JP2004038372A - Method for calculating current price of asset back security and its program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a rational and quick method for calculating the current price of an asset back security and its program. <P>SOLUTION: This method for calculating the current price of an asset back security to be issued with the gross assets of a cash flow based on assets as guarantee comprises a step 10 for making a computer accept the upper limit value of tranche indicating the range of the assets being the guarantee of a security and the lower limit value of the tranche, steps 20 and 30 for making an expected gross assets disappearing process indicating the expectation of the gross assets disappearing process with the gross assets disappearing process as a levy process based on a model obtained based on the past data of the gross assets disappearing process, and a step 40 for making an arithmetic means calculate the current price of the security based on the upper limit value of the tranche, the lower limit value of the tranche, and the expected gross assets disappearing process. <P>COPYRIGHT: (C)2004,JPO

Description

【００７４】
［数１］によって示される現在価値は、満期をｔとするアセットバック割引債の時刻ｔ＝０における現在価値（アセットバック割引債価格）であり、本発明に係るアセットバック証券の時価計算方法の評価の基本的な構成要素となる。なお、　本明細書の何れの数式においても、ファイナンスの分野における通常の記載にしたがい、通貨単位（通貨の次元）は明示していない。また、積分の表記は、測度論（積分論）における表記を用いている。
［数１］は、原資産の消滅額の累積である総資産消滅額Ｌ_ｔ（＝初期の総資産額Ａ−総資産残高）に対してのレンジフォワードであるプット型レンジフォワードになっている。［数１］は、消滅していく総資産消滅額Ｌ_ｔから表現している。ここで、トランシェの幅を定めるαとβは、単調増加な確率過程である総資産消滅額Ｌ_ｔの増加方向に対して正であり、α＞βの大小関係を有するものとする。
【００７５】
［リカバリー有り、プリペイメント無しのアセットバック証券の時価］
さらに、リカバリーを認める場合は、アセットバック証券の現在価格に対するリカバリー額の割合であるリカバリーレートをφ_ｔとすると、アセットバック利付債価格は［数２］のように表せる。リカバリーは、倒産などの場合に、貸した金全額を失うことなく、その一部が戻って来ることである。本発明に係るアセットバック証券の時価計算方法では、総資産がトランシェ内に入っている間の消滅していく総資産の一部がリカバリーし、アセットバック証券の所有者に戻ってくることを想定する場合を述べている。アセットバック利付債の価格は、アセットバック割引債価格に保有期間中クーポン支払いの条件をつけた利付債券価格のことをいう。ここで、リカバリー無しの場合は、利付債券価格は途中のより一般的に表記される利息キャッシュフローｃ（ｄｔ）で重み付けた割引債価格の合計として表される。したがって割引債の価格が求まれば容易にクーポン付きのアセットバック証券（利付債）価格が求まる。これに対し、リカバリーを認める場合においてクーポン付きのアセットバック証券（利付債）として表記した理由は、債券満期までにある時刻（ｔ）において発生するリカバリーによるキャッシュフローと利息として発生するキャッシュフローｃ（ｄｔ）の両方を、キャッシュフローとして同時に割引くことを考慮したためである。
【数２】

【００７６】
リカバリーが起こる場合は、総資産消滅額（元本消滅）のジャンプｄＬがトランシェ内に架かった場合に、そのトランシェにおける消滅分だけがリカバリーを受け取る権利の対象となり、そのうち実際のリカバリー額は、リカバリーレートφ_ｔを掛けた値になる。ここで、リカバリーレートφ_ｔとは、総資産消滅額ｄＬに対するリカバリー額の割合をいう。なお、ジャンプｄＬそれ自体は時刻ｔを明示的に含んでいないが、確率変数Ｌ_ｔの変分ｄＬが従う分布は時刻ｔの関数となっている。したがって、式の上ではｄｔが現れていないが、Ｅによる期待値計算の手順の中に時刻ｔについての積分が含まれるものとなる（例えば、後述する［数１６］におけるτによる積分）。
【００７７】
［プリペイメント有りのアセットバック証券の時価］
より一般的なのは、アセットバック証券において、アセットバック証券の担保であるキャッシュフローを支払予定期日より、支払責任者が早く払うプリペイメントがある場合である。一般に、倒産による元本消滅は、優先度が低い方から起こり、支払予定期限前に起こるプリペイメントおよびアセットバック証券の資産の満期が短いための償還は、優先度が高いトランシェから起こる。
【００７８】
総資産の減少総額に対するプリペイメント（前払い）の割合をφ_ｔとすると、アセットバック証券の利付債価格は、［数３］のように表せる。なお、リカバリーはプリペイメントの中に組み込むようにしてもよい。アセットバック利付債価格とは、アセットバック割引債価格に保有期間中のクーポンによる支払いの条件をつけた利付債券の価格をいう。
【数３】

【００７９】
ここで、［数３］中のＡは、アセットバック証券発行時の初期総資産額（原資産額）を表す。［数３］において、［数２］との違いは、アセットバック証券を担保する、キャッシュフローを支払うべき法人もしくは個人の倒産により、キャッシュフローが減少し、βで表されるトランシェの上限も侵食されていくことを考慮した点である。なお、本発明ではリカバリーレートのモデル化の例は省略したが、リカバリーレートも確率過程としてモデル化してもよい。
【００８０】
［数１］から［数３］により、アセットバック証券の時価評価のを適切に行なうためには、総資産消滅額Ｌ_ｔのモデル化を合理的に行なって適切に算出することが、必要である。さらに、リカバリーまで考慮する場合は、総資産（原資本）消滅額の確率微分ｄＬも考慮しなければならない。
【００８１】
次に、アセットバック証券の担保となる資産の総資産消滅額を、以下に述べるようなポアソン過程によるものと、単調増加レビー（Ｌｅｖｙ）過程によるものとによってモデル化し、演算手段によって計算する例を説明する。
【００８２】
［ポアソン過程の総資産消滅額］
まず、アセットバック証券の総資産消滅額Ｌ_ｔのモデル化を行なう上で解析的な計算が行いやすいモデル化の一例として、ポアソン過程の中心極限定理による拡散近似（ブラウン運動近似）による方法を説明する。
【００８３】
ポアソン過程とは、時間的に一定の率で偶然的に事象を生起させる過程である。ポアソン過程とは、最初は０から出発してしばらくは０のままで、あるとき突然大きさ１だけジャンプして、そしてしばらくは１のままで、またあるとき突然大きさ１だけジャンプして、そしてしばらくは……という、サンプルパスが右上がりの階段のような形を描く確率過程をいう。数学的な定義では、上で「しばらくは０のままで」、「しばらくは１のままで」という曖昧な表現を定式化することによって得られる。具体的には、１度ジャンプが起こってから次のジャンプが起こるまでの時間間隔が指数分布に従うとするものである。
【００８４】
本発明の一実施の形態では、アセットバック証券の担保となる総資産の消滅がポアソン過程によって起こるとする。例えば、アセットバック証券の担保となるキャッシュフローの支払義務者が倒産に陥りデフォルトを起こすとき、この倒産が偶然的に生起するものとして、ポアソン過程であるとしてとらえている。本発明では、多数の会社の倒産現象を扱う場合にポアソン過程を利用しても複雑になるため、拡散近似を利用してコンピュータによる計算の効率化を図っている。
【００８５】
中心極限定理とは、大きな母集団から抽出された標本の、標本平均の分布は標本の大きさが大きくなるとき、正規分布に近づくという定理である。本発明の一実施の形態では、アセットバック証券の担保となっている資産の過去の価値の推移に関わるデータを大量に抽出し、標本とし、この標本の推移が正規分布になるとモデル化している。
【００８６】
拡散近似（ブラウン運動近似）とは、大きな母集団から抽出された変動標本の、変動標本平均のプロセスは変動標本の大きさが大きくなるときブラウン運動に近づくという中心極限定理を確率変動に応用した近似方法をいう。本発明の一実施の形態では、［数５］のようなポアソン過程の拡散近似式では、モデルパラメータは一例として次のように定める。総資産消滅過程の過去のデータが、最小限のデータしか得られない場合、例えば、過去のデータとして、アセットバック証券の購入価格、満期、利率（クーポン）、格付ぐらいのデータしかない場合、ローン・プールを原資産とする場合で、初歩的なモデルを利用するのであれば、例えば、［数５］のポアソン過程の拡散近似式では、モデルパラメータは一例として次のように暫定的に決める。時間ｔは発行時点からの経過時間であり、ηは格付から決まる社債平均信用スプレッド、Ａは適当に現実的に大きい値で設定し、社債のスプレッドのボラティリティが社債市場で観測されるレベルぐらいに押さえるものとする。最後に、アセットバック証券のトランシェの幅は購入価格を実現するように決める。このとき、この分野に詳しいコンサルタントを利用するなどの方法も併用し、原資産の特性を研究して複雑な減衰経路をもつレビー過程を前提にしたモデルを利用し、得られる最小限のデータに合わせるようにモデルパラメータを調整することも可能である。
【００８７】
ここで、［数４］のように定めると、ポアソン過程の中心極限定理による拡散近似は［数５］のように表される。［数４］中のＬ_ｔは総資産消滅額であり、Ａは初期総資産額であり、ｈ_ｔは総資産消滅額Ｌ_ｔを初期総資産額Ａによって割った値である。［数５］のモデルの特徴の一つは、アセットバック証券の総資産消滅額Ｌ_ｔが正規分布によって表されるため、アセットバック証券の時価の解析が容易であり、解は解析解によって求まる点である。なお、［数５］中のηは消滅（倒産）危険率であり、ｄｗはブラウン運動を表し、ｔはアセットバック証券発行時から経過した時間である。
【数４】

【数５】

【００８８】
しかし、アセットバック証券の総資産消滅額Ｌ_ｔの動向をモデル化する上で、［数５］に示すようなポアソン過程の中心極限定理による拡散近似モデルを実際に用いると、アセットバック証券の瞬間資産消滅額ｄＬが、単調増加にならず、値が負になる確率が零ではないことが多い。よって、［数５］に示すモデル化をアセットバック証券の時価計算に用いることができるのは、瞬間資産消滅額ｄＬが、単調増加にならず、［数５］のモデル化による値が負になる確率をほぼ無視できる場合に限られる。
【００８９】
逆に、瞬間資産消滅額ｄＬが単調増加になる場合や、あるいは、瞬間資産消滅額ｄＬの値が負になる確率が零ではない点を考える必要があるとき（例えば、実際のアセットバック証券の資産消滅額を取り扱い、一度に一定額が消滅する可能性がある場合）には、［数５］におけるブラウン運動の連続パスによっては良好にモデル化することができない。したがって、このような場合には、ポアソン過程の中心極限定理による拡散近似よりは、後に述べる単調増加レビー過程を用いるモデルの方がより好ましい。
【００９０】
［単調増加レビー過程］
本発明のもう一つの実施の形態では、ポアソン過程の中心極限定理による拡散近似モデルを利用することなく、単調増加レビー過程を使い、アセットバック証券の時価計算のモデル化を試みる。レビー過程は、ポアソン過程も、ブラウン運動（ウィナー過程）も含む一般的な確率過程である。レビー過程は、一般に、確率連続な加法過程で、その見本関数が確率１でたかだか第１種不連続かつ右連続となっているものをいう。そのほかに、レビー過程には、時間的一様レビー過程でその見本関数が飛躍だけで変化し、そのレビー測度が有限なものである複合ポアソン過程や、コーシー過程（Ｃａｕｃｈｙ　ｐｒｏｃｅｓｓ）がある。
【００９１】
本発明においては、アセットバック証券の担保である資産の総資産消滅額Ｌ_ｔを、単調増加なレビー過程として取り扱う、つまり、時間とともに一方的に上がるだけの変化をする単調増加な確率過程として取り扱う。
【００９２】
本実施の形態の単調増加レビー過程では、解が解析解によって与えられるタイプを利用する。本実施の形態において利用するタイプは、レビー過程の中ではブラウン運動に次ぐ最も単純なタイプである。ブラウン運動が連続経路であるのに対し、本実施の形態において利用するタイプは数値が跳躍するジャンプのみで構成されるような複合ポアソン過程によるものとする。
一般に、金融工学では、株価や債券価格がブラウン運動に従うとして頻繁にモデル化される。ブラウン運動は瞬間の変動が正規分布に従う確率過程である。したがって、ブラウン運動によって株価や債券の価格をモデル化すると、ブラウン運動では株価や債券の価格が瞬間的に大きく動くことはない連続経路になる。
【００９３】
一般に、ブラウン運動を累積すると正規分布になる。これに対し、本実施の形態の解が解析解によって与えられるタイプは、累積すると拡張Γ分布になる。なお、本実施の形態の解が解析解で与えられるタイプのレビー過程は数多く存在し、拡張Γ分布となるのは最も単純な一つの例に過ぎず、本発明をこれに限定するものではない。本発明の一実施の形態に係るアセットバック証券の時価計算において用いる［数６］や、［数１１］（いずれも後に説明する）などの数式に用いるパラメータを適切に選べば、アセットバック証券の時価評価は、ポアソン過程のブラウン運動の正規分布を用いるより、容易かつ精緻にコンピュータの演算手段により計算ができる。
【００９４】
本実施の形態の解が解析解によって与えられるタイプは、アセットバック証券の担保となる資産の瞬間資産消滅額ｄＬが非心χ二乗分布に従い、総資産消滅額Ｌ_ｔが移動指数分布累積型Γ分布に従う確率過程を想定する。特に、初期時点からの総資産消滅額Ｌ_ｔの分布は、Γ分布になることが分かっている。
【００９５】
アセットバック証券の担保となる資産の瞬間資産消滅額ｄＬを表す非心χ二乗過程は非負に台を持つ確率過程である。また、正確なアセットバック証券の時価計算のモデルとしては、アセットバック証券の担保となる資産の総資産資産消滅額Ｌ_ｔの最大値が、アセットバック証券の発行時の資産の初期総資産額Ａになるところに、吸収壁を設ける必要がある。吸収壁とは、確率論における上限のことであり、例えば、アセットバック証券の担保となる原資産の減少は、元々の原資産（初期の総資産額）以上にはなることがないので、自然な上限ができる。この自然な上限を吸収壁とするのが普通である。
【００９６】
しかしながら、単調増加なレビー過程は、単調増加であるがゆえに、吸収壁に達するまでの確率分布が吸収壁の影響を受けないという性質があるので、吸収壁を無限遠に置いて分析を行なえる。特に、本発明におけるアセットバック証券の時価計算方法では、総資産資産消滅額Ｌ_ｔがトランシェ内に限定されるので、吸収壁は無視してもよい。
【００９７】
一般に、レビー過程は伊藤微分のような簡易な表記方法が存在しないため、［数６］のような積率母関数によって定義する。［数６］による総資産消滅額Ｌ_ｔの定義はあくまで一実施の形態であり、他のレビー過程を示す数式によって定義することもできるのは上述のとおりである。
【００９８】
積率母関数とは、確率分布の変換の一種である。本実施の形態では、［数６］のような積率母関数は、直接取り扱いにくい確率分布を取り扱う場合の計算テクニックの一種として使用される。
【数６】

