WO2004086183A2 - Procedes et systemes d'optimisation de risque lie au portefeuille, a facteurs et a objectifs multiples, fondee sur une analyse - Google Patents

Procedes et systemes d'optimisation de risque lie au portefeuille, a facteurs et a objectifs multiples, fondee sur une analyse Download PDF

Info

Publication number
WO2004086183A2
WO2004086183A2 PCT/US2004/008448 US2004008448W WO2004086183A2 WO 2004086183 A2 WO2004086183 A2 WO 2004086183A2 US 2004008448 W US2004008448 W US 2004008448W WO 2004086183 A2 WO2004086183 A2 WO 2004086183A2
Authority
WO
WIPO (PCT)
Prior art keywords
risk
security
matrix
portfolio
nonlinear
Prior art date
Application number
PCT/US2004/008448
Other languages
English (en)
Other versions
WO2004086183A3 (fr
Inventor
Anindya Chakraborty
Kete Charles Chalermkraivuth
Michael Craig Clark
Richard Paul Messmer
Carol Lynn Kiaer
Srinivas Bollapragada
Original Assignee
Ge Financial Assurance Holdings, Inc.
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Priority claimed from US10/390,709 external-priority patent/US7593880B2/en
Priority claimed from US10/390,689 external-priority patent/US20040186804A1/en
Priority claimed from US10/390,710 external-priority patent/US7640201B2/en
Application filed by Ge Financial Assurance Holdings, Inc. filed Critical Ge Financial Assurance Holdings, Inc.
Publication of WO2004086183A2 publication Critical patent/WO2004086183A2/fr
Publication of WO2004086183A3 publication Critical patent/WO2004086183A3/fr

Links

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • 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
    • G06Q40/00—Finance; Insurance; Tax strategies; Processing of corporate or income taxes
    • G06Q40/06—Asset management; Financial planning or analysis
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N20/00—Machine learning
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N5/00—Computing arrangements using knowledge-based models
    • G06N5/01—Dynamic search techniques; Heuristics; Dynamic trees; Branch-and-bound

