JP2003067565A - Integrated evaluation system and method for market risk and credit risk - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a new simulation system and method for conducting integrated evaluations on the market risk and credit risk of a portfolio. SOLUTION: A number of scenarios from the present time to a risk horizon are created based on a model of a default-free interest rate process, a model of a default process, a model of a stock-price process, and a model of an exchange rate, and the price of a portfolio and the price of an individual asset on the risk horizon are calculated for each of the scenarios created. Based on the calculated prices, a future price distribution of the portfolio and/or a future price distribution of the individual asset are determined to conduct integrated evaluations on the market risk and credit risk of the portfolio.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an integrated evaluation system and integrated evaluation method for portfolio market risk and credit risk, which financial institutions use for risk management of financial assets.

[0002]

[Prior Art] Companies, not only financial institutions, have portfolios of many financial assets.
We are constantly exposed to various risks such as market risk (interest rates, foreign exchange, stock prices, products ...), credit risk, liquidity risk, operation risk and so on. The importance of such quantitative evaluation of financial risk has been recognized by the first BIS regulation of 1988, and a risk evaluation model has been developed in the order of market risk and credit risk.

Here, the market risk is 1 for a portfolio composed of assets that are actively traded in the market.
It is the volatility of the present value of the portfolio, which is measured as a risk horizon (at the time of future risk assessment) of one day or at most about 10 days. The current market risk assessment model standard is RiskMetrics ^™ (JP
Morgan Bank's trademark), which models the rate of return of each asset, or more accurately the risk factor, with a multivariate normal distribution, which seems to be a good approximation in general.

On the other hand, it is called default that the issuer of financial instruments and the customer cannot fulfill the obligation, and the loss incurred by the default is collectively called credit risk. In addition to the direct loss that "the default of the customer will not receive the cash flow that was planned in the future", "the increase in the possibility of future default will reduce the asset value". Indirect losses are also included. The default, which is the source of credit risk, is a binary event such as ○ (no default) or × (default), so the rate of return of assets with credit risk cannot be approximated by a normal distribution. Moreover, typical risks
Since the horizon is long, 1 year, 2 years, etc., unlike market risk, the observation probability and pseudo-probability for price valuation (risk-neutral probability, forward-neutral probability, etc. are used only for asset price valuation, risk adjustment Probability) must be clearly distinguished and used.

Probably because of such problems, there is no standard credit risk evaluation model that can be called a standard. As a typical model, Cred
itMetrics ^™ (trademark of JP Morgan Bank)
And CRE by Credit Suisse Financial Products
There are DITRISK ^+, but these are basically models that evaluate only credit risk, and integration with market risk is done only in a provisional form. On the other hand, the present inventors proposed in Japanese Patent Application No. 2001-026147 a general framework for integrated evaluation of interest rate risk and credit risk.

[0006]

SUMMARY OF THE INVENTION In the present application, Japanese Patent Application No.
Based on the general framework for integrated evaluation of interest rate risk and credit risk proposed in No. 1-026147, stock price fluctuation risk and foreign exchange risk are added, and more practical market risk and credit risk It is intended to provide an integrated evaluation system and an integrated evaluation method. It is desirable that the integrated evaluation system and integrated evaluation method for market risk and credit risk solve the following problems in consideration of the characteristics of risks and the convenience of practical use described above. 1. All assets in one measure, such as the arbitrage-free model,
Please evaluate. There are various types of assets, but at present, their evaluation criteria are not always unified.
For example, marketable assets that are actively traded in the market are evaluated based on the market price (market value), while non-marketable assets such as loans are evaluated based on the loan amount (book value).
If the risk amount is calculated by measuring each individual asset with an easy-to-measure scale according to this convention, it becomes difficult to calculate the risk amount of the entire portfolio, which is the ultimate purpose of the risk evaluation model. To avoid this,
It is necessary to divide it to some extent and evaluate all assets with one unified scale. As a candidate for the unified measurement scale, market value evaluation based on a non-arbitrage model is a powerful candidate. However, in this case, the future point (risk horizon)
Since the market price will be evaluated in, it is necessary to clearly distinguish and use the observation probability and the pseudo-probability for price evaluation described above. 2. Portfolio effect, i.e. the correlation between asset prices
(In particular, here, correlation of default probability and default
It is possible to consider the effect of the correlation between probability and interest rate. A particularly important point in assessing the risk of a portfolio is the assessment of the effects of diversified investment. Even if it is a model that allows precise price valuation of individual assets, it cannot be an effective tool for portfolio risk management unless the effect of diversified investment considering the correlation of asset price fluctuations can be evaluated. 3. Be consistent with observed market prices. Since the present value calculated from the model does not match the current market price, contradiction arises because it is evaluated on a market value basis. In addition, since the market price should reflect all risks, it is not
It is highly desirable as it reflects all risks in some way in the assessment. 4. To be able to consider the term structure of interest rates and default rates.
In doing so, incorporate the results of the empirical analysis into the model as appropriate.
Be flexible enough to Since the risk horizon in assessing credit risk is considerably longer than in assessing market risk alone, it is necessary to consider them for variables with a term structure such as interest rates and default probabilities. Moreover, it is desirable that the results of those empirical analyzes can be incorporated as appropriate and reflected in risk assessment. For example, when analyzing the default data by rating, the default hazard rate h (t) (described later) is high at high rating.
There is a tendency to increase with t, and a tendency to decrease with t for low rating. These properties should, of course, be reflected in the model. 5. Appropriate asymmetry and non-normality of the return distribution of individual assets
What can be expressed in. As mentioned above, the default, which is the source of credit risk, is a binary phenomenon such as whether it is ○ or ×, so there is an extreme asymmetry in the rate of return on assets. Since this is one of the characteristics of credit risk, a model that can properly express this asymmetry is required when focusing on credit risk evaluation. 6. The distribution is specifically obtained. A model in which only a specified risk index, for example, the standard deviation or 95% VaR, is obtained as an output is unsuitable for risk evaluation. This is because if you do not take a comprehensive view of the distribution, you may mistake the magnitude of the risk. On the other hand, if the distribution of the price or the rate of return is specifically obtained as an output, an arbitrary risk index can be calculated from it, and the risk can be accurately grasped. 7. The computational load should not be too heavy. As will be described later, enormous calculation time is required for theoretically rigorous risk assessment under general conditions. Since risk assessment is a practical request, it is preferable that the calculation time be as short as possible. A short computation time is desirable, even if it would compromise theoretical rigor a little.

[0007]

An integrated evaluation system for market risk and credit risk according to the present invention comprises an input means for inputting data or an instruction, and an output means for outputting a calculation result based on the input data or instruction. Portfolio data storage means for storing data on the portfolio, including classification, quantity, market value, and transfer of each financial instrument that constitutes the portfolio to be evaluated, and classification, type, issuer, maturity date for each financial instrument・ Financial product attribute / rating data storage means that stores detailed data such as coupon rates and rating data, and market interest rates
Market data storage means for storing market data including time-series data of market prices and exchange rates of financial instruments including public and corporate bonds and stock, and default data including past bankruptcies by year and default rates by overseas year A default data storage means and a scenario that depicts the temporal change in interest rate fluctuations from the present to the risk horizon based on the default free interest rate process model expressed by an equation that describes future fluctuations in default free interest rates, Describes future scenarios of stock price fluctuations, as well as scenarios depicting temporal changes in the status of default occurrences from the present to risk horizon based on the default process model expressed by equations that describe future fluctuations in the default rate of occurrence. Based on the stock price fluctuation process model expressed by the equation Based on a scenario that depicts the temporal change in stock price fluctuations up to the rise, and the exchange rate fluctuation model expressed by an equation that describes future fluctuations in the exchange rate, the change over time in the exchange rate fluctuations from the present to the risk horizon And a future situation in which a large number of scenarios are drawn based on the data stored in at least one of the input or portfolio data storage means, financial instrument attribute / rating data storage means, market data storage means, and default data storage means. A risk horizon that calculates the portfolio price and individual asset price in the risk horizon by introducing the risk premium adjustment rate for each scenario generated by the scenario generating means and the future situation scenario generating means. Asset price calculation method and risk horizon Calculate future price distribution of portfolio and / or future price distribution of individual asset based on portfolio price and individual asset price in risk horizon calculated for each scenario by asset price calculation method Means for integrating market risk and credit risk based on the future portfolio price distribution and / or the individual asset future price distribution calculated by the future price distribution calculating means. Characterize.

This integrated evaluation system for market risk and credit risk is based on the future price distribution of the portfolio and / or the future price distribution of the individual asset calculated by the future price distribution calculation means. To output a practically necessary index by providing a risk index calculation means for calculating a risk index including standard deviation, VaR, CVaR, and lower partial product ratio of future prices and / or future rates of return in the horizon. You can

Similarly, the integrated evaluation system for market risk and credit risk is such that the future portfolio price distribution calculated based on the future price distribution calculation means and / or
Alternatively, it is possible to output a practically necessary index by providing a return index calculation means for calculating a return index including the expected return rate in the risk horizon based on the future price distribution of the individual asset.

In this integrated system for evaluating market risk and credit risk, the future situation scenario generation means according to the invention of claim 4 is domestic default free interest rate r (t) and hazard rate h _{j of} domestic company _j. (t), domestic company j
Stock price S _j (t) of the foreign partner country F, default free interest rate r _f (t), hazard rate h of company j of country F
_{f, j} (t), stock price S _{f, j} (t) of company j of country F, and future fluctuations of the exchange rate between home country and country F (counter currency conversion value of home country currency unit) V (t) Stochastic differential equation describing And μ ₀ (t), μ _j (t), μ _{s, j} (t), μ _{f, 0}
(t), μ _{f, j} (t), μ _{fs, j} (t) and μ _V (t) are r (t), h _j (t), S _j (t), r _f (t), h
drift of _{f, j} (t), S _{f, j} (t) and V (t), σ ₀
(t), σ _j (t), σ _{s, j} (t), σ _{f, 0} (t), σ
_{f, j} (t), σ _{fs, j} (t) and σ _V (t) are r (t), h _j (t), S _j (t), r _f (t) and h, respectively.
Volatility of _{f, j} (t), S _{f, j} (t) and V (t), all based on t and these random variables and the default state (whether or not defaulting) of all firms Then, a sample path from the current time t to the risk horizon T (T> t) is generated under the observation probability, and r (s), h _j (s), S _j (s), and r _f (s) And h _{f, j} (s) and S
_{f, j} (s) and V (s), the first processing means for obtaining t ≦ s ≦ T, and τ _j as the default time point of the company j not defaulting at time t, ,
Survival probability P _T {τ _j > T} under the observation probability of company j in risk horizon T Probability of defaulting by risk horizon T based on According to the second processing means for determining the default / non-default state of the company j in the risk horizon T for each sample path, a future situation scenario can be generated.

The invention according to claim 5 is the risk of calculating the portfolio price and the individual asset price in the risk horizon for the sample path generated by the future situation scenario generating means according to claim 4.・ It is one aspect of Horizon's asset price calculation method, and risk ・
For financial instruments issued by company j, which has already defaulted at horizon T, the price p ^def (T) at risk horizon T is calculated as the default time τ of company j.
_j , the financial product attribute / rating data including the maturity date, the interest payment date, the coupon rate, and the recovery rate, and the market data including the stock price of the company j and the price of the product up to the time τ _j . And the company j (j = 0) that is not defaulted in the risk horizon T
Is a default-free product, domestic default-free interest rate r (t), hazard rate h ^ _j (t) of domestic company j, stock price S _j (t) of domestic company j, and overseas counterparty Country Default free interest rate r _f (t) of country F, hazard rate h ^ _{f , j} (t) of company j of country F, stock price S _{f, j} (t) of company j of country F, and own country and partner country F Country of the exchange rate of the (national currency equivalent value of the partner country currency one unit) V (t), the variation of the risk horizon T and later under the risk-neutral probability P ^ or forward-neutral probability P ^s, the following stochastic differential equation Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And μ ₀ (t), μ _j (t), μ _{f, 0} (t) and μ
_{f, j} (t) are r (t), h _j (t), r _f (t),
And the drift of h _{f, j} (t), σ ₀ (t), σ _j (t), σ
_{s, j} (t), σ _{f, 0} (t), σ _{f, j} (t), σ _{fs, j}
(t) and σ _V (t) are r (t), h _j (t), and S, respectively.
_j (t), r _f (t), h _{f, j} (t), S _{f, j} (t) and V
volatility of (t), β ₀ (t), β _{n + j} (t), β
_{f, 0} (t), β _{f, nf + j} (t) and β _V (t) are z ₀ (t), z _{n + j} (t), z _{f, 0} (t) and z, respectively.
market price of risk of _{f, nf + j} (t), z _V (t), ξ _j
(t), ξ _{f, j} (t) is the risk premium adjustment rate, ρ
_jk (t), ρ ^ff _jk (t), ρ ^f _jk (t), ρ
_jV (t) and ρ ^f _jV (t) are correlation coefficients, both of which are generated based on t and these random variables and the default state (whether or not defaulting) of all companies. The cumulative cash flow X (t) of the financial product until time t is r (s) (T ≦ s) and h ^ _j (s) (T ≦ s)
By using the above-mentioned random variable including the above, the default / non-default state of the company j, and the financial instrument attribute data including the maturity date, the interest payment date, the coupon rate, and the recovery rate, Price p ^nondef (T) in horizon T is a risk neutral evaluation method (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^) or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information up to time T under the forward neutral probability P ^s is known, and v ₀ (T, T _i ^j ) is the maturity T _i. ^The risk-horizon asset price can be calculated by using the second processing means for calculating the default-free discount bond of ^j at the time T).

The invention according to claim 6 is a risk of calculating the portfolio price and individual asset price in the risk horizon for the sample path generated by the future situation scenario generating means according to claim 4. • Another aspect of the horizon asset price calculation method, which is a company that has already defaulted in the risk horizon T
The financial products issued by
The price p ^def (T) at is the default time τ _j of the company j, the financial product attribute / rating data including the maturity date, the interest payment date, the coupon rate and the recovery rate, the stock price of the company j, and the time τ of the product. and processing means a determined based on the market data, including the price of up to _j, firm j that do not default in the risk horizon T (j = 0 is the default-free) for the financial instruments issued by the will,
Domestic default-free interest rate r (t), hazard rate h ^ _j (t) of domestic company j, stock price S _j (t) of domestic company j, default-free interest rate r _f (t of overseas partner country F ), The hazard rate h ^ _{f, j} (t) of company j in country F, the stock price _{Sf, j} (t) of company j in country F, and the exchange rate between home country F and country F.
The following equation is the stochastic differential equation for the fluctuations of (the currency conversion value of one unit of the currency of the partner country) V (t) after the risk horizon T under the risk neutral probability P ^ or the forward neutral probability P ^s. Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And μ ₀ (t), μ _j (t), μ _{f, 0} (t) and μ
_{f, j} (t) are r (t), h _j (t), r _f (t), and the drift of h _{f, j} (t), σ ₀ (t), σ _j (t), σ, respectively.
_{s, j} (t), σ _{f, 0} (t), σ _{f, j} (t), σ _{fs, j}
(t) and σ _V (t) are r (t), h _j (t), and S, respectively.
_j (t), r _f (t), h _{f, j} (t), S _{f, j} (t) and V
volatility of (t), β ₀ (t), β _{n + j} (t), β
_{f, 0} (t), β _{f, nf + j} (t) and β _V (t) are z ₀ (t), z _{n + j} (t), z _{f, 0} (t) and z, respectively.
market price of risk of _{f, nf + j} (t), z _V (t), ξ _j
(t), ξ _{f, j} (t) is the risk premium adjustment rate, ρ
_jk (t), ρ ^ff _jk (t), ρ ^f _jk (t), ρ
_jV (t) and ρ ^f _jV (t) are correlation coefficients, each of which is based on t and these random variables and the default state (whether or not defaulting) of all companies). Letting τ _{j be} the default time point of the processing means B to be generated and the company j not defaulting in the risk horizon T, the risk neutral probability P of the company j at a certain future time point S, S> T for each sample path. ^ Or the probability of survival P ^ {τ _j under the forward neutral probability P ^s
> S} Probability of defaulting by time S based on According to the processing means C that determines the default / non-default state of the company j at time S for each sample path, and the cumulative cash flow X (t) of the product until time t,
The above random variables including r (s) (T ≦ s) and h _j (s) (T ≦ s), default / non-default state of company j, maturity date and interest payment date, coupon rate and recovery rate And the financial product attribute data including, the price p in the risk horizon T of the product.
^nondef (T) is a risk neutral evaluation method (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^), or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information up to time T under the forward neutral probability P ^s is known, and v ₀ (T, T _i ^j ) is the maturity T _i. (the price of the default-free discount bond of ^j at time T)), and the processing means D for calculating using the sample path generated by the processing means B and the processing means C. Horizon asset prices can be calculated.

