Information processing apparatus, information processing method and program product for fast computation of risk measures and risk contributions

An information processing apparatus includes: a rating threshold calculating unit obtaining a probability psk of transition from a rating s to a rating k stored in the storage device, and calculating a rating threshold θir at which the rating of a obligor i becomes a rating r or lower based on the obtained psk, a conditional probability calculating unit obtaining a constant ai of each obligor i stored in the storage device, and calculating a probability psir(z) that the rating of the obligor i becomes the rating r or lower under a condition that z is fixed based on the obtained ai, z calculated in the z calculating unit, and the θir calculated in the rating threshold calculating unit, and a wir calculating unit calculating a difference wir in exposure value between adjacent ratings of the obligor i based on an exposure value Vir of the obligor i when transition from a current rating to the rating r occurs, which is stored in the storage device.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is based upon and claims the benefit of priority of the prior Japanese Patent Application Nos. 2009-131204, filed on May 29, 2009, and 2010-095350, filed on Apr. 16, 2010, the entire contents of which are incorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

2. Description of the Related Art

Monte Carlo methods are widely used as calculating methods of VaR (Value at Risk, a percentile according to a confidence level α) of credit risk. Further, there are DM (Default Mode) method and MTM (Mark-to-Market/market price evaluation) method as methods of modeling credit risk. The DM method is a model creating a loss distribution using only the loss in the event of default, and is used for credit risk measurement for allowing credit, including medium and small companies and retails. On the other hand, the MTM method is a model creating a profit and loss distribution considering profit and loss other than in the event of default, and is used mainly for financial products (such as bonds for example) having a marketability that requires consideration of loss due to decline in market prices.

For either model of the DM method and the MTM method, methods of calculating VaR using a Monte Carlo method are known. For example, Japanese Laid-open Patent Publication No. 2009-32237 describes explanations of modeling of the DM method and the MTM method, and a calculating method of VaR with a Monte Carlo method. Further, besides Monte Carlo methods, for example, an approach to calculate VaR analytically with respect to the DM method is disclosed in “Analytical Evaluation Method for Credit Risk of Credit Portfolio Centered on Limiting Loss Distribution and Granularity Adjustment”, IMES Discussion Paper Series 2005-J-4, Institute for Monetary and Economic Studies, Bank of Japan, July 2005.

In late years, necessity of economic capital management (operating administration in the aspect of accounting management considering viewpoints such as what degree of risk, should be taken in what sector, profitability, and the like based on risk contributions of individual companies) is discussed both domestically and globally, and importance of breaking down the total VaR into “risk contributions of individual companies relative to VaR” is increasing.

However, in the above-described Monte Carlo methods, errors occur due to random numbers when calculating risk contributions, thereby causing problems such as unstable numeric values, discrepancy between the total VaR and the sum of the contributions of individual companies, taking a long time for calculation to obtain contributions in units which are too small, requiring a high-capacity memory, and so on.

On the other hand, there are proposed techniques to calculate VaR by an analytical approach. Using the analytical approach, it is possible to calculate risk contributions of individual companies accurately and rapidly. As described above, there are the DM method and the MTM method as methods of modeling credit risk, and the IMES Discussion Paper discloses an approach to calculate VaR analytically with respect to the DM method. On the other hand, the MTM method has many parameters as compared to the DM method and provides a complicated distribution, and thus has been considered to have a difficulty to calculate VaR analytically.

Now, the above-described problems will be described in more detail using drawings.FIG. 1is a diagram representing rating transitions and losses of a obligor i by the DM method. Reference symbol pidenotes a default rate according to the rating of the obligor i. As illustrated inFIG. 1, in the DM method, ones other than default can be grouped together, and thusFIG. 1can be expressed by a binomial distribution as illustrated inFIG. 2.FIG. 2is a diagram illustrating an example of expressingFIG. 1by a binomial distribution.

In the DM method, based on the rating transitions inFIG. 2, the probability distribution of loss in a portfolio is obtained as:

L is a loss of the portfolio,

Liis a loss by the obligor i,

Diis a random variable indicating the status of the obligor i (default or non-default),

xiis a random variable indicating the enterprise value of the obligor i,

l[•] is a defined function,

piis a default rate of the obligor i,

LGDiis a loss given default in exposure of the obligor i, and

EADiis an amount of exposure of the obligor i.

In practice, the default rate and the loss given default differ for every obligor included in the portfolio. Accordingly, even when an individual obligor can be expressed as inFIG. 2, the loss of the portfolio becomes a complicated probability distribution, for which it is not easy to obtain VaR.

Thus, conventionally, probability distributions and risk indicators (VaR) thereof have been calculated by simulation using a Monte Carlo method (generating random numbers corresponding to the random variable xifor the number of times of trial, and calculating in the order of Di→Li→L).

In late years, VaR calculating methods by analytical approximation such as granularity adjustment method have been known. Accordingly, it has become possible to stably and quickly obtain not only VaR but also risk contribution of each obligor.

are an expression for obtaining the risk contribution of the obligor i when obtaining VaR analytically in the DM method, and an expression for obtaining VaR analytically in the DM method.

On the other hand, in the MTM method, the diagram representing rating transitions and profits and losses of the obligor i becomes a polynomial distribution similar toFIG. 3, in which the number of parameters per obligor becomes large as compared to the DM method. Thus, there is a problem that the analytical approximation of VaR becomes quite difficult as compared to the DM method. Here,FIG. 3is a diagram representing rating transitions and profits and losses of the obligor i by the MTM method. Reference symbol pirdenotes a rating transition probability of the obligor i from the current rating to the rating r.

