Methodology for robust portfolio evaluation and optimization taking account of estimation errors

• US 20070288397A1
• Filed: 06/12/2006
• Published: 12/13/2007
• Est. Priority Date: 06/12/2006
• Status: Abandoned Application

• Current Assignee: NEC Europe Ltd (NEC Corporation)
• Original Assignee: NEC Europe Ltd (NEC Corporation)
• Inventors: Frahm, Gabriel, Jaekel, Uwe
• Application Number:
  US11/450,385
• Publication Number:
  US 20070288397A1
• Time in Patent Office: Days
• US Class Current: 705/36.R
• CPC Class Codes: G06Q 40/06 Asset management; Financial planning or analysis

Abstract

The present invention relates to a portfolio evaluation and optimization in the presence of estimation errors. The method for optimizing a portfolio with several financial instruments, comprises steps of: selecting constraints and optimality criteria for the portfolio; obtaining historical information for financial risk factors and selecting an appropriate model for simulating the risk factors of the portfolio by way of a generalized elliptical distribution. Similarly, the method for evaluating a portfolio with several financial instruments, comprises steps of: selecting evaluation criteria for the portfolio; obtaining historical information for financial risk factors and selecting an appropriate model for simulating the risk factors of the portfolio by way of a generalized elliptical distribution. The model selection is based on historical information. According to the present invention both estimation risk and market risk are considered by simulation. The risk factors are simulated by drawing parameters and paths given the above appropriate model and the observations. It is a preferable advantage of the present invention that historical data may contain missing values. Another preferable advantage of the present invention is that portfolio evaluation and optimization is possible for long-term investments with arbitrary financial instruments. The optimal portfolio strategy is determined by considering the selected constraints and the optimality criteria on the basis of the parameters and paths simulated.

Claims

1. A method for optimizing a portfolio with several financial instruments, the method comprising the steps of:

a) selecting constraints and optimality criteria for the portfolio;

b) obtaining historical information for financial risk factors;

c) selecting an appropriate model for simulating the risk factors of the portfolio by way of an elliptical distribution;

wherein the selection is based on the historical information;

d) considering both estimation risk and market risk by simulation;

e) selecting numerical accuracy criteria for the optimal portfolio composite;

f) simulating the risk factors by drawing a plurality of parameters and paths given the model and the observation (—

possibly containing missing values);

g) finding the optimal portfolio weights given the selected constraints and optimality criteria on the basis of the parameters and paths simulated;

h) proceeding the above simulation and finding of the optimal portfolio weights until said accuracy criteria are fulfilled.

3. The method according to claim 1, wherein the elliptical distributions for simulating the risk factors (X) of said financial instruments are represented at least in terms of an expected return vector (μ

), a dispersion matrix (Σ

), a generating variate (R) and random vector (U).

5. The method according to claim 1, wherein said missing values are simulated by means of standard techniques of multiple imputation and/or data augmentation.

7. The method according to claim 5, wherein the data augmentation is performed for missing historical information and for unknown future values.

9. The method according to claim 5, wherein the data augmentation is based on a Gibbs sampler.

11. The method according claim 3, wherein the realisation of the expected return vector (μ

) and the realisation of the dispersion matrix (Σ

) are posterior distributions obtained on the basis of said historical information and “

a priori”

information.

13. The method according to claim 11, wherein the a priori information is based on informative and/or non-informative priors.

16. The method according to claim 11, wherein the posterior return parameter (μ

) and the posterior dispersion matrix (Σ

) are conditioned on estimators ({circumflex over (μ

)}, {circumflex over (Σ

)}) for the expected return parameter and the dispersion matrix.

18. The method according to claim 16, wherein the estimators ({circumflex over (μ

)}, {circumflex over (Σ

)}) for the expected return parameter and the dispersion matrix are affine equivariant estimators.

21. The method according to claim 16, wherein the estimators for the expected return parameter ({circumflex over (μ

)}) and the dispersion matrix ({circumflex over (Σ

)}) are simulated by a matrix containing a d-dimensional random vectors uniformly distributed on a unit hypersphere (U;

=[U₁ . . . U_n]) and a matrix containing the generating variates (R) on the main diagonal (R;

=diag(R₁, . . . , R_n)).

23. The method according to any of claims 1, wherein the parameter of the generating variate (R) and/or the unit random vector (U) represent the market risk.

25. The method according to any of claim 1, wherein the parameter of the generating variate (R) and the expected return parameter (μ

) with the dispersion matrix (Σ

) can be simulated independently.

27. The method according to claim 3, wherein the posterior distribution of the dispersion matrix (Σ

) is a product of a nonsigular matrix (Λ

) and its transposed matrix (Λ

′

).