Definitions

  • the systems and methods of the invention relate to portfolio risk optimization.
  • the' portfolio optimization problem is defined by maximizing a return measure while minimizing a risk measure given a set of constraints.
  • classical Markowitz portfolio theory has been widely used as a foundation for portfolio optimization.
  • the framework has two major drawbacks that reduce its application to practical investment problems.
  • NLP nonlinear programming
  • the optimization problem has to be solved by a nonlinear programming (NLP) optimizer.
  • NLP nonlinear programming
  • general purpose nonlinear optimizers cannot generate an optimal solution within a reasonable amount of time.
  • problems with 30-50 asset classes reach the practical limit of a NLP optimizer.
  • Portfolio managers may use mean-variance optimization to determine broad asset allocations, but these solutions then must be further evaluated to determine an investment strategy that can be implemented, and this process generally leads to suboptimal solutions. With very large portfolio values, even small degradations in solution quality can have a significant impact on the calculated return.
  • the second drawback deals with the risk measure.
  • Variance measures the variation around mean. It is an accepted risk measure in a normal situation. Risk managers may also want to manage the portfolio to weather the occurrences of rare events with severe impact. Therefore, the downside risk, also called tail risk, has to be minimized.
  • the variance measure does not provide sufficient information about the tail risk when the distribution is not symmetrical about its mean (e.g., in a non-normal distribution situation). Asymmetric return distributions are common in practice. Therefore, a third measure, in addition to return and variance, is required to account for tail risk.
  • ALM asset-liability management
  • asset duration is approximately matched with liability duration to be within a pre-specif ⁇ ed target duration mismatch range.
  • Convexity is included in the analysis to improve accuracy.
  • key rate durations are used to capture the non-parallel movement of the yield curve.
  • the interactions between the risk factors require more integrated risk measures that provide the portfolio managers a better view of the portfolio total risk.
  • Experienced portfolio managers can manually adjust the constraints on risk sensitivities, i.e. key rate duration and convexity, to obtain a better risk/return portfolio by evaluating the risk measure after the optimization is completed. This iterative process may take approximately two weeks or more and yields suboptimal solutions.
  • Risk measures should provide additional information about the distribution of the portfolio values.
  • the portfolio managers want to manage the risk caused by rare events, i.e., downside risk.
  • a simulation technique is generally used to generate the distribution of the portfolio value based on a set of possible scenarios. The technique requires a significant amount of computation. Therefore, the simulation approach is mostly used to serve risk measurement rather than risk optimization purposes. Scenario-based optimization approach, which is based on the simulation technique, requires at least as much computational time as the simulation technique. Moreover, it is limited to only linear risk functions.
  • the invention addresses the above problems, as well as other problems, that are present in conventional techniques.
  • the invention provides a method for perfo ⁇ ning 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 ⁇ F's; defining the covariance matrix of ⁇ F; taking the Cholesky decomposition of the covariance matrix to generate a P transfomiation matrix; applying the P transfomiation matrix to Gamma to define a matrix Qk; detemiining the Eigenvalue decomposition of Qk to obtain a matrix of Eigenvectors N; and applying the matrix of Eigenvectors N and the P transformation matrix to evaluate the risk measures.
  • the invention provides 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 ⁇ F's; a fourth portion that defines the covariance matrix of -IF; a fifth portion that takes the Cholesky decomposition of the covariance matrix to generate a P transfomiation 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.
  • the invention provides 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 ⁇ F's; a fourth portion that defines the covariance matrix of ⁇ F; a fifth portion that takes the Cholesky decomposition of the covariance matrix to generate a P transfomiation 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 transfomiation matrix to evaluate the risk measures.
  • the invention provides a method for determining the allocation of securities in a portfolio, the method comprising: providing a collection of securities in a portfolio, each security being associated with associated attributes; providing risk factor data related to the portfolio; pooling the securities into a plurality of security clusters based on the attributes associated with each security and the risk factor data, each security being assigned to an security cluster, the pooling being performed using multivariate decision tree processing; processing the security clusters using a nonlinear programming optimizer to generate optimization results; and presenting the optimization results in a risk-return space for determination of a security allocation.
  • the invention provides a system for determining the allocation of securities in a portfolio, the system comprising: a security attribute portion, being at least one of an asset data portion and a liability data portion, containing security attributes that provides a collection of securities in a portfolio, each security being associated with associated attributes; a risk factor data portion that provides risk factor data related to the portfolio; a pooling portion that pools the securities into a plurality of security clusters based on the attributes associated with each security and the risk factor data, each security being assigned to an security cluster, the pooling being performed using multivariate decision tree processing; an optimization portion that processes the security clusters using a nonlinear programming optimizer to generate optimization results; a presentation , portion that presents the optimization results in a risk-return space for dete ⁇ riination of a security allocation.
  • the invention provides a computer readable medium for determining the allocation of securities in a portfolio, the computer readable medium comprising: a first portion that provides a collection of securities in a portfolio, each security being associated with associated attributes; a second portion that provides risk factor data related to the portfolio; a third portion that pools the securities into a plurality of security clusters based on the attributes associated with each security and the risk factor data, each security being assigned to a security cluster, the pooling being perfom ed using multivariate decision tree processing; a fifth portion that processes the security clusters using a nonlinear programming optimizer to generate optimization results; and a sixth portion that presents the optimization results in a risk-return space for detenriination of a security allocation.
  • the invention provides a method for determining an efficient frontier, which comprises a collection of security allocations in a portfolio, with multiple, conflicting objectives in a multi-factor portfolio problem, the method comprising providing a mathematical model of a relaxation of a problem; generating a sequence of additional constraints; and sequentially applying respective nonlinear risk functions to generate respective adjusted maximum return solutions to obtain an efficient frontier.
  • the invention provides a system for detemiining an efficient frontier, which comprises a collection of security allocations in a portfolio, with multiple, conflicting objectives in a multi-factor portfolio problem, the system comprising a model portion that provides a mathematical model of a relaxation of a problem; a constraint generation portion that generates a sequence of additional constraints; and a solution generation portion that sequentially applies respective nonlinear risk functions to generate respective adjusted maximum return solutions to obtain an efficient frontier.
  • the invention provides a computer readable medium for determining an efficient frontier, which comprises a collection of security allocations in a portfolio, with multiple, conflicting objectives in a multi-factor portfolio problem, the computer readable medium comprising: a first portion that provides a mathematical model of a relaxation of a problem; a second portion that generates a sequence of additional constraints; and a third portion that sequentially applies respective nonlinear risk functions 1 to generate respective adjusted maximum return solutions to obtain an efficient frontier.
  • Fig. 1 is a high level flowchart showing an optimization process in accordance with one embodiment of the invention
  • Fig. 2 is a flowchart showing the "problem simplification on risk measures" step of Fig. 1 in accordance with one embodiment of the invention
  • Fig. 3 is a flowchart showing the "nonlinear programming optimization using multivariate decision tree asset clusters" step of Fig. 1 in accordance with one embodiment of the invention
  • Fig. 4 is a flowchart showing the "sequential linear programming (SLP) optimization process" step of Fig. 1 in accordance with one embodiment of the invention
  • Fig. 5 is a diagram showing aspects of the initialization of the SLP process by solving a constrained relaxed LP problem