SUMMARY OF THE INVENTION

The present invention is made in view of such problems, and an object thereof is to allow obtaining VaR and the like quickly in analytical approximation in an MTM method.

Accordingly, an information processing apparatus of the present invention has: a z calculating unit calculating, based on a confidence level α stored in a storage device, a percentile of a risk factor z corresponding to α; a rating threshold calculating unit obtaining a probability pskof transition from a rating s to a rating k stored in the storage device, and calculating a rating threshold θirat which the rating of a obligor i becomes a rating r or lower based on the obtained psk; a conditional probability calculating unit obtaining a constant aiof each obligor i stored in the storage device, and calculating a probability psir(z) that the rating of the obligor i becomes the rating r or lower under a condition that z is fixed based on the obtained ai, z calculated in the z calculating unit, and the θircalculated in the rating threshold calculating unit using

With the information processing apparatus having such a structure, the probability psir(z) that the rating of the obligor i becomes the rating r or lower under a condition that z is fixed with respect to the probability pskof transition from the rating s to the rating k can be calculated as Expression (1). Further, with the information processing apparatus having such a structure, the difference wirin exposure value between adjacent ratings of the obligor i can be calculated as Expression (2) with respect to the exposure value Virwhen transition of the obligor i from a current rating to the rating r or lower occurs. The point that the psir(z) can be calculated as Expression (1) and the wircan be calculated as Expression (2) corresponds to that it is possible to express a polynomial distribution representing rating transitions and profits and losses of a obligor i in the MTM method by addition of binomial expressions. Therefore, a conditional expected value l(z) of a portfolio value V under the condition that z is fixed and a conditional variance v(z) of a portfolio value V under the condition that z is fixed can be easily (quickly) calculated, and VaR can be calculated analytically by the MTM method.

Further, an information processing method of the present invention executed by an information processing apparatus has: a z calculating step of calculating, based on a confidence level α stored in a storage device, a percentile of a risk factor z corresponding to α; a rating threshold calculating step of obtaining a probability pskof transition from a rating s to a rating k stored in the storage device, and calculating a rating threshold θirat which the rating of a obligor i becomes a rating r or lower based on the obtained psk; a conditional probability calculating step of obtaining a constant aiof each obligor i stored in the storage device, and calculating a probability psir(z) that the rating of the obligor i becomes the rating r or lower under a condition that z is fixed based on the obtained ai, z calculated in the z calculating step, and the θircalculated in the rating threshold calculating step using

p⁢⁢sir⁡(z)=N⁡(θir-ai⁢z1-ai)
where N is a cumulative probability function of a standard normal distribution; and a wircalculating step of calculating a difference wirin exposure value between adjacent ratings of the obligor i based on an exposure value Virof the obligor i when transition from a current rating to the rating r occurs, which is stored in the storage device, using wir=Vir−Vi(r+1).

Further, a program product of the present invention causes a computer to function as: a z calculating unit calculating, based on a confidence level α stored in a storage device, a percentile of a risk factor z corresponding to α; a rating threshold calculating unit obtaining a probability pskof transition from a rating s to a rating k stored in the storage device, and calculating a rating threshold θirat which the rating of a obligor i becomes a rating r or lower based on the obtained psk; a conditional probability calculating unit obtaining a constant aiof each obligor i stored in the storage device, and calculating a probability psir(z) that the rating of the obligor i becomes the rating r or lower under a condition that z is fixed based on the obtained ai, z calculated in the z calculating unit, and the θircalculated in the rating threshold calculating unit using

p⁢⁢sir⁡(z)=N⁡(θir-ai⁢z1-ai)
where N is a cumulative probability function of a standard normal distribution; and a wircalculating unit calculating a difference wirin exposure value between adjacent ratings of the obligor i based on an exposure value Virof the obligor i when transition from a current rating to the rating r occurs, which is stored in the storage device, using wir=Vir−Vi(r+1).

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Hereinafter, an embodiment of the present invention will be described based on the drawings.

Overview of this Embodiment

First, an overview of this embodiment will be described usingFIG. 4andFIG. 5. Incidentally, the processing illustrated below is executed by an information processing apparatus (computer) which will be described later.

As already described, a diagram representing rating transitions and profits and losses of a obligor i in the MTM method becomes a polynomial distribution likeFIG. 3. However, when a probability psirof transition of a obligor i from the current rating to a rating r or lower is obtained as described later corresponding to a transition probability pirof transition of the obligor i from the current rating to the rating r in a Merton model (enterprise value model), and a difference wirin exposure value between adjacent ratings of the obligor i is obtained as described later corresponding to an exposure value Virof the obligor when transition from the current rating to the rating r occurs, the polynomial distribution likeFIG. 3can be expressed by addition of binomial distributions as illustrated inFIG. 4.

Here,FIG. 5is a diagram for explainingFIG. 4in consideration of an enterprise value model.

In the MTM method, the information processing apparatus obtains the transition probability piraccording to the rating of a obligor i from a probability transition matrix stored in a storage device such as an HD, and a rating threshold θiris obtained so that the relation of
pir=E[l{θi(r−1)≦xi<θir}]=N(θir)−N(θi(r−1)
holds true, using

p⁢⁢si⁢⁢r=∑k=1r⁢pi⁢⁢kθi⁢⁢r=N-1⁡(p⁢⁢si⁢⁢r),
where psiris a probability of transition of the obligor i to the rating r or lower,

N(•) is a cumulative probability function of a standard normal distribution, and

N−1(•) is an inverse function of NH.

Incidentally, here the smaller the r is, the lower the rating it denotes.

Using θir, the function indicating a transition to rating BB can be represented as42inFIG. 5.

