CREDIT RISK ESTIMATION SYSTEM AND METHOD

Abstract: A method creates an adjusted credit risk transition matrix and related matrices. The method uses the steps of obtaining a real probability measure (<i>P</i>) transition matrix, and adjusting the (<i>P</i>) transition matrix to be consistent with default probabilities under martingale measure (<i>Q</i>) by making the last column (default probabilities) in the (<i>P</i>) transition matrix consistent with default probabilities calculated under (<i>Q</i>). The method further scales the other entries in the transition matrix (<i>P</i>) to compensate for the adjustment while retaining the relative weights among non-default classes from the (<i>P</i>) 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 (<i>P</i>) 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's™ 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, systc ms and methods for estimating credit risk and credit migration associated with credit in struments 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 (/') transition matrix to be consistent with default probabilities under martingale measure (O) computed, for instance, from bond market prices by making a column of default probabilities in the (/') transition matrix consistent with default probabilities calculated under (O) • scaling the other entries in the (/') 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 mav adjust the (P) transition matrix in accordance with the following R = {R ] R } - are all credit classes R - corresponds to default state / F = p, / 0 ,t, ) - adjusted (P) transition matrix for the time interval [/„,/, ] p ι J (t (] ,t i ) - transition probability from credit class R to credit class R p K j (t o , t, ) = 0 if j < K and /7 A A (/„ ,/, ) = 1 W '( A ) r,= N^(.7,) 1-N(z, +r n ) υ, = -*_ , λ - obtained ( ) probability of default for the credit class R (from the obtained (P) transition matrix) q t - (0) probability of default for the credit class R G - a set of credit classes (can be the whole set R = {R ] R κ } or some subset I G I - denotes the number of elements in G , if G is empty, put = o N( ) - standard normal cumulative density function Alternatively, the method may adjust the (P) transition matrix in accordance with the following T n p = || p" j (t υ ,f )\\ - adjusted (P) transition matrix for the time interval [/„,/,] for obligor n P" j ( l o' ( \ ) - obligor's n transition probability to migrate from credit class /?, to credit class R ; p K n ] (t 0 ,f) = 0 if J<K and A (t 0 ,t,) = l r - a ft : C" *-o ( 'o »'ι var M r; = N-' (y;) i i -- Nv ((: υ z; + ) - P. K p" κ - obtained (P) probability of default for the obligor n over interval [t 0 , t, ] if it is in the credit class R ι at time t (1 q" - (Q) probability of default for the obligor /; over interval [to, /, ] if it is in the credit class R at time /„ r t - instantaneous risk-free interest rate at time / y" - obligor n stock log-return over interval [t 0 , t, ] a" - obligor // 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 mav 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 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 [t 0 , t, ] and the (O) default probabilities over the same period [t 0 , t, J are obtained • an adjusted (P) transition matrix and (0 transition matrix over [t 0 , t, J are created • a first series of forward (0 transition matrices for standard observation times / = /, ,t, ,/ 3 ,. is built from the (0 transition matrix and an initial term structure of default probabilities under (J over [t 0 ,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 ot 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 (/ • *) 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(t h } ,t h ) for time intervals [t 0 J,], ,[t h ,,/ corresponding to given maturities /,./, f • credit spread s,(/,) for the highest credit class R l and for maturity equal to the first bucket point I on the corporate zero curve for each currency • incremental spreads Λs(/ 1 )= s (/,)- s ; ,(/,), t = 2 λ ' - 1 , for maturity / (by credit class and currency) • the difference Λs, ] (l h )= s A ,( _ s < ι ( between the spreads for the lowest credit rating for the longest and shortest maturity (bv 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 υn forward rates /('„ . /(',.':) /(' Λ I ) • credit spreads s ; (/, ) = s_ , (/, ) J - A (t, ) for maturity /, , / = 2, ...A " - 1 (by currency) • credit spread s κ fι ) = s l / fΔ.s, ft h ) for the lowest credit rating and longest maturity l h (by currency) • the rest of the spreads for maturity Λ by partitioning the interval [o, .s ; ft h j] proportionally to s (/,) bv takirnz Λ (t h )= s (t^-^ — — (by currency) • spreads for other maturities C, l h , by linear interpolation between .s (/,) and - (t 6 ) (by currency) • corporate zero curves at risk h oπzon by aαding 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 DESCRI PTION 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 1 , 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 1 , and 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 (0 In the preferred embodiment, the (P) transition matrix is adjusted by making a column of default probabilities consistent with default probabilities calculated under (0 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 mav be implemented in computer software as part of a credit risk system using compatible hardware The adjusted (/ • ") transition matrix mav be used in the pricing of credit swaps Credit swap pricing mav be incorporated into the credit risk model as it is consistent with the metnodology used in the model I his description has tour Parts Part 1 describes assignee s basic credit risk model, CreditVaR , which utilizes deterministic interest and foreign exchange rates Each obligor s standardized equity returns are decomposed into weighted average ot 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 Part 1 has Appendix 1 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 mboc iment, 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 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 probabihtv 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 I - Basic CreditVaR Meth 3doIoςy 1. Introduction CredιtVaR 1 M 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 CreditMetπcs 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 CreditMetπcs - Technical Document, April 2, 1997, J P Morgan & Co Incorporated that is hereby incorporated bv reference and available at www creditmetπcs 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 pr babilities and correlations These include yield curves, spread curves foreign exchange rates, transition probabilities from one credit rating to