JP2004005626A - Bond investment analysis/credit risk quantitative analysis system - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a system capable of measuring the term structure of bankruptcy probability by the sort as the combination of the term structure of bankruptcy probability, rating, and categories of business of bond issuing companies, and recovery rate for each rating, and grasping theoretical price (average price) and expected loss amount (credit spread) of bonds issued by the companies in fields related to investment analysis/financial commodities, and credit risk evaluation/management. <P>SOLUTION: This system comprises a data storage means to store price data of government and corporate bonds, attribute data of the government and corporate bonds for each name, data of the categories of business of corporate bond issuing companies, and data of financial attributes of the corporate bond issuing companies, a calculation condition designating means, and an operation means to determine bond investment analysis or credit risk measuring analysis based on data inputted from the data storage means in accordance with calculation conditions designated by the calculation condition designating means. The operation means is provided with a bond theoretical price operating means to operate bond theoretical prices of each name, and an expected loss amount operating means to operate an expected loss amount for each name. <P>COPYRIGHT: (C)2004,JPO

Description

（キャッシュ・フロー関数波線Ｃ_ｉｔ（ｓ_ｔ）、確定的キャッシュ・フロー関数Ｃ_ｉｔ（ｓ_ｔ）、期待損失額Ｌ_ｉｔ（ｓ_ｔ）、割引率Ｄ_ｔ（ｓ_ｔ）は０≦ｓ_ｔ≦ｓ_ａＭで定義された関数）。
【０００８】
また、上記の演算式において、確率的割引率に割り引かれた価格変動部分ε_ｉｔの分散共分散構造を、
１）償還期間ｓ_{ｉＭ（ｉ）}が短くなると債券価格の変動が小さくなること。
２）銘柄間の償還期間差が小さいものほど連動性が高いこと。
という、現実に市場で観察される債券価格変動特性を考慮して、以下の分散共分散構造をもたせることができる。

この分散共分散構造において、ａ_ｉｕｔに関して、償還期間が長い債券ほど分散が大きく、償還期間が近い債券価格同士は相関が大きいことを表現するために、下記のような定式化を行うことができる。

本発明による債券投資分析・信用リスク計量分析システムの他の態様において、債券理論価格演算手段は、分析対象の債券銘柄群すべてのキャッシュ・フローの発生時点を考慮し、さらに予め把握することが困難な銘柄属性により起因された価格変動がｎ期間前の価格変動に残存しているとして考えた債券価格モデルに基づいて債券理論価格を算出することを特徴とする。
【０００９】
この債券理論価格演算手段において、分析対象の債券銘柄群すべてのキャッシュ・フローの発生時点を考慮し、さらに予め把握することが困難な銘柄属性により起因された価格変動がｎ期間前の価格変動に残存しているとして考えた下記のような債券価格モデルの演算式に基づいて債券理論価格を算出することができる。

但し、Ｐ_ｉｔ（０）は第ｉ債券のｔ時点での市場価格、
第ｉ債券の第ｊキャッシュ・フローの発生時点は

（キャッシュ・フロー関数波線Ｃ_ｉｔ（ｓ_ｔ）、確定的キャッシュ・フロー関数Ｃ_ｉｔ（ｓ_ｔ）、期待損失額Ｌ_ｉｔ（ｓ_ｔ）、割引率Ｄ_ｔ（ｓ_ｔ）は０≦ｓ_ｔ≦ｓ_ａＭで定義された関数）
また、この債券理論価格演算手段において、利子延滞や元本の返済不能など将来に発生するキャッシュ・フローに不確実性がある場合のキャッシュ・フロー及び期待損失額を、下記の演算式に基づいて算出することができる。

ここで、ｓ_ｊｔ＝ｓ_ｉｊｔでない時、キャッシュ・フロー関数波線Ｃ_ｉｔ（ｓ_ｉｊｔ）＝０
第１項の［１−ｈ_ｉｔ（ｓ_ｉｊｔ）］は、ｔ＋ｓ_ｉｊｔ時点までにこの企業が倒産しない確率
第２項の［ｈ_ｉｔ（ｓ_ｉｊｔ）−ｈ_ｉｔ（ｓ_{ｉｊ−１ｔ}）］は、この企業がｔ＋ｓ_{ｉｊ−１，ｔ}時点とｔ＋ｓ_ｉｊｔ時点の間に倒産する確率
第ｉ社債発行企業がｔ＋ｓ_{ｉｊ−１，ｔ}時点とｔ＋ｓ_ｉｊｔ時点の間で倒産しなかった場合ｈ_ｉｔ（ｓ_ｉｊｔ）＝０で、ｔ＋ｓ_ｉｊｔ時点でクーポンＣ_ｉｔ（ｓ_ｉｊｔ）が支払われる。
第ｉ社債発行企業がｔ＋ｓ_{ｉｊ−１，ｔ}時点とｔ＋ｓ_ｉｊｔ時点の間で倒産した場合ｈ_ｉｔ（ｓ_ｉｊｔ）＝１で回収すべき金額Ｖ_ｉｔの内、回収率γ_ｔによりＶ_ｉｔγ_ｔが回収される。
【００１０】
上記の回収率γ_ｔの演算手段は、下記の企業の格付ｋに依存するような演算式に基づいて回収率を算出することができる。
γ_ｔ＝γ_ｔ（ｋ（ｉ））
また、上記の倒産時に回収すべき金額の演算手段は、下記のような演算式に基づいて回収すべき金額を算出することができる。
Ｖ_ｉｔ＝１００
本発明による債券投資分析・信用リスク計量分析システムにおける債券投資分析または信用リスク計量分析を算出する演算手段は、さらに、個別企業の倒産確率の期間構造を演算する倒産確率期間構造演算手段と、債券の将来時点で発生するキャッシュ・フローに対する割引率の期間構造を演算する割引率期間構造演算手段と、発行企業の属性に含まれる格付・業種・財務指標のいずれかのカテゴリー毎あるいは任意のカテゴリーの組合せ毎の倒産確率の期間構造と回収率を演算するカテゴリー別倒産確率期間構造・回収率演算手段とを有する。
【００１１】
ここで、上記の倒産確率期間構造演算手段は、下記のような演算式に基づいて倒産確率の期間構造を算出することができる。

ここで、ｈ_ｉｔ（ｓ_ｔ）は倒産確率、
ｚ_ｉｊｔは、社債発行企業の業種や財務指標、格付けなどの属性、

ｓ_ｔ ^＊＝ｓ_ｔ ^１／ｍでｍは、例えば４、２、１。
【００１２】
また、上記の割引率期間構造演算手段は、第ｉ債券のｔ時点での価格変動構造を推定するために、割引率を期間の関数と考えた割引関数の確率プロセスを表した下記の演算式に基づいて、割引率の期間構造を演算することができる。

ここで、上線Ｄ_ｔ（ｓ_ｔ）は平均割引関数
未知パラメータδ_ｊｔは、銘柄全てに対して共通。
【００１３】
さらに、前記割引率期間構造演算手段は、さらなる態様として、第ｉ債券のｔ時点での価格変動構造を推定するために、割引率を期間の関数と考えた割引関数の確率プロセスを表した下記の演算式に基づいて、割引率の期間構造を演算することができる。

ここで、上線Ｄ_ｔ（ｓ_ｔ）は平均割引関数、
割引率は個別銘柄の属性ｚ_ｉ１ｔ、…、ｚ_ｉｑｔに依存して異なる、
すなわち、Ｄ_ｔ（ｓ_ａｊｔ）→Ｄ_ｉｔ（ｓ_ａｊｔ）、
ｐ次の多項式の係数は、ｑ個の銘柄属性に依存する関数。
【００１４】
さらに、前記割引率期間構造演算手段は、さらなる態様として、第ｉ債券のｔ時点での価格変動構造を推定するために、割引率を期間の関数と考えた割引関数の確率プロセスを表した下記の演算式に基づいて、割引率の期間構造を演算することができる。

ここで、期間ｓ_ｔを、

と変換した場合、期間ｓ_ｔの定義域［０，∞）が［０，１）になり、

から、

となる。ここで、κ、αは、平均割引関数の形状を決めるパラメータである。また、多項式近似した場合、

となる。ここで、上線Ｇ_ｔ（ｓ_ｔ ^＊）は平均割引関数、未知パラメータδ_ｊｔは、銘柄全てに対して共通。
【００１５】
さらに、前記割引率上線Ｇ_ｔ（ｓ_ｔ ^＊）は、さらなる態様として、第ｉ債券のｔ時点での価格変動構造を推定するために、割引率を期間の関数と考えた割引関数の確率プロセスを表した下記の演算式に基づいて、割引率の期間構造を演算することができる。

