirit and scope of the invention as defined in the following claims.

Claims (19)

1. A method of creating an adjusted credit risk transition matrix of probabilities of credit migration comprising the steps of:
~ obtaining a transition matrix under probability measure (P), ~ adjusting the (P) transition matrix to be consistent with default probabilities under martingale measure (Q) by making a column of default probabilities in the (P) transition matrix consistent with default probabilities calculated under (Q), and ~ scaling the other entries in the adjusted (P) transition matrix to compensate for the adjustment while retaining the relative weights among non-default classes from the obtained (P) transition matrix in the adjusted (P) transition matrix.

2. A credit risk model for the analysis of complex credit instrument, comprising:
an obtained transition matrix under probability measure (P), and an adjusted (P) transition matrix based on the obtained (P) transition matrix and consistent with default probabilities under measure (Q) by making a column of default probabilities in the adjusted (P) transition matrix consistent with default probabilities calculated under (Q), wherein the other entries in the adjusted (P) transition matrix are scaled to compensate for the adjustment while retaining the relative weights among non-default classes from the obtained (P) transition matrix in the adjusted (P) transition matrix.

3. The credit risk model of claim 2, further comprising a Monte Carlo simulation engine to generate valuation scenarios based on probability distribution computed using the adjusted (P) transition matrix.

4. The model of claim 2 or 3, wherein the model is implemented in computer software on a computer readable medium.

5. The model of claim 4, wherein the software is for use on compatible hardware.

6. The model of claim 2 or 3, wherein the adjusted (P) transition matrix is used to price credit swaps.

7. The method of claim 1, wherein the step of adjusting the obtained (P) transition matrix is performed in accordance with the following:
R = {R1,..., R K} - are all credit classes R K - corresponds to default state T P = ¦¦P i,j(t0, t1)¦¦ - adjusted (P) transition matrix for the time interval [t0, t1]
p1,j(t0,t1) - transition probability from credit class R i to credit class R j P K, j(t0, t1) = 0 if j < K and p K,K (t0, t1) = 1 ~ i = N-1 (P i.K) Z i = N-1 (q i) ~ i,K - obtained (P) probability of default for the credit class R i (from the obtained (P) transition matrix) q i - (Q) probability of default for the credit class R i G - a set of credit classes (can be the whole set R = {R1,...,R K} or some subset ¦G¦ - denotes the number of elements in G ; if G is empty, put C0 = 0 N() - standard normal cumulative density function.

8. The method of claim 1, wherein the step of adjusting the (P) transition matrix is performed in accordance with the following:
T ~ = ¦¦ P ~,j(t0, t1)¦¦ - adjusted (P) transition matrix for the time interval [t0,t1] for obligor n p~, j(t0,t1) - obligor's n transition probability to migrate from credit class R i to credit class R j P~,j(t0, t1) = 0 if j < K and p ~,K (t0, t1) =1 Z~ - N-1 (q~) ~~,K - obtained (P) probability of default for the obligor n over interval [t0,t1] if it is in the credit class Ri at time t0 q~ - (Q) probability of default for the obligor n over interval [t0,t1] if it is in the credit class R1 at time t0 r t - instantaneous risk-free interest rate at time t y~ - obligor n stock log-return over interval [t0, t1]
.alpha.~ - obligor n drift function of time t N() - standard normal cumulative density function.

9. A method of pricing a credit swap, comprising the steps of:
~ Obtaining a (P) transition matrix over a risk horizon [t0,t1] and the (Q) default probabilities over the same period [t0,t1];
~ Creating an adjusted (P) transition matrix and (Q) transition matrix over [t0,t1];
~ Building a first series of forward (Q) transition matrices for standard observation times t=t1,t2,t3,... from the (Q) transition matrix and an initial term structure of default probabilities under Q over [t0,t1] ;

~ Translating the first series to produce a second series, having premium payment dates as the observation times by working with 'fractional' powers of transition matrices for subintervals;
~ Deriving a third series of transition matrices from the second series for co-migration on credit classes for pairs of obligors;
~ Multiplying cash flow for each payment date by a risk-free discount factor and by probability contained in the third series and summing the computed series for all payment dates; and ~ Subtracting the sum from a payoff function for fixed payments.

