JP4430854B2 - Price calculation device and program for financial products or insurance products using copula functions - Google Patents

Description

【００５９】
コピュラ関数は、それを介して、多変量分布を1変量の周辺分布と多変量の相互依存構造とに分解して考えることが出来るようになる点が優れている。ここで言う相互依存関係は、通常良く使われる相関係数ではなく、本発明では、ケンダールの順位相関（発生順位相関や信用順位相関）と呼ばれるもので表現する。本発明では、コピュラ関数は、該被保障参照体ごとにコンティンジェンシー発生の判定の基準となる閾値を表す一次元一様分布に従う確率変数である発生閾値確率変数から、複数の被保障参照体を同時に計算する場合の、それぞれ対応するコンティンジェンシー発生の基準となる閾値を表す多次元一様分布に従う確率変数である発生閾値多次元確率変数に変換する。特に信用リスク保障の場合には、債務者ごとにデフォルトの判定の基準となる閾値を表す一次元一様分布に従う確率変数であるデフォルト閾値確率変数から、複数の債務者を同時に計算（考慮）する場合のデフォルトの判定の基準となる閾値を表す多次元一様分布に従う確率変数であるデフォルト閾値多次元確率変数に変換する。この結果、本発明では、まとめて取り扱われるすべてのリスク参照単位の相互依存関係や、バスケットを構成するすべての債務者の信用力の相互依存関係を表現することができ、３つ以上のリスク参照単位の相関関係を考慮した金融保険商品や、３者以上の債務者の信用力の相関関係を考慮したバスケットタイプ金融商品の価格計算を簡便に計算できるようになる。
【００６０】
損失率とは、コンティンジェンシーの事象が発生した時に金融商品もしくは保険商品の購入者に売却者から支払われる金額を、商品ごとに契約時に設定される想定元本もしくは保険金額（または支払限度額）で割った値である。例えばコンティンジェンシーとして保険リスクを考える場合の損失率とは、１事故当たりの損害率であり、例えばコンティンジェンシーとして信用リスクを考える場合の損失率とは、貸出債権や社債などが債務不履行に陥った後、元本のうち回収できなかった割合のことである。特に信用リスクを考える場合、（1−損失率）を回収率と言う。特に金融商品の回収率とは、金融商品を発行した債務者がデフォルトになったときに金融商品の購入者に支払われる金額を、金融商品の発行時に履行すると契約した債務額によって割った値である。
【００６１】
順位相関とは、順位の差や順位の大小関係等の順位に関する相関を表す統計量である。代表的な順位相関として、ケンダールの順位相関とスピアマンの順位相関が挙げられる。
【００６２】
発生順位相関係数とは、二つのリスク参照単位間の、コンティンジェンシーの起こりやすさの相互依存度合いを順位相関という相関係数の形で表した数値である。本発明では、順位相関の一態様として、ケンダールの順位相関を利用している。しかし、二つの発生強度の相関を表す順位相関であれば、ケンダールの順位相関に限定しなくてもよい。例えば、スピアマンの順位相関なども考えられる。本発明で用いられるケンダールの順位相関は、二つのリスク参照単位の発生強度の依存関係を示したものである。二つのリスク参照単位に対する二つの発生強度の確率変数があり、一方が大きな値をとるとき他方も大きな値をとる場合、互いの依存関係は強いと言える。逆に、発生強度の確率変数の一方が大きい値をとるとき他方が小さい値をとる場合、互いの依存関係は弱いと言える。本発明の発生順位相関係数は、リスク参照単位間の発生強度の依存関係の強弱を数値で表したものである。発生強度に関する二つの時系列データＸ_tとＹ_tを考え、時間が経つにつれてＸ_tが増加し、かつＹ_tも増加する確率Ｐ_up ^up、Ｘ_tが減少し、かつＹ_tも減少する確率Ｐ_down ^down、逆に、Ｘ_tが増加し、かつＹ_tが減少する確率Ｐ_up ^down、Ｘ_tが減少し、かつＹ_tが増加する確率Ｐ_down ^up、Ｘ_tが増加または減少し、かつＹ_tが増加も減少もしない確率Ｐ_tie、Ｙ_tが増加または減少し、かつＸ_tが増加も減少もしない確率Ｐ^tieを使い、本発明で用いられる信用力順位相関として、ケンダールの順位相関およびスピアマンの順位相関は数式２のように定義される。
【数２】

【００６３】
信用力順位相関（または信用力順位相関係数）とは、発生順位相関係数を、信用リスクを保障する金融商品の場合の呼称である。つまり、２者の債務者間の信用力を順位相関係数という形で表したものである。この場合も、２者の債務者間の信用力の相関を表す順位相関であれば、ケンダールの順位相関に限定されず、例えば、スピアマンの順位相関なども考えられる。
【００６４】
発生順位相関件数や信用力順位相関に掛けられる定数とは、発生順位相関（あるいは信用力順位相関；ケンダールの順位相関）と、発生相関パラメータ（あるいは、信用力相関パラメータ；θ_ij）との関係を示す定数である。数学的な証明により、数式３のような発生順位相関（あるいは信用力順位相関；ケンダールの順位相関）と、（あるいは、信用力相関パラメータ；θ_ij）との定数（２／９）を介する関係が知られている。この関係の証明は、「An Introduction to Copulas(1999), Roger B.Nelsen, Lecture Notes in Statistics 139, Springer.」の文献の129ページの例５．２において説明されている。
【数３】

【００６５】
本発明の複数のリスク参照単位を対象とする金融商品や保険商品の価格計算方法や、バスケットタイプ金融商品の価格計算方法によると、従来のリスク参照単位間の発生強度の相関や債務者間の信用力の相関を考慮していないときと比べて、対象となるリスク参照単位やバスケットに組み込まれる債務（債権）を負う債務者が３以上となる場合であっても、これらの間の発生強度や信用力の変化の相関を考慮しているため、これら金融商品や、保険商品、バスケットデフォルトスワップなどのバスケットタイプ金融商品の価格(プレミアム）の計算を合理的にできるようになる。
【００６６】
以上のように提供される本発明において、それぞれの特徴をバスケットタイプ金融商品の分野におけるより具体的な概念に適用して考えることが出来る。具体的には、「被保障参照体とコンティンジェンシーとの組からなるリスク参照単位」を「担保資産（または参照資産）」と、「発生強度」を「デフォルト強度」と、「発生相関パラメータ」を「信用力相関パラメータ」と、「発生相関強度」を「信用力相関デフォルト強度」と、「発生閾値確率変数」を「デフォルト閾値確率変数」と、「発生閾値多次元確率変数」を「デフォルト閾値多次元確率変数」と、「複数リスクを対象とした商品」を「バスケットタイプ金融商品」と、「履歴データ」を「市場データ」と限定することにより、本発明の方法を用いてバスケットタイプ金融商品における価格の計算を行なうことが可能となる。
【００６７】
つまり、本発明においては、バスケットタイプ金融商品の担保資産となる金融商品を発行する債務者に関する、予め記憶手段に保存されている市場データに基づいて、該債務者のデフォルトの起こりやすさを示すデフォルト強度を演算手段が計算するデフォルト強度計算ステップと、該市場データに基づいて、該債務者間の信用力の相関を示す信用力相関パラメータを演算手段が計算する信用力相関パラメータ計算ステップと、該債務者ごとにデフォルトの判定の基準となる閾値を表す一次元一様分布に従う確率変数であるデフォルト閾値確率変数から、複数の債務者を同時に計算する場合のデフォルトの判定の基準となる閾値を表す多次元一様分布に従う確率変数であるデフォルト閾値多次元確率変数に変換するコピュラ関数モデルを用いて、該信用力相関パラメータおよび該債務者ごとのデフォルト強度に基づいて、該債務者間の信用力の相関を用いる信用力相関デフォルト強度を演算手段が計算する信用力相関デフォルト強度計算ステップと、該信用力相関デフォルト強度と、予め記憶手段に保存されている債務者のデフォルト時における該金融商品の回収率とに基づいて、バスケットタイプ金融商品の価格を演算手段が計算する価格計算ステップとを含むバスケットタイプ金融商品の価格計算方法も実現可能である。
【００６８】
また、本発明によって、バスケットタイプ金融商品の担保資産となる金融商品を発行する債務者に関する市場データをコンピュータが受け付ける市場データ受付ステップと、該債務者のデフォルト時における該金融商品の回収率をコンピュータが受け付ける回収率受付ステップと、該市場データに基づいて、該債務者のデフォルトの起こりやすさを示すデフォルト強度を演算手段が計算するデフォルト強度計算ステップと、該市場データに基づいて、該債務者間の信用力の相関を示す信用力相関パラメータを演算手段が計算する信用力相関パラメータ計算ステップと、該債務者ごとにデフォルトの判定の基準となる閾値を表す一次元一様分布に従う確率変数であるデフォルト閾値確率変数から、複数の債務者を同時に計算する場合のデフォルトの判定の基準となる閾値を表す多次元一様分布に従う確率変数であるデフォルト閾値多次元確率変数に変換するコピュラ関数モデルを用いて、該信用力相関パラメータおよび該債務者ごとのデフォルト強度に基づいて、該債務者間の信用力の相関を用いる信用力相関デフォルト強度を演算手段が計算する信用力相関デフォルト強度計算ステップと、該信用力相関デフォルト強度と、該回収率とに基づいて、バスケットタイプ金融商品の価格を演算手段が計算する価格計算ステップとを含むバスケットタイプ金融商品の価格計算方法が実現可能である。
【００６９】
さらに、本発明によって、前記コピュラ関数モデルが、前記債務者ごとに前記デフォルト閾値確率変数の全ての積を演算手段が計算する第１ステップと、前記債務者中から第１債務者および第２債務者を演算手段が選択する第２ステップと、選択された第１債務者および第２債務者のそれぞれの前記デフォルト閾値確率変数の積を演算手段が計算する第３ステップと、第１債務者および第２債務者のそれぞれの前記デフォルト閾値確率変数の該積と、第１債務者および第２債務者の信用力相関パラメータとの積を演算手段が計算する第４ステップと、該第３ステップおよび該第４ステップを、前記債務者中の全ての債務者の組み合せについて演算手段が計算し、その全ての債務者の組み合せの積と信用力相関パラメータとの積の和を演算手段が計算する第５ステップと、該第１ステップによって計算された値と、該第５ステップによって計算された値との積を演算手段が計算して、複数の債務者の前記デフォルト閾値多次元確率変数を演算手段が計算する第６ステップとを含むことを特徴とするバスケットタイプ金融商品の価格計算方法が実現可能である。
【００７０】
加えて、本発明によって、前記信用力相関パラメータ計算ステップが、前記債務者中から、第１債務者と第２債務者の２者を選択するステップと、前記市場データに基づいて、該第1債務者の信用力が増加し、かつ該第２債務者の信用力が増加する第１確率を演算手段が計算するステップと、前記市場データに基づいて、該第1債務者の信用力が減少し、かつ該第２債務者の信用力が減少する第２確率を演算手段が計算するステップと、前記市場データに基づいて、該第1債務者の信用力が増加し、かつ該第２債務者の信用力が減少する第３確率を演算手段が計算するステップと、前記市場データに基づいて、該第1債務者の信用力が減少し、かつ該第２債務者の信用力が増加する第４確率を演算手段が計算するステップと、前記市場データに基づいて、該第１確率および該第２確率の和から、該第３確率および該第４確率の和を引いて信用力順位相関を演算手段が計算するステップと、該信用力順位相関に対して定数を掛け、前記信用力相関パラメータを演算手段が計算するステップとを含むバスケットタイプ金融商品の価格計算方法が実現可能である。
【００７１】
さらに加えて、本発明によって、前記価格計算ステップが、前記信用力相関デフォルト強度に基づいて、バスケットタイプ金融商品の担保資産となる金融商品のうち、該バスケットタイプ金融商品の満期日までに最初のデフォルトが起こる確率を演算手段が計算するステップと、前記回収率から最初のデフォルトによって回収できない非回収率を演算手段が計算するステップと、該最初のデフォルトが起こる確率と、非回収率との積の現在価値を計算し、バスケットタイプ金融商品の価格を演算手段が計算するステップとを含むバスケットタイプ金融商品の価格計算方法が実現可能である。
【００７２】
本発明によれば、シミュレーションなどの計算量が膨大になる手法を用いることがなく、解析的な計算のみを行なうので、計算時間を短縮することができる。これは、価格の再計算値を頻繁に行なう必要があるバスケットタイプ金融商品のような金融商品には大変重要である。
【００７３】
上記したバスケットタイプ金融商品の価格計算方法は、損害保険分野にも適用することができる。上述のバスケットタイプ金融商品の価格に相当するのが純保険料である。個別債務者のデフォルト強度に当たるのが、事故（コンティンジェンシー）の起こる頻度を表すパラメータである。また、例えば、複数の工場を所有する企業に対する火災保険を考える場合、各債務者間の信用力の相関を示す信用力相関パラメータに該当するのが、それらの工場で火災が起こるかどうかの相互依存関係となる。そして、債権の回収率に対応するのが、（１−損害率）となる。また、債務者に対応するのは、損害保険の分野では、建物、自動車といった保険契約によって保障される対象である被保険対象となる。従って、バスケットデフォルトスワップの価格付けの論理的枠組みは被保険対象（リスク参照単位）が複数存在し、かつ、それらの対象についての事故が起こる頻度が独立とはいえない場合に適応することが可能となる。そして、このような損害保険の純保険料の計算を同じ論理的枠組みの中で行なうことができる。本発明は、発明者のこのような金融商品と保険商品の関係についての知見に基づき、具体性を有する範囲においてこれらを包含する概念を提供するものである。
【００７４】
入力手段とは、キーボードやマウスなどのオペレータの操作による入力手段のほか、フレキシブルディスクなどの記録媒体による入力、あるいはインターネットを含む電気通信回線やＬＡＮ（ローカルエリアネットワーク）などの電気信号による入力を受け付ける手段を含む。
【００７５】
記憶手段とは、文字や数字などのデータや、プログラムなどを記憶するための手段である。例えば、コンピュータ内のランダムアクセスメモリ、ハードディスクなどである。
【００７６】
演算手段とは、プログラムを実行するための任意の計算手段のことである。例えば、一般に中央処理装置すなわちＣＰＵといわれているものや、特定の計算に専用設計等さえれたＤＳＰ（デジタル信号処理）手段等であるがこれらに限定されるものではない。
【００７７】
コンピュータとは、上記のような入力手段と、記憶手段と、演算手段とを含む電子計算機をいう。コンピュータは、本発明の目的に合わせて適当な手段として自身を作用させる為の命令を実行し、入力手段からデータを受け付け、記憶手段からデータを呼び出し、演算処理を行う。さらにコンピュータは、その結果に応じて、計算結果を表示したり、記憶手段に記録したり、他のコンピュータに自ら送信したり、他のコンピュータの求めに応じて送信したり、他の装置を制御したりする。必要に応じて適当なネットワークや通信手段によって他の装置と通信を行なうものであっても良い。例えば、上記の命令の実行に当たり、コンピュータの演算手段が本発明の発生強度を計算する場合には、本発明の発生強度計算ステップをコンピュータの演算装置が実行することによって、コンピュータが発生強度計算手段として動作する。つまり、コンピュータが発生強度計算手段を含むものとなる。この例のように、コンピュータのいずれかの手段の動作として記載されている本発明の任意のステップは、コンピュータを構成する適当な手段が当該動作を実行することによって、コンピュータがその動作の手段を含むものとして動作可能なことを意味する。つまり、コンピュータが当該動作手段を含んで構成されているものとする。
【００７８】
コンピュータが読み取り可能な記録媒体とは、プログラムやデータを記録し、コンピュータが読み取り可能な媒体のことである。例えば、フレキシブルディスク、ＣＤ−ＲＯＭ、ＭＯ、ＲＯＭチップなどのことである。
【００７９】
【発明の実施の形態】
以下に、本発明の金融商品や保険商品の価格計算方法の実施の形態を図を用いながら詳細に説明する。以後の本実施形態の説明に当たり、被保障参照体として債務者を想定し、コンティンジェンシーとして債務不履行（デフォルト）を想定する具体例（即ち、リスク参照体として、バスケットタイプ金融商品であるクレジットデリバティブ）の債務発生順位相関係数を信用リスクを保障する金融商品を用いて、より具体的な例によって説明し、必要に応じて一般の場合の特徴や要素の名称を記載する。本発明をこの例のみに限定するものでない。
【００８０】
図１は、バスケットデフォルトスワップのスキームを示す概念図である。本実施の形態では、バスケットタイプ金融商品（クレジットデリバティブ）の一例として、バスケットデフォルトスワップ（ＢＤＳ）を利用し、説明する。バスケットデフォルトスワップとは、図１に示したバスケットデフォルトスワップのスキーム１中の「ＢＤＳの買い手２」と「ＢＤＳの売り手３」の２者間で取り交わされる契約のことをいう。
【００８１】
この契約の内容は、バスケットデフォルトスワップの参照資産（担保資産、被保障参照体）となる複数の債券（以下、「バスケット」とも呼ぶ）の保有者（以下、「ＢＤＳの買い手２」とも呼ぶ）が、ＢＤＳの売り手３に対して定期的にスワップ取引の対価であるプレミアムを支払う代りに、ＢＤＳの契約終了時である満期までの間に、バスケット内からデフォルト（コンティンジェンシー）が起きた場合、デフォルトした債券（リスク参照単位）からＢＤＳの買い手２が回収できなかった額だけＢＤＳの売り手３から保障されるというものである。さらに、一般的なＢＤＳは、バスケット内の最初のデフォルト発生時点において、ＢＤＳの契約は終了し、ＢＤＳの買い手２から売り手３へ支払われるプレミアムの支払も停止することが多い。例外的に、参照資産の保有者とＢＤＳの買い手が一致しない場合も有り得る。
【００８２】
図１に示すように、実線矢印はバスケット内の最初のデフォルト発生以前のキャッシュフローであり、点線矢印は、バスケット内の最初のデフォルト発生後のキャッシュフローである。最初のデフォルト発生前では、参照資産（担保資産）となる複数の債券のバスケットを発行する債務者からクーポンの総額を、債権の保有者（ＢＤＳの買い手２）が受け取る。ＢＤＳの買い手２から、ＢＤＳの売り手３に対して、プレミアムを支払う。最初のデフォルト後、デフォルトした債券を発行する債務者からデフォルトした債券の回収可能額を受け取る。なお、回収可能額は額面と回収率との積で表される。次に、ＢＤＳの売り手３からＢＤＳの買い手２は、デフォルトした債券の回収不能額を受け取る。なお、デフォルトした債券の回収不能額は額面と（１−回収率）との積で表される。本発明では、（１−回収率）を非回収率と呼ぶものとする。
【００８３】
本発明のバスケットタイプ金融商品の価格計算方法では、バスケットの参照資産（担保資産）となる金融商品を発行する債務者のデフォルト事象の生起（コンティンジェンシーの生起）はポアソン分布に従う確率変数によって代替する。すなわち、複合ポアソン過程の最初の飛躍時刻がデフォルト時刻を表す確率変数とする。デフォルトを表すポアソン過程の飛躍の起こる頻度を示すパラメーターをデフォルト強度（λ_t ⁱ 、発生強度）と呼び、デフォルト強度自身が確率変動することによって、各債務者の信用力の変化の将来の不確実性（リスク）を表現することになる。本発明のデフォルト強度（λ_t ⁱ ）が大きいと、デフォルトが起こりやすい（信用力が低い、コンティンジェンシーの発生強度が大きい）ことを示し、デフォルト強度（λ_t ⁱ ）が小さいと、デフォルトが起こりにくい（信用力が高い）ことを示す。
【００８４】
本発明では、デフォルト時刻の確率的表現として、一様分布に従う確率変数とデフォルト強度（λ_t ⁱ ）の関数との大小関係により示し、数式４として表す。数式４中のＵ_iは[０，１]区間の一様分布に従う確率変数であり、これを債務者ｉのデフォルト閾値確率変数と呼ぶ。（Ｕ₁，Ｕ₂，・・・，Ｕ_i，・・・，Ｕ_N）をベクトルで定めたものをデフォルト閾値多次元確率変数と呼ぶ。
【数４】