ここで、上線Ｇ_ｔ（ｓ_ｔ ^＊）は平均割引関数、
割引率は個別銘柄の属性ｚ_ｉ１ｔ、…、ｚ_ｉｑｔに依存して異なる、
すなわち、Ｇ_ｔ（ｓ_ｔ ^＊）→Ｇ_ｉｔ（ｓ_ｔ ^＊）、
ｐ次の多項式の係数は、ｑ個の銘柄属性に依存する関数。
本発明による債券投資分析・信用リスク計量分析システムの他の態様において、債券理論価格演算手段が、分析対象の債券銘柄群すべてのキャッシュ・フローの発生時点を考慮し、さらに予め把握することが困難な銘柄属性により起因された価格変動がｎ期間前の価格変動に残存しているとして考えた債券価格モデルに基づいて債券理論価格を算出するとき、平均割引関数の演算手段において、個別銘柄の属性ｚ_ｉ１ｔ、…、ｚ_ｉｑｔの１つに、予め把握することが困難な銘柄属性の代理変数として、求めた価格推定残差のｎ期間前の値を用いることができる。
【００１６】
一方、本発明による債券投資分析・信用リスク計量分析方法は、データ記憶手段に記憶された国債の価格データと、社債の価格データと、国債及び社債の銘柄属性データと、社債発行企業の格付けデータと業種データと、社債発行企業の財務的な属性データとから、算出条件である算出期間と銘柄と銘柄属性と企業属性と算出モデルとを指定するステップと、該指定された算出条件に従って債券の将来時点で発生するキャッシュ・フローに対する割引率の期間構造のパラメータを演算するステップと、該指定された算出条件に従って倒産確率の期間構造と回収率のパラメータを演算するステップと、求めたパラメータを用いて、個別銘柄の債券理論価格及び期待損失額と、個別企業の倒産確率と割引率の期間構造と、発行企業の属性に含まれる格付・業種・財務指標のいずれかのカテゴリー毎あるいは任意のカテゴリーの組合せ毎の倒産確率の期間構造と回収率とを演算するステップとを有する。
【発明の実施の形態】
本発明では、入力情報として、ｔ時点での国債価格とその銘柄属性（クーポン・レート、償還期間など）、同じ時点での社債価格とその銘柄属性（クーポン・レート、償還期間など）及び社債発行企業の属性（格付け、業種、財務指標など）の情報を用いる。これは、クーポン・レート、償還期間などの同じ銘柄属性を持つ社債価格の対国債価格との差は、その社債発行企業の信用リスクが市場で評価されたことによる。信用リスクは、基本的に契約が約定どおり実行されないリスクである。社債の信用リスクは、基本的に利子延滞や元本の返済不能であるが、投資家にとり企業の倒産にまで至らなくても信用不安の高まりによる債券価格の下落、すなわち、信用スプレッド（対国債価格との差）の増大が信用リスクである。もっとも、投資家の銘柄選好の背景には、信用リスクのみならず市場の流動性とも関係がある。すなわち、投資家は必ずしも償還時点まで保有するのでなく、途中で売却する際の換金容易性としての流動性も銘柄選好の理由となることから、国債と社債間で価格差が生じる場合もあるが、殆どが信用リスクによるものである。すなわち、本発明では、ｔ時点での国債価格とその銘柄属性（クーポン・レート、償還期間など）、同じ時点での社債価格とその銘柄属性（クーポン・レート、償還期間など）及び社債発行企業の属性（格付け、業種、財務指標など）情報とを入力情報とし、前記発行企業の属性に含まれる格付け、業種、財務指標の組み合わせ分類別、及び、個々の前記分類別、個々の企業の倒産確率の期間構造と前記社債発行企業の前記格付け、業種、財務指標などの属性毎の回収率とを計量する。
以下に本発明の実施の形態を、図面を参照して詳細に説明する。まず、本発明による債券投資分析・信用リスク計量分析システムを実施するためのハードウェア構成について、図１を用いて説明する。この債券投資分析・信用リスク計量分析システムは、システム全体を統括的に制御する、プログラムされた主制御部１に記憶装置２が接続されている。主制御部１には、また、入出力制御部３を介してキーボードやマウス等のポインティングデバイスからなる入力装置４、入力データのモニタ等に用いる表示装置５、及び分析結果や計量結果を出力する出力装置６が接続されている。
【００１７】
主制御部１は、ＯＳ（Ｏｐｅｒａｔｉｎｇ　ｓｙｓｔｅｍ）等の制御プログラム、債券投資分析・信用リスク計量分析を実行する演算の手順を規定した演算プログラム、及び所要データを格納するための内部メモリを有し、これらの演算プログラム等により、債券投資分析または信用リスク計量分析を算出する演算手段を実現している。
【００１８】
記憶装置２は、ハードディスクやフレキシブルディスク、光ディスク等のストレージ手段であり、債券投資分析・信用リスク計量分析を実行するために入力するデータを蓄積するデータベース２１と、上記した演算手段を実現するために主制御部１に読み込まれる種々の演算プログラムを蓄積する演算プログラムファイル２２を格納する。なお、記憶装置２は図１では一つのブロックとして表しているが、複数のストレージ手段を用いてもよいし、ネットワークを介して接続されていてもよい。
本実施の形態では、国債と社債の価格データ、銘柄属性データ及び社債発行企業の属性データをデータベース２１から入力し、演算プログラムファイル２２から主制御部１に読み込まれた演算プログラムにより所定の演算を実行し、個々の社債の理論価格（期待価格）、期待損失額、個別企業の倒産確率の期間構造、発行企業の属性に含まれる格付け、業種、財務指標の組み合わせグループ毎の倒産確率の期間構造と回収率、発行企業の属性に含まれる格付け、業種、財務指標の個々の倒産確率の期間構造と回収率を演算結果として出力装置６に出力する。演算手段は、後述する演算式によって実施される。
次に、データベース２１に蓄積されているデータの明細について説明する。日付、国債（長期国債、超長期国債など）、社債（一般社債、電力債、ＪＲ債、ＪＴ債など）といった債券種別や回号、市場価格（終値、売気配値、買気配）、クーポン・レート、発行日、償還日、初回利払日、利払月、利払日、経過利息、残存年数、発行額、指標銘柄フラグ、格付け（日本格付投資情報センター、日本格付研究所などの格付評価会社から付与されている格付）、業種、財務指標（自己資本比率、負債比率、売上高利益率など）などである。なお、図３は、データベース２１に蓄積されているデータの明細の一例を示した図表である。
以下に、まず本発明による債券投資分析または信用リスク計量分析を演算手段を用いて実現するための理論的なベースについて説明する。
【００１９】
現在時点をｔ時点と考え、ｔ時点での市場ではＮ_ｔ銘柄取引されていると考える。この時、第ｉ債券についてｔ時点で観測できる要素は、
１）ｔ時点の第ｉ債券の市場価格：Ｐ_ｉｔ（０）
２）クーポン、償還時の額面単価を含めたキャッシュ・フロー関数（キャッシュ・フローの発生時点を、期間の関数と考える）：Ｃ_ｉｔ
３）クーポン・レート、償還期間などの銘柄属性：Ｚ_ｉｔ＝｛ｚ_ｉｋｔ：ｋ＝１，…，ｑ｝
である。日本市場で取り引きされている債券の多くは、利付債であり、銘柄ごとにクーポンなどのキャッシュ・フローの発生時点が異なる。それを明確に示すために、ｔ時点からみた第ｉ債券の第ｊキャッシュ・フローの発生時点を、
【００２０】
【数１】

と表現する。そして、市場で取引きされているＮ_ｔ銘柄のキャッシュ・フローの発生時点のすべてを小さい順に並べたものを、
【００２１】
【数２】

された関数とし、償還時点の額面１００円もＣ_ｉｔ（ｓ_ａｊｔ）の中に含まれていると仮定する。従って、第ｉ債券のｔ時点の市場価格Ｐ_ｉｔ（０）は、将来のｔ＋ｓ_ａｊｔ時点で発生するキャッシュ・フローがＣ_ｉｔ（ｓ_ａｊｔ）、このキャッシュ・フローに対する割引率をＤ_ｔ（ｓ_ａｊｔ）、償還期間をＭとしたとき、
【００２２】
【数３】

と表現できる。但し、ｓ_ａｊｔ＝ｓ_ｉｊｔでない時、Ｃ_ｉｔ（ｓ_ａｊｔ）＝０である。実際の市場価格は、様々な要因によって変動し、確率変数の実現値とみなされる。企業が発行した一般社債には、通常、利子延滞や元本の返済不能など将来に発生するキャッシュ・フローに不確実性がある。第ｉ社債発行企業がｔ＋ｓ_ｉｊｔ時点までに倒産する確率は、企業の格付ｋに依存するとして、
【００２３】
【数４】

とし、第ｉ社債発行企業がｔ＋ｓ_{ｉｊ−１，ｔ}時点とｔ＋ｓ_ｉｊｔ時点の間に倒産した場合、元本の回収はｔ＋ｓ_ｉｊｔ時点で行われるとする。回収率γ_ｔ（ｋ（ｉ））は、将来の回収時点ｔ＋ｓ_ｉｊｔや同じ格付けでも企業が違えば異なると考えられる。例えば、ここでは企業の格付ｋに依存すると仮定する。第ｉ社債発行企業がｔ＋ｓ_{ｉｊ−１，ｔ}時点とｔ＋ｓ_ｉｊｔ時点の間で倒産しなかった場合、ｈ_ｉｔ（ｓ_ｉｊｔ）＝０であり、ｔ＋ｓ_ｉｊｔ時点でクーポンＣ_ｉｔ（ｓ_ｉｊｔ）が支払われる。一方、第ｉ社債発行企業がｔ＋ｓ_{ｉｊ−１，ｔ}時点とｔ＋ｓ_ｉｊｔ時点の間で倒産した場合、ｈ_ｉｔ（ｓ_ｉｊｔ）＝１で元本の内１００γ_ｔ（ｋ（ｉ））が回収される。
従って、ｋ格付の第ｉ企業発行社債のキャッシュ・フロー関数波線Ｃ_ｉｔ（ｓ_ｉｊｔ）は、
【００２４】
【数５】

となる。ここで、ｓ_ｊｔ＝ｓ_ｉｊｔでない時、キャッシュ・フロー関数波線Ｃ_ｉｔ（ｓ_ｉｊｔ）＝０である。上式の第１項の［１−ｈ_ｉｔ（ｓ_ｉｊｔ）］は、ｔ＋ｓ_ｉｊｔ時点までにこの企業が倒産しない確率であり、第２項の［ｈ_ｉｔ（ｓ_ｉｊｔ）−ｈ_ｉｔ（ｓ_{ｉｊ−１ｔ}）］は、この企業がｔ＋ｓ_{ｉｊ−１，ｔ}時点とｔ＋ｓ_ｉｊｔ時点の間に倒産する確率である。この式を以下のように、将来に発生するキャッシュ・フローが確定的である部分Ｃ_ｉｔ（ｓ_ｉｊｔ）と期待損失額Ｌ_ｉｔ（ｓ_ｉｊｔ）の部分に分離した形に変形すると、
【００２５】
【数６】

となる。
【００２６】
数３で表現した第ｉ債券のｔ時点の債券価格モデルに関して、将来のキャッシュ・フローがＣ_ｉｔ（ｓ_ａｊｔ）が確定的な国債などの債券を考える場合、確率変数である市場価格との関係で割引率Ｄ_ｔ（ｓ_ａｊｔ）が確率変数となる。すなわち、国債の市場価格の実現は、その背後にある確率的な割引率Ｄ_ｔ（ｓ_ａｊｔ）の実現と同等である。しかし、数３の左辺の確率変数である債券価格は、１個に対して右辺には将来のキャッシュ・フローに対応して割引率がＭ個あり、債券価格と割引率は、１対１対応していない。従って、債券価格に対して以下の割引関数の確率プロセスが対応する。
【００２７】
【数７】

すなわち、債券価格が市場で実現することは、割引関数の確率プロセスの１つのパスが実現したとみる。従って、数３で表現した第ｉ債券のｔ時点の債券価格モデルにおいて、割引率が確率変数であり、普通社債のように信用リスクがある債券の場合には、キャッシュ・フローも確率変数となる。倒産生起プロセスｈ_ｉｔ（ｓ_ｔ）、回収率プロセスγ_ｔ（ｋ（ｉ））、割引率のプロセスＤ_ｔ（ｓ_ｔ）は独立とし、分析対象のＮ_ｔ銘柄すべてのキャッシュ・フロー発生時点を考慮して、キャッシュ・フロー関数波線Ｃ_ｉｔ（ｓ_ｔ）、確定的キャッシュ・フロー関数Ｃ_ｉｔ（ｓ_ｔ）、期待損失額Ｌ_ｉｔ（ｓ_ｔ）、割引率