another for different credit rating systems (Moodv 8 states Moody 1 8 states, S&P 8 states) correlations between market indices The CreditVaR methodology mav be implemented both analvticallv and as a Monte Carlo simulation It calculates two risk measures standard deviation (in the analytic and simulation engines) and pcrcentile 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 anv statistic including ones deemed useful in risk measurement In particular I ' alite-at-Risk for anv percentile level together with a confidence band, can be computed (Another risk statistic that can be produced from SE is avera e 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 anv Monte Carlo method With additional computation, both engines AE and SE produce marginal standard deviation by each obligor 1 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 anv 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 Engιne(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 curv e, 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 preprocessing 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 t us model the only parameters which affect the risk of a portfolio are the two already men loned namely the transition probabilities for each obligor and the correlations between asse 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 onlv parameters to be estimated are these correlations It should be noted that interest rates and spreads ai e 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 ot some benchmark countrv-industrv 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 quahtv 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 covaπance matrix, samples of standardised multivaπate 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, 1 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+^ 1 The interface for Windows , could be a combination of Microsoft Access 7 /Microsoft Excel 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 covaπance 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 anc 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 everv 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 • Covaπances between the market indices underlying obligors' capitalization 3 Spot and forward ( 1 year) exchange rates 4 Market Data exposure data (PortEfoho) • For bonds principal amou it of the bond, coupon rate or spread, seniority (defined by recovery rate and its standa rd deviation), maturity of the bond, currency, etc » 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 bv obligor • Total present value of the bonds bv 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%o confidence intervals for the VaR values Marginal standard deviation and marginal VaR for a given obligor / are defined as T' D '-'total ohli σr, ■ > where STD, tot,al, ,' VaR l,oS,al, - are portfolio standard deviation and VaR,^ and STD t,ot,al. ob .h. ^or l ' ^ a ^, n i u i , M , K „ r " 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 / position value A AΛW > multiplied by / „„, w DeltaSTD .. Ptihlll-nr, ' _ VaR,,^ Delta VaR ob .. or P'Miyor, ( PtMiςυr, DeltaSTD Mιχ r and Deltal aR uM satisfy following equations [3] TD^ = ∑ IJeltaSTD ohlι ^ , yaR IMdl = ∑ D lta\ ' aR Mκorι In other words. DeltaSTD M ( DeltaVaR M ) is an obligor / 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 peicentages - capitalization of the company 9? Current zero-rate curves for comp rung present values of the bonds (fragment) Forward ( 1 year) zero-rate curves tor computing forward values of the bonds ( " fragment) Note forward curves were obtained from the current curves by bootstrapping method Table of foreign exchange rates (spot and forward) Covanance matrix of country-industry indices (fragment, actual size 211 by 211) Transition probabilities of credit π ting migrations (Moody-8 rating system) Note Any transition probability matrix may be used (not necessarily 8-state) Example 1 Sample bond portfolio "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 Portfolio Percentile (10° o ) = -105932.78 Portfolio Percentile (5°) = -297306.87 Portfolio Percentile ( ) -812778.79 Portfolio Percentile ( 0 . 51 ) -970228.53 Portfolio Percentile (0.11,) -4912661.85 Example 2 (actual program output) Exposure data (fragment of input, actual size is 1863 exposures, 207 obligors) 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 where r is standardized return of the market component of r and r s is standardized return of the idiosyncratic component / u and Λ, are independent From this follows that coefficients β and β satisfy condition A, + Λ = ι (2) We also assume that return of the market component r is linear combination of standardized country/industry index returns r , which are independent of r 6 1, = ^ + + β„r„ (3) cor(r r ) = 0 / = 1 , ., n Equations ( 1 ), (3) can be obtained bv 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 r s This representation is consistent with ( l )-(3) but based on country/industry weighting coefficients We show below how this representation can be obtained from ( l )-(3) Denote by σ , σ : - standard deviations of the firm A equity return and index / return respectively, p t] correlation between index returns r and r (ι,j = 1, ...,n) Then from (3) one has r = β,r,+... + β n r π β σ,r,+.. β n <? r σ. or where if,,^,...,^ are given by Note, that H', + W, + . + H> = Coefficients n , 1 ,ιι , ,,...,ιι ι ιι are called country/industry weights and can be expressed in percents. From equation (4) follows that Denote Then from (6) we have Using ( 1 ), (4), (8) one can express standardized equity return in terms of country/industry wemhts as w.σ. w σ i (9) = A, — 2 — =- • l + A^ σ Hence, to compute this representation one needs to have following parameters country/industry weights, index standard deviations and correlations, specific coefficient β (coefficient β for the market component can be computed from (2) as β ι = ) In the current implementation CreditVaR I derives parameters β s and β from "specific risk percent" s defined as a fraction of total equity return changes explained bv firm-specific movements A, = Vi - s : , A = * ( 10) 2.2 Correlation between two obligors Equations ( 1 ), (3) as well as equation (9) allow to compute correlations between equity returns of anv two firms Let r , are standardized equity returns of two different firms B and C respectively and where σ„ ( 1 1 ) Then correlation between r" and / ' is 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 .v* = 0 87 Country/industry weights W ΪD.G.VRL = 0 5 ' W "l}.