Fast and accurate method for estimating portfolio CVaR risk

• US 8,355,976 B2
• Filed: 01/18/2011
• Issued: 01/15/2013
• Est. Priority Date: 01/18/2011
• Status: Expired due to Fees

• Current Assignee: International Business Machines Corporation
• Original Assignee: International Business Machines Corporation
• Inventors: Ghosh, Soumyadip, Huang, Pu, Subramanian, Dharmashankar, Xu, Jie
• Primary Examiner(s): JOHNSON, GREGORY L
• Application Number:
  US13/008,123
• Publication Number:
  US 20120185406A1
• Time in Patent Office: 728 Days
• Field of Search: None
• US Class Current: 705/36.R
• CPC Class Codes: G06Q 40/06 Asset management; Financial planning or analysis

Abstract

A method, system and computer program product for measuring a risk of an asset portfolio. The system estimates a β-level CVaR (Conditional Value-at-Risk) of the asset portfolio by modeling interdependencies between assets in the asset portfolio. The modeling is based on Gaussian copula model.

Claims

1. A method for measuring a risk of a portfolio including n number of assets, the method comprising:

estimating, by a computing system, a β

-level CVaR (Conditional Value-at-Risk) of the portfolio by calculating

2. The method according to claim 1, further comprising:

capturing an interdependency between assets in the portfolio using a Gaussian copula model.

3. The method according to claim 2, wherein the Gaussian copula model is represented by n marginal Cumulative Distribution Functions (CDF) F_i(·

) and a n×

n matrix Σ

_Z, wherein F_i(·

) is a marginal CDF of a potential loss of an asset i, and Σ

_Z is a correlation matrix that captures interdependencies among asset losses.

4. The method according to claim 3, wherein the estimating the β

-level CVaR further includes steps of;

applying a matrix decomposition technique on the correlation matrix Σ

_Z to decompose the correlation matrix Σ

_Z as Σ

_Z=U^TDU, where D is a diagonal matrix with non-negative diagonal entries, and U is a unitary matrix;

generating J number of sample points V¹, . . . , V^J from a standard n-dimensional multivariate normal distribution whose mean value is zero and whose correlation matrix is an n×

n identify matrix, and creating J number of points Z¹, . . . , Z^J, by multiplying D^c and U^T to V¹, . . . , V^J as Z^j=D^1/2U^TV^j, where a sample point index j ranges from 1 to J;

creating J number of points X¹, . . . , X^J by calculating X_i^j=F_i<sup>−

1</sup>(Φ

(Z_i^j)), where the asset i ranges from 1 to n, the sample point index j ranges from 1 to J, Z_i^j is the i-th entry of Z^j, X_i^j is an i-th entry of X^j, F_i<sup>−

1</sup>(·

) is the inverse function of F_i(·

), and Φ

(·

) is the univariate standard normal CDF;

computing empirical losses L¹, . . . , L^J as L^j=a^TX^j, where the index j ranges from 1 to J, then sorting L¹, . . . , L^J in an ascending order, denoting the sorted L¹, . . . , L^J with L⁽¹⁾≦

. . . ≦

L^(J), and determining a largest integer K such that J−

K≧

J(1−

β

) so that K=max{j|J−

j≧

J(1−

β

), j=1, . . . , J};

estimating a β

-level VaR of total portfolio loss L as L^(K));

dividing points X¹, . . . , X^J into two groups, a first group and a second group;

dividing points V¹, . . . , V^J into two groups, a third group and a fourth group, according to the divided points X¹, . . . , X^J;

finding a hyper-plane that separates the third group and the fourth group, and representing the separating hyper-plane as f(x)=k^Tx−

b=0, where k is a unit normal vector (k^Tk=1) of the hyper-plane, |b| (the absolute value of b) is a distance from an origin (0,0) to the hyper-plane;

computing a shifting amount Δ

Z as Δ

Z=bD^1/2U^Tk;

shifting the points Z¹, . . . , Z^J by Δ

Z, and creating points Y¹, . . . , Y^J from the shifted points as Y_i^j=F_i<sup>−

1</sup>(Φ

(Z_i^j+Δ

Z_i)), where the asset i ranges from 1 to n, the index j ranges 1 to J;

computing a set of likelihood ratios w¹, . . . , w^J as

5. The method according to claim 4, wherein the matrix decomposition technique includes a singular value decomposition technique.