【００２８】
【数８】

で表わす。第ｉ債券のｔ時点での市場価格Ｐ_ｉｔ（０）は、
【００２９】
【数９】

と表現される。上線Ｄ_ｔ（ｓ_ａｊｔ）は、割引関数Ｄ_ｔ（ｓ_ａｊｔ）の平均値である。第ｉ債券のｔ時点での価格変動構造を推定するには、割引率を期間の関数と考えた割引関数の確率プロセスを表した割引関数モデルが必要であり、下記で定式化する。
【００３０】
【数１０】

ここで、未知パラメータδ_ｊｔは、銘柄全てに対して共通である。なお、割引率が個別銘柄の属性ｚ_ｉ１ｔ、…、ｚ_ｉｑｔに依存して異なる、すなわち、Ｄ_ｔ（ｓ_ａｊｔ）→Ｄ_ｉｔ（ｓ_ａｊｔ）と考え、平均割引関数として、
【００３１】
【数１１】

を仮定することもできる。ここで、ｐ次の多項式の係数は、ｑ個の銘柄属性に依存する関数である。
【００３２】
さらに、第ｉ債券のｔ時点での価格変動構造を推定するために、割引率を期間の関数と考えた割引関数の確率プロセスを表した割引関数モデルのさらなる関数として、下記で定式化する。
【数１２】

ここで、期間ｓ_ｔを、
【数１３】

と変換した場合、期間ｓ_ｔの定義域［０，∞）が［０，１）になり、
【数１４】

から、
【数１５】

となる。ここで、κ、αは、平均割引関数の形状を決めるパラメータである。また、多項式近似した場合、
【数１６】

となる。ここで、上線Ｇ_ｔ（ｓ_ｔ ^＊）は平均割引関数、未知パラメータδ_ｊｔは、銘柄全てに対して共通である。なお、割引率が個別銘柄の属性ｚ_ｉ１ｔ、…、ｚ_ｉｑｔに依存して異なる、すなわち、Ｇ_ｔ（ｓ_ａｊｔ）→Ｇ_ｉｔ（ｓ_ａｊｔ）と考え、平均割引関数として、
【００３３】
【数１７】

を仮定することもできる。ここで、ｐ次の多項式の係数は、ｑ個の銘柄属性に依存する関数である。
【００３４】
次に、確率的割引率に割り引かれた価格変動部分ε_ｉｔの分散共分散構造の定式化にあたって、現実に市場で観察される次のような債券価格変動特性を考慮している。
１）償還期間ｓ_{ｉＭ（ｉ）}が短くなると債券価格の変動が小さくなること。
２）銘柄間の償還期間差が小さいものほど連動性が高いこと。
つまり、各銘柄のε_ｉｔの分散が、償還時点に近づくにつれて小さくなること、また、各銘柄間の償還期間差が大きいものほど、ε_ｉｔの連動性が低くなることを考慮している。具体的には、ε_ｉｔに対して、次の分散共分散構造を仮定する。
【００３５】
【数１８】

ここで、ａ_ｉｕｔに関して、例えば
【００３６】
【数１９】

を仮定する。数１８と数１９は、償還期間が長い債券ほど分散が大きく、償還期間が近い債券価格同士は相関が大きいことを表現する。また、キャッシュ・フローが発生する２時点間に関して、期間が長いほど割引関数の相関が小さくなるように、Φ_ｉｕｔに関して次式を仮定する。
【００３７】
【数２０】

ここで、Φ_ｉｕｔは、ｓ_ａｊｔとｓ_ａｒｔ時点に発生するクーポンを割り引く割引率の共分散に対応する。
数１８の定式化で重要な点は、倒産確率が小さくなるにつれて、期待損失額が小さくなり、信用リスクが無い債券の価格変動に近づく構造となっていることと、同一企業が発行した債券でも償還期間が長い長期債の方がより大きな分散をもつという構造が自然に導入されている点である。なお、平均割引関数上線Ｇ_ｔ（ｓ_ｔ ^＊）、もしくは、上線Ｇ_ｉｔ（ｓ_ｔ ^＊）の場合には、期間ｓ_ｔを数１３により変換したｓ_ｔ ^＊を用いる。
わが国では社債を発行している企業の倒産件数が現時点では少なく、倒産確率をデータとして直接入手できないため、倒産確率を得るには何らかの倒産確率を表現したモデルを考える必要がある。ここでは、同じ格付けの企業でも、業種、財務体質により倒産確率が異なると考えられることから、企業がｔ時点からｓ_ｉｊｔ期間内に倒産する確率を定式化するにあたり、社債発行企業の業種や財務指標、格付けなどの属性を考慮した関数として、次のｐ次の多項式を考える。
【００３８】
【数２１】

ここで、ｓ_ｔ ^＊＝ｓ_ｔ ^１／ｍでｍは、例えば４、２、１などである。数２１の未知パラメータ

ルを第１モデルと呼ぶことにする。
次に、ｔ時点における銘柄間のクロス・セクション・モデルとしての第１モデルを動学化（時系列化）したモデルについて説明する。社債の市場価格は、事前に把握できる銘柄属性だけでなく、予め直接把握が困難な銘柄属性からも起因する。個々の企業の倒産確率に影響を与える企業属性に明示的に把握できる属性だけでなく、直接把握が困難な属性も含まれる。そこで、次に説明するモデルは、予め明示的に直接把握することが困難な銘柄属性を時系列属性としてモデルの中に導入するものである。実際の現象をより表現するため、様々な銘柄属性を纏めたインプリシットな銘柄属性として、時系列的属性を用いた債券価格モデルである。すなわち、予め把握することが困難な銘柄属性により起因された価格変動がｎ期間前の価格変動に残存しているとして考えるモデルである。例えば、このタイプのモデルとして、以下の２つのモデルを例示する。
（１）タイプ１
第ｉ債券のｔ時点の確率的割引関数Δ_ｔ（ｓ_ｔ）がｔ−ｎ時点の割引関数に依存すると考えたもので、平均割引関数上線Ｄ_ｔ（ｓ_ｔ）からの乖離部分、
【００３９】
【数２２】

に対して、以下の時系列モデル
【００４０】
【数２３】

を仮定する。従って、数２２と数２３から割引関数モデルは、
【００４１】
【数２４】

で表現される。次に、確定的なキャッシュ・フローＣ_ｉｔ（ｓ_ｔ）と期待損失額Ｌ_ｉｔ（ｓ_ｔ）の関数は、
【００４２】
【数２５】

を満たすことに注意して、
【００４３】
【数２６】

とおくと、数９に対応して、第ｉ社債のｔ時点での債券価格Ｐ_ｉｔ（０）は、
【００４４】
【数２７】

ここで、
【００４５】
【数２８】

である。ε_ｔ−ｎ（ｓ_ａｔ−ｎ）は、ｔ−ｎ期間前の時点で第１モデルを推定した際のε_ｔ−ｎである。なお、ψ_ｉｔ（ｓ_ａｔ）の分散共分散構造は、数１８〜数２０と同じである。数２７と数２８で表現された債券価格モデルを第２モデルと呼ぶことにする。
（２）タイプ２
ｎ期間前の価格推定残差ε_ｔ−ｎ（ｓ_ａｔ−ｎ）を予め識別されない様々な属性を纏めて表現したものと考え、割引率が個別銘柄の属性ｚ_ｉ１ｔ、…、ｚ_ｉｑｔに依存して異なる平均割引関数
【００４６】
【数２９】

の第ｑ属性とする。この属性は時間的に変化し、この変数の導入により個別銘柄ごとの微妙な平均割引関数の変化による価格変動を把握できるものと考えられる。数１０の平均割引関数の銘柄属性の１つに、ｎ期間前の価格推定残差ε_ｔ−ｎ（ｓ_ａｔ−ｎ）を用いた他は、当モデルは、モデル１と同じ構造である。当モデルを第３モデルと呼ぶことにする。なお、平均割引関数モデルとして、数１６で定式化した上線Ｇ_ｔ（ｓ_ｔ ^＊）、もしくは、数１７の上線Ｇ_ｉｔ（ｓ_ｔ ^＊）を用いる場合には、期間ｓ_ｔを数１３により変換したｓ_ｔ ^＊を用いる。
以上の各モデルより、債券の個別銘柄毎の理論価格（平均価格）・倒産確率、割引率、業種や各属性毎の倒産確率・回収率が算出される。次に、モデルの各未知

回収率γ_ｔ（ｋ（ｉ））、分散共分散構造のρ_ｔ、σ_ｔ ^２）を推定する方法について説明する。まず、信用リスクの無い国債のような債券から割引関数モデルの未知パラメータδ_１ｔ，…，δ_ｐｔを推定する。期待損失額Ｌ_ｉｔを０とした数９から、以下の式が導出される。
【００４７】
【数３０】

ここで、
【００４８】
【数３１】

である。数３０で表されたモデルの対数尤度１は、
【００４９】
【数３２】

である。対数尤度を最大化することで、未知パラメータを求めることができる。数３２に関して最大となる条件は、
【００５０】
【数３３】

である。よって、σ_ｔ ^２の最尤推定値は、
【００５１】
【数３４】

で求められる。山線σ_ｔ ^２を数３２に代入すると、最大対数尤度は、
【００５２】
【数３５】

となる。数１０の平均割引関数における次数や属性を選択するには、ＡＩＣ（Ａｋａｉｋｅ　Ｉｎｆｏｒｍａｔｉｏｎ　Ｃｒｉｔｅｒｉｏｎ）やＳＢＣ（Ｓｃｈｗａｒｔｚ　Ｂａｙｓｉａｎ　Ｃｒｉｔｅｒｉｏｎ）などの統計量を評価基準として用いる。数３０のモデルのＡＩＣ_１は、
【００５３】
【数３６】

となる。数３２で未知パラメータの最尤推定値を求めるには、非線形最適化のアルゴリズムにより求めることができる。なお、平均割引関数モデルとして、数１６で定式化した上線Ｇ_ｔ（ｓ_ｔ ^＊）、もしくは、数１７の上線Ｇ_ｉｔ（ｓ_ｔ ^＊）の場合には、期間ｓ_ｔを数１３により変換したｓ_ｔ ^＊を用いて数３０から数３６の計算がなされる。
次に、信用リスクの無い国債のような債券から以上の方法で得られた割引関数モデルを用いて、数９から、以下の式が導出される。
【００５４】
【数３７】

