# US 8,788,391 B2 · Numerical modelling apparatus

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- Number: US 8,788,391 B2
- Document ID: US8788391B2
- Status: Granted
- Kind: B2
- Application number: 12/946,878
- Filed: 2010-11-16
- Published: 2011-07-07
- Granted: 2014-07-22
- Earliest claimed: 2010-01-05
- Applicants: Michael Mura
- Inventors: Michael Mura
- Web page: https://patentgrep.com/patent/US8788391B2

## CPC

- G06Q 40/06: Asset management; Financial planning or analysis
- G06Q 10/04: Forecasting or optimisation specially adapted for administrative or management purposes, e.g. linear programming or "cutting stock problem" (market predictions or forecasting for commercial activities G06Q30/0202)
- G06Q 40/04: Trading; Exchange, e.g. stocks, commodities, derivatives or currency exchange
- G06Q: INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR

## Abstract

A numerical modeling apparatus and method of performing numerical modeling are described. An input unit may receive information relating to set of assets. processor unit may provide Risk Relation Matrix V having elements that represent relationship of risk related to respective pair of the assets. The Risk Relation Matrix V may be decomposed into eigenvectors and eigenvalues according to V=E·Λ·E′, where E is set of eigenvectors of the risk matrix V in columns, Λ is the corresponding diagonal eigenvalue matrix, and E′ is the transpose of E. Components of risk vectors may be derived in terms of unit independent risks by the corresponding row of the matrix product E·Λ1/2 relating to respective assets. An output unit may output the risk vector components of the risk vectors as risk vector dataset.

## Claims

### Claim 1

numerical modelling apparatus, the apparatus comprising:
an input unit arranged to receive signals containing data having information relating to a set of assets;
processor unit coupled to the input unit and arranged to:
a) provide matrix V having plurality of elements, wherein each of the elements represents relationship of risk related to respective pair of the assets, and wherein each element is given by scalar product of two risk vectors, such that each of the assets has an associated risk vector according to the elements of the matrix;
b) decompose the matrix V into eigenvectors and eigenvalues according to:
V=E·Λ·E′
wherein E is set of eigenvectors of the matrix V in columns, Λ is the corresponding diagonal eigenvalue matrix, and E′ is the transpose of E;
c) provide dataset of portfolio weights w for each of the assets according to:
w∝E·Λ−1/2·1
where E is set of eigenvectors of the matrix V in columns and Λ is the diagonal eigenvalue matrix and 1 is column vector all of whose elements are equal to one, wherein the portfolio weightings w are provided as column vector from:
D
=
w
′
·
E
·
Λ
1
/
2
·
1
w
′
·
V
·
w
where w represents column vector containing portfolio weights, E is the eigenvector matrix and Λ is corresponding diagonal eigenvalue matrix of the matrix V for the set of assets, and D is an Objective Function relating to the set of assets; and
d) derive components of each of the risk vectors in the basis of unit independent risks by the corresponding row of the matrix product E·Λ1/2 relating to each of the assets; and
an output unit coupled to the processor unit and arranged to output the components of each of the risk vectors as risk vector dataset onto tangible computer-readable recording medium or display device.

### Claim 2 (depends on 1)

The apparatus of claim 1, wherein the input unit is arranged to receive risk information relating to the set of assets and provide the risk information to the processor unit.

### Claim 3 (depends on 2)

The apparatus of claim 2, wherein the input unit is arranged to receive the matrix as the risk information and to provide the matrix to the processor unit.

### Claim 4 (depends on 2)

The apparatus of claim 2, wherein the input unit is arranged to receive the risk vectors as the risk information and the processor unit is arranged to provide the matrix from the risk vectors.

### Claim 5 (depends on 1)

The apparatus of claim 1, wherein the output unit is arranged to output the portfolio weightings w relating to the set of assets.

### Claim 6 (depends on 1)

The apparatus of claim 1, wherein the processor unit is arranged to provide dataset of expected returns R for each of the assets as vector according to:
R∝E·Λ1/2·1
where E is set of eigenvectors of the matrix V in columns and Λ is the corresponding diagonal eigenvalue matrix and 1 is column vector all of whose elements are equal to one.

### Claim 7 (depends on 6)

The apparatus of claim 6, wherein the output unit is arranged to output the expected returns R relating to the set of assets.

### Claim 8 (depends on 1)

The apparatus of claim 1, wherein the output unit is arranged to output the value of the Objective Function as Diversification Measure relating to the set of assets.

### Claim 9 (depends on 1)

The apparatus of claim 1, wherein the input unit is arranged to receive one or more constraints relating to the set of assets, and the processor unit is arranged to optimise the Objective Function D by variation of the portfolio weights w subject to the constraints.

### Claim 10 (depends on 1)

The apparatus of claim 1, wherein the processor unit is further arranged to collect the portfolio weights w at plurality of time intervals.

### Claim 11 (depends on 10)

The apparatus of claim 10, wherein the processor unit is further arranged to provide Price Index based on the price information and the collected weights relating to the set of assets.

### Claim 12 (depends on 11)

The apparatus of claim 11, wherein the output unit is arranged to output the Price Index as Benchmark Index relating to the set of assets.

### Claim 13

A method of performing numerical modeling of investment risk relating to financial assets using computer apparatus, the method comprising:
receiving signals at an input unit of the computer apparatus, the signals containing data having information relating to set of financial assets;
providing, by processor unit of the computer apparatus, matrix V, wherein the matrix V comprises plurality of elements, wherein each of the elements represents relationship of risk related to respective pair of the financial assets, and wherein each element is given by scalar product of two risk vectors, such that each of the financial assets has an associated risk vector according to the elements of the risk relation matrix;
automatically decomposing, by the processor unit of the computer apparatus, the matrix V into eigenvectors and eigenvalues according to:
V=E·Λ·E′
wherein E is set of eigenvectors of the matrix V in columns, Λ is the corresponding diagonal eigenvalue matrix, and E′ is the transpose of E;
providing, by the processor unit of the computer apparatus, dataset of portfolio weights w for each of the assets according to:
w∝E·Λ−1/2·1
where E is set of eigenvectors of the matrix V in columns Λ is the diagonal eigenvalue matrix and 1 is column vector all of whose elements are equal to one, wherein the portfolio weightings w are provided as column vector from:
D
=
w
′
·
E
·
Λ
1
/
2
·
1
w
′
·
V
·
w
where w represents column vector containing portfolio weights, E is the eigenvector matrix and Λ is corresponding diagonal eigenvalue matrix of the matrix V for the set of assets, and D is an Objective Function relating to the set of assets; and
deriving, by the processor unit of the computer apparatus, components of each of the risk vectors in the basis of unit independent risks by the corresponding row of the matrix product E·Λ1/2 relating to each of the financial assets; and
outputting, by the computer apparatus, the components of each of the risk vectors as a risk vector data set corresponding to investment risks relating to respective financial assets onto a tangible computer-readable recording medium or display device configured for use by user to evaluate portfolio of at least portion of the financial assets.

### Claim 14

non-transitory tangible computer-readable recording medium having recorded thereon instructions to cause a computer apparatus to perform operations comprising:
receiving signals at an input unit of the computer apparatus, the signals containing data having information relating to set of assets;
providing, by a processor unit of the computer apparatus, matrix V, wherein the matrix V comprises plurality of elements, wherein each of the elements represents relationship of risk related to respective pair of the assets, and wherein each element is given by scalar product of two risk vectors, such that each of the assets has an associated risk vector according to the elements of the matrix;
decomposing, by the processor unit of the computer apparatus, the matrix V into eigenvectors and eigenvalues according to:
V=E·Λ·E′
wherein E is set of eigenvectors of the risk matrix V in columns, Λ is the corresponding diagonal eigenvalue matrix, and E′ is the transpose of E;
providing, by the processor unit of the computer apparatus, dataset of portfolio weights w for each of the assets according to:
w∝E·Λ−1/2·1
where E is set of eigenvectors of the matrix V in columns and Λ is the diagonal eigenvalue matrix and 1 is column vector all of whose elements are equal to one, wherein the portfolio weightings w are provided as column vector from:
D
=
w
′
·
E
·
Λ
1
/
2
·
1
w
′
·
V
·
w
where w represents column vector containing portfolio weights, E is the eigenvector matrix and Λ is corresponding diagonal eigenvalue matrix of the matrix V for the set of assets, and D is an Objective Function relating to the set of assets; and
deriving by the processor unit of the computer apparatus, components of each of the risk vectors in the basis of unit independent risks by the corresponding row of the matrix product E·Λ1/2 relating to each of the assets; and
outputting, by the computer apparatus, the components of each of the risk vectors as a risk vector dataset onto tangible computer-readable recording medium or display device.