The invention according to claim 7 is the risk of calculating the portfolio price and the individual asset price in the risk horizon for the sample path generated by the future situation scenario generating means according to claim 4. A further alternative of the horizon asset price calculation means, for financial instruments issued by company j that has already defaulted in risk horizon T, the price p ^def (T) at risk horizon T is Based on the default time τ _j of the financial instrument, the financial instrument attribute / rating data including the maturity date, the interest payment date, the coupon rate and the recovery rate, and the market data including the stock price of the company j and the price of the instrument up to the time τ _j. The processing means A to be obtained by the above, and the company j that is not defaulted in the risk horizon T
For financial instruments issued by (j = 0 is default free), domestic default free interest rate r (t), hazard rate h ^ _j (t) of domestic company j, and stock price S of domestic company j
_j (t), default free interest rate r of overseas partner country F
_f (t), hazard rate of company j in country F h ^ _{f, j} (t), F
Stock price S _{f, j} (t) of company j in the country and exchange rate between home country and country F of the country (home currency conversion value of one unit of country currency) V
The fluctuation of (t) after the risk horizon T under the risk neutral probability P ^ or the forward neutral probability P ^s ,
The stochastic differential equation Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And μ ₀ (t), μ _j (t), μ _{f, 0} (t) and μ
_{f, j} (t) are r (t), h _j (t), r _f (t),
And the drift of h _{f, j} (t), σ ₀ (t), σ _j (t), σ
_{s, j} (t), σ _{f, 0} (t), σ _{f, j} (t), σ _{fs, j}
(t) and σ _V (t) are r (t), h _j (t), and S, respectively.
_j (t), r _f (t), h _{f, j} (t), S _{f, j} (t) and V
volatility of (t), β ₀ (t), β _{n + j} (t), β
_{f, 0} (t), β _{f, nf + j} (t) and β _V (t) are z ₀ (t), z _{n + j} (t), z _{f, 0} (t) and z, respectively.
market price of risk of _{f, nf + j} (t), z _V (t), ξ _j
(t), ξ _{f, j} (t) is the risk premium adjustment rate, ρ
_jk (t), ρ ^ff _jk (t), ρ ^f _jk (t), ρ
_jV (t) and ρ ^f _jV (t) are correlation coefficients, both of which are generated based on t and these random variables and the default state (whether or not to default) of all companies. The cumulative cash flow X (t) of the product until time t is defined by the above random variables including r (s) (T ≦ s) and h ^ _j (s) (T ≦ s), and the default / non-default of company j. By expressing by using the default state and the financial instrument attribute data including the maturity date, the interest payment date, the coupon rate, and the recovery rate, the price p ^nondef (T) of the instrument at the risk horizon T is calculated as the risk neutral value. Evaluation method (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^), or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information up to time T under the forward neutral probability P ^s is known, and v ₀ (T, T _i ^j ) is the maturity T _i. The price of the default-free discount bond of ^j at time T)) is used to directly or analytically calculate the above-mentioned expected value by using the joint distribution of random variables obtained by analytical calculation from the above stochastic differential equation. Processing unit E for calculating the price p ^nondef (T).

The invention according to claim 8 is another mode of the future situation scenario generating means in the integrated evaluation system for market risk and credit risk in the present invention, which is domestic default / free interest rate r (t), domestic Hazard rate h _j (t) of company j, stock price S _j (t) of domestic company j, overseas partner country F
Default free interest rate r _f (t) of country, hazard rate h _{f, j} (t) of company j in F country, stock price S _{f, j} of company j in country F
(t) and the following stochastic differential equation that describes the future fluctuation of the exchange rate of the home country and the F country (country currency conversion value of the country's currency) V (t) And a ₀ , a _j , a _{f, 0} , a _{f, j} , σ ₀ ,
σ _j , σ _{f, 0} , σ _{f, j} , σ _{s, j} , σ _{fs, j} , σ
_V is a non-negative constant, μ _{s, j} , μ _{fs, j} , μ _V is a constant,
b ₀ (t), b _j (t), b _{f, 0} (t), b _{f, j} (t) are deterministic functions of time t, ρ _jk , ρ ^ff _jk , ρ ^f _jk ,
ρ _jV and ρ ^f _jV are constants, and a sample path from the current time t to the risk horizon T (T> t) is generated under the observation probability, and r (s) and h _j (s) S _j (s)
And r _f (s), h _{f , j} (s), S _{f, j} (s) and V
(S), t ≦ s ≦ T, the first processing means, and when the default time of the company j that has not defaulted at time t is τ _j , the company j in the risk horizon T is sampled for each sample path. Survival probability P under the observation probability of
_T {τ _j > T} Probability of defaulting by risk horizon T based on And a second processing unit that determines the default / non-default state of the company j in the risk horizon T for each sample path.

The invention according to claim 9 is the risk of calculating the portfolio price and the individual asset price in the risk horizon for the sample path generated by the future situation scenario generating means according to claim 8.・ It is one aspect of Horizon's asset price calculation method, and risk ・
For financial instruments issued by company j, which has already defaulted at horizon T, the price p ^def (T) at risk horizon T is calculated as the default time τ of company j.
_j , the financial product attribute / rating data including the maturity date, the interest payment date, the coupon rate, and the recovery rate, and the market data including the stock price of the company j and the price of the product up to the time τ _j . And the company j (j = 0) that is not defaulted in the risk horizon T
Is a default-free) financial instrument issued by a domestic default-free interest rate r (t), domestic company j
Hazard rate h ^ _j (t), stock price S _j (t) of domestic company j,
Default free interest rate r _f (t) of overseas partner country F,
Hazard rate h ^ _{f, j} (t) of company J of country F, stock price _{Sf, j} (t) of company j of country F, and exchange rate between home country and country F (country currency of one country The converted value) V (t) of the risk-neutral probability P ^ or the forward-neutral probability P ^s after the risk horizon T is calculated by the following stochastic differential equation. Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And a ₀ , a _j , a _{f, 0} , a _{f, j} , σ ₀ ,
σ _j , σ _{f, 0} , σ _{f, j} , σ _{s, j} , σ _{fs, j} , σ
_V is a non-negative constant, μ _{s, j} , μ _{fs, j} , μ _V is a constant,
b ₀ (t), b _j (t), b _{f, 0} (t), b _{f, j} (t) are deterministic functions of time t, β ₀ (t), β _{n + j} (t), β
_{f, 0} (t), β _{f, nf + j} (t) and β _V (t) are z ₀ (t), z _{n + j} (t), z _{f, 0} (t) and z, respectively.
_{f, nf + j} (t), z _V (t) is the market price of risk and is a deterministic function of time t, and ξ _j (t) and ξ _{f, j} (t) are risk premium adjustment rates and are deterministic of time t Functions, ρ _jk , ρ ^ff
_jk , ρ ^f _jk , ρ _jV , ρ ^f _jV are correlation coefficients and are constants from -1 to 1, and the cumulative cash flow X (t) of the financial instrument until time t
, The above random variables including r (s) (T ≦ s) and h ^ _j (s) (T ≦ s), the default / non-default state of the company j, the maturity date and the interest payment date and the coupon rate, and By using the financial product attribute data including the recovery rate and the expression, the price p in the risk horizon T of the product
^nondef (T) is a risk neutral evaluation method (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^) or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information up to time T under the forward neutral probability P ^s is known, and v ₀ (T, T _i ^j ) is the maturity T. second processing means for calculating based on the price of the default-free discount bond of _i ^j at time T).

The invention according to claim 10 is the risk of calculating the portfolio price and the individual asset price in the risk horizon for the sample path generated by the future situation scenario generating means according to claim 8.・
Another method of calculating the asset price of the horizon, for a financial instrument issued by the company j that has already defaulted in the risk horizon T, the price p ^def (T) in the risk horizon T is set as the default of the company j. Calculated based on time τ _j , financial product attribute / rating data including maturity date, interest payment date, coupon rate and recovery rate, and market data including the stock price of company j and the price of the product up to time τ _j Company j (j = 0) that is not defaulted in processing means A and risk horizon T
Is a default-free) financial instrument issued by a domestic default-free interest rate r (t), domestic company j
Hazard rate h ^ _j (t), stock price S _j (t) of domestic company j,
Default free interest rate r _f (t) of overseas partner country F,
Hazard rate h ^ _{f, j} (t) of company J of country F, stock price _{Sf, j} (t) of company j of country F, and exchange rate between home country and country F (country currency of one country The converted value) V (t) of the risk-neutral probability P ^ or the forward-neutral probability P ^s after the risk horizon T is calculated by the following stochastic differential equation. Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And a ₀ , a _j , a _{f, 0} , a _{f, j} , σ ₀ ,
σ _j , σ _{f, 0} , σ _{f, j} , σ _{s, j} , σ _{fs, j} , σ
_V is a non-negative constant, μ _{s, j} , μ _{fs, j} , μ _V is a constant,
b ₀ (t), b _j (t), b _{f, 0} (t), b _{f, j} (t) are deterministic functions of time t, β ₀ (t), β _{n + j} (t), β
_{f, 0} (t), β _{f, nf + j} (t) and β _V (t) are z ₀ (t), z _{n + j} (t), z _{f, 0} (t) and z, respectively.
_{f, nf + j} (t), z _V (t) is the market price of risk and is a deterministic function of time t, and ξ _j (t) and ξ _{f, j} (t) are risk premium adjustment rates and are deterministic of time t Functions, ρ _jk , ρ ^ff
_jk , ρ ^f _jk , ρ _jV , and ρ ^f _jV are correlation coefficients and are constants from -1 to 1, and a processing means B for generating a large number of sample paths and a company not defaulting in the risk horizon T If τ _j is the default time point of _j , then some future time point S, S> T,
The risk-neutral probability P ^ or the survival probability P ^ {τ _j > S} under the forward-neutral probability P ^s of the company j in Probability of defaulting by time S based on According to the processing means C that determines the default / non-default state of the company j at time S for each sample path, and the cumulative cash flow X (t) of the product until time t,
The above random variables including r (s) (T ≦ s) and h _j (s) (T ≦ s), default / non-default state of company j, maturity date and interest payment date, coupon rate and recovery rate And the financial product attribute data including, the price p in the risk horizon T of the product.
^nondef (T) is a risk neutral evaluation method (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^), or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information up to time T under the forward neutral probability P ^s is known, and v ₀ (T, T _i ^j ) is the maturity T _i. processing unit D for calculating using the sample path generated by the processing unit B and the processing unit C based on the price of the default-free discount bond of ^j at time T).

The invention according to claim 11 is a risk of calculating the portfolio price and the individual asset price in the risk horizon for the sample path generated by the future situation scenario generating means according to claim 8.・
With respect to a financial instrument issued by the company j that has already defaulted in the risk horizon T, which is another aspect of the horizon asset price calculation means, the price p ^def (T) in the risk horizon T is Based on default time τ _j , financial product attribute / rating data including maturity date, interest payment date, coupon rate and recovery rate, and market data including the stock price of company j and the price of the product up to time τ _j. Requested processing means A and companies not defaulting in risk horizon T
For financial instruments issued by (j = 0 is default free), domestic default free interest rate r (t), hazard rate h ^ _j (t) of domestic company j, and stock price S of domestic company j
_j (t), default free interest rate r of overseas partner country F
_f (t), hazard rate of company j in country F h ^ _{f, j} (t), F
Stock price S _{f, j} (t) of company j in the country and exchange rate between home country and country F of the country (home currency conversion value of one unit of country currency) V
The fluctuation of (t) after the risk horizon T under the risk neutral probability P ^ or the forward neutral probability P ^s ,
The stochastic differential equation Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And a ₀ , a _j , a _{f, 0} , a _{f, j} , σ ₀ ,
σ _j , σ _{f, 0} , σ _{f, j} , σ _{s, j} , σ _{fs, j} , σ
_V is a non-negative constant, μ _{s, j} , μ _{fs, j} , μ _V is a constant,
b ₀ (t), b _j (t), b _{f, 0} (t), b _{f, j} (t) are deterministic functions of time t, β ₀ (t), β _{n + j} (t), β
_{f, 0} (t), β _{f, nf + j} (t) and β _V (t) are z ₀ (t), z _{n + j} (t), z _{f, 0} (t) and z, respectively.
_{f, nf + j} (t), z _V (t) is the market price of risk and is a deterministic function of time t, and ξ _j (t) and ξ _{f, j} (t) are risk premium adjustment rates and are deterministic of time t Functions, ρ _jk , ρ ^ff
_It is _assumed that _jk , ρ ^f _jk , ρ _jV , and ρ ^f _jV are correlation coefficients and are constants from −1 to 1, and the cumulative cash flow X (t) of the product until time t is r
The above random variables including (s) (T ≦ s) and h _j (s) (T ≦ s), the default / non-default state of the company j, the maturity date and the interest payment date, the coupon rate and the recovery rate. Financial product attribute data including
^nondef (T) is a risk neutral evaluation method (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^), or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information up to time T under the forward neutral probability P ^s is known, and v ₀ (T, T _i ^j ) is the maturity T _i. The price of the default-free discount bond of ^j at time T)) is used to directly or analytically calculate the above-mentioned expected value by using the joint distribution of random variables obtained by analytical calculation from the above stochastic differential equation. Processing unit E for calculating the price p ^nondef (T).

The integrated valuation method for market risk and credit risk according to the present invention is based on a default-free interest rate process model expressed by an equation that describes future changes in the default-free interest rate, and the interest rate from the present to the risk horizon. Based on a scenario depicting changes in fluctuations over time and a default process model expressed by an equation that describes future fluctuations in default occurrence rates, we have drawn changes over time in default occurrences from present to risk horizon. Based on the scenario and the stock price fluctuation process model expressed by the equation that describes the future fluctuation of stock price fluctuation, the scenario that depicts the temporal change of stock price fluctuation from the present to the risk horizon and the future fluctuation of the exchange rate Based on the exchange rate fluctuation model expressed by the described equation, risk horizon And a scenario that depicts the change over time in the exchange rate fluctuations based on at least one of the input or stored portfolio data, financial instrument attributes / rating data, market data, and default data. Step and the first
For each scenario generated by the steps of
The second step of introducing the risk premium adjustment rate to calculate the portfolio price and individual asset price in the risk horizon, and the risk horizon portfolio price and the risk horizon value calculated for each scenario in the second step A third step of obtaining a future price distribution of the portfolio and / or a future price distribution of the individual asset based on the individual asset price, and the future portfolio price distribution calculated by the third step and / Or based on the future price distribution of individual assets,
It is characterized by integrating and evaluating market risk and credit risk.

