Financial portfolio risk management

• US 20030055765A1
• Filed: 06/25/2002
• Published: 03/20/2003
• Est. Priority Date: 06/25/2001
• Status: Abandoned Application

• Current Assignee: Qinetiq Limited (QinetiQ Group PLC)
• Original Assignee: Qinetiq Limited (QinetiQ Group PLC)
• Inventors: Bernhardt, Mark
• Application Number:
  US10/178,784
• Publication Number:
  US 20030055765A1
• Time in Patent Office: Days
• US Class Current: 705/36
• CPC Class Codes: G06Q 40/06 Asset management; Financial planning or analysis

Abstract

A method for selecting a portfolio w consisting of N assets of prices p₁ each having a history of T+1 returns at time intervals i, (uncompounded returns over the previous t time steps) comprising the steps of;

a) defining a series of vectors {p₁, p₂ to p_T+1} to represent the price increments p for portfolio w for a given number of time steps t over a period T+1;

b) optionally removing any deterministic trends identified in step a);

c) calculating using support vector algorithms a linear combination of the vectors defined in step b), of maximal length and which is as near as possible perpendicular to each vector P_i in the series for optimal alpha parameters between C⁻ and C⁺

d) defining the portfolio w by the expression: $w=\sum {\alpha }_i^{*}p_i$

Some suitable algorithms and constraints for the algorithms are proposed. The invention further comprises computer programs for performing the invention when installed on suitable computer systems, and computer readable data.

Claims

1. A method for selecting a portfolio w consisting of N assets of prices p₁ each having a history of T+1 returns at time intervals i, (uncompounded returns over the previous t time steps) comprising the steps of;

a) defining a series of vectors {p₁, p₂ to p_T+1} to represent the price increments p for portfolio w over a historic time period T at time intervals i;

b) optionally removing any deterministic trends identified in step a);

c) calculating using support vector algorithms a linear combination of the vectors defined in step a), of maximal length and which is as near as possible perpendicular to each vector p_i in the series for optimal alpha values between C⁻

and C⁺

d) defining the portfolio w by the expression;

$w=\sum <br><br>{\mathrm{\alpha <br><br>}}_i^{*}<br><br>p_i$

3. A method as claimed in claim 1 wherein step c) involves;

i) applying the regression SVM algorithm;

Minimise

$L=\frac{1}{2}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>w^2<br><br>+C^{+}<br><br>\underset{i}{\sum <br><br>}<br><br>{\left({\mathrm{\xi <br><br>}}_i^{+}\right)}^{\mathrm{\lambda <br><br>}}+C^{-}<br><br>\underset{i}{\sum <br><br>}<br><br>{\left({\mathrm{\xi <br><br>}}_i^{-}\right)}^{\mathrm{\lambda <br><br>}}$

subject to the constraints

w.1=1, w·

p_i+ξ

⁻

_i≧

0, Σ

⁺_i−

w·

p_i≧

0, ξ

⁺_i≧

0 and ξ

⁻

_i≧

0

ii) implementing the SVM algorithm of step c) for λ

=1 and/or ξ

=2 and transforming the solution into its Lagrangian dual; and

iii) solving the solution to the Lagrangian dual of step ii) for optimal alpha values between C⁻

and C⁺.

5. A method as claimed in claim 3 wherein in step ii) the SYM algorithm is solved for λ

=1.

10. A program for a computer configured to perform the method of claim 1 based on data input including inter alia data selected from N, p, t, T. and/or C.

11. A computer readable storage media carrying a program as claimed in claim 10.

16. A system for selecting a portfolio w consisting of N assets, the system comprising;

a computer;

a database accessible by the computer and comprising data including prices p_i of a plurality of assets and a history of returns on those assets over a known time period T+1 at time intervals i;

interface means for permitting a user to access the computer and to input data selecting N assets from the database;

software resident on the computer for causing the computer to define a portfolio, the software utilising the method of claim 1;

means for providing to the user a visual representation of the defined portfolio w.

19. A system as claimed in claim 16 wherein the database is provided on a server separate from the computer but accessible by the computer via a telecommunications network

20. A system as claimed in claim 19 wherein there is a plurality of computers each having access to the database server via a telecommunications network.

21. A system as claimed in claim 16 wherein the interface means comprises one or more computer peripherals selected from;

a keyboard;

a computer mouse, tracker ball or touch sensitive panel;

a graphical user interface;

a touch sensitive display screen;

voice recognition technology.

22. A system as claimed in claim 16 wherein the means for providing a visual representation is selected from a printer and/or a display monitor.