ΛCTO = 0 2 ' W ω.FOOD = ° 3 Obligor C ("29715") Specific percent s c = 0.88 Country/industry weights W y \ U,S c G,\RL o 2 , w = 08 Let us assume following standard deviations and correlations between market indices According to (7) = 003 2 x 05 2 + 002 2 x 02 2 + 004 2 x 03 2 + 2 x 003 x 002 x 05 x 02 x 04 + 2x003x004x05x03x02 + 2x002x004x02x03x01 = 0000471 σ n = Vθ 000471 = 0022 σ,: =00 L χ02- +002- χ08- + 2x001x002x02x08x03 = 0000279 From (10), (11), (12) β s B = 087 ; oB _ nB W 1D GΛ7?/ σ ID GMlL _ n. Λ Q 05x003 PID.GNRI ~ PM - — υ tv = 033, 0022 B n ID AUTO u ID AUTO 02 x 002 βf ID, AUTO β HM = 049 009, 5 0022 OOD = A, W ""* > ?" > ™ > = 04 ^^ = 027 σ a 0022 A =088, A = β w <scΛi yσ ιτcw = 047 _____ s asm = 006 σ r 0017 r w t T Hfiv 0" - ? R ;V 08 X 002 βus w ,v = A, "" ' = 047 = 044 σ r 0017 Finally, compute correlation between r' and r' usιng(13) cor(r" , A + 009x006x002 + 009 044χ004 + 2x027x006χ001 + 2 027χ044 002 = 0017 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) Denote by where ( t is correlation matrix between indices and I κ is K x K identity matrix Then correlation matrix C of equity returns equals to Let us assume that matrix H is obtained by Cholesky decomposition of the matrix =H U H[, (16) Denote by H=^ f H (17) (H is λ ' χ(// + A ' ) matrix) Then , = Hw are Cholesky decompositions of C and C respectively For simulation of a sample of K dimensional normal vectors R with zero vector of means and covanance matrix C (standard deviations of coordinates of R equal 1 ) we first simulate n + K dimensional vector of standard independent normal variables x = {x, ,...,xj , XT. = (x[ ,...x^ J , and then compute R as R = Hx = rV T H ( 19) From ( 18) follows that R has distribution N(θ, ) cov(Tή = E(HXX T H T ) = HE(XX T )H T = HI π+κ τ = HI nt K H T = 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 seme spread over it, which depends on the credit rating of the issuer The notation is as follows P - price of note, / , - expected future reference rate v, - spot long-term yields, used to discount the cash-flows at time / s - stated credit spread, fixed under terms of the FRN m - maturity of note in years // - coupon payment frequency Assuming flat structure of credit spreads, the price of the note per 100 of face value, is l OOt ' / + s ) n 100 J = Σ π + r, + } ■ „.„ /- " Here, r, is forward reference rate for the period between /-th and (i l)-sl coupon payments Assuming the reference rate is annually compounded, the way to compute forward reference rate r, is the following where R, is the rate on the reference curve for time point t, Within the current framework of CreditVaR I model, e will take existing current and forward corporate curves as a proxy for ',, 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 recoverv models depending on the sign of exposure to a bank, including recoverv rate 0 if the value of the swap to the bank is negative 6) A bank's credit events are not modelled d e , the bank's credit rating is constant) The notation is as follows V - value of the swap to the bank, Bι - value of the bond modelling the cash flow received by the bank, Bi - value of the bond modelling the cash flow paid by the bank, r/ - recovery rate applied if bank receives in the event of default by counterparty, r - recovery rate applied if bank pays in the event of default by counterparty, 0 - notional pincipal in swap agreement The value of the swap to the bank then is 1 = B, - B 2 Bi and B 2 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 recoverv amount is calculated as follows If V 0, the recovered amount is r / *V, If V 0, the amount paid is r 2 *V Here, we don't take into consideration the event of default or credit rating migration by the bank Recovery rates r / and r-. are determined from legal considerations at the time of default We can use Beta distribution again to simulate recovery rates r/ and r ^ (in current implementation, we take r/ r 2 ) Alternatively, r? 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 commitm ;nt, LIBOR plus LC fee is paid on the Letter of Credit part (throughout its lifetime), and an Urused 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 -tιme 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 / be the total (authorized) loan commitment amount, / - the amount drawn (in loans), LC - the amount in Letters of Credit, II - unused portion of the loan commitment We have that / = L + LC + U Let / be LIBOR and s be spread over LIBOR, so (I s) is the interest paid on the loan, lc - the fee paid on the letter of credit, uf - the fee paid on the unused 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 wav interest is paid), facility fee and unused portion fee are valued as bonds with no principal repayment where y is the annuahzed rate taken from the corresponding current corporate curve used to discount cash flows (depends on credit rating of the ob gor in the CreditVaR framework), n is the frequency of coupon payments, and m is the numbe of years to maturity Forward value ( FV ) is computed similarh , 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 (DD), or 'average commitment usage" if the obligor changes credit rating (see Credit Metrics Technical Document, p 45 for appropriate information) Credit rating Average commitment usage (ED) Aaa 0 1% Aa 1 6% A 4 6% Baa 20 0% Ba 46 8% B 63 7% Caa 75 0% 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 \ - Y U„ = U- \ -x L +U - ll LC. LC - new used amount T - U new is proportionally distributed between loan amount L new and letters of credit amount LC Upgrade In this case X > Y New unused, loan and letter of credit portions are computed as Λ_=max(Λ-(//_- /),0) ^_ = / ' -/,_-//_ 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 tc new =o, U new =T-L new Forward value of the loan commitment in the case of upgrade or downgrade is computed as In the case of default the loan commitment value is L nvw x R , where R is the recovery rate sampled from the Beta distribution (consistent with CreditVaR I framework), and total forward value is FV total .= new χR + ( \L-L new/ ) Part 2 - CreditVaR Methodology: Credit Migration Process Under Two Probability Measures As described in Part 1, 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 (0) 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 O 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 In the CreditVaR framework it is assumed that credit migration process forms a discrete Markov chain with fixed moments of time T = {(/„ <(l < ... < C/ ; < ...