[0019]

BEST MODE FOR CARRYING OUT THE INVENTION The basic idea of the integrated valuation system and integrated valuation method for portfolio market risk and credit risk according to the present invention is that the risk of the portfolio currently held is given at a given future time (risk・ The portfolio price in the horizon is regarded as uncertainty, and the portfolio price distribution in the risk horizon is calculated. Based on this, a scale for risk evaluation and return evaluation is calculated, and quantitative portfolio management is performed. It is the one to try.

FIG. 1 is a diagram showing a framework of an integrated evaluation system for portfolio market risk and credit risk according to the present invention, and FIG. 2 is a diagram for implementing an integrated evaluation system for portfolio market risk and credit risk. It is a figure showing an example of hardware constitutions. First, FIG.
The hardware configuration will be described using.

As shown in FIG. 2, the integrated evaluation system of market risk and credit risk of a portfolio executes an arithmetic operation with a control device for controlling the entire system such as fetching a command to decode a program and issuing a command to execute the program. A CPU 11 including an arithmetic unit for storing data, and a ROM storing a program for starting the device when the power is turned on.
12, a RAM 13 that temporarily stores data and programs that function as a main storage device, an input device 15 such as a keyboard and a mouse that inputs data, and a display or printer that displays or outputs data. It has an output device 16, an interface circuit 14 for controlling input / output of data, and an external storage device 17 for storing data and programs.

The external storage device 17 has a calculation program file 21, a portfolio data file 22, a financial product attribute / rating data file 23, and market data for executing the integrated market risk and credit risk evaluation system of the present invention. File 24,
A default data file 25 is included.

The calculation program file 21 includes a future situation scenario generating means, a risk horizon asset price calculating means, a future asset price distribution calculating means, and a risk in the integrated market risk and credit risk evaluation system of the present invention. It includes programs such as a future situation scenario generation tool 32, an asset value evaluation tool 33, a return evaluation tool 51, and a risk evaluation tool 52 for realizing index calculation means and return index calculation means.

The portfolio data file 22 contains
Data related to the portfolio to be evaluated, for example, categories of financial product attributes that compose the portfolio (category such as bonds, loans, stock prices, swaps, forward exchange contracts),
Data such as the quantity, market value, and transfer of each financial product is stored.

Financial product attribute / rating data file 23
Includes detailed data about each financial product, such as corporate code for bonds, bond type, issuer, interest payment date, redemption date, number of interest payments, issue date, currency, country code, coupon rate, etc. Company code, currency, interest rate, interest payment date, number of payments, maturity date, collection rate, etc. Company code, currency, country code, etc. for stocks, company code, currency, coupon rate, interest payment date, number of payments for swaps , Maturity date, swap type, etc. for forward exchange contracts, the company code, delivery date, delivery currency, delivery amount, receipt currency, receipt amount, etc. are stored, and internal rating data for each financial product is stored. There is.

In the market data file 24, public bond price time series (domestic / overseas), public bond attribute (domestic / overseas), risk-free interest rate time series, exchange rate time series, stock price time series, stock index time series, etc. The data of is stored.

The default data file 25 stores yearly data of the number of domestic bankruptcies in the past and overseas default rates.

Although FIG. 2 shows a configuration for implementing the integrated market risk / credit risk evaluation system for a portfolio according to the present invention by a single computer, for example, data stored in the external storage device 17 is shown. Etc. are stored in another database server and the database server is connected to the computer operated by the user and the communication line, the necessary data is downloaded at the user's request and the calculation is performed on the user's computer. Alternatively, the present invention may be executed using a client-server system, such that all arithmetic processing is performed on the server side connected by a communication line and the result is displayed on a computer operated by the user. It is possible. Further, the market data, portfolio data, default data, and financial product attribute / rating data are not stored as data files in the external storage device 17 in advance, but input device 15 is used.
May be input by some means (for example, a keyboard, a mouse, an optical reader), or may be downloaded from another data source via a communication line.

Next, the framework of the integrated evaluation system for market risk and credit risk of a portfolio according to the present embodiment will be described with reference to FIG. In the present embodiment, in order to perform an integrated evaluation of market risk and credit risk, the following three main functions, that is, (1) occurrence of a probabilistic scenario for expressing future uncertainty,
It has (2) evaluation of future value of portfolio and (3) evaluation of distribution of future value of portfolio. In order to realize these three functions, the system according to the present embodiment has the following four tools shown in FIG.
In other words, future situation (default free interest rate, hazard rate, stock price, exchange rate) scenario generation tool 32,
An asset value evaluation tool 33, a return evaluation tool 51, and a risk evaluation tool 52 are provided (the details of these tools will be described later in the section of Examples). The above tools are basic equations (stochastic differential equations)
The calculation or simulation is performed based on the default free interest rate, hazard rate, stock price, and exchange rate modeled as a stochastic process by 311.

Next, the outline of the procedure for the integrated evaluation of the market risk and credit risk of the portfolio according to the present embodiment will be described with reference to (1) to (3) in FIG. Based on the current data of portfolio 313 and basic equation 311 (default free interest rate process model, hazard rate process model, stock price process model, exchange rate model represented by stochastic differential equation), future situation scenario generation tool 32 ( (See Fig. 2) causes a number of scenarios that illustrate changes over time in current fluctuations in interest rates, default probabilities, stock prices, and exchange rates from risk to horizon and the occurrence of defaults. With the introduction of the risk premium adjustment rate 312 for each scenario generated in step 1, the asset value evaluation tool 33 (see FIG. 2) is used to calculate the price of each asset in the risk horizon. The sum of all asset prices is the portfolio price. Calculate the portfolio price distribution in step (1) and calculate the future portfolio price distribution. If the price is aggregated for each asset, the price distribution for each asset can be obtained. Based on the price distribution obtained in step 1, the risk assessment tool 52 (see FIG. 2) is used to calculate the standard deviation of the future price and / or future rate of return in the risk horizon of the portfolio and individual assets, VaR (Value-at- Risk), C
Risk indicators such as VaR (Conditional-VaR), lower partial product ratio, marginal standard deviation of individual assets, marginal VaR, marginal CVaR are calculated. At the same time, a return evaluation tool 51 (see Fig. 2)
According to, the return index such as expected return rate of portfolio and individual assets is calculated.

In the following, the use of two types of probability measures, which is one of the features in the framework of the integrated evaluation system for market risk and credit risk of portfolios according to the present embodiment, will be explained, and the default process will be further described. To explain the important hazard rate. Next, the basic equations that describe the entire system are introduced in order, the price evaluation method for financial instruments using them is shown, and finally the general calculation procedure for calculating future price distribution is shown. In the following, the probability measure actually observed is represented by P,
Consider the probability space (Ω, F, P). F = {F _t ; t ≧
0} is the filtration generated from the stochastic structure given to the model. Also, the risk-neutral probability measure P
Suppose there is only one ^. [Observation Probability and Risk Neutral Probability] One of the features of the framework of the integrated evaluation system for market risk and credit risk of portfolios according to the present embodiment is that the observation probability and the pseudo-probability for price evaluation are clearly distinguished. This is the point to use.

Consider the price of an asset, where t is the current time. Generally, the cash flows that the owner of the asset receives in the future are considered to be random variables. According to the arbitrage theory used for valuing derivatives, the price of the asset at time t is the conditional expected value of the discounted present value of future cash flows under the risk-neutral probability (“information up to time t”). Is the expected value when is known). In other words, the price reflects the distribution of random variables such as future cash flows and interest rates used for discounts. However, what is reflected there is the distribution under the risk-neutral probability, not the distribution in the real world. This also applies to prices at future time points T, T> t, and the conditional expected value under risk-neutral probability (expected value when "information up to time T" is known) is time T. It becomes the price in. By the way, since the condition attached to the expected value, “information up to time T”, is information that is actually observed by time T, it is considered that they naturally occur according to the probability (observation probability) in the real world. To be Considering in this way, after all, in order to obtain the price distribution at the future time T at the time t, the following (1)
It turns out that the procedure of (3) must be taken. (1) "Information up to time T" is generated based on the information up to the current time t and the observation probability. (2) Assuming that “information up to time T” is known, time T
The conditional expected value of the discounted present value of the subsequent cash flows is obtained under the risk-neutral probability (that value is a sample of future value). (3) The above procedure is repeated a sufficient number of times to statistically process a large number of obtained samples (future value).

However, the conventional market risk evaluation model
Then, these two probabilities must be used distinctly
There was no One of the reasons is risk horizon or
The difference between the two probabilities is
Small, no problem if one thinks it is an approximation of the other
This is because the. However, in the credit risk assessment,
The situation has changed because the horizon has become very long, yearly.
To do. For example, from the prices of government bonds and corporate bonds,
Estimate the default probability for each rating (implied default
(Fault probability is calculated) and the observation value is set with a low rating.
It is known that the probability of default exceeding
It This implied default probability is the risk
It is a neutral probability, and the difference between this and the observation probability can never be ignored.
I can't come. For this reason, the frame according to the present embodiment is
In the framework, these two probabilities are clearly distinguished and used.
The basic policy is to do. [Hazard rate] Next, to describe the default process
As a preparation, the hazard rate will be described. Current time is t
= 0, the time defaulted by company j under the observation probability P
τ_jThen, τ_jIs a positive random variable. τ_j> T and
Under these conditions, the hazard rate h _jDetermine (t) by the following formula 1.
Mean

[0034]

[Equation 1] Here, P _t is the conditional probability of P given the information F _t . The hazard rate is the conditional strength that firms that do not default until a certain time t will default at the next moment (t, t + dt). Therefore, it cannot exceed 1, but the hazard rate h _j (t) defined by the equation 1 in the continuous time model may be greater than 1 because it is strong.

When h _j (t) is a deterministic function of t, the probability that company j is alive from time 1 to time T, T> t is given by the following formula 2.

[0036]

[Equation 2] To write Default Company j, tau _j may be the stochastic behavior clearly in, but since the number 2 holds between the tau _j and h _j (t), the period of h _j (t) The structure may be handled.

When h _j (t) is a random variable, h
_{When j} (s), t ≦ s ≦ T is specifically given, that is, when F _t is given, the following formula 3 is obtained.

[0038]

[Equation 3] H _j (t, T) is a cumulative hazard rate. The survival probability is τ
_{It is} given by the following equation 4 under the condition that _j > t.

[0039]

[Equation 4] Here, E _t [·] is a conditional expected value operator when F _t is given. [Fundamental equation] The basis of this model is a stochastic differential equation that describes future changes in default free interest rates, hazard rates, stock prices, and exchange rates (reference numeral 3 in Fig. 1).
11). Here, for each of them, the observation probability P
Based on the equations below, we describe the method of constructing the equations under pseudo-probabilities such as risk-neutral probability and forward-neutral probability.

[1. Interest Rate Process and Hazard Process] First, the stochastic differential equation that follows the interest rate and the hazard rate will be described.
Assume n companies. The default free spot rate at time t is represented by r (t) = h ₀ (t), the hazard rate of company j is represented by h _j (t), j = 1, ..., N, and h
(t) = (h ₀ (t), h ₁ (t), ..., H _n (t)). Under the observation probability P, it is assumed that these follow Equations 5 and 6 below.

[0041]

## EQU5 ## dr (t) = μ ₀ (r (t), t) dt + σ ₀ (r
(t), t) dz ₀ (t), t ≧ 0

[0042]

## EQU6 ## dh _j (t) = μ _j (h (t), t) dt + σ _j (h
(t), t) dz _j (t), t ≧ 0, j = 1, ..., n where (z ₀ (t), z ₁ (t), ..., z _n (t)) Is
In the (n + 1) -dimensional standard Brownian motion under P, the following expression 7 is established.

[0043]

Dz _j (t) dz _k (t) = ρ _jk (t) dt,
j, k = 0, 1, · · ·, although the n hazard ratio mentioned above that the term structure is observed, the E obtained from the number _{6 [h j (t)]} , well in fact their observations To match the parameters μ _j , σ in equation 6
By setting _j , the results of the empirical analysis can be reflected. Specific examples will be described later.

Next, a sufficiently large positive number T^*For z₀
Market price β of risk related to (t) ₀introduce (t)
And z derived from the following equation 8₀^ (T) is equivalent to P
Standard Brownian motion under the risk-neutral probability measure P ^
It

[0045]

[Equation 8] Also, r (t) follows the following equation 9 under P ^.

[0046]

## EQU9 ## dr (t) = μ₀^ (R (t), t) dt + σ
₀(r (t), t) dz₀^ (T), 0 ≦ t ≦ T^* However, μ₀^ (R (t), t) = μ₀(r (t), t) -β₀(t)
σ₀(r (t), t) Next is the hazard rate. References Artzner, P. and F. Delbaen
 (1995), “Default risk insurance and incomplete m
arkets, ”by Mathematical Finance, 5 187-195
And stop time τ_jIs a probability process h under the probability measure P
_j(t), even under the probability measure P ^ that is equivalent to P
τ_jStrength process h_j^ (T) exists. So P ^
Hazard rate h under_jThe following number for ^ (t)
Risk premium adjustment rate that satisfies 10_jIntroduce (t)
And assume that this is a deterministic function of time t.

[0047]

H _j ^ (t) = h _j (t) + ξ _j (t), j
= 1, ..., n By the above-mentioned Equation 6, Equation 7, Equation 9, and Equation 10, h ^ (t) = (r (t), h ₁ ^ under the risk neutral probability P ^
The behavior of (t), ..., H _n ^ (t)) is described.

[2. Stock Price Process] References Jarrow, RA and SM Turnbull (1996), “Prici
ng derivatives on financial securities subject to
The stock price process is added to the default free interest rate process and the hazard process described above with reference to credit risk, ”Journal of Finance, 50, 53-86.

It is assumed that the stock price S _j (t) of the company j at time t complies with the following expression 11.

[0050]

[Equation 11] For the sake of simplicity, it is assumed that the stock price becomes zero at the moment the default occurs in Equation 11, and if S _j (t) = 0, then S
_{Let j} (s) = 0 and s ≧ t. In this case, under certain conditions, the market price β of the risk for z _{n + j} (t)
_{n + j} (t) exists, Is the standard Brownian motion under the risk-neutral probability measure P ^ that is equivalent to P, and S _j (t) is the following equation 1 under P ^.
Follow 2.

[0051]

[Equation 12] [3. Exchange Process] References Amin, KI and RA Jarrow (1991), “Pricing f
foreign currency options under stochastic interest
rates ”, Journal of International Money and Financ
e, 10, 310-329 (hereinafter abbreviated as "Amin and Jarrow")
Refer to for further exchange process. Here, it is assumed that the default free interest rate process, the hazard process, and the stock price process are established in the other country as well as in Japan.

Exchange rate (passing time of one unit of currency of partner country)
(Converted value) is V (t), the default free
T rate_f(t) = h_{f, 0}(t), number of companies in the partner country
N _f, Company j, j = 1, ..., n_f, Default
Time τ_{f, j}, Its hazard rate Then, under the observation probability P,
It is assumed that the following equation 16 is followed.

[0053]

## EQU13 ## dr _f (t) = μ _{f, 0} (r _f (t), t) dt + σ
_{f, 0} (r _f (t), t) dz _{f, 0} (t), t ≧ 0

[0054]

Dh _{f, j} (t) = μ _{f, j} (h _f (t), t) dt
+ Σ _{f, j} (h _f (t), t) dz _{f, j} (t), t ≧ 0, j
= 1, ..., n _f

[0055]

[Equation 15]

[0056]

[Equation 16] Is a (2n + 2n _f +3) -dimensional standard Brownian motion under P, and the following expressions 17 to 21 are established.

[0057]

[Equation 17]

[0058]

[Equation 18]

[0059]

[Formula 19]

[0060]

[Equation 20]

[0061]

[Equation 21] Here, dz _V (t) dz _V (t) = dt.