【００８５】
バスケットを構成する複数の債務者の、異なるデフォルト時刻を考えるので、債務者ごとにデフォルト強度（λ_t ⁱ ）を考え、債務者ごとに一様分布に従う確率変数を考える。債務者ごとに考えた、一様分布に従う確率変数の間に信用力順位相関（ケンダールの順位相関）を持たせることにより、債務者間の信用力変化の依存構造を表現する。
【００８６】
さらに、本発明では、一様分布に従う1次元の確率変数と多次元の一様分布に従うコピュラ関数を定めることにより、債務者が複数の場合の信用力変化の相互依存関係を具体的に表現する。
【００８７】
従来用いられているようなバスケットを構成するすべての債務者の信用力変化の相互依存関係を考慮しない場合は、すべての債務者の信用力変化は互いに独立であり、この場合の個別債務者（個別のリスク参照単位）のデフォルト強度（λ_t ⁱ ）は、バスケットを構成するすべての債務者の信用力変化の相互依存関係を考慮する場合の信用力相関デフォルト強度（Λ）とは異なる。
【００８８】
前述の数式４のようなデフォルト時刻の確率的表現と同等の表現として次の様なものがある。現在までデフォルトしていないという条件のもとで、ある未来の時刻までデフォルトが起こらない確率は、強度に関する期待値を計算することによって数式５のように表現することができる。
【数５】

【００８９】
債務者間の信用力変化の相関を考慮しない場合、すなわち債務者の信用力は互いに独立であると仮定して良い場合には、債務者ｉの強度をλ_t ⁱとすると、時刻ｔまで、バスケットの中の一つの債務者もデフォルトしない確率（一つのリスク参照単位もコンティンジェンシーに遭遇しない確率）は、数式６のように示される。
【数６】

【００９０】
しかし、数式６とちがって現実の経済は、複数の債務者の信用力は互いに独立と仮定できない場合が一般的であり、その場合には数式６は成り立たない。そこで、コピュラ関数を利用することにより、本発明では数式６に相当する関係式を導く。本発明では、コピュラ関数を変形することにより、バスケットを構成するすべての債務者の信用力変化の相互依存関係を考慮する場合の各債務者のデフォルト強度（λ_t ⁱ ）を表現する。
【００９１】
現実の経済事象を鑑み、鋭意研究の結果、本発明者は、Ｎ者のバスケットを構成するすべての債務者の信用力の相互依存関係を表現するコピュラ（Copula）関数Ｃとして、数式７を提案する。
【数７】

【００９２】
数式７の関数形を定めるために必要なパラメーターは信用力相関パラメータθ_ijであり、これはバスケットを構成する債務者ｉと債務者ｊの信用力変化の依存度合い（信用力の相関）から推定されるものである。そこで信用力相関パラメーターθ_ijを演算手段によって計算するために、バスケットを構成するすべての債務者の中から異なる２者を選び出し、その２者の間の信用力の依存度合いを表現する信用力順位相関（ケンダールの順位相関）を市場データに基づいて計測することにより、該当する信用力相関パラメーターθ_ijを決定する。
【００９３】
数式８によって示される信用力順位相関（ケンダールの順位相関）とは、債務者ｉの信用力が上昇したときに、債務者ｊの信用力が上昇したのか下降したのかを調べ、その関連性を数値化したものであり、パラメーターθ_ijとは数式９のような関係にある。数学的な証明は、「An Introduction to Copulas(1999), Roger B.Nelsen, Lecture Notes in Statistics 139, Springer.」の文献の129ページの例５．２において説明されている。ここで、数式９における信用力順位相関（ケンダールの順位相関）と、信用力相関パラメータ（θ_ij）との関係を示す定数（２／９）を、本発明では信用力順位相関の定数と呼ぶこととする。
【数８】

【数９】

【００９４】
なお、数式８によって示される信用力順位相関は、債務者間の信用力の依存関係の強弱を数値で表したものである。市場データから得られる信用力に関する二つの時系列データＸ_tとＹ_tを考え、時間が経つにつれてＸ_tが増加し、かつＹ_tも増加する確率Ｐ_up ^up、Ｘ_tが減少し、かつＹ_tも減少する確率Ｐ_down ^down、逆に、Ｘ_tが増加し、かつＹ_tが減少する確率Ｐ_up ^down、Ｘ_tが減少し、かつＹ_tが増加する確率Ｐ_down ^up、Ｘ_tが増加または減少し、かつＹ_tが増加も減少もしない確率Ｐ_tie、Ｙ_tが増加または減少し、かつＸ_tが増加も減少もしない確率Ｐ^tieを使い、本発明で用いられる信用力順位相関（ケンダールの順位相関）は数式８のように定義する。市場データから得られる信用力に関する二つの時系列データＸ_tとＹ_tとしては、債務者の発行する債券の利回りを示す利回りデータや、債務者の発行する債券の利回りと、国債の利回りとの差を示すイールド・スプレッドデータ、債務者の信用格付示す債務者信用格付データ、債務者の発行する債券の信用格付を示す債券信用格付データなどを含ませることが考えられるが、その他の市場から得られる信用力に関するデータでもよく、限定されるものではない。これらの信用力に関するデータの例で、信用力の増減を示すために次のようなことが考えられる。信用力が増加するときとは、利回りデータ中の利回りが減少するときや、イールド・スプレッドデータ中の差が減少するとき、債務者信用格付データ中の信用格付が上昇するとき、債券信用格付データ中の信用格付が上昇するときなどであり、この中の一つを用いてもよいし、複数を用いてよいし、その他のときと組み合せてもよい。
【００９５】
信用力が減少するときとは、利回りデータ中の利回りが増加するときや、イールド・スプレッドデータ中の差が増加するとき、債務者信用格付データ中の信用格付が下降するとき、債券信用格付データ中の信用格付が下降するときなどであり、この中の一つを用いてもよいし、複数を用いてよいし、その他のときと組み合せてもよい。
【００９６】
バスケットを構成するすべての債務者の信用力変化の相互依存関係を考慮した後の個別債務者の信用力デフォルト強度（Λ）は、数式７によって示されるコピュラ関数Ｃを変形して求めることができる。この様に求めた順位相関考慮後の信用力相関デフォルト強度（Λ）を使い、バスケット内で最初に起こるデフォルト時刻の分布が分かり、バスケットデフォルトスワップの価格計算を解析的に計算できるようになる。
【００９７】
順位相関考慮後の信用力相関デフォルト強度（Λ）は、数式１０のように示される。
【数１０】

【００９８】
数式１０に示される、信用相関考慮後の個別債務者の信用力相関デフォルト強度を表す数式に、本発明において提案された数式７に示されるコピュラ関数を組み合わせても、解析的な計算を行うことができ、かつ、必要なパラメーターを容易に演算手段によって計算することが出来ることが本発明が優れている点である。
【００９９】
なお、本実施の形態におけるバスケットタイプ金融商品の価格計算方法を、バスケットデフォルトスワップ（ＢＤＳ）の価格計算に利用した例では次の様に行う。
【０１００】
まず、（ｉ）バスケット内の最初のデフォルトが発生するまで定期的（または１回のみ）に払いつづけるプレミアムすべての現在価値と、（ｉｉ）バスケット内の最初のデフォルトが発生した後に支払う保障額の現在価値が等しくなるように、プレミアム（価格）を計算する。ただし、デフォルト発生後の、その債務者の額面に対する回収率（Ｌ）を外生的に与える。（なお、この回収率は統計上得てもよい。また回収率は、すべての債務者に対して一定としてもよいし、債務者ごとに異なる値を設定してもよい。）
【０１０１】
さらに、プレミアムの支払いは1度の前払いと仮定し、プレミアムをｃとする。（なお、プレミアムの支払は定期的に支払うこととしてもよい。）(ｉ)プレミアム支払い側の現在価値は、ｃである。(ｉｉ)デフォルト生起後の保障側の現在価値は、保障額の支払は満期Ｔに行われると仮定し、ＢＤＳの満期をＴとすると、数式１１のように表される。
【数１１】

【０１０２】
ただし、数式１１のΛ_t ⁱは、相関考慮後の債務者ｉの信用力相関デフォルト強度を表す時間の確定的な関数であり、ｒは信用リスクフリーの金利を表す定数である。
【０１０３】
バスケットデフォルトスワップのプレミアムｃは、どの債券も額面価格は同一であり、かつ、どの債券の回収率もＬと仮定すると、数式１２のように示され、演算手段により計算できるようになる。
【数１２】