10. A method of pricing a credit swap, comprising the steps of:
~ Obtaining a (P) transition matrix over a risk horizon [t0,t1] and the (Q) default probabilities over the same period [t0,t1];
~ Creating an adjusted (P) transition matrix and (Q) transition matrix over [t0,t1];
~ Building a first series of forward (Q) transition matrices for standard observation times t = t1,t2,t3,... from the (Q) transition matrix and an initial term structure of default probabilities under Q over [t0,t1];
~ Deriving a third series of transition matrices from the first series for co-migration on credit classes for pairs of obligors;
~ Translating the third series to produce a second series, having premium payment dates as the observation times by working with 'fractional' powers of transition matrices for subintervals;

~ Multiplying cash flow for each payment date by a risk-free discount factor and by probability contained in the second series and summing the computed series for all payment dates; and ~ Subtracting the sum from a payoff function for fixed payments.

11. A method of pricing a credit swap, comprising the steps of:
~ Obtaining a (P) transition matrix over a risk horizon [t0,t1] and the (Q) default probabilities over the same period [t0,t1];
~ Creating an adjusted (P) transition matrix and (Q) transition matrix over [t0,t1];
~ Building a second series of forward (Q) transition matrices for premium payment dates from the (Q) transition matrix and an initial term structure of default probabilities under Q over [t0,t1];
~ Deriving a third series of transition matrices from the second series for co-migration on credit classes for pairs of obligors;
~ Multiplying cash flow for each payment date by a risk-free discount factor and by robability contained in the third series and summing the computed series for all payment dates; and ~ Subtracting the sum from a payoff function for fixed payments.

12. A system for pricing a credit swap, comprising:

~ Means for obtaining a (P) transition matrix over a risk horizon [t0,t1] and the (Q) default probabilities over the same period [t0,t1];
~ Means for creating an adjusted (P) transition matrix and (Q) transition matrix over [t0,t1];
~ Means for, from the adjusted (P) transition matrix and an initial term structure of default probabilities under Q over [t0,t1], building a first series of forward transition matrices for standard observation times t=t,t1, t2, t3,...;
~ Means for translating the first series to produce a second series, having premium payment dates as the observation times by working with 'fractional' powers of transition matrices for subintervals;
~ Means for deriving a third series of transition matrices from the second series for co-migration on credit classes for pairs of obligors;
~ Means for discounting cash flow for each payment date over the period by a risk-free discount factor and probability contained in the third series, and summing the computed series for all payment dates; and ~ Means for subtracting the sum from a payoff function for fixed payments.

13. A system for pricing a credit swap, comprising the steps of:
~ Means for obtaining a (P) transition matrix over a risk horizon [t0,t1] and the (Q) default probabilities over the same period [t0,t1];
~ Means for creating an adjusted (P) transition matrix and (Q) transition matrix over [t0,t1];
~ Means for building a first series of forward (Q) transition matrices for standard observation times t=t1,t2,t3,... from the (Q) transition matrix and an initial term structure of default probabilities under Q over [t0,t1];

~ Means for deriving a third series of transition matrices from the first series for co-migration on credit classes for pairs of obligors;
~ Means for translating the third series to produce a second series, having premium payment dates as the observation times by working with 'fractional' powers of transition matrices for subintervals;
~ Means for multiplying cash flow for each payment date by a risk-free discount factor and by probability contained in the second series and summing the computed series for all payment dates; and ~ Means for subtracting the sum from a payoff function for fixed payments.

14. A system for pricing a credit swap, comprising the steps of:
Means for obtaining a (P) transition matrix over a risk horizon [t0, t1] and the (Q) default probabilities over the same period [t0, t1];
~ Means for creating an adjusted (P) transition matrix and (Q) transition matrix over [t0, t1];
~ Means for building a second series of forward (Q) transition matrices for premium payment dates from the (Q) transition matrix and an initial term structure of default probabilities under Q over [t0, t1];
~ Means for deriving a third series of transition matrices from the second series for co-migration on credit classes for pairs of obligors;
~ Means for multiplying cash flow for each payment date by a risk-free discount factor and by probability contained in the third series and summing the computed series for all payment dates; and ~ Means for subtracting the sum from a payoff function for fixed payments.