Next, consider the measure conversion. First, regarding the hazard rates h _{f, j} (t) and h _{f, j} ^ (t), it is assumed that the same equation as in the equation 10 holds. Details are omitted here, but if the same calculation as in the literature Amin and Jarrow is performed here, under certain conditions, as in the stock price process, And the market price of the risk with respect to z _V (t) And β _V (t) exist, and under the risk-neutral probability measure P ^, r _f (t), h _{f, j} ^ (t), S _{f, j} (t), V
It is derived that (t) follows the following formulas 22, 23, 24, and 25.

[0063]

[Equation 22] dr_f(t) = μ_{f, 0}^ (R_f(t), t) dt +
σ_{f, 0}(r_f(t), t) dz_{f, 0}^ (T), 0≤t≤T^* However, μ_{f, 0}^ (R_f(t), t) dt = μ_{f, 0}(r_f(t), t)-
β_{f, 0}(t) σ_{f, 0}(r_f(t), t)

[0064]

(23) h _{f, j} ^ (t) = h _{f, j} (t) + ξ
_{f, j} (t), j = 1, ..., N _f

[0065]

[Equation 24]

[0066]

[Equation 25] Is the (2n + 2n _f +3) -dimensional standard Brownian motion under P ^, and the correlation structure is the same as z (t).

The basic equations in this framework have been completed. In summary, the equations that the random variable follows under the observation probability P are the equations 5, 6, 11 and 13 to 16, and the equations under the risk neutral probability P ^ are the equations 9 and 10. , Number 12, and number 22 to number 25. z (t)
And the correlation structure of z ^ (t) are given by Equations 17 to 21.

All of these equations have a high degree of freedom, and it is possible to obtain models of different properties simply by changing the setting of concrete equations. Note that the numbers 12, 24,
Equation 25 is an equation that is only needed to price stock / forex derivatives at a future time. For portfolios that include domestic and foreign equities but not equities / exchange derivatives, the future value is
5, calculated from equation 16 and equations 12, 24 and 25 are not used. [Conditional independence] The hazard process given by Eq.
Substituting in, we obtain the distribution function of τ _j . But n
When considering the defaults of companies, only the marginal distribution of each τ _j can be obtained from the equation 4, and (τ ₁ , τ ₂ ,
,, τ _n ) simultaneous distribution cannot be obtained. Therefore, in order to solve this problem, we introduce the assumption of conditional independence.

Simultaneous distribution of default times at time t
The function P_t{τ₁> T₁, ・ ・ ・, Τ _j> T_n}, T_j
If ≧ t, then conditional independence means T ≧ max_jt_j
Information up to F_TGiven, the default time τ_j
Are independent of each other, that is,
To stand up.

[0070]

[Equation 26] It has to be noted that the default time τ _j cannot be specified given the information F _T. t =
At 0, taking the expected values of both sides of Expression 26, the following simultaneous distribution function of Expression 27 is obtained.

[0071]

[Equation 27] Equation 2 used Equation 3 in the second equation.

In the case of conditional independence, ordinary independence Unlike, the effect of the correlation of h _j (t) can be considered. Regarding the concrete image of conditional independence,
See the simulation section below. [Assessment of present value] Here, after briefly touching on the general theory of price valuation of financial instruments, the price valuation of discount bonds with default risk will be described as a typical example of price valuation considering credit risk. The basic idea of price valuation of discount bonds is Jarrow, RA and SM Turnbull (1995),
“Pricing derivatives on financial securities sub
jectto credit risk ”, Journal of Finance, 50, 53-8
Follow the framework of 6 (abbreviated as "Jarrow and Turnbull" below).

The arbitrage-free price of a financial instrument is obtained by using the risk-neutral valuation method or the forward-neutral valuation method used for price valuation of derivatives. The price p (t, T) at the time t, t <T, of the product that can receive the cash flow X at the maturity T is calculated by the following formula 28.

[0074]

[Equation 28] Where E_t^ [・] Is a condition under the risk-neutral probability measure
Expected value operator, E _t ^T[・] Is forward neutral probability measurement
Conditional expectation operator under degrees, v₀(t, T) is differential
At the price of the discounted bond with maturity T at the time t
is there. For example, European call ops for stock j
In case of X = (S_j(T) -K, 0)⁺ As a result, Equation 28 may be calculated. Where S_j(T) is
Stock price at maturity T of option, K is exercise price,
(A, B)⁺= Max (A, B).

As an example of price evaluation, default risk
I would like to elaborate on some of the discount bonds. Issued by company j
Consider the i-th discount bond. For simplicity, that bond
The holder of the_i ^jPreviously issued default
If not alive_i ^j1 yen per day, default will occur
If maturity T_i ^jTo δ_jSuppose you only receive yen. Furthermore
And the recovery rate δ_j(0 ≦ δ_j<1) is constant. This assumption
Is the time τ for company j_j, Τ_j<T_i ^jDefaults to
If that time τ_jTo δ_jv₀(τ_j, T_i ^j) Only received
The total amount of the profits is the default-free interest rate r (s), τ
_j≤s≤T_i ^j, And τ_jTo T_i ^jOperating up to
The same and consistent with the Jarrow and Turnbull settings
It With this assumption, the default
The cash flow to take is τ_jAvoid relying directly on
Can greatly simplify the problem.

The cash flow of this discount bond at maturity T _i ^j is According to the risk-neutral valuation method, the price v _j (t, T _i ^j ) of the discount bond at time t, t <T _i ^j is
Under the condition that τ _j > t, the following equation 29 can be written.

[0077]

[Equation 29] Is. If r (t) and h _j (t) are independent, then number 29 is Ja
Formula obtained by rrow and Trunbull Matches Here, P _t ^ {τ _j > T _i ^j } is a company j
Is the probability of survival under the risk-neutral probability measure of P ^. r
When (t) and h _j (t) are not independent, the random variables H ₀ (t, T _i ^j ) and H _j ^ (t, T _i ^j ) are the expected values of Equation 29.
We must also consider the correlation of.

By the way, the expected value of the equation 29 has the same form as the equation 27, which is the simultaneous distribution function of the default time. Therefore, the default free interest rate h ₀ (t) = r
If we define the default time τ ₀ such that (t) is the hazard rate, the expected value of Equation 29 is (τ ₀ ,
τ _j ) can be associated with the joint distribution function.

If the forward neutral valuation method is used, the price formula of the discount bond can be expressed in a rather simple manner. According to the forward-neutral valuation method (a method of performing price valuation of risk assets using a forward-neutral probability measure (a probability measure such that the forward price of risk assets becomes martingale) instead of the risk-neutral probability measure P ^), The above-mentioned discount bond price can be written as the following Expression 30 under the condition that τ _j > t.

[0080]

[Equation 30] P ^T {τ _j > T} is the survival probability under the forward neutral probability measure P ^T , which is equivalent to the risk neutral probability measure P ^. Equation 30 is simpler than Equation 29 because it is not necessary to evaluate the correlation between interest rates and hazard rates.

Similar to the discussion under the risk-neutral probability measure P ^, the risk premier adjustment rate ξ _j satisfying the following equation 31 for the hazard rate h _j ^T (t) under the forward-neutral probability measure P ^T _: Suppose there is ^T (t).

[0082]

[Equation 31] Further, ξ _j ^T (t) is a deterministic function of time t and does not depend on T, and this is represented by ξ _j (t). Then,
Under the condition of τ _j > t, the following equation 32 can be written.

[0083]

[Equation 32] At time t, v ₀ (t, s), v _j (t, s), P
_{Since t} {τ _j > s} and s ≧ t are observable quantities, the risk premium adjustment rate ξ _j (t) can be specifically obtained from the following Expression 33.

[0084]

[Expression 33] Incidentally, z ₀ (s) risks of market price β ₀ (s) is v ₀ (t, s) is deduced from the term structure, for it, specifically described in the section of Examples described later.

If ξ _j ^T (s) = ξ _j (s) is obtained,
Hazard rate h _j ^T (s) under P ^T from Eqs. 6 and 31
Therefore, the prices of other assets with credit risk can also be obtained by the forward neutral valuation method. In general,
It is easier to perform price valuation using the forward neutral valuation method than the risk neutral valuation method. However, since the forward neutral probability measure P ^T depends on T, some caution is required.

As shown in the equation 33, ξ _j (s) is the time t
It is calculated so as to be consistent with the market price of financial instruments observed in. Since the market price also includes market evaluations for risks other than interest rate risk and credit risk, ξ _j (s) obtained in this way also incorporates the effects of other risks on prices. . This ξ _j (s)
Is also used to assess future value, which means that the impact of other risks is assessed in the future, although not explicitly. The method of using the risk premium adjustment rate ξ _j (s) to achieve consistency with the market price is an effective means for integrating with other risks, especially those that affect the market price. right. [Distribution of Future Value of Portfolio] As a specific example, consider the distribution of future value of a portfolio composed of discount bonds and stocks. Let the risk horizon be T and T> t. From Equations 30 and 32, the price of the above-mentioned discount bond at time T is given by Equation 34 below.

[0087]

[Equation 34] v_j(T, T_i ^j) Is the random variable v₀(T, T_i ^j), P
_T{τ_j> T_i ^j}, Τ _jIs a random variable that depends on.
Also, the price of the stock issued by company j is obtained from
It If you write this in a very formal way,
It becomes 5.

[0088]

[Equation 35] From the expressions 34 and 35, the future value of this portfolio is given by the following expression 36.

[0089]

[Equation 36] Here, w _s ^j is the holding amount of the stock of the company j, w _i ^j is the holding amount of the i-th discount bond issued by the company j, N _j is the number of discount bonds issued by the company j, and j = 0 represents a default-free company (typically a country or government). [Overview of Model and General Calculation Procedure] From basic equations, random variables v ₀ (T, T _i ^j ), P _T {τ _j > T _i ^j }, τ
_If the future joint distribution of _j is obtained, the future value distribution of the portfolio can be obtained from the formula 36 in principle. However, since the definition function 1 _A is included in Equation 36, even if v ₀ (T, T _i ^j ) and P _T {τ _j > T _i ^j } follow a distribution that is easy to handle, the future value distribution is It cannot be calculated analytically. Therefore, generally, Monte Carlo simulation is used to numerically obtain the distribution.

Specific procedure for obtaining future value distribution
Are shown below in (1) to (9). (1) Number 5, Number 6, Number 11, Number 13 to Number 16, Number 17 to
Based on Equation 21, sample path under observation probability P Generate. However, T is a risk horizon. (2) From the number 3, the domestic and overseas companies of each sample pass
P_T{τ_j> T} is calculated and by risk horizon T
Independently decide whether to default (this is a condition
It is a concrete meaning of independence. ). (3) r (T) generated in (1), r_f(T) is the initial value
Then, from Equations 9 and 22, the service under the risk-neutral probability P ^ is
Sample path r (s), r_f(s), T ≦ s ≦ max _{i, j}T
_i ^jGenerated by a sufficient number, and the discount rate v₀(T, s),
v_{f, 0}Find (T, s). (4) When company j has not defaulted to T
Is h generated in (1)_j(T) (or h_{f, j}(T))
As an initial value,
H_j(s), T ≦ s ≦ max_iT_i ^j(Or h
_{f, j}(s), T ≦ s ≦ max_iT_i ^{f, j}) Is enough
Cause ξ_j(s) (or ξ_{f, j}(s))
From number 32 (5) If the portfolio includes equity derivatives
The derivative company j defaults by T
If not, S generated in (1)_j(T) (S
_{f, j}(T)) as an initial value
Sample path S_j(s), T ≦ s ≦T ^j(S_{f, j}(s),
T ≦T ^{f, j}) Is generated in sufficient number. However,T
^j(T ^{f, j}) Is the stock deed for company j (foreign company j)
The longest maturity of the revertive. (6) When the portfolio includes foreign exchange derivatives
If V (T) generated in (1) is used as the initial value,
From sample path V (s), T ≦ s ≦T ^VJust enough
generate. However,T ^VIs the longest maturity of foreign exchange derivatives
It is a period. (7) In the case of a portfolio consisting of discount bonds and stocks
From the equations 34 to 36, v_j(T, T_i ^j), S
_jCalculate (T) and π (T). Other gold in portfolio
If interest / stock / exchange derivatives are included,
Those prices are the sample price generated in (3) to (6).
The value is calculated as appropriate. (8) If the number of scenarios is insufficient, return to (1)
It If the sufficient number of scenarios is reached, proceed to (9). (9) Aggregate the obtained scenarios to obtain individual assets and ports
Find the folio price distribution.

Since there is a correlation between each random variable,
Actually, the sample paths of (3) to (6) are generated at the same time, and in (7), price evaluation is performed using them. This sample path must occur under the risk-neutral probability.

As described above, reference numeral 3 in FIG.
Since there is a considerable degree of freedom in setting the default free interest rate, hazard rate, stock price, and exchange rate probabilistic model shown in 11, various models can be obtained by devising this part.

By the way, as shown in the above procedure, in order to obtain the distribution of future value under a general setting, the Monte Carlo simulation must be used with a double nested structure. The first stage simulations are (1) to (2), and the second stage simulations are (3) to (6). If the number of scenarios required at each stage is very large, it takes much time to calculate, which is unrealistic. If one scenario takes 1 hour, the calculation will take 10,000 hours or more, which is well over a year). In order to avoid this problem, modeling that extends the embodiment of the Gaussian model proposed in Japanese Patent Application No. 2001-026147 will be described next. [Gaussian Model] An embodiment using the Gaussian model will be described below. The Gaussian model saves a great deal of computation time because the prices of various assets are obtained in a closed form.

[1. Basic Equation in Gaussian Model] The current time is t = 0. In this model, the basic equation that the domestic default free interest rate r (t) under the observation probability P follows is assumed to be the following Expression 37.

[0095]

(37) dr (t) = (b ₀ (t) −a ₀ r (t)) dt + σ
₀ dz ₀ (t), t ≧ 0 Similarly, it is assumed that the basic equation followed by the hazard rate h _j (t) of the domestic company j under the observation probability P is the following Expression 38.

[0096]

(38) dh _j (t) = (b _j (t) −a _j h _j (t)) dt
+ Σ _j dz _j (t), t ≧ 0; j = 1, ..., N Similarly, the stock price S _j of the domestic company j under the observation probability P
The basic equation that (t) follows is assumed to be the following Expression 39.

[0097]

[Formula 39] Similarly, the basic equation that the default free interest rate r _f (t) of the overseas partner country under the observation probability P follows is assumed to be the following formula 40.

[0098]

D r _f (t) = (b _{f, 0} (t) −a _{f, 0} r
_f (t)) dt + σ _{f, 0} dz _{f, 0} (t), t ≧ 0 Similarly, overseas partner company j under observation probability P
The following equation 41 is assumed to be the basic equation that the hazard rate h _{f, j} (t) of follows.

[0099]

(41) dh_{f, j}(t) = (b_{f, j}(t) -a_{f, j}h
_{f, j}(t)) dt + σ_{f, j}dz_{f, j}(t), t ≧ 0; j
= 1, ..., n_f Similarly, overseas j companies under the observation probability P
Stock price S_{f, j}The basic equation that (t) follows is
I assume.

[0100]

[Equation 42] Similarly, the exchange rate between the home country and the overseas partner country under the observation probability P (home currency conversion value of one unit of the partner currency) V
The basic equation that (t) follows is assumed to be the following Equation 43.

[0101]

[Equation 43] In the above formulas 37 to 43, a ₀ , a _j ,
a _{f, 0} , a _{f, j} , σ ₀ , σ _j , σ _{s, j} , σ _{f, 0} , σ
_{f, j} , σ _{fs, j} , σ _V are non-negative constants, μ _{s, j} , μ _{f
s, j} and μ _V are constants, b ₀ (t), b _j (t), b
_{f, 0} (t) and b _{f, j} (t) are deterministic functions at time t, and the correlation coefficients are all constants.