6. The method according to claim 4, wherein U^TU is equal to UU^T which is equal to I_n, the I_n being an n×

n identify matrix.

7. The method according to claim 4, wherein the first group includes X^j'"'"'s that satisfy a^TX^j≧

L^(K), and the second group includes remainders.

8. The method according to claim 4, wherein the third group includes V^j'"'"'s whose corresponding X^j'"'"'s belong to the first group, and the fourth group includes remaining V^j'"'"'s.

22. The method according to claim 1, wherein a^T represents a transpose of column vector a, a is [a₁, . . . , a_n]^T, where a_i (i=1, . . . , n) denotes a number of shares invested in an asset i and Y^j is a shifted sample point.

9. A system for measuring a risk of a portfolio including n number of assets, the system comprising:

at least one memory device; and

at least one processor connected to the memory device,

wherein the processor is configured to;

estimate a β

-level CVaR (Conditional Value-at-Risk) of the portfolio, by calculating

10. The system according to claim 9, wherein the processor is configured to:

capture an interdependency between the assets in the portfolio using a Gaussian copula model.

11. The system according to claim 10, wherein the Gaussian copula model is represented by n marginal Cumulative Distribution Functions (CDF) F_i(·

) and a n×

n matrix Σ

_Z, wherein F_i(·

) is a marginal CDF of a potential loss of an asset i, and Σ

_Z is a correlation matrix that captures interdependencies among asset losses.

12. The system according to claim 11, wherein to estimate the β

-level CVaR of the portfolio, the processor is further configured to;

apply a matrix decomposition technique on the correlation matrix Σ

_Z to decompose the correlation matrix Σ

_Z as Σ

_Z=U^TDU, where D is a diagonal matrix with non-negative diagonal entries, and U is a unitary matrix;

generate J number of sample points V¹, . . . , V^J from a standard n-dimensional multivariate normal distribution whose mean value is zero and whose correlation matrix is an n×

n identify matrix, and creating J number of points Z¹, . . . , Z^J, by multiplying D^c and U^T to V¹, . . . , V^J as Z^j=D^1/2U^TV^j, where a sample point index j ranges from 1 to J;

create J number of points X¹, . . . , X^J by calculating X_i^j=F_i<sup>−

1</sup>(Φ

(Z_i^j)), where the asset i ranges from 1 to n, the sample point index j ranges from 1 to J, Z_i^j is the i-th entry of Z^j, X_i^j is an i-th entry of X^j, F_i<sup>−

1</sup>(·

) is the inverse function of F_i(·

), and Φ

(·

) is the univariate standard normal CDF;

compute empirical losses L¹, . . . , L^J as L^j=a^TX^j, where the index j ranges from 1 to J, then sorting L¹, . . . , L^J in an ascending order, denoting the sorted L¹, . . . , L^J with L⁽¹⁾≦

. . . ≦

L^(J), and determine a largest integer K such that J−

K≧

J(1−

β

) so that K=max{j|J−

j≧

J(1−

β

), j=1, . . . , J};

estimate a β

-level VaR of total portfolio loss L as L^(K);

divide points X¹, . . . , X^J into two groups, a first group and a second group;

divide points V¹, . . . , V^J into two groups, a third group and a fourth group, according to the divided points X¹, . . . , X^J;

find a hyper-plane that separates the third group and the fourth group, and representing the separating hyper-plane as f(x)=k^Tx−

b=0, where k is a unit normal vector (k^Tk=1) of the hyper-plane, |b| (the absolute value of b) is a distance from an origin (0,0) to the hyper-plane;

compute a shifting amount Δ

Z as Δ

Z=bD^1/2U^Tk;

shifting the points Z¹, . . . , Z^J by Δ

Z, and creating points Y¹, . . . , Y^J from the shifted points as Y_i^j=F_i<sup>−

1</sup>(Φ

(Z_i^j+Δ

Z_i)), where the asset i ranges from 1 to n, the index j ranges 1 to J;

compute a set of likelihood ratios w¹, . . . , w^J as

13. The system according to claim 12, wherein the matrix decomposition technique includes a singular value decomposition technique.

14. The system according to claim 12, wherein U^TU is equal to UU^T which is equal to I_n, the I_n being an n×

n identify matrix.