Here, using43and44inFIG. 5,42ofFIG. 5can be expressed as43-44ofFIG. 5. That is,43and44ofFIG. 5are forms similar to Diused in the DM method, and corresponds to the binomial distribution. This facilitates calculation of an expected value and a distribution. Besides the rating BB, it can be expanded as follows by expressions using Sirand Dir.

Siris a random variable denoting the status of a obligor i (whether Riis r or not),

Riis a rating of the obligor i after rating transition,

Diris a random variable indicating the status of the obligor i (whether or not Riis equal to or lower than rating r),

Viis a value of exposure after rating transition of the obligor i, and

Viris a value of exposure when the rating of the obligor i is the rating r.

In calculation with the enterprise value model, generally an expressing method of xilike the following expression for example is used.
xi=√{square root over (ai)}z+√{square root over (1−ai)}ζi

z is a risk factor,

aiis a constant representing the dependence of an enterprise value xion z, and

ζiis a factor not expressed by z in an enterprise value xiof the obligor i.

In the above expression, a conditional probability when the random variable z is fixed to a certain value as45inFIG. 5is considered. Here, an expected value l(z) and a variance v(z) of the portfolio value V are needed for analytical approximation of VaR. However, expressing a polynomial distribution by addition of binomial distributions as described above is also possible here, and it is possible to obtain l(z) and v(z) by calculation based on a probability psir(z) of transition of the obligor to the rating r or lower.

For example, in the case of multifactor with the model being expanded or in the case where it is not necessarily be Corr(ζi, ζj)=0, particularly calculation of v(z) and v′(z) becomes difficult, but the possibility of expression by binomial distributions facilitates calculation. Describing more specifically, it is difficult to perform the calculations as in the supplemental theories4,5of the IMES Discussion Paper in the DM method as the polynomial distributions as they are similarly for the MTM method, but expression by addition of binomial distributions facilitates these calculations.

As described above, the conditional expected value l(z) and the variance v(z) of the portfolio value and derivatives of them, which are required for calculating granularity adjustment, can be calculated also in the case of polynomial distributions of the MTM method. Consequently, a portfolio VaR (percentile qα according to a confidence level α) and a risk contribution of an individual company can be calculated.

As the analytical approximation as above, besides the above ones, saddle point approximation and the like are known. In the saddle point approximation and the like, numerical integration may be required other than the analytical method (closed-form). In this embodiment, the numerical integration is not necessary as described above, and a VaR and an individual company risk contribution can be calculated by only the closed-form. That is, according to this embodiment, it is possible to rapidly calculate the VaR and rapidly and stably calculate the individual company risk contribution.

Further, according to this embodiment, a granularity adjustment method can be applied to the MTM method. In calculation by the granularity adjustment method, dividing into a term that can be ignored and a term that cannot be ignored is possible in a limiting loss distribution (assuming that there is no credit concentration), and thus it is useful in view of grasping the influence of credit concentration.

Incidentally, when a VaR obtained by a Monte Carlo method can be obtained, this VaR may be assumed as a regular value, and the individual company risk contribution can be corrected by multiplying a correction ratio c as shown by the following expressions so that a VaR obtained by analytical approximation matches a VaR obtained by the Monte Carlo method.

c=VaR by Monte Carlo/VaR by analytical approximation
Individual risk contribution of a obligor i c·(individual company risk contribution of the obligor obtained by the analytical approximation)

FIG. 6is a diagram illustrating an example of a hardware structure of an information processing apparatus. As illustrated inFIG. 6, an information processing apparatus1includes, as hardware components, an input device11, a display device12, a recording medium drive device13, a ROM (Read Only Memory)15, a RAM (Random Access Memory)16, a CPU (Central Processing Unit)17, an interface device18, and an HD (Hard Disk)19.

The input device11is formed of a keyboard, a mouse, and/or the like operated by an operator (or user) of the information processing apparatus1, and is used for inputting various operation information or the like to the information processing apparatus1. The display device12is formed of a display or the like used by the user of the information processing apparatus1, and is used for displaying various information (or screen) and the like. The interface device18is an interface connecting the information processing apparatus1to a network or the like.

A program product related to flowcharts which will be described later is provided to the information processing apparatus1by, for example, a recording medium14such as a CD-ROM, or is downloaded via a network or the like. The recording medium14is set to the recording medium drive device13, and the program product is installed from the recording medium14to the HD19via the recording medium drive device13.

The ROM15stores a program and the like which are read first when the power of the information processing apparatus1is turned on. The RAM16is a main memory of the information processing apparatus1. The CPU17reads a program from the HD19as necessary, stores it in the RAM16, and executes this program, so as to provide part of functions which will be described later, or executes a flowchart or the like which will be described later. The RAM16stores, for example, a table and/or the like which will be described later.

FIG. 7is a diagram illustrating a software structure of the information processing apparatus.

As illustrated inFIG. 7, the information processing apparatus1includes, as software components, a z calculating unit21, a pssrcalculating unit22, a psircalculating unit23, a θircalculating unit24, a conditional probability calculating unit25, a wircalculating unit26, a conditional expected value calculating unit27, a conditional variance calculating unit28, a qα calculating unit29, an E[Vi] calculating unit30, a risk contribution calculating unit31, and a portfolio VaR calculating unit32.

The z calculating unit21obtains a confidence level α as a parameter from the RAM16or the like for example, and calculates the percentile of the risk factor z corresponding to the obtained confidence level α using
z=N−1(1−α).
Incidentally, the parameter may be inputted by the operator of the information processing apparatus1via the input device11or the like and stored in the RAM16or the like, or may be stored in advance in the HD19or the like. The same applies to the parameters below.