}, U , ≥ 0 In sections 1-5 of this Part we present mathematical models of the migration process and define probabilities under P and (J measures to migrate from one credit class at time l' t to another credit class at time //, , Theoretical results of sections 1-5 are applied in sections 6-8 to compute P and (J 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 B ~ ,..., B l> ) is a standard /) -dimensional Brownian motion on a probability space (Ω,F,f) (B B Z ,...,B D are independent), where F = {/-^,/>θ} is a standard filtration of B Let I = (/',/ " I j be a set of all market indices Denote bv V m the index l m value at time / l 0 It is assumed that V m satisfies stochastic differential equation (SDE) of the foim dl '" = I ' (μ"'dt + θ^dB ϋ"dB + ... - θ df ) , / > 0 (11) V n m = v m , m = \,2,...,M , where μ ™ , 0 ™ are piecewise continuous deterministic functions of / / = 1,2,..., D Let us fix some time interval [t 0 ,',] corresponding to two consecutive time steps from T t 0 = £/,, t,=U, ,, l≥O Denote x" = Ir(r, m )-ln(r, o m ) be the index l m log-return for the time interval [/„,/], t„ ≤t <t, Then from Ito's lemma and (11) follows that for t Q ≤t < t, with initial condition x™ =0 In a matrix form (12) can be written as where " = (*,' <fj *,= *" W«D dβ, =( ,..., )T Let us consider N obligors (issuers) S = {.S '1 ,^ 2 ,...,^ j, and let H" be an obligor S" stock price at time t We assume that H" satisfies SDE dH" = H," (α,V/ + σ, A + σ_dB +... + σ" D dB° ) , / > 0 , (14) H Q n =h n = l,2,...,N, where a" , σ", are piecewise continuous deterministic functions of time, / = 1,2,..., D Then for the stock log-return y" = ln(H" )- ln(H," ) one obtains d.s o L - (1.5) + }σ; 1 ^ σ; 2 £; + ... + σ; D \ Similar to ( 1 3) this can be written as (1 6) where ^ ...^r. * = fe Kl 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 (2.1) n = l,2,...,N, where a" and \, m are uncorrelated covε" , v"')= 0 , and Eε")=0, m-\,2,...,M, // = 12,...,N In the matrix form (22) where A = (A °- ■; , W °J , A is a vector of idiosyncratic components which are uncorrelated with index returns (23) (here and below expected value /.() and covanance cov() are conditional to σ -algebra Ό It is assumed that ιαnk(θ,) = M ≤D for all /e[t 0 .tι] (Note, that if rαnk(θ,)< \l < D for some / = / then it can be shown that there exists time interval [/ " t]ςz [r α /,], e [t 7] and index subset Iczl, ϊ≠0, such that for all indices in I their log-returns can be represented as linear functions of index I I log-returns with time dependent deterministic coefficients Then index set I can be reduced to subset IT on [f ) Then from (18), (19) it follows that the (M XM)- matrix θβ is nonsingular on [/„,/,] and (24) Indeed from (13, (16) follows that (2.5) (2.6) Then (2.2) and (2.5) give cov(y,,x,) = cov(β + β,x, + ε,.x,) Therefore, and (2.4) holds. Equations (2.2) and (2.4) also give the covariance matrix between equity specific components: cov(ε, ,ε t ) = cov(y t -/?,"- βx, ,y t - ft' - βx,) In the special case of constant volatility matrices / and σ one has {,-t )σ[l D -0 T (θO τ Yυ)y τ , where J D is (D X D) identity matrix In this case equity specific components ε" are uncorrelated if and only if Ojσ = λ, where λ is a diagonal matrix. This means that orthogonal projections of the rows of stock volatility matrix σ on the linear subspace, which is an orthogonal compliment in R π of the linear space generated by rows of 0 , are orthogonal Note, that in th s case covanance matrix of stock returns equals 3. Credit migration model under probability measure P It is assumed that at any time / e T each obligor S" e S belongs to one of credit classes R = {/^ R λ }, where R k corresponds to default Suppose that credit migration process forms a Markov chain on R with discrete times T and absorbing state R κ Denote by R" a credit class of obligor S " at time // e T Similar to section 1 , we fix a time interval [/ 0 ,t, ] corresponding to two consecutive time steps from T /,, = (l, , ' 1 = 1-1, , , / > 0 Suppose that R" = R, and p" (/„ — > t, ) is a probability of S" to migrate to credit class R at time /, (3 1) Denote by 7 ;( /„ →',H ( ' → / .)| K K (3 2) a transition matrix of S" for the time interval [/„,/, J Then For each nondefault state R t and probabilities ^(t,, — > t,), j = \,...,K define thresholds /?," ; (t,j,t ι ), j = \,...,K - 1 , for the standard normal distribution as follows /; 2 (/ 1J - /,)=N(/7; 1 (/,„/ 1 ))-N(/; : (/ ϋ ,t l )) Λ.,('»→',)= :(Wι))- ,( / „,/,)) (33) where N() is standard normal cumulative density function Let it, = Z, ' (\, - / ( ',)) De a vector of standardized stock returns with correlation matrix given by (34) where Z v , is a (N x N) diagonal matrix with stock returns standard deviations (35) ( var() is a variance under /-" measure conditional to σ -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 /, is defined by the value of its normalized stock return u," and thresholds h" (t 0 ,tf) Assumption: Given R," = R, , R," = R if and only if »," e (/»,", (/„,/, ), ?,",,, (t o ,?, )], where /, y = l,2,...,tf , Λ I " () (/ ϋ ,t 1 ) = ∞ , A (/„,/, ) = -» 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 R" = R, , Λ'" e S , // = 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 / v (x, , ,,..., x^ ) their joint density function Then the joint probability that at time /, obligors will be in credit classes R J , R , ..., R respectively is given by IR; 'υ = R,'i ,...,R, ' V o = R ' ) (3 6) 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 O Let r t be an instantaneous risk-free interest rate at time / We assume that r t 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 dl 7 = 1 (/ dt + θ"dB] + B dB + ... + θ" n 'dB ) , t > 0 , (4.1) J7 = r m , w = l,2 V/, c/H = //," (r,dt + σ,",c/A + σ.\ dB; + ... + σ;' D 6/R," ) , / > 0 , (4.2) where are Brownian motions under the probability measure (J defined by the Radon-Nikodym derivative — Jp = exp r f l - ∑ ^J f r ' A? * " - 1 -7 J f Iir l^v '' (4.4) |Γ,| : =(Γ;) 2 +(Γ ) : +... + (A , where vector-function r, =(r,',r,- r") solves equations θ r] + ^ r + ... + #,>," = //,"' - /; , m = 1,2 M (4.5) σ ,ι , + σ, ,τ ~ ÷ . - , - r n ,N (4.6) In the matrix form: (4.7) where μ, = (//'-/,..., μ) 1 - r r ) , α, =(α,'-r, α, N -r . We assume that solution r, of (47) exists for all te [t u ,t,] Similar to (15), (18) one computes index and stock log-returns from (41), (42) for the time interval [/„,/,] = [U, l lti ] x, = , + joΛ / </</,, -v, = 0 (4.8) y, = b,+ ]σ s dB s , /„</</,, y, o = 0 (4.9) where «,=fø ^ *,=fe *; V F }[ ' . - i. +&J -÷fcrf .v Let E ρ () cov ρ (.,.) be an expected value and covariance matrix under the probability measure 0, conditional to σ -algebra F, Then from (4.8), (4.9) follows that Z^(x,) = «, (410) r cov Q (x, , x, ) = cov(x t , X, ) = jθ d.s (412) cov Q ( -, , , ) = cov(y, , (413) cov ϋ {y,,y, ) = cov(y, ,_>>, ) = jσtf s (414) In particular. var c ( )=var( ) (415) for all // = 1,2,..., N Moreover, cov Q (ε, , x, ) = cov(ε, , x, ) = 0 , (416) cov Q (ε t ,ε,) = cov(ε t ,ε,), (417) but I° {ε,)= b -β,ά,- A' ≠0 (418) 5. Credit migration model under ( 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 0 measure 5 ? Let obligor S" <= S be in the credit class R" = R, at time /„ According to the assumption in section 3 its credit rating at time t, will be R" = R, if and only if w," <= ( ?,", (/„,/, ),Λ7_, (t 0 ,t, )J, y = 1,2,..., ' , where thresholds " (t ύ ,tf) are derived from the standard normal distribution. Then probability under O measure for obligors S S 2 ,...,S to migrate from the credit classes R,' = R R = R at time t.. to the credit classes R, 1 = R, ,...,R; V = R, at time /, respectively is (5.1) As follows from the definition of »r, = (u',' ,..., n', v J and (4 15) (5.2) where (5.3) and C"(t ,t) is a constant given by |(r -a[')js V» Then for the vector it', = (if, 1 ιϊ, j using (34), (414), (415) one has cov ρ (ύ', , it', ) = cov Q (if, , If, ) = -L(y, ) , -A ,, )/ ,',) = Z l l ,cov°0-,^ f )Z l l , (5.4) From equations (49), (5.3), (54) it follows that the random vector if, = (if, 1 , ... , ιt', v j has a normal distribution under the probability measure (J with zero vector of means, covariance matrix cov'(if, ,n, )= cov(u, , u, ) and probability density function / (x,,x, v ) Then w" is a standard normal variable for each // = 1,...,N and (5.5) = ?(»ϊ-,7 e (Λ" (/„./,)- -,7 (/., . ). Λ,^ .(/o. -Cft,,,/,)} = N(/,V 1 (/, ) ,t 1 )-r,;(/ 1 „t 1 ))-N(^(/, )) / 1 )-r;(/ 11 .t l )). Therefore, similar to transition probabilities p"_ } (t i — /,), transition probabilities </,",(/,, — ,) under martingale measure O are defined bv shifted thresholds &",('«.'. )= *; (/„./,)- (5.6) Note that transition probabilities q" (t t , — » /,) depend on // and can be different for different obligors Equations (55), (56) show the relationship that exists between two transition matrices 7; p (/„ - /,) = || Λ Λ ; (/„ → /, and T?(f„ → /, ) = J^ (/„ → /, Similar to (36) for the 0 probability of joint migrations we have =c = ( / ? ;, / , ) , ,/ ; i i ,( t 1 t l... e( J / 0 , , ^7, ( / n 1 } = O e fe Λ (/„,/, ih ., (/„,/, ){ ( 5 ?) J ■•• J/ /V (* P *: x )dxβx 2 ...dx sl', y,('o 'l ) JΛ ,('o 'l ) 6. Construction of transition matrix under O -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 O transition matrix for the period [/„,/,], t, = t 0 + 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 O 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 /-* transition matrix and those determined by the given 0 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 0 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 (J transition matrix over the risk horizon Note that the last column of such O transition matrix will coincide with the given O default probabilities First, we introduce some general notation For obligor S" e S and observation times / > , s, / e T , let q"( s → l) denote the O default probability of Λ' π over the period [s,/] given its initial rating of R at time s , I e , < 7 ( s → /) = 0(1^ = D I R; = R ) , / = 1,2,..., A' Recall that the default state R k = D is absorbing so that q k (s → f ) = I Put K = K - 1 , and package them in a -dimensional vector we have q" (s — > /) > 0 , where 0 denotes the K -dimensional zero vector Now suppose we are given a P transition matrix T P " = p" " for each individual obligor S" , respectively, and 0 default probabilities q"(/„ -> /, ) over the risk interval [t υ , t, ] Put <7," = c l" (/ , - ', ) and r," = N "1 (q ) , 1 < / < K Let ft.-*;}* } ) + ( σ ; : ): + ... + ( σ ; ) Δ - as in Section 5 Define transition matrix !][ ' = \\ p" s (/,,,/, ) || by />;,(',„',) = O≠K) (61) where υ = - = ^- — — , p k (/„,/, ) = 0 if / < K and /; A A (/„,/, ) = 1 We have adjusted 1 " /',";. the last column of the given matrix /," so that the new P default thresholds satisfy N (p" k (/,,,/,))= -' +( " for i <, K , ic, they differ from the (J default thresholds z" by constant C" 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, ι e , / (/„,/,) p" ' '' ' = ' " for all 1,/ <K (62) I-/>Λ ( ' , „ / , ) ι-R Now, we apply the theory of change of measure to / " „' to obtain the desired (J transition matrix over the risk horizon With A =|| / ," (/„,/, )JI of (61), let h".(t { ,t) be its thresholds In particular, we have h; (t l „t ) = N l (p k (t a )) = z"+C < (6.3) Let 7^,- = (I ^AC IM ' I ) !! be the corresponding transition matrix under 0 , by (55) and (56), its (ι,j) -th entries are defined by , n ,,/,) = /V(^ 1 (/ I1 ,/ 1 ))-N(^; j (/ ( ,,/,)), /. =1,2 K (64) where t Cf," ; (/„,/, ) = /» (/,„/,) - C" (/„,/, ) Using (63), (64) and N (-∞) = 0 , we have f (/,„',) = /V(/ (/„,/,)-( ' ;')- /V(- χ ) = N(r,") = </,"(/„ →Λ ) for each / = 1,2,.... K Thus, by construction, the last column of T' fits exactly the given default probabilities under O over the period /,,,/, ] In summary, a suitable ad|ustment is made to the given P transition matrix such that its last column is compatible with the given O 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 ( ' is then subtracted from other non-default thresholds of the modified /' matrix to obtain the corresponding thresholds, and therefore the entries, of the desired O 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 (J default probabilities q"(t„ → t, ) are also obligor independent so that q"(t„ —>/,) = q(t„ — > t ) Let q be the / -th component of q(/„ —»/,), and put z,= N-'(//,), f = ΛΓ' ( £ - Here, for ('„, we may want to average the differences it, -z : over only the / -th terms for which q t and p ~ t k are not too small (so that the thresholds are away from - -c), also, instead of taking the average, one can choose (',, which gives the least mean square error Similar to (61), define the new transition matrix T p = || p t (/„,/,) || by where υ = ^ — , p k (/„,/,) = 0 if j < K and / λ (t n A ) = 1 Therefore, the last 1 - A λ column of the given matrix T P is adjusted in such way that the new P default thresholds satisfy k (/„,/,)) = r, +C„ , le, they differ from the O default thresholds r by constant C„ independent of / Similar to (62) we have A,('. '|) _ for all ι,j<K -/>,*.-('„.',) !