15. Software on a computer readable medium for pricing a credit swap, comprising instructions to:
~ Obtain a (P) transition matrix over a risk horizon [t0,t1] and the (Q) default probabilities over the same period [t0,t1];
~ Create an adjusted (P) transition matrix and (Q) transition matrix over [t0,t1];
~ Build a first series of forward (Q) transition matrices for standard observation times t=t1,t2,t3,... from the (Q) transition matrix and an initial term structure of default probabilities under Q over [t0,t1];
~ Translate the first series to produce a second series, having premium payment dates as the observation times by working with 'fractional' powers of transition matrices for subintervals;
~ Derive a third series of transition matrices from the second series for co-migration on credit classes for pairs of obligors;
~ Multiply cash flow for each payment date by a risk-free discount factor and by probability contained in the third series and sum the computed series for all payment dates; and ~ Subtract the sum from a payoff function for fixed payments.

16. A method of modeling stochastic interest rates, credit spreads and foreign exchange rates comprising the steps of:
~ Computing obligors' credit ratings at risk horizon based on country/industry market index logreturns and idiosyncratic components;

~ Computing base zero interest rates for all currencies and given maturities t0,t1,...,t b from base forward rates f(t0,t1), f(t1,t2),...,f(t b-1,t b) corresponding to time intervals [t0,t1], [t1,t2],...,[t b-1,t b];
~ Computing credit spreads s i(t1)=s i-1 (t1)+.DELTA.s i (t1) for maturity t1 and credit classes i=2,...K-1 (by currency);
~ Computing credit spread s K-1(t b)=s K-1(t b) + .DELTA.s K-1, (t b) for the lowest credit rating and longest maturity t b (by currency);
~ Computing the rest of the spreads for maturity t b by partitioning the interval [0,s K-1(t b)]
proportionally to s i(t1) by taking (by currency);
~ Computing spreads for other maturities t2,t3,...,t b-1 by linear or any other type of interpolation between s i(t1) and s i(t b) (by currency);
~ Computing the corporate zero curves at risk horizon by adding the appropriate spread to the base curves (by currency); and ~ Computing foreign exchange rates.

17. A credit risk model for the analysis of portfolios of credit instruments, comprising:
~ Means for computing obligors' credit ratings at risk horizon based on country/industry market index logreturns and idiosyncratic components;

~ Means for computing base zero interest rates for all currencies and given maturities t0,t1,...,t b from base forward rates f(t0,t1), f(t1,t2),...,f(t b-1, t b) corresponding to time intervals [t0,t1], [t1,t2],...,[t b-1,t b];
~ Means for computing credit spreads s i(t1)=s i-1, (t1) + .DELTA.s i(t1) for maturity t1 and credit classes i = 2,...K -1 (by currency);
~ Means for computing credit spread s K-1, (t b) = s K-1(t1) + .DELTA.s K-1, (t b) for the lowest credit rating and longest maturity t b (by currency);
~ Means for computing the rest of the spreads for maturity t b by partitioning the interval [0,s K-1(t b)] proportionally to s i(t1) by taking (by currency);
~ Means for computing spreads for other maturities t2,t3,...,t b-1, by linear or any other type of interpolation between s i(t1) and s i(t b) (by currency);
~ Means for computing the corporate zero curves at risk horizon by adding the appropriate spread to the base curves (by currency); and ~ Means for computing foreign exchange rates.

18. The credit risk model of claim 17, wherein, all risk factors (interest and foreign exchange rates, market index returns, idiosyncratic components) have a joint probability distribution so that credit migration process and interest and foreign exchange rates are correlated.

19. The credit risk model of claim 18, further comprising a Monte Carlo simulation engine to generate valuation scenarios based on Monte Carlo simulations of risk factors.