2. A method for selecting a portfolio w consisting of N assets of prices p_i each having a history of T+1 returns at time intervals i, (uncompounded returns over the previous t time steps) comprising the steps of;

a) defining a series of vectors {q₁, q₂ to q_T+1} to represent the time evolution of a price increment q₁ for each asset in the portfolio;

b) optionally removing any deterministic trends identified in step a);

c) calculating using support vector algorithms a linear combination of the vectors defined in step a), of maximal length and which is as near as possible perpendicular to each vector q_i in the series for optimal alpha values;

d) determining from the solutions to step c), optimal solutions for a series of vectors α

_i* where;

w=Σ

_ia_i*q_i

4. A method as claimed in claim 2 wherein step c) involves;

i) applying the regression SVM algorithm;

Minimise

$L=\frac{1}{2}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>w^2<br><br>+C^{+}<br><br>\underset{i}{\sum <br><br>}<br><br>{\left({\mathrm{\xi <br><br>}}_i^{-}\right)}^{\mathrm{\lambda <br><br>}}+C^{-}<br><br>\underset{i}{\sum <br><br>}<br><br>{\left({\mathrm{\xi <br><br>}}_i^{-}\right)}^{\mathrm{\lambda <br><br>}}$

subject to the constraints

w·

q_i+ξ

⁻

_i≧

0, ξ

⁺_i−

w.q₁≧

0, ξ

⁺_i≧

0 and ξ

⁻

_i≧

0

ii) implementing the SVM algorithm of step c) for λ

=1 and/or λ

=2 and transforming the solution into its Lagrangian dual; and

iii) solving the solution to the Lagrangian dual of step ii) for optimal alpha values between C⁻

and C⁺ subject to the constraint Σ

α

_i=1.

6. A method as claimed in claim 4 wherein in step ii) the SVM algorithm is solved for λ

=1.

12. A program configured to perform the method of claim 2 based on input data including inter alia data selected from N, q, t, T and/or C.

13. A computer readable storage media carrying a program as claimed in claim 12.

17. A system for selecting a portfolio w consisting of N assets, the system comprising;

a computer;

a database accessible by the computer and comprising data including prices p_i of a plurality of assets and a history of returns on those assets over a known time period T+1 at time intervals i;

interface means for permitting a user to access the computer and to input data selecting N assets from the database;

software resident on the computer for causing the computer to define a portfolio, the software utilising the method of claim 2;

means for providing to the user a visual representation of the defined portfolio w.

23. A system as claimed in claim 17 wherein the database is provided on a server separate from the computer but accessible by the computer via a telecommunications network

24. A system as claimed in claim 23 wherein there is a plurality of computers each having access to the database server via a telecommunications network.

25. A system as claimed in claim 17 wherein the interface means comprises one or more computer peripherals selected from;

a keyboard, a computer mouse, tracker ball or touch sensitive panel;

a graphical user interface;

a touch sensitive display screen;

voice recognition technology.

26. A system as claimed in claim 17 wherein the means for providing a visual representation is selected from a printer and/or a display monitor.

7. A method for selecting a portfolio w consisting of N assets of prices p_i each having a history of T+1 returns at time intervals i, (uncompounded returns over the previous t time steps) comprising the steps of;

a) defining a vector x_i of T+1 returns on an asset p_i over a historic time period T at time intervals i;

b) select a minimum desired threshold return value r where

w.x_i−

r+ξ

_i≧

0

wherein ξ

_i are positive (non-zero) slack variables reflecting the amount the portfolio w historically fell short of the desired value of r,

c) optimise the problem in step b) by applying the Langrangian function

minimize

$L=\frac{1}{2}<br><br>\text{\hspace{1em}<br><br>}<br><br>{<br><br>w<br><br>}^2+\frac{C}{p}<br><br>\underset{i=1}{\overset{T}{\sum <br><br>}}<br><br>{\mathrm{\xi <br><br>}}_i^p$

where ξ

^p represents the non-zero slack variables of step b) to a power p and C is a weighting constant;

d) transforming the function of c) to the dual Langrangian and solving the quadratic programming problem for dual variables α

where p=1 and/or p=2;

e) determining from the solutions to step d), a portfolio w where;

$w=\underset{i=1}{\overset{T}{\sum <br><br>}}<br><br>{\mathrm{\alpha <br><br>}}_i<br><br>x_i$

8. A method as claimed in claim 7 further comprising;

after step a), identifying an overall mean level of return R for portfolio w from the expression

$\frac{1}{T}<br><br>\underset{i}{\overset{T}{\sum <br><br>}}<br><br>w·<br><br>x_i=R$

and apply in extrapolation of x_i according to the expression

x_i|→

x_i+λ

−

μ

where μ

is the mean returns vector (based on R) for the historical price data and λ

is the vector of predicted future returns.

9. A method as claimed in claim 7 wherein step d) involves maximising the quadratic equations respectively;

and

14. A program configured to perform the method of claim 7 based on input data including inter alia data selected from N, p, t, T, r, x and/or C.

15. A computer readable storage media carrying a program as claimed in claim 14.

18. A system for selecting a portfolio w consisting of N assets, the system comprising;

a computer;

a database accessible by the computer and comprising data including prices p_i of a plurality of assets and a history of returns on those assets over a known time period T+1 at time intervals i;

interface means for permitting a user to access the computer and to input data selecting N assets from the database;

software resident on the computer for causing the computer to define a portfolio, the software utilising the method of claim 7;

means for providing to the user a visual representation of the defined portfolio w.

27. A system as claimed in claim 18 wherein the database is provided on a server separate from the computer but accessible by the computer via a telecommunications network

28. A system as claimed in claim 27 wherein there is a plurality of computers each having access to the database server via a telecommunications network.

29. A system as claimed in claim 18 wherein the interface means comprises one or more computer peripherals selected from;

a keyboard;

a computer mouse, tracker ball or touch sensitive panel;

a graphical user interface;

a touch sensitive display screen;

voice recognition technology.

30. A system as claimed in claim 18 wherein the means for providing a visual representation is selected from a printer and/or a display monitor.