-/ . _ 64 7. Forward transition matrices under (J 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 O -measure Such series will be useful, for example, when pricing credit derivatives via risk neutral valuation Recall that for each obligor S" eS we assume that its credit migration process observed at times in T forms a Markov chain on states R under O -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 = { ιt tl < //, < ••• < u B ) (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 [//„,//,], [//,,//,], , [ιι„ ,,//,,] With such series, the transition matrix over any period [ιt ι ,ιt h ] with a<b is given by the product of transition matrices over [ιt l ,ιi l ,], , [tt h ,,//,,], respecting the order Let T () = {/,=/„ +/H I / = 0,1,2,... ,7. } be a set of 'standard' observation times, where H is the risk horizon, we assume that T cT For obligor S" e S , suppose we are given a term structure of default probabilities under 0 -measure By appropriately interpolating if necessary, this means that we are given the 0 default probabilities satisfying 0<q n (/„→/,)<q"(t„→t 2 )<"-<q' , (t„ →/,)<e, (71) where e 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 [/,,/ 6 ] define vectors (\ f (t h →/,) as follows q;,(' k →',) = q"(>„ →',) (72a) for / e {>, > + 1 ,7} Potentially these vectors may have negative components because M ' can have negative entries However if these vectors satisfy 0 = q;,( , →f)≤q f (< h →L t )≤--- < -<\" L →> ! )<e, (72b) then we refer to them as the time f (forward) default probabilities obtained from the initial term siπictiii of default probabilities (71) using tiansition man ix M Note that the equality in the first relation of (72b) is equivalent to the condition that the first K components of the last column of XI equals q"(/,, — > /,,), 1 e , it matches the initially given default probabilities for the period [/,,,/,,] We wish to construct certain series of (J transition matrices A<Λ→/A /„ C '(/, →Λ), , r?(f, , →/,) (73) on T, such that the following proposition is satisfied with Z, = l^(t, , — > f ) Proposition: Given initially a term structure of time /„ default probabilities satisfying (71), there is a set of invertible matrices Z t , Z ^ , , Z l such that for each b , 0 < b < L , the time l h forward default probabilities obtained from (71) using M = ZZ ^ ---Z b satisfy (72b), 1 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 A which was produced from the given data for [/,,,/,] in Section 6 q"(', → / ,) For the initial interval [/,,/,], let >, = 1° , its last column. by (67), matches the given default probabilities for the interval If Y is invertible and condition (72b) already holds for M = > ' , , then we set P^(t →- /,) = !, More generally, consider the following 1- parameter family of matrices Z.(a) = aX,+(\-a)Y., 0<α< (74a) where with </, =<7,"(t„ →t, )< 1 and / • ' = 1 -</,"(/, -»/,) for / = 1,2 K The matrices Z,(α) all have the same last column and in particular is independent of a and equal to that of Y t Choose the "smallest" a , call it α,, in (74a) such that vectors q" (t ] →, ι) defined by (72a) with invertible hi = Zfa) satisfy (72b) with b = \ there (in practice, we fix a positive integer / > 1 from beginning, and r jstπct values of a in the set A(/) = {() 1 / 2 / (I -\) /,l} Then «, is chosen to be the smallest a in A(/) for which condition (72b) holds ) That such α, exists is clear since for a - 1 , the condition is satisfied Thus, we have the time /, forward default probabilities 0<q"(/, →/ <q"(/, →Λ)<- q π (/, →/,)<e (75) with respect to M - Z (a l ), here, subscript M is omitted We set l^ (/,,— /,) = Z, (or,) Now, for subsequent periods [/,,/, ,], / _ 1, we proceed recursively to construct invertible transition matrices Using the time /, forward default probabilities (75) as given, we can obtain in a similar way an invertible transition matrix Z, on [/,,/,] whose last column is compatible with the (forward) default probability q"(/, — /,) and whose other entries have, in each row, the same relative weights as those of T^ (or, as before, have the weights distributed according to some preference) Note that the time /, forward default probabilities obtained from the time t, default probabilities (75) using Z, of [/,,/,] will be the same as the ones obtained from the time / default probabilities (71) using Z. Z., on [/,,,/,] Repeating the procedure for intervals [/,,/ ], [t , ,t 4 ] and so on, we construct a desired series of transition matrices, (73), on T, 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 t,, 0 < / < L , in that the cumulative transition matrix Z. Z ^ ---Z. t on the interval [/„,/,] has (the first K components of) the last column matching the given q " (t , — > t, ) Remark There are many choices for such series W r e 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 O transition matrices in place of the initial term structure of 0 default probabilities 8. Joint process under 0 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 T„ 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 = { //„ < //, < • • • //„ } ( T includes nonstandard' time steps corresponding to payment dates, times to instrument s maturity, etc , which need not belong to T„ ) 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 \u k , u k \ 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 [/ , , / , , J and suppose [n, v] z [t , , t ιt is a proper subinterval, put θ = (v - n)/(t , - / , ) Now consider the matrix Z" defined by Z" = ∑ ((θ)), (Z - /) λ , (8 1 ) where ((θ)) < = θ(θ - 1 )• ■ ■ (() - A + I ) / A-' , as in the Taylor expansion around the identity matrix / When series (8 1 ) converges we mav truncate it after a certain number of terms to evaluate /" with a small error Note that a typical transition matrix, especially over a period that is not too long, is near ' / 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 tar Starting with a P transition matrix on the risk horizon and an initial term structure of (J 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 O 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 0 transition matrices can be used in evaluating credit swaps A credit swap on a corporate boi d 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 O -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 geneπcally in the form the sum is over an exhaustive set {E } of all possible, mutually exclusive events, I L is the indicator function of event E , and / ; 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 X ι r for the fixed payments side (buyer) Denoting by τ t . and τ u , 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 τ c and τ u For example, E = ( event that u, , < τ B < it, and ι c > ιι, ) where //, , < it, are two specified times, usually a pair o r consecutive premium payment times Depending on E , f E is a function involving notional amount or forward values of the reference facility, a strike price, recovery rates of B and " , etc and depends on whether premium payments are made in advance or in arrears Both E and f L . 