なお、数式１２は数式１１より導かれる。
【０１０４】
【数１３】

なお、数式１３は数式１０において、すべてのλｕ_iがｕによらず一定値をとると仮定することにより導かれる。
【０１０５】
【数１４】

なお、数式１４は数式７と同じである。
【０１０６】
数式１３と数式１４とのｕ_iに関する関係は、数式１５に示すとおりである。
【数１５】

【０１０７】
次に、数式１２〜１４を用いて計算する本発明に係るバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法の一実施の形態を図２を用いて説明する。
図２は、本発明に係るバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法の一実施の形態である。
【０１０８】
[デフォルト強度（λ_t ⁱ ）の計算]
バスケットタイプ金融商品（クレジットデリバティブ）の担保資産となる金融商品を発行する債務者に関する、予め記憶手段に保存されている市場データに基づいて、債務者のデフォルトの起こりやすさを示すデフォルト強度（λ_t ⁱ）を演算手段が計算する（ステップ１０）。デフォルト強度（λ_t ⁱ）は、担保資産（参照資産）となる金融商品を発行する債務者のデフォルトになりやすさを示すので、デフォルト強度は、定数でもよいし、時間の確定的な関数であっても良いし、確率過程であっても良い。デフォルト強度（λ_t ⁱ ）は、市場データから計算することが望ましい。例えば、担保資産が債券であるとき、債務者の発行する債券と国債の利回りとの差を示すイールド・スプレッドの大きさに基づいて、デフォルト強度を計算してもよい。ここで、 市場データとは、債務者の発行する債券の利回りを示す利回りデータと、債務者の発行する債券の利回りと、国債の利回りとの差を示すイールド・スプレッドデータと、債務者の信用格付示す債務者信用格付データと、債務者の発行する債券の信用格付を示す債券信用格付データとからなるグループから選択される一以上のデータでよいし、株価など、その他の債務者の信用に係るデータでもよい。
【０１０９】
[信用力相関パラメータ（θ_ij）の計算]
予め記憶手段に保存されている市場データに基づいて、数式９と数式８に示される信用力順位相関（ケンダールの順位相関）を用いて、債務者間の信用力の相関を示す信用力相関パラメータ（θ_ij）を演算手段が計算する（ステップ２０）。
なお、信用力相関パラメータ（θ_ij）の演算手段による計算をさらに具体的に示すと、図４のようになる。図４は、数式８に示される計算を具体化したフローチャートである。
【０１１０】
債務者中から、第１債務者ｉと第２債務者ｊの２者を演算手段が選択し２者の債務者の組み合せを作成し（ステップ２１）、第1債務者ｉの信用力が増加し、かつ第２債務者ｊの信用力が増加する第１確率（Ｐ_up ^up）を演算手段が計算する（ステップ２２）。次に、第1債務者ｉの信用力が減少し、かつ第２債務者ｊの信用力が減少する第２確率（Ｐ_down ^down）を演算手段が計算する（ステップ２３）。第1債務者ｉの信用力が増加し、かつ第２債務者ｊの信用力が減少する第３確率（Ｐ_up ^down）を演算手段が計算する（ステップ２４）。第1債務者ｉの信用力が減少し、かつ第２債務者ｊの信用力が増加する第４確率（Ｐ_down ^up）を演算手段が計算する（ステップ２５）。第１債務者の信用力が増加したり減少し、かつ第2債務者の信用力が変化しない第５確率（Ｐ_tie）を計算する（ステップ２６）。第２債務者の信用力が増加したり減少し、かつ第１債務者の信用力が変化しない第６確率（Ｐ^tie）を計算する（ステップ２７）。そして、このように求めた第１確率〜第６確率を用いて、数８の信用力順位相関を計算する（ステップ２８）。なお、第５確率（Ｐ_tie）、第６確率（Ｐ^tie）がともに０である場合には、より簡易に計算が可能となる。つまり、この場合に数式２の分母が１となることにあわせて、ステップ２６〜ステップ２８を、第１確率および該第２確率の和（Ｐ_up ^up＋Ｐ_down ^down）から、第３確率および該第４確率の和（Ｐ_up ^down＋Ｐ_down ^up）を引いて信用力順位相関｛（Ｐ_up ^up＋Ｐ_down ^down）−（Ｐ_up ^down＋Ｐ_down ^up）｝を演算手段が計算するものとしてもよい。何れの場合であっても、さらに、信用力順位相関（数式８）に対して、数式９に基づく定数（９／２）を掛け、信用力相関パラメータ（θ_ij）を演算手段が計算する（ステップ２９）。ステップ２１からステップ２９を、演算手段はすべての債務者の組み合せについて計算し、すべての債務者間の組み合せの信用相関パラメータθ_ijが計算される。
【０１１１】
本発明において、信用力が増加するときとは、利回りデータ中の利回りが減少するときと、イールド・スプレッドデータ中の差が減少するときと、債務者信用格付データ中の信用格付が上昇するときと、債券信用格付データ中の信用格付が上昇するときとからなるグループから選択される一以上のときであってもよいし、他の場合のときを組み合せてもよい。
信用力が減少するときとは、利回りデータ中の利回りが増加するときと、イールド・スプレッドデータ中の差が増加するときと、債務者信用格付データ中の信用格付が下降するときと、債券信用格付データ中の信用格付が下降するときとからなるグループから選択される一以上のときであってもよいし、他の場合のときを組み合せてもよい。
【０１１２】
[信用力相関デフォルト強度（Λ）の計算]
関数モデル（数式１０および数式１４）を用いて、信用力相関パラメータおよび債務者ごとのデフォルト強度（λ_t ⁱ ）に基づいて、債務者間の信用力の相関を用いる信用力相関デフォルト強度（Λ_t ⁱ）を演算手段が計算する（ステップ３０）。
【０１１３】
信用力相関デフォルト強度（Λ_t ⁱ）を演算手段が計算する方法を図５に示されるフローチャートを用いて、数式１２〜数式１４に基づき、より具体的に説明する。デフォルト閾値確率変数Ｕに対応する債務者ごとのｕの積（ｕ₁ｕ₂ｕ₃…ｕ_N）を演算手段が計算する（ステップ３１）。債務者中から第１債務者（ｉ）および第２債務者（ｊ）を演算手段が選択する（ステップ３２）。選択された第１債務者および第２債務者に対応する積（（１−ｕ_i）（１−ｕ_j））を演算手段が計算する（ステップ３２）。第１債務者ｉおよび第２債務者ｊに対応する積（（１−ｕ_i）（１−ｕ_j））と、第１債務者ｉおよび第２債務者ｊの信用力相関パラメータ（θ_ij）との積（θ_ij（１−ｕ_i）（１−ｕ_j））を演算手段が計算する（ステップ３４）。バスケット中の全ての債務者の組み合せ（θ_ij（１−ｕ_i）（１−ｕ_j））について演算手段が計算する（ステップ３５）。全ての債務者の組み合せについて和（Σθ_ij（１−ｕ_i）（１−ｕ_j））を演算手段が計算する（ステップ３６）。コピュラ関数（数式１４）Ｃを演算手段が計算する（ステップ３７）。なお、デフォルト閾値多次元確率変数に対応するのが、数式１４の左辺部分（ｕ₁，ｕ₂，……，ｕ_i，……，ｕ_N）である。コピュラ関数Ｃから、数式１５を介して、数式１３を用いて、信用力相関デフォルト強度（Λ_t ⁱ）を演算手段が計算する。
【０１１４】
[バスケットタイプ金融商品（クレジットデリバティブ）の価格ｃの計算]
信用力相関デフォルト強度（Λ_t ⁱ）と、予め記憶手段に保存されている債務者のデフォルト時における金融商品の回収率（Ｌ）とに基づいて、数式１２を用いて、バスケットタイプ金融商品（クレジットデリバティブ）の価格（プレミアム）ｃを演算手段が計算する（ステップ４０）。
【０１１５】
バスケットタイプ金融商品（クレジットデリバティブ）の価格（プレミアム）ｃを演算手段が計算する方法を図６を用いて、より詳細に説明する。
信用力相関デフォルト強度（Λ_t ⁱ）に基づいて、バスケットタイプ金融商品（クレジットデリバティブ）の担保資産となる金融商品のうち、最初のデフォルトが起こる確率を演算手段が計算する（ステップ４１）。回収率（Ｌ）から最初のデフォルトによって、回収できない非回収率（１−Ｌ）を演算手段が計算する（ステップ４２）。数式１２に基づいて、最初のデフォルトが起こる確率と、非回収率との積の現在価値を計算し、バスケットタイプ金融商品（クレジットデリバティブ）の価格（ｃ）を演算手段が計算する（ステップ４３）。
【０１１６】
[他の実施の形態]
図３は本発明に係るバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法の他の実施の形態を示すフローチャートである。図３によって示されるフローチャートは、市場データと回収率Ｌを予め記憶手段に保存しておくのではなく、コンピュータの入力手段によって入力される点が、図２によって示されるフローチャートと異なる。その他の点は図２に示される形態と同じである。
【０１１７】
つまり、図３によって示される形態では、市場データを入力手段が受付け、記憶手段に保存し（ステップ１００）、回収率Ｌを入力手段が受付け記憶手段に保存する（ステップ１１０）。市場データに基づく、デフォルト強度（λ_t ⁱ ）の計算ステップ１２０は、図２のステップ１０に相当する。市場データに基づく、信用力相関パラメータ（θ_ij）の計算ステップ１３０は図２のステップ２０に相当する。コピュラ関数モデル（Ｃ）を用いて、デフォルト強度（λ_t ⁱ ）および信用力相関パラメータ（θ_ij）に基づく、信用力相関デフォルト強度（Λ_t ⁱ）の計算ステップ１４０は図２のステップ３０に相当する。信用力相関デフォルト強度（Λ_t ⁱ）および回収率（Ｌ）に基づく、バスケットタイプ金融商品（クレジットデリバティブ）の価格計算ステップ１５０は、図２のステップ４０に相当する。
【０１１８】
入力手段とは、キーボードやマウスなどのオペレータの操作による入力手段のほか、フレキシブルディスクなどの記録媒体による入力、あるいはインターネットを含む電気通信回線やＬＡＮ（ローカルエリアネットワーク）などの電気信号による入力を受け付ける手段を広く含む。そして、市場データは、インターネットを含む電気通信回線を媒介して、任意のウェブサイトから入力手段によって入力し、記憶手段に保存してもよい。
【０１１９】
【実施例】
次に、本発明に係るバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法の効用を説明するために実施例を挙げる。本実施例では、債務者数３のバスケットのバスケットデフォルトスワップの価格計算方法を挙げ、従来の債務者間の信用力の相関を考慮しない場合と、本発明の債務者間の信用力の相関（依存度合い）を考慮した場合との差を説明することによって、本発明のバスケットタイプ金融商品（クレジットデリバティブ）の価格計算の効用を認識できるようにする。
【０１２０】
[比較例：従来の債務者間の信用力相関を考慮しない場合]
債務者３者のデフォルト強度（λ）は市場データに基づいて表１に示すものとする。デフォルト生起後の回収率は、どの債務者についても同様にＬ＝0.2(20%)とする。信用リスクフリー金利ｒは、期間構造を考慮せず、５年間一定でｒ＝0.01(1.0%)とする。デフォルトスワップ契約の満期はT=５年とする。
【表１】

【０１２１】
以上の仮定のもと、参照資産が一つの場合のデフォルトスワップのプレミアム（価格）ｃは、数式１２の信用力相関デフォルト強度Λの代わりに、信用力相関を考慮しないλを用いる数式１６を用いて計算される。数式１６を用いた参照資産が一つの場合（バスケットではない場合）のデフォルトスワップのプレミアム（価格）の計算結果は表２のようになる。
【数１６】

【０１２２】
【表２】

【０１２３】
もし、債務者３者が発行する債券の信用リスクを別々にヘッジするのであれば、そのコストは全部で、 0.01879+0.02249+0.02617=0.06745(6.745%) となる。
【０１２４】
さらに、債務者間の信用力の相関を考慮しないとき、これらの３社からなるバスケットを参照資産とするバスケットデフォルトスワップのプレミアムは、数式１６を変形した数式１７によって計算される。
【数１７】

【０１２５】
数式１７に、表１のデフォルト強度λを代入すると、数式１８のようになり、バスケットデフォルトスワップのプレミアムが計算される。
【数１８】

【０１２６】
数式１８によって計算されるプレミアムは0.06549(6.549%)となり、バスケットにしない債務者３社の債券を別々にヘッジしたとき（表２のプレミアムの和を計算した0.06745（6.745%）のとき）と比べ、バスケット内の最初のデフォルトのみの保障となっている分だけ割安であることが分かる。想定元本を100億円とすると、バスケットにしないときのプレミアムに対して、バスケットにしたが債務者間の信用相間を考慮しないときのプレミアムは1960万円安くなる。
【０１２７】
[本発明の例：債務者間の信用相関を考慮する場合]
次に、本発明の実施例として、債務者間の信用相関を考慮する場合のバスケットタイプ金融商品（クレジットデリバティブ）の価格計算をおこなう。
仮定条件は、比較例と同様なものとする。つまり、債務者３者のデフォルト強度（λ）は市場データに基づいて表１に示すものとする。デフォルト生起後の回収率は、どの債務者についても同様にL=0.2 (20%)とする。信用リスクフリー金利ｒは、期間構造を考慮せず、５年間一定でｒ＝0.01(1.0%)とする。デフォルトスワップ契約の満期はT=５年とする。
【０１２８】
次に、債券を発行する債務者３社の信用力変化の相関を表３のように仮定する。
【表３】

この場合のバスケットデフォルトスワップのプレミアムは、相関考慮後の個別債務者の強度Λは、数式１３に数式１４を代入することによって得られる数式１９によって計算される。
【０１２９】
【数１９】

数式１９を、数式１２から得られる数式２０へ代入することにより、債務者間の信用相関を考慮したプレミアムが計算される。
【数２０】

【０１３０】
数式２０を計算すると、ｃ＝0.06447(6.447%)と算出され、相関を考慮しない場合に比べて割安になる。想定元本を100億円とすると、比較例のバスケットにしたが債務者間の信用相間を考慮しないときのプレミアムに比べ、本発明の債務者間の信用力相関を考慮するバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法によるプレミアムの額は1020万円安くなる。
【０１３１】
さらに、信用力相関パラメータθ_ijが表４の場合、バスケットデフォルトスワップのプレミアムは、0.06601(6.601%)となり、相関を考慮しない場合に比べて割高になる。想定元本を100億円とすると、比較例のバスケットにしたが債務者間の信用相間を考慮しないときのプレミアムに比べ、本発明の債務者間の信用力相関を考慮するバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法によるプレミアムの額は520万円高くなる。
【表４】

