EP1272950A2 - Credit risk estimation system and method - Google Patents

Abstract

A method creates an adjusted credit risk transition matrix and related matrices. The method uses the steps of obtaining a real probability measure (<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 _](t₀,f) = 0 if J<K and _A(t₀,t,) = l r - a ft :

C" *-o ( 'o »'ι ^varM r; = N-' (y;)

i i -- Nv ((: υ z; + )

- P. K

p"_κ - obtained (P) probability of default for the obligor n over

interval [t₀, 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 [t₀, 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 [t₀, t, ] and the (O) default probabilities over the

same period [t₀, t, J are obtained

• an adjusted (P) transition matrix and (0 transition matrix over [t₀, 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 [t₀,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 [t₀J,], ,[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_hj]

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 - (t₆)

(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 Access⁷ /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^,_ni_ui ,_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 r₆

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_nr_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 ^WΪD.G.VRL = ^{0 5} ' ^W"l}.ΛCTO = ^{0 2} ' ^Wω.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)

= 003² x 05² + 002² x 02² + 004² x 03² + 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 ^W1D GΛ7?/ ^σ ID GMlL _ n. _ΛQ 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^yσιτcw _{= 047}_____ s asm = 006 σ_r 0017

r ^wt 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_UH[, (16)

Denote by

H=^^fH (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^TH

( 19)

From ( 18) follows that R has distribution N(θ, )

cov(Tή = E(HXX^TH ^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, - B₂

Bi and B₂ 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 r₂*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/ r₂) 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 [t₀,',] corresponding to two consecutive time steps from T

t₀ = £/,, 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"_DdB° ) , / > 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[t₀.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^τYυ)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" (t₀,tf) Assumption: Given R," = R, , R," = R if and only if »," e (/»,", (/„,/, ), ?,",,, (t_o,?, )], where

/, y = l,2,...,tf , Λ_I"₍₎(/_ϋ,t₁) = ∞ , 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, '^Vo = 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;'_D6/R," ) , / > 0 ,

(4.2)

where

are Brownian motions under the probability measure (J defined by the Radon-Nikodym derivative

— Jp = exp ^rf^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_sdB_s , /„</</,, y,_o = 0

(4.9) where

«,=fø ^

*,=fe *;^VF

}[_'. - 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_, (t₀,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 S² ,...,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(/,V₁(/,₎,t₁)-r,;(/₁„t₁))-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, (^/n₁ }

= O e fe _Λ (/„,/, _ih ., (/„,/, ){ ₍5 ?)

J ■•• J/_/V(*_P*: x )dxβx₂...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, = t₀ + 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," = ^cl" (/ , - ', ) 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 _tCf,"_; (/„,/, ) = /» (/,„/,) - 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<qⁿ(/„→/,)<q"(t„→t₂)<"-<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_,,t₄] 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^re 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_rJ, Q(E) (83)

L

Given a term structure of interest rates, the risk free discounting, d_rf_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

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 / = /, ,/-,, t₃,... 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^ransιtιon 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-J₃,... from the (0 transition matrix and an initial term

structure of default probabilities under 0 over [t₀,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_hj= 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 t₂) f(⁽ _{b I}O corresponding to time

intervals [/„,/,], [t,,t₂], ,[/,_,, 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