Methods and systems for analytical-based multifactor multiobjective portfolio risk optimization

• US 20040186804A1
• Filed: 03/19/2003
• Published: 09/23/2004
• Est. Priority Date: 03/19/2003
• Status: Abandoned Application

• Current Assignee: GE Financial Assurance Holdings Incorporated (General Electric Company), General Electric Company
• Original Assignee: General Electric Company
• Inventors: Chalermkraivuth, Kete Charles, Chakraborty, Anindya
• Application Number:
  US10/390,689
• Publication Number:
  US 20040186804A1
• Time in Patent Office: Days
• US Class Current: 705/36
• CPC Class Codes: G06Q 40/03 Credit; Loans; Processing thereof
• CPC Class Codes: G06Q 40/06 Asset management; Financial planning or analysis
• CPC Class Codes: G06Q 40/08 Insurance

Abstract

The invention provides systems and methods for performing a risk measure simplification process through matrix manipulation. The method includes defining the change in risk factors; defining portfolio risk sensitivities as Delta and Gamma; restating the change in risk factors in Delta-Gamma formulation, the Delta-Gamma formulation having the factors ΔF'"'"'s; defining the covariance matrix of ΔF; taking the Cholesky decomposition of the covariance matrix to generate a P transformation matrix; applying the P transformation matrix to Gamma to define a matrix Q_k; determining the Eigenvalue decomposition of Q_k to obtain a matrix of Eigenvectors N; and applying the matrix of Eigenvectors N and the P transformation matrix to evaluate the risk measures.

Claims

1. A method for performing a risk measure simplification process through matrix manipulation, the method comprising:

defining the change in risk factors;

defining portfolio risk sensitivities as Delta and Gamma;

restating the change in risk factors in Delta-Gamma formulation, the Delta-Gamma formulation having the factors Δ

F'"'"'s;

defining the covariance matrix of Δ

F;

taking the Cholesky decomposition of the covariance matrix to generate a P transformation matrix;

applying the P transformation matrix to Gamma to define a matrix Q_k;

determining the Eigenvalue decomposition of Q_k to obtain a matrix of Eigenvectors N; and

applying the matrix of Eigenvectors N and the P transformation matrix to evaluate the risk measures.

2. The method of claim 1, wherein applying the matrix of Eigenvectors N and the P transformation matrix to evaluate the risk measures includes determining a total transform operator:

L=N^TP in order to obtain stored transforms;

δ

*, Γ

* &

Δ

F*; and

wherein transformed variables are defined as;

Δ

F*=LΔ

Fδ

_k*=(L^T)<sup>−

1</sup>δ

_k and(L^T)<sup>−

1</sup>Γ

_kL<sup>−

1</sup>=Γ

_k*.

3. The method of claim 1, wherein higher order moments of risk measures are reduced to a complexity of O(m).

5. The method of claim 1, wherein Delta and Gamma are respectively defined as:

6. The method of claim 1, wherein restating the change in risk factors in Delta-Gamma formulation, the Delta-Gamma formulation having the factors Δ

F'"'"'s, includes using the relationship;

Δ

V_k=δ

_k^TΔ

F+½

Δ

F^TΓ

_kΔ

F.

7. The method of claim 1, wherein defining the covariance matrix of Δ

F includes defining the covariance matrix as;

8. The method of claim 1, wherein taking the Cholesky decomposition of the covariance matrix to generate a P transformation matrix includes using the expression:

PΣ

P^T=I.

9. The method of claim 1, wherein applying the P transformation matrix to Gamma to define a matrix Q_k includes defining Q_k as:

Q_k=(P<sup>−

1</sup>)^TΓ

_k(P<sup>−

1</sup>).

10. The method of claim 1, wherein determining the Eigenvalue decomposition of Q_k to obtain a matrix of Eigenvectors N includes using the relationships:

N^TQ_kN=Γ

_k*N^TN=I=NN^T

where Γ

*, being the Gamma transform, is now diagonal and N is the orthogonal Eigenvector matrix by orthogonality.

4. The method of 1, wherein defining the change in risk factors is performed using m risk factors, and the change in each risk factor is defined by:

11. A system for performing a risk measure simplification process through matrix manipulation, the system comprising:

a first portion that defines the change in risk factors;

a second portion that defines Delta and Gamma;

a third portion that restates the change in risk factors in Delta-Gamma formulation, the Delta-Gamma formulation having the factors Δ

F'"'"'s;

a fourth portion that defines the covariance matrix of Δ

F;

a fifth portion that takes the Cholesky decomposition of the covariance matrix to generate a P transformation matrix;

a sixth portion that applies the P transformation matrix to Gamma to define a matrix Q_k;

a seventh portion that determines the Eigenvalue decomposition of Q_k to obtain a matrix of Eigenvectors N; and

an eighth portion that applies the matrix of Eigenvectors N and the P transformation matrix to evaluate the risk measures.