【０１３２】
以上のように、本発明に係るバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法によると、バスケットに組み込まれる債券を発行する債務者間の信用力の相関を考慮するため、バスケットタイプ金融商品（クレジットデリバティブ）の価格の計算がより合理的に計算されることが分かる。つまり、債務者間の信用力の相関パラメータが正の場合はプレミアム（価格）が高く計算されるのを防ぎ、逆に、債務者間の信用力の相関パラメータが負の場合はプレミアム（価格）が低く計算されるのを防ぐ。
【０１３３】
[さらに他の実施の形態]
また、コピュラ関数モデルを用いて発生相関パラメータおよび該リスク参照単位ごとの発生強度から発生閾値多次元確率変数を生成する形態として、コンピュータプログラムによって生成される乱数（または疑似乱数）から多次元一様分布に従う乱数（または疑似乱数）を生成し、それを使ってモンテカルロシミュレーションにより直接計算を行う手法もある。
【０１３４】
[他の利用分野]
以上のような本発明のバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法は損害保険分野にも適用することができる。上述のバスケットデフォルトスワップの価格に相当するのが純保険料である。個別債務者のデフォルト強度に当たるのが、事故の起こる頻度を表すパラメータである。また、例えば、複数の工場を所有する企業に対する火災保険を考える場合、各債務者間の信用力の相関を示す信用力相関パラメータに該当するのが、それらの工場で火災が起こるかどうかの相互依存関係となる。そして、債権の回収率に対応するのが、（１−損害率）となる。また、債務者に対応するのは、損害保険の分野では、建物、自動車といった保険契約によって保障される対象である被保険対象となる。従って、バスケットデフォルトスワップの価格付けの論理的枠組みは被保険対象が複数存在し、かつ、それらの対象についての事故が起こる頻度が独立とはいえない場合に適応することとなる。そして、このような損害保険の純保険料の計算を同じ論理的枠組みの中で行なうことができる。
【０１３５】
【発明の効果】
上記したところから明らかなように、本発明によれば、複数リスクを対象とした金融商品もしくは保険商品の価格計算方法と、この方法をコンピュータを用いて実行するためのプログラムが提供される。本発明の複数リスクを対象とした商品の価格計算方法によると、従来の発生強度の相関を考慮していないときと比べ、３以上のコンティンジェンシーがある場合でも、コンティンジェンシー発生の相関を考慮しているため、バスケット・デフォルト・スワップなどの複数リスクを対象とした商品の価格計算を合理的にできる様になる。また、他の相関を考慮する方法と比べ、パラメータ推定の計算も含め、計算が容易になる。
本発明をバスケットタイプ金融商品に具体的に適用すれば、バスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法と、この方法をコンピュータを用いて、実行するためのプログラムが提供される。本発明のバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法によると、従来の債務者間の信用力の相関を考慮していないときと比べ、バスケットに組み込まれる債務（債券）を負う３者以上の債務者がある場合でも、債務者間の信用力変化の相関を考慮しているため、バスケットデフォルトスワップなどのバスケットタイプ金融商品（クレジットデリバティブ）の価格(プレミアム）の計算を合理的にできるようになる。
【図面の簡単な説明】
【図１】バスケットタイプ金融商品（クレジットデリバティブ）の一例として挙げるバスケットデフォルトスワップ（ＢＤＳ）のスキームを表した概念図である。
【図２】本発明に係るバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法の一実施の形態を表したフローチャートである。
【図３】本発明に係るバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法の他の実施の形態を表したフローチャートである。
【図４】本発明に係るバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法の信用力相関パラメータを計算するステップの一実施の形態を表したフローチャートである。
【図５】本発明に係るバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法のコピュラ関数モデルの一実施の形態を表したフローチャートである。
【図６】本発明に係るバスケットタイプ金融商品（クレジットデリバティブ）の価格計算方法の価格（プレミアム）の計算方法の一実施の形態を表したフローチャートである。
【図７】本発明に係るリスク参照単位の概念を説明する概念図である。
【符号の説明】
１ バスケットデフォルトスワップ（ＢＤＳ）のスキーム
２ ＢＤＳの買い手
３ ＢＤＳの売り手
４ バスケット
１０ デフォルト強度計算ステップ
２０ 信用力相関パラメータ計算ステップ
３０ 信用力相関デフォルト強度計算ステップ
４０ 価格計算ステップ
１００ 市場データ受付ステップ
１１０ 回収率受付ステップ
１２０ デフォルト強度計算ステップ
１３０ 信用力相関パラメータ計算ステップ
１４０ 信用力相関デフォルト強度計算ステップ
１５０ 価格計算ステップ
２０１、２１１ 被保障参照体
２０２、２１２ コンティンジェンシー（火災）
２０３、２１３ コンティンジェンシー（地震）
２０４、２０５、２１４、２１５ リスク参照単位[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a price calculation method that reflects a correlation with financial products or insurance products for a plurality of risks. More particularly, the present invention relates to a price calculation method capable of reflecting the correlation between different risks in the calculation of the price of a basket type financial product and the calculation of the net premium of non-life insurance.
[0002]
[Prior art]
2. Description of the Related Art In recent years, financial products and insurance products for avoiding various risks faced by financial institutions and business corporations have become widespread. For example, financial derivatives called credit derivatives are traded between financial institutions and business corporations or between financial institutions as a means of avoiding credit risk. Specifically, there are basket-type financial products that are useful as risk hedging instruments in the credit risk management of loans held by banks and corporate bonds held by institutional investors. An example of such a basket type financial product is a basket type credit derivative. This is a financial derivative that guarantees the default damage from a collection of multiple reference assets (collateral assets) called baskets. A basket default swap (hereinafter abbreviated as BDS) is known as a representative of basket type credit derivatives.
[0003]
This means that the holder of multiple bonds (referred to as a basket) (= BDS buyers) that serve as reference assets pays a premium to the BDS seller on a regular basis. If a default occurs from within the basket, only the amount that the BDS buyer could not recover from the initial defaulted bond is guaranteed by the BDS seller. The BDS contract is closed when the first default occurs. In exceptional cases, the owner of the reference asset may not necessarily match the buyer of the BDS.
[0004]
Here, the basket type credit derivative includes not only the first default but also financial derivatives such as the second default in the target aggregate, which are triggered by a predetermined order default. Although it refers to the same product, it is called differently from the basket default swap in order to clearly express the number of default occurrences as a derivative. That is, products that guarantee loss due to the first default in the basket are first to default, products that guarantee loss due to the second default are second to default, and products that guarantee loss due to the third default are third.・ This is called toe default.
[0005]
Credit risk refers to the occurrence of defaults on loans held by banks, etc. (default), and defaults on bonds held by institutional investors (collectively referred to as contingencies). Refers to uncertainty.
[0006]
This section describes conventional price calculation techniques for financial products that are useful as a means of avoiding credit risk.
In the case of calculating the price of a product for a plurality of credit risks as represented by BDS, a rational calculation method using historical data on information that can be observed in the market has not been established. The reason is not only the frequency of occurrence of each contingency, but also the interdependency (correlation) of the frequency of occurrence of these different contingencies. This is because correlation estimation work using a plurality of the history data is considerably difficult. For example, when calculating the premium (price) of a BDS, it is not sufficient to evaluate the credit risk of the multiple bonds that make up the basket, and the correlation of the credit risk between the debtors who issue these bonds Need to be measured, but this is difficult. On the other hand, the necessity of obtaining this correlation itself is clear from the fact that chain failures are likely to occur in related industries during a recession. As described above, there is a problem that it is difficult to appropriately measure the correlation of credit risk between obligors in order to appropriately calculate the BDS premium.
[0007]
In addition, two approaches that have been used in the past as methods for evaluating financial products that contain credit risk are described.
As one of the traditional methods for evaluating financial derivatives that incorporate credit risk, information specific to the company (financial information) is excluded, and only information that can be observed in the financial market is the occurrence of default or credit rating changes. There is an exogenous approach that affects uncertainty. This method is similar to the pricing theory of derivative financial instruments such as interest rate derivatives, and emphasizes consistency with the interest rate model popular in the market, and is widely adopted especially in Europe and the United States.
[0008]
As another conventional evaluation method, corporate value is expressed based on financial information unique to the company, and uncertainty (risk) of default occurrence or credit rating change depends only on the financial information specific to the company. There is also an endogenous approach. This method is similar to the famous Black-Scholes model as an option price evaluation model, and is a representative example in which the software developed by the US company KMV employs this method.
[0009]
Among conventional price calculation methods for credit risk, focusing on the method of modeling the correlation of occurrence frequency, a method using a multidimensional normal distribution can be mentioned. . The multi-dimensional normal distribution method has been known as a pricing method that takes into account the creditworthiness correlation between issuers of the bonds that make up the basket of the basket default swap, and is used in the pricing of interest rate derivatives, etc. The method is diverted. When credit risk is targeted in this method, either the above-mentioned exogenous approach or endogenous approach can be adopted.
[0010]
First, the case where an exogenous approach is adopted and a multidimensional normal distribution is used will be described.
In this case, the uncertainty of default occurrence or credit rating change is expressed by an intensity function model, and the intensity (occurrence intensity) that expresses the occurrence frequency changes while affecting each other. It is expressed by a random variable (multidimensional Brownian motion) that follows a normal distribution. That is, the correlation of the change in the generated intensity is expressed by the correlation that the corresponding Brownian motion is embodied.
[0011]
Such a method using a multidimensional normal distribution has been conventionally known as a pricing method that takes into account the correlation of credit risk between issuers of bonds that make up the BDS basket. The method used in is diverted. However, there are many cases where an analytical solution cannot be constructed, and in that case, evaluation is performed by Monte Carlo simulation.
[0012]
In addition, when a method using a multidimensional normal distribution is adopted, it is necessary to simultaneously and consistently estimate the parameter of the intensity of occurrence of individual contingencies and the correlation of the change of the intensity of occurrence. However, in this method, if three or more contingencies are targeted, even if they are the same kind of contingencies, the calculation becomes complicated and technically difficult because they become multivariate parameters.
[0013]
Furthermore, since it is unnatural both theoretically and theoretically to take negative values, it is also known to adopt a model that does not allow negative values. Therefore, it is difficult not only to make an analytical evaluation method, but it is also difficult to estimate parameters related to the correlation.
[0014]
Next, a case where an endogenous approach is adopted and a multidimensional normal distribution is used will be described.
As a particular problem when adopting the endogenous approach, it is impossible to measure the corporate value itself in the market, so it is fundamentally difficult to measure the correlation of creditworthiness changes from market data. Therefore, in practice, stock prices are sometimes used as an alternative to corporate value. However, in order to obtain the correlation of changes in creditworthiness from the correlation coefficient of stock prices, there is a problem that complicated calculation is required. For this reason, it is rare to use an endogenous approach to calculate the price of complex credit derivatives.
[0015]
An evaluation method using Gaussian copula (also called Gaussian Copula. Normal Copula) has recently been developed as a new method that can be used for both the endogenous and exogenous approaches. However, this method is not suitable for constructing analytical solutions. Therefore, it is technically difficult to evaluate a product having complicated billing conditions such as second to default and third to default.
[0016]
In addition, we will describe the price calculation technology that has been used in the past for insurance products that have been used as a means to hedge (avoid) insurance risk in the insurance field. Insurance risk refers to uncertainty related to the occurrence of natural disasters and fires faced by business corporations or individuals covered by insurance.
With the liberalization of insurance, the individual needs of business corporations or individuals, as seen in insurance products in the form of guaranteeing the overlap of damage caused by multiple disasters, which are usually covered by different insurance lines. In response, insurance companies can design and sell insurance that combines various insurance items. Even in existing insurance products before that, as a product that covers multiple risks, for example, disasters of multiple insurance types by adding special contracts such as a flood risk guarantee special contract, earthquake risk guarantee special contract, liability guarantee special contract etc. to fire insurance Products that guarantee loss due to Conventionally, for those insurances with special provisions, an insurance population is prepared for each special provision and the net premium rate is calculated.
[0017]
In recent years, products called integrated risk management programs that simultaneously support insurance risks such as disasters and liability, as well as financial risks such as stock price and bond decline risk, have begun to be sold. This product, also called balance sheet protection or holistic cover, is tailor-made and designed to suit the nature of the customer's balance sheet. In this way, in the case of individual insurance products that meet the risk characteristics unique to the insured, it is difficult to set the insurance population. It was necessary to specifically model each of multiple risks in order to calculate However, until now, it has been difficult to model and evaluate the interdependence of multiple risks, so the occurrence intensity between contingencies (unexpected situations or accidents) belonging to each risk It was common to calculate net premiums by reorganizing the classification of product contingencies so that the correlation was as small as possible, and assuming that the correlation was zero and using the reproducibility of the Poisson distribution. . However, contingency classifications that are substantially uncorrelated have limitations on the number of classifications that can be classified.
[0018]
Conventional methods for evaluating insurance products that target multiple insurance risks in consideration of correlation include two Poisson random variables representing the frequency of occurrence of each of the two risks. There is a method of decomposing each into two independent random variables using the relation number. Therefore, if the linear correlation coefficient between two risks can be estimated based on the historical data about the insurance risk in question, it is possible to calculate the net premium for the insurance product that covers the two risks. However, when this method is used with three or more risks, there is a problem that the correspondence between the three Poisson random variables and the matrix of the linear correlation coefficient becomes complicated and the calculation becomes difficult.
[0019]
[Problems to be solved by the invention]
The present invention considers the correlation of the occurrence intensity of a plurality of risk reference units belonging to each risk in a price calculation method for financial products or insurance products for a plurality of risks such as basket default swap. It was made in view of the problem of calculation in some cases. Introducing a new copula function to provide an analytical price calculation method for products targeting multiple risks, taking into account the correlation of occurrence intensity when there are three or more contingencies For the purpose. It is another object of the present invention to provide a technique for relatively easily estimating various parameters such as (rank) correlation coefficient necessary for this price calculation from market data in a manner consistent with this price calculation method. . In addition, it aims at providing the computer program which enables the above calculation.
[0020]
[Means for Solving the Problems]
In order to solve the above-described problem, in the present invention, historical data relating to a risk reference unit composed of a pair of a secured reference object and a contingency is stored in advance in a storage unit, and a product that refers to the risk reference unit Based on the billing conditions stored in advance in the storage means, an occurrence intensity calculation step in which the calculation means calculates an occurrence intensity indicating the likelihood of occurrence of contingency for each risk reference unit, and based on the history data Represents an occurrence correlation parameter calculation step in which the calculation means calculates an occurrence correlation parameter indicating a correlation between occurrence intensities for different risk reference units, and a threshold value used as a criterion for determining contingency occurrence for each risk reference unit. An occurrence threshold random variable, which is a random variable following a one-dimensional uniform distribution, is assigned to multiple risk reference units. In the case of calculating together, using a copula function model that converts the threshold value that is the basis for the occurrence of the corresponding contingency into an occurrence threshold multidimensional random variable that is a random variable that follows the multidimensional uniform distribution that each represents, A generation correlation strength calculation step in which a calculation means calculates a generation correlation strength representing a correlation of the generation strength between the risk reference units from the generation correlation parameter and the generation strength for each risk reference unit, and the generation correlation strength is stored in advance. A plurality of risks including a price calculation step in which a calculation means calculates a price of a product for a plurality of risks based on a loss rate for each risk reference unit stored in the means and a billing condition of the product. A price calculation method for financial products or insurance products is provided.
[0021]
Further, as another aspect of the price calculation method of the product of the present invention, historical data stored in advance in a storage means relating to a risk reference unit composed of a set of a secured reference object and a contingency, and the risk reference unit A generation intensity calculation step in which the calculation means calculates a generation intensity indicating the likelihood of occurrence of contingency for each risk reference unit based on the billing conditions stored in advance in the storage means of the commodity to be referred to, and the history An occurrence correlation parameter calculation step in which the calculation means calculates an occurrence correlation parameter indicating a correlation between occurrence intensities for different risk reference units based on the data, and a criterion for determining the occurrence of contingency for each risk reference unit The occurrence threshold random variable, which is a random variable that follows a one-dimensional uniform distribution When calculating together, using a copula function model that converts the threshold value that is the basis for occurrence of the corresponding contingency into a generation threshold multidimensional random variable that is a random variable that follows the multidimensional uniform distribution that each represents, An occurrence threshold multidimensional random variable generation step for generating a large number of realization values of the occurrence threshold multidimensional random variable from the occurrence correlation parameter and the occurrence intensity for each risk reference unit; and an actual value of the occurrence threshold multidimensional random variable; Including a price calculation step in which the calculation means calculates the price of the product for multiple risks based on the loss rate for each risk reference unit stored in the storage means in advance and the billing conditions of the product, A price calculation method for financial products or insurance products for multiple risks is provided.
[0022]
Further, as another aspect of the price calculation method of the product of the present invention, a risk reference unit reception step in which a computer receives a risk reference unit consisting of a pair of a secured reference object and a contingency, and for each risk reference unit, A product receiving step in which a computer receives a billing condition for a product that refers to the risk reference unit, and an occurrence intensity indicating the likelihood of occurrence of the contingency for each risk reference unit is stored in the storage means in advance. Based on the history data for the reference unit, the calculation means calculates a generation intensity parameter indicating the correlation between the generation intensity for the different risk reference units based on the history data. Occurrence correlation parameter calculation step and contingency generation for each risk reference unit The threshold value that is the standard for the occurrence of the corresponding contingency when calculating the occurrence threshold random variable that is a random variable that follows the one-dimensional uniform distribution and represents the threshold value that serves as the criterion for determination, for multiple risk reference units. Using the copula function model that converts to the occurrence threshold multidimensional random variable that is a random variable that follows the multidimensional uniform distribution represented by the, from the occurrence correlation parameter and the occurrence intensity for each risk reference unit, between the risk reference units A generation correlation strength calculation step for calculating a generation correlation strength representing a correlation of the generation strength, the generation correlation strength, a loss rate for each risk reference unit stored in the storage unit in advance, and claiming the product And a price calculation step in which the calculation means calculates the price of the product for the multiple risks based on the conditions. Financial instruments or insurance price calculation method of the product was is provided.
[0023]
Further, as another aspect of the price calculation method for the product of the present invention, risk rating history data for calculating risk rating history data by replacing the occurrence intensity with a risk rating by statistical or qualitative processing based on the history data. A calculation step, wherein the generation correlation parameter calculation step is a step in which the calculation means calculates a generation correlation parameter indicating a correlation between generation intensities for different risk reference units based on the risk rating history data. A price calculation method for financial products or insurance products for multiple risks is provided.
[0024]
Here, statistical or qualitative processing means calculation of a moving average over an appropriate period with respect to historical data or function processing using other auxiliary data (for example, when historical data is market data for corporate bonds). The total amount of bonds issued and the sales and ratio of the issuing company), and the classification into finite classes based on these results.
[0025]
In addition, as another aspect of the price calculation method of the product of the present invention, a risk reference unit reception step in which a computer receives a risk reference unit composed of a pair of a secured reference object and a contingency, and the risk reference unit is referred to A product acceptance step in which the computer accepts billing conditions for the product to be performed for each risk reference unit, a generation strength acceptance step in which the computer accepts an occurrence strength indicating the likelihood of occurrence of contingency for each risk reference unit, and the risk reference A loss rate reception step in which the computer receives a loss rate for each unit, an occurrence correlation parameter reception step in which the computer receives an occurrence correlation parameter indicating a correlation between the occurrence intensities of the risk reference units, and a contingency for each risk reference unit Represents the threshold value that is used as a criterion for occurrence Probability according to the multidimensional uniform distribution that represents the threshold value that is the basis for the occurrence of the corresponding contingency when the occurrence threshold random variable that is a random variable according to the uniform distribution is calculated for multiple risk reference units. An occurrence correlation strength representing a correlation of occurrence strength between the risk reference units from the occurrence correlation parameter and the occurrence strength for each risk reference unit using a copula function model that is converted into an occurrence threshold multidimensional random variable that is a variable. The calculation means calculates the price of the product for multiple risks based on the generated correlation strength calculating step, the generated correlation strength, the loss rate, and the billing conditions of the product. A method for calculating a price of a financial product or insurance product for a plurality of risks is provided.
[0026]
In addition, as a specific aspect in the price calculation method of the commodity of the present invention, the copula function model is configured such that all of the occurrence threshold random variables for the plurality of risk reference units included in the plurality of risks to be targeted. A first step in which the computing means calculates a product; a second step in which the computing means selects a first risk reference unit and a second risk reference unit from among the plurality of risk reference units; and the selected first risk reference. A third step of calculating a product of the occurrence threshold random variable of each of the unit and the second risk reference unit; and the occurrence threshold random variable of each of the first risk reference unit and the second risk reference unit A fourth step in which a computing means calculates a product of the product and the generated correlation parameter of the first risk reference unit and the second risk reference unit; and the third step And the fourth step are calculated by the computing means for all combinations of the risk reference units, and the computing means calculates the sum of the values calculated by all the combinations of the risk reference units. And calculating means calculates a product of the value calculated in the first step and the value calculated in the fifth step, and the calculating means calculates the occurrence threshold multidimensional random variable of a plurality of risk reference units. And a sixth step to calculate.
[0027]
In addition to these, as a more specific aspect in the price calculation method of the commodity of the present invention, the occurrence correlation parameter calculation step includes: a first risk reference unit included in the plurality of risk reference units included in the target multiple risks; Selecting a risk reference unit and a second risk reference unit, and based on the history data or the risk rating history data, the contingency of the first risk reference unit is increased. The calculating means calculates a first probability that the contingency of the second risk reference unit is likely to occur, and based on the history data or the risk rating history data, The likelihood of a tungency is reduced and the likelihood of a contingency of the second risk reference unit is reduced A second probability is calculated by the calculating means, and the contingency of the first risk reference unit is increased based on the history data or the risk rating history data, and the second risk reference The calculation means calculates a third probability that the contingency of the unit decreases, and the contingency of the first risk reference unit is generated based on the history data or the risk rating history data. Calculating the fourth probability that the contingency of the second risk reference unit increases and the probability that the contingency of the second risk reference unit increases, and based on the history data or the risk rating history data, The likelihood of contingency of one risk reference unit increases or decreases, and the second risk reference unit The step of calculating the fifth probability that the likelihood of the intungity does not change and the probability that the contingency of the first risk reference unit is likely to occur based on the history data or the risk rating history data. The calculation means calculates a sixth probability that the contingency of the second risk reference unit does not change and the probability of occurrence of the contingency increases or decreases, and the occurrence order based on the values from the first probability to the sixth probability It is possible to carry out by the step of calculating the correlation coefficient by the calculating means and the step of calculating the generated correlation parameter by multiplying the occurrence rank correlation coefficient by a constant.
[0028]
In addition to these, as another aspect of the price calculation method for the product of the present invention, the price calculation step calculates a probability corresponding to the billing condition of the product based on the generated correlation strength, and Calculating a basic statistic including an expected value and variance based on the probability and the value of the loss rate, and calculating a price of a product for a plurality of risks. A characteristic price calculation method for financial products or insurance products targeting multiple risks is provided.
[0029]
In addition to these, a program for causing a computer to execute any of the steps described above is provided as another category in the present invention.
[0030]
In addition to the above, in the present invention, the historical data includes bond price data of bonds issued by the debtor, coupon rate data of bonds issued by the debtor, and maturity of bonds issued by the debtor. Maturity period data on period, yield data indicating the yield of bonds issued by the debtor, yield spread data indicating the difference between the yield of bonds issued by the debtor and the yield or swap rate of government bonds, and Obligor credit rating data indicating the credit rating of the debtor, bond credit rating data indicating the credit rating of the bond issued by the debtor, and stock price time series data representing the stock price issued by the debtor in time series Stock market index time series data representing stock market indices in time series, bond price, coupon rate, maturity period data A combination of data selected from the group consisting of: yield data, yield spread data, debtor credit rating data, bond credit rating data, stock price time series data, and stock market index time series data. The model may include one or more data selected from the group consisting of mathematical model data calculated by the calculation means. However, other market data is not limited as long as it represents the creditworthiness of the debtor.
[0031]
When the historical data is one or more data selected from the group consisting of the yield data, the yield spread data, the obligor credit rating data, and the bond credit rating data, When the power increases, when the yield in the yield data decreases, when the difference in the yield spread data decreases, and when the credit rating in the obligor credit rating data increases , One or more times selected from the group consisting of when the credit rating in the bond credit rating data rises, and when the creditworthiness decreases when the yield in the yield data increases When the difference in the yield spread data increases, when the credit rating in the obligor credit rating data decreases, and when the bond credit One or more when the credit rating in the attached data is selected from the group consisting of a time of falling, and can also be.
[0032]
The definitions in the present invention of major words used in the present invention will be described below.
A contingency is an unforeseen event or accident that causes a company to continue to operate. For example, it refers to earthquakes, wind disasters, water disasters, fires, default (default), rating decline, etc.
Risk refers to the uncertainty of contingency occurrence.
[0033]
The above contingencies are contingencies related to credit risk such as default (default), rating downgrade, etc., or situations or accidents that cause damage to fixed assets, which are situations or accidents that cause damage to financial assets. And contingencies related to insurance risks such as natural disasters such as earthquakes and bad weather and fires.
[0034]
Default is one of the contingencies described above, and refers to the state in which a debt to a financial instrument issued by a debtor defaults. For example, before the maturity of a bond issued by a debtor, the debtor goes bankrupt, and the coupon paid at the time of bond issuance or the amount redeemed at the maturity date is paid to the bond purchaser. The state that cannot be done.
[0035]
A secured reference is a resource or economic entity that has economic value that is referenced in a financial or insurance product as a potential contingency. For example, when a fire is selected as a contingency, it indicates a building or factory where a fire can occur. When earthquakes, wind disasters, and water disasters are selected as contingencies, this refers to an assembly of buildings, factories, etc. that are included in areas where natural disasters may occur (if necessary, they may be further divided). When contingency is selected as default (default), rating down, etc., it refers to debtors with credit risk or securities issued by the debtors, stocks, convertible bonds, structured bonds and other securities.
[0036]
The reference asset is one of the above-mentioned secured reference objects, and refers to an asset that is referred to as a security target. Among these secured references, it is an asset that can be a basic variable that determines the price of financial or insurance products. Speaking of basket default swaps (BDS), the reference asset is a bond that is a collateral asset.
[0037]
The risk reference unit is a concept in which one secured reference object and one contingency that is a cause of occurrence are regarded as one set as an object to be referred to by a financial product or an insurance product. It refers to a formal set such as (guaranteed reference object, contingency). A conceptual diagram showing this is shown in FIG. For example, the first reference factory and the second factory of company A are referred to as guaranteed reference bodies 201 and 211, respectively, and the risk reference unit when the fires 202 and 212 that can occur in these factories are considered as contingencies is expressed formally. And (Company A's first factory, fire) 204 and (Company A's second factory, fire) 214, and adding an earthquake as a contingency, (Company A's first factory, earthquake) ) 205 and (company A's second factory, earthquake) 215 are added.
[0038]
The reason for considering such a risk reference unit is to clearly specify which contingency is likely to occur in which secured reference entity when considering the interdependence (correlation) of the frequency of occurrence. It is necessary to do. That is, it is not sufficient to specify only the contingency and only the secured reference object, and it is necessary to specify the assumed contingency using a combination of both.
[0039]
The product billing conditions are contractual conditions regarding the amount of money given to the purchaser and supplier of financial products or insurance products, and are the main factors that determine the specifications of the product. Even if the contingency is the same, for example, when default (default) is selected as the contingency, there may be a plurality of product claim conditions as follows. That is, conditions that guarantee only the primary loss from a collection of multiple reference assets (collateral assets) called baskets, conditions that guarantee only the secondary loss, or characteristics of the initial credit risk through securitization Is a condition regarding the priority order for coupon payment or redemption when a plurality of new financial products having completely different characteristics are formed. Even if the contingency is the same, for example, when fire is selected as the contingency, there may be a plurality of billing conditions for the product. In other words, conditions that guarantee only physical damage such as the disappearance of a building due to a fire, conditions that guarantee a fire profit loss due to the operation of the factory being stopped due to the occurrence of a fire, There are conditions that guarantee damages to fire for injuries of people and things inside, and conditions that guarantee damages due to fire caused by the occurrence of an earthquake.
[0040]
The probability corresponding to the billing condition is, for example, that (corporate bond, default) is a risk reference unit, and a basket default swap that covers a pool of corporate bonds collected and pooled is a product. Is the probability of the event “the first default occurs” against the claim condition “compensates for the loss caused by the first default”. That is, the occurrence probability of the event specified by the billing condition.
[0041]
The historical data refers to past quantity data including the current time related to the risk reference unit, and mathematical model data such as parameters and indexes calculated by the arithmetic means from the quantity data using a mathematical model. For example, when default or downgrading is selected as a contingency for a secured reference object, not only market price data but also external credit rating agencies (external rating agencies) are included in the historical data. Ratings (one of the risk ratings described below), etc. are included.
[0042]
The generation intensity indicates the ease of occurrence of contingency in the secured reference object. The generation intensity may be a constant, a deterministic function of time, or a stochastic process. It is desirable if it is based on data about the contingency.
[0043]
Default intensity (λ_t ⁱ) Is one of the above occurrence strengths, and the contingency is default (default). Indicates the “easiness of becoming” that is the default for debtors that issue financial products that serve as collateral assets (reference assets). The default intensity is preferably calculated from market data. For example, when the collateral asset is a bond, the default strength may be calculated based on the yield spread that indicates the difference between the bond issued by the debtor and the yield of the government bond.
[0044]
The risk rating is the contingency of these secured references when the secured references are different from each other or when it is difficult to directly observe the changes in the intensity of the risk reference unit over time. It is an indicator that integrates or substitutes the likelihood of occurrence. Usually, it is used in the form of symbols (A, B, C, etc.) having an order of around 10 or integers. Like the occurrence intensity, the risk rating is an index for explaining the occurrence probability. For example, the risk rating can be relatively correlated with the occurrence intensity such that the occurrence probability is smaller as the risk rating by an integer is smaller.
[0045]
Risk rating history data is a set of risk ratings, and represents the change in the intensity of occurrence of the referenced reference over time as a value (risk rating) that is different from the value of the intensity of occurrence (risk rating). It is a thing. For example, when fire is selected as a contingency for a secured reference object, an index indicating the likelihood of fire is numerically expressed in the form of time-series data based on historical data on fire occurrence. The result of a regular risk survey may be used. For example, if default is selected as the contingency for a secured reference entity, debt obligations or debt securities issued by debtors, stocks, convertible bonds, structured bonds, etc. An index (credit spread or rating) indicating the probability of default (default) is quantified and expressed in the form of time-series data.
[0046]
A basket-type financial product is one or more default or rating transitions (more than one) that occur in a collection of multiple reference assets called baskets, such as basket-type credit derivatives and asset backed security (Asset Backed Security). In general, this refers to a financial product whose payment is determined by the trigger of the occurrence intensity or risk rating of the risk reference body. In the present invention, since the basket type financial product price calculation method of the present invention can also be applied to the property and casualty insurance field, the basket type financial product widely includes property and casualty insurance products.
[0047]
As explained in the section of “Prior Art”, basket-type credit derivatives in the example of basket-type financial products are a collection of reference assets (collateral assets) that are called multiple loans, corporate bonds, etc., called baskets. The financial derivative that guarantees the default damage. A reference asset corresponding to a secured reference object in the present invention is treated as an aggregate, and is a financial product for avoiding losses that occur, and issuance of structured bonds through securitization as well as in the form of financial derivative transactions It is a product that is also becoming popular. In general, loan assets and corporate bonds are often used as reference assets (collateral assets). However, these assets are not limited to these, and are receivables that can estimate future regular cash flows with certainty. If so, it is possible to design products that use them as reference assets (collateral assets). For example, the issuance of receivables after securitization such as mortgages, auto loans, real estate rental income, patent usage fees, etc. has been devised.
[0048]
The basket default swap (BDS) refers to the holder of a plurality of bonds (hereinafter also referred to as “baskets”) serving as reference assets (collateral assets) for the basket default swap (hereinafter also referred to as “BDS buyers”). Instead of paying premiums to regular sellers, if the default occurred from within the basket by the end of the BDS contract, the BDS buyer could not be recovered from the first defaulted bond Only the amount is guaranteed by the BDS seller. After a default occurs from within the basket, the BDS contract is closed (terminated). In exceptional cases, the owner of the reference asset may not necessarily match the buyer of the BDS.
[0049]
Unless otherwise noted, BDS guarantees the first default as described above, but guarantees losses due to defaults in a predetermined order, such as the second default in the aggregate. It is called second to default or third to default. In order to clearly distinguish the BDS targeting the first default from these, it may be particularly called a first-to-default.
[0050]
The Integrated Risk Management Program (IRMP) is an insurance product that comprehensively covers financial risks such as stock price and exchange rate fluctuation risks, and insurance risks such as natural disasters and liability. . Typical products of this IRMP include single trigger and double trigger. Single trigger is a product that is paid when the sum of the loss due to insurance risk and the loss due to financial risk exceeds the set amount, and some payments are capped and are also called stop loss reinsurance contracts. Yes. A double trigger is a product that is paid only when both the insurance risk and financial risk losses are met and both conditions are met, and is also called a dual trigger. Further, similar products with three or more target risks are called multi-triggers.
[0051]
The price of the product in the financial product or the insurance product here is a premium paid by the person who purchases the product or a plurality of times regularly or their total value. For basket-type financial products, the premium of such financial products is called price.
[0052]
An intensity function model is a mathematical model that defines the intensity (intensity) that represents the probability of occurrence of contingency in unit time, and regards the first jump time of the Poisson process with this intensity as the contingency occurrence time. It is. In general, a stochastic process in which the number of event occurrences follows a Poisson distribution is called a Poisson process, but the characteristic of the Poisson process is that it jumps at most once in a very short time interval. It is suitable for modeling infrequent contingencies such as fire, default (default), and rating downgrade. A debtor is an economic entity such as a natural or legal person who has an obligation to make a payment in accordance with the gist of the debt, or a group thereof.
[0053]
Market data includes bond price data for bonds issued by debtors, coupon rate data for bonds issued by debtors, maturity data on the maturity of bonds issued by debtors, and bonds issued by debtors. Yield data indicating the yield of the debtor, yield spread data indicating the difference between the yield of bonds issued by the debtor and the yield or swap rate of the government bond, debtor credit rating data indicating the debtor's credit rating, and debtor Bond credit rating data indicating the credit rating of the bonds issued by the issuer, stock price time series data representing the stock price issued by the debtor in time series, stock market index time series data representing the stock market index in time series, and Bond price, coupon rate, maturity data, yield data, yield spread data, debtor credit rating data, bond credit Even one or more data selected from the group consisting of mathematical model data calculated by the arithmetic means by a mathematical model from a combination of data selected from the group consisting of supplementary data, stock price time series data and stock market index time series data It may be data that represents the creditworthiness of other obligors. Here, the swap rate refers to a rate on the fixed interest rate side in an interest rate swap that is a contract of a relative transaction for exchanging cash flows of the same currency under a predetermined condition over a certain period.
[0054]
The stock price time series data is data that takes time series of prices that are the closing price of the stock in the stock market, the closing price of the stock exchange, the average value of the day, week, month, half year, and year.
[0055]
The stock market index time-series data is an index that represents the movement of the stock price of the entire stock market or a part (for each industry) of the stock market. For example, various Nikkei averages (Nikkei average 225, etc.) and indices such as TOPIX.
[0056]
Generation correlation parameter (θ_ij) Is a parameter indicating a correlation of occurrence intensity between different risk reference units in a plurality of risk reference units. To calculate the occurrence correlation parameter, the creditworthiness correlation parameter is calculated by selecting two different risk reference units and calculating the Kendall rank correlation expressing the degree of dependence of the two occurrence strengths.
[0057]
The creditworthiness correlation parameter is a name for an occurrence correlation parameter when the secured reference body is a bond of a company or the like and the contingency is the default or rating change of the bond. That is, in the basket type financial product, the parameter indicates the correlation between the creditworthiness of the debtor i and the debtor j constituting the basket. Credit strength correlation parameter (θ_ij) Is estimated from the dependency of the debtor and the debtor's creditworthiness that make up the basket. In order to calculate the credit correlation parameter, Kendall rank correlation (in the present invention, which selects two different parties out of all the obligors constituting the basket and expresses the degree of dependence of the creditworthiness between the two parties. The creditworthiness correlation parameter is calculated by measuring the credit rank correlation).
[0058]
A copula function is a functional (Formula 1) from a marginal distribution function of a single variable random variable to a multivariate simultaneous distribution function of a multivariate random variable. Alternatively, the copula function is a default judgment when multiple obligors are calculated (considered) simultaneously from a default threshold random variable that is a random variable that follows a one-dimensional uniform distribution that represents a threshold used as a default judgment criterion. It can also be said to be a function (see Equation 4) that converts to a default threshold multidimensional random variable that is a random variable that follows a multidimensional uniform distribution representing a reference threshold.
[Expression 1]