The pssrcalculating unit22obtains a rating transition matrix (probability psrof transition from the current rating s to the rating r) illustrated inFIG. 8as a parameter from the RAM16or the like for example, obtains a probability pssrof transition from the current rating s to the rating r or lower based on the obtained probability psrof transition from the current rating s to the rating r, and substitutes it in the corresponding part of a table as illustrated inFIG. 9. Here,FIG. 8is a diagram illustrating an example of a table storing the rating transition matrix.FIG. 9is a diagram illustrating an example of a table storing probabilities pssrof transition from the rating s to the rating r or lower.

The psircalculating unit23calculates the probability psirof transition of a obligor i to the rating r or lower based on the probability pssrof transition from the rating s to the rating r or lower as illustrated inFIG. 9calculated in the pssrcalculating unit22. Describing more specifically, the psircalculating unit23obtains the current rating of the obligor i from rating information by obligor as illustrated inFIG. 17which will be described later, obtains the corresponding probability pssrof transition from the rating s to the rating r or lower fromFIG. 9(for example, obtains the portion surrounded by a dotted line inFIG. 9) based on the obtained rating information, and substitutes it in a table as illustrated inFIG. 10(for example, in the part surrounded by a dotted line inFIG. 9).FIG. 10is a diagram illustrating an example of a table storing probabilities psirof transition of the obligor i to the rating r or lower.

Here, when there are plural rating systems, there exist tables corresponding toFIG. 8andFIG. 9matching the number of rating systems. In this case, the psircalculating unit23obtains a value with reference toFIG. 8andFIG. 9of the corresponding rating system for each obligor.

The ƒircalculating unit24obtains the probability psirof transition of a obligor i to the rating r or lower as illustrated inFIG. 10calculated in the psircalculating unit23from the RAM16or the like, calculates a rating threshold θirat which the rating of the obligor i becomes the rating r or lower based on the obtained probability psirusing
θir=N−1(psir)
and substitutes it in the corresponding part of a table as illustrated inFIG. 11.FIG. 11is a diagram illustrating an example of a table storing rating thresholds θirat which the rating of the obligor i becomes the rating r or lower.

The conditional probability calculating unit25obtains a value aias a parameter given in units of obligors from the RAM16or the like for example, and calculates a probability psir(z) that the rating of a obligor i changes to the rating r or lower under the condition that the risk factor z is fixed based on the obtained value aigiven in units of obligors, the percentile of the risk factor z corresponding to the confidence level α calculated in the z calculating unit21, and θircalculated in the θircalculating unit24using

p⁢⁢si⁢⁢r⁡(z)=N⁡(θi⁢⁢r-ai⁢z1-ai)
and substitutes it in the corresponding part of a table as illustrated inFIG. 14.

Incidentally, here the value aigiven in units of obligors is calculated in units of obligors using a correlation matrix illustrated inFIG. 12and correlation calculation information by obligor illustrated inFIG. 13(a specific calculating method is shown in the IMES Discussion Paper or the like, and thus details are omitted in this embodiment).

FIG. 14is a diagram illustrating an example of a table storing probabilities psir(z) that the rating of a obligor i becomes the rating r or lower under the condition that the risk factor z is fixed (conditional probability given that the risk factor z is fixed).

Further, the conditional probability calculating unit25similarly calculates a first derivative psir′(z) of the probability psir(z) that the rating of a obligor i becomes the rating r or lower under the condition that the risk factor z is fixed using

p⁢⁢si⁢⁢r′⁡(z)=-ai1-ai⁢n⁡(θi⁢⁢r-ai⁢z1-ai)
and substitutes it in the corresponding part of a table as illustrated inFIG. 15.

FIG. 15is a diagram illustrating an example of a table storing first derivatives psir′(z) of probabilities that the rating of a obligor i becomes the rating r or lower under the condition that the risk factor z is fixed (conditional probability given that the risk factor z is fixed).

Further, the conditional probability calculating unit25similarly calculates a second derivative psir″(z) of the probability psir(z) that the rating of a obligor i becomes the rating r or lower under the condition that the risk factor z is fixed using

p⁢⁢si⁢⁢r″⁡(z)=-ai1-ai·θi⁢⁢r-ai⁢z1-ai⁢n⁡(θi⁢⁢r-ai⁢z1-ai)
and substitutes it in the corresponding part of a table as illustrated inFIG. 16.

FIG. 16is a diagram illustrating an example of a table storing second derivatives psir″(z) of probabilities that the rating of a obligor i becomes the rating r or lower under the condition that the risk factor z is fixed (conditional probability given that the risk factor z is fixed).

The wircalculating unit26calculates a difference wirin value between adjacent ratings of a obligor i based on the current rating of the obligor and the exposure value Virwhen transition to the rating r occurs using
wir=Vir−Vi(r+1)
as illustrated inFIG. 17as a parameter, and substitutes the calculated wirin the corresponding position of a table as illustrated inFIG. 18.

FIG. 17is a diagram illustrating an example of a table storing current ratings of obligors i and exposure values Virwhen transition to the rating r occurs.FIG. 18is a diagram illustrating an example of a table storing differences wirin value between adjacent ratings of obligors i.

The conditional expected value calculating unit27calculates a conditional expected value l(z) of a portfolio value V and a conditional expected value li(z) of the portfolio value V of each obligor based on the difference wirin value between adjacent ratings of a obligor i as illustrated inFIG. 18calculated in the wircalculating unit26, and the probability psir(z) that the rating of the obligor i becomes the rating r or lower under the condition that the risk factor z is fixed as illustrated inFIG. 14calculated in the conditional probability calculating unit25using

li⁡(z)=∑r⁢wi⁢⁢r⁢p⁢⁢si⁢⁢r⁡(z)⁢⁢l⁡(z)=E⁡[V❘z]=∑i⁢li⁡(z)Expression⁢⁢3
and substitutes the li(z) in the corresponding part of a table as illustrated inFIG. 19for example.