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 0 It is equal to ∑d r J, Q(E) (83) L Given a term structure of interest rates, the risk free discounting, d r f L , of cashflows f h can be approximated reasonably in a straightforward manner Therefore, the remaining key ingredient in pricing credit swaps is to evaluate 0(E), where E usually are some joint events pertaining to the default times τ c and τ H In practice, all the probabilities O(E) appearing in pricing formula (83) 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 //„ < τ n < ιι k , and //,, < τ c < if ) Let 7„ = (R, ,R j ) be the initial ratings of (B,C) at time //„ and B k the rating of B at time u k (and similarly for (', , etc) Then, we have 0(E) = 0(B k , =DX = |/„) = ∑O(A =D,B l l =D,C . l =R J \I u ) = ∑Q( =D\C _ =R ] )Q(B k l =D,C . l =R J |7„) -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 [n k ^,n k ] and [//„,// t _,] Hence O(E) are easily attained from the transition matrices This completes the final piece of computation in pricing credit swaps 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-pπcing 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 / for a bond maturing at t + . Note that T is the term to maturity, not an absolute time point Let / " (/, 7, , l ) be the forward rate seen at time / for a contract starting at t ->- f and maturing at / + /, We assume that the zero curve matuπtv buckets are the same for all currencies and all credit ratings, and we denote the maturity terms bv # = {/,,/, t h }, where / = 0 </, </, •••<. t h For convenience, we denote the zero curve as seen at time h by y = (>',, , y h ) = (y(hj ι ),y(h, l ^ ),---,y(h b )) and the one seen at time zero by y" = (_',", _-",..., = (y(0,t,),y( ,t ^ ),•••, '(0,t Λ )) The zero rate for a term between two bucket points is determined by linear interpolation (If t < t, or t t h then the zero rate is set to v, or y h , 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 = (v,,x : ,...,x = (/(/?,/„,/, ),/(/?,/,,/, ), ■ ••, f(h,t h ,,/ ) x can be given by x, = ', and x = y - ' } '' ' ' ' for / > 1 (1) Conversely, Written in matrix forms, Denoting the above transition matrix by T, then x = T ' \ where the inverse matrix is given explicitly as The following method is proposed to generate interest rate scenarios Let x° = (x", v ',..., °j = (/(0,/„,/,),/(0, /,,/,), • •• , f(0,' h ,,/ / ,)) be the forward rates seen at time 0 We assume that the forward rates \ are lognormally distributed and that the log relative changes V V, V ξ =(ξ,,ξ,,...,ξ Λ ) = (In— -,ln— - In—) follow a multivariate normal distribution r,' r,' x h ' N(μ,Ω), where μ = (//,, ., μ h ) are drifts and Ω 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 ξ =(ξ,,ξ, ξ A ) from the distribution as defined above, obtain x = (x," exp ξ χ , ... , x exp ξ h ) , then a Dply equation ( 1 ) to get continuously compounded zero rates y 2.2. Simulation of Spreads Given a credit rating ^, , / = 1 K - 1 , let s, be the spread curve for 7^ over that of the base curve \ Thus, + s, is the zero curve for 7^ 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 anv time point / 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 , (t, ) for the highest credit rating for (typically Aaa) and for maturity equal to the first bucket point on the zero curve ( /, ) incremental spreads A , (/, ) = s # (/, ) — s, , (/, ) , / = 2 K - 1 for maturity t, As k , (t h ) = s k , (t h ) - s k , (/, ), l 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, (/, ) = , (/, ) + As, (/, ) Lognormal assumption on As, (t, ) ensures that spreads are increasing when credit rating is decreasing, I e s (/, ) > s ; , (/, ) for / = 2, 2 Obtain the s , f< h ) = ^ k t, ^ + Δs, , (t h ) We assume that As λ , (t h ) has lognormal distribution This way we make sure that the spread curve doesn't decrease with maturity for anv scenario 3 Obtain the rest of the spreads for maturity //, by partitioning the interval [0, s k ft h )\ proportionally to s (/, ) by taking s (l h ) = s (ι )-^ — — This way, we get spreads that f' increase when credit rating declines for maturity lh 4 Obtain the interior of the spread curves by linear interpolation between s, (/, ) and s, (t Λ ) The corporate zero curves at risk horizon are obtained by adding the appropriate spread to the base curves 2.3. Simulation of F\ rates The FX rates at risk horizon are assumed to be lognormally 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 paπicipation 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 Obhgoi 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 currentlv 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: CLAIMS We claim 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 (7-*), • adjusting the (P) transition matrix to be consistent with default probabilities under martingale measure (0 by making a column of default probabilities in the (P) transition matrix consistent with default probabilities calculated under (0, 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 (7-") 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 (0 by making a column of default probabilities in the adjusted (P) transition matrix consistent with default probabilities calculated under (0, 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 matnx 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 (/') transition matrix is used to price credit swaps 7 The method of claim 1, wherein the step of adjusting the obtained (7') transition matrix is performed in accordance with the following R = {/^ | R k \ - are all credit classes R, - corresponds to default state T 1' - p. J) - adjusted (/') transition matrix for the time interval [/,,,/,] P, j ( I,,',) - transition probability from credit class R to credit class R P P ""O ( "" t) J fi≠K) / ,(/„,/,) = 0 if ι - K and p k k (/„,/,) = 1 I I - o " , =' -'(/7 lA ) z =N- q,) \-M(z,+C,) υ, = _ p, k - obtained (P) probability of default for the credit class R, (from the obtained (P) transition matrix) q, - (0 probability of default for the credit class It G - a set of credit classes (can be the whole set R = {/?, R k } or some subset I G I - denotes the number of elements in G , if G is empty, put „=0 N( ) - standard normal cumulative density function The method of claim 1, wherein the step of adjusting the (7*) transition matrix is performed accordance with the following l n )\\ - adjusted (P) transition matrix for the time interval [/„,/,] for obligor // P ι ' ' ,( >t \ ) - obligor ' s n transition probability to migrate from credit class R t to credit class R, p k " ] (t„t { ) = 0 xf j<K and AA(OA) = 1 /(', - «;)* N ' (A) 7" λ - obtained (/') probability of default for the obligor // over interval [/ ,,/, ] if it is in the credit class R at time t , q" - (0 probability of default for the obligor // over interval [/„,/, ] if it is in the credit class R at time / /, - instantaneous risk-free interest rate at time / y" - obligor /; stock log-return over interval [/„,/, ] a" - obligor // drift function of time / N() - standard normal cumulative