[0059]
The copula function is superior in that it allows the multivariate distribution to be decomposed into a univariate marginal distribution and a multivariate interdependent structure. The interdependency mentioned here is not a correlation coefficient that is usually used, but in the present invention, it is expressed by what is called Kendall's rank correlation (occurrence rank correlation or credit rank correlation). In the present invention, the copula function includes a plurality of secured reference objects from an occurrence threshold random variable that is a random variable that follows a one-dimensional uniform distribution that represents a threshold that is a criterion for determining the occurrence of contingency for each secured reference object. Are simultaneously converted to occurrence threshold multidimensional random variables, which are random variables that follow a multidimensional uniform distribution representing thresholds that are the basis for the occurrence of corresponding contingencies. In particular, in the case of credit risk guarantee, multiple obligors are calculated (considered) simultaneously from a default threshold random variable that is a random variable that follows a one-dimensional uniform distribution that represents the default threshold for each obligor. A default threshold multi-dimensional random variable that is a random variable that follows a multi-dimensional uniform distribution representing a threshold serving as a default determination criterion in the case is converted. As a result, in the present invention, it is possible to express the interdependence of all risk reference units handled together and the interdependence of the creditworthiness of all obligors constituting the basket, so that three or more risk references can be expressed. It becomes possible to easily calculate the price of financial insurance products that take into account unit correlations and basket type financial products that take into account the correlation between creditworthiness of three or more debtors.
[0060]
Loss rate refers to the amount paid by the seller to the purchaser of financial or insurance products when a contingency event occurs, and the notional amount or insurance amount (or payment limit amount) set at the time of contract for each product. ) Divided by. For example, the loss rate when considering insurance risk as a contingency is the loss rate per accident. For example, the loss rate when considering credit risk as a contingency is the default of loans and corporate bonds. The ratio of the principal that could not be recovered after falling. When considering credit risk in particular, (1-loss rate) is called the recovery rate. In particular, the recovery rate of a financial product is the amount paid to the purchaser of a financial product when the debtor who issued the financial product becomes the default divided by the amount of debt contracted to be fulfilled when the financial product is issued. is there.
[0061]
The rank correlation is a statistic representing a correlation related to rank, such as a difference in rank or a magnitude relationship between ranks. Typical rank correlations include Kendall rank correlation and Spearman rank correlation.
[0062]
The occurrence rank correlation coefficient is a numerical value representing the degree of interdependence of the likelihood of contingency between two risk reference units in the form of a correlation coefficient called rank correlation. In the present invention, Kendall's rank correlation is used as an aspect of rank correlation. However, it is not necessary to limit to the Kendall rank correlation as long as it is a rank correlation representing a correlation between two generation intensities. For example, Spearman's rank correlation is also conceivable. The Kendall rank correlation used in the present invention indicates the dependency relationship between the occurrence intensities of two risk reference units. If there are two occurrence intensity random variables for two risk reference units, and one takes a large value and the other takes a large value, it can be said that the mutual dependence is strong. Conversely, when one of the occurrence intensity random variables takes a large value and the other takes a small value, it can be said that the mutual dependency is weak. The occurrence rank correlation coefficient of the present invention is a numerical value representing the strength of the dependency of occurrence intensity between risk reference units. Two time-series data on the intensity of occurrence X_tAnd Y_tX over time_tIncreases and Y_tThe probability P of increasing_up ^up, X_tDecreases and Y_tThe probability P of decreasing_down ^downConversely, X_tIncreases and Y_tThe probability P of decreasing_up ^down, X_tDecreases and Y_tThe probability P of increasing_down ^up, X_tIncreases or decreases and Y_tProbability P does not increase or decrease_tie, Y_tIncreases or decreases and X_tProbability P does not increase or decrease^tieAs a creditworthiness rank correlation used in the present invention, Kendall rank correlation and Spearman rank correlation are defined as Equation 2.
[Expression 2]

[0063]
The creditworthiness rank correlation (or creditworthiness rank correlation coefficient) is a name in the case of a financial product that guarantees credit risk. In other words, it represents the creditworthiness between two obligors in the form of a rank correlation coefficient. Also in this case, as long as it is a rank correlation that represents the correlation of creditworthiness between two obligors, it is not limited to Kendall's rank correlation, and for example, Spearman's rank correlation is also conceivable.
[0064]
The number of occurrence rank correlations and the constant multiplied by creditworthiness rank correlation are occurrence rank correlation (or creditworthiness rank correlation; Kendall rank correlation) and occurrence correlation parameter (or creditworthiness correlation parameter; θ_ij). By mathematical proof, the occurrence rank correlation (or creditworthiness rank correlation; Kendall rank correlation) as shown in Equation 3 and (or the creditworthiness correlation parameter; θ_ij) And a constant (2/9) is known. The proof of this relationship is illustrated in Example 5.2 on page 129 of the document "An Introduction to Copulas (1999), Roger B. Nelsen, Lecture Notes in Statistics 139, Springer."
[Equation 3]

[0065]
According to the price calculation method for financial products and insurance products that target a plurality of risk reference units of the present invention, and the price calculation method for basket type financial products, the correlation of the occurrence intensity between conventional risk reference units and between debtors Even when there are 3 or more obligors with debts (credits) embedded in the target risk reference unit or basket compared to when the correlation of creditworthiness is not considered, the intensity of occurrence between these And the correlation of changes in creditworthiness are taken into account, it becomes possible to rationally calculate the price (premium) of these financial products, insurance products, basket-type financial products such as basket default swaps.
[0066]
In the present invention provided as described above, each feature can be applied to a more specific concept in the field of basket type financial products. Specifically, “risk reference unit consisting of a pair of secured reference object and contingency” is “collateral assets (or reference assets)”, “occurrence intensity” is “default intensity”, and “occurrence correlation parameter” "Credit strength correlation parameter", "occurrence correlation strength" is "credit strength correlation default strength", "occurrence threshold random variable" is "default threshold random variable", and "occurrence threshold multidimensional random variable" is " By limiting the “default threshold multidimensional random variable”, “product targeting multiple risks” to “basket type financial product”, and “history data” to “market data”, the method of the present invention is used. It is possible to calculate the price of a type financial product.
[0067]
That is, in the present invention, the default probability of the debtor is shown based on the market data previously stored in the storage means regarding the debtor issuing the financial product that is the collateral asset of the basket type financial product. A default strength calculating step in which the calculating means calculates a default strength; a credit strength correlation parameter calculating step in which the calculating means calculates a credit strength correlation parameter indicating a correlation of creditworthiness between the obligors based on the market data; A threshold value used as a default judgment criterion when a plurality of debtors are calculated simultaneously from a default threshold random variable that is a random variable that follows a one-dimensional uniform distribution representing a threshold value used as a default judgment standard for each debtor. Using a copula function model to convert to a default threshold multidimensional random variable that is a random variable that follows a multidimensional uniform distribution A credit strength correlation default strength calculating step in which a computing means calculates a credit strength correlation default strength using a credit strength correlation between the debtors based on the credit strength correlation parameter and the default strength for each debtor; A price calculation step in which the calculation means calculates the price of the basket type financial product based on the credit strength correlation default strength and the recovery rate of the financial product at the default of the debtor stored in the storage means in advance. A price calculation method for basket-type financial products is also feasible.
[0068]
In addition, according to the present invention, a market data receiving step in which a computer receives market data relating to a debtor issuing a financial product that is a collateral asset of a basket type financial product, and a recovery rate of the financial product at the default time of the debtor A recovery rate receiving step received by the user, a default strength calculating step in which a calculation means calculates a default strength indicating the likelihood of default of the debtor based on the market data, and the debtor based on the market data. A creditworthiness correlation parameter calculation step in which the calculation means calculates a creditworthiness correlation parameter indicating a creditworthiness correlation between the creditworthiness parameters, and a random variable that follows a one-dimensional uniform distribution that represents a threshold value used as a default judgment criterion for each obligor. Default when calculating multiple obligors simultaneously from a certain default threshold random variable Based on the creditworthiness correlation parameter and the default strength for each obligor, using a copula function model that converts to a default threshold multidimensional random variable that is a random variable that follows a multidimensional uniform distribution that represents a threshold that is a criterion for determining Based on the credit strength correlation default strength calculating step for calculating the credit strength correlation default strength using the correlation of creditworthiness between the debtors, the credit strength correlation default strength, and the recovery rate, the basket It is possible to realize a price calculation method for a basket type financial product that includes a price calculation step in which the calculation means calculates the price of the type financial product.
[0069]
Furthermore, according to the present invention, the copula function model includes a first step in which the computing means calculates all products of the default threshold random variables for each obligor, and the first obligor and the second obligor from among the obligors. A second step in which the calculating means selects a person; a third step in which the calculating means calculates a product of the respective default threshold random variables of the selected first obligor and second obligor; A fourth step in which a computing means calculates a product of the product of the default threshold random variable of each of the second obligors and the creditworthiness correlation parameters of the first obligor and the second obligor; the third step; The calculation means calculates the fourth step for all combinations of obligors in the obligor, and calculates the sum of the products of the combinations of all obligors and the creditworthiness correlation parameter. The calculation means calculates a product of the fifth step to calculate, the value calculated by the first step, and the value calculated by the fifth step, and the default threshold multidimensional random variable of a plurality of obligors It is possible to realize a price calculation method for a basket-type financial product, characterized in that it includes a sixth step in which the calculation means calculates.
[0070]
In addition, according to the present invention, the creditworthiness correlation parameter calculating step selects the first obligor and the second obligor from the obligors, and the first data is based on the market data. The calculating means calculates a first probability that the creditworthiness of the debtor increases and the creditworthiness of the second debtor increases, and the creditworthiness of the first debtor decreases based on the market data. And calculating the second probability that the creditworthiness of the second debtor decreases, and based on the market data, the creditworthiness of the first debtor increases, and the second debtor The calculating means calculates a third probability that the creditworthiness of the borrower will decrease, and based on the market data, the creditworthiness of the first debtor decreases and the creditworthiness of the second debtor increases. A step of calculating the fourth probability by the calculating means, and based on the market data; A step of calculating a creditworthiness rank correlation by subtracting the sum of the third probability and the fourth probability from the sum of the first probability and the second probability; It is possible to realize a price calculation method for a basket-type financial product, which includes multiplying a constant and calculating the creditworthiness correlation parameter by a calculation means.
[0071]
In addition, according to the present invention, the price calculation step is performed by the first maturity date of the basket type financial product among the financial products to be collateralized assets of the basket type financial product based on the credit strength correlation default strength. The product of the step in which the computing means calculates the probability that a default will occur, the step in which the computing means calculates the non-recovery rate that cannot be recovered from the recovery rate by the first default, and the probability that the first default will occur and the non-recovery rate. The method of calculating the price of the basket type financial product including the step of calculating the present value of the product and the step of calculating the price of the basket type financial product by the calculation means is feasible.
[0072]
According to the present invention, calculation time can be shortened because only analytical calculation is performed without using a method such as a simulation that requires a large amount of calculation. This is very important for financial products such as basket-type financial products that require frequent recalculation of prices.
[0073]
The above-described price calculation method for basket type financial products can also be applied to the non-life insurance field. Net insurance premiums correspond to the prices of the basket-type financial products described above. The parameter that represents the frequency of accidents (contingencies) corresponds to the default strength of individual debtors. In addition, for example, when considering fire insurance for a company that owns multiple factories, the credit correlation parameter that indicates the correlation of creditworthiness among each obligor corresponds to whether or not a fire occurs at those factories. It becomes a dependency. And (1-damage rate) corresponds to the collection rate of claims. In addition, in the field of property and casualty insurance, the obligor is insured, which is the object guaranteed by insurance contracts such as buildings and automobiles. Therefore, the logical framework for basket default swap pricing can be applied when there are multiple insured objects (risk reference units) and the frequency of accidents for those objects is not independent. It becomes. And the calculation of such net premiums for non-life insurance can be done within the same logical framework. The present invention provides a concept encompassing these in a specific range based on the inventors' knowledge about the relationship between financial products and insurance products.
[0074]
The input means accepts input by means of an operator's operation such as a keyboard or mouse, input by a recording medium such as a flexible disk, or input by an electric signal such as a telecommunication line including the Internet or a LAN (local area network). Including means.
[0075]
The storage means is means for storing data such as characters and numbers, programs, and the like. For example, a random access memory or a hard disk in a computer.
[0076]
The calculation means is any calculation means for executing a program. For example, a central processing unit, that is, a CPU, or a DSP (digital signal processing) means designed exclusively for a specific calculation is not limited to these.
[0077]
The computer refers to an electronic computer including the above input means, storage means, and arithmetic means. The computer executes instructions for operating itself as an appropriate means in accordance with the object of the present invention, receives data from the input means, calls data from the storage means, and performs arithmetic processing. In addition, the computer displays the calculation result, records it in the storage means, sends it to another computer itself, sends it as requested by the other computer, and controls other devices according to the result. To do. If necessary, communication with other devices may be performed by an appropriate network or communication means. For example, when the calculation means of the present invention calculates the generation intensity of the present invention in executing the above instruction, the calculation intensity calculation means of the computer is executed by the computer calculation device executing the generation intensity calculation step of the present invention. Works as. That is, the computer includes the generated intensity calculation means. As in this example, any step of the present invention described as the operation of any means of the computer is performed by the appropriate means constituting the computer executing the operation so that the computer It means that it can operate as an inclusion. That is, it is assumed that the computer includes the operation means.
[0078]
The computer-readable recording medium is a medium that records a program and data and is readable by the computer. For example, a flexible disk, a CD-ROM, an MO, a ROM chip, etc.
[0079]
DETAILED DESCRIPTION OF THE INVENTION
Hereinafter, embodiments of the price calculation method for financial products and insurance products of the present invention will be described in detail with reference to the drawings. In the following description of the present embodiment, a specific example in which a debtor is assumed as a secured reference body and default (default) is assumed as a contingency (that is, a credit derivative that is a basket type financial product as a risk reference body). ) Debt generation rank correlation coefficient is explained with a more specific example using a financial product that guarantees credit risk, and general features and element names are described as necessary. The invention is not limited to this example.
[0080]
FIG. 1 is a conceptual diagram showing a basket default swap scheme. In the present embodiment, a basket default swap (BDS) will be used and described as an example of a basket type financial product (credit derivative). The basket default swap refers to a contract exchanged between the “BDS buyer 2” and the “BDS seller 3” in the basket default swap scheme 1 shown in FIG.
[0081]
The contents of this agreement are the holders (hereinafter also referred to as “BDS buyer 2”) of multiple bonds (hereinafter also referred to as “baskets”) that serve as reference assets (collateral assets, secured reference bodies) for basket default swaps. However, instead of paying the BDS seller 3 on a regular basis as a consideration for swap transactions, a default (contingency) occurs from within the basket before the expiration of the BDS contract. The amount of money that the BDS buyer 2 could not recover from the default bond (risk reference unit) is guaranteed from the BDS seller 3. In addition, a typical BDS often terminates the BDS contract at the time of the first default occurrence in the basket and stops paying premiums paid from the BDS buyer 2 to the seller 3. In exceptional cases, the owner of the reference asset may not match the buyer of the BDS.
[0082]
As shown in FIG. 1, the solid line arrow is the cash flow before the first default occurrence in the basket, and the dotted line arrow is the cash flow after the first default occurrence in the basket. Prior to the first default, the bond holder (BDS buyer 2) receives the total amount of coupons from a debtor who issues a plurality of bond baskets serving as reference assets (collateral assets). A premium is paid from BDS buyer 2 to BDS seller 3. After the initial default, a recoverable amount of the default bond is received from the debtor issuing the default bond. The recoverable amount is represented by the product of the face value and the recovery rate. Next, BDS seller 3 to BDS buyer 2 receive an uncollectible amount of the default bond. Note that the default uncollectible amount of bonds is represented by the product of the face value and (1-recovery rate). In the present invention, (1-recovery rate) is referred to as a non-recovery rate.
[0083]
In the price calculation method for basket-type financial products of the present invention, the occurrence of default events (occurrence of contingencies) of debtors issuing financial products that serve as reference assets (collateral assets) for baskets is replaced by random variables that follow Poisson distribution. To do. That is, the first jump time of the composite Poisson process is a random variable representing the default time. A parameter that indicates the frequency of the Poisson process jump that represents the default is the default intensity (λ_t ⁱ ), And the default strength itself varies with probability, which represents the future uncertainty (risk) of changes in creditworthiness of each obligor. The default intensity (λ_t ⁱ ) Is large, it means that the default is likely to occur (low creditworthiness, high contingency occurrence strength), and default strength (λ_t ⁱ ) Is small, the default is unlikely to occur (high creditworthiness).
[0084]
In the present invention, as a probabilistic representation of the default time, a random variable and a default intensity (λ_t ⁱ ) And is expressed as Expression 4. U in Equation 4_iIs a random variable that follows a uniform distribution in the interval [0, 1], and is called a default threshold random variable of the obligor i. (U₁, U₂, ..., U_i, ..., U_N) Is defined as a default threshold multidimensional random variable.
[Expression 4]

[0085]
Since different default times for multiple obligors that make up the basket are considered, the default strength (λ_t ⁱ ) And random variables that follow a uniform distribution for each obligor. The dependency structure of creditworthiness change among obligors is expressed by giving creditworthiness rank correlation (Kendall rank correlation) between random variables that are considered for each obligor and follow a uniform distribution.
[0086]
Furthermore, in the present invention, by defining a one-dimensional random variable that follows a uniform distribution and a copula function that follows a multi-dimensional uniform distribution, the interdependence of creditworthiness changes in the case of multiple debtors is specifically expressed. .
[0087]
If we do not consider the interdependence of changes in creditworthiness of all obligors that make up the basket as used in the past, the change in creditworthiness of all obligors will be independent of each other, and in this case the individual obligors ( Default intensity (λ) for individual risk reference units_t ⁱ ) Is different from the credit strength correlation default strength (Λ) when considering the interdependence of changes in creditworthiness of all obligors that make up the basket.
[0088]
The following expression is equivalent to the probabilistic expression of the default time as shown in Equation 4 above. The probability that the default does not occur until a certain future time under the condition that it has not been defaulted up to now can be expressed as Equation 5 by calculating an expected value related to the intensity.
[Equation 5]

[0089]
If the correlation of creditworthiness changes between debtors is not considered, that is, if it can be assumed that the creditworthiness of debtors is independent of each other, the strength of debtor i is_t ⁱThen, the probability that one obligor in the basket will not default until time t (the probability that one risk reference unit will not encounter contingency) is expressed as in Equation 6.
[Formula 6]

[0090]
However, unlike Equation 6, in the actual economy, it is common that the creditworthiness of a plurality of obligors cannot be assumed to be independent from each other. In this case, Equation 6 does not hold. Thus, by using a copula function, a relational expression corresponding to Expression 6 is derived in the present invention. In the present invention, by changing the copula function, the default strength of each obligor (λ when considering the interdependence of changes in creditworthiness of all obligors constituting the basket is considered._t ⁱ ).
[0091]
As a result of diligent research in light of actual economic events, the present inventor has proposed Equation 7 as a copula function C that expresses the interdependence of the creditworthiness of all debtors that make up the basket of N parties. To do.
[Expression 7]

[0092]
The parameter necessary to determine the function form of Equation 7 is the creditworthiness correlation parameter θ_ijThis is estimated from the degree of dependence (credit strength correlation) on the change in creditworthiness of debtor i and debtor j constituting the basket. Therefore, the creditworthiness correlation parameter θ_ijIn order to calculate the value by means of computing means, two different parties are selected from all the debtors making up the basket, and the creditworthiness rank correlation (Kendall rank correlation) expressing the degree of creditworthiness dependency between the two parties Is measured based on market data, the relevant creditworthiness correlation parameter θ_ijTo decide.
[0093]
The creditworthiness order correlation (Kendall's rank correlation) expressed by Equation 8 is a study of whether the creditworthiness of obligor j has increased or decreased when obligor i's creditworthiness has increased, and It is a numerical value and the parameter θ_ijIs in the relationship as in Equation 9. The mathematical proof is described in Example 5.2 on page 129 of the document "An Introduction to Copulas (1999), Roger B. Nelsen, Lecture Notes in Statistics 139, Springer." Here, the creditworthiness rank correlation (Kendall rank correlation) in Equation 9 and the creditworthiness correlation parameter (θ_ijIn the present invention, the constant (2/9) indicating the relationship with () is referred to as a credit rank correlation constant.
[Equation 8]

[Equation 9]

[0094]
In addition, the creditworthiness rank correlation shown by Formula 8 expresses the strength of the dependency relationship of creditworthiness between obligors as a numerical value. Two time series data on creditworthiness obtained from market data X_tAnd Y_tX over time_tIncreases and Y_tThe probability P of increasing_up ^up, X_tDecreases and Y_tThe probability P of decreasing_down ^downConversely, X_tIncreases and Y_tThe probability P of decreasing_up ^down, X_tDecreases and Y_tThe probability P of increasing_down ^up, X_tIncreases or decreases and Y_tProbability P does not increase or decrease_tie, Y_tIncreases or decreases and X_tProbability P does not increase or decrease^tieAnd the creditworthiness rank correlation (Kendall rank correlation) used in the present invention is defined as shown in Equation 8. Two time series data on creditworthiness obtained from market data X_tAnd Y_tYield data showing the yield of bonds issued by the debtor, yield spread data showing the difference between the yield of bonds issued by the debtor and the yield of government bonds, debtor credit rating showing the credit rating of the debtor Data, bond credit rating data indicating the credit rating of bonds issued by debtors, etc. can be included, but it may be data on creditworthiness obtained from other markets, and is not limited. In the example of data regarding these creditworthinesses, the following can be considered to show the increase or decrease in creditworthiness. When creditworthiness increases, when the yield in the yield data decreases, when the difference in the yield spread data decreases, when the credit rating in the debtor credit rating data increases, the bond credit rating data It is when the credit rating of the inside rises, etc. One of these may be used, a plurality may be used, or may be combined with other times.
[0095]
When creditworthiness decreases, when the yield in the yield data increases, when the difference in the yield spread data increases, when the credit rating in the debtor credit rating data decreases, the bond credit rating data This is when the credit rating inside falls, and one of them may be used, a plurality of them may be used, or a combination with other times may be used.
[0096]
The creditor default strength (Λ) of each individual obligor after taking into account the interdependence of creditor changes of all obligors constituting the basket can be obtained by modifying the copula function C expressed by Equation 7. . Using the credit strength correlation default strength (Λ) after taking into account the rank correlation, the distribution of the default time that occurs first in the basket is known, and the price calculation of the basket default swap can be calculated analytically.
[0097]
The credit strength correlation default strength (Λ) after considering the rank correlation is expressed as Equation 10.
[Expression 10]

[0098]
Analytical calculation can be performed even if the copula function shown in Formula 7 proposed in the present invention is combined with the formula showing the credit strength correlation default strength of the individual obligor after taking into account the credit correlation shown in Formula 10. The present invention is excellent in that the necessary parameters can be easily calculated by the calculation means.
[0099]
In the example in which the price calculation method of the basket type financial product in the present embodiment is used for the price calculation of the basket default swap (BDS), it is performed as follows.
[0100]
First, (i) the present value of all premiums that continue to be paid periodically (or only once) until the first default in the basket occurs, and (ii) the amount of security paid after the first default in the basket occurs The premium (price) is calculated so that the present values are equal. However, the recovery rate (L) for the face value of the debtor after the occurrence of default is given exogenously. (Note that this recovery rate may be obtained statistically. In addition, the recovery rate may be constant for all debtors, or a different value may be set for each debtor.)
[0101]
Further, the premium payment is assumed to be made in advance and the premium is c. (Note that premium payment may be made periodically.) (I) The present value of the premium payer is c. (ii) The present value of the guarantee side after the occurrence of default is expressed as Equation 11 assuming that the payment of the guarantee amount is made at the maturity T and the maturity of the BDS is T.
## EQU11 ##

[0102]
Where Λ in Equation 11_t ⁱIs a deterministic function of time representing the default strength of creditor correlation of obligor i after considering correlation, and r is a constant representing the interest rate free of credit risk.
[0103]
The premium c of the basket default swap is expressed as Equation 12 assuming that the face value of all bonds is the same and the recovery rate of any bond is L, and can be calculated by the calculation means.
[Expression 12]

Equation 12 is derived from Equation 11.
[0104]
[Formula 13]

Note that Formula 13 is the same as Formula 10 except that all λu_iIs assumed to take a constant value regardless of u.
[0105]
[Expression 14]

Equation 14 is the same as Equation 7.
[0106]
U of Equation 13 and Equation 14_iThe relationship is as shown in Formula 15.
[Expression 15]

[0107]
Next, an embodiment of a price calculation method for a basket type financial product (credit derivative) according to the present invention, which is calculated using Formulas 12 to 14, will be described with reference to FIG.
FIG. 2 shows an embodiment of a price calculation method for basket type financial products (credit derivatives) according to the present invention.
[0108]
[Default intensity (λ_t ⁱ )
Default intensity (λ) that indicates the probability of default of obligors based on market data stored in the storage means in advance for obligors who issue financial instruments that serve as collateral assets for basket-type financial instruments (credit derivatives)_t ⁱ) Is calculated by the computing means (step 10). Default intensity (λ_t ⁱ) Indicates the ease of default for debtors issuing financial instruments that serve as collateral assets (reference assets), so the default strength may be a constant or a deterministic function of time, It may be a stochastic process. Default intensity (λ_t ⁱ ) Is preferably calculated from market data. For example, when the collateral asset is a bond, the default strength may be calculated based on the yield spread that indicates the difference between the bond issued by the debtor and the yield of the government bond. Here, market data refers to yield data that indicates the yield of bonds issued by the debtor, yield spread data that indicates the difference between the yield of bonds issued by the debtor and the yield of government bonds, and creditworthiness of the debtor. It may be one or more data selected from the group consisting of debtor credit rating data indicating the rating and bond credit rating data indicating the credit rating of the debt issued by the debtor. Such data may be used.
[0109]
[Creditability correlation parameter (θ_ij)
A creditworthiness correlation parameter indicating a correlation of creditworthiness between obligors using the creditworthiness rank correlation (Kendall rank correlation) shown in Formula 9 and Formula 8 based on market data stored in the storage means in advance. (Θ_ij) Is calculated by the computing means (step 20).
The creditworthiness correlation parameter (θ_ijMore specifically, the calculation by the calculation means in FIG. 4 is as shown in FIG. FIG. 4 is a flowchart embodying the calculation shown in Equation 8.
[0110]
From the obligor, the calculation means selects the first obligor i and the second obligor j to create a combination of the two obligors (step 21), and the creditworthiness of the first obligor i increases. And the first probability that the creditworthiness of the second debtor j increases (P_up ^up) Is calculated by the computing means (step 22). Next, the second probability that the creditworthiness of the first obligor i decreases and the creditworthiness of the second obligor j decreases (P_down ^down) Is calculated by the computing means (step 23). The third probability that the creditworthiness of the first obligor i increases and the creditworthiness of the second obligor j decreases (P_up ^down) Is calculated by the computing means (step 24). The fourth probability that the creditworthiness of the first obligor i will decrease and the creditworthiness of the second obligor j will increase (P_down ^up) Is calculated by the computing means (step 25). The fifth probability that the creditworthiness of the first debtor increases or decreases and the creditworthiness of the second debtor does not change (P_tie) Is calculated (step 26). The sixth probability (P) that the creditworthiness of the second debtor increases or decreases and the creditworthiness of the first debtor does not change^tie) Is calculated (step 27). Then, using the first to sixth probabilities thus obtained, the creditworthiness rank correlation of Equation 8 is calculated (step 28). The fifth probability (P_tie), Sixth probability (P^tie) Are both 0, the calculation can be performed more easily. That is, in this case, in accordance with the fact that the denominator of Equation 2 is 1, steps 26 to 28 are performed using the first probability and the sum of the second probabilities (P_up ^up+ P_down ^down) From the third probability and the sum of the fourth probability (P_up ^down+ P_down ^up) Minus creditworthiness rank correlation {(P_up ^up+ P_down ^down)-(P_up ^down+ P_down ^up)} May be calculated by the calculation means. In any case, the creditworthiness rank correlation (Formula 8) is further multiplied by a constant (9/2) based on Formula 9, and the creditworthiness correlation parameter (θ_ij) Is calculated by the computing means (step 29). From step 21 to step 29, the calculating means calculates all combinations of obligors, and the credit correlation parameter θ of all combinations among obligors._ijIs calculated.
[0111]
In the present invention, when creditworthiness increases, when the yield in the yield data decreases, when the difference in the yield spread data decreases, and when the credit rating in the obligor credit rating data increases And one or more times selected from the group consisting of when the credit rating in the bond credit rating data rises, or a combination of other cases.
When creditworthiness decreases, when the yield in the yield data increases, when the difference in the yield spread data increases, when the credit rating in the debtor credit rating data decreases, and when bond credit It may be one or more times selected from the group consisting of when the credit rating in the rating data is lowered, or may be combined with other cases.
[0112]
[Calculation of creditworthiness default strength (Λ)]
Using the functional model (Equation 10 and Equation 14), the credit strength correlation parameter and the default strength (λ_t ⁱ ) Based on the creditworthiness correlation default strength (Λ_t ⁱ) Is calculated by the computing means (step 30).
[0113]
Credit strength default strength (Λ_t ⁱ) Will be described in more detail based on Equations 12 to 14, using the flowchart shown in FIG. The product of u for each obligor corresponding to the default threshold random variable U (u₁u₂u_Three... u_N) Is calculated by the computing means (step 31). The computing means selects the first debtor (i) and the second debtor (j) from the debtors (step 32). The product corresponding to the selected first and second obligors ((1-u_i) (1-u_j)) Is calculated by the computing means (step 32). The product corresponding to the first obligor i and the second obligor j ((1-u_i) (1-u_j)) And the creditworthiness correlation parameter (θ of the first obligor i and the second obligor j)_ij) Product (θ_ij(1-u_i) (1-u_j)) Is calculated by the computing means (step 34). The combination of all obligors in the basket (θ_ij(1-u_i) (1-u_j)) Is calculated by the computing means (step 35). Sum (Σθ) for all debtor combinations_ij(1-u_i) (1-u_j)) Is calculated by the computing means (step 36). The computing means calculates the copula function (Formula 14) C (step 37). The default threshold multidimensional random variable corresponds to the left side portion (u₁, U₂, ......, u_i, ......, u_N). From the copula function C, the credit strength correlation default strength (Λ_t ⁱ) Is calculated by the calculation means.
[0114]
[Calculation of basket type financial product (credit derivative) price c]
Credit strength default strength (Λ_t ⁱ) And the recovery rate (L) of the financial product at the default time of the debtor stored in the storage means in advance, the price (premium) c of the basket type financial product (credit derivative) using Equation 12 Is calculated by the computing means (step 40).
[0115]
A method in which the calculation means calculates the price (premium) c of the basket type financial product (credit derivative) will be described in more detail with reference to FIG.
Credit strength default strength (Λ_t ⁱ), The computing means calculates the probability that the first default will occur among the financial products that serve as collateral assets for the basket type financial product (credit derivative) (step 41). From the recovery rate (L), the computing means calculates the non-recovery rate (1-L) that cannot be recovered by the first default (step 42). Based on Equation 12, the present value of the product of the probability of the first default and the non-recovery rate is calculated, and the calculation means calculates the price (c) of the basket type financial product (credit derivative) (step 43). .
[0116]
[Other embodiments]
FIG. 3 is a flowchart showing another embodiment of the price calculation method for basket type financial products (credit derivatives) according to the present invention. The flowchart shown in FIG. 3 is different from the flowchart shown in FIG. 2 in that the market data and the collection rate L are not stored in the storage means in advance but are input by the input means of the computer. The other points are the same as the embodiment shown in FIG.
[0117]
That is, in the form shown in FIG. 3, the input means accepts market data and stores it in the storage means (step 100), and the input means saves the collection rate L in the acceptance storage means (step 110). Default intensity based on market data (λ_t ⁱ ) Calculation step 120 corresponds to step 10 in FIG. Credit correlation parameter (θ based on market data)_ij) Calculation step 130 corresponds to step 20 in FIG. Using the copula function model (C), the default intensity (λ_t ⁱ ) And creditworthiness correlation parameters (θ_ij) Based creditworthiness default strength (Λ_t ⁱ) Calculation step 140 corresponds to step 30 in FIG. Credit strength default strength (Λ_t ⁱ) And the recovery rate (L), the price calculation step 150 of the basket type financial product (credit derivative) corresponds to step 40 in FIG.
[0118]
The input means accepts input by means of an operator's operation such as a keyboard or mouse, input by a recording medium such as a flexible disk, or input by an electric signal such as a telecommunication line including the Internet or a LAN (local area network). Widely includes means. The market data may be input from an arbitrary website by an input unit via a telecommunication line including the Internet and stored in a storage unit.
[0119]
【Example】
Next, an example will be given to explain the utility of the price calculation method for basket type financial products (credit derivatives) according to the present invention. In this embodiment, the price calculation method of the basket default swap of the basket with 3 obligors is given, and the correlation between the creditworthiness between obligors of the present invention and the case where the conventional creditworthiness correlation between obligors is not considered ( The utility of the price calculation of the basket type financial product (credit derivative) of the present invention can be recognized by explaining the difference from the case of considering the degree of dependence).
[0120]
[Comparative example: When the creditworthiness correlation between conventional debtors is not considered]
The default strength (λ) of the three debtors is shown in Table 1 based on market data. The recovery rate after default occurrence is L = 0.2 (20%) for all obligors. The credit risk-free interest rate r is assumed to be r = 0.01 (1.0%) for a fixed period of 5 years without considering the period structure. The default swap contract has a maturity of T = 5 years.
[Table 1]

[0121]
Based on the above assumption, the premium (price) c of the default swap when there is one reference asset is obtained by using Equation 16 using λ that does not consider the creditworthiness correlation instead of the creditworthiness correlation default strength Λ of Equation 12. Is calculated. The calculation result of the premium (price) of the default swap when there is one reference asset (when it is not a basket) using Equation 16 is as shown in Table 2.
[Expression 16]

[0122]
[Table 2]

[0123]
If the credit risk of bonds issued by the three debtors is hedged separately, the total cost is 0.01879 + 0.02249 + 0.02617 = 0.06745 (6.745%).
[0124]
Further, when the correlation of creditworthiness among obligors is not taken into consideration, the premium of the basket default swap that uses the basket of these three companies as a reference asset is calculated by Expression 17 which is a modification of Expression 16.
[Expression 17]

[0125]
Substituting the default strength λ of Table 1 into Equation 17, Equation 18 is obtained and the basket default swap premium is calculated.
[Expression 18]

[0126]
The premium calculated by Equation 18 is 0.06549 (6.549%), compared to when hedged separately from the debtors of three obligors that are not in the basket (when 0.06745 (6.745%), which calculates the sum of premiums in Table 2). You can see that it is cheaper by the guarantee of only the first default in the basket. If the assumed principal amount is 10 billion yen, the premium when the basket is not considered but the credit between the obligors is 19.6 million yen lower than the premium when the basket is not used.
[0127]
[Example of the present invention: When considering credit correlation between obligors]
Next, as an embodiment of the present invention, the price of a basket type financial product (credit derivative) is calculated when considering the credit correlation between debtors.
The assumed conditions are the same as in the comparative example. That is, the default strength (λ) of the three debtors is shown in Table 1 based on market data. The recovery rate after occurrence of default is set to L = 0.2 (20%) for all obligors. The credit risk-free interest rate r is assumed to be r = 0.01 (1.0%) for a fixed period of 5 years without considering the period structure. The default swap contract has a maturity of T = 5 years.
[0128]
Next, as shown in Table 3, the correlation between the creditworthiness changes of the three debtors issuing bonds is assumed.
[Table 3]

The premium of the basket default swap in this case is calculated by Equation 19 obtained by substituting Equation 14 into Equation 13 for the individual obligor strength Λ after correlation.
[0129]
[Equation 19]

By substituting Equation 19 into Equation 20 obtained from Equation 12, a premium considering the credit correlation between debtors is calculated.
[Expression 20]

[0130]
When Equation 20 is calculated, c = 0.06447 (6.447%) is calculated, which is cheaper than when the correlation is not considered. Assuming that the notional amount is 10 billion yen, compared to the premium when the credit basket between the obligors is not considered, but the basket-type financial product that considers the creditworthiness relationship between the obligors of the present invention ( The amount of premium by the price calculation method of (credit derivative) will be reduced by 10.2 million yen.
[0131]
Furthermore, the creditworthiness correlation parameter θ_ijIn Table 4, the basket default swap premium is 0.06601 (6.601%), which is higher than when the correlation is not considered. Assuming that the notional amount is 10 billion yen, compared to the premium when the credit basket between the obligors is not considered, but the basket-type financial product that considers the creditworthiness relationship between the obligors of the present invention ( The amount of premium by the price calculation method of credit derivatives will be 5.2 million yen higher.
[Table 4]

[0132]
As described above, according to the price calculation method for a basket type financial product (credit derivative) according to the present invention, the basket type financial product ( It can be seen that the calculation of the price of (credit derivative) is more rationally calculated. In other words, if the correlation parameter of creditworthiness between debtors is positive, the premium (price) is prevented from being calculated high. Conversely, if the correlation parameter of creditworthiness between debtors is negative, the premium (price) To prevent it from being calculated low.
[0133]
[Still another embodiment]
As a form of generating an occurrence threshold multidimensional random variable from an occurrence correlation parameter and an occurrence intensity for each risk reference unit using a copula function model, a multidimensional uniform is generated from a random number (or pseudorandom number) generated by a computer program. There is also a method in which random numbers (or pseudo-random numbers) according to the distribution are generated and directly calculated by using Monte Carlo simulations.
[0134]
[Other fields of use]
The above-described price calculation method for basket-type financial products (credit derivatives) according to the present invention can also be applied to the non-life insurance field. The net premium is equivalent to the basket default swap price mentioned above. A parameter that represents the frequency of accidents corresponds to the default strength of individual debtors. In addition, for example, when considering fire insurance for a company that owns multiple factories, the credit correlation parameter that indicates the correlation of creditworthiness among each obligor corresponds to whether or not a fire occurs at those factories. It becomes a dependency. And (1-damage rate) corresponds to the collection rate of claims. In addition, in the field of property and casualty insurance, the obligor is insured, which is the object guaranteed by insurance contracts such as buildings and automobiles. Therefore, the logical framework for basket default swap pricing will be applied when there are multiple insured objects and the frequency of accidents for those objects is not independent. And the calculation of such net premiums for non-life insurance can be done within the same logical framework.
[0135]
【The invention's effect】
As is apparent from the above, according to the present invention, there are provided a price calculation method for financial products or insurance products targeting a plurality of risks, and a program for executing this method using a computer. According to the price calculation method for products targeting multiple risks according to the present invention, the correlation between occurrences of contingencies can be obtained even when there are 3 or more contingencies compared to the case where the correlation between the occurrences is not considered. This makes it possible to rationally calculate prices for products that target multiple risks such as baskets, defaults, and swaps. In addition, the calculation including the calculation of the parameter estimation becomes easier as compared with other methods considering the correlation.
If the present invention is specifically applied to a basket type financial product, a price calculation method for a basket type financial product (credit derivative) and a program for executing this method using a computer are provided. According to the price calculation method for basket-type financial products (credit derivatives) of the present invention, three or more parties with debts (bonds) embedded in the basket, compared with the case where the correlation of creditworthiness between debtors is not considered. Even if there are multiple debtors, it is possible to reasonably calculate the price (premium) of basket-type financial products (credit derivatives) such as basket default swaps because the correlation of creditworthiness changes among debtors is considered. become.
[Brief description of the drawings]
FIG. 1 is a conceptual diagram showing a scheme of a basket default swap (BDS) given as an example of a basket type financial product (credit derivative).
FIG. 2 is a flowchart showing an embodiment of a price calculation method for a basket type financial product (credit derivative) according to the present invention.
FIG. 3 is a flowchart showing another embodiment of a price calculation method for basket type financial products (credit derivatives) according to the present invention.
FIG. 4 is a flowchart showing an embodiment of steps for calculating a creditworthiness correlation parameter of a price calculation method for a basket type financial product (credit derivative) according to the present invention.
FIG. 5 is a flowchart showing an embodiment of a copula function model of a basket type financial product (credit derivative) price calculation method according to the present invention.
FIG. 6 is a flowchart showing an embodiment of a price (premium) calculation method of a price calculation method for basket type financial products (credit derivatives) according to the present invention.
FIG. 7 is a conceptual diagram illustrating the concept of a risk reference unit according to the present invention.
[Explanation of symbols]
1 Basket default swap (BDS) scheme
2 BDS buyers
3 BDS sellers
4 baskets
10 Default strength calculation step
20 Credit rating correlation parameter calculation step
30 Credit strength correlation default strength calculation step
40 Price calculation step
100 Market data reception step
110 recovery rate acceptance step
120 Default strength calculation step
130 Credit rating correlation parameter calculation step
140 Credit strength correlation default strength calculation step
150 Price calculation step
201, 211 Guaranteed reference object
202, 212 Contingency (fire)
203, 213 Contingency (earthquake)
204, 205, 214, 215 Risk reference unit

Claims (7)

A generation intensity calculation means, a generation correlation parameter calculation means, a generation correlation strength calculation means, a copula function model calculation means, and a price constituted by using at least the calculation means and the storage means in a computer having calculation means and storage means A price calculation device for financial products or insurance products for multiple risks, comprising:
The generated intensity calculation means stores history data stored in advance in the storage means relating to the risk reference unit consisting of a pair of a secured reference object and a contingency, and stores in the storage means in advance for a product that refers to the risk reference unit. Is generated from the storage means , and based on the history data and the billing condition data , a generation intensity indicating the likelihood of occurrence of contingency for each risk reference unit is calculated, The generated intensity data is stored in the storage means as a generated intensity table,
The occurrence correlation parameter calculation means receives the history data from the storage means , calculates an occurrence correlation parameter indicating a correlation between occurrence intensities for different risk reference units based on the history data, and generates the occurrence correlation Parameter data is stored in the storage means as an occurrence correlation parameter table ;
The generated correlation strength calculation means calls the generated correlation parameter data and the generated strength data from the generated correlation parameter table and the generated intensity table in the storage means, and refers to the generated correlation parameter and the risk reference The generation correlation strength reflecting the correlation of the generation strength between the risk reference units is calculated from the generation strength for each unit by operating the copula function model calculation means, and the generated correlation strength data is output. ,
The copula function model calculating means includes:
A calling means for calling out the occurrence intensity data and the occurrence correlation parameter for the selected risk reference unit;
Multiplication means for calculating and outputting multiplication data of two data;
Summing means including storage means and addition means;
The first risk reference unit and a second risk reference unit different from the first risk reference unit are selected, and the occurrence intensity data and the occurrence correlation parameter for the selected two risk reference units are selected. The second factor which is the data of the first factor that is the data of the probability of occurrence of the contingency for the first risk reference unit and the data of the probability of the occurrence of the contingency for the second risk reference unit is called by the means Product data and generated correlation parameter data are calculated by operating the multiplication means at least twice, and the multiplication data from the multiplication means is changed while changing the combination of the first and second risk reference units. The first summation data is accumulated by the summing means, and is output.
The generated correlation strength calculation means includes:
Multiplication means for calculating and outputting multiplication data of two data;
Summing means including storage means and addition means;
A division means for calculating and outputting division data of two data;
The risk reference unit for which the occurrence correlation strength is calculated is specified, the first risk reference unit and the second risk reference unit different from the first risk reference unit are selected, and the first risk reference is selected. When the unit and the second risk reference unit are risk reference units that are not subject to calculation, the data of the probability of occurrence of contingency for the first and second risk reference units is defined as the first factor and the second factor, When the first risk reference unit or the second risk reference unit is a risk reference unit to be calculated, a value that is twice the probability that no contingency occurs for the risk reference unit to be calculated is excluded from 1. The first factor, the second factor, and the occurrence phase of the first risk reference unit and the second risk reference unit with the value data as the first factor or the second factor The product of the parameters is calculated to obtain the sum to obtain the second sum data, and the dividing means is operated to operate the copula function model calculating means for the data that becomes the numerator including the second sum data. The ratio data is calculated by dividing by the denominator data including the first total data obtained by
The multiplication means multiplies the ratio data by the occurrence intensity data of the risk reference unit to be calculated to calculate the occurrence correlation intensity data for the risk reference unit to be calculated,
The price calculation means receives the generated correlation strength data from the generated correlation strength calculation means, receives loss rate data for each risk reference unit and billing condition data of the product from the storage means, Based on the occurrence correlation strength , the loss rate for each risk reference unit , and the billing conditions of the product, the price of the product for multiple risks is calculated and the price data is output. multiple risk price calculation system of financial instruments or insurance products targeting. A risk reference unit receiving means, a product receiving means, a generated intensity calculating means, and a generated correlation parameter calculating means configured using at least the calculating means and the storage means in a computer having at least an input means, a calculating means, and a storing means; A price calculation device for financial products or insurance products for multiple risks, comprising an occurrence correlation strength calculation means, a copula function model calculation means and a price calculation means,
The risk reference unit accepting means accepts a designated input or data of a risk reference unit consisting of a set of a secured reference object and a contingency by the input means ,
The product accepting means accepts designation input or data of a billing condition of a product referring to the risk reference unit for each risk reference unit by the input means,
The occurrence intensity calculation means receives the data of the risk reference unit from the risk reference unit reception means, and saves the occurrence intensity indicating the likelihood of occurrence of the contingency for each risk reference unit in the storage means in advance. Calculated based on the historical data about the risk reference unit being, and storing the data of the occurrence intensity in the storage means as an occurrence intensity table,
The occurrence correlation parameter calculation means receives the history data from the storage means , calculates an occurrence correlation parameter indicating a correlation between occurrence intensities for different risk reference units based on the history data, and generates the occurrence correlation Parameter data is stored in the storage means as an occurrence correlation parameter table;
The generated correlation strength calculation means calls the generated correlation parameter data and the generated strength data from the generated correlation parameter table and the generated intensity table in the storage means, and refers to the generated correlation parameter and the risk reference The generation correlation strength reflecting the correlation of the generation strength between the risk reference units is calculated from the generation strength for each unit by operating the copula function model calculation means, and the generated correlation strength data is output. ,
The copula function model calculating means includes:
A calling means for calling out the occurrence intensity data and the occurrence correlation parameter for the selected risk reference unit;
Multiplication means for calculating and outputting multiplication data of two data;
Summing means including storage means and addition means;
The first risk reference unit and a second risk reference unit different from the first risk reference unit are selected, and the occurrence intensity data and the occurrence correlation parameter for the selected two risk reference units are selected. The second factor which is the data of the first factor that is the data of the probability of occurrence of the contingency for the first risk reference unit and the data of the probability of the occurrence of the contingency for the second risk reference unit is called by the means Product data and generated correlation parameter data are calculated by operating the multiplication means at least twice, and the multiplication data from the multiplication means is changed while changing the combination of the first and second risk reference units. The first summation data is accumulated by the summing means, and is output.
The generated correlation strength calculation means includes:
Multiplication means for calculating and outputting multiplication data of two data;
Summing means including storage means and addition means;
A division means for calculating and outputting division data of two data;
The risk reference unit for which the occurrence correlation strength is calculated is specified, the first risk reference unit and the second risk reference unit different from the first risk reference unit are selected, and the first risk reference is selected. When the unit and the second risk reference unit are risk reference units that are not subject to calculation, the data of the probability of occurrence of contingency for the first and second risk reference units is defined as the first factor and the second factor, When the first risk reference unit or the second risk reference unit is a risk reference unit to be calculated, a value that is twice the probability that no contingency occurs for the risk reference unit to be calculated is excluded from 1. The first factor, the second factor, and the occurrence phase of the first risk reference unit and the second risk reference unit with the value data as the first factor or the second factor The product of the parameters is calculated to obtain the sum to obtain the second sum data, and the dividing means is operated to operate the copula function model calculating means for the data that becomes the numerator including the second sum data. The ratio data is calculated by dividing by the denominator data including the first total data obtained by
The multiplication means multiplies the ratio data by the occurrence intensity data of the risk reference unit to be calculated to calculate the occurrence correlation intensity data for the risk reference unit to be calculated,
The price calculating means receives the data of the generated correlation strength from the generated correlation strength calculating means, receives the data of the loss rate for each risk reference unit from the storage means, and the product claim condition from the product receiving means Receiving the data, calculating the price of the product for multiple risks based on the generated correlation strength, the loss rate for each risk reference unit, and the billing condition of the product , and outputs, price computing device of financial instruments or insurance products that target multiple risk. In a computer having at least computing means and storage means for calculating and outputting risk rating history data by replacing the generated intensity with risk rating by statistical or qualitative processing based on the history data , the computing means and the storage means A risk rating history data calculation unit configured using at least a storage unit ;
The occurrence correlation parameter calculation means calculates an occurrence correlation parameter indicating a correlation between occurrence intensities for different risk reference units based on the risk rating history data from the risk rating history data calculation means, and generates an occurrence correlation parameter. The price calculation apparatus for financial products or insurance products for a plurality of risks according to claim 1 or 2 , wherein said data is stored in said storage means as an occurrence correlation parameter table . Risk reference unit accepting means, product accepting means, occurrence intensity accepting means, and loss rate accepting means constituted by using at least the computing means and the storage means in a computer having at least input means, computing means, and storage means A price calculation device for financial products or insurance products for a plurality of risks, comprising a generated correlation parameter receiving means, a generated correlation strength calculating means, a copula function model calculating means, and a price calculating means,
The risk reference unit accepting means accepts a designated input or data of a risk reference unit consisting of a set of a secured reference object and a contingency by the input means ,
The product accepting means receives, based on the data of the risk reference unit received from the risk reference unit accepting means , designation input or data of a billing condition of a product that refers to the risk reference unit by the input means. Are accepted every time ,
The occurrence intensity accepting unit is configured to input a generation intensity designation input or data indicating the likelihood of contingency for each risk reference unit based on the data of the risk reference unit received from the risk reference unit accepting unit. Received by the input means, and stored in the storage means as a generated intensity table ,
The loss rate accepting unit is a unit that accepts a designation input or data of a loss rate for each risk reference unit by the input unit based on the data of the risk reference unit received from the risk reference unit accepting unit ,
The occurrence correlation parameter accepting unit is configured to input designation input or data of an occurrence correlation parameter indicating a correlation of occurrence intensities between the risk reference units based on the data of the risk reference unit received from the risk reference unit accepting unit. Is received by the means and stored in the storage means as an occurrence correlation parameter table ,
The generated correlation strength calculation means calls the generated correlation parameter data and the generated strength data from the generated correlation parameter table and the generated intensity table in the storage means, and refers to the generated correlation parameter and the risk reference The generation correlation strength reflecting the correlation of the generation strength between the risk reference units is calculated from the generation strength for each unit by operating the copula function model calculation means, and the generated correlation strength data is output. ,
The copula function model calculating means includes:
A calling means for calling out the occurrence intensity data and the occurrence correlation parameter for the selected risk reference unit;
Multiplication means for calculating and outputting multiplication data of two data;
Summing means including storage means and addition means;
The first risk reference unit and a second risk reference unit different from the first risk reference unit are selected, and the occurrence intensity data and the occurrence correlation parameter for the selected two risk reference units are selected. The second factor which is the data of the first factor that is the data of the probability of occurrence of the contingency for the first risk reference unit and the data of the probability of the occurrence of the contingency for the second risk reference unit is called by the means Product data and generated correlation parameter data are calculated by operating the multiplication means at least twice, and the multiplication data from the multiplication means is changed while changing the combination of the first and second risk reference units. The first summation data is accumulated by the summing means, and is output.
The generated correlation strength calculation means includes:
Multiplication means for calculating and outputting multiplication data of two data;
Summing means including storage means and addition means;
A division means for calculating and outputting division data of two data;
The risk reference unit for which the occurrence correlation strength is calculated is specified, the first risk reference unit and the second risk reference unit different from the first risk reference unit are selected, and the first risk reference is selected. When the unit and the second risk reference unit are risk reference units that are not subject to calculation, the data of the probability of occurrence of contingency for the first and second risk reference units is defined as the first factor and the second factor, When the first risk reference unit or the second risk reference unit is a risk reference unit to be calculated, a value that is twice the probability that no contingency occurs for the risk reference unit to be calculated is excluded from 1. The first factor, the second factor, and the occurrence phase of the first risk reference unit and the second risk reference unit with the value data as the first factor or the second factor The product of the parameters is calculated to obtain the sum to obtain the second sum data, and the dividing means is operated to operate the copula function model calculating means for the data that becomes the numerator including the second sum data. The ratio data is calculated by dividing by the denominator data including the first total data obtained by
The multiplication means multiplies the ratio data by the occurrence intensity data of the risk reference unit to be calculated to calculate the occurrence correlation intensity data for the risk reference unit to be calculated,
The price calculating means receives the billing condition data of the product from the product receiving means, receives the loss rate data from the loss rate receiving means, and receives the generated correlation strength data from the generated correlation strength calculating means. Receiving the generated correlation strength, the loss rate, and the billing conditions of the product, calculating the price of the product for multiple risks, and outputting the price data Price computing device of financial instruments or insurance products targeting. The generation correlation parameter calculation means includes:
The calculation means that selects two of the first risk reference unit and the second risk reference unit from among the plurality of risk reference units included in the target multiple risks, and outputs data of the selection result. A first means configured ;
The selection result is received from the first means , the contingency of the first risk reference unit is increased based on the history data or the risk rating history data, and the second risk reference unit Calculating a first probability of increasing the probability of occurrence of contingency, and outputting data of the first probability ;
The selection result is received from the first means , the contingency of the first risk reference unit is reduced based on the history data or the risk rating history data, and the second risk reference unit Calculating a second probability that the contingency is less likely to occur and outputting data of the second probability;
The selection result is received from the first means , the contingency of the first risk reference unit is increased based on the history data or the risk rating history data, and the second risk reference unit Calculating a third probability that the contingency is less likely to occur and outputting data of the third probability ;
The selection result is received from the first means , the contingency of the first risk reference unit is reduced based on the history data or the risk rating history data, and the second risk reference unit Calculating a fourth probability of increasing the probability of occurrence of contingency and outputting data of the fourth probability ;
The selection result is received from the first means, and the contingency of the first risk reference unit is increased or decreased based on the history data or the risk rating history data, and the second risk Calculating a fifth probability that the probability of occurrence of contingency of the reference unit does not change and outputting data of the fifth probability ;
Receiving the selection result from the first means, and based on the history data or the risk rating history data, the contingency of the first risk reference unit does not change, and the second risk reference unit Calculating a sixth probability that the contingency is increased or decreased , and outputting data of the sixth probability ;
Receiving the data of the first probability to the sixth probability from the second means to the seventh means , calculating an occurrence rank correlation coefficient based on the values from the first probability to the sixth probability, and generating the occurrence rank phase; An eighth means configured by the computing means for outputting the data of the relation number ;
Receiving the occurrence rank correlation coefficient data from the eighth means , multiplying the occurrence rank correlation coefficient by a constant, calculating the occurrence correlation parameter, and generating the occurrence correlation parameter data as the occurrence correlation parameter; The financial means or insurance product for multiple risks according to any one of claims 1 to 3, further comprising: ninth means configured by the computing means stored in the storage means as a table. Price computing device. The price calculation means includes:
A tenth means configured by the computing means for calculating a probability corresponding to the billing condition of the product based on the generated correlation strength and outputting data of the probability;
Data on the probability corresponding to the billing conditions for the product is received from the tenth means, and based on the probability and the value of the loss rate, a basic statistic including expected value and variance is calculated, and a plurality of risks are calculated. calculate the price of a product intended for, outputs and of the basic statistics data and said price data, the preceding claims, characterized in that it comprises a first 11 means constituted by said calculating means 4 A price calculation device for financial products or insurance products that targets multiple risks. A computer program for causing a computer to function as a price calculation apparatus including each means according to any one of claims 1 to 5 .

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