Where z₀(t), z_{f, 0}Regarding (t)
Market price of risk β₀(t), β_{f, 0}(t) and h
_j(t), h_{f, k}(t), j = 1, ..., N, k = 1,
..., n _fRisk premium adjustment rate for, ξ
_j(t), ξ_{f, k}(t) are deterministic at time t
Suppose it is a function. Then, the risk-neutral probability P ^
Below, the domestic default free interest rate r (t) is
Follow 44.

[0103]

## EQU44 ## dr (t) = (φ ₀ (t) −a ₀ r (t)) dt + σ
₀ dz ₀ ^ (t), t ≧ 0 Similarly, under the risk-neutral probability P ^, the hazard rate h _j (t) of the domestic company j follows Formula 45 below.

[0104]

H _j ^ (t) = h _j (t) + ξ _j (t), j =
1, ..., N Similarly, under the risk-neutral probability P ^, the stock price S _j (t) of the domestic company j follows the following formula 46.

[0105]

[Equation 46] Similarly, under the risk-neutral probability P ^, the default free interest rate r _f (t) of the overseas partner country follows Equation 47 below.

[0106]

D r _f (t) = (φ _{f, 0} (t) −a _{f, 0} r
_f (t)) dt + σ _{f, 0} dz _{f, 0} ^ (t), t ≧ 0 Similarly, under the risk-neutral probability P ^, the hazard rate h _{f, j} (t) of the overseas company j is According to number 48 of.

[0107]

[Formula 48] h_{f, j}^ (T) = h_{f, j}(t) ＋ ξ_{f, j}(t),
    j = 1, ..., n_f Similarly, shares of overseas company j under the risk-neutral probability P ^
Value S_{f, j}(t) follows the following formula 49.

[0108]

[Equation 49] Similarly, under the risk-neutral probability P ^, the exchange rates V (t) of the home country and the partner country follow the following expression 50.

[0109]

[Equation 50] The default-free interest rate process used here is the document Jul.
l, J. and A. White, (1990), “Pricing interest-rat
e-derivative securities, ”Review of Financial Stu
dies, 3, 573-592, is an extended Vasicek model. Literature Inui, K. and M. Kijima (1998), “A markovia
n framework in multi-factor Heath-Jarrow-Morton mo
dels, ”Journal of Financial and Quantitative Anal
According to ysis, 33, 423-440, the following formulas 51 and 52 can be made consistent with the term structure of the default free interest rate at the current time t = 0.

[0110]

[Equation 51]

[0111]

[Equation 52] However, in the equations 51 and 52, f₀(0, t), f
_{f, 0}(0, t) is the domestic and foreign default at the current time.
It is a term structure with a forward-free rate. z₀
(t) and z_{f, 0}Each risk market price for (t)
β₀(t) and β _{f, 0}(t) is given by the following equation. Hazard under observation probability P and risk-neutral probability P ^
The process is formally the same as the extended Vasicek model.
Specifically, the assumption regarding the risk premium adjustment rate is used.
Then, the following expression 53 is derived from the expressions 38 and 45.
However, ξ_jIt is assumed that (t) is differentiable by t.

[0112]

[Equation 53] Reference Kijima, M. (1998), “A Gaussian term structure
According to the model for valuing corporate bonds subject to credit risk, ”, Mathematical Finance, 8, 229-2 47, φ _j (t ) Is the following number 54
You can do it like.

[0113]

[Equation 54] In general, these are F _t -adaptive stochastic processes and are not deterministic functions at time t.

[2. Fundamental equations in Gaussian model
[Analysis solution] Below, only the basic part of the handling of mathematical formulas
Indicates. First, the solution of Eq. 38 and its cumulative hazard
The rate is given by the following formula. h_j(t) and H_j(t, T) is a Gauss-Markov process
And follows a normal distribution. h _jAverage of (t) m_j(t) and variance s
_j ^Two(t) and covariance s_jk(t) and H_jAverage M of (t, T)
_j(t) and variance S_j ^Two(t) and covariance S_jk(t) is that
It is given by the following formula. Formula 37, Formula 40, Formula 41, which have the same formula as Formula 38,
The solutions of the equations 44, 47 and 53 are also given in the same manner.

We will touch on stock prices and exchange rates. Number 39
The solution of is given by the following equation. The solution of Equation 42 is similarly given.

The solution of equation 43 is given by the following equation. Intuitively, it seems that the stock price declines as the hazard rate rises, but the effects are not taken into consideration in Equations 39 and 42. In the general expression, in Equation 11, μ
This effect can be taken into account by making _{s, j} (t) dependent on h _j (t).

On the other hand, the expected rates of return of Eqs. 46, 49, and 50 are random variables that follow a normal distribution. For example, the solution of Eq. 46 is a random variable (H ₀ ^ (0, t), H _j ^ (0, t), z _{s, j} ^
(t)) is a random variable. These are only used in the valuation of equity / forex derivatives and do not appear explicitly. [3. Period structure of hazard rate in Gaussian model]
A concrete method for reflecting the results of the empirical analysis of the term structure of default probability in the model is shown. In this model, assuming that the period structure of the hazard rate follows a three-parameter Weibull distribution, it is expressed as the following Expression 55.

[0118]

[Equation 55] The Weibull distribution is a distribution well known in the field of survival analysis, and its hazard rate is given by h (t) = λγtγ ^-1 . λ is a scale parameter and γ is a shape parameter. h (t) increases with t when γ> 1, is constant regardless of t when γ = 1, and decreases with t when 0 <γ <1. Here, a three-parameter Weibull distribution with a positional parameter m added is used. If λ _j , γ _j , and m _j included in the above equation 55 are estimated and used from the actual values of the default data, the results of the empirical analysis can be incorporated into the risk assessment. Regarding the default data analysis using the three-parameter Weibull distribution, the literature "Estimation and optimization in financial models" Koji Inui and Yukio Muromachi (2000) Asakura Shoten and the literature "Practice of hazard function estimation" Teruki Yoshi Suzuki (2000) Nissay Basic Research Institute Report, 1
See the disclosure on page 6, 101-113. At this time, the following formula is established. [3. Price Evaluation in Gaussian Model] Below is a simple analytic solution for commodity prices such as discount bonds.

[3-1. Price Evaluation of Discount Bond] First, the price of a discount bond with a default-free maturity T at time t is given by the following formula. Next, the price of the above-mentioned discount bond with default risk at the time t is given by the following formula from Expression 29. The yield curves of government bonds (= default-free bonds) and corporate bonds calculated from the model are consistent with the market prices of government bonds and corporate bonds observed at t = 0 by giving φ _j (t) appropriately. Become. Specifically, φ ₀ (t) and φ _j (t) are given by equations 51 and 54.

The price expression can be expressed more simply by using the forward neutral evaluation method. In this case, the price of the discount bond with default risk is calculated by the following formula 56 from formulas 30 and 32.
Given in.

[0121]

[Equation 56] It is also possible to express the discount bond price using φ _j (t) estimated from the market without using L _j (t, T), and that expression is more essential in the price valuation of credit derivatives. Target.

[3-2. Fixed Rate Bond Price Evaluation] Company j
Coupon C (constant) issued by T, interest payment date T = (t ₁ , t ₂ , ..., t _M ), t _j > t, j =
The price p _j (t, T; C) of fixed-rate bond with 1, 1, ..., M, maturity t _M , and recovery rate δ at time t is given by the following formula. In particular, in the case of default-free fixed rate bonds, it is given by the following formula 57.

[0123]

[Equation 57] [3-3. Price Evaluation of Floating Rate Bonds] Consider the following floating rate bonds issued by company j. First, the interest payment date
T = (t ₁ , t ₂ , ..., t _M ), t _j > t, j = 1,
..., M, maturity is t _M , and recovery rate is δ. The coupon rate for the period (t _j , t _{j + 1} ] is C (t _j , q)
Then, C (t, q) is assumed as the following equation, where α and β are constants. R (t, q) is the interest rate (zero coupon yield) for period q at time t. Here, the following equation (58) is assumed.

[0124]

[Equation 58] Then, it can be written that C (t, q) = α′r (t) + β ′. Further, particularly, the previous interest payment date is set to t ₀ , t ₀ ≦ t. t
_{When 0} = t, the contribution of the coupon generated at t is not included in the bond price.

First, when the floating rate bond is default-free, the price at time t can be written as the following formula 59.

[0126]

[Equation 59] However, C (t ₀ , q) is a known coupon rate at time t, p ₀ (t, t ₁ , t ₂ ; α ′, β ′) = p ₀ (t,
t ₁ , t ₂ ; α, β, q) is the time of the default-free discount bond in which C (t ₁ , q) (t ₂ −t ₁ ) is redeemed at the time t ₂ , t ₂ > t ₁ . The price at t is given by: Next, if there is a default risk on this floating rate bond,
The price at time t can be written as: However, C (t ₀ , q) is a known coupon rate at time t, p _j (t, t ₁ , t ₂ ; α ′, β ′, δ) = p
_j (t, t ₁ , t ₂ ; α, β, q, δ) is at time t, t ₂ >.
At t ₁ , the price of the discount bond with default risk at which C (t ₁ , q) (t ₂ −t ₁ ) is redeemed at time t,
It is given by the following formula. In the case of a floating interest rate with α ≠ 0, a term appears that shows the contribution of the correlation between the default free interest rate and the hazard rate, unlike the price formula for discount bonds.

[3-4. Interest rate swap] Consider a plain vanilla swap that exchanges fixed and variable interest rates.

Consider price evaluation in the case of default free. The side receiving the fixed interest rate is A, and the side receiving the floating interest rate is B. The interest payment date is T = (t ₁ , t ₂ , ..., t
_M ), t _j > t, j = 1, ..., M, maturity is t _M , and recovery rate is δ. The fixed interest rate is R, the floating interest rate paid to t _{j + 1} is C (t _j , q) (t _{j + 1} −t _j ), and C (t,
Let q) be a constant and assume the following. Further, particularly, the previous interest payment date is set to t ₀ , t ₀ ≦ t.
When t ₀ = t, the coupon contribution generated at t is not included in the bond price. At this time, the price of the swap seen from A at time t is given by the following formula. p ₀ (t, T; R) is the number 57, and p ₀ (t, T; α, β) is the number 5
Obtained from 9.

Price evaluation is difficult when there is a default risk. Several methods have been proposed, but the most plausible method among them is Duffie D. and M. Huang (19
96), “Swap rates and credit quality” Journal of F
Probably disclosed by inance, 51, 921-949. But,
Several additional assumptions are needed to find analytical solutions along these models.

[3-5. Price Evaluation of Stock Option] The price at the European call option time t with the maturity T> t and the exercise price K> 0 for the stock of the company j is given by the following formula. v _j (t, T; δ _j = 0) is the price of the discount bond issued by company j with a recovery rate of zero and maturity T at time t, and is as follows. The price of the European put option with maturity T and exercise price K at time t is given by the following equation. p (t, T; K) = Kv _j (t, T; δ _j = 0) φ (−z +
[sigma] _x ) -S (t) [phi] (-z) [3-6. Forward exchange contracts] Credit risk is not considered in this forward exchange contract and the next currency option. The current time is t,
Value F at t of a transaction for exchanging 1 unit (1 dollar) of foreign currency with domestic currency V _c (yen) at future time points T, T> t
Find (t, T, V _c ).

First, future times T, T at time t>
t, forward price (forward exchange rate) H (t,
T) is given by the following equation. However, the current currency exchange contracts are past t ₀ , t ₀ <t,
The contract rate is H (t,
Not the same as T). The value of the contract at the current time t is given by the following formula.

[0132] _{F (t, T, V c} ) = V c v 0 (t, T) -V
(t) v _{f, 0} (t, T) Of course, F (t, T, V _c ) = 0 is Vc = H.
This is the case of (t, T).

[3-7. Currency option] Maturity T, exercise
Currency of price K European call option time
The price at t, t <T is given by the following formula. Currency with maturity T and exercise price K European put op
The price at the time t, t <T of the option is given by the following formula
To be [4. Simultaneous minutes of default time in Gaussian model
Fabric Function and Monte Carlo Simulation] Risk Ho
Find the portfolio price distribution in Rise T
In addition to the above price evaluation formula,
Know the simultaneous distribution of default and non-default status of assets
It is necessary to use the simultaneous distribution function of the default time.
Given. In this model, the default time τ
_jEquation 4 and τ that represent the survival probability of _j, J = 1, ..., n,
27 and the simultaneous distribution function of are respectively the following 6
It can be expressed as 0.

[0134]

[Equation 60] In Equation 60, the contribution of the hazard rate correlation is S
It is evaluated by _jk (0, T), j ≠ k.

More specifically, n that constitutes a portfolio
It suffices that the combination of the default (×) and non-default (○) of individual assets and the occurrence probabilities be obtained in all cases, but if the combination is designated, the occurrence probability is immediately obtained from the equation (60). Therefore, theoretically, the price distribution of the future portfolio can be calculated analytically.

However, this is almost impossible to realize for a large portfolio of assets n. I mean,
This is because the total number of combinations of default and non-default is 2 ⁿ , and when n becomes large, it rapidly increases. For this reason, we have no choice but to rely on Monte Carlo simulation for the part that seeks the joint distribution of future default / non-default states. In this general calculation procedure, this simulation is (1)-
It corresponds to (2). In particular, (1) will be described.

Normally, a natural number n of an appropriate size is set and the time step Δt = (T−t) / n is set, and the expressions 37 to 43 are set.
And the standard Brownian motion z (t), the time t
Each random variable may be generated in increments of Δt until the time T, using the value in the above as the initial value. By the way, in this Gaussian model, the conditional distribution at the times s, s ≧ t when the value of the random variable is given at the time t, is easily obtained. There is no need to discretize and calculate with. For example,
The conditional distribution according to (r (s), h _j (s), logV (s)) is a trivariate normal distribution of mean m ^T (s) and variance-covariance matrix Σ (s). Given. Using the above equation, (r (s), h _j (s), logV (s))
It is easy to calculate a sample of. The same applies to other random variables.

[5. Overview of integrated valuation system for market risk and credit risk] Until now, mainly the simulation and integrated valuation method for integrated valuation, which is the most important process in the integrated valuation system for market risk and credit risk of portfolios. I explained from the theoretical side. Here, from a more practical aspect, a procedure for executing the entire integrated evaluation system of market risk and credit risk will be described with reference to FIGS.

FIG. 3 is a diagram showing an embodiment of the overall processing flow of the integrated evaluation system for market risk and credit risk (hereinafter abbreviated as "integrated evaluation system") according to the present invention. In FIG. 3, steps S1 to S8 are steps for preparing for the simulation, and steps S9 to S13 correspond to steps shown in FIG. Therefore, steps S1 to S8 may be executed in the order of the steps shown in FIG. 3, but they do not necessarily have to be executed in this order. On the other hand, steps S9 to S13
Is basically executed according to the flow shown in FIG.

Next, each step shown in FIG. 3 will be described in more detail. First, set the processing reference date (S
1). The processing reference date is the date when risk measurement is started in the integrated evaluation system, and the processing reference date is always registered when the system starts.

Next, the steps of fetching attribute data (S2) and data integration (S3) will be described with reference to FIG. In step S2, financial product attribute data required in the integrated evaluation system is registered in each data file. The financial product attribute data file to be registered is a financial product attribute (bond) data file 231,
Financial product attribute (financing) data file 232, financial product attribute (stock) data file 233, financial product attribute (swap) data file 234, financial product attribute (exchange contract) data file 235, company-specific in-house rating information data file 236 There are 6 types.

Financial product attribute (bond) data file 23
1 includes, for example, bond code (in-house code), company code (stock code), bond type (discount bond, fixed-rate bond, floating-rate bond, type), stock name, holding flag (holding or not holding). Type), Issuer (Type of government bond, municipal bond, government-guaranteed bond, financial bond, general corporate bond, yen-denominated bond), coupon rate, interest payment date, redemption date, number of interest payments, first interest payment date, issue date , Currency (yen, dollar, euro, type), country code,
The data of coefficient α, floating rate bond spread β, floating rate bond base interest rate, and recovery rate are imported.

Financial product attribute (financing) data file 23
2 includes, for example, a loan code (in-house code), a company code (debtor code), a currency (types of yen, dollar, and euro), a coupon rate, an interest payment date, the number of interest payments, a maturity date, and a recovery rate. Capture data.

Data file 2 of financial product attribute (stock)
For example, stock code (in-house code), company code, currency, and country code data are fetched in 33.

The financial product attribute (swap) data file 234 contains, for example, a swap code (in-house code),
Company code, currency (type of yen, dollar, euro), coupon rate, interest payment date, number of interest payments, maturity date, swap type (fixed payment variable receipt, variable payment fixed receipt, type),
Take in the data of coefficient α and floating rate bond spread β.

The financial product attribute (exchange contract) data file 235 contains, for example, an exchange contract code (in-house code),
Data of company code (counter party), delivery date (execution date), delivery currency, delivery amount, receipt currency, and receipt amount are imported.

Internal rating data file 236 for each company
Includes the internal code, internal rating, and rating (for example, the corporate rating information from an external credit bureau is entered as reference information),
Capture the update date data.

In step S3, a rating is assigned to each financial product from the financial product property data registered in each of the financial product property data files 231 to 235 and the data registered in the in-house rating data file 236. , Are integrated into one financial product attribute / rating data file 23 (see FIG. 2) so that they can be easily handled by the integrated evaluation system.

In step S4, the data of the portfolio managed by the integrated evaluation system is taken into the portfolio data file 22 (see FIG. 2). The data to be captured is divided into data regarding the base date balance and data regarding the organizational hierarchy.

The data regarding the reference date balance is, for example, reference date, portfolio code,
Attribute classification (bond, loan, stock price, swap, exchange contract, type), in-house code (code corresponding to attribute classification), company code (for example, code based on the stock code of the Japan Securities Dealers Association's "in-store quote information") ), Quantity (par value for bonds, principal for loans, share holding for stock prices, notional amount for swaps, notional amount for swaps, foreign exchange (dollar or Euro)), market price (market price per 100 yen for bonds, market price per share for stock prices, restructuring cost per unit amount of notional amount (yen, dollar, euro) for swaps, currency exchange) If it is a reservation, the difference between the present value of yen and the foreign currency (after yen conversion) (foreign currency 1
(Per unit amount). Further, a portfolio code and a portfolio name are taken as portfolio name data, a domestic / overseas division (domestic / overseas type), a company code, a company name, and a country code and a country name are taken as company name data. Furthermore, it is possible to capture the data when there is a change in the portfolio (trade, interest payment, redemption).

As the organizational hierarchy information, a portfolio code, an attribute classification (bond, loan, stock price, swap, exchange reservation type) and an in-house code are taken in.

In step 5, the market data necessary for the integrated evaluation system is converted into the market data file 2
4 and the default data file 25 (see FIG. 2). As market data, risk-free interest rate time-series data file 241, exchange rate time-series data file 242, stock price time-series data file 243, stock index time-series data file (not shown, mainly used for stress tests), bond price Time series data files 245a-b, bond data files 245c-
Set d (these files are shown in FIGS. 5, 6 and 7).

Risk-free interest rate time series data file 241
For example, the reference date, currency (yen, dollar, euro,
Type), time series of interest rates (overnight interest rate,
1M-LIBOR, 3M-LIBOR, etc.) are taken in. The exchange rate time-series data file 242 is loaded with, for example, reference date, conversion category (type of yen / dollar, yen / euro,) and exchange rate time-series data. In the stock price time series data file 243, for example, as the stock price time series domestic data file 243a, time series data of the reference date, stock code, company code, and stock price (PanR
olling, TSE 64K, BVD, etc.) will be imported. Further, as the stock price time series overseas data file 243b, the reference date, country code, stock code, company code,
Time series data of stock prices is captured. The stock index time series data file 244 includes time series data of the reference date, country code, and stock index (for example, Nikkei 225 in Japan, S & P500 in the US, FT100 in the UK, DAX in Germany, CAD40 in France, etc.). It is captured.

The public and corporate bond price time series (domestic) data file 245a includes, for example, the reference date, the issue code (for example, the issue code of the "Securities Quote Issue Information" of the Japan Securities Dealers Association), and the market price of the issue quote. Series data is imported. Bond data time series (overseas) data file 245b
Includes, for example, the reference date, CUSIP (code name), and time series time value data (NTT Telecross, CNN).
CoBond, etc.) is incorporated. The bond code, the brand name, the redemption date, the coupon rate, the interest payment month, and the interest payment date data are loaded into the public and corporate bond attribute (store-quoted brand) data file 245c. Data of CUSIP, brand name, redemption date, coupon rate, interest payment month, and interest payment date are taken into the public and corporate bond attribute (overseas) data file 245d.

In the default data file 25, as a domestic bankruptcy count file 25a, a reference year, a rating, the number of applicable companies, and the number of bankruptcies per year are fetched from, for example, data of an external credit bureau. In addition, the year, overseas rating, and default rate (for example, Moodys yearly default rate) are captured as the overseas yearly default rate data file 25b.

In step S6, a basic equation 311 (default free interest rate process model, hazard rate process model represented by stochastic differential equations) for generating a scenario depicting temporal changes in future situations in the integrated evaluation system, Estimate the parameters (average regression level, average regression power, volatility, etc.) applied to the stock price process model and the exchange rate process model).

As shown in FIG. 5, from the data of the risk-free interest rate time series data file 241, for example, the interest rate parameter a ₀ (mean regression power), σ ₀ (volatility −), m ₀ (mean regression level) shown in Expression 37 are shown. ) Is estimated. From the data of the exchange rate time-series data file 242, for example, the exchange parameter μ (drift) and
Estimate (volatility). From the data of the stock price time-series data file 243, for example, the stock price parameter μ _{s, j} (drift) and σ _{s, j} (volatility) shown in Formula 39 are estimated. Default data file 25
From the rating definition information based on the a and b data and the manually input in-house rating data file 236, for example,
Hazard parameter a _j (mean regression force) and σ
Estimate _j (volatility), λ _j (scale parameter), γ _j (shape parameter), and m _j (position parameter) that appear in Equation 55.

As the parameter estimation method, for example, "maximum likelihood method" or "least square method" is used. That is, the parameters are estimated so as to "best match" the observed data. The estimation method is divided depending on how to define this "best match". In the "maximum likelihood method," we consider that "the probability that the observed data will be obtained is the highest" is "the one that works best" (a certain series of observed data was obtained, the probability that that data is obtained is I think it was higher than the probability that any other data could be obtained). Since the probability (called the likelihood) that a given observation data is obtained changes depending on the parameter, the parameter that maximizes the probability (= likelihood) is obtained. On the other hand, in the "least squares method", "the sum of squares of the error between the observed data and the theoretical value is the smallest" is considered to be "the best match". Since the theoretical value changes depending on the parameter, the parameter that minimizes the sum of squared error is sought.

In step S7, the interest rate term structure is estimated. As shown in FIG. 6, public and corporate bond price time series data files 245a and b and public and corporate bond attribute data files 245.
Based on the data of c and d and the internal rating data from the financial product attribute / rating data file 23, the interest rate term structure according to the internal rating and the default free interest rate term structure on the processing reference date are calculated by, for example, the Vasicek and Fong method. Ask the theoretical price of the bond to "best match" with the market price on the processing date. Specifically, the interest rate term structure is expressed using several parameters, and a parameter value is calculated so that the theoretical price of the bond price and the market price are "best". (The method of obtaining the parameters depends on the interest rate period structure model used, but for example, in the Vasicek and Fong method, the parameters are obtained by exploratory multivariate regression analysis.) The obtained parameter values, in other words, the interest rate period structure Based on the equations 51 and 54, φ ₀ (t) and φ _j (t)
And they will be used for current and future price valuations.

In step S8, the correlation between risk factors is calculated. As shown in FIG. 7, the analysis period is set, risk-free interest rate time series data file 241, exchange rate time series data file 242, stock price time series (domestic)
Based on the data of the data file 243a and the stock price time series (overseas) data file 243b, a correlation coefficient matrix between risk factors (interest rate, foreign exchange, stock price) is calculated and decomposed into a lower triangular matrix by Cholesky decomposition. The financial product attribute / rating data file 23 is used to select an arbitrary stock price. Similarly, the correlation coefficient between the hazard rate and (interest rate / exchange / stock price) is calculated from the time-series data of the interest rate term structure of government bonds and corporate bonds (not shown) and the time-series data of interest rates / exchange / stock prices. To calculate. The decomposed result (CR value) is used to generate a random number having a correlation necessary for calculating a future price distribution. The use of correlated random numbers makes it possible to express the diversified investment effect.

As described above, by performing steps S1 to S8, preparations for simulation of the portfolio by the integrated evaluation system are completed. Next, the integrated risk calculation process of step S9 will be described with reference to FIG. The content of the integrated risk calculation process has already been described, and therefore the process procedure will be mainly described here. In order to perform the simulation, first, the simulation conditions are set. It is necessary to enter the following items (a) to (f). (A) Risk factor Specify which of interest rate, hazard rate, foreign exchange, and stock price is stochastically fluctuated. For example, the market risk can be evaluated if three other than the hazard rate are stochastically changed and the default is not taken into consideration. Moreover, credit risk can be evaluated by avoiding stochastic fluctuations in interest rates, exchange rates, and stock prices, and further considering defaults. If all are stochastically varied and defaults are taken into account, integrated risk can be evaluated. (B) Number of scenarios Select the number of times to repeat the simulation. (C) Number of divisions The number of divisions sets how many periods the risk horizon will be divided into. For example, in the case of dividing the risk horizon for one year into two, first create a scenario after six months, and then create a scenario after one year. It is set for the case where it is necessary to consider that the situation after half a year (that A company defaults by then) influences the situation after that (probability of default of affiliated companies of A company). (D) It is the time of risk / horizon evaluation (for example, specified by the year elapsed from the processing reference date). (E) Set the percentage point when calculating the percentage point VaR. (F) Monte Carlo simulation method Specify a random number generation method (for example, select from three types: unprocessed, symmetrical variable method, symmetrical variable method + quadratic sampling method.) After simulation conditions are set, Set reference date, portfolio data file 22, financial product attribute / rating data file 23, interest rate parameter / hazard rate parameter / stock price parameter / exchange parameter obtained in step 6, interest rate term structure parameter obtained in step S7 The integrated risk calculation is performed based on the Cholesky decomposition result (CR value) obtained in step S8, and the risk factor value distribution and the future price distribution of the portfolio are output.

In step S10, risk factor value statistics, risk amount (for each asset) (standard deviation, skewness, kurtosis, percentile point), risk amount are calculated based on the risk factor value distribution and future price distribution obtained in step S9. (For each portfolio), Risk Contribution (RC), marginal risk amount, and expected return are required. In addition, Risk Contribut
The ion is a value when the risk amount of the portfolio (for example, standard deviation, VaR, T-VaR, etc.) is allocated to each asset.

At step S11, as shown in FIG.
The frequency distribution section width is calculated based on the risk factor value distribution and the future price distribution obtained in step S9, and the portfolio value frequency and the risk factor frequency are output.

In step S12, the risk amount (for each portfolio), risk amount (for each asset), RC, marginal risk amount, expected return obtained in step S10, portfolio data file 22, interest rate period structure parameter, financial product attribute, The risk adjusted return is calculated based on the rating data file 23, and the excess profit is output.

In step 13, the optimization execution conditions (required return rate of portfolio, incorporation ratio for each asset, incorporation amount) are set based on the future price distribution obtained in step S9 and the portfolio data file 22. Optimize the amount of assets held for each portfolio. As a result, the optimized quantity and the optimized T-VaR are obtained.

By executing steps S1 to S8 and S9, it is possible to perform a stress test using the same system, which is different from the gist of the present invention.

[0167]

[Example] The result of obtaining the price distribution of a bond (government bond / corporate bond) portfolio with a number n of issues n years later using a Gaussian model is briefly shown as a numerical example. Before showing the results, the risk indicators used below will be briefly explained. VaR: The difference between the expected value of the price and the 100 (1-α) -percentage point of the distribution, which is obtained from the price distribution in the risk horizon. Risk horizon and α, 0 <α <
It is a value that is determined only when 1 is specified. The same applies to the other risk indicators mentioned here, such as T-VaR and RC. -T-VaR: The difference between the expected price value and the average value below 100 (1-α) -percentage point in the risk horizon. RC (risk contribution): A value when a risk amount of a portfolio (for example, standard deviation, VaR, T-VaR, etc.) is allocated to each asset, and is calculated based on simulation results. It has the property that the sum of RC of all assets becomes the risk amount of the portfolio. RC ratio: A value obtained by dividing RC by the present value. RC measure: A value obtained by dividing the expected rate of return by the RC ratio.
Used in terms of return / risk. (RC, R
C ratio and RC measure are not general terms. ) [1. Prerequisites for Numerical Examples] Table 1 shows the attributes (rating, face value, coupon rate, recovery rate) of each bond in the portfolio considered below.

[0168]

[Table 1] All interest on bonds is payable in half a year, and cash flows generated during the period are managed at default free interest rates. (There are various ways to handle cash flows generated during the period, including the case of default. I treated it quite simply here, and I assume that if you do default, you will receive half of what you did not.)

Rating is 6 of Aaa, Aa, ..., B
Hazard ratio parameters are assigned in stages.
Use rating-specific values estimated from ody's Investors Service default data. The parameters of the hazard rate used are shown in the table.

[0170]

[Table 2] However, in the basic equation, a _j = 0.2, j = 1, ...
.., n, As for the current interest rate term structure by government bond and rating, refer to the data on the website of JP Morgan and refer to the forward rate f _j (0, t) as a quadratic function of the remaining term t. Give in. Ratings are in parentheses. Specific coefficient values are shown in Table 3.

[0171]

[Table 3] The recovery rate of the bonds that make up the forward rate given here is 0.4 (the recovery rate of the bonds included in the portfolio can be set independently of the recovery rate of the bonds that give the forward rate). The parameters of the default-free interest rate process are a ₀ = 0.018, b ₀ (t) = b ₀ = 0.0.
54a ₀ , σ ₀ is variously changed to 0.0%, 0.5%, 1.0%, and the like. Interest rates and hazard rates are uncorrelated. The correlation coefficient of the hazard rate by rating is
Set as follows. Both row and column are Aaa, Aa, A, Baa,
They are Ba case and B case. A diagonal component of 0.8 indicates a correlation between the same ratings. The number of sample paths is 50,000.

[2. Calculation result] [2-1. Integrated evaluation of credit risk and market risk] Figure 11
Shows the future value distribution of the portfolio after T = 1 year. Figure 11 (a) shows when only credit risk is evaluated (σ ₀
= 0.0%), and Fig. 11 (b) shows the distribution when the credit risk and the interest rate risk are integratedly evaluated (σ ₀ = 1.0%).

The distribution in FIG. 11 (a) is multi-peaked, but the largest right peak is the distribution when no asset defaults, and the three small peaks on the left are mainly B It reflects the distribution when the rated assets F, N, O, P (see Table 1) default. There is also a small lump on the left,
These are defaults for other high-value assets,
Corresponding to the default occurrence of multiple assets. Such a multimodal distribution is characteristic of credit risk (when the number of assets increases when the asset size is relatively uniform,
These Komines overlap, resulting in a fat tail distribution with the skirt extending gently to the left as a whole).

In FIG. 11 (b), the peaks are diffused and overlapped due to interest rate risk, resulting in a large single peak. Table 4 shows the risk indexes of the cases of FIGS. 11A, 11B and 11C and the case (d) described later in comparison.

[0175]

[Table 4] Table 4 shows that interest rate risk has a non-negligible effect on future price distribution.

Add more stocks to this portfolio
Fig. 11 (c) shows the case (σ ₀= 1.0%). Addition
The shares obtained are 6 stocks of companies B, D, F, H, J and L (table
1), all initial price 2, μ_s= 0.2, σ_s=
At 0.1, the correlation is zero. In this FIG. 11 (c),
The distribution is wider than the risk of stock price fluctuations compared to Fig. 11 (b).
ing. Port stocks of companies that already have bonds
In terms of risk assessment, adding to the folio is
Hold more market risk and
Will significantly increase credit risk exposure
It As a result, the amount of additional stock seen from the whole is not so large.
I'm not sure, but VaR, which is dominated by credit risk,
The value of T-VaR is pushed up greatly.

[2-2. Interest rate risk hedging] Next, Table 1
The following shows an example of hedging the interest rate risk of a portfolio (duration 3.61) consisting of 20 bonds of The following three interest rate swaps are included in the portfolio.・ Swap A: Maturity 2 years, half year payment, 6 months floating interest rate fixed interest payment (5.28% per year), notional principal 30.4 ・ Swap B: Maturity 4 years, half year payment 6 months floating interest rate fixed Interest payment (5.32% per year), notional amount 30.0 ・ Swap C: 6-year maturity, half-year payment, 6-month floating interest rate fixed interest payment (5.35% per year), notional amount 20.0 Has a present value close to zero, and the portfolio duration is set to almost zero when three are incorporated. Interest rate swaps are evaluated as if they are default-free.

FIG. 12 shows the future price distribution when Swap A, Swap B, and Swap C are added to the bond portfolio in this order. In addition, the above interest rate swap is 3
Table 4 lists the risk indicators for case (d), which were both incorporated.
Shown in. From Figure 12 and Table 4, the effect of interest rate risk hedging by interest rate swaps can be seen. In case (d) where the duration is almost zero (third figure from the top in FIG. 12), interest rate risk is not considered.
It has a shape similar to the multi-peak shape of. Reflecting this, in Table 4, the risk index of case (d) is quite close to the value of case (a).

[2-3. Risk / Return Analysis of Individual Assets] Finally, we will introduce the results of risk / return analysis of individual assets (in this model, future value distributions of portfolios and individual assets can be specifically obtained, so calculation of any risk index is possible). Moreover, since the return index (expected return rate) measured on the same scale as the risk index is obtained, the risk / return analysis can be performed in a very consistent manner. In a model that evaluates the amount of risk on an amount basis, risk / return analysis cannot be performed unless a return index is calculated using a logic that is completely different from that model).

FIG. 13 shows the RC for 95% T-VaR.
(Risk contribution, risk contributio
n) and present value distribution. 95% T-VaR (abbreviated) calculated from the future value distribution of individual assets and R shown here
Comparing the values of C, RC has dropped to less than half of T-VaR on average and down to about 10% for some assets, which is the effect of the diversified investment effect. In the following, this RC is adopted as the risk amount.

FIG. 14 shows the R for 95% T-VaR.
C ratio and expected rate of return are shown for each asset, which is, so to speak, a risk / return diagram. All assets with an expected rate of return of around 4% are rated B, because the coupon was set low for credit risk. Figure 15 shows
The RC measure of each asset is shown. As a rule of thumb, the higher the RC measure, the better the asset.
In this case, G government bonds and bonds with high rating and low face value are relatively good. The RC measure of a bond, such as an H bond, which has a high face value (exposure, amount of loss when defaulted), even if it has a high rating, becomes slightly lower. What is generally bad is B-rated assets. Also, Ba
The reason why the RC measure of I-bonds of the grade is extremely low is that the face value is large. In this way, the RC measure for T-VaR used here is an index that reflects not only the rating and coupon rate but also the size of the exposure.
It seems to be effective in practice.

[0182]

The present invention provides a simulation model capable of integrated evaluation of market risk and credit risk such as interest rate risk, stock price fluctuation risk and foreign exchange risk. Specifically, the problem to be solved by the invention is that all assets can be evaluated on one scale, the effect of diversified investment can be evaluated, and the period of interest rates and default rates that are consistent with observed market prices. Clear issues such as structure consideration, asymmetry and non-normality of return distribution of individual assets, price distribution in risk horizon can be obtained concretely, calculation load is not so heavy, etc. , It is superior to the conventional model in that it can perform integrated evaluation of market risk and interest rate risk in a natural form, is very simple and easy to use, and is theoretically very clear and consistent.

[Brief description of drawings]

FIG. 1 is a block diagram showing an outline of an integrated evaluation system for market risk and credit risk of a portfolio according to the present invention.

FIG. 2 is a block diagram showing a hardware configuration for implementing an integrated evaluation system for market risk and credit risk of a portfolio according to the present invention.

FIG. 3 is a flowchart showing a processing procedure of an integrated evaluation system for market risk and credit risk of a portfolio according to the present invention.

FIG. 4 is a block diagram showing a process in a step of data integration.

FIG. 5 is a block diagram showing a process in a parameter estimation step.

FIG. 6 is a block diagram showing a process in a step of estimating an interest rate period structure.

FIG. 7 is a block diagram showing a process in a step of calculating correlation between risk factors.

FIG. 8 is a block diagram showing processing in integrated risk calculation, evaluation, and analysis steps.

FIG. 9 is a block diagram showing processing in steps of frequency distribution section width calculation and risk adjusted return calculation.

FIG. 10 is a block diagram showing processing in an optimization step.

FIG. 11 is a diagram showing a future value distribution of a portfolio after one year according to an example.

FIG. 12 is a diagram showing a future value distribution incorporating a portfolio swap according to an embodiment.

FIG. 13: R for 95% T-VaR, according to examples
It is a figure which shows C and distribution of present value.

FIG. 14: R for 95% T-VaR, according to an example
It is the figure which showed Cratio and expected return rate by asset.

FIG. 15: RCmeasure by asset according to an embodiment
It is the figure which showed.

[Explanation of symbols]

11 CPU 12 ROM 13 RAM 14 Interface circuit 15 Input device 16 Output device 17 External storage device 21 Calculation program file 22 Portfolio data file 23 Financial product attribute / rating data file 24 Market Data File 25 Default data file

   ─────────────────────────────────────────────────── ─── Continued front page    (72) Inventor Yukio Muromachi             1-1-1 Yurakucho, Chiyoda-ku, Tokyo             Inside the Nissay Research Institute

Claims (12)