【００９９】
ここで、［数６］中のＥ_ｔ［ｅθ^ｄＬ］は、確率変数ｅθ^ｄＬの期待値を表す。ν、λ_ｔはこのレビー過程を記述するパラメータである。初期時点から確率積分して得られるΓ分布の式中でνはΓ関数の引数であり、Γ分布を構成する指数分布の実数重複度を表している。λ_ｔはその構成要因の指数分布パラメータでこの指数分布をジャンプ間隔とするポアソン過程のハザードレートにあたる。ただし、瞬間消滅資産額ｄＬは、Ｌ_ｔが単調増加であるために、ν，λ_ｔ’＞０になる。また、λ_ｔ’＝（ｄλ_ｔ）／（ｄｔ）である。レビー伊藤の定理により、無限分割可能分布（非心χ二乗分布）のキューミュラント母関数は、［数７］のように無数のジャンプ（跳躍とも、非連続パスともいう。不連続な値の飛びをいう。）の和によって表すことができる。キューミュラント母関数は、積率母関数の対数をとったものである。
【数７】

【０１００】
［数７］が示すことは、総資産消滅額Ｌｔのジャンプ幅ｄＬ＝ｈのハザードレートが［数８］のように表せるということである。ハザードレートｈは、つぎの瞬間にジャンプと呼ばれる不連続な値の飛びを生じる確率を示す。通常、ハザードレートは、時間の関数として表現される。
【数８】

【０１０１】
［数６］によって示される確率過程の積率母関数の微分定義式を積分すると、［数９］のように表される。ここで、積率母関数微分定義式は、積率母関数の満たす微分方程式をいう。
【数９】

【０１０２】
［数９］は原点に（λ_ｔ／λ_Ｔ）だけ確率集中させた指数分布をν回畳込んだ分布を右にζだけ移動させた分布になっている。ここで、ν回畳込んだ分布とは、同じ指数分布に従う独立なν個の確率変数の和が従う確率分布のことである。ただしこの場合νは整数である必要はなく、この概念を連続化させたものになっている。
【０１０３】
なお、［数９］を逆ラプラス変換することにより、［数１０］で示される総資産消滅額Ｌ_ｔの確率Ｐｒ_ｔが得られる。総資産消滅額Ｌ_ｔの確率Ｐｒ_ｔは、［数１０］の右辺のように、ガンマ分布を二項分布で加重平均した形の累積確率密度関数になっている。
【数１０】

重要なことはνが正整数の場合、和分（ｓｕｍ）は有限になることである。
【０１０４】
ここで、λ_ｔをモデル化する際の関係式を［数１１］に表す。［数１１］は、［数１２］によって示される積率母関数の性質に基づいて、［数９］から導かれたものである。［数１１］の導き方は、瞬間資産消滅額ｄＬのような直接取り扱いにくい確率分布を［数６］に示される積率母関数でまず定義し、［数１２］と［数９］により、［数９］右辺をθについて微分することにより、θ→０の極限値を計算し、［数１１］を導く。
【数１１】

【数１２】

【０１０５】
総資産消滅額Ｌ_ｔの期待値である期待総資産消滅過程Ｅ_ｔ［Ｌ_ｔ｜Ｌ_ｔ＝ζ］を、例えば、ロジスティック曲線などの成長曲線によってモデル化することができる。ロジスティック曲線とは、成長曲線の一種である。一般的には、母集団の大きさｙが時間ｔの関数ｙ＝ｋ／（１＋ｅ^−ｋｂｔ）によって与えられる。ただし、ｋ、ｂは正の定数である。より一般的には、ｙ＝ｋ／（１＋ｅ^{ｃｆ（ｔ）}）の形の関数の表す曲線をいう。ただし、ｃは定数、ｆ（ｔ）は時間ｔの関数である。ただし、本発明における成長曲線はこの一般式ｙ＝ｋ／（１＋ｅ^{ｃｆ（ｔ）}）と同じ形である必要はなく、この一般式の変形であってもよいし、指数曲線でもよい。さらには、ｔａｎｈ（ａｔ）のような関数式でもよい。
【０１０６】
本発明の一実施の形態においては、ｙを総資産消滅額Ｌ_ｔとして、現実の経済から得られるアセットバック証券の担保となる資産の消滅に係るデータに基づいて、コンピュータが、ｃｆ（ｔ）の関数を具体的に定めるのに必要なパラメータを非線型回帰推定することにより計算することが好ましい。
【０１０７】
本発明の一実施の形態として、金利と、アセットバック証券の担保となる総資産消滅額は独立であると仮定すると、［数１０］に示す分布関数を用い、通常の割引債価格をＤｔ（Ｔ）とすると、アセットバック割引債価格は、［数１］に基づき［数１３］のように表される。
【数１３】

【０１０８】
アセットバック証券という金融商品を時価評価する際の最小単位が割引債であるので、アセットバック証券の時価を計算するために、アセットバック割引債の価格を計算する。ただし、一般に、アセットバック割引債の元本（総資産）償還はアセットバック証券の契約条件の制約を受ける。すなわち、アセットバック証券の担保となる総資産総残高がトランシェ内か外かによって変化する。
【０１０９】
［数１３］に基づき、［数１３］のアセットバック割引債の期待値計算を行なうと、ｍａｘ（，）を取り除くことができ、［数１４］のようになる。一般には、νが正整数の場合、和分の上限はνになるので簡便なコンピュータプログラムによってアセットバック割引債の期待値計算を行うことができて、評価することができる。［数１４］は、リカバリー条件なし、プリペイメント条件なしのアセットバック割引債の価格である［数１］に相当するものである。
【数１４】

【０１１０】
同様に、プリペイメント条件付アセットバック利付債価格は、［数１５］、［数１６］の積分計算をおこなえばよい。これも［数１４］のように解析解を求めることは可能である。［数１５］は、リカバリー条件あり、プリペイメント条件ありのアセットバック割引債の価格である［数１］に相当するものである。
【数１５】