density function A method of pricing a credit swap, comprising the steps of • Obtaining a (P) transition matrix over a risk horizon [/„,/, ] and the (0 default probabilities over the same period [/„,/, ], • Creating an adjusted (P) transition matrix and (0 transition matrix over [t^t, ], • Building a first series of forward (0 transition matrices for standard observation times / = /, ,/-,, t 3 ,... from the (0 transition matrix and an initial term structure of default probabilities under O over [/„,/, ], • Translating the first series to produce a second seπes, 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 A method of pricing a credit swap, comprising the steps of • Obtaining a (P) transition matrix over a risk horizon [/,,,/, ] and the (0 default probabilities over the same period [/„,/, ], • Creating an adjusted (P) transition matrix and (0 transition matπx over [/„,/, ], • Building a first series of forward (0 transition matrices for standard observation times / = /, ,/,,/ , .. from the (0 transition matrix and an initial term structure of default probabilities under O over [/„,/, ], • 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 bv 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 A method of pricing a credit swap, compπsing the steps of • Obtaining a (P) transition matrix over a risk horizon [/ ,, /, ] and the (0 default probabilities over the same period [/ , /, ], • Creating an adjusted (P) transition matrix and (0 transition matrix over [/ , / , • Building a second series of forward (0 transition matrices for premium payment dates from the (0 transition matrix and an initial term structure of default probabilities under (J over [/„, /, ] , • 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 pavment date bv 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 A system for pricing a credit swap, comprising • Means for obtaining a (P) transition matrix over a risk horizon [/,,, /, ] and the (0 default probabilities over the same period [/„, /, ], • Means for creating an adjusted (P) transition matrix and (0 transition matrix over [A,, / , ], • Means for, from the adjusted (P) transition matrix and an initial term structure of default probabilities under O over [t υ ,t,J, building a first series of forward transition matrices for standard observation times / = /, ,/,,/,,... , • 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 t r ansιtιon 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 A system for pricing a credit swap, comprising the steps of • Means for obtaining a (P) transition matrix over a risk horizon [/,,,/, ] and the (0 default probabilities over the same period [t„,t, ], • Means for creating an adjusted (P) transition matrix and (0 transition matrix over • Means for building a first series of forward (0 transition matrices for standard observation times t = t, J-J 3 ,... from the (0 transition matrix and an initial term structure of default probabilities under 0 over [t 0 ,t,], 85 • Means for deriving a third series of transition matrices from the first seπes 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 seπes for all payment dates, and • Means for subtracting the sum from a payoff function for fixed payments A system for pricing a credit swap, comprising the steps of • Means for obtaining a (P) transition matrix over a risk horizon [/,,, /, ] and the (0 default probabilities over the same period [/„, /, ] , • Means for creating an adjusted (7') transition matrix and (0 transition matrix over ['. , A L • Means for building a second series of forward (0 transition matrices for premium payment dates from the (0 transition matrix and an initial term structure of default probabilities under 0 over [/„,/, ], • 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 '.umming 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 [/,,, /, ] and the (0 default probabilities over the same period [/,,, /, ] , • Create an adjusted (P) transition matrix and (0 transition matrix over [/,,, /, ], • Build a first series of forward (0 transition matrices for standard observation times / = /, ,/, , / from the (0 transition matrix and an initial term structure of default probabilities under (J over [/,,, /, ] , • 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 seπes and sum the computed series for all payment dates, and • Subtract the sum from a payoff function for fixed payments 16 A method of modelling stochastic interest rates, credit spreads and foreign exchange rates comprising the steps of • Computing obligors ' credit ratings at risk honzon :>ased on country/industry market index logreturns and idiosyncratic components, • Computing base zero interest rates for all currencies and given maturities t t) ,t t b from base forward rates /(t„J, )./('ιJ : ) f{' h I O corresponding to time intervals • Computing credit spreads s,(t,)= s, ,(/,) + As, (/,) for maturity /, and credit classes / = 2,.. K - 1 (by currency), • Computing credit spread s λ i (t h )= λ ff) + Δ.s k t (/ h ) for the lowest credit rating and longest maturity t h (by currency), • Computing the rest of the spreads for maturity l h by paπitioning the interval [O, s k ft )] propoπionally to s,(t,) by taking s (t h j= s (t )- -± — — (by currency), • Computing spreads for other maturities /,,/, t h , by linear or any other type of interpolation between \(/,) and s,(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 A credit risk model for the analysis of poπfohos 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 t„J, t h from base forward rates f{t ϋ ,tf),f{t t 2 ) f( ( b I O corresponding to time intervals [/„,/,], [t,,t 2 ], ,[/,_,, t , • Means for computing ere lit spreads v, (t, ) = s ι , (t, ) + Δs, (t, ) for maturity t, and credit classes / = 2, X - 1 by currency), • Means for computing credit spread s k , (t b J = s K ft t ) + As k ft h ) for the lowest credit rating and longest maturity t b (by cuπency), • Means for computing the rest of the spreads for maturity t b by paπitioning the s (t ) interval [ ,s k ft h )] proportionally to s (/, ) by taking sft b ) = sft λ ' ' h (by currency), • Means for computing spreads for other maturities t,,/ 3 t b , by linear or any other type of interpolation between s (/, ) and s (t h ) (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

Inventors: CROUHY MICHEL, IM JOHN, NUDELMAN GREGORY, TCHERNITSER ALEXEI, CROUHY, MICHEL, IM, JOHN, NUDELMAN, GREGORY, TCHERNITSER, ALEXEI
Applicants: CANADIAN IMP BANK OF COMMERCE, CROUHY MICHEL, IM JOHN, NUDELMAN GREGORY, TCHERNITSER ALEXEI, CANADIAN IMPERIAL BANK OF COMMERCE, CROUHY, MICHEL, IM, JOHN, NUDELMAN, GREGORY, TCHERNITSER, ALEXEI
Application year: 2000
Main classification: G06F17
IPC codes: G06Q40