15. The system according to claim 12, wherein the first group includes X^j'"'"'s that satisfy a^TX^j≧

L^(K), and the second group includes remainders.

16. The system according to claim 12, wherein the third group includes V^j'"'"'s whose corresponding X^j'"'"'s belong to the first group, and the fourth group includes remaining V^j'"'"'s.

23. The system according to claim 9, wherein a^T represents a transpose of column vector a, a is [a₁, . . . , a_n]^T, where a_i (i=1, . . . , n) denotes a number of shares invested in an asset i and Y^j is a shifted sample point.

17. A computer program product for measuring a risk of an asset portfolio including n number of assets, the computer program product comprising a storage medium readable by a processing circuit and storing instructions run by the processing circuit for performing a method, the method comprising:

estimating a β

-level CVaR (Conditional Value-at-Risk) of the portfolio by calculating

18. The computer program product according to claim 17, wherein the method further comprises:

capturing an interdependency between the assets in the portfolio using a Gaussian copula model.

19. The computer program product according to claim 18, wherein the Gaussian copula model is represented by n marginal Cumulative Distribution Functions (CDF) F_i(·

) and a n×

n matrix Σ

_Z wherein F_i(·

) is a marginal CDF of a potential loss of an asset i, and Σ

_Z is a correlation matrix that captures interdependencies among asset losses.

20. The computer program product according to claim 19, wherein the estimating the β

-level CVaR further includes steps of;

applying a matrix decomposition technique on the correlation matrix Σ

_Z to decompose the correlation matrix Σ

_Z as Σ

_Z=U^TDU, where D is a diagonal matrix with non-negative diagonal entries, and U is a unitary matrix;

generating J number of sample points V¹, . . . , V^J from a standard n-dimensional multivariate normal distribution whose mean value is zero and whose correlation matrix is an n×

n identify matrix, and creating J number of points Z¹, . . . , Z^J, by multiplying D^c and U^T to V¹, . . . , V^J as Z^j=D^1/2U^TV^j where a sample point index j ranges from 1 to J;

creating J number of points X¹, . . . , X^J by calculating X_i^j=F_i<sup>−

1</sup>(Φ

(Z_i^j)) where the asset i ranges from 1 to n, the sample point index j ranges from 1 to J, Z_i^j is the i-th entry of Z^j, X_i^j is an i-th entry of X^j, F_i<sup>−

1</sup>(·

) is the inverse function of F_i(·

) and Φ

(·

) is the univariate standard normal CDF;

computing empirical losses L¹, . . . , L^J as L^j=a^TX^j, where the index j ranges from 1 to J, then sorting L¹, . . . , L^J in an ascending order, denoting the sorted L¹, . . . , L^J with L⁽¹⁾≦

. . . ≦

L^(J) and determining a largest integer K such that J−

K≧

J(1−

β

) so that K=max{j|J−

j≧

J(1−

β

), j=1, . . . , J};

estimating a β

-level VaR of total portfolio loss L as L^(K);

dividing points X¹, . . . , X^J into two groups, a first group and a second group;

dividing points V¹, . . . , V^J into two groups, a third group and a fourth group, according to the divided points X¹, . . . , X^J;

finding a hyper-plane that separates the third group and the fourth group, and representing the separating hyper-plane as f(x)=k^Tx−

b=0, where k is a unit normal vector (k^Tk=1) of the hyper-plane, |b| (the absolute value of b) is a distance from an origin (0,0) to the hyper-plane;

computing a shifting amount Δ

Z as Δ

Z=bD^1/2U^Tk;

shifting the points Z¹, . . . , Z^J by Δ

Z, and creating points Y¹, . . . , Y^J from the shifted points as Y_i^j=F_i<sup>−

1</sup>(Φ

(Z_i^j+Δ

Z_i)), where the asset i ranges from 1 to n, the index j ranges 1 to J;

computing a set of likelihood ratios w¹, . . . , w^J as

21. The computer program product according to claim 20, wherein the matrix decomposition technique includes a singular value decomposition technique.

24. The computer program product according to claim 17, wherein a^T represents a transpose of column vector a, a is [a₁, . . . , a_n]^T, where a_i (i=1, . . . , n) denotes a number of shares invested in an asset i and Y^j is a shifted sample point.