FIG. 19is a diagram illustrating an example of a table storing conditional expected values l(z) of portfolio values V, conditional variances of the portfolio values V, and the like.

Further, the conditional expected value calculating unit27similarly calculates a first derivative l′(z) of a conditional expected value l(z) of a portfolio value V and a first derivative li′(z) of a conditional expected value li(z) of the portfolio value V of each obligor using

li′⁡(z)=∑r⁢wi⁢⁢r⁢p⁢⁢si⁢⁢r′⁡(z)l′⁡(z)=ⅆⅆz⁢E⁡[V❘z]=∑i⁢li′⁡(z)
and substitutes the li′(z) in the corresponding part of the table as illustrated inFIG. 19for example.

Further, the conditional expected value calculating unit27similarly calculates a second derivative l″(z) of a conditional expected value l(z) of a portfolio value V and a second derivative li″(z) of a conditional expected value li(z) of the portfolio value V of each obligor using

li″⁡(z)=∑r⁢wi⁢⁢r⁢p⁢⁢si⁢⁢r″⁡(z)l″⁡(z)=ⅆ2ⅆz2⁢E⁡[V❘z]=∑i⁢li″⁡(z)
and substitutes the li″(z) in the corresponding part of the table as illustrated inFIG. 19for example.

The conditional variance calculating unit28calculates a conditional variance v(z) of a portfolio value V and a partial distribution vi(z) of each obligor of v(z) based on the difference wirin value between adjacent ratings of a obligor i as illustrated inFIG. 18calculated in the wircalculating unit26, and the probability psir(z) that the rating of the obligor i becomes the rating r or lower under the condition that the risk factor z is fixed as illustrated inFIG. 14calculated in the conditional probability calculating unit25using

vi⁡(z)=Vi⁢∂∂Vi⁢Var⁡[V❘z]⁢⁢v⁡(z)=12⁢∑i⁢vi⁡(z)Expression⁢⁢(4)
and substitutes the vi(z) in the corresponding part of the table as illustrated inFIG. 19for example.

Further, the conditional variance calculating unit28similarly calculates a first derivative v′(z) of a conditional variance v(z) of a portfolio value V and a partial distribution vi′(z) of each obligor of v′(z) using

vi′⁡(z)=Vi⁢∂∂Vi·ⅆⅆz⁢Var⁡[V❘z]v′⁡(z)=12⁢∑i⁢vi′⁡(z)
and substitutes the vi′(z) in the corresponding part of the table as illustrated inFIG. 19for example.

The qα calculating unit29calculates a percentile qα or the like of a portfolio value according to the confidence level α based on the conditional expected value li(z), the first derivative li′(z), and the second derivative li″(z) of a portfolio value V of each obligor calculated in the conditional expected value calculating unit27and stored in the table as illustrated inFIG. 19, and the vi(z) and the first derivative vi′(z) calculated in the conditional variance calculating unit28and stored in the table as illustrated inFIG. 19using

The E[Vi] calculating unit30obtains a rating transition matrix (probability psrof transition from the current rating s to the rating r) illustrated inFIG. 8as a parameter from the RAM16or the like for example and obtains an exposure value Virof a obligor when transition from the current rating to the rating r occurs as illustrated inFIG. 17as a parameter from the RAM16or the like for example, and calculates an expected value E[Vi] of an exposure value of a obligor i based on the obtained probability psrof transition from the current rating to the rating r and the obtained current rating of the obligor i and exposure value Virwhen transition to the rating r occurs using

E⁡[Vi]=∑r⁢pi⁢⁢r⁢Vi⁢⁢r
and substitutes the calculated E[Vi] in the corresponding position of a table as illustrated inFIG. 20.

FIG. 20is a diagram illustrating an example of a table storing expected values of exposure values by obligor, individual company risk contributions, and so on.

The risk contribution calculating unit31calculates a risk contribution (individual company risk contribution) of a obligor i based on the E[Vi] calculated in the E[Vi] calculating unit30and stored in the table as illustrated inFIG. 20and the qαicalculated in the qα calculating unit29and stored in the table as illustrated inFIG. 19using
risk contribution of a obligori=qαi−E[Vi]
for example and substitutes the calculated risk contribution of the obligor i in the corresponding part of the table as illustrated inFIG. 20.

The portfolio VaR calculating unit32calculates a portfolio VaR based on the risk contribution of a obligor i calculated in the risk contribution calculating unit31and stored in the table as illustrated inFIG. 20using

Portfolio⁢⁢VaR=q⁢⁢α-∑i⁢E⁡[Vi]=∑i⁢⁢risk⁢⁢contribution⁢⁢of⁢⁢an⁢⁢obligor⁢⁢i
for example and substitutes the calculated risk contribution of the obligor i in the corresponding part of the table as illustrated inFIG. 20.

Hereinafter, processing of the information processing apparatus will be described using flowcharts.

FIG. 21is a flowchart illustrating an example of processing to calculate the percentile of the risk factor z corresponding to the confidence level α.

In step S10, the z calculating unit21obtains a confidence level α as a parameter from the RAM16or the like for example, and calculates the percentile of the risk factor z corresponding to the obtained confidence level α using
z=N−1(1−α).

FIG. 22is a flowchart illustrating an example of processing to calculate the rating threshold θirat which the rating of a obligor i becomes the rating r or lower.