ここで、
【００５５】
【数３８】

である。θ_ｔ＝（β_ｔ ^＊，γ_ｔ（ｋ（ｉ）），ρ_ｔ）とすると、数３７で表されたモデルの対数尤度１は、
【００５６】
【数３９】

である。対数尤度を最大化することで、未知パラメータθ_ｔを求めることができる。数３９に関して最大となる条件は、
【００５７】
【数４０】

である。よって、σ_ｔ ^２の最尤推定値は、
【００５８】
【数４１】

で求められる。山線σ_ｔ ^２を数３９に代入すると、最大対数尤度は、
【００５９】
【数４２】

となる。数２１の倒産確率関数における次数や属性を選択するには、既に述べたＡＩＣやＳＢＣなどの統計量を評価基準として用いる。数３７のモデルのＡＩＣ_ｔは、
【００６０】
【数４３】

となる。数３７で未知パラメータの最尤推定値を求めるには、非線形最適化のアルゴリズムにより求めることができる。
【００６１】
以上で、本発明による債券投資分析・信用リスク計量分析システムのベースとなる算出モデルの理論的な説明を終える。次に、本実施の形態における演算のステップについて、図１、図２、図４、及び図５を参照して説明する。
【００６２】
まず、図２は本発明による債券投資分析・信用リスク計量分析システム全体のフローを示した図である。ステップＢ１において、債券投資分析・信用リスク計量分析を実行するために、分析結果を求める前提となる算出条件が入力される（ステップＢ１）。ここでは、算出期間、銘柄、銘柄属性、企業属性、算出モデルが指定される。これらの算出条件に従って、データベース２１から抽出されたデータが主制御部１に読み込まれる。次に、読み込まれたデータと演算プログラムファイル２２から読み込まれる演算プログラムの実行により、割引率の期間構造などのパラメータが演算される（ステップＢ２）。さらに、同様にして、倒産確率の期間構造や回収率などのパラメータが演算される（ステップＢ３）。これらの求められたパラメータに基づいて、債券理論価格及び期待損失額の演算が実行される（ステップＢ４）。ステップＢ４で求めた債券理論価格及び期待損失額に基づいて、計量分析結果、具体的には倒産確率の期間構造、債券の将来時点で発生するキャッシュ・フローに対する割引率の期間構造、社債発行企業の属性に含まれる格付・業種・財務指標のいずれかのカテゴリー毎あるいは任意のカテゴリーの組合せ毎の倒産確率の期間構造と回収率、が出力される（ステップＢ５）。
【００６３】
図４は、国債のように信用リスクのない債券の割引関数モデルを計算するフローを示したフローチャートである。まずステップＣ１で入力される算出条件に基づき入力されたデータに従って、債券のキャッシュ・フローの発生時点が確認される。具体的には、数２の計算が実行される（ステップＣ２）。次に入力された価格データと銘柄属性データに基づいて相互のデータの関係が設定され（ステップＣ３）、数３０のｙ_ｔとＸ_ｔを求め、数３２に基づいて、対数尤度を最大化することで数３０に登場する未知のパラメータβ_ｔを求めるルーチンに入る（ステップＣ４、Ｃ５、Ｃ６）。そのパラメータに基づいて割引率の期間構造を算出する（ステップＣ７）。
【００６４】
図５は、債券発行企業の倒産確率の期間構造、格付や業種など各種企業属性を組合わせた分類別の倒産確率の期間構造、格付毎の回収率、企業が発行した債券の理論価格（平均価格）、及び期待損失額（信用スプレッド）を求めるためのフローを示したフローチャートである。まず、債券の価格と企業属性と図４のフローで求めた信用リスクのない国債のような債券から得られた割引関数モデルの設定データが入力される（ステップＤ１）。次に図示は省略したが、図４のステップＣ２とＣ３と同様に、債券のキャッシュ・フローの発生時点が確認され、次に入力された価格データと銘柄属性データに基づいて相互のデータの関係が設定される。次に、数３７、数３９に基づいて、数３７における未知のパラメータγ_ｔ（ｋ（ｉ））、

れらの求められたパラメータに基づいて、倒産確率の期間構造及び回収率が算出される（ステップＤ５）。ステップＤ６では、ステップＤ５で算出された倒産確率の期間構造及び回収率に基づいて、債券理論価格、期待損失額、価格推定残差が算出され、モデル１ではここで計算が終了する（ステップＤ７）。なお、モデル２、モデル３では、ステップＤ６で算出された価格推定残差をも入力データとし、モデル１と同じ計算ステップで未知パラメータを求める。
【００６５】
なお、図６、図７、図８及び図９は、本発明による債券投資分析・信用リスク計量分析システムを用いて、分析対象銘柄数・回収率・ρ_ｔ・倒産確率関数モデルのパラメータ、各種企業属性を組合わせた分類別の倒産確率の期間構造、企業が発行した債券の理論価格（平均価格）・期待損失額（信用スプレッド）をそれぞれ出力した例を示す図である。
【００６６】
【発明の効果】
以上のように本発明によれば、債券の価格、銘柄属性、債券発行企業の属性情報を入力することで、債券発行の企業の倒産確率の期間構造、格付や業種など各種企業属性を組み合わせた分類別の倒産確率の期間構造、及び格付け毎の回収率を計量でき、かつ、企業が発行した債券の理論価格（平均価格）、及び、期待損失額（信用スプレッド）を計量分析することができることから、保険・銀行・証券会社や投資顧問会社などの金融機関、及び、株式や債券に資金投資する事業法人や個人において実施される債券投資手法、債券投資システム、金融商品、信用リスク管理システム、信用リスク・デリバティブの評価といった投資分析・金融商品、信用リスク評価・管理に関係する分野全般における、債券発行の企業の倒産確率の期間構造、格付や業種など各種企業属性を組み合わせた分類別の倒産確率の期間構造、及び格付け毎の回収率を計量でき、かつ、企業が発行した債券の理論価格（平均価格）、及び、期待損失額（信用スプレッド）を計量分析することができる。
【図面の簡単な説明】
【図１】本発明を実施するためのハードウェア構成を示したブロック図である。
【図２】本発明における演算のステップを示すフローチャートである。
【図３】データベースの内容の例を示す図表である。
【図４】割引関数モデルを計算するフローチャートである。
【図５】債券発行の企業の倒産確率の期間構造、格付や業種など各種企業属性を組み合わせた分類別の倒産確率の期間構造、及び格付け毎の回収率、企業が発行した債券の理論価格（平均価格）、及び、期待損失額（信用スプレッド）を計算するフローチャートである。
【図６】分析対象銘柄数、回収率、ρ_ｔ、倒産確率関数モデルのパラメータの出力例を示した図である。
【図７】各種企業属性を組み合わせた分類別の倒産確率の期間構造の出力例を示した図である。
【図８】企業が発行した債券の理論価格（平均価格）の出力例を示した図である。
【図９】企業が発行した債券の期待損失額（信用スプレッド）の出力例を示した図である。[0001]
TECHNICAL FIELD OF THE INVENTION
The present invention relates to a bond investment method, a bond investment system, a financial product, and a credit risk management performed by a financial institution such as an insurance, bank, securities company, investment advisory company, and a business corporation or an individual investing in stocks and bonds. Related to investment analysis and financial products such as systems, credit risk and derivative evaluation, and credit risk evaluation and management.
[0002]
[Prior art]
Since the late 1990s, the amount of corporate bonds issued has increased sharply as an important means of financing companies. In addition, corporate bonds are gaining in importance as a valuable investment target in securities investment strategies against the background of sluggish stock prices and ultra-low interest rates. On the other hand, in recent years, credit risks of companies have been rapidly becoming apparent. Therefore, for investors, the risk of corporate bankruptcy, the principal recovery rate in the event of bankruptcy, and the decline in bond prices due to increased credit uncertainty even if bankruptcy does not occur, that is, credit spread (expected loss) In order to grasp the increase, it is indispensable to obtain a bankruptcy probability, a period structure and a recovery rate, and a proper price (theoretical price) and a proper expected loss when making a decision on buying and selling.
Important issues related to the traditional bond price model and the method of measuring credit risk information include: (1) whether the bond price model is a normative model or a statistical model (financial econometric model); The criteria or (3) what data is used as the information source. In recent years, when modeling bond prices, they are often prescriptively modeled as "interest rate" derivatives.
[0003]
The normative model is a model that presupposes an ideal situation such as efficiency and nonarbitration in the market, and pre-requires the mechanism of the market price and the status of the market formation. In addition, because of the relationship with the actual money market, the way of grasping on the basis of interest rates has become general. However, what actually exists in the bond market is the price, and the bond market is not in such an ideal situation. Also requires a realistic response. As pointed out in the practical world, models that deviate from actual phenomena become desk theory. That is, a price model that is in line with Mark to the reality is required.
[0004]
The bond price model as a derivative of "interest rate" takes the approach of deriving a formal price model, even if interest rates do not always correspond to actual interest rate fluctuations, and estimating parameters according to the realized market price of the bond . Many of the normative bond price models are based on a stochastic process of instantaneous spot rates. In the case of interest-bearing bonds, they are priced as the sum of the discount rates of discounted bonds at the time of cash flow, which are valued at their present value. This approach uses only the instantaneous spot rate stochastic process to represent the term structure of the discount rate, and the medium- to long-term interest rate is expressed as the conditional expectation of the instantaneous spot rate accumulated over the future period. Therefore, it is expressed as a function of the instantaneous spot rate. That is, all discount rates (interest rates) are deterministic functions of ultra-short-term interest rates, such as overnight call rates. The model is merely a formula that gives a conversion formula from the ex post market price to a parameter. The model itself as a formula has no scientific value and does not express phenomena such as actual bond market price fluctuations. That is, to create a bond price model, it is important to model the market price of the bond from the beginning, not the interest rate. At present, all corporate bonds in Japan are interest-bearing bonds, and there are no discount bonds.
The quantitative analysis method of credit risk information such as bankruptcy probability and recovery rate includes the following sources: cumulative bankruptcy probability by rating provided by rating companies and credit information companies, rating transition probability provided by rating companies, past bankruptcy companies. Some are based on data, stock prices, stock returns, bond prices, etc. There are very few cases where rating agencies measure the recovery rate other than providing historical data.
[0005]
In recent years, data on bankruptcy has been accumulating in Japan due to the default of corporate bond issuers, but the number is extremely small compared to the US market at present. Under such circumstances, corporate bond price and stock price data, including credit risk forecasts, based on various corporate information of many investors will be an important source of credit risk information. The corporate value approach is famous as a method of obtaining the probability of bankruptcy. In Japan, most bankruptcy probability measurement methods used are based on this approach. This approach is derived from Merton's (1974) stock price determination theory, and is a method of assuming bankruptcy when the corporate value falls below the debt level, and formulating the process of changing corporate value and estimating the probability of bankruptcy. Is what you do. This approach estimates unknown parameters in a corporate value change process model from stock price change characteristics. However, stocks have no maturity, so information on the expected bankruptcy time of investors, such as when the company will go bankrupt in the future, is weak. It is considered that the degree of influence by various factors is large. On the other hand, bonds have a maturity and, in some cases, the bond issuer has issued multiple stocks with different maturities, so the price of these bonds will give investors More detailed information about expected information can be grasped. And since corporate bonds are debts to the company, if the company goes bankrupt, the funds are repaid to the bond investors in preference to the shareholders, so the principal recovery rate is also extremely important for the bond investors, As in the case of the bankruptcy probability period structure, the bond price largely reflects credit risk information on the recovery rate.
The present invention has been made in view of the above situation, and an object of the present invention is to provide financial institutions such as insurance, banks, securities companies and investment advisory companies, and business corporations that invest in stocks and bonds. In the fields related to investment analysis, financial products, and credit risk evaluation and management, such as bond investment methods, bond investment systems, financial products, credit risk management systems, and credit risk derivative evaluations performed by individuals, By inputting the issue attributes and the attribute information of the bond issuing company, the term structure of the bankruptcy probability of the bond issuing company, the term structure of the bankruptcy probability by classification combining various corporate attributes such as rating and industry, and collection by rating We provide a system that can measure the rate and understand the theoretical price (average price) and expected loss (credit spread) of bonds issued by companies. It is to.
[0006]
[Means for Solving the Problems]
In order to solve the above-mentioned problems, the bond investment analysis / credit risk measurement analysis system according to the present invention provides government bond price data, corporate bond price data, government bond and corporate bond attribute data, and corporate bond issuer rating data. Data storage means for storing the financial data of the bond issuance company, business type data, and calculation condition specifying means for specifying the calculation period, brand, brand attribute, company attribute, and calculation model. Calculating means for calculating a bond investment analysis or a credit risk measurement analysis based on data input from the data storage means according to calculation conditions specified by the means. The calculating means is a bond theoretical price calculating means for calculating the theoretical bond price of the individual issue, and an expected loss amount calculating means for calculating the expected loss amount of the individual issue. Characterized by comprising and.
[0007]
In the above-mentioned bond investment analysis / credit risk measurement analysis system, the bond theoretical price calculation means calculates the bond theoretical price based on the calculation formula of the calculation model in consideration of the generation point of the cash flow of all the bond issues to be analyzed below. Is calculated.

(Cash flow function wavy line C_it(S_t), Deterministic cash flow function C_it(S_t), Expected loss L_it(S_t), Discount rate D_t(S_t) Is 0 ≦ s_t≤s_aMFunction defined in).
[0008]
Further, in the above equation, the price fluctuation portion ε discounted to the stochastic discount rate_itThe covariance structure of
1) Redemption period s_{iM (i)}The shorter the is, the smaller the change in bond prices.
2) The smaller the difference between the redemption periods between issues, the higher the linkage.
Considering the bond price fluctuation characteristics actually observed in the market, the following variance-covariance structure can be provided.

In this dispersion covariance structure, a_itIn order to express that a bond with a longer redemption period has a larger variance and a bond price with a shorter redemption period has a larger correlation, the following formulation can be performed.

In another embodiment of the bond investment analysis / credit risk measurement analysis system according to the present invention, the bond theoretical price calculation means considers the time of occurrence of the cash flows of all the bond issues to be analyzed, and is difficult to grasp in advance. The present invention is characterized in that a theoretical bond price is calculated based on a bond price model which is considered to have price fluctuations caused by various brand attributes remaining in price fluctuations n periods ago.
[0009]
In this bond theoretical price calculation means, the price fluctuation caused by the security attribute which is difficult to grasp in advance is considered in consideration of the time of occurrence of the cash flows of all the security issues to be analyzed, and the price fluctuation before n periods is changed. The theoretical bond price can be calculated based on the following formula for calculating the bond price model, which is considered as remaining.

Where P_it(0) is the market price of the i-th bond at time t,
The j-th cash flow of the i-th bond is

(Cash flow function wavy line C_it(S_t), Deterministic cash flow function C_it(S_t), Expected loss L_it(S_t), Discount rate D_t(S_t) Is 0 ≦ s_t≤s_aMFunction defined in
In addition, the theoretical bond price calculation means calculates the cash flow and expected loss amount in the case where there is uncertainty in the future cash flow such as interest overdue or unrecoverable principal, based on the following calculation formula. Can be calculated.

Where s_jt= S_ijtIf not, the cash flow function wavy line C_it(S_ijt) = 0
Item [1-h]_it(S_ijt)] Is t + s_ijtProbability that this company will not go bankrupt by time
The second term [h_it(S_ijt) -H_it(S_ij-1t)] Means that this company is t + s_{ij-1, t}Time point and t + s_ijtProbability of going bankrupt during time
The company issuing the i-th bond is t + s_{ij-1, t}Time point and t + s_ijtIf you did not go bankrupt during the time_it(S_ijt) = 0, t + s_ijtCoupon C at time_it(S_ijt) Will be paid.
The company issuing the i-th bond is t + s_{ij-1, t}Time point and t + s_ijtIn case of bankruptcy between time points_it(S_ijt) = 1 the amount V to be collected_itOf which, the recovery rate γ_tBy V_itγ_tIs collected.
[0010]
The above recovery rate γ_tCan calculate the recovery rate based on an arithmetic expression that depends on the rating k of the following company.
γ_t= Γ_t(K (i))
Further, the calculating means of the amount to be collected at the time of bankruptcy can calculate the amount to be collected based on the following arithmetic expression.
V_it= 100
The calculation means for calculating the bond investment analysis or the credit risk measurement analysis in the bond investment analysis / credit risk measurement analysis system according to the present invention further comprises: a bankruptcy probability period structure calculation means for calculating the period structure of the bankruptcy probability of an individual company; Means for calculating the term structure of the discount rate for cash flows generated in the future at the future point of time, and for each category or any category of rating, industry, or financial indicators included in the attributes of the issuer It has a bankruptcy probability period structure and recovery rate calculating means for each category for calculating a bankruptcy probability period structure and a recovery rate for each combination.
[0011]
Here, the bankruptcy probability period structure calculating means can calculate the bankruptcy probability period structure based on the following arithmetic expression.

Where h_it(S_t) Means bankruptcy probability,
z_ijtAre attributes such as the type of corporate bond issuer, financial indicators, and ratings,

s_t ^*= S_t ^{1 / m}And m is, for example, 4, 2, 1.
[0012]
Further, the above-mentioned discount rate period structure calculating means calculates the following equation, which represents a probability process of a discount function in which the discount rate is considered as a function of the period, in order to estimate the price fluctuation structure of the i-th bond at time t. , The period structure of the discount rate can be calculated.

Where the overline D_t(S_t) Is the average discount function
Unknown parameter δ_jtIs common to all brands.
[0013]
Further, as a further aspect, the discount rate period structure calculating means expresses a probability process of a discount function in which a discount rate is considered as a function of a period in order to estimate a price fluctuation structure at time t of the i-th bond. , The period structure of the discount rate can be calculated.

Where the overline D_t(S_t) Is the average discount function,
Discount rate is attribute z of individual brand_i1t, ..., z_iqtDepends on different,
That is, D_t(S_ajt) → D_it(S_ajt),
The p-order polynomial coefficient is a function that depends on q brand attributes.
[0014]
Further, as a further aspect, the discount rate period structure calculating means expresses a probability process of a discount function in which a discount rate is considered as a function of a period in order to estimate a price fluctuation structure at time t of the i-th bond. , The period structure of the discount rate can be calculated.

Where period s_tTo

Is converted to_tBecomes [0,1) in the domain [0, ∞) of

From

It becomes. Here, κ and α are parameters that determine the shape of the average discount function. Also, when polynomial approximation,

It becomes. Where the overline G_t(S_t ^*) Means average discount function, unknown parameter δ_jtIs common to all brands.
[0015]
Furthermore, the discount rate line G_t(S_t ^*As a further aspect, in order to estimate the price fluctuation structure at the time point t of the i-th bond, discounting is performed based on the following arithmetic expression that represents a stochastic process of a discount function that considers a discount rate as a function of a period. The period structure of the rate can be calculated.

Where the overline G_t(S_t ^*) Is the average discount function,
Discount rate is attribute z of individual brand_i1t, ..., z_iqtDepends on different,
That is, G_t(S_t ^*) → G_it(S_t ^*),
The p-order polynomial coefficient is a function that depends on q brand attributes.
In another embodiment of the bond investment analysis / credit risk measurement analysis system according to the present invention, the bond theoretical price calculation means considers the time of occurrence of cash flows of all the bond issues to be analyzed, and is difficult to grasp in advance. When calculating the theoretical price of a bond based on the bond price model considered that the price fluctuation caused by the various stock attributes remains in the price fluctuation n periods ago, the average discount function calculation means z_i1t, ..., z_iqtAs one of the above, the value of the obtained price estimation residual n periods before can be used as a proxy variable of a brand attribute that is difficult to grasp in advance.
[0016]
On the other hand, the bond investment analysis / credit risk measurement / analysis method according to the present invention includes the following: the bond data, the bond data, the bond attribute data of the bond and the bond, and the rating data of the bond issuing company stored in the data storage means. And business data, and financial attribute data of the bond issuing company, a step of designating a calculation period, a brand, a brand attribute, a company attribute, and a calculation model, which are calculation conditions, and a step of designating a bond according to the specified calculation condition. Calculating a parameter of a term structure of a discount rate with respect to a cash flow generated at a future time point; calculating a parameter of a term structure of a bankruptcy probability and a recovery rate according to the designated calculation condition; The theoretical bond price and expected loss amount of individual issues, the period structure of the bankruptcy probability and discount rate of individual companies, and the attributes of issuers And a step of calculating the a term structure of default probability of each combination of one of categories each or any category rating, industry and financial indicators recovery.
BEST MODE FOR CARRYING OUT THE INVENTION
In the present invention, as input information, the government bond price and its brand attributes (coupon rate, redemption period, etc.) at time t, the bond price and its brand attributes (coupon rate, redemption period, etc.) at the same time, and bond issuance Use information on company attributes (ratings, industries, financial indicators, etc.). This is because the difference between the bond price and the JGB price for bonds with the same security attributes, such as coupon rate and redemption period, is due to the credit risk of the bond issuer being evaluated in the market. Credit risk is basically the risk that a contract will not be executed as promised. The credit risk of corporate bonds is basically due to interest arrears and repayment of principal, but for investors, the decline in bond prices due to increased credit uncertainty even without corporate bankruptcy, that is, credit spread ( The increase in the price difference) is credit risk. However, investors' preference for stocks is related not only to credit risk but also to market liquidity. In other words, investors do not necessarily hold shares until the redemption point, and liquidity as the ease of cashing out when selling in the middle is also a reason for stock preference, so there may be a price difference between government bonds and corporate bonds. Mostly due to credit risk. That is, in the present invention, the government bond price and its brand attributes (coupon rate, redemption period, etc.) at time t, the bond price and its brand attributes (coupon rate, redemption period, etc.) at the same time and the bond issuing company's Attribute (rating, business type, financial index, etc.) information is used as input information, and the classification, combination of rating, business type, and financial index included in the attributes of the issuing company, and each of the above classifications, the probability of bankruptcy of individual companies And the collection rate for each attribute such as the rating, business type, financial index, etc. of the bond issuing company.
Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings. First, a hardware configuration for implementing a bond investment analysis / credit risk measurement analysis system according to the present invention will be described with reference to FIG. In this bond investment analysis / credit risk measurement / analysis system, a storage device 2 is connected to a programmed main control unit 1 that controls the entire system as a whole. The main control unit 1 also outputs an input device 4 including a pointing device such as a keyboard and a mouse via the input / output control unit 3, a display device 5 used for monitoring input data, and outputs an analysis result and a measurement result. The output device 6 is connected.
[0017]
The main control unit 1 has a control program such as an OS (Operating @ system), a calculation program that defines a calculation procedure for executing bond investment analysis / credit risk measurement analysis, and an internal memory for storing required data. These calculation programs and the like implement a calculation means for calculating bond investment analysis or credit risk measurement analysis.
[0018]
The storage device 2 is a storage device such as a hard disk, a flexible disk, or an optical disk. The storage device 2 stores a database 21 for storing data to be input for executing bond investment analysis and credit risk measurement analysis, and realizes the arithmetic unit described above. An arithmetic program file 22 for storing various arithmetic programs read by the main control unit 1 is stored. Although the storage device 2 is shown as one block in FIG. 1, a plurality of storage units may be used, or the storage device 2 may be connected via a network.
In the present embodiment, price data of JGBs and corporate bonds, brand attribute data, and attribute data of bond issuers are input from the database 21, and predetermined calculations are performed by a calculation program read from the calculation program file 22 into the main control unit 1. Execution, theoretical price (expected price) of each corporate bond, expected loss, term structure of bankruptcy probability of individual company, rating included in the attributes of the issuing company, combination of industry, financial indicators Term structure of bankruptcy probability for each group And the collection rate, the rating included in the attributes of the issuing company, the type of business, and the term structure of the bankruptcy probability of each financial index and the collection rate are output to the output device 6 as calculation results. The calculation means is implemented by a calculation formula described later.
Next, the details of the data stored in the database 21 will be described. Dates, bond types such as government bonds (long-term government bonds, super-long-term government bonds, etc.), corporate bonds (general corporate bonds, electric power bonds, JR bonds, JT bonds, etc.), serial numbers, market prices (closing price, bid prices, bid prices), coupons, Rate, issuance date, redemption date, first interest payment date, interest payment month, interest payment date, accrued interest, remaining life, issuance amount, indicator brand flag, rating (Rating evaluation by Japan Credit Rating Information Center, Japan Credit Rating Institute, etc.) These are ratings assigned by the company, industries, and financial indicators (equity ratio, debt ratio, profit margin on sales, etc.). FIG. 3 is a chart showing an example of details of data stored in the database 21.
First, a theoretical base for realizing the bond investment analysis or the credit risk measurement analysis according to the present invention by using the calculation means will be described.
[0019]
Consider the current point in time as t, and the market at t_tWe think that the stock is traded. At this time, the elements that can be observed at time t for the i-th bond are:
1) Market price of bond i at time t: P_it(0)
2) Cash flow function including coupon and face value at the time of redemption (cash flow generation time is considered as a function of period): C_it
3) Brand attributes such as coupon rate and redemption period: Z_it= ｛Z_ikt: K = 1, ..., q｝
It is. Most of the bonds traded in the Japanese market are interest-bearing bonds, and the point at which cash flows such as coupons occur differs for each issue. To clearly show this, the point in time when the j-th cash flow of the i-th bond from the point
[0020]
(Equation 1)

Is expressed as And N which is traded in the market_tAll of the issues at the time of cash flow generation are listed in ascending order,
[0021]
(Equation 2)

And the par value of 100 yen at the time of redemption is also C_it(S_ajt). Therefore, the market price P of the ith bond at time t_it(0) is the future t + s_ajtThe cash flow generated at the time is C_it(S_ajt), The discount rate for this cash flow is D_t(S_ajt), When the redemption period is M,
[0022]
(Equation 3)

Can be expressed as Where s_ajt= S_ijtWhen not, C_it(S_ajt) = 0. The actual market price fluctuates due to various factors and is regarded as the realization of a random variable. General corporate bonds issued by companies generally have uncertainties in future cash flows, such as interest overdue or principal repayment. The company issuing the i-th bond is t + s_ijtThe probability of going bankrupt by the time depends on the rating k of the company,
[0023]
(Equation 4)

And the i-th issuer is t + s_{ij-1, t}Time point and t + s_ijtIf the bankruptcy occurs during the time, the principal is collected at t + s_ijtLet it be done at the time. Recovery rate γ_t(K (i)) is the future collection time t + s_ijtAnd even the same rating may be different for different companies. For example, suppose here that it depends on the rating k of the company. The company issuing the i-th bond is t + s_{ij-1, t}Time point and t + s_ijtH, if you did not go bankrupt_it(S_ijt) = 0 and t + s_ijtCoupon C at time_it(S_ijt) Will be paid. On the other hand, the company issuing the i-th_{ij-1, t}Time point and t + s_ijtIf bankruptcy occurs during the time, h_it(S_ijt) = 1 and 100γ of the principal_t(K (i)) is collected.
Accordingly, the cash flow function dashed line C of the i-th corporate bond with a k rating_it(S_ijt)
[0024]
(Equation 5)

It becomes. Where s_jt= S_ijtIf not, the cash flow function wavy line C_it(S_ijt) = 0. [1-h] of the first term of the above equation_it(S_ijt)] Is t + s_ijtThe probability that this company will not go bankrupt by the time point, [h_it(S_ijt) -H_it(S_ij-1t)] Means that this company is t + s_{ij-1, t}Time point and t + s_ijtThe probability of going bankrupt during the time. This equation is expressed as follows, where the future cash flows are deterministic_it(S_ijt) And expected loss L_it(S_ijtIf you transform it into a separate shape
[0025]
(Equation 6)

It becomes.
[0026]
With respect to the bond price model at the time point t of the i-th bond expressed by Expression 3, the future cash flow is C_it(S_ajt) Is a determinate bond such as a government bond, the discount rate D is related to the market price which is a random variable._t(S_ajt) Is a random variable. In other words, the realization of the market price of government bonds depends on the stochastic discount rate D_t(S_ajt). However, the bond price, which is a random variable on the left side of Equation 3, has M discount rates on the right side corresponding to future cash flows, and the bond price and discount rate have a one-to-one correspondence. I haven't. Therefore, the following stochastic process of the discount function corresponds to the bond price.
[0027]
(Equation 7)

That is, it is considered that realization of the bond price in the market is realized by one path of the stochastic process of the discount function. Accordingly, in the bond price model at the time t of the i-th bond expressed by Expression 3, the discount rate is a random variable, and in the case of a bond having a credit risk such as a straight corporate bond, the cash flow is also a random variable. . Bankruptcy initiation process h_it(S_t), Recovery rate process γ_t(K (i)), discount rate process D_t(S_t) Is independent and N_tThe cash flow function dashed line C takes into account the cash flow occurrence time of all issues._it(S_t), Deterministic cash flow function C_it(S_t), Expected loss L_it(S_t),Discount rate

[0028]
(Equation 8)

Expressed by Market price P at time t of the i-th bond_it(0) is
[0029]
(Equation 9)

Is expressed as Overline D_t(S_ajt) Is the discount function D_t(S_ajt). In order to estimate the price fluctuation structure at the time point t of the i-th bond, a discount function model that represents the probability process of the discount function that considers the discount rate as a function of the period is necessary, and is formulated below.
[0030]
(Equation 10)

Where the unknown parameter δ_jtIs common to all brands. Note that the discount rate is the attribute z of the individual brand._i1t, ..., z_iqt, Ie, D_t(S_ajt) → D_it(S_ajt), And as an average discount function,
[0031]
[Equation 11]

Can also be assumed. Here, the coefficient of the p-order polynomial is a function that depends on q brand attributes.
[0032]
Further, in order to estimate the price fluctuation structure at the time point t of the i-th bond, the following formula is formulated as a further function of the discount function model that represents the probability process of the discount function in which the discount rate is considered as a function of the period.
(Equation 12)

Where period s_tTo
(Equation 13)

Is converted to_tBecomes [0,1) in the domain [0, ∞) of
[Equation 14]

From
(Equation 15)

It becomes. Here, κ and α are parameters that determine the shape of the average discount function. Also, when polynomial approximation,
(Equation 16)

It becomes. Where the overline G_t(S_t ^*) Means average discount function, unknown parameter δ_jtIs common to all brands. Note that the discount rate is the attribute z of the individual brand._i1t, ..., z_iqt, Ie, G_t(S_ajt) → G_it(S_ajt), And as an average discount function,
[0033]
[Equation 17]

Can also be assumed. Here, the coefficient of the p-order polynomial is a function that depends on q brand attributes.
[0034]
Next, the price fluctuation part ε discounted by the stochastic discount rate_itIn formulating the variance-covariance structure, we consider the following bond price fluctuation characteristics actually observed in the market.
1) Redemption period s_{iM (i)}The shorter the is, the smaller the change in bond prices.
2) The smaller the difference between the redemption periods between issues, the higher the linkage.
That is, ε of each stock_itIs smaller as the redemption point approaches, and the larger the redemption period difference between each issue,_itIt is considered that the interlocking of the is reduced. Specifically, ε_it, The following variance-covariance structure is assumed.
[0035]
(Equation 18)

Where a_itFor example,
[0036]
[Equation 19]