29. The method according to claim 11, wherein the posterior distributions of expected return and dispersion matrix (μ

, Σ

) are based on the estimators of the expected return and the dispersion matrix ({circumflex over (Σ

)}).

31. The method according to claim 11, wherein the posterior distribution of the dispersion matrix (Σ

) is simulated with the steps of;

(i) simulating a random sample UR;

where U;

=[U₁ . . . U_n] is a matrix containing n columns of d-dimensional random vectors uniformly distributed on the unit hypersphere and R;

=diag(R₁, . . . , R_n) contains n generating variates on the main diagonal;

(ii) calculating the inverse of the estimator of the dispersion matrix ({circumflex over (Σ

)}(UR)<sup>−

1</sup>), and

(iii) multiplication of {circumflex over (Λ

)} from the left and from the right.

33. The method according to claim 11, wherein the posterior distribution of the expected return (μ

) is simulated after the simulation of the posterior distribution of the dispersion matrix (Σ

) based on said simulated matrix (U) and the matrix containing the generating variates (R) on the main diagonal and the symmetric square root of the posterior distribution of the dispersion matrix (Σ

).

35. The method according to claim 3, wherein the simulation of the risk factors by way of a vector X comprises the steps of:

(i) simulating a realization of a posterior distribution of the dispersion matrix (Σ

|{circumflex over (μ

)}, {circumflex over (Σ

)}));

(ii) simulating a realization of a posterior distribution of the return or location (μ

|{circumflex over (μ

)}, {circumflex over (Σ

)})) by taking the symmetric root of the realization of the posterior distribution of the dispersion matrix (Σ

|{circumflex over (Σ

)}) into account;

(iii) simulating new realizations of the generating variate (R) and U^(d) to obtain a possible realizations of the vector X and

(iv) calculating the corresponding trajectory or path of stock prices based on the estimates of the location, the dispersion matrix ({circumflex over (μ

)}, {circumflex over (Σ

)}) and the vector X.

(v) repeating steps (i) to (iv) until there is a sufficiently large number of simulated trajectories which fulfill the predetermined accuracy criteria.

37. The method according to claim 1, wherein the optimization criteria is at least one of the group consisting of, optimization of an expected utility function, performance and risk measures like, e.g. Value at Risk (VaR), Return on Investment (RoI), shape ratio, and multi-objective decision criteria.

39. A computer system for carrying out the method according to claims 1.

41. A storage medium for storing a computer program to accomplish the method according to claim 1.

2. A method for evaluation a portfolio with several financial instruments, the method comprising the steps of:

a) providing weights for a given portfolio;

a′

) selecting evaluation criteria for the portfolio;

b) obtaining historical information for financial risk factors;

c) selecting an appropriate model for simulating the risk factors of the portfolio by way of an elliptical distribution;

wherein the selection is based on the historical information;

d) considering both estimation risk and market risk by simulation;

e) selecting a numerical accuracy criteria for the given portfolio composite;

f) simulating the risk factors by drawing parameters and paths given the model and the observation (—

possibly containing missing values);

g) evaluating the given portfolio by the selected evaluation criteria on the basis of the parameters and paths simulated;

h) proceeding the above simulation and evaluation algorithm until said accuracy criteria are fulfilled.

4. The method according to claim 2, wherein the elliptical distributions for simulating the risk factors (X) of said financial instruments are represented at least in terms of an expected return vector (μ

), a dispersion matrix (Σ

), a generating variate (R) and random vector (U).

6. The method according to claim 2, wherein said missing values are simulated by means of standard techniques of multiple imputation and/or data augmentation.

8. The method according to claim 6, wherein the data augmentation is performed for missing historical information and for unknown future values.

10. The method according to claim 6, wherein the data augmentation is based on a Gibbs sampler.

12. The method according claim 4, wherein the realisation of the expected return vector (μ

) and the realisation of the dispersion matrix (Σ

) are posterior distributions obtained on the basis of said historical information and “

a priori”

information.

14. The method according to claim 12, wherein the a priori information is based on informative and/or non-informative priors.

15. The method according to claim 2, wherein the assessing in step g) is based on confidence intervals and hypothesis tests.

17. The method according to claim 12, wherein the posterior return parameter (μ

) and the posterior dispersion matrix (Σ

) are conditioned on estimators ({circumflex over (μ

)}, {circumflex over (Σ

)}) for the expected return parameter and the dispersion matrix.

19. The method according to claim 17, wherein the estimators ({circumflex over (μ

)}, {circumflex over (Σ

)}) for the expected return parameter and the dispersion matrix are affine equivariant estimators.