In step S20, the pssrcalculating unit22obtains a rating transition matrix (probability psrof transition from the current rating s to the rating r) illustrated inFIG. 8as a parameter from the RAM16or the like for example, obtains a probability pssrof transition from the current rating s to the rating r or lower based on the obtained probability psrof transition from the current rating s to the rating r, and substitutes it in the corresponding part of a table as illustrated inFIG. 9.

In step S21, the psircalculating unit23calculates the probability psirof transition of a obligor i to the rating r or lower based on the probability pssrof transition from the rating s to the rating r or lower as illustrated inFIG. 9calculated in step S20, and substitutes it in the corresponding part of a table as illustrated inFIG. 10.

In step S22, the θircalculating unit24obtains the probability psirof transition of a obligor i to the rating r or lower as illustrated inFIG. 10calculated in step S21from the RAM16or the like, calculates a rating threshold θirat which the rating of the obligor i becomes the rating r or lower based on the obtained probability psirusing
θir=N−1(psir)
and substitutes it in the corresponding part of a table as illustrated inFIG. 11.

FIG. 23is a flowchart illustrating an example of processing to calculate the conditional probability or the like.

In step S30, the conditional probability calculating unit25obtains a value aias a parameter given in units of obligors from the RAM16or the like for example, and calculates a probability psir(z) that the rating of a obligor i changes to the rating r or lower under the condition that the risk factor z is fixed based on the obtained ai, the percentile of the risk factor z corresponding to the confidence level α calculated in the z calculating unit21, and θircalculated in the θircalculating unit24using

p⁢⁢si⁢⁢r⁡(z)=N(θi⁢⁢r-ai⁢z1-ai)
and substitutes it in the corresponding part of a table as illustrated inFIG. 14.

In step S31, the conditional probability calculating unit25similarly calculates a first derivative psir′(z) of the probability psir(z) that the rating of a obligor i becomes the rating r or lower under the condition that the risk factor z is fixed using

p⁢⁢si⁢⁢r′⁡(z)=-ai1-ai⁢n⁡(θi⁢⁢r-ai⁢z1-ai)
and substitutes it in the corresponding part of a table as illustrated inFIG. 15.

In step S32, the conditional probability calculating unit25similarly calculates a second derivative psir″(z) of the probability psir(z) that the rating of a obligor I becomes the rating r or lower under the condition that the risk factor z is fixed using

p⁢⁢si⁢⁢r″⁡(z)=-ai1-ai·θi⁢⁢r-ai⁢z1-ai⁢n⁡(θi⁢⁢r-ai⁢z1-ai)
and substitutes it in the corresponding part of a table as illustrated inFIG. 16.

FIG. 24is a flowchart illustrating an example of processing to calculate the difference wirin value between adjacent ratings of a obligor i.

In step S40, the wircalculating unit26calculates a difference wirin value between adjacent ratings of a obligor I based on the current rating of the obligor I and the exposure value Virwhen transition to the rating r occurs using
wir=Vir−Vi(r+1)
as illustrated inFIG. 17as a parameter, and substitutes the calculated wirin the corresponding position of a table as illustrated inFIG. 18.

FIG. 25is a flowchart illustrating an example of processing to calculate the percentile qα of a portfolio value or the like according to the confidence level α.

In step S50, the conditional expected value calculating unit27calculates a conditional expected value l(z) of a portfolio value V and a conditional expected value li(z) of the portfolio value V of each obligor based on the difference wirin value between adjacent ratings of a obligor I as illustrated inFIG. 18calculated in the processing ofFIG. 24, and the probability psir(z) that the rating of the obligor I becomes the rating r or lower under the condition that the risk factor z is fixed as illustrated inFIG. 14calculated in the processing ofFIG. 23using

li⁡(z)=∑r⁢wi⁢⁢r⁢p⁢⁢si⁢⁢r⁡(z)l⁡(z)=E⁡[V❘z]=∑i⁢li⁡(z)
and substitutes the li(z) in the corresponding part of a table as illustrated inFIG. 19for example.

Further, the conditional expected value calculating unit27similarly calculates a first derivative l′(z) of a conditional expected value l(z) of a portfolio value V and a first derivative li′(z) of a conditional expected value li(z) of the portfolio value V of each obligor using

li′⁡(z)=∑r⁢wi⁢⁢r⁢p⁢⁢si⁢⁢r′⁡(z)l′⁡(z)=ⅆⅆz⁢E⁡[V❘z]=∑i⁢li′⁡(z)
and substitutes the li′(z) in the corresponding part of the table as illustrated inFIG. 19for example.

Further, the conditional expected value calculating unit27similarly calculates a second derivative l″(z) of a conditional expected value l(z) of a portfolio value V and a second derivative li″(z) of a conditional expected value li(z) of the portfolio value V of each obligor using

li″⁡(z)=∑r⁢wi⁢⁢r⁢p⁢⁢si⁢⁢r″⁡(z)l″⁡(z)=ⅆ2ⅆz2⁢E⁡[V❘z]=∑i⁢li″⁡(z)
and substitutes the li″(z) in the corresponding part of the table as illustrated inFIG. 19for example.

In step S51, the conditional variance calculating unit28calculates a conditional variance v(z) of a portfolio value V and a partial distribution vi(z) of each obligor of v(z) based on the difference wirin value between adjacent ratings of a obligor i as illustrated inFIG. 18calculated in the processing ofFIG. 24, and the probability psir(z) that the rating of the obligor i becomes the rating r or lower under the condition that the risk factor z is fixed as illustrated inFIG. 14calculated in the processing ofFIG. 23using

vi⁡(z)=Vi⁢∂∂Vi⁢Var⁡[V❘z]v⁡(z)=12⁢∑i⁢vi⁡(z)
and substitutes the vi(z) in the corresponding part of the table as illustrated inFIG. 19for example.