Is assumed. Equations (18) and (19) express that the longer the redemption period is, the greater the variance is, and the closer the redemption period is, the greater the correlation between bond prices is. In addition, with respect to the time between two points when cash flows occur, the longer the period, the smaller the correlation of the discount function becomes._itAssume the following equation:
[0037]
(Equation 20)

Where Φ_itIs s_ajtAnd s_artThis corresponds to the covariance of the discount rate that discounts coupons that occur at the time.
The important point in the formulation of Equation 18 is that as the probability of bankruptcy decreases, the expected loss decreases and the structure approaches the price fluctuation of bonds without credit risk. The point is that a structure in which long-term bonds with longer maturities have greater variance is naturally introduced. Note that the average discount function line G_t(S_t ^*) Or overline G_it(S_t ^*), The period s_tIs converted by Expression 13 into s_t ^*Is used.
At present, the number of bankruptcies of companies that issue corporate bonds is small and the probability of bankruptcy is not directly available as data. Therefore, in order to obtain the probability of bankruptcy, it is necessary to consider a model expressing some kind of bankruptcy probability. Here, even companies with the same rating are considered to have different bankruptcy probabilities depending on the type of business and financial constitution._ijtIn formulating the probability of bankruptcy during the period, the following p-order polynomial is considered as a function that takes into account attributes such as the type of business of corporate bonds, financial indicators, and ratings.
[0038]
(Equation 21)

Where s_t ^*= S_t ^{1 / m}And m is, for example, 4, 2, 1, or the like. Equation 21 unknown parameters

Let's call this model the first model.
Next, a description will be given of a model in which the first model as a cross-section model between brands at time t is dynamically (time-series). The market price of corporate bonds is caused not only by stock attributes that can be grasped in advance, but also by stock attributes that are difficult to grasp directly in advance. This includes not only the attributes that can be explicitly grasped as the company attributes that affect the bankruptcy probability of individual companies, but also those that are difficult to grasp directly. Therefore, the model described below introduces brand attributes that are difficult to grasp directly in advance as time-series attributes in the model. It is a bond price model that uses time-series attributes as implicit issue attributes that summarize various issue attributes in order to express actual phenomena more. In other words, the model considers that price fluctuations caused by brand attributes that are difficult to grasp in advance remain in price fluctuations n periods ago. For example, the following two models are exemplified as models of this type.
(1) Type 1
Probabilistic discount function Δ at time t for bond i_t(S_t) Is considered to depend on the discount function at time t−n._t(S_t),
[0039]
(Equation 22)

For the following time series model
[0040]
[Equation 23]

Is assumed. Therefore, the discount function model is obtained from Equations 22 and 23 as follows:
[0041]
(Equation 24)

Is represented by Next, the definite cash flow C_it(S_t) And expected loss L_it(S_t) Function is
[0042]
(Equation 25)

Note that
[0043]
(Equation 26)

In other words, the bond price P at the time point t of the i-th corporate bond corresponds to Equation 9._it(0) is
[0044]
[Equation 27]

here,
[0045]
[Equation 28]

It is. ε_t-n(S_at-n) Is ε when the first model is estimated at a time point before the TN period._t-nIt is. Note that ψ_it(S_atThe variance-covariance structure in ()) is the same as in Expressions 18 to 20. The bond price model represented by Expressions 27 and 28 will be referred to as a second model.
(2) Type 2
Price estimation residual ε before n period_t-n(S_at-n) Is considered to represent various attributes that are not identified beforehand, and the discount rate is the attribute z of the individual brand._i1t, ..., z_iqtDepends on different average discount function
[0046]
(Equation 29)

Is the q-th attribute. This attribute changes over time, and it is thought that the introduction of this variable makes it possible to grasp the price fluctuation due to the subtle change in the average discount function for each individual issue. One of the brand attributes of the average discount function in Equation 10 is the price estimation residual ε before n periods._t-n(S_at-nThis model has the same structure as model 1 except for the use of (1). This model is called a third model. Note that, as an average discount function model, the overline G_t(S_t ^*) Or the upper line G of Equation 17_it(S_t ^*), The period s_tIs converted by Expression 13 into s_t ^*Is used.
From the above models, the theoretical price (average price) / bankruptcy probability, discount rate, bankruptcy probability / collection rate for each industry and each attribute are calculated for each individual issue of the bond. Next, each unknown

Recovery rate γ_t(K (i)), ρ of the variance-covariance structure_t, Σ_t ²) Will be described. First, the unknown parameter δ of the discount function model is calculated from bonds such as government bonds without credit risk._1t,…, Δ_ptIs estimated. Expected loss L_itThe following equation is derived from Equation 9 where is 0.
[0047]
[Equation 30]

here,
[0048]
(Equation 31)

It is. The log likelihood 1 of the model represented by Equation 30 is
[0049]
(Equation 32)

It is. By maximizing the log likelihood, unknown parameters can be obtained. The condition that is maximum with respect to Equation 32 is
[0050]
[Equation 33]

It is. Therefore, σ_t ²The maximum likelihood estimate of is
[0051]
(Equation 34)

Is required. Mountain line σ_t ²Substituting into Equation 32, the maximum log likelihood is
[0052]
(Equation 35)

It becomes. In order to select the order and attribute in the average discount function of Expression 10, statistics such as AIC (Akaike Information Criterion) and SBC (Schwartz Baysian Criterion) are used as evaluation criteria. AIC of model 30₁Is
[0053]
[Equation 36]

It becomes. In order to obtain the maximum likelihood estimation value of the unknown parameter in Expression 32, it is possible to obtain the maximum likelihood estimation value by an algorithm of nonlinear optimization. Note that, as an average discount function model, the overline G_t(S_t ^*) Or the upper line G of Equation 17_it(S_t ^*), The period s_tIs converted by Expression 13 into s_t ^*Is used to calculate Equations 30 to 36.
Next, the following equation is derived from Equation 9 using the discount function model obtained from a bond such as a government bond without credit risk by the above method.
[0054]
(37)

here,
[0055]
[Equation 38]

It is. θ_t= (Β_t ^*, Γ_t(K (i)), ρ_t), The log likelihood 1 of the model represented by Equation 37 is
[0056]
[Equation 39]

It is. By maximizing the log likelihood, the unknown parameter θ_tCan be requested. The condition that is maximal with respect to Equation 39 is
[0057]
(Equation 40)

It is. Therefore, σ_t ²The maximum likelihood estimate of is
[0058]
(Equation 41)

Is required. Mountain line σ_t ²Substituting into Equation 39, the maximum log likelihood is
[0059]
(Equation 42)

It becomes. In order to select the order and the attribute in the bankruptcy probability function of Equation 21, the statistics such as AIC and SBC described above are used as evaluation criteria. AIC of model of number 37_tIs
[0060]
[Equation 43]

It becomes. In order to obtain the maximum likelihood estimated value of the unknown parameter in Expression 37, the maximum likelihood value can be obtained by a non-linear optimization algorithm.
[0061]
This concludes the theoretical description of the calculation model serving as the basis for the bond investment analysis and credit risk measurement analysis system according to the present invention. Next, the calculation steps in the present embodiment will be described with reference to FIGS. 1, 2, 4, and 5. FIG.
[0062]
First, FIG. 2 is a diagram showing a flow of the entire bond investment analysis / credit risk measurement analysis system according to the present invention. In step B1, a calculation condition that is a premise for obtaining an analysis result is input in order to execute bond investment analysis and credit risk measurement analysis (step B1). Here, the calculation period, brand, brand attribute, company attribute, and calculation model are specified. Data extracted from the database 21 is read into the main control unit 1 according to these calculation conditions. Next, by executing the read data and the calculation program read from the calculation program file 22, parameters such as the period structure of the discount rate are calculated (step B2). Further, similarly, parameters such as the period structure of the bankruptcy probability and the recovery rate are calculated (step B3). The calculation of the theoretical bond price and the expected loss amount is executed based on these determined parameters (step B4). Based on the theoretical bond price and expected loss obtained in step B4, the results of econometric analysis, specifically, the term structure of the bankruptcy probability, the term structure of the discount rate for future cash flows of the bond, the bond issuer The term structure and recovery rate of bankruptcy probabilities are output for each of the categories of the rating, industry, and financial index included in the attribute of, or for each combination of arbitrary categories (step B5).
[0063]
FIG. 4 is a flowchart showing a flow for calculating a discount function model of a bond having no credit risk such as a government bond. First, the generation point of the cash flow of the bond is confirmed according to the data input based on the calculation conditions input in step C1. Specifically, the calculation of Equation 2 is performed (Step C2). Next, a mutual data relationship is set based on the input price data and brand attribute data (step C3), and y in Expression 30 is set._tAnd X_t, And the logarithmic likelihood is maximized based on Equation 32 to obtain an unknown parameter β appearing in Equation 30._t(Steps C4, C5, C6). The term structure of the discount rate is calculated based on the parameters (step C7).
[0064]
Figure 5 shows the term structure of the probability of bankruptcy of a bond issuing company, the term structure of bankruptcy probabilities by classification that combines various corporate attributes such as rating and industry, the recovery rate for each rating, and the theoretical price of bonds issued by companies (average 9 is a flowchart showing a flow for obtaining a price) and an expected loss (credit spread). First, setting data of a discount function model obtained from a bond, such as a government bond having no credit risk, obtained by the flow of FIG. 4 and a corporate attribute and a bond attribute are input (step D1). Next, although not shown in the drawing, as in steps C2 and C3 in FIG. 4, the time point at which the cash flow of the bond is generated is confirmed, and the relationship between the mutual data based on the price data and the brand attribute data input next is confirmed. Is set. Next, the unknown parameter γ in Equation 37 is calculated based on Equations 37 and 39._t(K (i)),

The period structure and the recovery rate of the bankruptcy probability are calculated based on the obtained parameters (step D5). In step D6, the theoretical bond price, expected loss amount, and residual price estimate are calculated based on the term structure and the recovery rate of the bankruptcy probability calculated in step D5, and the calculation ends in the model 1 (step D7). ). In the models 2 and 3, the price estimation residual calculated in step D6 is also used as input data, and unknown parameters are obtained in the same calculation steps as in model 1.
[0065]
FIGS. 6, 7, 8 and 9 show the number of issues to be analyzed, the recovery rate, and ρ using the bond investment analysis / credit risk measurement / analysis system according to the present invention._t・ Examples of the output of the bankruptcy probability function model parameters, the bankruptcy probability period structure for each category combining various company attributes, the theoretical price (average price) and expected loss (credit spread) of bonds issued by companies FIG.
[0066]
【The invention's effect】
As described above, according to the present invention, by inputting the bond price, the brand attribute, and the attribute information of the bond issuing company, the term structure of the bankruptcy probability of the bond issuing company, various corporate attributes such as rating and industry are combined. The ability to measure the term structure of bankruptcy probabilities for each category, the recovery rate for each rating, and the quantitative analysis of the theoretical price (average price) and expected loss (credit spread) of bonds issued by companies From, financial institutions such as insurance, banks, securities companies and investment advisory companies, as well as corporate corporations and individuals investing in stocks and bonds, bond investment methods, bond investment systems, financial products, credit risk management systems, In the investment analysis and financial products, such as credit risk and derivative valuation, and in all fields related to credit risk valuation and management, the term structure and The term structure of bankruptcy probabilities by category that combines various corporate attributes such as financial and business types, the recovery rate for each rating, and the theoretical price (average price) of bonds issued by a company, and the expected loss (credit Spread) can be quantitatively analyzed.
[Brief description of the drawings]
FIG. 1 is a block diagram showing a hardware configuration for implementing the present invention.
FIG. 2 is a flowchart showing calculation steps according to the present invention.
FIG. 3 is a chart showing an example of the contents of a database;
FIG. 4 is a flowchart for calculating a discount function model.
Figure 5: Bankruptcy probability period structure of a bond-issuing company, bankruptcy probability period structure by classification combining various company attributes such as rating and industry, collection rate for each rating, theoretical price of bonds issued by a company ( 9 is a flowchart for calculating an average price) and an expected loss (credit spread).
FIG. 6: Number of issues to be analyzed, recovery rate, ρ_tFIG. 7 is a diagram showing an output example of parameters of a bankruptcy probability function model.
FIG. 7 is a diagram showing an output example of a period structure of bankruptcy probabilities by classification obtained by combining various company attributes.
FIG. 8 is a diagram showing an output example of a theoretical price (average price) of a bond issued by a company.
FIG. 9 is a diagram illustrating an output example of an expected loss (credit spread) of a bond issued by a company.

Claims (17)

Data storage means for storing government bond price data, corporate bond price data, government bond and corporate bond brand attribute data, rating data and industry data of corporate bond issuing companies, and financial attribute data of corporate bond issuing companies, Calculation condition specifying means for specifying a calculation period, a brand, a brand attribute, a company attribute, and a calculation model; and a bond investment analysis based on data input from the data storage means in accordance with the calculation conditions specified by the calculation condition specifying means. Or a bond investment analysis / credit risk measurement analysis system having calculation means for calculating a credit risk measurement analysis, wherein the calculation means includes: a bond theoretical price calculation means for calculating a bond theoretical price of an individual issue; A bond investment analysis / credit risk measurement / analysis system including an expected loss calculation means for calculating an expected loss. 2. The bond investment according to claim 1, wherein the bond theoretical price calculation means calculates the bond theoretical price based on an arithmetic expression of a calculation model that takes into consideration a cash flow generation time point of all of the bond issue groups to be analyzed below. 3. Analysis and credit risk analysis system.

(The cash flow function wavy line C _it ( _st ), the deterministic cash flow function C _it ( _st ), the expected loss amount L _it ( _st ), and the discount rate D _t ( _st ) are 0 ≦ _st ≦. function defined in _saM ). In the arithmetic expression according to claim 2, the variance-covariance structure of the price fluctuation portion ε _it discounted by the stochastic discount rate is
1) The shorter the redemption period _{siM (i), the} smaller the change in bond price.
2) The smaller the difference between the redemption periods between issues, the higher the linkage.
3. The bond investment analysis / credit risk measurement analysis system according to claim 2, wherein the following variance-covariance structure is provided in consideration of bond price fluctuation characteristics actually observed in the market.

In the variance-covariance structure according to claim 3, with respect to a it, the following formula is used to express that bonds with a longer redemption period have a larger variance and bond prices with a _shorter redemption period have a _higher correlation. The bond investment analysis / credit risk measurement / analysis system according to claim 3, which performs the following.

The bond theoretical price calculation means considers the time of occurrence of the cash flows of all the bond issues to be analyzed, and furthermore, the price change caused by the stock attribute which is difficult to grasp in advance is changed to the price change n periods ago. 2. The bond investment analysis and credit risk measurement analysis system according to claim 1, wherein the bond bond theoretical price is calculated based on a bond price model considered as remaining. In the bond theoretical price calculation means, the price fluctuation caused by the security attribute which is difficult to grasp in advance is considered in consideration of the time of occurrence of cash flows of all the security issues to be analyzed. 6. The bond investment analysis / credit risk measurement analysis system according to claim 5, wherein the bond bond theoretical price is calculated based on the following bond price model arithmetic expression considered as remaining.

(The cash flow function wavy line C _it ( _st ), the deterministic cash flow function C _it ( _st ), the expected loss amount L _it ( _st ), and the discount rate D _t ( _st ) are 0 ≦ _st ≦. function defined by _saM ) The bond theoretical price calculation means calculates a cash flow and an expected loss amount in the case where there is uncertainty in a future cash flow such as interest delinquency or inability to repay the principal based on the following calculation formula. The bond investment analysis / credit risk measurement / analysis system according to any one of claims 2 to 5.

_Here, when it is not _s jt ₌ s _ijt, cash flow function wavy line _{C it (s i jt) =} 0
The first term of _{[1-h it (s ijt} )] is _{t + s ijt} to the point on the probability second term the company is not bankrupt _{[h it (s ijt) -h} it (s ij-1t)] is the company _{t + s ij-1, t} the time and _{t + s} probability bankrupt during the _ijt time the i bonds issuers is _{t + s ij-1, t} the time and _{t + s ijt} if not bankrupt between time points _h it _{(s ijt} ) = 0, coupon C _it (s _ijt ) is paid at t + s _ijt .
The i bonds issuers are among the _{t + s ij-1, t} the time and _{t + s ijt} when bankruptcy _h it _{(s ijt)} between the time = amount _{V it} to be recovered in 1, the recovery rate γ _{t V it} γ _t Is collected. The bond investment analysis / credit risk measurement / analysis system according to claim 7, wherein the calculation means of the recovery rate _γt calculates the recovery rate based on an operation formula depending on the following rating k of the company.
γ _t = γ _t (k (i)) 8. The bond investment analysis / credit risk measurement / analysis system according to claim 7, wherein the calculating means for calculating the amount to be collected at the time of bankruptcy calculates the amount to be collected based on the following arithmetic expression.
V _it = 100 The calculating means further includes a bankruptcy probability period structure calculating means for calculating a bankruptcy probability period structure of an individual company, and a discount rate period structure calculation for calculating a discount rate period structure for a cash flow generated at a future time of a bond. Means and bankruptcy probability period structure and recovery rate by category that calculates the bankruptcy probability period structure and recovery rate for each category of rating, industry, and financial indicators included in the attributes of the issuing company or for any combination of categories The bond investment analysis / credit risk measurement / analysis system according to claim 1, further comprising a calculation unit. 11. The bond investment analysis / credit risk measurement analysis system according to claim 10, wherein said bankruptcy probability period structure calculating means calculates a bankruptcy probability period structure based on the following arithmetic expression.

Here, _hit (s _t ) is the bankruptcy probability,
z _ijt is an attribute such as a business type, a financial index, and a rating of a bond issuing company;

s _t ^* = m in _s ^{t 1 / m,} for example, 4, 2, 1. The discount rate period structure calculating means estimates the price fluctuation structure of the i-th bond at time t based on the following calculation formula expressing the probability process of the discount function considering the discount rate as a function of the period. The bond investment analysis / credit risk measurement / analysis system according to claim 10, wherein a term structure of the discount rate is calculated.

Here, the _overline D _t (s _t ) is the average discount function unknown parameter δ _jt is common to all brands. The discount rate period structure calculating means estimates the price fluctuation structure of the i-th bond at time t based on the following calculation formula expressing the probability process of the discount function considering the discount rate as a function of the period. The bond investment analysis / credit risk measurement / analysis system according to claim 10, wherein a term structure of the discount rate is calculated.

Here, the overline D _t (s _t ) is an average discount function,
The discount rate is individual stocks of attribute _{z i1t,} ..., depending on the _{z iqt} different,
In other _{words, D t (s ajt) →} D it (s ajt),
The p-order polynomial coefficient is a function that depends on q brand attributes. The discount rate period structure calculating means estimates the price fluctuation structure of the i-th bond at time t based on the following calculation formula expressing the probability process of the discount function considering the discount rate as a function of the period. The bond investment analysis / credit risk measurement / analysis system according to claim 10, wherein a term structure of the discount rate is calculated.

Here, the period _st is

If you were converted, the domain of the period _{s t} [0, ∞) is in [0, 1),

From

It becomes. Here, κ and α are parameters that determine the shape of the average discount function. Also, when polynomial approximation,

It becomes. Here, the _overline G _t (s _t ^* ) is the average discount function, and the unknown parameter δ _jt is common to all brands. The discount rate period structure calculating means estimates the price fluctuation structure of the i-th bond at time t based on the following calculation formula expressing the probability process of the discount function considering the discount rate as a function of the period. The bond investment analysis / credit risk measurement / analysis system according to claim 10, wherein a term structure of the discount rate is calculated.

Here, the overline G _t ( _st ^* ) is an average discount function,
The discount rate is individual stocks of attribute _{z i1t,} ..., depending on the _{z iqt} different,
That is, _Gt ( _st ^* ) → _Git ( _st ^* ),
The p-order polynomial coefficient is a function that depends on q brand attributes. _{3. The} means for calculating the average discount function, wherein one of the attributes z _i1t ,..., Z _iqt of the individual brand is used as a proxy variable of the brand attribute which is difficult to grasp in advance. 6. The bond investment analysis / credit risk measurement / analysis system according to claim 5, wherein a value before n periods is used. From the government bond price data, corporate bond price data, government bond and corporate bond attribute data, bond issuing company rating data and industry data, and corporate bond issuing company financial attribute data stored in the data storage means. Specifying a calculation period, a calculation period, a brand, a brand attribute, a company attribute, and a calculation model which are calculation conditions; and a parameter of a period structure of a discount rate for a cash flow generated at a future time point of the bond according to the specified calculation condition. Calculating the parameters of the period structure of the bankruptcy probability and the recovery rate according to the designated calculation conditions; and using the obtained parameters, the theoretical bond price and expected loss amount of the individual issue, Bankruptcy probability and discount rate term structure, and rating, industry and financial indicators included in the attributes of the issuer for each category or voluntary Step and bonds investment analysis, credit risk quantitative analysis method with which to calculate the the term structure of default probability of each combination of categories recovery.

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