  • Fig. 6 is a diagram showing aspects of an iteration of the SLP process by calculating the tangent plane to the nonlinear risk function, adding a new constraint by adjusting the tangent plane by the step size ⁇ , and solving the resulting problem to obtain a new solution;
  • Fig. 7 is a diagram showing aspects of the calculated risk value versus return in accordance with one embodiment of the invention.
  • Fig. 8 is a diagram illustrating further aspects of an efficient frontier in three- dimensional space in accordance with one embodiment of the invention.
  • Fig. 9 is a block diagram showing a problem simplification system in accordance with one embodiment of the invention.
  • Fig. 10 is a block diagram showing a multivariate decision tree (MVDT) system in accordance with one embodiment of the invention.
  • MVDT multivariate decision tree
  • Fig. 11 is a block diagram showing a sequential linear programming system in accordance with one embodiment of the invention.
  • any te ⁇ ri in the singular may be inteipreted to be in the plural, and alternatively, any te ⁇ ri in the plural may be interpreted to be in the singular.
  • a "security” or “securities” means a financial instrument, which might illustratively be either investment security (e.g. bonds and/or stocks) or insurance products (e.g. a life insurance policy and/or guarantee investment contracts), for example, as well as a wide variety of other financial instruments.
  • the proposed analytical-based optimization approach achieves higher computational efficiency by utilizing analytical forms of risk measures in conjunction with mathematical transformations to simplify formulas for computation without losing accuracy, in accordance with one embodiment of the invention.
  • the risk measures may be developed from a multifactor risk framework. The optimization results are presented in a multidimensional risk-return space.
  • the portfolio risk optimization problem may be reformulated with additional risk measures and may be solved either by using (1) multivariate decision trees in conjunction with a nonlinear programming (NLP) optimizer; or (2) sequential linear programming (SLP) process.
  • NLP nonlinear programming
  • SLP sequential linear programming
  • Fig. 1 is a high-level flowchart showing aspects of an optimization process.
  • Fig. 1 shows that two different optimization processes (300, 400) may be used for solving a refo ⁇ riulated optimization problem.
  • One optimization approach uses multivariate decision tree asset clustering.
  • the other optimization approach uses sequential linear programming (SLP) approach.
  • Fig. 1 shows that a problem simplification process 200 may be perfo ⁇ ned in accordance with additional aspects of the invention.
  • the process of Fig. 1 starts with the analysis of risk factors. This can be done through risk factor data.
  • the data can be either historical data or risk factor scenarios provided by a scenario generation subprocess.
  • a valuation subprocess risk sensitivities and return measures of both assets and liabilities are evaluated.
  • the problem simplification method may be added to improve the computational efficiency.
  • the process of Fig. 1 starts with the data collection and processing of various types of data, as shown in step 130.
  • the input data might include risk factor data 100, asset data 110 and/or liability data 120.
  • the initial data collection and processing that is performed corresponds to the particular multifactor multi-objective portfolio risk optimization framework 10 that is applied in a particular situation.
  • the particular multifactor multi-objective portfolio risk optimization framework 10 that is chosen depends on the nature of the evaluation being performed, the nature of the various inputs (100, 110 and 120) and the particular outputs that are desired, for example.
  • the multifactor multi -objective portfolio risk optimization framework 10 that is chosen possesses a variety of model parameters 20 ⁇
  • the process includes the computation of risk sensitivities and risk evaluation in step 140. Further, the process involves the evaluation of returns in step 150.
  • the processing of both step 140 and step 150 provides the processed data to populate the model parameters 20' of the multifactor risk optimization framework 10. Accordingly, the model parameters 20' are populated in step 20.
  • the process of Fig. 1 may include step 200.
  • Step 200 provides for the problem simplification of risk measures, i.e., further to the computation of risk sensitivities and the evaluation of risk in step 140. Further details of step 200 are described below. It is appreciated that the processing of step 200 may be used in the situation where the risk measure is particularly complex, for example.
  • the process of Fig. 1 populates the model parameters of the multifactor risk optimization fi-amework 10, as noted above.
  • the inventive technology includes two different optimization approaches in the optimization step 30.
  • One optimization approach includes the use of nonlinear programming optimization using multivariate decision tree processing in step 300, i.e., so as to result in security clustering. This optimization approach is described in Section C below.
  • Another different optimization approach includes the use of sequential linear programming (SLP) of step 400.
  • SLP sequential linear programming
  • the SLP optimization process is described in Section D below. It is noted that either of the multivariate decision tree processing of step 300 or the SLP processing of step 400 may or may not be used in conjunction with the problem simplification on risk measures processing of step 200.
  • the risk factors are the representations, i.e., proxies, of the underlying risk exposures that affect the variation of the security value. Examples of risk exposures are interest rate, foreign exchange, prepayment, 'credit, and liability risk, for example. More than one factor can be used to represent an individual risk exposure. For example, key rates on the yield curve are used to capture the term structure risk exposure.
  • the change in the value of the security may be approximated by the Taylor series expansion to second order given by:
  • ⁇ Fj the change in value of the i risk factor, where i ranges from 1 to m
  • ⁇ F j the change in value of the/ risk factor, where j ranges from 1 to m
  • i ranges from 1 to m
  • risk sensitivities may be defined as the first and second-partial derivative of the security value with respect to the risk factors.
  • Equivalent measures for fixed- income securities are duration and convexity.
  • risk sensitivity measures There are variations of risk sensitivity measures. First, we can define as the percentage change of the security value with respect to change in the risk factor. Delta (or partial duration) and gamma (or partial convexity) can be written as:
  • Monetary delta and monetary gamma may be defined as the following:
  • Equation (1) may be re- ritten as,
  • w k rw F £ F j ( 4 )
  • the portfolio value and the change in the portfolio value is a summation of the security value and the change in the individual security value respectively.
  • the change in the portfolio value may then be written as:
  • W k the weight assigned to the security k
  • the first measure is the variance (or standard deviation).
  • the analytical fomi of the variance is given by:
  • VAR value at risk
  • VAR(q) ⁇ + w g ⁇ (12)
  • Portfolio optimization problems can often be expressed as:
  • w is a vector representing the fi-actions of the portfolio that are invested in each asset
  • g is a linear function, usually return measure
  • / is a vector of non-linear functions, typically risk measures
  • A is a set of linear inequality constraints
  • / is a set of linear equality constraints
  • Risk measure ⁇ or/,(w) ⁇ target ⁇ for jo 1, 2, ..., n andp ⁇ q;
  • the efficient frontier can be identified.
  • VAR is included as a measure of downside risk
  • the efficient frontier is a surface in a three-dimensional space, as shown in Fig. 8. Further risk measures may be added by adding yet further dimensions.
  • the efficient frontier might be two-dimensional, three dimensional, or more than three- dimensional, i.e., hypersurface.
  • the optimization problem that is formulated above cannot be solved by an LP optimizer any longer since the risk measures are nonlinear.
  • An NLP optimizer cannot be applied directly into practice due to computational limit.
  • the portfolio managers want to have more granular asset selection strategies, rather than broad asset allocation.
  • the NLP optimizer reaches the practical runtime limit at about 30-50 asset classes, and even then, iteration to determine the efficient frontier is prohibitive.
  • the inventive technology provides two different independent methods: (1) multivariate decision trees in conjunction with a nonlinear programming (NLP) optimizer to solve problem (P2), or (2) sequential linear programming (SLP) algorithm to solve. roblem (PI). Further, either of these methods may be used with an inventive risk measure "problem simplification" process.
  • Fig. 2 is a flowchart showing further details of the risk measure simplification process.
  • the process of Fig. 2 uses the risk factor data 100, the asset data 110, and the liability data 120, as described above. As shown in Fig. 2, the process starts in step 200 and passes to step 210.
  • the framework for an individual security 7c' can be either asset or liability
  • the value of the security is assumed a function of multiple risk factors:
  • V k f(F practiceF 2 ,...,F n; )
  • the main quantity of interest is the change in the portfolio value, which was described in Equation (5) as:
  • the weights W k are the unknown decision variables. Thus, one can see that from the optimization perspective the computational intensity of the problem depends both on l m the numbers of risk factors, as well as V, the numbers of individual securities.
  • VAR value at risk
  • the various measures of risk are actually functions of higher order moments of the main analytical form and the various measures of risk can involve computations of order 0(m 6 ) and beyond.
  • a simplification procedure to reduce the complexity and subsequently increase computational efficiency can have substantial benefits in processing time.
  • the objective here is to apply a set of nonsingular linear transformations, first on the covariance structure of the various risk factors (i.e., essentially, doing a Principal Component transfomiation) and then apply this transform on the matrix of gamma (i.e. convexity) and then perform an Eigenvalue decomposition that provides us with a diagonalized form.
  • gamma i.e. convexity
  • Eigenvalue decomposition that provides us with a diagonalized form.
  • step 220 the process defines Delta and Gamma and restates the problem in Delta-Gamma formulation as defined in Equations (2) and (3).
  • ⁇ and T we will use ⁇ and T to represent monetary delta and monetary gamma as defined in Equations (3). That is, define Delta and Gamma as:
  • step 225 the process defines the covariance matrix of ⁇ F.
  • the covariance matrix of ⁇ F is defined by:
  • step 230 Given the above, in step 230, we take the Cholesky
  • step 240 the "P" transfo ⁇ ri is applied on T k to obtain Qk-
  • step 250 the process detennines the Eigenvalue decomposition of Q k to get the matrix of eigenvectors N. That is, consider the Eigenvalue decomposition of Q :
  • N T Q k N r k * (20)
  • T where is T a new defined matrix of T, is now diagonal and N is the orthogonal Eigenvector matrix by orthogonality.
  • AV P 1 ⁇ V where ⁇ V is a vector
  • wi a vector of weight w /£ defined earlier.
  • V k in the denominator is incorporated into the ⁇ and ⁇ accordingly.
  • the cross- terms take the following shape:
  • step 270 of the process of Fig. 2 the stored transfonris may be used to evaluate all the risk measures.
  • the order of computational complexity for the Cholesky and Eigenvalue decompositions as described in Steps (1) and (3) above are quoted from Press et al. ,1992, (Press et al: Numerical Recipes in C ,Cambridge University Press,2nd Edn 1992), as follows:
  • the problem simplification method is perfo ⁇ xied using an illustrative problem simplification system 1300 as shown in Fig. 9.
  • the problem simplification system 1300 includes components to perfo ⁇ n the problem simplification process as described above.
  • the problem simplification system 1300 performs a risk measure simplification process through matrix manipulation.
  • the problem simplification system 1300 includes a first portion 1310 that defines the change in risk factors; a second portion 1320 that defines Delta and Gamma; a third portion 1330 that restates the change in risk factors in Delta-Gamma formulation, the Delta-Gamma fomiulation having the factors ⁇ Fs; and a fourth portion 1340 that defines the covariance matrix of ⁇ F.
  • the problem simplification system 1300 includes a fifth portion 1350 that takes the Cholesky decomposition of the covariance matrix to generate a P transfomiation matrix; a sixth portion 1360 that applies the P transfomiation matrix to Gamma to define a matrix Q k ; and a seventh portion 1370 that dete ⁇ nines the Eigenvalue decomposition of Qk to obtain a matrix of Eigenvectors N. Additionally, the problem simplification system 1300 includes an eighth portion 1380 that applies the matrix of Eigenvectors N and the P transfomiation matrix to evaluate the risk measures.
  • the problem simplification system 1300 includes a processing portion 1390 that coordinates the processing of the various components of the problem simplification system 1300, i.e., so as to perform the features of the invention, as described above.
  • a suitable interface 1392 i.e., such as a bus, may be used to connect the various components of the problem simplification system 1300.
  • the problem simplification system 1300 may be in the form of a general purpose computer and/or may be disposed on a computer readable medium, for example, so as to be accessed and implemented on a general purpose computer, for example.
  • Fig. 3 shows step 300 of Fig. 1 in further detail.
  • the challenge here is to group the set of securities in such a fashion that each group be as homogeneous as possible with respect to the risk function being measured.
  • one embodiment of the inventive technology uses multiple target multivariate decision trees to arrive at logical groups of the securities such that pooled measures of these can be used as proxies to original securities to serve as inputs to the NLP solver.
  • a "volatility target” is considered.
  • Va ⁇ ance( ⁇ Vk ) Va ⁇ ance( ⁇ Vk )
  • multivariate decision trees are extensions of the popular univariate classification and regression tree approach, but have more than one response variable.
  • the application of this approach is pertinent to cases where the responses themselves co-vary with each other and hence cannot be treated separately.
  • the inventive technology provides a variation from known multivariate decision trees processing.
  • the main change provided is to devise a matrix analog of the split criterion on which nodes are split at each level.
  • a matrix analog of the split criterion on which nodes are split at each level.
  • we mention one commonly used analog which is based on deviance.
  • N in the tree deviance is defined by Larsen et al. (2002) (Larsen, David R and Speckman, Paul L, "Multivariate Regression Trees for, analysis of abundance data", 2002) as:
  • a node N is a subset of the indices ⁇ 1, ..., n ⁇ .
  • the deviance of a node N is defined as
  • Vi is proportional to Variance ⁇ / ), .
  • V t V is a constant matrix independent of i
  • V For all practical purposes we choose V; to be equal and estimate it with the sample covariance matrix, which provides us with the known classical matrix form of Least Squares Error.
  • Fig. 3 is a flowchart showing the multivariate decision tree process in accordance with one embodiment of the invention.
  • the process starts in step 300 and passes to step 330.
  • step 330 the process incorporates the "problem simplification" results from step 200 as discussed above, in accordance with one embodiment of the invention. However, it is appreciated that the problem simplification step 200 may not be needed depending on the number of asset classes, for example.
  • step 340 the process computes the variance( ⁇ V k ) of a Volatility Target.
  • step 350 the process uses the Volatility and Asset Yields as two concurrent target variables.
  • the process runs a
  • MVDT algorithm to create clusters as homogeneous as possible based on these two concurrent target variables.
  • step 360 the process computes the pooled. measures for each group, i.e., on all variables which form an input to the NLP solver, which is used.
  • step 370 the process implements an NLP solver to compute optimum results, as is desired.
  • Fig. 10 is a block diagram showing a multivariate decision tree system 1340.
  • the multivariate decision tree system 1340 includes a pooling portion 1310, an optimization portion 1320, and a presentation portion 1330.
  • the multivariate decision tree system 1340 may assist in determining the allocation of securities in a portfolio, as described above.
  • the system may input asset data 110 that provides a collection of securities in a portfolio, each security being associated with associated attributes. Further, the multivariate decision tree system 1340 may input risk factor data 100 that provides risk factor data related to. the portfolio.
  • the pooling portion 1310 pools the securities into a plurality of security clusters based on the attributes associated with each security and the risk factor data, each security being assigned to a security cluster. The pooling is perfo ⁇ ried using multivariate decision tree processing.
  • the optimization portion 1320 processes the security clusters using a nonlinear programming optimizer to generate optimization results. Further, the presentation portion 1330 presents the optimization results in a risk-retum space for determination of a security allocation in a desired manner, i.e., such as on a monitor.
  • the multivariate decision tree system 1340 includes a processing portion 1340 that coordinates the processing of the various components of the multivariate decision tree system 1340, i.e., so as to perfo ⁇ n the features of the invention, as described above.
  • a suitable interface 1342 i.e., such as a bus, may be used to connect the various components of the multivariate decision tree system 1340.
  • the multivariate -decision tree system 1340 may be in the form of a general purpose computer and/or may be disposed on a computer readable medium, for example, so as to be accessed and implemented on a general purpose computer, for example.
  • a sequential linear programming (SLP) technique may be used in place of the multivariate decision tree processing.
  • SLP sequential linear programming
  • non-linear functions there are typically non-linear functions/ These non-linear functions are typically be related to risk, but could also arise from other sources.
  • the technique provides for the nonlinear functions /to be transforaied into constraints. In general, it should be appreciated that non-linear constraints would result in an intractable problem.
  • the invention provides for a sequence of proxy constraints which are linear. These constraints are used to obtain the efficient frontier between the multiple objectives of the problem.
  • Sequential linear programming has been used for problems with nonlinear, but convex constraints, by first relaxing the problem and eliminating the nonlinear constraints, and then successively building a set of linear constraints that approximate each nonlinear constraint in the region of the optimal solutions along the efficient frontier.
  • Fig. 4 is a flowchart showing step 400 in further detail, in accordance with one embodiment of the invention. As shown in Fig. 4, the method starts in step 400 and passes to step 410.
  • step 410 the process fo ⁇ rulates a relaxed linear programming problem that does not include any of the nonlinear measures. This problem is entered into a set of candidate problems in step 410, where, initially, it is the only candidate problem.
  • step 415 the process dete ⁇ nines whether the candidate list is empty. If the candidate list is empty in step 415, the process passes to step 417 and the process ends. If the candidate list is not empty in step 415, then the process passes to step 420. In step 420, any problem is randomly selected from the . candidate list, and is designated the current problem.
  • Fig. 5 illustrates the first such point, w 0 , i.e., the point 502.
  • Fig. 5 also shows a plurality of linear constraints that go to fomi a feasible region (512, 514, 516, 518, and 520). If the problem is not feasible, then another candidate problem must be selected. That is, the process passes from step 435 back to step 415. For example, the problem will not be feasible if the plurality of constraints do not form a feasible region, which could eventually occur if it is attempted to reduce a risk measure too far.
  • step 440 the nonlinear measures are evaluated at the optimal point, yielding a point on the efficient frontier in the risk/return space. For example, several such points are shown in Fig. 7.
  • step 445 a determination is made whether it is desired to improve any of the risk measures. For example, in accordance with one embodiment of the invention, a desired lower bound could be provided to the process. Alternatively, the SLP process could continue to improve each risk measure until the problems fail at step 435. If yes in step 445, then the. process passes to step 450.
  • step 450 for each risk measure to be improved, the desired granularity of the efficient frontier is used to determine a step size, and the process uses the gradient of the nonlinear measure, together with the step size, to add a constraint to the current problem, creating a new problem, which is added to the candidate list.
  • the step size one could simply use a small value that is granular relative to the nonlinear function value at the current solution. For example, the current value of the nonlinear function is 10,000.
  • the step size can be determined at 10. This simple method would require a large number of iterations, which is computationally intensive. One can improve the computational efficiency with step size determination methods.
  • the improved method calculates the distance between the current nonlinear function value (i.el risk level) and the target value (i.e. minimum risk value).
  • the preliminary step size is given by the distance divided by the desired number of steps. Then, the preliminary step size is adjusted with infonnation obtained by testing the terrain around the current solution, in accordance with one embodiment of the invention.
  • Fig. 6 shows aspects of an iteration of the SLP process of Fig. 4.
  • Fig. 6 shows that the feasible region 602 of the relaxed linear program lies below and to the left of the constraints.
  • the curve 604 represents the contour of the nonlinear risk measure that passes through the optimal solution 606 to the relaxed LP.
  • the process detenriines a tangent plane 608 at the optimal solution, and uses the tangent plane 610 as a new constraint, i.e., after shifting the tangent plane from line 608. That is, the line 610 is the new constraint that is added, parallel to the plane 608, but moved a distance ⁇ toward a lower-risk solution.
  • the SLP process as described above will define the efficient frontier.
  • risk contours are likely to be convex in the range of interest.
  • the step size is sufficiently small, one can easily check to see if the nonlinear function is convex in the region of interest.
  • the risk measures are evaluated for the new solution, - ⁇ , then the function is not convex. In this case, it may be useful to reduce step size.
  • Fig. 7 is a graph illustrating solutions provided by the SLP process in a two dimensional space by solving a trade-off problem between one return and one risk measure.
  • Fig. 8 is a graph showing a three dimensional efficient frontier provided by the SLP process described above.
  • two risks are included in the analysis, i.e., risk 1 and risk 2.
  • the two risks are plotted against retum. It should of course be appreciated that more than two risks may well be used, but that such does not readily lend itself to graphical representation. However, such additional risks may of course be shown mathematically so as to result in an efficient frontier, as described above.
  • Fig. 11 is a block diagram showing a sequential linear programming system 1440.
  • the sequential linear programming system 1440 determines an efficient frontier, which comprises a collection of allocations in a portfolio, in a situation with multiple, conflicting objectives in a multi-factor portfolio problem.
  • the sequential linear programming system 1440 includes a model portion 1410, a constraint generation portion 1420 and a solution generation portion 1430, which may be used to practice the invention as described above.
  • the model portion 1410 may provide a mathematical model of a relaxation of a problem.
  • the constraint generation portion 1420 generates a sequence of additional constraints.
  • the solution generation portion 1430 sequentially applies respective nonlinear risk functions to generate respective adjusted maximum return solutions to obtain an efficient frontier, in accordance with one embodiment of the invention.
  • the sequential linear programming system 1440 includes a processing portion 1440 that coordinates the processing of the various components of the sequential linear programming system 1440, i.e., so as to perfomi the features of the invention, as described above.
  • a suitable interface 1442 i.e., such as a bus, may be used to connect the various components of the sequential linear programming system 1440.
  • the sequential linear programming system 1440 may be in the form of a general purpose computer and/or may be disposed on a computer readable medium, for example, so as to be accessed and implemented on a general purpose computer, for example.
  • the analytical-based multiple risk factor optimization approach uses analytical fomis for the calculation of risk measures.
  • the proposed approach uses not only risk measures that capture risk caused by the variation of the portfolio value around mean, measured by the variance or standard deviation, but also additional information about the distribution of the portfolio value.
  • Skewness and Value at Risk (VAR) are additional risk measures that can be used to control the portfolio downside risk.
  • the SLP algorithm overcomes the computational hurdle by solving the nonlinear problem with an LP optimizer.
  • the SLP algorithm efficiently finds optimal (or ⁇ -optimal) solutions to a class of nonlinear optimization problems with minimal computational effort. In the case of convexity, optimality is guaranteed. In the case of non-convexity, we provide a method for ensuring a good, fast solution.
  • the analytical- based optimization with the SLP algorithm provides a breakthrough for solving ALM optimization problems.
  • the proposed approach overcomes the hurdle faced by the classical Markowitz portfolio optimization and traditional ALM approaches.
  • Typical ALM portfolio management requires solving the optimization problems at the asset rather than asset class levels. This kind of optimization problem exceeds the practical limit of a NLP optimizer.
  • the SLP algorithm provides a better solution than the methods currently in use.
  • Today a traditional optimization approach is widely used for solving ALM optimization problems.
  • the approach solves for an optimal solution by controlling mismatches between asset- and liability-duration and convexity.
  • a trial and e ⁇ or method is used to obtain an improved solution by adjusting the constraints on key rate duration mismatches.
  • this approach yields a sub-optimal solution since the portfolio manager losses sight of the portfolio total risk.
  • portfolio optimization can only be done at the coarsest possible level of granulation, or must rely on linear estimates of portfolio risk, which are incomplete.
  • Solution approaches are computationally intensive, and generally still rely heavily on the experience of the users to tweak them into usable form.
  • the analytical-based optimizer provides significant improvement on speed over the simulation approach.
  • the multi-objective optimization based on multiple risk measures provides efficient portfolios in a three dimensional space.
  • a second risk measure for example Value at Risk (VaR)
  • VaR Value at Risk
  • the new chart provides portfolio managers a view on the surface of efficient frontier that results from the trade-off between a return measure and two risk measures. In essence, it provides also a trade-off between two risk measures. In other words, a portfolio manager who wants to minimize the tail risk may have to assume more variance risk.
  • Figs. 9-11 show illustrative operating systems. It is appreciated that the systems of the invention or portions of the systems of the invention may be in the form of a "processing machine," such as a general purpose computer, for example.
  • processing machine is to be understood to include at least one processor that uses at least one memory.
  • the at least one memory stores a set of instructions.
  • the instructions may be either permanently or temporarily stored in the memory or memories of the processing machine.
  • the processor executes the instructions that are stored in the memory or memories in order to process data.
  • the set of instructions may include various instructions that perfomi a particular task or tasks, such as those tasks described above in the flowcharts. Such a set of instructions for performing a particular task may be characterized as a program, software program, or simply software.
  • the processing machine executes the instructions that are stored in the memory or memories to process data.
  • This processing of data may be in response to commands by a user or users of the processing machine, in response to previous processing, in response to a request by another processing machine and/or any other input, for example.
  • the processing machine used to implement the invention may be a general purpose computer.
  • the processing machine described above may also utilize any of a wide variety of other technologies including a special purpose computer, a computer system including a microcomputer, mini-computer or mainframe for example, a programmed microprocessor, a micro-controller, a peripheral integrated 1 circuit element, a CSIC (Customer Specific Integrated Circuit) or ASIC (Application Specific Integrated Circuit) or other integrated circuit, a logic circuit, a digital signal processor, a programmable logic device such as a FPGA, PLD, PLA or PAL, or any other device or airangement of devices that is capable of implementing the steps of the processes of the various embodiments of the inventions.
  • each of the processors and/or the memories of the processing machine may be located in geographically distinct locations and connected so as to communicate in any suitable manner.
  • each of the processor and/or the memory may be composed of different physical pieces of equipment. Accordingly, it is not necessary that the processor be one single piece of equipment in one location and that the memory be another single piece of equipment in another location. That is, it is contemplated that the processor may be two pieces of equipment in two different physical locations. The two distinct pieces of equipment may be connected in any suitable manner. Additionally, the memory may include two or more portions of memory in two or more physical locations.
  • processing as described above is performed by various components and various memories.
  • the processing performed by two distinct components as described above may, in accordance with a further embodiment of the invention, be perfomied by a single component.
  • the processing performed by one distinct component as described above may be perfomied by two distinct components.
  • the memory storage perfomied by two distinct memory portions as described above may, in accordance with a further embodiment of the invention, be performed by a single .memory portion.
  • the memory storage performed by one distinct memory portion as described above may be performed by two memory portions.
  • various technologies may be used to provide communication between the various processors and/or memories, as well as to allow the processors and/or the memories of the invention to communicate with any other entity; i.e., so as to obtain further instructions or to access and use remote memory stores, for example.
  • Such technologies used to provide such communication might include a network, the Internet, Intranet, Extranet, LAN, an Ethernet, or any client server system that provides communication, for example.
  • Such communications technologies may use any suitable protocol such as TCP/IP, UDP, or OSI, for example.
  • a set of instructions is used in the processing of the invention.
  • the set of instructions may be in the fom of a program or software.
  • the software may be in the form of system software or application software, for example.
  • the software might also be in the form of a collection of separate programs, a program module within a larger program, or a portion of a program module, for example
  • the software used might also include modular programming in the fbim of object oriented programming.
  • the software tells the processing machine what to do with the data being processed.
  • the instructions or set of instructions used in the implementation and operation of the invention may be in a suitable fonn such that the processing machine may read the instructions.
  • the instructions that fonn a program may be in the form of a suitable programming language, which is converted to machine language or object code to allow the processor or processors to read the instmctions. That is, written lines of programming code or source code, in a particular programming language, are converted to machine language using a compiler, assembler or interpreter.
  • the machine language is binary coded machine instmctions that are specific to a particular type of processing machine, i.e., to a particular type of computer, for example. The computer understands the machine language.
  • any suitable programming language may be used in accordance with the various embodiments of the invention.
  • the programming language used may include assembly language, Ada, APL, Basic, C, C++, COBOL, dBase, Forth, Fortran, Java, Modula-2, Pascal, Prolog, REXX, Visual Basic, and/or JavaScript, for example.
  • assembly language Ada
  • APL APL
  • Basic Basic
  • C C
  • C++ C++
  • COBOL COBOL
  • dBase Forth
  • Fortran Fortran
  • Java Modula-2
  • Pascal Pascal
  • Prolog Prolog
  • REXX REXX
  • Visual Basic Visual Basic
  • JavaScript JavaScript
  • the instmctions and/or data used in the practice of the invention may utilize any compression or encryption technique or algorithm, as may be desired.
  • An encryption module might be used to encrypt data.
  • files or other data may be decrypted using a suitable decryption module, for example.
  • the invention may illustratively be embodied in the fonn of a processing machine, including a computer or computer system, for example, that includes at least one memory.
  • a processing machine including a computer or computer system, for example, that includes at least one memory.
  • the set of instructions i.e., the software for example, that enables the computer operating system to perform the operations described above may be contained on any of a wide variety of media or medium, as desired.
  • the data that is processed by the set of instructions might also be contained on any of a wide variety of media or medium. That is, the particular medium, i.e., the memory in or used by the processing machine, utilized to hold the set of instmctions and/or the data used in the invention may take on any of a variety of physical forms or transmissions, for example.
  • the medium may be in the form of paper, paper transparencies, a compact disk, a DVD, an integrated circuit, a hard disk, a floppy disk, an optical disk, a magnetic tape, a RAM, a ROM, a PROM, a EPROM, a wire, a cable, a fiber, communications channel, a satellite transmissions or other remote transmission, as well as any other medium or source of data that may be read by the processors of the invention.
  • the memory or memories used in the processing machine that implements the invention may be in any of a wide variety of fomis to allow the memory to hold instmctions, data, or other info ⁇ nation, as is desired.
  • the memory might be in the form of a database to hold data.
  • the database might use any desired anangement of files such as a flat file anangement or a relational database airangement, for example.
  • a user interface includes any hardware, software, or combination of hardware and software used by the processing machine that allows a user to interact with the processing machine.
  • a user interface may be in the fonn of a dialogue screen for example.
  • a user interface may also include any of a mouse, touch screen, keyboard, voice reader, voice recognizer, dialogue screen, menu box, list, checkbox, toggle switch, a pushbutton or any other device that allows a user to receive information regarding the operation of the processing machine as it processes a set of instmctions and/or provide the processing machine with information.
  • the user interface is any device that provides communication between a user and a processing machine.
  • the information provided by the user to the processing machine through the user interface may be in the fomi of a command, a selection of data, or some other input, for example.
  • a user interface is utilized by the processing machine that performs a set of instmctions such that the processing machine processes data for a user.
  • the user interface is typically used by the processing machine for interacting with a user either to convey information or receive info ⁇ nation from the user.
  • the user interface of the invention might interact, i.e., convey and receive information, with another processing machine, rather than a human user. Accordingly, the other processing machine might be characterized as a user.
  • a user interface utilized in the system and method of the , invention may interact partially with another processing machine or .processing machines, while also interacting partially with a human user.

Landscapes

  • Engineering & Computer Science (AREA)
  • Theoretical Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Business, Economics & Management (AREA)
  • Software Systems (AREA)
  • Physics & Mathematics (AREA)
  • Development Economics (AREA)
  • Finance (AREA)
  • Computing Systems (AREA)
  • Mathematical Physics (AREA)
  • Evolutionary Computation (AREA)
  • Data Mining & Analysis (AREA)
  • General Engineering & Computer Science (AREA)
  • Accounting & Taxation (AREA)
  • Artificial Intelligence (AREA)
  • Game Theory and Decision Science (AREA)
  • Strategic Management (AREA)
  • Computational Linguistics (AREA)
  • Human Resources & Organizations (AREA)
  • Operations Research (AREA)
  • Economics (AREA)
  • Marketing (AREA)
  • Entrepreneurship & Innovation (AREA)
  • Technology Law (AREA)
  • General Business, Economics & Management (AREA)
  • Computer Vision & Pattern Recognition (AREA)
  • Medical Informatics (AREA)
  • Financial Or Insurance-Related Operations Such As Payment And Settlement (AREA)
  • Management, Administration, Business Operations System, And Electronic Commerce (AREA)

Abstract

L'invention concerne des systèmes et des procédés permettant de mettre en oeuvre un processus de simplification de mesure de risque au moyen d'une manipulation de matrice. Le procédé de l'invention consiste: à définir le changement dans les facteurs de risque; à définir les sensibilités de risque lié au portefeuille sous l'appellation Delta et Gamma; à redéfinir le changement dans les facteurs de risque dans la formulation Delta Gamma, cette dernière présentant les facteurs ?F; à définir la matrice de covariance de ?F; à utiliser la décomposition de Cholesky de la matrice de covariance afin de générer une matrice de transformation P; à appliquer la matrice de transformation P à Gamma afin de définir une matrice Qk; à déterminer la décomposition de valeur propre de Qk afin d'obtenir une matrice de vecteurs propres N; et à appliquer la matrice de vecteurs propres N et la matrice de transformation P afin d'évaluer les mesures de risque. L'invention concerne également des systèmes et des procédés permettant de déterminer l'attribution de titres dans un portefeuille. Le procédé de l'invention consiste: à fournir un ensemble de titres dans un portefeuille, chaque titre étant associé à des attributs associés; à fournir des données de facteur de risque associées au portefeuille; à mettre en commun les titres sous forme d'une pluralité de groupements de titres en fonction des attributs associés à chaque titre et des données de facteur de risque, chaque titre étant attribué à un groupement de titres, la mise en commun étant effectuée au moyen d'un traitement par arbre décisionnel à variables multiples; à traiter les groupements de titres au moyen d'un optimiseur de programmation non linéaire afin de générer des résultats d'optimisation; et à présenter les résultats d'optimisation dans un espace retour de risque afin de déterminer une attribution de titre. L'invention concerne en outre des systèmes et des procédés permettant de déterminer une frontière efficace. Le procédé de l'invention met en oeuvre un ensemble d'attributions de titres dans un portefeuille, et des objectifs conflictuels multiples dans un problème de portefeuille à facteurs multiples. Ledit procédé consiste: à utiliser un modèle mathématique de la relaxation d'un problème (410); à générer une séquence de contraintes supplémentaires (440); et à appliquer de manière séquentielle des fonctions de risques non linéaires respectives afin de générer des solutions de retour maximum ajustées respectives (606) de manière à obtenir une frontière efficace (450).
PCT/US2004/008448 2003-03-19 2004-03-19 Procedes et systemes d'optimisation de risque lie au portefeuille, a facteurs et a objectifs multiples, fondee sur une analyse WO2004086183A2 (fr)

Applications Claiming Priority (6)

Application Number Priority Date Filing Date Title
US10/390,710 2003-03-19
US10/390,709 2003-03-19
US10/390,709 US7593880B2 (en) 2003-03-19 2003-03-19 Methods and systems for analytical-based multifactor multiobjective portfolio risk optimization
US10/390,689 2003-03-19
US10/390,689 US20040186804A1 (en) 2003-03-19 2003-03-19 Methods and systems for analytical-based multifactor multiobjective portfolio risk optimization
US10/390,710 US7640201B2 (en) 2003-03-19 2003-03-19 Methods and systems for analytical-based multifactor Multiobjective portfolio risk optimization

Publications (2)

Publication Number Publication Date
WO2004086183A2 true WO2004086183A2 (fr) 2004-10-07
WO2004086183A3 WO2004086183A3 (fr) 2005-06-09

Family

ID=33102134

Family Applications (1)

Application Number Title Priority Date Filing Date
PCT/US2004/008448 WO2004086183A2 (fr) 2003-03-19 2004-03-19 Procedes et systemes d'optimisation de risque lie au portefeuille, a facteurs et a objectifs multiples, fondee sur une analyse

Country Status (1)

Country Link
WO (1) WO2004086183A2 (fr)

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US8187081B2 (en) 2008-06-25 2012-05-29 Aristocrat Technologies Australia Pty Limited Gaming system and a method of gaming

Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US6021397A (en) * 1997-12-02 2000-02-01 Financial Engines, Inc. Financial advisory system
US6173276B1 (en) * 1997-08-21 2001-01-09 Scicomp, Inc. System and method for financial instrument modeling and valuation
US6278981B1 (en) * 1997-05-29 2001-08-21 Algorithmics International Corporation Computer-implemented method and apparatus for portfolio compression
US6393409B2 (en) * 1997-10-31 2002-05-21 Morgan Stanley Dean Witter & Co. Computer method and apparatus for optimizing portfolios of multiple participants
US6473084B1 (en) * 1999-09-08 2002-10-29 C4Cast.Com, Inc. Prediction input
US6546375B1 (en) * 1999-09-21 2003-04-08 Johns Hopkins University Apparatus and method of pricing financial derivatives

Patent Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US6278981B1 (en) * 1997-05-29 2001-08-21 Algorithmics International Corporation Computer-implemented method and apparatus for portfolio compression
US6173276B1 (en) * 1997-08-21 2001-01-09 Scicomp, Inc. System and method for financial instrument modeling and valuation
US6393409B2 (en) * 1997-10-31 2002-05-21 Morgan Stanley Dean Witter & Co. Computer method and apparatus for optimizing portfolios of multiple participants
US6021397A (en) * 1997-12-02 2000-02-01 Financial Engines, Inc. Financial advisory system
US6473084B1 (en) * 1999-09-08 2002-10-29 C4Cast.Com, Inc. Prediction input
US6546375B1 (en) * 1999-09-21 2003-04-08 Johns Hopkins University Apparatus and method of pricing financial derivatives

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US8187081B2 (en) 2008-06-25 2012-05-29 Aristocrat Technologies Australia Pty Limited Gaming system and a method of gaming
US9330522B2 (en) 2008-06-25 2016-05-03 Aristocrat Technologies Australia Pty Limited Gaming system and a method of gaming
US9767654B2 (en) 2008-06-25 2017-09-19 Aristocrat Technologies Australia Pty Limited Gaming system and a method of gaming

Also Published As

Publication number Publication date
WO2004086183A3 (fr) 2005-06-09

Similar Documents

Publication Publication Date Title
US7593880B2 (en) Methods and systems for analytical-based multifactor multiobjective portfolio risk optimization
US7640201B2 (en) Methods and systems for analytical-based multifactor Multiobjective portfolio risk optimization
US20040186804A1 (en) Methods and systems for analytical-based multifactor multiobjective portfolio risk optimization
Pages et al. An optimal Markovian quantization algorithm for multi-dimensional stochastic control problems
Briani et al. Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory
Hochreiter et al. Financial scenario generation for stochastic multi-stage decision processes as facility location problems
Liu et al. Practical deep reinforcement learning approach for stock trading
Fécamp et al. Risk management with machine-learning-based algorithms
EP3971814A1 (fr) Programmation quadratique partiellement en nombres entiers quantiques et interface utilisateur graphique pour optimisation de portefeuille
Lopes et al. Parsimony inducing priors for large scale state-space models
Jayasekara et al. On convex multiobjective programs with application to portfolio optimization
Panda et al. Long term electrical load forecasting: an empirical study across techniques and domains
Chen et al. Adaptive model design for Markov decision process
Gagliardini et al. Extracting statistical factors when betas are time-varying
WO2004086183A2 (fr) Procedes et systemes d'optimisation de risque lie au portefeuille, a facteurs et a objectifs multiples, fondee sur une analyse
Bastürk et al. A neural network with shared dynamics for multi-step prediction of value-at-risk and volatility
Wang et al. Applying Least Squares Support Vector Machines to Mean‐Variance Portfolio Analysis
Okhrin et al. Estimation of optimal portfolio weights
JP2004534305A (ja) 整合性のあるリスク予測と集積因子モデルを作成する方法および装置
Layfield et al. Explainability and Interpretability in Decision Trees and Agent based Modelling when Approximating Financial Time series. A matter of balance with performance
Desmoulins‐Lebeault Gram–Charlier Expansions Portfolio Selection in Non‐Gaussian Universes
Vijayalakshmi Pai et al. Integrated Metaheuristic Optimization Of 130–30 Investment‐Strategy‐Based Long–Short Portfolios
Fan Uncertainty Quantification on Financial Valuation Model
de Melo et al. A genetic algorithm for investment tracking with stochastic model predictive control
Canestrelli et al. Inquiries on the applications of multidimensional stochastic processes to financial investments

Legal Events

Date Code Title Description
AK Designated states

Kind code of ref document: A2

Designated state(s): AE AG AL AM AT AU AZ BA BB BG BR BW BY BZ CA CH CN CO CR CU CZ DE DK DM DZ EC EE EG ES FI GB GD GE GH GM HR HU ID IL IN IS JP KE KG KP KR KZ LC LK LR LS LT LU LV MA MD MG MK MN MW MX MZ NA NI NO NZ OM PG PH PL PT RO RU SC SD SE SG SK SL SY TJ TM TN TR TT TZ UA UG US UZ VC VN YU ZA ZM ZW

AL Designated countries for regional patents

Kind code of ref document: A2

Designated state(s): BW GH GM KE LS MW MZ SD SL SZ TZ UG ZM ZW AM AZ BY KG KZ MD RU TJ TM AT BE BG CH CY CZ DE DK EE ES FI FR GB GR HU IE IT LU MC NL PL PT RO SE SI SK TR BF BJ CF CG CI CM GA GN GQ GW ML MR NE SN TD TG

121 Ep: the epo has been informed by wipo that ep was designated in this application
122 Ep: pct application non-entry in european phase