Further, the conditional variance calculating unit28similarly calculates a first derivative v′(z) of a conditional variance v(z) of a portfolio value V and a partial distribution vi′(z) of each obligor of v′(z) using

vi′⁡(z)=Vi⁢∂∂Vi·ⅆⅆz⁢Var⁡[V❘z]v′⁡(z)=12⁢∑i⁢vi′⁡(z)
and substitutes the vi′(z) in the corresponding part of the table as illustrated inFIG. 19for example.

Here, the order of the processing of step S50and the processing of step S51is not in question. Step S51may be earlier than step S50, or may be later than step S50. Further, they may be performed simultaneously.

In step S52, the qα calculating unit29calculates a percentile qα or the like of a portfolio value according to the confidence level α based on the conditional expected value li(z), the first derivative li′(z), and the second derivative li″(z) of a portfolio value V of each obligor calculated in step S50and stored in the table as illustrated inFIG. 19, and the vi(z) and the first derivative vi′(z) calculated in step S51and stored in the table as illustrated inFIG. 19using

FIG. 26is a flowchart illustrating an example of processing to calculate the expected value E[Vi] of the exposure value of a obligor i.

In step S60, the E[Vi] calculating unit30obtains a rating transition matrix (probability psrof transition from the current rating s to the rating r) illustrated inFIG. 8as a parameter from the RAM16or the like for example and obtains an exposure value Virof a obligor i when transition from the current rating to the rating r occurs as illustrated inFIG. 17as a parameter from the RAM16or the like for example, and calculates an expected value E[Vi] of an exposure value of a obligor i based on the obtained probability psrof transition from the current rating to the rating r and the obtained current rating of the obligor i and exposure value Virwhen transition to the rating r occurs using

E⁡[Vi]=∑r⁢pi⁢⁢r⁢Vi⁢⁢r
and substitutes the calculated E[Vi] in the corresponding position of a table as illustrated inFIG. 20.

FIG. 27is a flowchart illustrating an example of processing to calculate the individual company risk contribution and the portfolio VaR.

In step S70, the risk contribution calculating unit31calculates a risk contribution (individual company risk contribution) of a obligor i based on the E[Vi] calculated in the processing ofFIG. 26and stored in the table as illustrated inFIG. 20and the qαicalculated in the processing ofFIG. 25and stored in the table as illustrated inFIG. 19using
risk contribution of a obligori=qαi−E[Vi]
for example and substitutes the calculated risk contribution of the obligor i in the corresponding part of the table as illustrated inFIG. 20.

In step S71, the portfolio VaR calculating unit32calculates a portfolio VaR based on the risk contribution of a obligor i calculated in step S70and stored in the table as illustrated inFIG. 20using

Portfolio⁢⁢VaR=q⁢⁢α-∑i⁢E⁡[Vi]=∑i⁢⁢risk⁢⁢contribution⁢⁢of⁢⁢an⁢⁢obligor⁢⁢i
for example and substitutes the calculated risk contribution of the obligor i in the corresponding part of the table as illustrated inFIG. 20.

As described above, the information processing apparatus of this embodiment has a technical characteristic in that it calculates analytical approximation of VaR of the portfolio value V in the MTM method using the difference wirin exposure value between adjacent ratings. Hereinafter, this point will be described in more detail.

First, calculation of VaR by the Monte Carlo method in each of the DM method and the MTM method will be described.

Expressions for performing a simulation by the Monte Carlo method in the DM method are shown below.

L is a loss of the portfolio,

Liis a loss by the obligor i,

Diis a random variable indicating the status of the obligor i (default or non-default),

xiis a random variable indicating the enterprise value of the obligor i,

l[•] is a defined function,

xi<θiis whether the enterprise value xiof a obligor i becomes lower than the default threshold θior not

LGDiis a loss given default in exposure of the obligor i, and

EADiis an amount of exposure of the obligor i,

piis a default rate of the obligor i

N(•) is a cumulative probability function of a standard normal distribution, and

N−1(•) is an inverse function of N(•).

Expressions for performing a simulation by the Monte Carlo method in the MTM method are shown below.

V is the value of a portfolio,

Viis a value of exposure after rating transition of a obligor i

Viris a value of exposure when the obligor i has a rating r,

Riis the rating of the obligor i after rating transition,

Siris a random variable indicating the status of the obligor i (whether Ri is r or not),

xiis a random variable indicating the enterprise value of the obligor i,

pikis a probability of transition of the obligor to a rating k,

psiris a probability of transition of the obligor to a rating r,

N(•) is a cumulative probability function of a standard normal distribution, and

N−1(•) is an inverse function of N(•).

Note that the smaller the r is, the lower the rating it denotes in the above-described expressions.

As shown in Expression (5), in the DM method, the information processing apparatus generates random numbers corresponding to the random variable xifor the number of times of trials and calculates in the order of Di→Li→L, to thereby calculate a probability distribution and a risk indicator (namely, VaR) thereof. Further, as shown in Expression (6), in the MTM method, the information processing apparatus generates random numbers corresponding to the random variable xifor the number of times of trials and calculates in the order of Sir→Vi→V, to thereby calculate a probability distribution and a risk indicator (namely, VaR) thereof.