[Claims] 1. An input means for inputting data or an instruction, an output means for outputting a calculation result based on the input data or instruction, and a classification / quantity / quantity of each financial instrument constituting a portfolio to be evaluated. Portfolio data storage that stores portfolio data such as market prices and transfers, and finance that stores detailed data such as classification, type, issuer, maturity date, coupon rate, and rating data for each financial instrument. Product attribute / rating data storage means, market interest rate / market price of financial products including public and corporate bonds and stocks
Market data storage means for storing market data including time-series data of exchange rates, default data storage means for storing default data including past bankruptcies by year and default rate by overseas year, and default / free interest rate Describes future scenarios of default occurrence rates and scenarios depicting temporal changes in interest rate fluctuations from the present to risk horizon based on a default-free interest rate process model expressed by equations that describe future fluctuations Based on the default process model expressed by the equation, there is a scenario that depicts the temporal change in the default occurrence situation from the present to the risk horizon and a stock price fluctuation process model that is expressed by the equation that describes the future fluctuation of stock price fluctuations. Based on the present, we will draw a temporal change of stock price fluctuation from present to risk horizon. And the scenario that depicts the temporal change of the exchange rate fluctuation from the present to the risk horizon based on the exchange rate fluctuation model expressed by the equation that describes the future fluctuation of the exchange rate. A future situation scenario generation means and a future situation scenario generation means that are generated in large numbers based on the data stored in at least one of the storage means, the financial product attribute / rating data storage means, the market data storage means, and the default data storage means. Risk Horizon's asset price calculation method that calculates the price of the portfolio and individual asset price in the risk horizon by introducing the risk premium adjustment rate for each scenario generated by the risk horizon. For each scenario by means of asset price calculation A future price distribution calculation means for obtaining the future price distribution of the portfolio and / or the future price distribution of the individual assets based on the price of the portfolio and the individual asset price in the issued risk horizon, and Market risk and credit, characterized by integrating and evaluating market risk and credit risk based on the future portfolio price distribution and / or the future price distribution of individual assets calculated by the price distribution calculation means. Integrated risk assessment system. 2. The future price and / or future price in the risk horizon based on the future portfolio price distribution and / or the individual asset future price distribution calculated based on the future price distribution calculation means. The integrated evaluation system for market risk and credit risk according to claim 1, further comprising a risk index calculation means for calculating a risk index including standard deviation, VaR, CVaR, and a lower partial product rate of the return rate. 3. A return index including an expected rate of return in risk horizon based on a future portfolio price distribution calculated based on the future price distribution calculation means and / or a future price distribution of individual assets. The integrated evaluation system for market risk and credit risk according to claim 1, comprising return index calculation means for calculating. 4. The future situation scenario generation means includes domestic default / free interest rate r (t), hazard rate h _j (t) of domestic company j, stock price S _j (t) of domestic company j, and Default free interest rate r _f (t) of country F, hazard rate h _{f, j} (t) of company j of country F, stock price S of company j of country F
Stochastic differential equations that describe future fluctuations of _{f, j} (t) and the exchange rate of the home country and the F country (country currency conversion value of the home country currency unit) V (t) And μ ₀ (t), μ _j (t), μ _{s, j} (t), μ _{f, 0} (t), μ
_{f, j} (t), μ _{fs, j} (t) and μ _V (t) are r (t), h _j (t), S _j (t), r _f (t), h _{f, j} ( t), drift of S _{f, j} (t) and V (t), σ ₀ (t), σ _j (t), σ _{s, j} (t), σ _{f, 0} (t), σ _{f, j} (t), σ _{fs, j} (t) and σ _V (t) are r (t), h _j (t), S _j (t), r _f (t), h _{f, j} (t), Volatility of S _{f, j} (t) and V (t), both of which are based on t and these random variables and the default state (whether or not defaulting) of all companies A sample path from the current time t to the risk horizon T (T> t) is generated, and r (s) and h
_j (s), S _j (s), r _f (s), h _{f, j} (s) and S _{f, j}
(s) and V (s), the first processing means for obtaining t ≦ s ≦ T and the default time point of the company j not defaulting at time t are τ _j , the risk Survival probability P _T {τ _j > T} under the observation probability of company j in Horizon T Probability of defaulting by risk horizon T based on 2. The integrated evaluation system for market risk and credit risk according to claim 1, further comprising: second processing means for determining a default / non-default state of the company j in the risk horizon T for each sample path. 5. The risk horizon asset price calculation means for calculating the portfolio price and individual asset price in the risk horizon for the sample path generated by the future situation scenario generation means is a risk horizon For financial instruments issued by company j that has already defaulted at T, the price p ^def (T) at risk horizon T is calculated as the default time τ _{j of} company j, the maturity date and the interest payment date, and the coupon rate and recovery. Financial product attribute / rating data including rate and company j
First processing means to be obtained based on the stock price of the product and the market data including the price of the product up to the time τ _j , and the issuance of a company j (j = 0 is default free) not defaulting in the risk horizon T For domestic products, domestic default / free interest rate r (t), hazard rate h ^ _j (t) of domestic company j, stock price S of domestic company j
_j (t), default free interest rate r of overseas partner country F
_f (t), hazard rate of company j in country F h ^ _{f , j} (t), F
Stock price S _{f, j} (t) of company j in the country and exchange rate between home country and country F of the country (home currency conversion value of one unit of country currency) V
The fluctuation of (t) after the risk horizon T under the risk neutral probability P ^ or the forward neutral probability P ^s ,
The stochastic differential equation Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And μ ₀ (t), μ _j (t), μ _{f, 0} (t) and μ _{f, j} (t)
Are r (t), h _j (t), r _f (t), and h _{f, j, respectively.}
drift of (t), σ ₀ (t), σ _j (t), σ _{s, j} (t), σ _{f, 0} (t), σ _{f, j} (t), σ _{fs, j} (t) and σ _V (t) is r (t), h _j (t), S _j (t), r _f (t), h _{f, j} (t), S _{f, j} (t) and V (t), respectively. , Β ₀ (t), β _{n + j} (t), β _{f, 0} (t), β _{f, nf + j} (t), and β _V (t) are z ₀ (t) and z, respectively.
_{n + j} (t), z _{f, 0} (t), z _{f, nf + j} (t), z _V (t) market price of risk, ξ _j (t), ξ _{f, j} (t) are risk premium adjustment rates , Ρ _jk (t), ρ ^ff _jk (t), ρ ^f _jk (t), ρ
_jV (t) and ρ ^f _jV (t) are correlation coefficients, both of which are generated based on t and these random variables and the default state (whether or not to default) of all companies. Cumulative cash flow X (t) of the financial product until time t
, The above random variables including r (s) (T ≦ s) and h ^ _j (s) (T ≦ s), the default / non-default state of the company j, the maturity date and the interest payment date and the coupon rate, and By using the financial product attribute data including the recovery rate and the expression, the price p in the risk horizon T of the product
^nondef (T) is a risk neutral evaluation method (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^) or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information until the time T is known under the forward neutral probability P ^s , v ₀ (T, T _i ^j ) is the maturity T _i. 2) Calculated based on the price of the default-free discount bond of ^j at time T)
The integrated evaluation system for market risk and credit risk according to claim 4, further comprising: 6. The risk horizon asset price calculation means for calculating the portfolio price and individual asset price in the risk horizon for the sample path generated by the future situation scenario generation means is a risk horizon For financial instruments issued by company j that has already defaulted at T, the price p ^def (T) at risk horizon T is calculated as the default time τ _{j of} company j, the maturity date and the interest payment date, and the coupon rate and recovery. Financial product attribute / rating data including rate and company j
Processing means A based on the stock price of the product and the market data including the price of the product up to time τ _j, and the finance issued by company j (j = 0 is default free) not defaulting in risk horizon T For products, domestic default free interest rate r (t), hazard rate h ^ _j (t) of domestic company j, and stock price S of domestic company j
_j (t), default free interest rate r of overseas partner country F
_f (t), hazard rate of company j in country F h ^ _{f, j} (t), F
Stock price S _{f, j} (t) of company j in the country and exchange rate between home country and country F of the country (home currency conversion value of one unit of country currency) V
The fluctuation of (t) after the risk horizon T under the risk neutral probability P ^ or the forward neutral probability P ^s ,
The stochastic differential equation Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And μ ₀ (t), μ _j (t), μ _{f, 0} (t) and μ _{f, j} (t)
Are r (t), h _j (t), r _f (t), and h _{f, j, respectively.}
drift of (t), σ ₀ (t), σ _j (t), σ _{s, j} (t), σ _{f, 0} (t), σ _{f, j} (t), σ _{fs, j} (t) and σ _V (t) is r (t), h _j (t), S _j (t), r _f (t), h _{f, j} (t), S _{f, j} (t) and V (t), respectively. , Β ₀ (t), β _{n + j} (t), β _{f, 0} (t), β _{f, nf + j} (t), and β _V (t) are z ₀ (t) and z, respectively.
_{n + j} (t), z _{f, 0} (t), z _{f, nf + j} (t), z _V (t) market price of risk, ξ _j (t), ξ _{f, j} (t) are risk premium adjustment rates , Ρ _jk (t), ρ ^ff _jk (t), ρ ^f _jk (t), ρ _jV (t), and ρ ^f _jV (t) are correlation coefficients, all of which are t and these random variables and all companies. When the processing means B for generating a large number of sample paths based on the default state (function of whether or not) and the default time point of the company j not defaulting in the risk horizon T are τ _j , each sample path is future time S, S> T, the probability of survival P ^ {τ _j> S} under risk-neutral probability P ^ or forward-neutral probability P ^s of companies j in Probability of defaulting by time S based on According to the processing means C for determining the default / non-default state of the company j at time S for each sample path, and the cumulative cash flow X (t) of the product until time t.
, The above random variables including r (s) (T ≦ s) and h ^ _j (s) (T ≦ s), the default / non-default state of the company j, the maturity date and the interest payment date and the coupon rate, and By using the financial product attribute data including the recovery rate and the expression, the price p in the risk horizon T of the product
^nondef (T) is a risk neutral evaluation method (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^), or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information until the time T is known under the forward neutral probability P ^s , v ₀ (T, T _i ^j ) is the maturity T _i. 5. The processing means D for calculating using the sample path generated by the processing means B and the processing means C on the basis of (the price at time T of the default-free discount bond of ^j ). Integrated market and credit risk assessment system described. 7. The risk horizon asset price calculation means for calculating the portfolio price and individual asset price in the risk horizon for the sample path generated by the future situation scenario generation means is a risk horizon For financial instruments issued by company j that has already defaulted at T, the price p ^def (T) at risk horizon T is calculated as the default time τ _{j of} company j, the maturity date and the interest payment date, and the coupon rate and recovery. Financial product attribute / rating data including rate and company j
Processing means A based on the stock price of the product and the market data including the price of the product up to time τ _j, and the finance issued by company j (j = 0 is default free) not defaulting in risk horizon T For products, domestic default free interest rate r (t), hazard rate h ^ _j (t) of domestic company j, and stock price S of domestic company j
_j (t), default free interest rate r of overseas partner country F
_f (t), hazard rate of company j in country F h ^ _{f, j} (t), F
Stock price S _{f, j} (t) of company j in the country and exchange rate between home country and country F of the country (home currency conversion value of one unit of country currency) V
The fluctuation of (t) after the risk horizon T under the risk neutral probability P ^ or the forward neutral probability P ^s ,
The stochastic differential equation Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And μ ₀ (t), μ _j (t), μ _{f, 0} (t) and μ _{f, j} (t)
Are drifts of r (t), h _j (t), r _f (t), and h _{f, j} (t), σ ₀ (t), σ _j (t), σ _{s, j} (t), σ _{f, 0} (t), σ _{f, j} (t), σ _{fs, j} (t) and σ _V
(t) is r (t), h _j (t), S _j (t), r
Volatility of _f (t), h _{f, j} (t), S _{f, j} (t) and V (t), β ₀ (t), β _{n + j} (t), β _{f, 0} (t), β _{f , Nf + j} (t) and β _V (t) are z ₀ (t) and z, respectively.
_{n + j} (t), z _{f, 0} (t), z _{f, nf + j} (t), z _V (t) market price of risk, ξ _j (t), ξ _{f, j} (t) are risk premium adjustment rates , Ρ _jk (t), ρ ^ff _jk (t), ρ ^f _jk (t), ρ _jV (t), and ρ ^f _jV (t) are correlation coefficients, and all are t and these random variables and all companies. The cumulative cash flow X (t) of the product until time t is r (s) (T ≦ s) and h ^. _j (s)
The above random variables including (T ≦ s) and the default of company j /
The risk horizon T of the product is expressed by using the non-default state and the financial product attribute data including the maturity date, the interest payment date, the coupon rate, and the recovery rate.
Price p ^nondef (T) at (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^), or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information until the time T is known under the forward neutral probability P ^s , v ₀ (T, T _i ^j ) is the maturity T _i. The price of the default-free discount bond of ^j at time T)) is used to directly or analytically calculate the above-mentioned expected value by using the joint distribution of random variables obtained by analytical calculation from the above stochastic differential equation. ^And processing means E for calculating the price p ^nondef (T)
The integrated evaluation system for market risk and credit risk according to claim 4, comprising: 8. The future situation scenario generating means includes domestic default / free interest rate r (t), hazard rate h _j (t) of domestic company j, stock price S _j (t) of domestic company j, and Default free interest rate r _f (t) of country F, hazard rate h _{f, j} (t) of company j of country F, stock price S of company j of country F
_{f, j} (t) and the following stochastic differential equations that describe future fluctuations of the exchange rate between the home country and the F country (country currency conversion value of the home country currency unit) V (t) And a ₀ , a _j , a _{f, 0} , a _{f, j} , σ ₀ , σ _j , σ _{f, 0} , σ _{f, j} , σ _{s, j} , σ _{fs, j} , σ _V are non-negative Constants, μ _{s, j} , μ _{fs, j} , μ _V are constants, b ₀ (t), b _j (t), b _{f, 0} (t), b _{f, j} (t) are deterministic at time t , Ρ _jk , ρ ^ff _jk , ρ ^f _jk , ρ _jV , and ρ ^f _jV are constants), the sample path from the current time t to the risk horizon T (T> t) under the observation probability is Generated, r (s), h _j (s), S _j (s), r _f (s), and h _{f
, J} (s) and S _{f, j} (s) and V (s), t ≦ s ≦ T,
When the default time point of the company j that has not defaulted at time t is τ _j , the risk
Survival probability P _T {τ _j > T} under the observation probability of company j in Horizon T Probability of defaulting by risk horizon T based on 2. The integrated evaluation system for market risk and credit risk according to claim 1, further comprising: a second processing unit that determines a default / non-default state of the company j in the risk horizon T for each sample path. 9. The risk horizon asset price calculation means for calculating the portfolio price and individual asset price in the risk horizon for the sample path generated by the future situation scenario generation means is a risk horizon For financial instruments issued by company j that has already defaulted at T, the price p ^def (T) at risk horizon T is calculated as the default time τ _{j of} company j, the maturity date and the interest payment date, and the coupon rate and recovery. Financial product attribute / rating data including rate and company j
First processing means to be obtained based on the stock price of the product and the market data including the price of the product up to the time τ _j , and the issuance of a company j (j = 0 is default free) not defaulting in the risk horizon T For domestic financial products, domestic default free interest rate r (t), hazard rate h ^ _j (t) of domestic company j, stock price S of domestic company j
_j (t), default free interest rate r of overseas partner country F
_f (t), hazard rate of company j in country F h ^ _{f, j} (t), F
Stock price S _{f, j} (t) of company j in the country and exchange rate between home country and country F of the country (home currency conversion value of one unit of country currency) V
The fluctuation of (t) after the risk horizon T under the risk neutral probability P ^ or the forward neutral probability P ^s ,
The stochastic differential equation Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And a ₀ , a _j , a _{f, 0} , a _{f, j} , σ ₀ , σ _j , σ _{f, 0} , σ _{f, j} , σ _{s, j} , σ _{fs, j} , σ _V are non-negative Constants, μ _{s, j} , μ _{fs, j} , μ _V are constants, b ₀ (t), b _j (t), b _{f, 0} (t), b _{f, j} (t) are deterministic at time t , Β ₀ (t), β _{n + j} (t), β _{f, 0} (t), β _{f, nf + j} (t), and β _V (t) are z ₀ (t) and z, respectively.
_{n + j (t), z} f, 0 (t), z f, nf + j (t), deterministic function of time t in the market price of the risk of _{z V (t), ξ j} (t), ξ f, j (t) is a risk premium adjustment rate and is based on a deterministic function at time t, ρ _jk , ρ ^ff _jk , ρ ^f _jk , ρ _jV , ρ ^f _jV are correlation coefficients and are constants from -1 to 1) It is assumed that the cumulative cash flow X (t) of the financial instrument until time t is r (s) (T ≦ s) and h ^ _j (s) (T ≦
s), the default variable / non-default state of the company j, and the financial instrument attribute data including the maturity date, the interest payment date, the coupon rate, and the recovery rate. Price p ^nondef (T) in risk horizon T is a risk neutral evaluation method (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^) or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information up to time T under the forward neutral probability P ^s is known, and v ₀ (T, T _i ^j ) is the maturity T. the second to be calculated on the basis of the _i default-free discount bond price at time T of the ^j)
9. The integrated evaluation system for market risk and credit risk according to claim 8, further comprising: 10. The risk horizon asset price calculation means for calculating the portfolio price and individual asset price in the risk horizon for the sample path generated by the future situation scenario generation means is a risk horizon For financial instruments issued by company j that has already defaulted at T, the price p ^def (T) at risk horizon T is calculated as the default time τ _{j of} company j, the maturity date and the interest payment date, and the coupon rate and recovery. Financial product attribute / rating data including rate and company j
Processing means A based on the stock price of the product and the market data including the price of the product up to time τ _j, and the finance issued by company j (j = 0 is default free) not defaulting in risk horizon T For products, domestic default free interest rate r (t), hazard rate h ^ _j (t) of domestic company j, and stock price S of domestic company j
_j (t), default free interest rate r of overseas partner country F
_f (t), hazard rate of company j in country F h ^ _{f, j} (t), F
Stock price S _{f, j} (t) of company j in the country and exchange rate between home country and country F of the country (home currency conversion value of one unit of country currency) V
The fluctuation of (t) after the risk horizon T under the risk neutral probability P ^ or the forward neutral probability P ^s ,
The stochastic differential equation Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And a ₀ , a _j , a _{f, 0} , a _{f, j} , σ ₀ , σ _j , σ _{f, 0} , σ _{f, j} , σ _{s, j} , σ _{fs, j} , σ _V are non-negative Constants, μ _{s, j} , μ _{fs, j} , μ _V are constants, b ₀ (t), b _j (t), b _{f, 0} (t), b _{f, j} (t) are deterministic at time t , Β ₀ (t), β _{n + j} (t), β _{f, 0} (t), β _{f, nf + j} (t), and β _V (t) are z ₀ (t) and z, respectively.
_{n + j (t), z} f, 0 (t), z f, nf + j (t), deterministic function of time t in the market price of the risk of _{z V (t), ξ j} (t), ξ f, j (t) is a risk premium adjustment rate and is based on a deterministic function at time t, ρ _jk , ρ ^ff _jk , ρ ^f _jk , ρ _jV , ρ ^f _jV are correlation coefficients and are constants from -1 to 1) Assuming that the processing means B that generates a large number of sample paths and the default time point of the company j that is not defaulted in the risk horizon T are τ _j , the company j at a certain future time point S, S> T for each sample path. Of the risk-neutral probability P ^ or the survival probability P ^ {τ _j > S} under the forward-neutral probability P ^s of Probability of defaulting by time S based on According to the processing means C for determining the default / non-default state of the company j at time S for each sample path, and the cumulative cash flow X (t) of the product until time t.
, The above random variables including r (s) (T ≦ s) and h ^ _j (s) (T ≦ s), the default / non-default state of the company j, the maturity date and the interest payment date and the coupon rate, and By using the financial product attribute data including the recovery rate and the expression, the price p in the risk horizon T of the product
^nondef (T) is a risk neutral evaluation method (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^), or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information until the time T is known under the forward neutral probability P ^s , v ₀ (T, T _i ^j ) is the maturity T _i. 9. The processing means D for calculating using the sample path generated by the processing means B and the processing means C based on (the price at time T of the default-free discount bond of ^j ). Integrated market and credit risk assessment system described. 11. The risk horizon asset price calculation means for calculating the portfolio price and individual asset price in the risk horizon for the sample path generated by the future situation scenario generation means is a risk horizon For financial instruments issued by company j that has already defaulted at T, the price p ^def (T) at risk horizon T is calculated as the default time τ _{j of} company j, the maturity date and the interest payment date, and the coupon rate and recovery. A financial instrument attribute / rating data including a rate, a processing means A obtained based on the stock price of the company j and market data including the price of the product up to the time τ _j, and a company j not defaulting in the risk horizon T For financial products issued by (j = 0 is default free), domestic default -Free interest rate r (t), hazard rate of domestic companies _j h ^ j (t), stock price S of domestic companies j
_j (t), default free interest rate r of overseas partner country F
_f (t), hazard rate of company j in country F h ^ _{f, j} (t), F
Stock price S _{f, j} (t) of company j in the country and exchange rate between home country and country F of the country (home currency conversion value of one unit of country currency) V
The fluctuation of (t) after the risk horizon T under the risk neutral probability P ^ or the forward neutral probability P ^s ,
The stochastic differential equation Is the (n + n _f +3) -dimensional standard Brownian motion under the risk-neutral probability P ^ or the forward-neutral probability P ^s , and And a ₀ , a _j , a _{f, 0} , a _{f, j} , σ ₀ , σ _j , σ _{f, 0} , σ _{f, j} , σ _{s, j} , σ _{fs, j} , σ _V are non-negative Constants, μ _{s, j} , μ _{fs, j} , μ _V are constants, b ₀ (t), b _j (t), b _{f, 0} (t), b _{f, j} (t) are deterministic at time t , Β ₀ (t), β _{n + j} (t), β _{f, 0} (t), β _{f, nf + j} (t), and β _V (t) are z ₀ (t) and z, respectively.
_{n + j (t), z} f, 0 (t), z f, nf + j (t), deterministic function of time t in the market price of the risk of _{z V (t), ξ j} (t), ξ f, j (t) is a risk premium adjustment rate and is based on a deterministic function at time t, ρ _jk , ρ ^ff _jk , ρ ^f _jk , ρ _jV , ρ ^f _jV are correlation coefficients and are constants from -1 to 1) It is assumed that the cumulative cash flow X (t) of the product up to the time t is generated by the above-mentioned random variable including r (s) (T ≦ s) and h ^ _j (s) (T ≦ s), and the company By using the default / non-default state of j and the financial instrument attribute data including the maturity date and the interest payment date and the coupon rate and the recovery rate, the price p ^nondef (T ) Is the risk-neutral evaluation method (However, E _T ^ [] is a conditional expected value operator when the information up to time T is known under the risk-neutral probability P ^), or the forward neutral evaluation method (However, E _T ^s [] is a conditional expected value operator when the information until the time T is known under the forward neutral probability P ^s , v ₀ (T, T _i ^j ) is the maturity T _i. The price of the default-free discount bond of ^j at time T)) is used to directly or analytically calculate the above-mentioned expected value by using the joint distribution of random variables obtained by analytical calculation from the above stochastic differential equation. ^And processing means E for calculating the price p ^nondef (T)
The integrated evaluation system for market risk and credit risk according to claim 8, comprising: 12. A scenario, which depicts the temporal change in interest rate fluctuations from the present to the risk horizon, based on a default free interest rate process model expressed by an equation that describes future fluctuations in default free interest rates, and defaults. Based on the default process model expressed by the equation that describes the future fluctuation of the occurrence rate, the scenario that depicts the time-dependent change of the default occurrence situation from the present to the risk horizon and the future fluctuation of the stock price fluctuation are described. Based on the stock price fluctuation process model expressed by the equation, there is a scenario that depicts the temporal change of stock price fluctuation from the present to the risk horizon and the exchange rate fluctuation model expressed by the equation that describes the future fluctuation of the exchange rate. Based on the current scenario to the risk horizon, , A first step of generating a large number based on input or stored portfolio data, financial instrument attribute / rating data, market data, and / or default data, and individual steps generated by the first step. Introducing the risk premium adjustment rate to the scenario, the second step of calculating the portfolio price and individual asset price in the risk horizon, and the risk horizon calculated for each scenario by the second step. A third step of obtaining a future price distribution of the portfolio and / or a future price distribution of the individual asset based on the price of the portfolio and the individual asset price in The price distribution of the portfolio and / or the future of individual assets Based on the rated distribution, and evaluating by integrating the credit risk and market risk, integrated method for evaluating the credit risk and market risk.