Integrative method for modeling multiple asset classes
First Claim
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1. A method for combining two or more risk models to create a risk model with wider scope than its constituent parts. The method insures that the combined risk model is consistent with the component models from which it is formed. The method consists of the following steps:
- letting C1 denote a class of algorithms for constructing estimates of a covariance matrices from time histories of data;
letting C2 denote a class of asset classes;
for x in C2 let C3(x),denoting a class of multi-factor risk models for x;
for y in C3(x) denoting its parts as follows;
factor exposures X(y,t) at time t;
factor returns f(y,t) at time t; and
specific covariance matrix D(y,t) at time t;
factor covariance matrix F(y,t) at time t;
giving the following components;
two or more asset classes x1, . . . , xn, et x denote an asset class which is a union of these given asset classes;
for each asset class x1 giving a risk model yi in C3(xi);
letting Y(t) be such that the decomposition;
which results in residuals φ
(t) such that correlation (φ
1,φ
j) is nearly zero if i≠
j; and
constructing a risk model for x as follows;
forming X(t)=diag(X(y1,t), . . . , X(yn,t));
forming D(t)=diag(D(y1,t), . . . , D(yn,t));
applying a method from C1 to estimate a covariance matrix G(t) from a history of the g(t)s; and
applying an optionally different method from C1 to estimate a covariance matrix Φ
(t) from a history the φ
(t)s;
Then X(t)[Y(t)G(t)Y(t)t+Φ
(t)φ
(t)]X(t)t+D(t) is a risk model for x. Insure the risk model is consistent with each component, asset class risk model as follows;
Let F1(t) be the block diagonal matrix obtained from Y(t)G(t)Y(t)′
+Φ
(t) by setting all elements to zero except those of the blocks corresponding to each asset class. Each such block represents the covariance among the factors explaining risk for a particular asset class. Let F2(t) be the block diagonal matrix whose blocks contain the asset class factor covariance matrices, F(y1,t) in the same order as they appear in F1(t);
the off-block diagonal elements are zero. Given a real symmetric positive semi-definite matrix M, let M1/2 denote a square root of M so that M1/2(M1/2)=M. There may be several choices for M1/2. Let M−
1/2 denote the inverse of M1/2, or in the event that the inverse does not exist, let M−
1/2 be the pseudoinverse. Then X(t)(F21/2F1−
1/2(Y(t)G(t)Y(t)′
+Φ
(t))(F21/2F1−
1/2)′
)X(t)′
is a risk model that is consistent with the component asset class models.
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Abstract
The invention provides a method and apparatus for combining two or more risk models to create a risk model with wider scope than its constituent parts. The method insures that the newly formed risk model is consistent with the component models from which it is formed.
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Citations
3 Claims
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1. A method for combining two or more risk models to create a risk model with wider scope than its constituent parts. The method insures that the combined risk model is consistent with the component models from which it is formed. The method consists of the following steps:
-
letting C1 denote a class of algorithms for constructing estimates of a covariance matrices from time histories of data;
letting C2 denote a class of asset classes;
for x in C2 let C3(x),denoting a class of multi-factor risk models for x;
for y in C3(x) denoting its parts as follows;
factor exposures X(y,t) at time t;
factor returns f(y,t) at time t; and
specific covariance matrix D(y,t) at time t;
factor covariance matrix F(y,t) at time t;
giving the following components;
two or more asset classes x1, . . . , xn, et x denote an asset class which is a union of these given asset classes;
for each asset class x1 giving a risk model yi in C3(xi);
letting Y(t) be such that the decomposition;
which results in residuals φ
(t) such that correlation (φ
1,φ
j) is nearly zero if i≠
j; and
constructing a risk model for x as follows;
forming X(t)=diag(X(y1,t), . . . , X(yn,t));
forming D(t)=diag(D(y1,t), . . . , D(yn,t));
applying a method from C1 to estimate a covariance matrix G(t) from a history of the g(t)s; and
applying an optionally different method from C1 to estimate a covariance matrix Φ
(t) from a history the φ
(t)s;
Then X(t)[Y(t)G(t)Y(t)t+Φ
(t)φ
(t)]X(t)t+D(t) is a risk model for x.Insure the risk model is consistent with each component, asset class risk model as follows;
Let F1(t) be the block diagonal matrix obtained from Y(t)G(t)Y(t)′
+Φ
(t) by setting all elements to zero except those of the blocks corresponding to each asset class. Each such block represents the covariance among the factors explaining risk for a particular asset class.Let F2(t) be the block diagonal matrix whose blocks contain the asset class factor covariance matrices, F(y1,t) in the same order as they appear in F1(t);
the off-block diagonal elements are zero.Given a real symmetric positive semi-definite matrix M, let M1/2 denote a square root of M so that M1/2(M1/2)=M. There may be several choices for M1/2. Let M−
1/2 denote the inverse of M1/2, or in the event that the inverse does not exist, let M−
1/2 be the pseudoinverse.Then X(t)(F21/2F1−
1/2(Y(t)G(t)Y(t)′
+Φ
(t))(F21/2F1−
1/2)′
)X(t)′
is a risk model that is consistent with the component asset class models.
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2. A method for combining two or more risk models to create a risk model with wider scope than its constituent parts, comprising the steps of:
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denoting a class of algorithms for constructing estimates of a covariance matrices from time histories of data;
denoting a class of asset classes;
denoting a class of multi-factor risk models; and
constructing risk models for each asset class as follows;
applying a method to estimate a covariance matrix from a history; and
combining asset class risk models to form a risk model with broad coverage that is consistent with each asset class model. - View Dependent Claims (3)
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Current AssigneeBarra Incorporated
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Original AssigneeBarra Incorporated
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InventorsStefek, Daniel, Goldberg, Lisa Robin, Hui, Ken Chorlam, Scheffler, Scott Steven, Torre, Nicolas Goodrich
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Granted Patent
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Time in Patent OfficeDays
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Field of Search
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US Class Current703/2
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CPC Class CodesG06Q 40/03 Credit; Loans; Processing t...G06Q 40/08 InsuranceY10S 706/925 BusinessY10S 706/932 Mathematics, science, or en...