【数１６】

【０１１１】
［アセットバック証券の時価計算］
本発明に係るアセットバック証券の時価計算をコンピュータによって行なうための方法についての一実施の形態を、図面を参照しながら、次に説明する。
図５に示すように、本発明のアセットバック証券の時価計算方法は、総資産額と、トランシェの上限値を定めるβと、トランシェの下限値を定めるαとをコンピュータの入力手段によって受け付ける（ステップ１０）。トランシェとはアセットバック証券の担保となる将来のキャッシュフローの総資産残高の契約規程範囲のことである。なお、トランシェは総資産額における範囲を示している。トランシェと増減が逆となる総資産消滅額の範囲を定めるために用いた、α＞βの関係を有するαとβを用いると、トランシェの上限と下限は、（トランシェの上限）＝（初期の総資産残高）−β、（トランシェの上限）＝（初期の総資産残高）−αとそれぞれ定めることが出来る（図３参照）。
【０１１２】
ロジスティック曲線に基づき、総資産消滅額Ｌ_ｔの平均経路νλ_ｔを演算手段が計算する（ステップ２０）。
ステップ２０の計算例としては、図７に示すようなフローチャートが挙げられる。まず、ロジスティック曲線モデルの選択をコンピュータの入力手段が受付ける（ステップ２１）。このロジスティック曲線モデルは、アセットバック証券の対象となる担保から得られる過去の総資産消滅額のデータから、適宜適当なものを選択することができる。ロジスティック曲線モデルとしては、現実の総資産消滅額を表すのに適当な成長曲線であればよく、ｔａｎｈ（ａｔ）や、ｙ＝ｋ／（１＋ｅ^{ｃｆ（ｔ）}）の形の関数を表す曲線や、母集団の大きさｙが時間ｔの関数ｙ＝ｋ／（１＋ｅ^−ｋｂｔ）、指数関数でもよい。
【０１１３】
例えば、ロジスティック曲線としてｔａｎｈ（ａｔ）が選択されたとき、時刻ｔに対して、総資産消滅額Ｌ_ｔの過去のデータと、ロジスティック曲線ｔａｎｈ（ａｔ）との数値との差を演算手段が計算する（ステップ２２）。
次に、総資産消滅額Ｌ_ｔの過去のデータと、ロジスティック曲線ｔａｎｈ（ａｔ）の数値との差に基づき、最小二乗法によって、差を最小にするパラメータａを演算手段が非線形回帰推定計算する（ステップ２３）。そして、パラメータａに基づきロジスティック曲線ｔａｎｈ（ａｔ）を演算手段が計算する（ステップ２４）。こうして求めた総資産消滅額Ｌ_ｔの過去のデータにフィットするロジスティック曲線ｔａｎｈ（ａｔ）を総資産消滅額Ｌ_ｔの増加経路とみなし、その増加経路を各時刻ｔでの平均経路νλ_ｔとする。つまり、演算手段によって計算された該ロジスティック曲線を、演算手段が、総資産消滅過程の平均経路とする（ステップ２５）。
【０１１４】
次に、平均経路νλ_ｔに基づき、総資産消滅額Ｌ_ｔがレビー過程としてモデル化し得られる［数１１］に示すように期待総資産消滅過程Ｅ［Ｌ_ｔ］を演算手段が計算する（ステップ３０）。ステップ３０の一例としては、図８に示すようなフローチャートが挙げられる。期待総資産消滅過程Ｅ_ｔ［Ｌ_ｔ］を表すレビー過程モデル式の選択をコンピュータが受付ける（ステップ３１）。期待総資産消滅過程Ｅ_ｔ［Ｌ_ｔ］の例としては、［数１１］が挙げられる。しかし、［数６］の定義が異なれば、［数１１］も異なる。
さらに、選択された期待総資産消滅過程Ｅ_ｔ［Ｌ_ｔ］を表すレビー過程モデル式に平均経路νλ_ｔを代入することにより、期待総資産消滅過程Ｅ_ｔ［Ｌ_ｔ］を演算手段が計算する（ステップ３２）。
【０１１５】
トランシェの上限値を定めるβと、トランシェの下限値を定めるαと、期待総資産消滅過程Ｅ［Ｌ_ｔ］とに基づき、［数１４］に基づき、任意の時点でのアセットバック証券の時価Ｂ＝Ｄ（ｔ）を演算手段が計算する（ステップ４０）。
【０１１６】
次に、本発明のアセットバック証券の時価計算方法の他の実施の形態として、図６のフローチャートを説明する。図６のフローチャートと、図５のフローチャートとの違いは、プリペイメント率を考慮している点である。
まず、トランシェの上限値を定めるβと、トランシェの下限値を定めるαと、プリペイメント率φ_ｔをコンピュータが受け付ける（ステップ１１０）。そして、記憶手段に保存されている所与のロジスティック曲線に基づき、総資産消滅額Ｌ_ｔの平均経路νλ_ｔを演算手段が計算する（ステップ１２０）。総資産消滅額Ｌ_ｔの平均経路νλ_ｔに基づき、総資産消滅額Ｌ_ｔがレビー過程であるとして、期待総資産消滅過程Ｅ［Ｌ_ｔ］を演算手段が計算する（ステップ１３０）。そして、トランシェの上限値を定めるβと、トランシェの下限値を定めるαと、期待総資産消滅過程Ｅ［Ｌ_ｔ］と、プリペイメント率φ_ｔとに基づき、［数１５］を用いて、任意の時点でのアセットバック証券の時価Ｂ＝Ｄ（ｔ）を演算手段が計算する（ステップ１４０）。
【０１１７】
【実施例】
本発明に係るアセットバック証券の時価計算方法の実施例を次に説明する。アセットバック証券の時価計算方法における数式に出てくるパラメータの推定方法は、入手可能な情報によりそれぞれの場合に合わせて工夫することができる。
次の事例はローン不良化の実績が類似の過去データから想定できると仮定できる場合のアセットバック証券の時価計算方法である。アセットバック証券の担保となる全原資産の平均的な資産消滅額Ｌ_ｔはほぼロジスティック曲線ｙ＝ｔａｎｈ（ａｔ）に従うと仮定する。
【０１１８】
実体経済から得られる資産消滅額Ｌ_ｔに関する過去データをもとに演算手段によって、非線型回帰推定を行なう。非線形回帰推定の際に同時に行なう誤差解析からパラメータνを演算手段によって計算できる。ここで過去のデータとしては、１９９７年４月から２００１年１０月の大手金融機関によるローン不良債権化シナリオのデータを用いた。
資産消滅額に関する過去のデータなどの情報が少ない場合は、［数１４］または［数１５］によって示されるモデルで時間に依存するパラメータを定数にするなどして、推定計算をすることができる。
【０１１９】
図９は、再評価時点におけるアセットバック証券の利回りの変化を示した図である。縦軸は、再評価時点におけるアセットバック証券の利回りを表し、横軸は発行時点からの再評価するまでの時間（年）を表している。この事例ではアセットバック証券の担保はローンであり、契約規定範囲であるトランシェは、βで規定される上限が残高１２億６千万円で、αで規定される下限が８億円であるとしたアセットバック利付債（証券）を想定した。このアセットバック証券は５年満期であり、クーポンは４．３５％であった。ここで、本発明に係るアセットバック証券の時価計算方法と、従来のモンテカルロシミュレーションによるものと、従来の大手情報会社であるベンダー評価モデルによるものとを比較した。なお、本発明に係るアセットバック証券の時価計算方法では、アセットバック証券の担保となる資産の平均減衰データから各種パラメータを計算した。モンテカルロシミュレーションは、アセットバック証券の担保となる資産の各構成銘柄の倒産シナリオに基づき、将来時点で再評価した際の２２０００回の平均値を示した。
【０１２０】
ここで、乱数の代わりに市場データを用いた、従来の大手情報ベンダーの評価モデルは、アセットバック証券の担保となる総資産消滅過程の個々の原資産が均一なものであるとしてボトムアップ的に単純化したものである。すなわち、それぞれの均一資産の減少過程が独立的にポアソン過程に従うものとしてモデル化しており、利回りからこのモデルの特徴を殆ど決めてしまう基本パラメータ（ハザードレート）を計算することを特徴としている。従来のボトムアップ方式によって解析的なモデル化ができるのは限定的であり、前述の従来の例が希少例として挙げられる。しかし、このような従来例では、ボトムアップ方式で解析解を求めようとするので、自由度があるモデルパラメータを少なくしないといけない。なぜなら、アセットバック証券の担保となる資産数が増加すると、コンピュータの演算時間がかかりすぎて現実的でなくなるからである。この理由のため、従来例のモデルパラメータは、例えば、契約期間、個別の倒産危険率、資産数などのパラメータに限定される。このように、従来例は単純にできているため、総資産消滅過程の平均経路は単純な指数関数となり、現実的な経済現象と乖離してしまう。
【０１２１】
一方、本発明のアセットバック証券の時価計算方法は、アセットバック証券の担保となる総資産を直接モデル化するというトップダウン方式で総資産消滅過程をモデル化し、自由度が高くなっているので、前記の従来例とは異なり、証券の担保となる資産数が増加しても、解析的な答えを出すことができ、しかも、コンピュータの演算時間が従来と比べかかりにくく、効率的になっている。
【０１２２】
図９に示すように、本発明に係るアセットバック証券の時価計算方法による再評価時点でのアセットバック証券の利回りの変化は、従来のモンテカルロシミュレーションを用いたアセットバック証券の利回りの変化とほぼ同じであることが分かる。この点で、モンテカルロシミュレーションと同等の予測精度が得られることがわかる。計算機実験であるモンテカルロシミュレーションに比べ、本実施態様によるのアセットバック証券の時価計算方法は、解析的に行われるために、計算速度が速く、効率的であり、また精緻な方法である。また、従来の解析的な手法であるベンダー再評価モデルでは、計算速度は早いものの、その結果を左右するパラメータの決定は、発行時の利回りのみに基づいており、過去データに基づくシミュレーションからは図９に示すように乖離してしまっている。これは、個々のパラメータを適切に設定しないと最終的な結果が適切に得られないという、ボトムアップ型のベンダー再評価モデルの問題を示している。本発明の評価方法にあっては、こういったボトムアップ手法をとらずにレビー過程を用いた新規な方法を用いることにより、モンテカルロシミュレーションに匹敵する予測精度と、解析的手法による計算速度とを両立することが可能となった。
【０１２３】
さらに、再評価の時間（年）（ｒｅｖａｌｕｉｎｇ　ｔｉｍｅ（ｙｅａｒ））の１．８年から２．０年において、ベンダーモデルによるアセットバック証券の利回り変化のカーブと、本発明のアセットバック証券の時価計算方法によるアセットバック証券の利回りの変化のカーブを比較すると、図９から明らかなように、本発明ではアセットバック証券の利回りが急激に上昇している。これは、本発明のアセットバック証券の時価計算方法において、総資産消滅過程をレビー過程としたためである。レビー過程はジャンプ発生頻度を自由に調整できるので、総資産消滅過程において消滅額が一時期に集中してジャンプし、過去のデータに基づいて１９９８年頃の実体経済の変動を反映して、アセットバック証券の利回りが急激に変化している。ところが、従来の標準ポアソン過程を前提とするベンダーモデルでは、総資産消滅過程はジャンプ頻度を考慮していないため、利回りが急激に上下することはない。
【０１２４】
なお、本実施の形態の説明において説明した例以外の場合であっても、任意の総資産消滅過程がレビー過程を利用して理論的に高い精度で近似できることがわかっている。
【０１２５】
【発明の効果】
上記したところから明らかなように、本発明のアセットバック証券の時価計算方法によると、合理的かつ迅速なアセットバック証券の時価計算ができるようになる。また、アセットバック証券の時価の解析解による計算がより合理的なものとなり、解析解に基づく高速数値計算も可能となる。
【図面の簡単な説明】
【図１】一般的なアセットバック証券のトランシェの構造を示す模式図である。
【図２】一般的なアセットバック証券のトランシェの構造と資本市場との関係を示す模式図である。
【図３】一般的なアセットバック証券の総資産残高の変移を示す模式図である。
【図４】一般的なアセットバック証券の総資産残高の変移を示す模式図である。
【図５】本発明に係るアセットバック証券の時価計算方法の一実施の形態を示すフローチャートである。
【図６】本発明に係るアセットバック証券の時価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７】本発明に係るアセットバック証券の時価計算方法のロジスティック曲線の計算の一実施の形態を示すフローチャートである。
【図８】本発明に係るアセットバック証券の時価計算方法の期待総資産消滅過程の一実施の形態を示すフローチャートである。
【図９】本発明に係るアセットバック証券の時価計算方法と、従来のモンテカルロシミュレーションを用いるアセットバック証券の時価計算方法と、従来のベンダー再評価モデルを用いるアセットバック証券の時価計算方法による再評価時点におけるアセットバック証券の利回りの変化を比較した比較図である。
【符号の説明】
Ｄ〜（ｔ）　　時刻ｔに発生するキャッシュフローの現在価値
Ｅ_０［・］　　Ｅ_０［Ｘ］は、確率変数Ｘの期待値（平均値）
ｍａｘ　　　　　ｍａｘＸは集合Ｘの最大要素
ｍｉｎ　　　　　　　　　　　　ｍｉｎＸは集合Ｘの最小要素
ｒ_ｔ　　ディスカウントレート（割引率）
α　　　ヨーロピアンコールオプションの組み合せにおける行使価格の低い方の価格、総資産残高におけるトランシェの下限を定める値
β　　　ヨーロピアンコールオプションの組み合せにおける行使価格の高い方の価格、総資産残高におけるトランシェの上限を定める値
Ｂ　　　アセットバック利付債価格
ｔ　　　アセットバック証券発行時からの時間
Ｔ　　　満期
Ｌ_ｔ　　時刻ｔにおける元本量
Ｌｔ　　　　　　　時刻Ｔにおける元本量
ｃ（ｄｔ）次の瞬間に支払われるクーポン
ψ_ｔ　　リカバリーレート
ｄＬ　リカバリーを考慮した際の元本量Ｌ_ｔの確率微分
Ａ　　　　初期総資産額
φ　　　　資産消滅割合
η　　　　ハザードレート、消滅・倒産危険度
ｄｗ　ブラウン運動、ウィナー過程
Ｅ_ｔ［・］　　　積率母関数
θ　　　積率ジェネレーター
ν　　　Γ関数の引数、畳込み重複度
λ_ｔ　　指数分布の引数、ポアソンパラメター
ｈバー　　　　　ジャンプ幅
ζ　　　現時点での資産消滅額
Ｐｒ_ｔ　累積確率密度関数
ｚ　　　確率密度関数の引数（確率変数の上限を与える）
λ_Ｔ指数分布の引数、ポアソンパラメター
λ_Ｔνλ_Ｔのν乗
λ_ｔν　λ_ｔのν乗
κ　　　　和分の添字
ｚ　　　　積分の添字、確率密度関数の引数（確率変数・元本消滅量の上限を与える）
ι　　　和分の添字
Ｄ_ｔ（Ｔ）　　　通常の割引債価格
Ｄ〜ｔ，ζ　　　　　　　アセットバック割引債価格
τ　　　積分の添字、キャシュフローのタイミング
ｃτ　τ時点で支払われるクーポン
ν_ｔ，ζ（τ）瞬間元本償還率
１０　受付ステップ
２０　ロジスティック曲線計算ステップ
３０　期待総資産消滅過程計算ステップ
４０　アセットバック証券の時価計算ステップ
１２０　ロジスティック曲線計算ステップ
１３０　期待総資産消滅過程計算ステップ
１４０　アセットバック証券の時価計算ステップ[0001]
TECHNICAL FIELD OF THE INVENTION
The present invention relates to a method of calculating the market value of asset-backed securities (ABS: Asset Backed Security, also referred to as asset-backed securities) and a computer program thereof.
[0002]
[Prior art]
In recent years, accounting standards have changed in Japan as well, and it has been required to evaluate the market value of assets owned by companies. As a result, the need for valuation of asset-backed securities has increased.
[0003]
Here, asset-backed securities are generally issued based on the assets held by a company, and are used to separate assets such as receivables and real estate held by the company from the company, and to be issued using cash flow generated from the assets as a source. Security. For example, asset-backed securities are issued based on cash flows of various assets such as consumer loans such as mortgages, car loans, and credit card bonds, bonds, bank loans, and accounts receivable. It should be noted that a loan based on a mortgage such as a mortgage may be classified as a MBS (Mortgage Backed Security, mortgage-backed securities) which is not included in the ABS. CMO (Collateralized Mortgage Obligation), CBO (Collateralized Bonded Obligation), CLO (Collateralized Loan Obligation), etc. are also liquid assets based on the cash flow income of any underlying asset, and are therefore A.
[0004]
Since the asset-backed securities collateral the cash flows generated by the assets, after the issuance of the asset-backed securities, cash flows may not be generated from the individual underlying assets secured for some reason after the issuance of the asset-backed securities. At this time, the market value of the asset-backed securities will be lower than when it was issued.
[0005]
One reason cash flows are not generated is, for example, in mortgage-backed asset-backed securities, where a mortgage-payer's individual defaults or pays off before the due date. This is caused by a decrease in cash flow for future interest rates. As a result, the market value of asset-backed securities will decrease.
[0006]
Now, in order to issue asset-backed securities, a special purpose company (or a special purpose company, special, company, company, SPC) is first established to separate assets from companies. The company transfers the assets to its special purpose company.
The SPE issues securities backed by the transferred assets and sells them to investors. Since the assets are separated from the company, even if the original company goes into bankruptcy or the like, if the assets held by the special purpose company are healthy, investors can receive the payment of the securities with confidence.
[0007]
In other words, asset-backed securities can be said to be securities invested not in the creditworthiness of the original company but in the creditworthiness of the target asset. For example, in the case of asset-backed securities backed by loan receivables, even if the company that held the receivables goes bankrupt, if the loan receivables themselves are good, the investors will be paid for the securities through a special purpose company be able to.
[0008]
Assets separated from companies include accounts receivable, notes receivable, bonds (corporate bonds), loans (mortgages in real estate, mortgages, car loans, credit card loans), lease receivables, commercial paper (CP), etc. There are securitizations by issuing asset-backed securities for a wide range of assets. However, asset-backed securities with no high-grade reassurance are generally complex with many types of assets to be pledged, and their market value is difficult to assess as described below.
[0009]
One of the reasons market valuation of asset-backed securities is difficult is that the market for determining the market value of assets cannot be evaluated because the market for reselling asset-backed securities is incomplete. The resale market is a market in which securities issued by an issuer can be bought and sold again by arbitrary persons, and is also referred to as a secondary market or a secondary market.
[0010]
In addition, another reason that the market value of asset-backed securities is difficult is that detailed information about the individual mark-to-market of the underlying assets included in the asset-backed securities would be required if the asset-backed securities were to be revalued elaborately. It is mentioned. For example, asset-backed securities based on mortgages require credit information on defaults of those who are obliged to pay individual loans embedded as collateral, prepayment information, and the like. Generally, such information is difficult to obtain or has a high cost to obtain.
[0011]
Furthermore, if you know who among the mortgage borrowers has gone bankrupt and who is left, or if you know the remaining borrower's current income, gambling habits, etc., the market value is determined. A good analysis of However, let's evaluate the value of securities (detailed later) that are actually issued by dividing the securities backed by mortgages (loan bonds) such as CMOs into several kinds of different parts called several tranches Even so, in most cases, such details are not disclosed, not even what percentage of the total loan set has become nonperforming as of that day. The information readily available to ordinary investors is even more limited, and the available information is often limited to the price at which the CMO was purchased and the interest rate of the bond.
[0012]
Another reason that the market value of asset-backed securities is difficult is that when trying to simulate the market value of each underlying asset that is the collateral of the asset-backed securities, the complex market value of each underlying asset by the Monte Carlo method Calculations are required, which can be time-consuming and complicated.
[0013]
If an investor who purchases an asset-backed security seeks detailed information on the current status of the individual assets included in the issuance of the asset-backed security, the investor will not be able to obtain detailed information on the individual assets at that time. Investigating is not an easy task. To date, few cases have disclosed detailed information on the underlying assets for the purpose of valuation.
[0014]
Conventionally, asset-backed securities, which are often used in practice, are valued based on forecasts of future changes in yield (yield) of government bonds with high creditworthiness, based on the credit rating at the time of issuance of asset-backed securities. This is a method of adding a yield spread to the yield of government bonds and predicting the yield of asset-backed securities. Calculate the present value (market value) of the asset-backed security based on the predicted yield of this asset-backed security. Here, the yield spread refers to the difference between the yield of government bonds and the yield of asset-backed securities. Generally, at the time of issuance, the yield (yield) of high-rated bonds such as government bonds tends to be higher than the yield (yield) of lower-rated bonds compared to government bonds. Is calculated assuming the existence of
[0015]
In such a conventional method of valuing the market value of asset-backed securities by adding the yield spread to the yield of government bonds, the logical basic structure of the asset-backed securities based on the timing of the collateral cash flow (see FIG. 1) Tranche) does not reflect the complexity. For this reason, the valuation of asset-backed securities may result in suspicious valuation that deviates from the real economy.
[0016]
Further, as a market valuation published by asset-backed securities, there is a market valuation published by a major information vendor. The market valuation of asset-backed securities announced by the conventional major information vendors is based on a system (bottom-up system) that individually models the bankruptcy and prepayment of asset components, etc. This is a method (simple evaluation model) for obtaining an analytical solution. This model is not a computer experimental method such as Monte Carlo simulation, but is the only conventional method that enables the market value evaluation of ABS by an analytical solution. However, in this model, as a premise, simplification has been made that the individual underlying assets that constitute the total assets are uniform. It is difficult to estimate input parameters from data. In addition, it is difficult to reflect the dependencies and correlations between the many stocks that make up the underlying assets, and the risk adjustments caused by them, in the results of mark-to-market.
[0017]
[Problems to be solved by the invention]
SUMMARY OF THE INVENTION The present invention has been made in view of the above-mentioned problems relating to the valuation of asset-backed securities, and provides a method for calculating the market value of asset-backed securities reasonably and quickly and a program therefor.
[0018]
[Means for Solving the Problems]
In order to achieve the above object, the present invention provides a method for calculating the market value of asset-backed securities, the method for calculating the market value of asset-backed securities issued with the total assets of cash flows based on assets as collateral. A step of the computer receiving the upper limit value of the tranche indicating the range of the amount of assets to be collateralized by the asset-backed securities and the lower limit value of the tranche in the storage means; Calculating the expected total asset extinction process indicating the expected value of the total asset extinction process by the arithmetic means based on the model that has been set, and setting the upper limit of the tranche stored in the storage means. Calculating the market value of the asset-backed securities based on the calculated value and the lower limit value and the calculated expected total asset extinction process. Characterized in that it comprises a flop.
[0019]
Another aspect of the present invention is a method of calculating the market value of asset-backed securities, which is a method of calculating the market value of asset-backed securities issued with the total assets of cash flows based on assets as collateral. The upper limit of the tranche indicating the range of the amount of assets to be secured by the security, the lower limit of the tranche, and the prepayment indicating the ratio of the amount of the extinguished assets due to payment before the due date to the total amount of extinguished assets. A step of receiving the total rate in the storage means by the computer and storing the total rate in the storage means based on the past data of the total asset disappearance process and a model obtained based on the data. A step of calculating the expected total asset extinction process indicating the expected value by the calculating means, and the upper and lower limits of the tranche stored in the storage means. Comprising a value, and calculated the expected total assets annihilation process, based on the said prepayment rate stored in the storage means; and the market value of the asset backed securities calculating means for calculating.
[0020]
As another aspect, the method for calculating the market value of asset-backed securities according to the present invention is a method for calculating the market value of asset-backed securities issued with the total assets of cash flows based on assets as collateral. A tranche upper limit value indicating a range of asset amounts to be collateral, a lower limit value of the tranche, and a prepayment indicating a ratio of an extinguishing amount of the assets to a total extinguishing amount of assets due to payment before the scheduled payment date. A computer accepting and storing in a storage means a recovery rate indicating a rate and a rate of recovery in the amount of asset disappearance when the cash flow payer defaults; and historical data of the total asset disappearance process And the expected model of the total asset extinction process, based on Calculating the expected total assets disappearance process, the upper limit value and the lower limit value of the tranche stored in the storage device, the calculated expected total assets disappearance process, and storing the expected total asset disappearance process in the storage device. Calculating a market value of the asset-backed securities based on the prepayment rate and the recovery rate.
[0021]
Based on the past data of the total asset disappearance process and a model obtained based on the data, the calculation means calculates the expected total asset disappearance process indicating the expected value of the total asset disappearance process with the total asset disappearance process as the Levy process. The step of: accepting the selection of a given logistic curve model stored in the storage means by the computer; and minimizing the difference based on the difference between the past data of the total asset disappearance process and the numerical value of the logistic curve model. Calculating means for estimating a parameter in the logistic curve model by nonlinear regression; calculating the logistic curve based on the parameter; and calculating the calculated logistic curve by averaging the total asset disappearance process. The steps taken by the route and the calculation means, and the selection of the Levy process model formula representing the expected total asset disappearance process Computer, and substituting the average path into a Levy process model formula representing the selected expected total asset disappearance process, and calculating the expected total asset disappearance process by a calculating means. Good.
[0022]
The Levy process may be a process selected from a group consisting of a Poisson process, a Wiener process, and a process combining the Poisson process and the Wiener process. In addition, the Levy process may be monotonically increasing.
[0023]
The present invention also provides a computer program for executing the method of calculating the market value of asset-backed securities on a computer. The asset-backed securities market value calculation program of the present invention is a asset-backed securities market value calculation program issued with the total assets of cash flows based on assets as collateral. The upper limit of the tranche indicating the range of the amount of assets, and the step of receiving the lower limit of the tranche by the computer and storing it in the storage means, based on the past data of the total asset disappearance process and a model obtained based on the data, Calculating the expected total asset extinction process indicating the expected value of the total asset extinction process by using the total asset extinction process as a Levy process; and calculating the upper and lower limits of the tranche stored in the storage unit. Calculating the market value of the asset-backed securities based on the expected total asset extinguishing process performed by the calculating means. To perform the steps.
[0024]
Another embodiment of the present invention provides a method for calculating the market value of asset-backed securities, which is a program for calculating the market value of asset-backed securities issued with the total assets of cash flows based on assets as collateral. The upper limit of the tranche that indicates the range of the asset value to be secured by the security, the lower limit of the tranche, and the ratio of the amount of the extinguished asset to the total extinguished asset due to payment before the scheduled payment date. A step of receiving the prepayment rate by the computer and storing it in the storage means; and, based on the past data of the total asset disappearance process and a model obtained based on the data, the total asset disappearance process is defined as a Levy process. Calculating the expected total asset extinction process indicating the expected value of the tranche stored in the storage means. Calculating a market value of the asset-backed securities based on the limit value and the lower limit value, the calculated expected total asset extinguishing process, and the prepayment rate stored in the storage means. Execute.
[0025]
Another embodiment of the present invention provides a method for calculating the market value of asset-backed securities, which is a program for calculating the market value of asset-backed securities issued with the total assets of cash flows based on assets as collateral. A tranche upper limit that indicates the range of assets that will be the security of the security, a lower limit of the tranche, and a prepayment that indicates the ratio of the amount of asset extinguished to the total asset extinguished by payment made before the scheduled payment date. A computer accepting and storing in a storage means a recovery rate indicating a rate and a rate of recovery in the amount of asset disappearance when the cash flow payer defaults; and historical data of the total asset disappearance process And based on the model required based on the total asset A calculating means for calculating an expected total asset disappearance process indicating an expected value of the process; an upper limit value and a lower limit value of the tranche stored in the storage means; the calculated expected total asset disappearance process; Calculating the market value of the asset-backed securities based on the prepayment rate and the recovery rate stored in the storage unit.
[0026]
Based on the past data of the total asset disappearance process and a model obtained based on the data, the calculation means calculates the expected total asset disappearance process indicating the expected value of the total asset disappearance process with the total asset disappearance process as the Levy process. The step of: accepting the selection of a given logistic curve model stored in the storage means by the computer; and minimizing the difference based on the difference between the past data of the total asset disappearance process and the numerical value of the logistic curve model. Calculating means for performing a non-linear regression estimation calculation on the parameters in the logistic curve model to be calculated, calculating the logistic curve based on the parameters, and calculating the calculated logistic curve in the total asset disappearance process. The average route, the steps performed by the calculation means, and the Levy process model The computer selects the expected total asset disappearance process, and calculating the expected total asset disappearance process by substituting the average path into a Levy process model formula representing the selected expected total asset disappearance process. Can be.
[0027]
The Levy process may be a process selected from a group consisting of a Poisson process, a Wiener process, and a process combining the Poisson process and the Wiener process. In addition, it is preferable to select a process in which the Levy process is monotonically increasing. Further, among such Levy processes, one that gives an analytical solution can be selected.
The present invention also provides a computer-readable recording medium on which any of the above programs is recorded.
As described above, according to the present invention, without using a method (a computer experimental method) that requires a long calculation time, such as Monte Carlo simulation, analytically (when a Levy process that gives an analytical solution is used) or By calculating the numerical value of a given mathematical expression to the extent of numerical integration, it is possible to more quickly calculate the market value of asset-backed securities, and to develop a high-speed numerical calculation algorithm. This is a significant advantage under market value accounting standards, which require more accurate mark-to-market.
[0028]
Hereinafter, main terms related to the method of calculating the market value of asset-backed securities according to the present invention will be described.
The input means includes input means operated by an operator such as a keyboard and a mouse, input from a recording medium such as a floppy (registered trademark) disk, or an electric signal such as an electric communication line including the Internet or a LAN (local area network). Means for receiving an input by the user.
[0029]
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.
[0030]
The calculation means is a numerical calculation means for executing a program. For example, it is generally called a central processing unit, that is, a CPU.
[0031]
The computer refers to an electronic computer including the input unit, the storage unit, and the calculation unit as described above. The computer executes a command for causing itself to act as an appropriate means according to the purpose of the present invention, receives data from the input means, calls the data from the storage means, and performs arithmetic processing by the arithmetic means. Further, the computer displays the calculation result according to the result, records it in the storage means, transmits the calculation to another computer by itself, transmits the calculation at the request of another computer, and controls other devices. Or If necessary, communication with other devices via an appropriate network or communication means may be performed. For example, in executing the above-mentioned instruction, when the computing means of the computer calculates the expected total asset disappearance process of the present invention, the computing device of the computer executes the step of calculating the expected total asset disappearance process of the present invention. Thereby, the computer operates as the expected total asset disappearance process calculation means. That is, the computer includes the expected total asset disappearance process calculation means. As in this example, any step of the present invention described as the operation of the computer itself, or the operation of any means of the computer, is performed by the appropriate means constituting the computer executing the operation. Means that it can operate as a means including the operation means in addition to the input means, the storage means, and the calculation means. That is, it is assumed that the computer includes the operation unit.
[0032]
The computer-readable recording medium is a medium on which programs and data are recorded and which can be read by a computer. For example, it is a floppy (registered trademark) disk, a CD-ROM, an MO, a ROM chip, or the like.
[0033]
Asset-backed securities are generally securities issued with the total amount of cash flow based on a certain asset as collateral. More specifically, asset-backed securities are issued in support of assets held by a company, separate assets such as receivables and real estate held by the company from the company, and are issued using cash flow generated from the assets as a source. Security.
[0034]
Assets separated from companies include accounts receivable, notes receivable, bonds (corporate bonds), loans (mortgages in real estate, mortgages, car loans, credit card loans), lease receivables, commercial paper (CP), etc. There are securitizations by issuing asset-backed securities for a wide range of assets. The assets serving as collateral for the asset-backed securities according to the present invention are not limited to these examples, and include a wide range of tangible assets, intangible assets, and the like as long as they generate cash flows. In addition, a loan based on a mortgage such as a mortgage may be classified as MBS (Mortgage Backed Security), which is not included in the ABS. In the present invention, the MBS and an example of the MBS are not included in the ABS. CMO (Collateralized Mortgage Obligation), CBO (Collateralized Bonded Obligation), CLO (Collateralized Loan Obligation), etc. are also liquid assets based on the cash flow income of any underlying asset, and are therefore A. Examples of assets backed by asset-backed securities include second-tier mortgages, home equity loans (revolving), mobile home loans, leisure mortgages, commercial real estate, lease bonds, car loans, credit card bonds, and medical fees. Rights, bonds to developing countries, junk bonds, LBO loans, and the like.
[0035]
The discount rate is the interest rate used to discount the future value of an asset-backed security to its present value.
The total asset extinguished amount is the amount extinguished for some reason among the total assets of the cash flow as collateral for the asset-backed securities. The reason for this is that, for example, in the case of mortgage-backed securities called CMOs, the repayment manager of the loan backed by the asset-backed securities has already repaid earlier than the expected payment date, so the cash equivalent to future interest rates There are cases such as prepayment where the flow disappears, and the repayment manager not defaulting on the debt and the subsequent cash flow disappears.
[0036]
The growth curve is a curve that shows the state of growth of an organism in the first place. There are two types, one related to the growth of the size of an individual and the other related to the increase in the number of individuals, that is, population growth. Usually, the horizontal axis indicates time, and the vertical axis indicates appropriate measurement values. In the case of average growth of the size of an individual or growth of an individual group, an S-shaped curve (sigmoid curve) is generally obtained in many cases. Typical examples of the growth curve include a logistic curve, and other examples include a Golpertz curve, a von Peltaranfiy curve, and a connection of two or three S-shaped curves.
[0037]
A logistic curve is a type of the above-mentioned growth curve. In general, the size y of the population is a function of time t, y = k / (1 + e).^-Kbt). Here, k and b are positive constants. More generally, y = k / (1 + e^{cf (t)}) May be referred to as a logistic curve (where c is a constant and f (t) is a function of time t). In the present invention, these are all included. In the method for calculating the market value of asset-backed securities according to the present invention, y indicates the transition of the total asset disappearance amount, and cf (t) is an explanatory function indicating the transition of the total asset disappearance amount with respect to time. However, the logistic curve is y = k / (1 + e)^{cf (t)}) Does not have to be a function expression, and may be arbitrarily modified.
[0038]
The past data in the disappearance process of the total assets disappearance amount refers to data on the increase and decrease of assets serving as collateral for asset-backed securities obtained from the real economy. This historical data differs for each asset-backed security, and also varies in quality (reliability) and quantity depending on the investors holding the security. The following (1) to (5) are examples assumed as past data of the extinction process of the total asset extinction amount. It is preferable to consider each case individually.
[0039]
[(1) When the past data in the process of disappearing total assets can obtain only minimum data]
Generally, past data during the process of disappearing total assets is most often obtained only with a minimum of data. This case refers to, for example, a case where only data such as the purchase price, maturity, interest rate (coupon), and rating of asset-backed securities can be obtained as past data in the process of disappearing total assets. In the case where a loan pool such as a mortgage loan is used as an underlying asset and a rudimentary model is used, for example, in a diffusion approximation equation of a Poisson process such as the following [Equation 5], model parameters are used as an example. It may be tentatively determined as follows. Time t is the time elapsed from the time of issuance, and η is the average credit spread of the bond determined by the rating. By setting A to an appropriate value that is realistic and large, it is assumed that the volatility of the spread of the corporate bond is suppressed to the extent observed in the actual corporate bond market. Finally, the width of the tranche of the asset-backed securities is determined so that the price calculated based on this Poisson process realizes the purchase price. At this time, using a method that uses a consultant who is familiar with this field, and studying the characteristics of the underlying asset and using a model that assumes a Levy process with a complicated attenuation path, It is also possible to adjust the model parameters to match.
[0040]
[(2) In addition to the case of (1), when the underlying asset information at the time of issuance of each asset constituting the asset-backed securities is available]
In this case, depending on the amount and quality of the obtained information, for example, it is a case where the total amount of individual underlying assets serving as collateral for asset-backed securities and simple information on the underlying asset composition can be additionally obtained. In the case of (1), A appropriately set is the total amount of the underlying asset, and as a sum of the diffusion approximation formulas obtained from the respective ratings according to the underlying asset composition, it is also possible to assemble the entire total asset disappearance process as a Levy process. It is.
[0041]
[(3) When a huge amount of past loan repayment data is available]
Large financial institutions can use the huge amount of past loan repayment data. If the accuracy of past loan repayment performance data is high, that is, the situation in which the data is collected is the same as the situation in which it is applied, for example, when the economic conditions are close, the individual asset The average attenuation path is calculated, and the total asset disappearance process is modeled as a Levy process using the Levy process that realizes the numerical value as the average route. If the accuracy of past loan repayment data is not very accurate, parameters are estimated from past data, assuming that the average path of the total asset disappearance process is a growth curve. Then, the growth curve is modeled using a Levy process that uses the average path of the total asset disappearance process.
[0042]
[(4) When the underlying assets and total amount of individual assets at the time of issuance of asset-backed securities and the tranche width are available]
The value of each underlying asset value of the asset-backed securities is directly used for market valuation, and the parameter (for example, ν) relating to the variance of the Levy process is calculated backward from the issue price. If the width of the tranche is not known, it is preferable to estimate from the rating information as much as possible to realize the rating at the time of issuance. If this is extremely difficult, the parameters for the variance of the Levy process are estimated from historical data, and the width of the tranche is estimated from the issue price.
[0043]
[(5) When the market value of asset-backed securities is calculated, the balance of each of the underlying assets used as collateral for asset-backed securities is known]
Further, when the market value of the asset-backed securities is evaluated, if the balance of each underlying asset serving as collateral for the asset-backed securities is known, the value is evaluated using the Levy process starting from the value. If it is not known, the current total asset balance is estimated by a maximum likelihood method or the like using a model based on the passage of time from the issuance time.
As can be seen from these past data examples (1) to (5), the use of a logistics curve (growth curve) for modeling the total asset extinction process of asset-backed securities is a very favorable exception. It is limited to a typical case. In any case, using the method described above, the total asset disappearance process is modeled based on past data on the total asset disappearance process.
[0044]
Next, the prepayment rate indicates the ratio φ of the extinction amount due to prepayment to the total asset extinction amount. Here, the prepayment is repayment before the deadline. For example, CMO, which is one of the mortgage-backed securities, shows that the repayment manager (borrower) of the loan backed by the asset-backed securities repays the principal and interest earlier than the expected payment date. Disappears.
[0045]
The recovery rate indicates a ratio of recovery, which is an asset returned to the total asset amount, to the total asset disappearance amount, among the disappearance amount of the total assets lost for some reason.
[0046]
The Levy process is generally a stochastic addition process in which the sample function has a probability of 1 and is at most discontinuous of the first kind and right-continuous. The Levy process is a kind of stochastic process, and is a fairly general stochastic process including a Poisson process and a Wiener process (Brownian motion). In addition, the Levy process includes a composite Poisson process in which the sample function changes only by a leap in a temporally uniform Levy process and a Levy measure of which is finite, and a Cauchy process.
[0047]
Here, the stochastic process is a concept that mathematically expresses a value or position that changes stochastically with time. In the field of financial engineering, stock and securities prices are usually modeled as following the Brownian movement. Brownian motion is a stochastic process in which instantaneous fluctuations follow a normal distribution. In the Brownian motion, the value or position does not move significantly at a moment (it fluctuates along a continuous path), but the Levy process is a stochastic process that includes such a jump at the moment (has a discontinuous path). Also, in the Brownian movement, when the price is modeled, the price may rise or fall, but in the Levy process, the price movement changes unilaterally with time (monotonically decreasing) or increasing only (monotonically increasing). Can also be modeled.
[0048]
In the method for calculating the market value of asset-backed securities according to the present invention, modeling is performed assuming that the transition process of the total asset depletion amount of the depleted portion of the total assets serving as collateral for the asset-backed securities is a monotonically increasing Levy process including jumps . The modeling of the Levy process does not need to be limited to a specific mathematical formula, and may be changed as appropriate in accordance with individual asset-backed securities. For example, the mathematical expression of the description of “Levy process” in “51 @ additive process” on pages 143 to 147 of “Iwanami Mathematics Dictionary (3rd Edition)” (edited by The Mathematical Society of Japan, published by Iwanami Shoten) , A model formula of the Levy process may be created as appropriate.
The expected total asset extinction process is the total asset extinction amount L_tShows the expected value of.
[0049]
As described above, according to the method for calculating the market value of asset-backed securities according to the present invention, it is possible to provide a method and program for calculating the market value of asset-backed securities that is reasonably quick.
[0050]
BEST MODE FOR CARRYING OUT THE INVENTION
Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.
[Asset-backed securities]
The asset-backed securities in the present invention generally include asset-backed securities, asset-backed security (Asset \ Backed \ Security), and are referred to as ABS for short. As described above, in the case of MBS and its MBS, It also includes certain CMOs, CLOs, CBOs, and the like.
[0051]
Asset-backed securities are issued on the basis of assets held by the company, separate assets such as receivables and real estate held by the company from the company, and are issued as collateral (backing, funding) with cash flows generated from the assets It is a security.
[0052]
Generally, in order to issue asset-backed securities, a legal entity separate from the company that is the originator of the asset (SPC = special purpose company or special purpose company, SPV, SP (Also called). The company that is the originator of the asset transfers the asset to its special purpose company. The SPE issues securities backed by the transferred assets and sells them to investors.
[0053]
Since the assets are separated from the company, even if the original company goes into bankruptcy or the like, if the assets held by the special purpose company are healthy, investors can receive the payment of the securities with confidence. In other words, asset-backed securities can be said to be securities invested not in the creditworthiness of the original company but in the creditworthiness of the target asset. For example, in the case of asset-backed securities backed by loan receivables, even if the company holding the receivables goes bankrupt, if the loan receivables themselves are good receivables, investors can pay the securities through a special purpose company. Can be received.
[0054]
Assets separated from companies include accounts receivable, notes receivable, bonds (corporate bonds), loans (mortgages in real estate, mortgages, car loans, credit card loans), lease receivables, commercial paper (CP), etc. Yes, securitization by issuing asset-backed securities with a wide range of assets as collateral.
[0055]
It should be noted that the assets serving as collateral for the asset-backed securities according to the present invention are not limited to these examples, and include a wide range of tangible assets, intangible assets, and the like as long as the assets generate cash flows. Examples of assets backed by asset-backed securities include second-tier mortgages, home equity loans (revolving), mobile home loans, leisure mortgages, commercial real estate, lease bonds, car loans, credit card bonds, and medical fees. Claims, bonds for developing countries, junk bonds, LBO loans, etc.
[0056]
[General asset decrease model]
The market value of asset-backed securities is reduced by the fact that the collateral assets do not generate the expected cash flow. The expected cash flows are not generated, for example, in mortgage-backed asset-backed securities, where the individual mortgage repayer may default or repay before the due date. Prepaid payments may reduce interest rate cash flows in the future. As a result, the market value of asset-backed securities will fall.
[0057]
As described in the background section, it is difficult to calculate the market value of asset-backed securities in a conventional manner because it is necessary to model the price fluctuations of the individual assets constituting the underlying asset and analyze the entire asset. Met. On the other hand, in the present invention, the value reduction process of the entire underlying asset serving as collateral for the asset-backed securities is directly modeled as one stochastic process (Levy process). That is, asset-backed securities that securitize and liquidate assets are considered as derivatives (options) of the total asset balance of collateral assets.
[0058]
In order to easily and accurately calculate the market value, in the present embodiment, a general asset decrease model is considered in calculating the market value of the asset-backed securities. The general asset depletion model is one of the models in which the price movement of an asset changes unilaterally with time.
[0059]
Generally, as asset-backed securities, bonds are issued as collateral based on the contract stipulation of the total asset balance of future cash flows (hereinafter referred to as “tranche”). In order to explain the tranche, a description will be given using a type of mortgage-backed securities called CMO (Collateralized Mortgage Obligation) shown in FIG.
[0060]
The CMO issues cash flows from the repayment of mortgages in several different types of tranches. For example, as shown in FIG. 1, a tranche is a first tranche for repayment of a short-term mortgage loan, a second tranche for repayment of a medium-term mortgage loan, with the expected interest and the principal of each loan combined as underlying assets Like the third tranche of long-term mortgage repayment and the fourth tranche of ultra-long-term mortgage repayment, they are broken down by the mortgage repayment cash flow period. Needless to say, the manner of dividing the tranches has a high degree of freedom such as distribution of principal and interest to each tranche and is diverse, and is not limited to the example of FIG.
[0061]
Assuming that the mortgage is repaid with equal principal and interest repayment, the balance of the assets will gradually decrease every month, and this repayment, which is the generated cash flow, will be paid first to the first tranche, and once the first tranche is paid off, Sequentially allocating to two tranches, and so on, results in several tranches with different maturities and repayment schedules.
[0062]
As shown in Figure 2, the shortest first tranche is for institutional investors such as banks, and the second and third tranches for medium and long terms are medium- and long-term institutional investors such as banks that issue medium- and long-term financial bonds. In addition, the longest fourth tranche is intended for pension managers and other appropriate investors who operate for a period similar to the repayment period according to the characteristics of the repayment period of each tranche. Recruitment (placement) can be performed. However, depending on the characteristics of the repayment period, it is not necessary to create a tranche, and a tranche based on another priority of the repayment period may be used.
[0063]
In this way, it is possible to securitize on a more favorable condition for the issuer of the asset-backed securities if the underlying assets serving as the collateral of the asset-backed securities are divided into tranches, rather than being sold as a whole. Dividing the underlying assets that are the collateral for asset-backed securities into tranches based on certain characteristics (eg, repayment period) is called segmentation arbitrage.
[0064]
In the case of asset-backed securities issued for a tranche, if the total assets outstanding as collateral for the asset-backed securities are equal to or greater than the tranche's upper limit, 100% of the principal of the asset-backed securities is guaranteed at that time However, if the total asset balance falls within the tranche (below the upper limit of the tranche), the principal disappears according to the rate of the fall. If it falls below the tranche, the asset-backed securities will have no value.
[0065]
This will be described with reference to the tranche of FIG. Assume that an institutional investor has purchased asset-backed securities backed by assets in the range of the tranche in FIG. The total assets outstanding as collateral for asset-backed securities are calculated based on the total asset extinction process L_tIncreases as shown by the curve indicated by γ.
[0066]
The total asset balance is equal to or greater than the upper limit of the tranche (the total asset disappearance amount L in FIG. 3)._tWhen β is equal to or less than β), the principal of the asset-backed securities backed by assets in the range of the tranche in FIG. 3 is guaranteed 100%. However, for some reason, the total asset disappearance amount L_tIs greater than or equal to β, that is, the total asset balance is equal to the total asset disappearance amount (L_t−β), the principal of the asset-backed securities backed by the tranche in FIG._t−β) disappears. Further, the total asset disappearance amount L_tIs greater than or equal to α, that is, if the total asset balance falls below the lower limit of the tranche, the total asset balance no longer covers the tranche (ie, the asset-backed securities), and the asset-backed securities covered by the tranche range Is no longer worth it.
[0067]
In general, asset-backed securities are valued as a range-forward combination of total asset balances with tranches at different exercise times. This component unit range forward is also called asset-backed discount bond. A range forward is a combination of European call options (a type of call option that can only be exercised at maturity, as opposed to an American call option that can be exercised before maturity), with call options with different strike prices, This is a transaction in which the lower exercise price is long (buy forward transaction) and the higher exercise price is short (sell forward transaction). Range forward is also called bull spread. In ABS, a range corresponds to the width of a tranche, and the difference between two strike prices, a high price (corresponding to β in FIG. 3) and a low price (corresponding to α in FIG. 3), corresponds to the tranche width. The forward contract is a forward contract. In other words, at some point in the future, the range-forward transaction in the European option, which is an option transaction for selling or buying the underlying asset at a predetermined price, and the ABS transaction for a certain tranche have the same payoff shape. That is, the same handling is possible.
[0068]
As another method of understanding, the disappearance of assets that generate cash flows serving as collateral for asset-backed securities can be considered as shown in the schematic diagram of FIG. Total assets disappearance L_tIs caused by the natural depletion of assets due to cash flow generation, prepayment caused by the cash flow payer paying before the due date, and payer default. In the case of default, recovery (debt collection) shall be considered. FIG. 4 shows a process in which the total asset balance decreases. This total asset is composed of multiple assets. Assume that the plurality of assets do not have a risk of default, but rather include various assets ranging from those that are paid off and prepayment are made to those that have default risk. Here, the priority refers to the order in which a predetermined value is unlikely to disappear when the principal disappears due to the bankruptcy of a corporation or an individual who should pay cash flows secured by asset-backed securities. . In other words, if the principal is extinguished due to the bankruptcy of the debt repayer, the value will be lost from the lower priority. In the following, a default risk is expressed as "low priority", and a prepayment is expressed as "high priority".
[0069]
In FIG. 4, the sum of the natural increase due to the cash flow generation and the increase due to the prepayment is the total asset disappearance amount L._tRatio to φ_tAnd At this time, the total asset disappearance amount L_tFor example, other than the natural increase and the total of the prepayment with respect to are increments by default, and the ratio is (1-φ_t). Then, at the time t, when the prepayment occurs, the asset is φ_tLt disappears, and when a default occurs, assets with lower priority (1-φ_t) ・ L_tIt disappears by the minute. A-φ_t・ L_t− (1-φ_t) ・ L_t= AL_tIs the total asset balance at time t.
[0070]
Total assets disappearance L_tMay include a seasonal term. This is because the frequency of prepayment varies depending on the season. For example, in the case where the target ABS is MBS (CLO) and is for a mortgage, for example, during a transfer period when the work location changes such as March, the prepayment often increases. There is an empirical rule that prepayments often increase in June and December of bonus months.
[0071]
Next, in order to appropriately represent the market value of the asset-backed securities, [Equation 1] (without prepayment), [Equation 2] (with recovery, without prepaidment) And three types of calculation models represented by [Equation 3] (with prepayment) are proposed. In addition, in [Equation 1] to [Equation 3], it is assumed that the recovery is considered and incorporated in the prepayment. Note that E₀Means the average after performing the probability integration on the term expressed as the random variable at the time of interest (t = 0). Also, c (dt) is a coupon paid at the next moment, and r_tIs the discount rate and is a function of time.
[0072]
Here, recovery (recovery of debt) means that when there is a total assets balance in a tranche, the total assets balance decreases for some reason and a part of the decreased assets is collected and incorporated into the total assets balance. Say. An example of an actual recovery is when the debtor of the loan backed by the asset-backed securities defaults due to bankruptcy, etc. The total value of the asset = the decrease in the total asset balance), and a part of it is returned to be incorporated into the total assets that are backed by the asset-backed securities.
[0073]
[Market value of asset-backed securities without prepayment]
As shown in FIG. 3, the total asset disappearance amount L of the asset-backed securities_tIs assumed to follow a monotonically increasing stochastic process, the present value of the cash flow at time t when the tranche section is [β, α] is represented by [Equation 1]. Here, β representing the upper limit of the tranche by the difference from the underlying asset and α similarly representing the lower limit are the total asset disappearance L which is a monotonically increasing stochastic process._tIs positive with respect to the increase direction (the decrease direction of assets), and has a magnitude relation of α> β. That is, L_tIs L_tWhen <β, 100% of the principal of the asset-backed securities backed by the cash flow in the tranche [β, α] is guaranteed. L_tIs β <L_t<Α, the principal of the asset-backed securities secured by the cash flow in the tranche [β, α] is (L_t−β) and (α−L_tOnly) is guaranteed. L_tIs α <L_tIn this case, the principal of the asset-backed securities collateralized by the cash flow in the tranche [β, α] will be extinguished.
(Equation 1)