20. The method according to claim 2, wherein a joint posterior distribution of the expected return parameter (μ

) and the dispersion matrix (Σ

) is approximated.

22. The method according to claim 17, wherein the estimators for the expected return parameter ({circumflex over (μ

)}) and the dispersion matrix ({circumflex over (Σ

)}) are simulated by a matrix containing a d-dimensional random vectors uniformly distributed on a unit hypersphere (U;

=[U₁ . . . U_n]) and a matrix containing the generating variates (R) on the main diagonal (R;

=diag(R₁, . . . , R_n)).

24. The method according to any of claims 2, wherein the parameter of the generating variate (R) and/or the unit random vector (U) represent the market risk.

26. The method according to any of claim 2, wherein the parameter of the generating variate (R) and the expected return parameter (μ

) with the dispersion matrix (Σ

) can be simulated independently.

28. The method according to claim 4, wherein the posterior distribution of the dispersion matrix (Σ

) is a product of a nonsigular matrix (Λ

) and its transposed matrix (Λ

′

).

30. The method according to claim 12, wherein the posterior distributions of expected return and dispersion matrix (μ

, Σ

) are based on the estimators of the expected return and the dispersion matrix ({circumflex over (Σ

)}).

32. The method according to claim 12, wherein the posterior distribution of the dispersion matrix (Σ

) is simulated with the steps of;

(i) simulating a random sample UR;

where U;

=[U₁ . . . U_n] is a matrix containing n columns of d-dimensional random vectors uniformly distributed on the unit hypersphere and R;

=diag(R₁, . . . , R_n) contains n generating variates on the main diagonal;

(ii) calculating the inverse of the estimator of the dispersion matrix ({circumflex over (Σ

)}(UR)<sup>−

1</sup>), and

(iii) multiplication of A from the left and from the right.

34. The method according to claim 12, wherein the posterior distribution of the expected return (μ

) is simulated after the simulation of the posterior distribution of the dispersion matrix (Σ

) based on said simulated matrix (U) and the matrix containing the generating variates (R) on the main diagonal and the symmetric square root of the posterior distribution of the dispersion matrix (Σ

).

36. The method according to claim 4, wherein the simulation of the risk factors by way of a vector X comprises the steps of:

(i) simulating a realization of a posterior distribution of the dispersion matrix (Σ

|({circumflex over (μ

)}, {circumflex over (Σ

)}));

(ii) simulating a realization of a posterior distribution of the return or location (μ

|({circumflex over (μ

)}, {circumflex over (Σ

)})) by taking the symmetric root of the realization of the posterior distribution of the dispersion matrix (Σ

|{circumflex over (Σ

)}) into account;

(iii) simulating new realizations of the generating variate (R) and U^(d) to obtain a possible realizations of the vector X and

(iv) calculating the corresponding trajectory or path of stock prices based on the estimates of the location, the dispersion matrix ({circumflex over (μ

)}, {circumflex over (Σ

)}) and the vector X.

(v) repeating steps (i) to (iv) until there is a sufficiently large number of simulated trajectories which fulfill the predetermined accuracy criteria.

38. The method according to claim 2, wherein the optimization criteria is at least one of the group consisting of, optimization of an expected utility function, performance and risk measures like, e.g. Value at Risk (VaR), Return on Investment (RoI), shape ratio, and multi-objective decision criteria.

40. A computer system for carrying out the method according to claims 2.

42. A storage medium for storing a computer program to accomplish the method according to claim 2.

43. A method for optimizing a portfolio comprising several financial instruments, the method comprising the steps of:

a) selecting constraints and optimality criteria for the portfolio;

b) obtaining historical information for financial risk factors;

c) selecting an appropriate model for simulating the risk factors of the portfolio by way of an elliptical distribution and specifying the parameters of a generating variate (R) of said elliptical distribution ;

wherein the selection is based on the historical information;

d) considering both estimation risk and market risk by simulation;

e) selecting a numerical accuracy criteria for the optimal portfolio composite;

f) finding affine equivariant estimators for a mean vector (μ

) and covariance matrix (Σ

);

g) simulating the risk factors, wherein possible paths are simulated by way of the generating variate (R);

h) generating possible realizations of the true covariance matrix and the true mean vector from the simulated sample errors by utilizing the equivariance property;

i) computing possible paths of different portfolio evolutions using the mean and covariance parameters obtained in step g)

j) simulating an portfolio outcome by drawing parameters and paths from the universe of models conditioned on the observations and the model;

k) finding the optimal portfolio weights given the selected constraints and optimality criteria on the basis of the parameters and paths simulated;

l) proceeding the above resampling and optimization algorithm until numerical accuracy criteria are fulfilled.