12. The system of claim 11, wherein the eighth portion, that applies the matrix of Eigenvectors N and the P transformation matrix to evaluate the risk measures, determines a total transform operator:

L=N^TP in order to obtain stored transforms;

δ

*, Γ

* &

Δ

F*.

13. A computer readable medium for performing a risk measure simplification process through matrix manipulation, the computer readable medium comprising:

a first portion that defines the change in risk factors;

a second portion that defines Delta and Gamma;

a third portion that restates the change in risk factors in Delta-Gamma formulation, the Delta-Gamma formulation having the factors Δ

F'"'"'s;

a fourth portion that defines the covariance matrix of Δ

F;

a fifth portion that takes the Cholesky decomposition of the covariance matrix to generate a P transformation matrix;

a sixth portion that applies the P transformation matrix to Gamma to define a matrix Q_k;

a seventh portion that determines the Eigenvalue decomposition of Q_k to obtain a matrix of Eigenvectors N; and

an eighth portion that applies the matrix of Eigenvectors N and the P transformation matrix to evaluate the risk measures.

14. The computer readable medium of claim 13, wherein the eighth portion, that applies the matrix of Eigenvectors N and the P transformation matrix to evaluate the risk measures, determines a total transform operator:

L=N^TP in order to obtain stored transforms;

δ

*, Γ

* &

Δ

F*.

15. The computer readable medium of claim 13, wherein the system reduces higher order moments of risk measures to a complexity of O(m).

16. A method for performing a risk measure simplification process through matrix manipulation, the method comprising:

defining the change in risk factors;

defining portfolio risk sensitivities as Delta and Gamma;

restating the change in risk factors in Delta-Gamma formulation, the Delta-Gamma formulation having the factors Δ

F'"'"'s;

defining the covariance matrix of Δ

F;

taking the Cholesky decomposition of the covariance matrix to generate a P transformation matrix;

applying the P transformation matrix to Gamma to define a matrix Q_k;

determining the Eigenvalue decomposition of Q_k to obtain a matrix of Eigenvectors N;

applying the matrix of Eigenvectors N and the P transformation matrix to evaluate the risk measures; and

wherein defining the change in risk factors is performed using m risk factors, and the change in each risk factor is defined by;

$\begin{array}{ccc}{F}_{m\times <br><br>1}=\left(\begin{array}{c}{F}_1\\ F_2\\ \mathrm{\dots <br><br>}\\ F_m\end{array}\right)& \Rightarrow <br><br>& \mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>F_{m\times <br><br>1}=\left(\begin{array}{c}\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>F_1\\ \mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>F_2\\ \mathrm{\dots <br><br>}\\ \mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>F_m\end{array}\right);<br><br>\end{array}$

wherein Delta and Gamma are respectively defined as;

$\begin{array}{ccc}{\mathrm{\delta <br><br>}}_k=\left(\begin{array}{c}\frac{\partial <br><br>V_k}{\partial <br><br>F_1}\\ \mathrm{\dots <br><br>}\\ \frac{\partial <br><br>V_k}{\partial <br><br>F_m}\end{array}\right);<br><br>\mathrm{and}& \text{\hspace{1em}<br><br>}& {\mathrm{\Gamma <br><br>}}_k=\left(\begin{array}{cc}\frac{{\partial <br><br>}^2<br><br>V_k}{\partial <br><br>F_1^2}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\dots <br><br>}& \text{\hspace{1em}<br><br>}\\ \text{\hspace{1em}<br><br>}& \frac{{\partial <br><br>}^2<br><br>V_k}{\partial <br><br>F_j<br><br>\partial <br><br>F_i}\\ \frac{{\partial <br><br>}^2<br><br>V_k}{\partial <br><br>F_i<br><br>\partial <br><br>F_j}& \text{\hspace{1em}<br><br>}\\ \text{\hspace{1em}<br><br>}& \frac{{\partial <br><br>}^2<br><br>V_k}{\partial <br><br>F_m^2}\end{array}\right).\end{array}$

17. The method of claim 16, wherein restating the change in risk factors in Delta-Gamma formulation, the Delta-Gamma formulation having the factors Δ

F'"'"'s, includes using the relationship;

18. The method of claim 16, wherein defining the covariance matrix of Δ

F includes defining the covariance matrix as;