[0074]
The present value represented by [Equation 1] is the present value (asset-back discount bond price) of the asset-back discount bond with a maturity of t at time t = 0, and the present value of the asset-back securities market value calculation method according to the present invention. It is a basic component of evaluation. In any formula in this specification, the currency unit (dimension of currency) is not specified according to the usual description in the field of finance. In addition, the notation of integral uses the notation in measure theory (integral theory).
[Equation 1] is the total asset extinction amount L which is the accumulation of the extinction amount of the underlying asset._tIt is a put-type range forward that is a range forward for (= initial total asset amount A-total asset balance). [Equation 1] is the disappearance amount L of the total assets which disappears._tIt is expressed from. Here, α and β that determine the width of the tranche are the total assets disappearance L which is a monotonically increasing stochastic process._tIs positive with respect to the increasing direction, and has a magnitude relationship of α> β.
[0075]
[Market value of asset-backed securities with recovery and no prepayment]
If recovery is permitted, the recovery rate, which is the ratio of the recovery amount to the current price of the asset-backed securities, is_tThen, the asset-backed bond price can be expressed as [Equation 2]. Recovery is the return of a portion of a loan without losing the full amount of the loan in case of bankruptcy. The method of calculating the market value of asset-backed securities according to the present invention assumes that part of the total assets that disappear while the total assets are in the tranche will recover and return to the owner of the asset-backed securities. You have stated that. The price of an asset-backed bond refers to the value of the interest-bearing bond with the condition of coupon payment during the holding period added to the price of the asset-backed discounted bond. Here, in the case where there is no recovery, the interest-bearing bond price is expressed as the sum of the discount bond prices weighted by the interest cash flow c (dt) which is more generally described. Therefore, if the price of the discount bond is obtained, the price of the asset-backed securities (interest-bearing bond) with the coupon can be easily obtained. On the other hand, in the case where recovery is permitted, the reason for expressing it as asset-backed securities with coupons (interest-bearing bonds) is that the cash flow due to recovery that occurs at a certain time (t) before the bond expiration and the cash flow c as interest This is because both dt) and dt) are considered to be discounted simultaneously as cash flows.
(Equation 2)

[0076]
In the event of recovery, if the jump dL of the total asset disappearance (the loss of principal) spans the tranche, only the disappearance in that tranche is subject to the right to receive recovery, and the actual recovery amount is the recovery amount. Rate φ_tMultiplied by. Here, recovery rate φ_tMeans the ratio of the recovery amount to the total asset disappearance amount dL. Although the jump dL itself does not explicitly include the time t, the probability variable L_tIs a function of time t. Therefore, although dt does not appear in the equation, the procedure for calculating the expected value by E includes integration for time t (for example, integration by τ in [Equation 16] described later).
[0077]
[Market value of asset-backed securities with prepayment]
More commonly, there is a prepayment in the asset-backed securities where the responsible person pays the cash flow, which is the collateral of the asset-backed securities, earlier than the expected payment date. In general, the loss of principal due to bankruptcy occurs from the lower priority, and prepayments that occur before the due date and redemptions due to short maturity of assets in asset-backed securities occur from higher priority tranches.
[0078]
The ratio of prepayment (prepayment) to the total decrease in total assets is φ_tThen, the interest-bearing bond price of the asset-backed securities can be expressed as [Equation 3]. The recovery may be incorporated in the prepayment. The asset-backed bond price is the price of the interest-bearing bond with the asset-backed discount bond price and the condition of coupon payment during the holding period.
(Equation 3)

[0079]
Here, A in [Equation 3] represents the initial total asset amount (underlying asset amount) when the asset-backed securities are issued. In [Equation 3], the difference from [Equation 2] is that the cash flow decreases due to the bankruptcy of a corporation or an individual who should pay cash flow to secure the asset-backed securities, and the upper limit of the tranche represented by β also erodes. This is a point that is taken into consideration. Although an example of modeling the recovery rate is omitted in the present invention, the recovery rate may be modeled as a stochastic process.
[0080]
According to [Equation 1] to [Equation 3], in order to appropriately perform the valuation of the market value of the asset-backed securities, the total asset depletion amount_tIt is necessary to perform the modeling of rationally and calculate appropriately. Furthermore, when considering even recovery, the stochastic derivative dL of the total asset (original capital) disappearance must also be considered.
[0081]
Next, an example will be described in which the total amount of disappearance of assets serving as collateral for asset-backed securities is modeled by a Poisson process as described below and a monotonically increasing Levy process, and is calculated by arithmetic means. explain.
[0082]
[Total loss of assets in Poisson process]
First, the total asset extinguished amount L_tAs an example of modeling that is easy to perform analytical calculation in modeling, a method based on diffusion approximation (Brownian motion approximation) using a central limit theorem of a Poisson process will be described.
[0083]
The Poisson process is a process in which an event occurs by chance at a constant rate over time. A Poisson process is a process that starts at 0, stays at 0 for a while, jumps suddenly by a magnitude 1 at some point, then stays at 1 for a while, and suddenly jumps by a magnitude 1 at a time. And for a while ... a stochastic process in which the sample path draws a shape like a staircase rising to the right. The mathematical definition is obtained by formulating the ambiguous expression above, "Leave it 0 for a while" and "Leave it a 1 for a while". Specifically, it is assumed that the time interval from the occurrence of one jump to the occurrence of the next jump follows an exponential distribution.
[0084]
In one embodiment of the present invention, it is assumed that the disappearance of the total assets serving as collateral for the asset-backed securities occurs by the Poisson process. For example, when the debtor of the cash flow, which is the collateral for asset-backed securities, goes bankrupt and defaults, it considers this bankruptcy to be a coincidence and a Poisson process. In the present invention, when a bankruptcy phenomenon of a large number of companies is handled, even if the Poisson process is used, the calculation becomes more complicated by using the diffusion approximation because the calculation becomes complicated.
[0085]
The central limit theorem is a theorem that the sample average distribution of a sample extracted from a large population approaches a normal distribution when the sample size increases. In one embodiment of the present invention, a large amount of data relating to the past value transition of the asset backed by the asset-backed securities is extracted and used as a sample, and the sample is modeled as having a normal distribution. .
[0086]
Diffusion approximation (Brownian motion approximation) is applied to stochastic variation using the central limit theorem that the process of averaging a variable sample of a variable sample extracted from a large population approaches Brownian motion when the size of the variable sample increases. Refers to an approximation method. In one embodiment of the present invention, in the diffusion approximation formula of the Poisson process such as [Equation 5], the model parameters are determined as follows as an example. If the past data of the total asset disappearance process is obtained only with minimal data, for example, if there are only past data such as the purchase price, maturity, interest rate (coupon), and rating of asset-backed securities, loan If the pool is used as the underlying asset and a rudimentary model is used, for example, in the diffusion approximation formula of the Poisson process of [Equation 5], the model parameters are provisionally determined as follows as an example. Time t is the elapsed time from the time of issuance, η is the average credit spread of the bond determined by the rating, A is set to a suitably large value, and the volatility of the bond spread is about the level observed in the bond market. Shall be suppressed. Finally, the tranche width of the asset-backed securities is determined to realize the purchase price. At this time, using a method that uses a consultant who is familiar with this field, and studying the characteristics of the underlying asset and using a model that assumes a Levy process with a complicated attenuation path, It is also possible to adjust the model parameters to match.
[0087]
Here, if defined as in [Equation 4], the diffusion approximation by the central limit theorem of the Poisson process is expressed as [Equation 5]. L in [Equation 4]_tIs the total assets disappearance, A is the initial total assets, h_tIs the total asset disappearance amount L_tIs divided by the initial total asset value A. One of the features of the model of [Equation 5] is that the total asset disappearance amount L of asset-backed securities_tIs represented by a normal distribution, so it is easy to analyze the market value of asset-backed securities, and the solution is obtained by the analytical solution. Here, η in [Equation 5] is an annihilation (bankruptcy) risk rate, dw represents Brownian motion, and t is a time elapsed from the time of issuance of the asset-backed securities.
(Equation 4)

(Equation 5)

[0088]
However, the total asset disappearance amount L of asset-backed securities_tIn modeling the trend of the above, if a diffusion approximation model based on the central limit theorem of the Poisson process as shown in [Equation 5] is actually used, the instantaneous asset disappearance dL of the asset-backed securities does not increase monotonically. The probability of a negative value is often non-zero. Therefore, the modeling shown in [Equation 5] can be used for calculating the market value of asset-backed securities because the instantaneous asset extinguishing amount dL does not increase monotonically, and the value obtained by modeling [Equation 5] becomes negative. Only when the probability of becoming
[0089]
Conversely, when the instantaneous asset disappearance dL increases monotonically, or when it is necessary to consider that the value of the instantaneous asset disappearance dL becomes non-negative (for example, the actual asset-backed securities In the case where the asset disappearance amount is handled and a certain amount may disappear at one time), it cannot be modeled well by the continuous path of the Brownian motion in [Equation 5]. Therefore, in such a case, a model using a monotonically increasing Levy process described later is more preferable than a diffusion approximation based on the central limit theorem of the Poisson process.
[0090]
[Monotonically increasing Levy process]
In another embodiment of the present invention, modeling of the market value calculation of asset-backed securities is attempted using a monotonically increasing Levy process without using a diffusion approximation model based on a central limit theorem of a Poisson process. The Levy process is a general stochastic process including both Poisson process and Brownian motion (Wiener process). The Levy process is generally an additive process that is stochastically continuous and whose sample function has a probability of 1 and is at most discontinuous of the first kind and right-continuous. In addition, the Levy process includes a composite Poisson process in which the sample function changes only by a leap in a temporally uniform Levy process and a Levy measure of which is finite, and a Cauchy process.
[0091]
In the present invention, the total asset extinguishing amount L of the asset that is the collateral of the asset-backed securities_tIs treated as a monotonically increasing Levy process, that is, a monotonically increasing stochastic process that changes only unilaterally with time.
[0092]
In the monotonically increasing Levy process of the present embodiment, a type in which a solution is given by an analytical solution is used. The type used in the present embodiment is the simplest type following the Brownian motion in the Levy process. While the Brownian motion is a continuous path, the type used in the present embodiment is based on a complex Poisson process in which the numerical value only includes a jump.
Generally, in financial engineering, stock prices and bond prices are frequently modeled as following the Brownian motion. Brownian motion is a stochastic process in which instantaneous fluctuations follow a normal distribution. Therefore, when stock prices and bond prices are modeled by the Brownian motion, the Brownian motion results in a continuous path in which the stock prices and the bond prices do not instantaneously change significantly.
[0093]
In general, the cumulative Brownian motion results in a normal distribution. On the other hand, the type in which the solution of the present embodiment is given by the analytical solution becomes an extended Γ distribution when accumulated. Note that there are many Levy processes of the type in which the solution of the present embodiment is given as an analytical solution, and the extended Γ distribution is only one of the simplest examples, and the present invention is not limited to this. . By appropriately selecting parameters used in mathematical expressions such as [Equation 6] and [Equation 11] used in the market value calculation of the asset-backed securities according to one embodiment of the present invention (all of which will be described later), The market value can be calculated more easily and more precisely by a computer than by using the normal distribution of the Brownian motion in the Poisson process.
[0094]
The type in which the solution of the present embodiment is given by the analytical solution is that the instantaneous asset disappearance dL of the asset serving as the collateral of the asset-backed securities follows a noncentral χ square distribution, and the total asset disappearance L_tIs a stochastic process that follows a moving exponential distribution cumulative 累積 distribution. In particular, the total asset disappearance amount L from the initial time_tHas been found to be a Γ distribution.
[0095]
The non-central χ-square process, which represents the instantaneous asset extinguishing amount dL of the asset serving as the security of the asset-backed securities, is a stochastic process having a non-negative base. In addition, as an accurate model for calculating the market value of asset-backed securities, the total asset extinguishing amount L_tIs necessary to provide an absorption wall where the maximum value of A becomes the initial total assets A of the assets at the time of issuance of the asset-backed securities. The absorption wall is the upper limit in the probability theory. For example, the decrease in the underlying asset as collateral for asset-backed securities does not exceed the original underlying asset (the initial total asset value). There can be an upper limit. Usually, this natural upper limit is used as the absorption wall.
[0096]
However, since the monotonically increasing Levy process is monotonically increasing, the probability distribution until it reaches the absorbing wall is not affected by the absorbing wall, so the analysis can be performed with the absorbing wall at infinity. . In particular, in the method for calculating the market value of asset-backed securities according to the present invention, the total asset asset disappearance amount L_tIs limited within the tranche, so the absorbing wall may be ignored.
[0097]
In general, the Levy process does not have a simple notation like the Ito derivative, and is defined by a moment generating function such as [Equation 6]. Total asset disappearance amount L by [Equation 6]_tIs merely an embodiment, and it can be defined by a mathematical expression indicating another Levy process as described above.
[0098]
A moment generating function is a type of conversion of a probability distribution. In the present embodiment, a moment generating function such as [Equation 6] is used as a kind of calculation technique when handling a probability distribution that is difficult to handle directly.
(Equation 6)

[0099]
Here, E in [Equation 6]_t[Eθ^dL] Is the random variable eθ^dLRepresents the expected value of. ν, λ_tAre parameters describing this Levy process. In the formula of the Γ distribution obtained by probability integration from the initial time point, ν is an argument of the Γ function, and represents the real number overlap of the exponential distribution constituting the Γ distribution. λ_tIs the hazard rate of the Poisson process with the exponential distribution as the jump interval using the exponential distribution parameter of the constituent factor. However, the instantaneously disappearing asset amount dL is L_tIs monotonically increasing, so that ν, λ_t'> 0. Also, λ_t’= (Dλ_t) / (Dt). According to Levy Ito's theorem, the Kumulant generating function of the infinitely divisible distribution (noncentral χ square distribution) is an infinite number of jumps (also referred to as a jump or a discontinuous path) as shown in [Equation 7]. Jumps). The queue-mulant generating function is the logarithm of the moment generating function.
(Equation 7)

[0100]
[Equation 7] indicates that the hazard rate of the jump width dL = h of the total asset disappearance amount Lt can be expressed as [Equation 8]. The hazard rate h indicates a probability of causing a discontinuous value jump called a jump at the next moment. Typically, hazard rates are expressed as a function of time.
(Equation 8)

[0101]
Integrating the differential definition of the moment-of-moment generating function of the stochastic process represented by [Equation 6] yields [Equation 9]. Here, the moment generating function differential definition expression refers to a differential equation satisfied by the moment generating function.
(Equation 9)

[0102]
[Equation 9] is based on (λ_t/ Λ_T), The distribution obtained by convolving the exponential distribution with probability concentration by ν times is shifted to the right by ζ. Here, the distribution convolved ν times means a probability distribution followed by the sum of independent ν random variables following the same exponential distribution. However, in this case, ν does not need to be an integer, and is a continuation of this concept.
[0103]
In addition, by inverse Laplace transform of [Equation 9], the total asset disappearance amount L represented by [Equation 10] is obtained._tOf the probability Pr_tIs obtained. Total assets disappearance L_tOf the probability Pr_tIs a cumulative probability density function in the form of a weighted average of a gamma distribution with a binomial distribution, as shown on the right side of [Equation 10].
(Equation 10)

The important thing is that if v is a positive integer, the sum is finite.
[0104]
Where λ_tIs represented by [Equation 11]. [Equation 11] is derived from [Equation 9] based on the property of the moment generating function shown by [Equation 12]. [Equation 11] is derived by first defining a probability distribution that is difficult to directly handle, such as the instantaneous asset disappearance amount dL, by the moment generating function shown in [Equation 6], and by using [Equation 12] and [Equation 9], [Equation 9] The limit value of θ → 0 is calculated by differentiating the right side with respect to θ, and Equation 11 is derived.
[Equation 11]

(Equation 12)

[0105]
Total assets disappearance L_tOf expected total assets E, which is the expected value of_t[L_t｜ L_t= Ζ] can be modeled by, for example, a growth curve such as a logistic curve. A logistic curve is a type of growth curve. Generally, the size y of the population is a function of time t, y = k / (1 + e).^-Kbt). Here, k and b are positive constants. More generally, y = k / (1 + e^{cf (t)}) Means a curve represented by a function. Here, c is a constant, and f (t) is a function of time t. However, the growth curve in the present invention is represented by the general formula y = k / (1 + e).^{cf (t)}) Does not have to be the same as the general formula, and may be a modification of this general formula or an exponential curve. Furthermore, a functional expression such as tanh (at) may be used.
[0106]
In one embodiment of the present invention, y is the total asset disappearance L_tThe computer performs nonlinear regression estimation of parameters necessary for specifically defining the function of cf (t) based on data on the disappearance of assets serving as collateral for asset-backed securities obtained from the real economy. It is preferable to calculate the following.
[0107]
As one embodiment of the present invention, assuming that the interest rate and the total amount of extinction of the total assets serving as collateral for the asset-backed securities are independent, the normal discount bond price is calculated as Dt ( T), the asset-backed discount bond price is expressed as [Equation 13] based on [Equation 1].
(Equation 13)

[0108]
Since the minimum unit at the time of valuation of a financial instrument called an asset-backed securities is a discount bond, the price of the asset-backed discount bond is calculated in order to calculate the market value of the asset-backed securities. However, in general, the redemption of the principal (total assets) of the asset-backed discount bonds is subject to the contract terms of the asset-backed securities. In other words, the total balance of total assets serving as collateral for asset-backed securities changes depending on whether it is inside or outside the tranche.
[0109]
When the expected value of the asset-backed discount bond of [Expression 13] is calculated based on [Expression 13], max (,) can be removed, and Expression [14] is obtained. In general, when ν is a positive integer, the upper limit of the sum is ν, so that the expected value of the asset-backed discount bond can be calculated and evaluated by a simple computer program. [Equation 14] corresponds to [Equation 1] which is the price of the asset-backed discount bond without the recovery condition and the prepayment condition.
[Equation 14]

[0110]
Similarly, the price of the asset-backed interest-bearing bond with prepayment condition may be calculated by integrating [Equation 15] and [Equation 16]. It is also possible to obtain an analytical solution as in [Equation 14]. [Equation 15] corresponds to [Equation 1] which is the price of the asset-backed discount bond with the recovery condition and the prepayment condition.
(Equation 15)

(Equation 16)

[0111]
[Market value calculation of asset-backed securities]
An embodiment of a method for calculating the market value of asset-backed securities by a computer according to the present invention will be described below with reference to the drawings.
As shown in FIG. 5, in the method for calculating the market value of asset-backed securities according to the present invention, the total asset amount, β that determines the upper limit of the tranche, and α that determines the lower limit of the tranche are received by the input means of the computer (step). 10). A tranche is the contractual scope of the total asset balance of future cash flows that will be collateral for asset-backed securities. The tranche indicates the range in the total assets. Using α and β, which have the relationship α> β, used to define the range of the total asset disappearance amount that is opposite to the tranche, the upper and lower limits of the tranche are (upper limit of the tranche) = (initial (Total asset balance) −β, (the upper limit of the tranche) = (initial total asset balance) −α (see FIG. 3).
[0112]
Based on the logistic curve, total assets disappearance L_tAverage path νλ_tIs calculated by the calculating means (step 20).
An example of the calculation in step 20 includes a flowchart as shown in FIG. First, the input means of the computer receives the selection of the logistic curve model (step 21). As the logistic curve model, an appropriate model can be appropriately selected from the past total asset extinguishment amount data obtained from the collateral targeted for the asset-backed securities. As the logistic curve model, any growth curve suitable for representing the actual total asset disappearance may be used, such as tanh (at) or y = k / (1 + e).^{cf (t)}) And the size y of the population is a function y = k / (1 + e) at time t.^-Kbt), An exponential function.
[0113]
For example, when tanh (at) is selected as the logistic curve, the total asset disappearance amount L_tThe calculation means calculates the difference between the past data and the numerical value of the logistic curve tanh (at) (step 22).
Next, the total asset disappearance amount L_tBased on the difference between the past data and the numerical value of the logistic curve tanh (at), the arithmetic means calculates the parameter a for minimizing the difference by nonlinear regression estimation by the least squares method (step 23). Then, the calculation means calculates the logistic curve tanh (at) based on the parameter a (step 24). Total asset disappearance amount L thus obtained_tThe logistic curve tanh (at) that fits the past data of_t, And the increased path is the average path νλ at each time t._tAnd That is, the logistic curve calculated by the calculating means is set as the average path in the total asset disappearance process by the calculating means (step 25).
[0114]
Next, the average path νλ_tBased on the total asset extinguished amount L_tCan be modeled as a Levy process, and as shown in [Equation 11], the expected total asset disappearance process E [L_t] Is calculated by the calculating means (step 30). An example of step 30 is a flowchart shown in FIG. Expected Elimination Process E_t[L_t] Is selected by the computer (step 31). Expected Elimination Process E_t[L_t] Is given by [Equation 11]. However, when the definition of [Equation 6] is different, [Equation 11] is also different.
In addition, the selected expected total assets disappearance process E_t[L_t], The average path νλ_t, The expected total asset disappearance process E_t[L_t] Is calculated by the calculating means (step 32).
[0115]
Β that determines the upper limit of the tranche, α that determines the lower limit of the tranche, and the expected total asset disappearance process E [L_tBased on [Equation 14], the calculating means calculates the market value B = D (t) of the asset-backed securities at an arbitrary time (step 40).
[0116]
Next, a flowchart of FIG. 6 will be described as another embodiment of the method for calculating the market value of asset-backed securities according to the present invention. The difference between the flowchart of FIG. 6 and the flowchart of FIG. 5 is that the prepayment rate is considered.
First, β that determines the upper limit of the tranche, α that determines the lower limit of the tranche, and the prepayment rate φ_tIs received by the computer (step 110). Then, based on a given logistic curve stored in the storage means, the total asset disappearance amount L_tAverage path νλ_tIs calculated by the calculating means (step 120). Total assets disappearance L_tAverage path νλ_tBased on the total asset extinguished amount L_tIs a Levy process, the expected total assets disappearance process E [L_t] Is calculated by the calculating means (step 130). Then, β that determines the upper limit of the tranche, α that determines the lower limit of the tranche, and the expected total asset disappearance process E [L_t] And the prepayment rate φ_tBased on the above, the calculating means calculates the market value B = D (t) of the asset-backed securities at an arbitrary time point using [Equation 15] (step 140).
[0117]
【Example】
An embodiment of the method for calculating the market value of asset-backed securities according to the present invention will be described below. The method of estimating the parameters appearing in the formula in the method of calculating the market value of asset-backed securities can be devised according to each case based on available information.
The following is an example of a method for calculating the market value of asset-backed securities when it is possible to assume that the performance of loan deterioration can be estimated from similar past data. Average amount of asset extinguishment L of all underlying assets used as collateral for asset-backed securities_tAssume approximately a logistic curve y = tanh (at).
[0118]
Asset depletion L obtained from the real economy_tNon-linear regression estimation is performed by arithmetic means based on past data regarding The parameter ν can be calculated by the calculating means from the error analysis performed simultaneously with the nonlinear regression estimation. Here, as the past data, data of a loan bad loan scenario by a major financial institution from April 1997 to October 2001 was used.
When there is little information such as past data on the amount of asset disappearance, the estimation calculation can be performed by making the time-dependent parameter a constant in the model represented by [Equation 14] or [Equation 15].
[0119]
FIG. 9 is a diagram showing a change in the yield of asset-backed securities at the time of revaluation. The vertical axis represents the yield of the asset-backed securities at the time of revaluation, and the horizontal axis represents the time (year) from the time of issuance until revaluation. In this case, the asset-backed securities are collateralized by a loan, and the tranche, which is the contract stipulated range, has an upper limit specified by β of 1,260 million yen and a lower limit specified by α of 800 million yen. Asset-backed bonds (securities). The asset backed securities had a maturity of five years and the coupon was 4.35%. Here, the method of calculating the market value of asset-backed securities according to the present invention was compared with a method based on a conventional Monte Carlo simulation and a method based on a vendor evaluation model which is a conventional major information company. In the method for calculating the market value of asset-backed securities according to the present invention, various parameters were calculated from average attenuation data of assets serving as collateral for asset-backed securities. Monte Carlo simulations show an average of 22,000 revaluations at a future time based on the bankruptcy scenario of each component of the asset backed by the asset-backed securities.
[0120]
Here, the conventional valuation model of a major information vendor using market data instead of random numbers is based on a bottom-up approach, assuming that the individual underlying assets in the process of disappearing total assets, which are collateral for asset-backed securities, are uniform. It is a simplification. In other words, the model is such that the process of decreasing each uniform asset independently follows the Poisson process, and the basic parameter (hazard rate) that largely determines the characteristics of this model is calculated from the yield. It is limited that analytical modeling can be performed by the conventional bottom-up method, and the above-described conventional example is a rare example. However, in such a conventional example, since an analytical solution is obtained by a bottom-up method, the number of model parameters having a degree of freedom must be reduced. This is because if the number of assets to be collateralized by the asset-backed securities increases, the computation time of the computer becomes too long to be realistic. For this reason, conventional model parameters are limited to parameters such as, for example, contract period, individual bankruptcy risk, and number of assets. As described above, since the conventional example is simply made, the average path of the total asset disappearance process becomes a simple exponential function, which deviates from a real economic phenomenon.
[0121]
On the other hand, the method for calculating the market value of asset-backed securities of the present invention models the disappearance process of total assets by a top-down method of directly modeling the total assets serving as collateral for asset-backed securities, and has a high degree of freedom. Unlike the conventional example described above, even if the number of assets serving as collateral for securities increases, an analytical answer can be obtained, and the computation time of the computer is less likely to be required than in the conventional case, and the efficiency is increased. .
[0122]
As shown in FIG. 9, the change in the yield of the asset-backed securities at the time of revaluation according to the market value calculation method of the asset-backed securities according to the present invention is almost the same as the change in the yield of the asset-backed securities using the conventional Monte Carlo simulation. It turns out that it is. In this regard, it can be seen that a prediction accuracy equivalent to that of the Monte Carlo simulation can be obtained. Compared with the Monte Carlo simulation which is a computer experiment, the method of calculating the market value of asset-backed securities according to the present embodiment is performed in an analytical manner, so that the calculation speed is fast, efficient, and an elaborate method. In addition, although the calculation speed is fast with the conventional analytical method of the vendor re-evaluation model, the parameters that determine the result are determined only based on the yield at the time of issuance. As shown in FIG. This indicates a problem with a bottom-up vendor revaluation model in which the final result cannot be obtained properly unless individual parameters are set properly. In the evaluation method of the present invention, by using a new method using a Levy process without taking such a bottom-up method, prediction accuracy comparable to Monte Carlo simulation and calculation speed by an analytical method are obtained. It is now possible to achieve both.
[0123]
Further, in the revaluation time (year) of 1.8 to 2.0 years, the curve of the yield change of the asset-backed securities by the vendor model and the market value calculation of the asset-backed securities of the present invention Comparing the curves of the change in the yield of the asset-backed securities by the methods, as is apparent from FIG. 9, the yield of the asset-backed securities sharply increases in the present invention. This is because, in the method for calculating the market value of asset-backed securities according to the present invention, the total asset disappearance process is a Levy process. In the Levy process, the frequency of jumps can be adjusted freely, so in the process of extinction of total assets, the amount of extinction jumps at a time, reflecting changes in the real economy around 1998 based on past data, and The yield is changing rapidly. However, in the conventional vendor model based on the standard Poisson process, the yield does not fluctuate sharply because the total asset disappearance process does not consider the jump frequency.
[0124]
Note that, even in cases other than the example described in the description of the present embodiment, it is known that an arbitrary total asset disappearance process can be approximated with high accuracy theoretically using the Levy process.
[0125]
【The invention's effect】
As is apparent from the above description, the method for calculating the market value of asset-backed securities according to the present invention allows the market value of asset-backed securities to be calculated reasonably and quickly. In addition, the calculation based on the analysis solution of the market value of the asset-backed securities becomes more rational, and high-speed numerical calculation based on the analysis solution becomes possible.
[Brief description of the drawings]
FIG. 1 is a schematic diagram showing the structure of a tranche of general asset-backed securities.
FIG. 2 is a schematic diagram showing a relationship between a tranche structure of general asset-backed securities and a capital market.
FIG. 3 is a schematic diagram showing a change in the total asset balance of a general asset-backed securities.
FIG. 4 is a schematic diagram showing a change in the total asset balance of general asset-backed securities.
FIG. 5 is a flowchart showing one embodiment of a method for calculating the market value of asset-backed securities according to the present invention.
FIG. 6 is a flowchart showing another embodiment of the method for calculating the market value of asset-backed securities according to the present invention.
FIG. 7 is a flowchart showing one embodiment of calculation of a logistic curve in the method for calculating the market value of asset-backed securities according to the present invention.
FIG. 8 is a flowchart showing one embodiment of a process of extinguishing expected total assets in the method for calculating the market value of asset-backed securities according to the present invention.
FIG. 9 shows a method of calculating the market value of asset-backed securities according to the present invention, a method of calculating the market value of asset-backed securities using a conventional Monte Carlo simulation, and a revaluation of the market value of asset-backed securities using a conventional vendor revaluation model. FIG. 9 is a comparison diagram comparing changes in yield of asset-backed securities at a time point.
[Explanation of symbols]
D ~ (t) Current value of cash flow generated at time t
E₀[・] E₀[X] is the expected value (average value) of the random variable X
max maxX is the largest element of the set X
min minX is the minimum element of the set X
r_tDiscount rate (discount rate)
α The lower exercise price of the combination of European call options and the value that sets the lower limit of the tranche in the total asset balance
β The higher exercise price in the combination of European call options and the value that sets the upper limit of the tranche in the total asset balance
B Asset-backed interest-bearing bond price
t @ Time since issuance of asset-backed securities
T maturity
L_t元 Principal amount at time t
Lt Principal amount at time T
c (dt) Coupon paid at the next moment
ψ_tRecovery rate
dL Principal amount L in consideration of recovery_tStochastic derivative of
A Initial total assets
φ Asset extinguishing ratio
η hazard rate, risk of disappearance or bankruptcy
dw @ Brownian motion, Wiener process
E_t[・] Moment generating function
θ product moment generator
Argument of ν Γ function, convolution multiplicity
λ_t引 数 Argument of exponential distribution, Poisson parameter
h bar jump width
資産 Current amount of asset disappearance
Pr_tCumulative probability density function
Argument of z probability density function (given the upper limit of random variable)
λ_TExponential distribution argument, Poisson parameter
λ_Tνλ_TTo the power of ν
λ_tν λ_tTo the power of ν
Subscript of κ sum
Subscript of z integral, argument of probability density function (given the upper limit of stochastic variable and principal disappearance)
Subscript of ι 和
D_t(T) Normal discount bond price
D to t, @ Asset back discount bond price
Subscript of τ integral, timing of cash flow
Coupon paid at cτ τ
ν_t,ζ (τ) Instant principal redemption rate
10 Reception step
20 Logistic curve calculation step
30 Expected total assets disappearance process calculation step
40 Market value calculation step for asset-backed securities
120 ° logistic curve calculation step
130 Step of calculating expected total assets disappearance process
140 Market value calculation step for asset-backed securities

Claims (13)

A method for calculating the market value of asset-backed securities issued with the total assets of cash flows based on assets as collateral,
A step in which the computer receives the total asset amount, the upper limit value of the tranche indicating the range of the asset value serving as the collateral of the asset-backed securities, and the lower limit value of the tranche, and saves the lower limit value in the storage means;
Based on the past data of the total asset extinction process and a model obtained based on the past data, the calculation means calculates the expected total asset extinction process indicating the expected value of the total asset extinction process with the total asset extinction process as the Levy process. Steps and
Calculating the market value of the asset-backed security by the calculating means based on the upper and lower limits of the tranche stored in the storage means and the calculated expected total asset extinguishing process; Method of calculation. A method for calculating the market value of asset-backed securities issued with the total assets of cash flows based on assets as collateral,
Total assets, tranche upper limit indicating the range of assets to be secured by the asset-backed securities, lower limit of the tranche, and total amount of assets extinguished due to payment before the scheduled payment date A step in which the computer receives a prepayment rate indicating a ratio to the amount of disappearance and stores the prepayment rate in a storage unit;
Based on the past data of the total asset extinction process and a model obtained based on the past data, the calculation means calculates the expected total asset extinction process indicating the expected value of the total asset extinction process with the total asset extinction process as the Levy process. Steps and
Calculate the market value of the asset-backed securities based on the upper and lower limits of the tranche stored in the storage means, the calculated expected total asset disappearance process, and the prepayment rate stored in the storage means. Means for calculating the market value of asset-backed securities. A method for calculating the market value of asset-backed securities issued with the total assets of cash flows based on assets as collateral,
Total asset amount, tranche upper limit value indicating the range of asset value to be collateral for asset-backed securities, lower limit value of the tranche, total asset extinguishment amount of assets extinguished due to payment before the scheduled payment date A computer receiving a prepayment rate indicating a percentage of the amount, and a recovery rate indicating a percentage recovered in the amount of disappearance of the asset when the cash flow payer defaults, and storing the same in a storage means;
Based on the past data of the total asset extinction process and a model obtained based on the past data, the calculation means calculates the expected total asset extinction process indicating the expected value of the total asset extinction process with the total asset extinction process as the Levy process. Steps and
The asset-backed securities based on the upper and lower limits of the tranche stored in the storage means, the calculated expected total asset disappearance process, the prepayment rate stored in the storage means, and the recovery rate. Calculating the market value of the asset-backed securities. Based on the past data of the total asset extinction process and a model obtained based on the past data, the calculation means calculates the expected total asset extinction process indicating the expected value of the total asset extinction process with the total asset extinction process as the Levy process. The steps are
Accepting a selection of a given logistic curve model stored in storage means by a computer;
Based on the difference between the past data of the total asset disappearance process and the numerical value of the logistic curve model, the calculating means calculates a parameter in the logistic curve model that minimizes the difference by nonlinear regression estimation.
A calculating means calculating a logistic curve based on the parameter;
Calculating the logistic curve by an average path of the total asset disappearance process and calculating means;
Receiving a selection of a Levy process model formula representing the expected total asset disappearance process;
Calculating the expected total asset disappearance process by a calculation means by substituting the average path into a Levy process model formula representing the selected expected total asset disappearance process. Method of calculating the market value of asset-backed securities described in. The asset according to any one of claims 1 to 4, wherein the Levy process is a process selected from a group consisting of a Poisson process, a Wiener process, and a process combining the Poisson process and the Wiener process. Market value calculation method for back securities. The method of claim 1, wherein the Levy process is monotonically increasing. A program for calculating the market value of asset-backed securities issued with collateral based on the total assets of cash flows based on assets,
A step in which the computer receives the total asset value, the upper limit value of the tranche indicating the range of the asset value serving as the collateral of the asset-backed securities, and the lower limit value of the tranche, and saves the lower value in the storage means;
Based on the past data of the total asset extinction process and a model obtained based on the past data, the calculation means calculates the expected total asset extinction process indicating the expected value of the total asset extinction process with the total asset extinction process as the Levy process. Steps and
Calculating the market value of the asset-backed securities by the calculating means based on the upper and lower limits of the tranche stored in the storage means and the calculated expected total asset disappearance process. A program for calculating the market value of asset-backed securities. A program for calculating the market value of asset-backed securities issued with collateral based on the total assets of cash flows based on assets,
The total amount of assets, the upper limit of the tranche that indicates the range of assets to be secured by the asset-backed securities, the lower limit of the tranche, and the total amount of disappearance of the assets due to payment before the scheduled payment date. A step in which the computer receives a prepayment rate indicating a ratio with respect to the amount of asset disappearance and stores the prepayment rate in a storage unit;
Based on the past data of the total asset extinction process and a model obtained based on the past data, the calculation means calculates the expected total asset extinction process indicating the expected value of the total asset extinction process with the total asset extinction process as the Levy process. Steps and
Calculating the market value of the asset-backed securities based on the upper and lower limits of the tranche stored in the storage means, the calculated expected total asset extinction process, and the prepayment rate stored in the storage means; Calculating the market value of asset-backed securities for performing the steps of: A program for calculating the market value of asset-backed securities issued with collateral based on the total assets of cash flows based on assets,
Total asset value, tranche upper limit value indicating the range of asset value to be secured by the asset-backed securities, lower limit value of the tranche, and total asset extinguishment amount of assets extinguished due to payment before the scheduled payment date A computer receiving a prepayment rate indicating a percentage of the amount, and a recovery rate indicating a percentage recovered in the amount of disappearance of the asset when the cash flow payer defaults, and storing the same in a storage means;
Based on the past data of the total asset extinction process and a model obtained based on the past data, the calculation means calculates the expected total asset extinction process indicating the expected value of the total asset extinction process with the total asset extinction process as the Levy process. Steps and
The asset backing based on the upper and lower limits of the tranche stored in the storage means, the calculated expected total asset disappearance process, the prepayment rate stored in the storage means, and the recovery rate. Calculating the market value of the security by the calculating means. Based on the past data of the total asset extinction process and a model obtained based on the past data, the calculation means calculates the expected total asset extinction process indicating the expected value of the total asset extinction process with the total asset extinction process as the Levy process. The steps are
Accepting a selection of a given logistic curve model stored in storage means by a computer;
Based on the difference between the past data of the total assets disappearance process and the numerical value of the logistic curve model, the calculating means performs a non-linear regression estimation calculation on the parameter in the logistic curve model that minimizes the difference,
A calculating means calculating a logistic curve based on the parameter;
Calculating the logistic curve by an average path of the total asset disappearance process and calculating means;
Receiving a selection of a Levy process model formula representing the expected total asset disappearance process;
10. A step of calculating said expected total asset disappearance process by a calculation means by substituting said average path into a Levy process model formula representing the selected expected total asset disappearance process. The program for calculating the market value of asset-backed securities described in. The asset according to any one of claims 7 to 10, wherein the Levy process is a process selected from a group consisting of a Poisson process, a Wiener process, and a process combining the Poisson process and the Wiener process. Market value calculation program for back securities. 12. The program according to claim 7, wherein the Levy process is monotonically increasing. A computer-readable recording medium on which the program according to claim 7 is recorded.

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