Next, analytical approximation of VaR will be described. For example, the IMES Discussion Paper discloses an approach of analytical approximation of VaR in the DM method. The IMES Discussion Paper describes approximation of a quantile qα(L) of a loss ratio distribution L with qα(E[L|Z])+Δqα(L). Incidentally, the IMES Discussion Paper describes as theory2that an α quantile qα(E[L|Z]) of a limiting loss distribution E[L|Z] is calculated by a conditional expected loss given a 1−a quantile of Z. Further, the IMES Discussion Paper describes as theory3a granularity adjustment term which is a difference Δqα(L) (=qα(L)−qα(E[L|Z])) between a true loss quantile and an a quantile of a limiting loss distribution. Then the IMES Discussion Paper describes that this granularity adjustment term Δqα(L) is represented as the following expression. Here, l(z)=E[L|Z=z], v(z)=VaR[L|Z=z].

Then the IMES Discussion Paper describes that a VaR is calculated as the quantile qα(L) of the loss ratio distribution L by calculating the above expression.

Incidentally, in the Monte Carlo method, the information processing apparatus uses

Vi=∑r⁢Sir⁢Vir
in the MTM method corresponding to the calculation of Li=DiLGDiEADiof the DM method as shown in Expression (6). Accordingly, the information processing apparatus calculates the above-described granularity adjustment term using this expression in the analytical approximation of the MTM method.

Here, l(z)=E[V|z], v(z)=VaR[V|z]. This calculation of VaR[V|z] is represented more specifically as the following expression.

Here, the amount of calculation of the part Corr, [SirVir, SjsVj|z] is proportional to the “square of (the number of obligors×the number of ratings)”. For example, when the number of obligors=10000 and the number of ratings=10, the calculation of correlation is needed to be performed the number of times equal to the square of (10000×10), that is, the tenth power of 10.

On the other hand, the information processing apparatus according to this embodiment calculates the a quantile qα of a portfolio value V using the difference wirin exposure value between adjacent ratings. At this time, the information processing apparatus calculates E[V|z] in Expression (3) or the like and VaR[V|z] in Expression (4) or the like for calculating Δqα, but also in this calculation, the amount of calculation in proportion to “the square of (the number of obligors×the number of ratings)” is needed. However, as described usingFIG. 15,FIG. 16, and the like of this embodiment, the information processing apparatus calculates N−1(psir(z)) and the like besides wir, psir(z), and ps′ir(z) for “the number of obligors×the number of ratings”, and stores the calculated values of correlations ρij for the amount of “the square of the number of obligors” as parameters in a storage device such as a memory or a cache. In this point, the information processing apparatus can read and reuse data of parameters from the storage device such as a memory when executing calculation of the amount in proportion to “the square of (the number of obligors×the number of ratings)” requiring these parameters. Accordingly, calculation efficiency increases significantly. Further, the information processing apparatus keeps frequently used data stored in a cache having much higher reading and writing speed than that of the memory. Thus, when these parameters are stored in the cache and the operation is increased in speed particularly using a function called SIMD (Single Instruction Multiple Data) of the CPU or the like, it is possible to prevent the reading speed of the memory from becoming a bottle neck, and reduce the idling time of the CPU or the like. That is, when the analytical approximation of VaR in the MTM method is obtained, the difference wirin exposure value between adjacent ratings can be used to increase calculation efficiency, and thus the VaR in the MTM method can be calculated rapidly.

Incidentally, vi(z) and vi′(z) are represented by the following expressions.

Here, v∞iand v′∞iare represented by the following expressions.

Further, vGiand v′Giare represented by the following expressions.

Parts of N2(x, y, ρ) in Expression (8) and Expression (9) are cumulative probability functions of a bivariate standard normal distribution. Further, t=min(r, s) (one with a lower rating out of r and s). Further, ρij=corr(xi, xj|z) (the part εicorresponding to the idiosyncratic risk of a obligor in xiis calculated as corr(εi, εj)=0).

Moreover, Δaαi(z) is represented by, more specifically, the following expression.

With the above-described structure, the information processing apparatus according to this embodiment uses the difference wirin exposure value between adjacent ratings so as to store N−1(psir(z)), ρij, and the like besides wir, psir(z), and ps′ir(z) as parameters in the cache, which can be reused repeatedly when performing the calculation of Expression (7) to Expression (10). Thus the calculation efficiency can be increased. Accordingly, the analytical approximation of VaR in the MTM method can be calculated rapidly.

As has been described above, according to this embodiment, it becomes possible to quickly calculate the conditional expected value l(z) and the variance v(z) of a portfolio value needed for calculating granularity adjustment, and the derivatives of them for polynomial distributions in the MTM method. Further consequently, the portfolio VaR (percentile qα according to the confidence level α) and risk contributions of individual companies can be calculated rapidly.

In short, according to this embodiment, a polynomial distribution representing rating transitions and profits and losses of a obligor i in the MTM method can be expressed by addition of binomial distributions, and thereby VaR and the like can be obtained quickly by analytical approximation even in the MTM method.

Note that the processing of this embodiment can be applied to calculation of a risk indicator similar to VaR, such as an expected shortfall, or the like.

Further, the above-described li(z), li′(z), li″(z), vi(z), vi′(z), and qαiare repeating of the same calculation pattern for each obligor. Accordingly, the conditional expected value calculating unit27, the conditional variance calculating unit28, the qα calculating unit29, and the like can use the function called SIMD (Single Instruction Multiple Data) mounted in the CPU17or the like (hereinafter referred to as a SIMD function) to calculate the li(z), li′(z), li″(z), vi(z), vi′(z), and qαirapidly.

In the foregoing, the preferred embodiment of the present invention has been described in detail, but the present invention is not limited to such a specific embodiment, and can be modified and changed in various ways within the scope of the spirit of the invention described in the claims that follow.

According to the present invention, it is possible to allow obtaining VaR and the like quickly in analytical approximation in an MTM method.

The present embodiments are to be considered in all respects as illustrative and no restrictive, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. The invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof.