JP2010524079A - Method and system for multiple portfolio optimization - Google Patents

Abstract

  A method and system for optimizing a plurality of portfolios, each portfolio including one or more interests in one or more tradable assets and receiving asset data associated with the plurality of portfolios And receiving an optimization constraint that includes at least one global constraint that defines the constraints that apply across the aggregation of multiple portfolios, and one or more objectives that are applied to the individual portfolio during the optimization A step of aggregating portfolio data optimized to generate aggregate optimized asset data; determining whether the aggregate optimized asset data satisfies a global constraint; and the at least one global constraint comprises: The optimized asset data only if it is satisfied. It may include a step of outputting.

Description

  The present invention relates to a method and system for optimizing multiple portfolios of tangible or intangible assets. More particularly, the present invention relates to a method and system for optimizing multiple portfolios while applying portfolio constraints.

This application is a continuation-in-part of U.S. Patent Application No. 11 / 730,750 entitled “Method and System For Multiple Portfolio Optimization” filed on April 3, 2007. , A continuation-in-part of U.S. Patent Application No. 10 / 640,630, filed on August 14, 2003, which is a continuation-in-part of U.S. Provisional Application No. 60 / 448,147 Claim priority. This application also claims the priority of US Provisional Application No. 60 / 907,525, filed April 5, 2007. The entire contents of each of the above applications are incorporated herein by reference.

  Asset managers, such as stocks and / or other asset portfolios, are often defined in terms of rate of return diversification, eg, historically or adjusted using known portfolio management techniques Strive to maximize return on overall investment while maintaining risk at a given level.

  With the original work of Harry Markowitz in 1952, mean-variance optimization has become a common tool for portfolio selection. A mean-variance efficient portfolio can be constructed by an optimizer that takes input from an appropriate risk model and alpha model. Such a portfolio helps to guarantee a higher expected return (eg, tax-excluded, various constraints) for a given level of risk.

  Risk is at the heart of modern portfolio theory. The standard deviation (eg, variance) of an asset's rate of return is often used to measure the risk associated with holding that asset. However, there may be other suitable measures or measures that are more appropriate than the standard deviation of the rate of return. The general definition of risk is the variability or variability in the rate of return of a single asset or portfolio, usually measured by standard deviation. ITG, the assignee of the present invention, has developed a set of risk models for portfolio managers and traders to measure, analyze, and manage risk in a rapidly changing market. (See, eg, application 10 / 640,630). These models can be used in combination with a portfolio optimizer, such as those described herein, for example, to generate, among other things, an average-dispersed efficient portfolio.

  According to modern portfolio theory, for any portfolio of assets (e.g., stocks and / or other assets), the portfolio assets that produce the maximum possible expected return at any given level of portfolio risk. There is an efficient frontier that represents various weighted combinations.

  In addition, the ratio of rate of return to volatility that can be useful in comparing two portfolios from a risk-adjusted rate of return is the Sharpe Ratio. This ratio was developed by Nobel Laureate William Sharpe. Generally, the higher the sharp ratio, the better. A high sharp ratio implies that the portfolio or asset (eg, equities) has achieved a good rate of return per risk unit. The sharp ratio can be used to compare different assets or different portfolios. Often, the sharp ratio is calculated by first subtracting the risk free rate from the portfolio rate of return and then dividing by the portfolio standard deviation. The historical average rate of return for an asset or portfolio can often lead to misrecognition and should not be considered alone when selecting an asset or comparing portfolio performance. The Sharpe ratio takes into account the potential impact of revenue variability on expected returns and enables an objective comparison of assets or portfolios that can vary significantly in terms of rate of return.

  By combining portfolios with a single risk factor, Sharp has simplified Markowitz's work. Sharp has developed elaborate inferences that have come to be known as the heretical idea of investment risk and reward-the Capital Asset Pricing Model (CAPM). According to CAPM, every investment carries two different risks. One is the risk of being in the market, what Sharp called “systematic risk”. This risk, also called “beta”, can be reduced by decentralization. The other risk, “non-systematic risk”, is specific to corporate assets. These risks can also be mitigated through appropriate decentralization. Sharp recognized that the expected return of a portfolio depends only on its “beta” and its relationship to the overall market. CAPM helps measure portfolio risk and the return that investors can expect in exchange for that risk.

  Portfolio optimization often involves the process of analyzing a portfolio and managing assets therein. In general, this is done to obtain the highest revenue at a given level of risk. Portfolio optimization can be performed regularly and periodically, eg, monthly, quarterly, semi-annual, or yearly. Similarly, a portfolio can be rebalanced at a desired or required frequency, and rebalancing is ultimately achieved by changing the composition of assets within the portfolio. The portfolio does not need to be rebalanced every time it is optimized, so it can be optimized as often as desired. When considering rebalancing decisions, the optimal portfolio is pursued, so the tax and / or transaction costs implications normally associated with buying and selling are also considered.

  In some existing portfolio optimizers, techniques such as “hill climbing” or linear / quadratic programming are used to find the optimal solution. However, when using these techniques, issues such as long (buy) / short (short sell), minimum position size, number of positions constraints, tax costs, and transaction costs generally cannot be accurately modeled. In addition, US Pat. No. 6,003,018 entitled “Portfolio Optimization By Means Of Resampled Efficient Frontiers” shows another optimizer method. The entire disclosure of US Pat. No. 6,003,018 is incorporated herein by reference. The present invention provides significant improvements over these and other optimizers.

  The assignee of the present invention has developed a portfolio optimizer, now called the ITGOpt® optimizer, which provides more accurate results than previously used optimization and rebalance systems. Use mixed integer programming (MIP) techniques to produce. In previous versions of ITGOpt®, the system performed an optimization in a single pass that considered all of the constraints and parameters simultaneously. In that version, it was possible to suppress or introduce features related to the trading of certain securities. In addition, general portfolio characteristics could be specified, including, for example, leverage, volume, and constraints on long versus short positions. Furthermore, the constraints can be applied to the entire portfolio or can be applied individually to the long side or the short side of the portfolio. In addition, previous versions of ITGOpt® combine a branch-and-bound algorithm with objective scoring of potential solutions, and in doing so, completeness of the results. By reducing the size of the problem without compromising (integrity), we avoid heuristics that tend to lead to false recognition.

  In addition, the traditional ITGOpt® optimizer was able to accurately model and analyze the implications related to tax law. For example, integer modeling of tax brackets and tax lots allows the ITGOpt (TM) optimizer to make a net tax liability without giving up large profitable shares. Makes it possible to minimize Traditional ITGOpt® is also suitable for high-in-first-out (HIFO), last-in-first-out (LIFO), or first-in-first-out (FIFO) accounting methods Is possible. In addition, traditional ITGOpt® was designed with an emphasis on the real-world complexity of sophisticated investment strategies. Traditional ITGOpt® optimizers have been able to deal with complex and / or non-linear problems that can arise in real-world fund management.

  In addition, the traditional ITGOpt® optimizer was able to factor in transaction costs arising from market effects when calculating its solution. The optimizer included a cost model, ACE®, for predicting market impact. Inclusion of ACE® allows users to evaluate implicit transaction costs along with optimization scenario risks and expected returns.

  Particularly in quantitatively managed portfolios, when rebalancing a portfolio, it is common to control buying and selling costs through constraints on the quantity of equity sold or the expected cost of buying and selling. One way to estimate the expected cost of buying and selling is to use a mathematical model such as ITG's ACE® (U.S. patent application 10/10 entitled `` System and method for using and optimizing transaction costs ''). No. 166,719, the entire contents of which application are hereby incorporated by reference).

  Constraints placed on the volume of equity traded or the expected cost are reasonably effective when applied to the management of a single portfolio. However, many portfolio managers manage multiple portfolios that may have overlapping holdings. Multiple portfolio rebalancing can cause problems when one stock found in two or more portfolios may want to buy or sell in two or more portfolios. For example, portfolios A and B may both contain the same interest in IBM shares, but when rebalancing portfolios A and B, it may be desirable to buy and sell IBM's interest in each portfolio. Administrators attempting to buy or sell the total number of interests in the multiple portfolio will count the single interest found in the multiple portfolio multiple times, resulting in a larger overall execution size and higher expected trading costs. Become. For example, the administrator knows that two IBM interests need to be bought and sold in portfolios A and B, and therefore buys and sells IBM's interests that exist in both portfolios A and B, instead of one share at a time. May be executed by equity. The result of this “duplication count” of shared interest is that the cost of executing each realized portfolio is greater than the cost that the portfolio manager would be willing to spend or expect.

  In addition, the conventional ITGOpt® optimizer used effective historical back-testing. ITGOpt® optimizer can closely track portfolios over time and considers the impact of stock splits, dividends, mergers, company splits, bankruptcies, and name changes Can do.

  In addition, the traditional ITGOpt® optimizer has been implemented to handle many funds and many users. Traditional ITGOpt® optimizers have included multi-user, client-server relational database management technology with an infrastructure to handle the demands and large number of transactions from many concurrent users.

  In addition, the traditional ITGOpt® optimizer has been systematically integrated with trade order management and accounting systems. The traditional ITGOpt (R) optimizer was built on relational database management technology and was easily combined with other databases. The conventional ITGOpt® optimizer could also generate a trade list for execution by a proprietary TOM system. In addition, the traditional ITGOpt (R) optimizer design allows for extended customization of reports to meet the needs of corporate sales and customers. In addition, custom report formats could be designed quickly and cost-effectively.

U.S. Patent Application No. 11 / 730,750 U.S. Patent Application No. 10 / 640,630 U.S. Provisional Application No. 60 / 448,147 US Provisional Application No. 60 / 907,525 U.S. Patent No. 6,003,018 U.S. Patent Application No. 10 / 166,719

V. K. Chopra, `` Mean-Variance Revisited: Near-Optimal Portfolios and Sensitivity to Input Variations '', Journal of investing, 1993 M. Best and R. Grauer, "On the Sensitivity of Mean-Variance Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results", Review of Financial Studies, 1991 V. K. Chopra and W. T. Ziemba, `` The Effect of Errors in Means, Variances and Covariances on Optimal Portfolio Choice '', Journal of Portfolio Management, 1993 J. G. Kallberg and W.T. Ziemba, "Mis-specification in Portfolio Selection Problems", Risk and Capital, G. Bamberg and A. Spreman, Lecture Notes in Economics and Mathematical Sciences, 1984

  There may be a variety of portfolio optimization systems and methods, including conventional versions of the ITGOpt® optimization system, but the art is an improvement on the above and / or other systems and processes. And there is a great need for processes.

  Various embodiments of the present invention make significant improvements to existing methods and systems.

  In accordance with embodiments of the present invention, improved systems and methods are provided for optimizing multiple portfolios of tangible or intangible assets, such as securities or stocks.

  In one embodiment of the present invention, a method for optimizing multiple portfolios is provided. Each portfolio includes one or more interests in one or more tradeable assets. The method receives asset data defining a plurality of portfolios, receives one or more individual portfolio optimization parameters corresponding to the plurality of portfolios, and receives one or more global optimization parameters. For each portfolio, optimizing the asset data based on the corresponding individual optimization parameters, aggregating the optimized asset data to generate aggregate optimized asset data; Determining whether the aggregate optimized asset data satisfies the global optimization parameter; and outputting the optimized asset data only when the global optimization parameter is satisfied.

  In another embodiment of the present invention, a computer readable storage medium having stored therein computer executable program code, the following operations: receiving asset data defining a plurality of said portfolios; For each portfolio, an operation for receiving asset data defining a portfolio, an operation for receiving one or more individual portfolio optimization parameters corresponding to the plurality of portfolios, an operation for receiving one or more global optimization parameters, and An operation for optimizing the asset data based on the corresponding individual optimization parameter, an operation for aggregating the optimized asset data to generate aggregated optimized asset data, and the aggregated optimized asset data Satisfies the global optimization parameter A computer-executable program for optimizing a plurality of portfolios by executing an operation for determining whether or not the global optimization parameter is satisfied, and outputting the optimized asset data only A computer readable storage medium is provided in which the code is stored.

  In another embodiment of the present invention, a system is provided for performing optimization of multiple portfolios of assets. The system receives asset data defining a plurality of portfolios, receives one or more individual portfolio optimization parameters corresponding to one or more of the portfolios, and receives one or more global optimization parameters An asset optimized to optimize each portfolio of the plurality of portfolios using the asset data and the corresponding one or more of the individual optimization parameters to generate aggregate optimized asset data Aggregating data, determining whether the aggregate optimized asset data satisfies the one or more global optimization parameters, and the optimization only if the one or more global optimization parameters are satisfied Client interface configured to output generalized asset data It can contain over nest.

  The above and / or other aspects, features, and / or advantages of various embodiments will be further understood in view of the following description in conjunction with the accompanying drawings. Various embodiments can include or exclude different aspects, features or advantages where applicable. In addition, various embodiments can combine one or more aspects, features, or advantages where applicable. The description of aspects, features or advantages of a particular embodiment should not be construed as limiting any other embodiment of the claimed invention.

  The accompanying drawings are provided by way of example without limiting the broad scope of the invention or various other embodiments.

6 is a flowchart illustrating a process according to some embodiments of the invention. 6 is another flowchart illustrating a process according to some embodiments of the invention. In particular, it illustrates a computer that can be used to perform the process steps of various embodiments of the present invention. In particular, it illustrates a computer system that can be used to implement the process steps of various embodiments of the present invention. FIG. 2 is a schematic diagram illustrating a database management structure according to some embodiments. For example, an exemplary graph of revenue (eg, in millions) versus risk (eg, in millions) to find an optimal portfolio. Revenue (e.g., in units of $ 1 million) vs. risk (e.g., in units of $ 1 million) showing a set of average-dispersion points, for example, out of the mean-dispersion efficient frontier, according to some exemplary embodiments of the invention FIG. It is a flowchart explaining the process of optimization of a multiple portfolio. 6 is a flowchart illustrating adjustment of constraints during the subsequent process of multiple portfolio optimization. FIG. 6 is a flow chart illustrating the process of optimizing multiple portfolios by “punishing” objectives or individual portfolios. 2 is a chart showing a comparison of optimization methods applying global constraints and objectives in Example 1. FIG. 2 is a chart showing a comparison of traded interests resulting from an optimization method applying global constraints and objectives in Example 1. 10 is a chart showing a comparison of optimization methods that allow crossing and apply global constraints and objectives in Example 2. 10 is a chart showing a comparison of optimization methods that do not allow crossing and apply global constraints and objectives in Example 2.

  Embodiments of the present invention may include one or more computers comprising one or more computers and / or computers, such as a local area network (LAN), a wide area network (WAN), the Internet, and / or another network. It can be implemented on a network. In various embodiments, one or more servers, client computers, application computers, and / or other computers may be utilized to implement one or more aspects of the present invention. Exemplary computers include, for example, a central processing unit, memory (such as RAM), digital data storage (such as a hard drive), input / output ports (such as parallel and / or serial ports), data input devices (such as keyboards). Etc.). The client computer, in some embodiments, includes browser software for interacting with the server, such as using hypertext transfer protocol (HTTP), to make requests to the server, such as over the Internet. Can do.

  In some embodiments, the system can utilize a relational database, such as utilizing a relational database management system (RDBMS) program to create, update, and / or manage the relational database. An RDBMS receives structured query language (SQL) statements entered by a user or included in an application program, creates, updates, and / or provides access to the database. Some exemplary RDBMS include the ORACLE database product line and the IBM DB2 product line. In some exemplary embodiments, as shown in FIG. 4, one or more client computers may be provided, such as a LAN-based system, for example. The client computer can include a suitable operating system, such as, for example, WINDOWS NT or another system. In some embodiments, the system is adapted to provide an object-based graphical user interface (GUI).

  In some embodiments, the system provides a multi-user client server system, as shown in FIG. In some embodiments, the system provides a hierarchical object-based portfolio control structure for managing a variant of the core set of strategies. In some embodiments, the based data can include assets held, sales, prices, corporate actions, and the like. In some embodiments, multiple risk models such as, for example, models available from BARRA, NORTHFIELD, and custom models may be utilized.

  In some embodiments, the portfolio includes data objects such as assets held, historical execution, universes, benchmarks, risk models, and / or market data. In some embodiments, the population of selected stocks can include all of the relatively active securities in, for example, related markets. For example, assuming that the U.S. market is the relevant market, then the selected stock population may in some embodiments be listed on the New York Stock Exchange, American Stock Exchange, NASDAQ Domestic Market, And about 8000 stocks, including several small stocks. Preferably, specific objects in the portfolio can be defined by attributes and / or parameters set by the user. In some embodiments, an example portfolio can be generated based on analytical data attributes, and a 3% S & P tracking portfolio with a population of 1000 Russells as of January 1, 2003. Is an illustrative example.

  In some embodiments, the portfolio database can include an attribute hierarchy, such as, for example, a 5-level hierarchy as illustrated in FIG. In some exemplary embodiments, lower levels may inherit higher level attributes. In addition, lower levels can preferably override inherited attributes.

  In some embodiments, the portfolio database can include characteristics that can be, for example, any stock-specific data. Preferably, the user can define the characteristics, such as using formulas and / or rules for generating new characteristics from other characteristics. As an illustrative example, a user may use an algebraic generation method such as “A = B + C × D”. As another illustrative example, a user may use a set membership method, such as “A + 1 if B <C and B> D”, for example. In some embodiments, for example, "remove sin stocks with p / e's> 10 and price <5" to allow the name to be removed from the population for compliance and / or other reasons. A filter can be provided. In some embodiments, the system can provide default values for unconditioned properties.

  In some embodiments, a user can create a customized report, such as a customized asset level report. Preferably, the report definition can be saved with a name (eg, in digital data storage).

  In some embodiments, any dimension of the portfolio “space” can be part of an objective function or constraint. In some embodiments, the system facilitates investigating trade-offs between any combination of, for example, expected revenue, risk / tracking, exposure, transaction costs, taxes, and positions / volumes / sizes. can do.

  In some embodiments, a population can be provided for one optimization, in which both sides (e.g., buy side and sell side) are subject to constraints imposed on each side separately and to the entire portfolio. Rebalanced due to constraints.

  In some embodiments, the user is provided with a graphical user interface that is presented to the user via a client computer. In some embodiments, the graphical user interface may allow for importing and / or exporting data and files, setting parameters, performing optimization, and / or receiving optimization results. In some embodiments, a user can generate or import a specific task schedule, in which, for example, data import and / or export can be automated and functions available within the user interface Will be available for batch processing.

  FIG. 3 shows an example of a computer configuration that may be used to implement computerized process steps, such as in process 100 and process 200 shown in FIGS. 1 and 2, for example. In some embodiments, the computer 320 includes a central processing unit (CPU) 322 that can communicate with a set of input / output (I / O) devices 324 via a bus 326. The I / O device 324 can include, for example, a keyboard, mouse, video monitor, printer, and / or other devices.

  CPU 322 may communicate with computer readable media (eg, conventional volatile or non-volatile data storage devices) 328 (hereinafter “memory 328”) via bus 326. The interaction between the CPU 322, I / O device 324, bus 326, and memory 328 can be similar to that known in the art.

  The memory 328 can include, for example, market and accounting data 330 that can include stock-related data such as stock prices and company-related data such as book prices. The memory 328 can also store software 338. Software 338 may include a plurality of modules 340 for performing process steps, such as the steps of process 100 and / or process 200 shown in FIGS. Conventional programming techniques can be used to implement these modules. The memory 328 can also store these and / or other data files.

  In some embodiments, the various methods described herein can be implemented via a computer program product for execution on one or more computer systems. For example, a series of computer instructions can be stored on a computer-readable medium (e.g., a diskette, CD-ROM, or ROM) or transmitted to a computer system via an interface device such as a modem. it can. The medium can be substantially tangible (eg, a communication line) and / or substantially intangible (eg, a wireless medium using microwaves, light, infrared, etc.). Computer instructions can be written in various programming languages and / or stored within a memory device, such as a semiconductor device (e.g., a chip or circuit), a magnetic device, an optical device, and / or other memory device. be able to. In various embodiments, the communication can use any suitable communication technology.

  1 and 2 illustrate process steps that can be performed in some exemplary embodiments of the invention. These two processes are exemplary, and various embodiments of the present invention can be applied in various processes.

  With respect to the exemplary process 100 shown in FIG. 1, in a first step 102, the process begins to evaluate an existing or new portfolio. Next, in a second step 104, the system receives information that applies to the optimization analysis. Next, in a third step 106, information is input to an optimization system such as an optimization engine. Next, in a fourth step 108, optimization algorithms and methods are executed via the optimization engine. Next, in a fifth step 110, the optimization results are provided to the user. Next, in a sixth step 112, the user acts based on the optimization result. For example, the user may rebalance the portfolio based on the results, for example.

  With respect to the exemplary process 200 shown in FIG. 2, in a first step 202, a user may enter portfolio data. In some embodiments, a user can generate a portfolio using a portfolio name editor. Preferably, the user can use a file import / export utility to load the required data, such as, for example, identifier maps, owned assets, benchmarks, populations, characteristics, and / or risk models. Preferably, the user can also define portfolio attributes, such as analysis date, benchmark, population, characteristics, risk model, etc., using a parameter editor. Preferably, the user can also delete the data.

  Next, at step 204, the user can identify and reconcile the missing data. In some embodiments, a user can collate data from multiple sources. In some embodiments, some potential issues may include asset status or identifier changes, lost or incorrect characteristics or risk data, and / or membership in benchmarks or populations, etc. it can. In some embodiments, the asset summary report may provide a high level problem notification. In some embodiments, loss data reports can be used for owned assets, benchmarks, populations, characteristics, and / or factor exposure. In some embodiments, the user can use a data editor to solve the problem.

  Next, in step 206, the user can specify a rebalance purpose. In some embodiments, a user can use a parameter editor to, for example, cash flow, objective function (e.g., alpha, risk aversion), risk constraint (e.g., two or more benchmarks, common And “inherent factors”, cash balance, turnover constraints, position size, number of positions, and / or buy / sell size constraints, and / or population characteristic filters, etc., can be selected. Preferably, the user can select user specific parameters for use in the process of the present invention. Preferably, the user can create a constraint matrix using a row / column bounds editor.

  Next, in step 208, the user can examine the current portfolio characteristics. In some embodiments, the user can receive a report for one or more of assets, populations, benchmarks, and / or final portfolios. Preferably, the user can receive summaries and details regarding accounting, characteristics, factor exposures, and / or buying and selling.

  Next, at step 210, the user can adjust parameters and constraints. In some embodiments, the user can perform this step via a parameter editor. Preferably, a row / column limit editor is provided.

  Next, at step 212, the user can optimize and generate a rebalanced portfolio. This step can utilize an optimization engine to optimize and generate the proposed portfolio / trading. Preferably, in that case, the user can inspect the proposed portfolio / buying, for example via a buying and selling summary screen or report, or a buying and selling detail report. Preferably, the user can then edit the proposed portfolio / trading as needed. Preferably, the user can then incorporate the proposed portfolio / trading into a particular execution.

  As indicated by arrow A2, the user can repeat steps 208-212 as needed, such as to continuously evaluate portfolios / trading and rebalance the portfolio.

  In some embodiments of the present invention, step 108 in the process shown in FIG. 1 and / or step 212 in the process shown in FIG. 2 may include an optimization method as described below. . To implement these methods, in some embodiments, an optimizer (eg, generated via software or the like) can include a software module or the like that accomplishes the steps described below.

  In some embodiments of the present invention, a portfolio optimizer can be provided that allows to identify an acceptable region of error. This has for example an optimizer that may propose to make changes or trades as a result of "noise" in various inputs that could potentially result in many related trades and different costs. It can be advantageous in that it helps to avoid. In some embodiments of the present invention, with an approximate understanding of how noise these inputs have, the system allows the portfolio manager to maintain how large an area is considered to be acceptable internally. Can be recognized.

In some embodiments, the optimizer may define a confidence region for the portfolio P ₀ on the efficient frontier that corresponds to the risk aversion degree γ. In some embodiments, this region is such that c _low * Risk (P ₀ ) <Risk (P) <c _high * Risk (P ₀ ) and Ret (P)> c * Ret (P _opt ) Include all portfolios P. Here, P _opt is a portfolio on an efficient frontier such that Risk (P _opt ) = Risk (P). In addition, c _low , c _high , and c are the optimal portfolio corresponding to risk reduction, relative average deviation of risk increase, and risk aversion γ and different vectors of return. Expected return. It can be assumed that the revenue vectors are normally distributed around their average value. In some embodiments, the user can set a particular confidence level by setting different values for the constants c _low , c _high , and c.

In the resampled efficient portfolio optimization of US Pat. No. 6,003,018 described above, the confidence region is calculated around the resampled efficient frontier portfolio P ₀ and a variance value for P ₀ is specified. Includes all portfolios that are less than or equal to the value associated with the confidence level. In that regard, the key to resampled efficient portfolio optimization is to calculate a resampled efficient frontier portfolio. The resampling process generates simulated revenue that provides an alternative input to the calculation of an efficient frontier portfolio. The resampled efficient frontier portfolio is the result of an averaging process across many possible efficient frontiers.

  On the other hand, in some embodiments of the present invention, a standard efficient frontier portfolio is used instead of a resampled efficient frontier portfolio. In particular, in contrast to a resampled efficient frontier portfolio, an efficient frontier portfolio can be defined as the portfolio with the highest expected return for a certain value of risk. In many cases, it is not appropriate to use a resampled efficient frontier. As an illustrative example only, consider two assets with a correlation coefficient of zero, expected returns of 10% and 20%, and a standard deviation of returns of 20%. The maximum revenue portfolio includes only the second asset, and its expected return is 20%. The resampled portfolio corresponding to the maximum revenue point on the resampled efficient frontier contains approximately 35% of the first asset and 65% of the second asset, with an expected return of approximately 16% Not too much.

  The resampled efficient frontier portfolio is constructed by taking the average of many portfolios acquired through simulation. Thus, in most cases, these portfolios contain a number of different assets. There are difficulties in using such a portfolio, especially if it is desirable to have an optimal portfolio that includes a limited number of assets taken from the population.

  Calculation of the trust region for the Mean-Variance Efficient Set in some embodiments:

I. Definitions and assumptions:
In some embodiments, the key parameters of ITGOpt's mean-dispersion model are the α-asset expected return vector and the Σ-asset return covariance matrix. These parameters can be estimated using historical data, analytical models, analyst predictions, or other methods.

VK Chopra, `` Mean-Variance Revisited: Near-Optimal Portfolios and Sensitivity to Input Variations, '' Journal of investing, 1993, the entire disclosure of which is incorporated herein by reference, Explains that the composition of an optimal portfolio can change significantly. M. Best and R. Grauer, “On the Sensitivity of Mean-Variance Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results”, Review of Financial Studies, 1991, the entire disclosure of which is incorporated herein by reference. The year describes, among other things, the impact of changes in the expected asset return vector on the average-diversified efficient frontier and the composition of the optimal portfolio. VK Chopra and WT Ziemba, The Effect of Errors in Means, Variances and Covariances on Optimal Portfolio Choice, Journal of Portfolio Management, 1993, and JG Kallberg and WT Ziemba, the entire disclosures of which are incorporated herein by reference. `` Mis-specification in Portfolio Selection Problems '', Risk and Capital, edited by G. Bamberg and A. Spreman, Lecture Notes in Economics and Mathematical Sciences, 1984, in particular, is expected in the utility function of investors. Describes the relative importance of revenue error, revenue specific variance and covariance error. The relative impact of errors in these parameters depends on the risk tolerance of investment. If the risk aversion parameter is not very high, the expected return error has a much more significant impact on the utility function than the error of other parameters. There are two possible ways to model the error of α.
Relative error model: r _i = α _i * (1+ d * z _i ), where r _i is the actual expected return of asset i, α _i is the estimated expected return of asset i, d is the standard deviation of the error, z _i is a normal random variable with mean 0 and standard deviation 1.
Absolute difference model: r _i = α _i + d * z _i , where r _i is the actual expected return of asset i, α _i is the estimated expected return of asset i, d is the standard deviation of the error, and z _i is the mean A normal random variable with 0 and standard deviation 1.

  According to the CAPM model, assets with higher returns have a higher risk or higher variance of returns. Therefore, the error in estimating the expected return should be proportional to the expected return value. Taking this last observation into account, the inventors consider a relative error model in some embodiments.

II. Confidence region for mean-disperse efficient set:
Consider the standard portfolio optimization problem that arises in some embodiments.

  Here, γ is a risk aversion parameter, α is a vector of estimated expected returns, h is a vector of position dollars, Risk (h) is a risk function, and Q is a set of realizable portfolios. When γ approaches infinity, problem (1) is equivalent to the following problem.

  When γ approaches 0, problem (1) is equivalent to the following problem.

  “T” represents the return versus risk tradeoff coefficient,

  Here, h (1), h (2), and h (3) are the optimal solutions of the problem (1), the problem (2), and the problem (3), respectively.

  Consider a modified optimization problem using the vector of actual expected returns.

Here, r is a vector of actual expected returns. Let h (r) be the optimal portfolio for problem (5). If the real revenue vector is r, the return on this portfolio is r ^T h (r). The inventors have found that the optimal portfolio h (α) for the revenue vector α, which has the same level of risk as h (r), has:

  The relative difference in revenue between portfolios h (r) and h (α) is a function of t, d, and z.

  The relative risk difference between portfolios h (1) and h (r) is a function of γ, d, and z.

The variable z is a normal random variable, so the expected relative revenue difference between the portfolios h (r) and h (α) is a function of t and d.
δ (t, d) = E _z (D (t, d, z)) (9)

  The optimal portfolio corresponding to the high risk aversion approachs the minimum diversification portfolio and is much less sensitive to the expected return vector error than the optimal portfolio corresponding to the low risk aversion. The function δ (t, d) is equal to 0 when t is 0 and increases as t increases. Similarly, the function δ (t, d) is equal to 0 when d is 0 and increases as d increases.

R _Up represents the expected relative increase in Risk.
R _Up (t, d) = E _z (-R (t, d, z) | R (t, d, z) <0) (10)
We also represent the expected relative decrease in Risk by R _Down .
R _Down (t, d) = E _z (R (t, d, z) | R (t, d, z) ≥0) (11)

The function Return (x) represents the mean-distributed efficient frontier for the vector of expected returns α and the risk function Risk (h), where the value of Return (x) is the return of the optimal portfolio with variance x is there. Now, for a given point (x ^* , Return (x ^* )) on the mean-dispersion efficient frontier corresponding to the trade-off coefficient t and the standard deviation d, the set of points Ω (t, d) Define.

  Intuitively, it is a set of mean-dispersion points that do not deviate much from the mean-dispersion efficient frontier than most optimal portfolios obtained for different realizations of the expected return vector.

III. Estimates of functions δ, R _Up and R _Down :
For example, for all possible combinations of 10 values of the trade-off factor t and 10 values of the standard deviation d of errors, using Monte Carlo simulation, for example, the function δ, R _Up , And R _Down can be estimated. The results of the study can be summarized in a table, for example. Table 1-Table 3 (Table 1-Table 3) below show the results summarized in several exemplary tables. For example, to calculate the function for specific values x and y of tradeoffs and standard deviations, the tradeoff values t1 and t2 such that t1 ≦ x ≦ t2 and d1 ≦ y ≦ d2, and the standard deviation Two values d1 and d2 can be found in the table.

IV. Set of constraints for the trust region:
For a given risk aversion parameter and standard deviation d, it is possible to find the optimal portfolio h ^* of problem (1). Now, the trade-off factor t corresponding to h ^* can be calculated and the following upper limits can be set for portfolio risk.
Risk (h) ≦ Risk (h ^* ) * (1 + R _Up (t, d)) (13)

  To set a lower bound on the expected return of a portfolio, the following problems must be solved:

Using this solution, constraints can be established as follows.
α ^T h ≧ α ^T h '* (1-δ (t, d)) (15)

Calculation of the sharp ratio in some embodiments:
In some embodiments, an optimizer is provided that can provide optimization of a portfolio of assets based on a sharp ratio as a measure of goodness. Preferably, the system can provide pre-maximization based on expected revenue and expected risk. Instead of just retrospectively using the sharp ratio, the embodiment can provide forward optimization based on the sharp ratio. Thus, some embodiments of the present invention may provide a unique form of portfolio optimization based on, for example, maximization of the sharp ratio.

I. Definitions and assumptions:
In some embodiments, the standard objective function is the following term multiplied by some factor across all portfolios h that belong to the set Q (which is defined by the constraints imposed on the portfolio): Is the maximization of the total.
α (h)-Expected return on the final portfolio.
Risk (h) —The variance of the earnings of the final portfolio (or the difference between the final portfolio and the benchmark portfolio) divided by the portfolio acquisition price (basis).
TC (h)-The transaction cost of a transaction that makes the current portfolio the final portfolio.
TaxCost (h)-Total tax liability after transition to the final portfolio.
Penalties (h)-Penalties for violating any soft constraints and realizing "almost-long-term" benefits.

  Preferably, the optimal portfolio selection problem seeks a portfolio that maximizes expected returns with relatively low values of Risk, TC, TaxCost, and Penalties. In that regard, a positive coefficient is placed before α and a negative coefficient is placed before all other terms. All terms except Risk can be grouped into one term A (h), which can be called “adjusted return”. This term represents the total revenue after taking all additional costs into account. The coefficient before the Risk term can be represented by -γ, where γ is a risk aversion parameter. This parameter can establish a trade-off between risk and return on the potential investment portfolio.

  In some embodiments, the alternative objective function may be maximization of the reward-to-variability ratio S of the potential investment portfolio hεQ.

  This ratio is known as a sharp ratio or a measure of sharpness. In this case, Risk is the variance in the revenue of the final portfolio. The revenue variance of the difference between the final portfolio and the benchmark portfolio is not used as a Risk for Sharp ratio. In addition, in the standard objective function, Risk is divided by portfolio acquisition value B, but in the sharp ratio, Risk should not be divided by B.

II. Finding the Sharpe Ratio in some embodiments:
In some embodiments, S ² can be maximized instead of S. Therefore, the square root of Risk in the denominator is removed.

First, A ² is replaced with its piece-wise linear approximation. In that regard, the lower and upper limits of A are found such that the value of A that maximizes S exists between these boundaries.

A. Boundary discovery algorithm:
In some embodiments, a boundary discovery algorithm is used. In some embodiments, the algorithm can include substantially the following:
・ Problem: find the optimal solution h ^* for max _{h∈Q} A (h);
Set UpperBound = A (h);
Set LowerBound = A (h ^* ) / 2;
Set S1 = A (h ^* ) / sqrt (Risk (h ^* ));
Set flag = 1;
・ Problem: find the optimal solution h ^* for max _{{h∈Q, A (h) == LowerBound}} Risk (h);
Set S2 = A (h ^* ) / sqrt (Risk (h ^* ));
・ If (S2 <S1), flag = 0;
・ While (flag) (Loops while flag is true) {
・ LowerBound / = 2.0;
・ S1 = S2;
・ Problem: find the optimal solution h ^* for max _{{h∈Q, A (h) == LowerBound}} Risk (h);
Set S2 = A (h ^* ) / sqrt (Risk (h ^* ));
・ If (S2 <S1), flag = 0;
-Otherwise UpperBound = LowerBound * 2.0;
・}

In most cases in the above algorithm, the LP (linear program) is solved three times. This algorithm ends with obtaining the LowerBound and UpperBound values of Lower and Upper of A, respectively. A set of links I ₁ , I ₂ , ..., I _n is defined to represent A and generate a piecewise linear approximation A2 for A ² , where I ₁ = LowerBound and I _n = UpperBound. For all i <n-1, I _{i + 1} = I _i (1 + b) is set, I _n ≤ I _n-1 (1 + b), and n is selected to satisfy these conditions It should be. Relative error values given in the approximation of A ² by A2 relative to (1 + e), is set as follows.

By way of example only, if e = 0.0002, the error in the approximation of A ² is at most 1.002, and the final error in S is at most 1.0001. In some embodiments, the value of e is a user-selected variable that can be selected, for example, via a computer input device.

  Now it is possible to find the maximum sharp ratio S. The initial value of S is set equal to the value of S1 obtained from the previous algorithm.

B. Sharp ratio discovery algorithm:
In some embodiments, a sharp ratio discovery algorithm is provided. In some embodiments, the algorithm can include substantially the following:
Set LastS = 0;
・ While (S-LastS> 0.001) (Loops while S-LastS> 0.001) {
• Problem: find the optimal solution h ^* for X = max _{h∈Q} [A2 (h)-S * S * Risk (h)];
Set LastS = S;
Set to S = sqrt (S * S + X / Risk (h ^* ))

In some embodiments, the algorithm outputs an optimal value for the sharp ratio S and a portfolio h ^* that achieves this ratio. In some embodiments, to reduce computation time, optimization starts at each iteration with the optimal solution obtained in the previous iteration.

III. Method convergence in embodiments:
The above “boundary discovery” algorithm creates a problem at the maximum possible value of adjusted revenue. In some embodiments, this number decreases by a factor of two for each step of the algorithm. In some embodiments, the algorithm ends when the best value of the sharp ratio corresponding to the current level of adjusted revenue is lower than the sharp ratio in the previous iteration. In most cases, the maximum value of the sharp ratio is achieved so that the adjusted revenue falls between the maximum adjusted revenue and half of the maximum adjusted revenue.

In the “sharp ratio discovery” algorithm, an updated guess of the maximum sharp ratio value S is used at each iteration. This is represented by S _i , h _i , and X _i corresponding to the values of S, h ^* , and X obtained in the i th iteration of the algorithm. For all portfolios h, X _i is the maximum value of the optimization problem at the i th iteration, so we get:

By substituting the optimal portfolio h _{i + 1} of the i + 1 iteration into inequality (1), we obtain:

  Now, the i + 1 iteration is

  here,

It is. Now substituting (4) for (3), we get

  Finally, put together (2) and (5) to get:

From the last inequality, it is possible to conclude the following properties of the algorithm:
Obtain X _i ≧ X _{i + 1} for all iterations i of the algorithm.
Risk (h _i ) ≧ Risk (h _{i + 1} ) for all iterations i of the algorithm.
If X _i / 2 ≦ X _{i + 1} , then Risk (h _i ) / 2 ≧ Risk (h _{i + 1} ).

  These characteristics indicate that the algorithm converges at an exponential rate to the optimum value of the sharp ratio.

Multi-portfolio optimization in some embodiments:
In some embodiments of the present invention, the optimization system handles, for example, a situation where a portfolio manager manages a portfolio for one or more clients and the client has a different portfolio of assets. be able to. In some embodiments, the system is adapted to allow a portfolio to be rebalanced on a large scale instead of simply rebalancing on a small scale (eg, a personal scale). For example, the system can rebalance on a large scale without having each individual make any trades individually. It should be noted that individual accounts may differ, but nevertheless they often have a common asset in their portfolio.

  In some embodiments, the system performs optimization on a smaller or individual basis (eg, on an account-by-account basis) and evaluates which results also meet multi-portfolio needs. Thus, some embodiments can basically optimize individual accounts and aggregate them. Based on this optimization, the system can generate results that provide an optimized portfolio across multiple accounts, reducing potential transaction costs, reducing the frequency of buying and selling required, And / or provide other benefits.

1. Definitions and assumptions:
The standard optimization problem in some embodiments includes maximizing an objective function Ω (h) over all portfolios h that belong to the constraint set Q defined by the constraints imposed on the portfolio. In multi-portfolio optimization, there are K portfolios, the objective function Ω _k (h _k ) and constraint set Q for all portfolios h _k , k = 1, ..., K. _k exists. In addition, the total portfolio Σ _{k = 1, K} h _k should satisfy the constraint set Q of the total portfolio.

h _k ^Opt , k = 1,..., K represents a portfolio such that h _k ^Opt εQ _k that maximizes the value of the objective function Ω _k . Many portfolio managers, for example, have the following multi-portfolio optimization problem: Σ _{k = 1, K} h _k ∈Q, and for all portfolios h _k , h _k ∈Q _k and Ω _k (h There is a problem of finding an optimal set of _K portfolios h ₁ ,..., h _K such that the value of _k ) approaches the optimal value Ω _k (h _k ^Opt ). In some embodiments, the following two different measures can be used for the distance between Ω _k (h _k ) and Ω _k (h _k ^Opt ):
• Relative scale: The minimum value of the following formula.

Absolute scale: The minimum value of the following formula.
Ω _k (h _k ^Opt )-Ω _k (h _k )

  If the portfolio manager wants to bring the value of the objective function for each portfolio close to its maximum value as a percentage, a relative measure can be used. Alternatively, absolute measures can be used if the portfolio manager wants these values to approximate in dollars.

II. Multi-portfolio optimization algorithm:
In some embodiments, the multi-portfolio optimization algorithm can substantially include:

In the first step of the algorithm, it is possible to find the optimal portfolio h _k ^Opt εQ _k for all k, k = 1,.

  In the second step of the algorithm, it is possible to distinguish the following two cases:

  Relative scale: maximizes the value of the scalar variable x under the following constraints:

Absolute scale: minimizes the value of the scalar variable y under the following constraints:

In the case of a relative scale, the value of Ω _k (h _k ^Opt ) is assumed to be positive. If it is negative, the value of variable x is minimized instead of maximized.

  In addition, the claimed invention can optimize multiple portfolios subject to global constraints. This optimization may require multiple rounds to reach an acceptable solution under given applicable constraints. One embodiment of the present invention is illustrated in FIG. 8, and in more detail in FIG.

  FIG. 8 is a flowchart illustrating an example of the method and system of the present invention. In process 800, the system receives data for a plurality of portfolios in step 802. The system then performs a check at step 804 to ensure that all necessary data is present and correct. The system then receives global constraints at step 806. These constraints are considered when optimizing the sum of multiple portfolios. Some global constraints that can be used include, without limitation, the total assets sold (percentage, quantity, amount), the total assets sold (percentage, quantity, amount), the total assets purchased (percentage, quantity) , Money), acceptable risk level, transaction costs, late comer, and crossing.

  Next, in step 808, the system receives parameters used in optimizing the individual portfolio. The system then optimizes the portfolios individually at step 810 using the individual optimization parameters for the individual portfolios. This newly optimized asset data is then aggregated in step 812 and the aggregated optimized asset data is checked in step 814 to determine if it is within the bounds of the global constraint. . If the aggregate optimized asset data meets global constraints, optimized asset data for each portfolio is displayed at step 816. However, if the aggregated optimized asset data fails to meet the global constraints, the constraints imposed on the individual portfolio are adjusted at step 818 and optimization re-execution begins at step 810. This process continues interactively until it is time for aggregate optimized asset data to meet global constraints.

  The following examples illustrate the following non-limiting aspects of the invention.

There are three portfolios (h), each with three securities (S) to be bought and sold.
・ H ₁ : S ₁ , S ₂ , S ₃
・ H ₂ : S ₁ , S ₂ , S ₄
・ H ₃ : S ₁ , S ₅ , S ₆

The optimization of these three portfolios is subject to the following global constraints (M _TOTAL ):
・ M _TOTAL-1 : S ₁ ≦ 100 shares are traded ・ M _TOTAL-2 : S ₂ ≦ 100 shares are traded ・ M _TOTAL-3 : Total trading cost ≦ $ 200

The optimization of these three portfolios is subject to the following individual constraints (M _i ): For purposes of explanation and examples, constraints are identified by the method: M _{PORTFOLIO-CONSTRAINT NUMBER} . In this example, the individual constraints are very similar to the global constraints, but this is just one example of an embodiment. Another embodiment may have individual portfolio constraints that do not resemble global constraints, either for objects or securities, or for quantities or equity. Another embodiment may have individual constraints that are similar or different in part or in whole to global constraints. In this example, the individual constraints are as shown below.
M _1-1 : S ₁ ≦ 100 shares are traded
M _1-2 : S ₂ ≦ 100 shares are traded
M _1-3 : Total trading cost ≤ $ 200
M _2-1 : S ₁ ≦ 100 shares are traded
M _2-2 : S ₂ ≦ 100 shares are traded
M _2-3 : Total trading cost ≤ $ 200
M _3-1 : S ₁ ≦ 100 shares are traded
M _3-2 : S ₂ ≦ 100 shares are traded
M _3-3 : Total trading cost ≤ $ 200

If each portfolio is optimized individually, assuming that all other aspects of the example are in line with the rules, possible results may be as follows:
・ H ₁
○ S ₁ = 100 interests are traded ○ S ₂ = 100 interests are traded ○ S ₃ = 100 interests are traded ○ Total trading cost = $ 100
・ H ₂
○ S ₁ = 100 interests are traded ○ S ₂ = 100 interests are bought and sold ○ S ₄ = 100 interests are bought and sold ○ Total trading cost = $ 10
・ H ₃
○ S ₁ = 100 interests are traded ○ S ₆ = 100 interests are bought and sold ○ S ₅ = 100 interests are bought and sold ○ Total trading cost = $ 10

These individual portfolio optimizations satisfy the constraints imposed on the individual portfolios, but aggregate asset data must still be checked to determine whether global constraints are met. The aggregate optimization data is as follows.
300 shares of S ₁ are bought and sold
200 shares of S ₂ are bought and sold
100 shares of S ₃ are bought and sold
S ₄ are buying and selling of 100 equity
S ₅ is buying and selling of 100 equity
100 total trade cost S ₆ of the equity is traded = $ 300

Therefore, the number of S ₁ and S ₂ interests traded across all of the portfolio exceeded the total number of interests that the portfolio manager intended to buy or sell. In addition, the cost of buying and selling across the portfolio as a whole exceeds the maximum planned buying and selling cost of the portfolio.

  Thus, if aggregate constraints can be imposed on multi-portfolio optimization, the system can adjust the constraints imposed on the individual portfolio and re-run the optimization.

FIG. 9 is a flowchart illustrating an example of how the present invention can adjust individual portfolio constraints. In step 902, it is determined how much each share (S _i ) was bought and sold in each portfolio during the last optimization round. In step 904, these individual equity quantities are summed to determine the total number of equity traded across all of the portfolio (S _TOTAL ).

In step 906, this aggregate optimized asset data is checked against applicable global constraints (S _TOTAL > M _TOTAL ). If the constraints are met, optimized asset data is displayed at step 908. In step 906, if the global constraints are not met, the system can adjust the constraints imposed on the individual portfolio optimization in the following manner. In step 910, a determination is made whether “latecomer” is allowed.

The “latecomer” allowance allows securities that have not been traded in any portfolio during the previous optimization round to be traded in this optimization round. If "latecomers" are not allowed, the system must perform a check at step 914 to see if each security has been traded during a previous optimization round. If a security has never been traded during the previous optimization round, in step 916, the maximum number of shares in that security that can be traded in the next optimization round (M _i ) is equal to zero. Is set. If a security has been traded during the previous optimization round, at step 912, the maximum number of shares of that security that can be traded in a particular portfolio during the next optimization round ( M _i ) is divided by the total number of securities interests traded across all portfolios during the previous optimization round and is imposed on the number of securities interests tradeable across all portfolios. Set equal to the number of shares of the securities traded in each portfolio during the previous optimization round [S _i * (M _TOTAL / S _TOTAL )] multiplied by the constraint.

Next, in step 918, the system performs a check to see if “crossing” is allowed. “Crossing” occurs when individual shares are purchased and sold in an optimized multiple portfolio. If “crossing” is allowed, then in step 926, optimization is re-executed using the adjusted constraints. If “crossing” is not allowed, in step 920, the total number of shares of the specified securities purchased (S _i-BOUGHT ) and the total number of shares of the specified securities sold over all of the portfolio to be optimized (S _i-SOLD ) must be determined.

In step 922, it is determined whether the total number of interests in the specific securities purchased (S _i-BOUGHT ) is greater than the total number of interests in the specific securities sold (S _i-SOLD ). If so, at step 928, the maximum number of interests in that security that can be sold during the next optimization round is set to zero. If the total number of shares purchased (S _i-BOUGHT ) is less than the total number of shares sold (S _i-SOLD ), then in step 924, at the end of the next optimization round. The maximum number of securities interests that can be purchased in is set to zero. In one embodiment, to prevent “crossing”, if the buy or sell side is set to zero, the adjusted constraints used during subsequent optimization rounds are set to zero. Not relevant to buy side or sell side only. Finally, optimization is re-executed using the adjusted constraints (step 926).

Referring back to the example started with reference to Figure 8, where the number of S ₁ and S ₂ shares sold and the cost of buying and selling both exceeded the limits intended by the trader, the constraints imposed on the individual portfolio are Now it is adjusted as follows.

Assuming that both “latecomer” and “crossing” are allowed, the adjustment can be as follows: Using the formula M _i = S _i * (M _TOTAL / S _TOTAL ) in FIG. 10, the following calculation yields the maximum number of shares of each security that can be traded in the next optimization round.
・ M _1-1 : S ₁ ≦ 33 interest is traded [100 * (100/300)]
・ M _1-2 : S ₂ ≦ 50 interests are bought and sold [100 * (100/200)]
・ M _1-3 : Total trading cost ≤ $ 200
・ M _2-1 : S ₁ ≦ 33 interest is traded [100 * (100/300)]
・ M _2-2 : S ₂ ≦ 50 interests are traded [100 * (100/200)]
・ M _2-3 : Total transaction cost ≤ $ 200
・ M _3-1 : S ₁ ≦ 33 interest is traded [100 * (100/300)]
・ M _3-2 : S ₂ ≦ 50 shares are traded [100 * (100/200)]
・ M _3-3 : Total trading cost ≤ $ 200

Thus, the next optimization round can result in:
・ H ₁
○ S ₁ = 33 interests are traded ○ S ₂ = 50 interests are traded ○ S ₃ = 100 interests are traded ○ Total trading cost = $ 61
・ H ₂
○ S ₁ = 33 interests are traded ○ S ₂ = 50 interests are traded ○ S ₄ = 100 interests are traded ○ Total trading cost = $ 61
・ H ₃
○ S ₁ = 33 interests are traded ○ S ₅ = 100 interests are traded ○ S ₆ = 100 interests are traded ○ Total trading cost = $ 78

The aggregate data from this optimization round satisfies all global constraints. Total trading costs were reduced to a lower level that met global constraints due to the difference in the number of shares traded for each security. The aggregate data that satisfies the constraints is shown below.
S ₁ = 99 interests are bought and sold
S ₂ = 100 stake is bought and sold
S ₃ = 100 stake is bought and sold
S ₄ = 100 stake is bought and sold
S ₅ = 100 stake is bought and sold
S ₆ = 100 stake is bought and sold Total trading cost = $ 200

  Therefore, by adjusting the constraints according to the claimed invention, the optimization of the present invention imposes constraints imposed on individual portfolio optimization to satisfy global constraints imposed on multiple portfolios overall. Can be adjusted.

III. Optimal solution and solution time:
The optimal set of portfolios h ₁ , ..., h _K calculated by the above algorithm is the relative or absolute distance from the value of Ω _k (h _k ^Opt ) to the value of the objective function Ω _k (h _k ) The maximum is minimized and the maximum is taken for all portfolios h ₁ , ..., h _K. This set also satisfies the constraints imposed on the total portfolio Σ _{k = 1, K} h _k . Thus, the solution meets the characteristics required by portfolio managers in multi-portfolio optimization.

In some embodiments, the time required to solve the first step of the algorithm is substantially equal to the time to find the optimal solution for all portfolios h _k , k = 1, ..., K . This is solved without the additional constraints imposed on the total portfolio. In the second step, another optimization problem is solved. However, this should not spend substantially longer than the resolution time in the first step. In some embodiments, it may use the fact that the optimal solution obtained in the first step is the optimal solution of the problem in the second step if the constraints imposed on the total portfolio are relaxed. it can. Thus, for a relaxed problem, a dual solution can be calculated first. This can then be used as the initial feasible solution to solve the problem that is paired with the problem in the second step of the algorithm. This approach can speed up finding the optimal solution in the second step of the algorithm.

IV. Objective-based algorithms for multi-portfolio optimization:
Referring to FIG. 10, in accordance with some embodiments of the present invention, further “hidden” optimization is achieved by adjusting the individual optimization objective function to make the numerical value of the function less desirable. It is possible to reach. This adjustment may also be referred to as “punishing” the individual optimization objective function.

Multi-portfolio optimization can be formulated as an objective function that is a linear product of multiple individual portfolio objectives subject to individual portfolio constraints and global constraints. The optimal value of the individual portfolio objective function when individually optimized can be expressed as Ω ^* _i , where i = 1, ..., n is the portfolio. The optimal value of the individual portfolio objective function when optimized with global constraints can be expressed as Ω _i ^start , where i = 1, ..., n is the portfolio. Thus, individual portfolio objective optimization is included within the range between Ω ^* _i and Ω _i ^start . This range is

Can be expressed as If Ω ^* _i = Ω _i ^start , an optimal solution is feasible (ie, does not violate global constraints).

Using these terms, a new slack variable, expressed as ΔΩ _i ,

Can be defined by The slack variable measures the deviation of each portfolio from the optimal value as a fraction of that range. This definition ensures that the slack variable is in the range 0 to 1, where 0 is the optimal value and 1 is just a realizable value. The slack variable does not depend on the specific purpose desired during the optimization.

  If individual portfolios that are part of a multi-portfolio have different types of objectives (eg, risk, tracking error, revenue, etc.), over- or under-sale of some individual portfolios can result. This situation is often suddenly caused by the size differences of multiple portfolios that are optimized together. In this situation, it may be necessary to appropriately increase or decrease or weight the purpose of the individual portfolios so that all portfolios are traded fairly.

  FIG. 10 is a flowchart illustrating multi-portfolio optimization that uses “punishing” individual portfolio objectives to satisfy global constraints and best meet global objectives. In process 1000, in step 1002, data for a plurality of portfolios is received. In step 1004, a check is performed to ensure that all necessary data is present and correct. In step 1006, global constraints are received.

  Global constraints are taken into account when optimizing the sum of multiple portfolios. Some global constraints used include, but are not limited to, total assets sold (percentage, quantity, amount), total assets sold (percentage, quantity, amount), ACE costs, total assets purchased ( (Percentage, quantity, amount), acceptable risk level, transaction costs, latecomers, and crossings.

  Next, in step 1008, constraints used in optimizing the individual portfolio can be received. In step 1010, the global objective of optimization is received.

  Global objectives are considered when optimizing the sum of multiple portfolios and allow managers to apply more comprehensive management strategies to their investment portfolios. Some global objectives relate to, without limitation, minimizing / maximizing the sum of risk, revenue, individual portfolio deviations, or minimizing / maximizing / equalizing individual portfolio deviations. Individual portfolio deviations can be expressed as slack variables.

  Next, in step 1012, individual portfolio objectives used in optimizing the individual portfolio are received. These objectives can relate to risk or revenue.

  In step 1014, the portfolio is individually optimized using the constraints and objectives imposed on the individual portfolio. This newly optimized asset data is then aggregated at step 1016 and the aggregated optimized asset data is checked at step 1018 to determine if it is within the bounds of the global constraint. If the aggregate optimized asset data satisfies the global constraint, a determination is made at step 1020 to check whether the solution best meets the global objective. If the solution meets the global objective, at step 1024, an optimized solution for the portfolio is displayed.

  If the aggregate optimized asset data fails to meet the global constraints or fails to meet the global objective, the individual objective function is “punished” in step 1022 and the optimization Re-execution begins at step 1014. As part of an effort to meet the global constraints imposed on optimization, the objective is penalized. Lagrangian relaxation techniques can be used. The quantity at which the objective function is “punished” is determined using Lagrangian Dual Problems, part of Lagrangian relaxation, a technique well known in the optimization field.

  Although it is not necessary to apply global objectives to multi-portfolio optimization, this option provides an additional level of control that has not been historically available in multi-portfolio optimization solutions. Two examples of global objectives are minimizing the total individual portfolio deviation and minimizing the worst individual portfolio deviation. These examples are described in further detail below.

Fairness between imbalanced size portfolios When optimizing objectives and constraints across multiple portfolios, portfolios of unbalanced sizes are combined together because all of the individual accounts in the multiple portfolio are linked by the cumulative number of interests traded. When optimized to, unintended results can occur. Specifically, if two or more portfolios are of different sizes and have a common asset to be bought and sold, a larger portfolio may adversely affect the buying and selling of smaller portfolios.

  For example, suppose that two portfolios A and B are purchased separately when they are optimized. Portfolio A is 10 times larger than Portfolio B, Portfolio A needs to purchase 100,000 shares, while Portfolio B only needs to purchase 10,000 shares. However, the purchase of 100,000 shares in Portfolio A will push up the price of IBM shares and adversely affect the purchase of a smaller 10,000 shares in Portfolio B.

  In addition, to establish fairness, the present invention, in some embodiments, allows for splitting transaction costs between portfolios in situations where crossing is allowed. For example, a large portfolio that buys and sells a large number of shares bears a lower transaction cost per share than a small portfolio that buys and sells a small number of shares. Therefore, as part of an effort to prevent negative impacts on smaller portfolios in situations where crossings are allowed and large portfolios benefit from crossings, the transaction costs associated with buying and selling small portfolios are , In whole or in part, can be redistributed into a larger portfolio.

  Thus, according to one embodiment of the present invention, an optimization system and method is used to reduce unintended results from size differences when unbalanced size portfolios are optimized together. Features can be included to ensure fairness.

  The fairness aspect can be reduced by optimizing the volume of cross-trading between portfolios while reducing trading costs and / or when larger portfolios are trading large quantities of the same asset. This can be achieved by reducing or eliminating trading in large portfolios. The trading cost per share of each security is set to depend on the total trading volume of the securities traded by all managers of the relevant portfolio. When modeling trading costs, the algorithm for distributing net trading volume among multiple portfolios to be optimized takes into account the total exchange value (wealth) of each portfolio, and thus the larger portfolio share. The cost of buying and selling is cheaper. One embodiment of the present invention can be described according to the following model.

  The part of the individual portfolio i model that relates to buying and selling costs can be presented as follows:

Where w ⁱ (a) is the allocation of asset a in portfolio i and b ⁱ (a) and s ⁱ (a) are the purchases of asset a (as part of the exchange value of portfolio i), respectively. And the amount sold, h ⁱ (a) is the initial asset of asset a in the portfolio, a is the population of assets, and i is the number of the individual portfolio.

  If crossing is allowed, some of the multiple portfolios that are optimized portfolios purchase securities from other portfolios. This type of transaction is considered “free” (ie, no additional transaction costs are incurred). Only the net balance of non-crossed equity must be purchased or sold in the open market. Thus, in general, each portfolio has transaction costs only for external purchases and sales (volume) of securities.

  The net total imbalance for all portfolios can be expressed by the following equation:

  This equation can also be rewritten in linear form using the following two inequalities:

  The transaction cost per equity depends on the net total volume v (a) of buying and selling. The transaction cost per equity can be found using:

  here,

It is. Based on this model, the allocation of trading costs per asset across the portfolio depends on the total trading volume of the asset, the exchange value of the portfolio, portfolio objectives, and portfolio constraints.

  Thus, the cost of buying and selling for a large portfolio is relatively cheaper than the cost of buying and selling for a small portfolio, so the transaction cost allocated across a size imbalanced portfolio is much larger. Can be allocated more to the portfolio. Any other priority can be imposed by introducing a trading cost utility function into the model.

  If crossing is not a “free” transaction, the model can be modified as follows:

Here, t _cross is a transaction cost per buying and selling of the cross buying and selling. However, when crossing is not allowed and multiple portfolios with imbalanced sizes buy and sell the same assets in the same direction, the smaller portfolios will become more expensive due to the increased transaction costs facilitated by buying and selling larger portfolios. It may be best to reduce or completely eliminate buying and selling.

  Take an example with two portfolios of imbalance in size with the following characteristics:

  A comparison between individual optimization and multi-portfolio optimization that considers fairness without crossing is as follows.

  Individual optimization of each portfolio requires the buying and selling of assets APC, BBY, and BIIB. However, individual optimization means that even though the amount of portfolio 1 is 10 times smaller than portfolio 2, smaller portfolio 1 will face the same price impact as larger portfolio 2 Do not consider the facts.

  Multi-portfolio optimization recognizes the price impacts perceived by Portfolio 1 and finds a different allocation for Portfolio 1 that does not require buying or selling APC, BBY, and BIIB. Different allocations result in slightly higher tracking errors, but the corresponding utility losses are more offset by the reduction in trading costs, as shown in the table below.

Minimizing the sum of individual portfolio deviations-Method A
The multi-portfolio global objective aimed at minimizing the sum of individual portfolio deviations is used to minimize the collective ΔΩ _{i of the} portfolio. This means that the sum of ΔΩ _i (the deviation of each portfolio from its optimal value that is a fraction between 0 and 1) is minimized until any other individual purpose is compromised. . Minimizing the deviation of each individual portfolio from its optimal value ensures that larger portfolios cannot be given priority over smaller portfolios due to unbalanced trading volume.

  The multi-portfolio global objective of minimizing the sum of individual portfolio deviations can be presented mathematically as follows:

here,
X _i is a set of constraints imposed on the allocation of individual portfolio i,
X _i is the distribution vector of individual portfolio i,
• Inequality E.3 presents global constraints (total number of equity sold or trading costs) across all portfolios.

Multiple portfolios that attempt to meet this global objective are partially optimized and checked for compliance with global constraints. The best solution for this purpose is that the optimization solution for each individual portfolio that can be expressed as Ω ^* _i = Ω _i ^start is also an optimization solution that can be realized under any global constraint. However, in situations where Ω ^* _i = Ω _i ^start does not occur, the individual purpose can be “punished” to minimize the total deviation across all portfolios.

Minimizing worst individual portfolio deviation-Method B
The multi-portfolio global objective of minimizing the worst individual portfolio deviation is used to minimize the worst individual ΔΩ _{i of the} portfolio. This means that the ΔΩ _{i of the} portfolio with the worst deviation (the deviation of each portfolio from its optimal value, which is a fraction between 0 and 1) is improved until any other individual purpose is compromised. Means. This is due to differences in the handling of large and small portfolios, where the large portfolio deviation is compromised as part of an effort to improve the small portfolio deviation when the small portfolio has the worst deviation. It is a safety measure against becoming a thing. In some cases, this global objective results in a situation where each portfolio has an equal ΔΩ _i .

The global objective of the multi-portfolio aiming to minimize the worst individual portfolio deviation (F ^* ) can be presented mathematically as follows.

For multi-portfolio optimization using objectives, the values of Ω ₁ ^start and Ω ₂ ^start are affected by E. and E.

Can be obtained by solving E.6 finds the minimum number of shares that must be traded to meet both portfolio requirements.

Multiple portfolios that attempt to meet this global objective are partially optimized and checked for compliance with global constraints. The best solution for this purpose is that the optimization solution for each individual portfolio that can be expressed as Ω ^* _i = Ω _i ^start is also an optimization solution that can be realized under any global constraint. However, in the situation where Ω ^* _i = Ω _i ^start does not occur, the next best solution is a solution with equal ΔΩ _i for each portfolio. To reach this solution, the individual objective can be “punished” to minimize the worst individual portfolio deviation.

(Example)
(Example of multi-portfolio using global objectives)
The proposed approach is now demonstrated in the following two examples. In each example, two long (buy-in) portfolios (n = 2) with S & P 500 as the “population” are optimized. The data used in this example is taken from the 31-Dec-2006 ITG-daily risk model file on December 31, 2006, and specific risk values are alpha and closing prices. It is.

(Example 1)
In the first example, the two portfolios have a cash allocation, and in the second example, these portfolios are rebalanced using different requirements.

  In the first embodiment, the related information is as follows.

  Portfolios 1 and 2 are limited by the total trading volume of 200,000 shares. range

Both portfolios were individually optimized to calculate the values Ω ₁ ^* and Ω ₂ ^* . The values of Ω ₁ ^start and Ω ₂ ^start were obtained by solving Problem E.10. The results of these calculations are as follows.

  A graphical representation of the multi-portfolio optimization results for these portfolios under methods A and B and heuristics is shown in FIGS. As shown in FIG. 11, the heuristic method prioritizes portfolio 1 over portfolio 2, so portfolio 1 is optimized to approximately its individual optimal value. However, it is also shown that Portfolio 2 deviates more than 20% from the solution of the unconstrained problem. Comparing this result with the results of methods A and B, it can be seen that by using global objectives, a better controlled method of multi-portfolio optimization can be implemented. FIG. 12 shows that the total number of interests traded in the portfolio is still in compliance with the global constraints of 200,000 interests, although the optimization is different. The data used in creating Figure 11 and Figure 12 can be found in Table 4 (Table 9), Table 5 (Table 10), Table 6 (Table 11), and Table 7 (Table 12) below. Can do.

(Example 2)
Example 2 is a more complex scenario involving global constraints, individual constraints, global objectives, and individual objectives.

  In the second embodiment, the related information is as follows.

  Portfolios 1 and 2 are limited by the total cost of buying and selling of $ 40000.00. range

Both portfolios were individually optimized to calculate the values Ω ₁ ^* and Ω ₂ ^* . The values of Ω ₁ ^start and Ω ₂ ^start were obtained by solving Problem E.10. The results of these calculations are as follows.

  The results of individual portfolio optimization do not meet global constraints. Specifically, the global constraint that limits the total cost of buying and selling to $ 40000.00 is not met. Individual purposes are therefore “punished”.

  In addition, this example considers the possibility that crossing can be a constraint that affects the total cost of buying and selling during portfolio optimization. Specifically, in this example, the buying and selling costs are more expensive if crossing is not allowed. The parentheses surrounding the numbers in the trading cost column indicate that it is a crossing, and if the crossing is not allowed, an increase cost corresponding to that is generated.

  FIG. 13 shows the portfolio deviation after optimization using methods A and B in an environment where crossing is allowed. FIG. 14 shows portfolio deviations after optimization using methods A and B in an environment where crossing is not allowed. It can be seen that the deviation is larger when crossing is not allowed. This was due, at least in part, to the increased cost of non-crossing trades that subsequently reduced the number of trades that could be made in the portfolio before the global trade cost constraint was exceeded.

  The data used to create FIG. 13 can be found in Table 8 (Table 15) and Table 9 (Table 16), and the data used to create FIG. 17) and can be found in Table 11 (Table 18).

  While exemplary embodiments of the present invention have been described herein, the present invention is not limited to the various embodiments described herein and will be understood by one of ordinary skill in the art based on the present disclosure. Any and all embodiments having such modifications, omissions, combinations (eg, aspects of the various embodiments), adaptations, and / or alterations. Limitations in the claims should be construed broadly based on the language used in the claims, and are not limited to the examples described in this specification or during the performance of the application, and such examples are not exclusive. Should be interpreted. For example, in the present disclosure, the word “preferably” is non-exclusive and means “preferably but not limited to”. Means-plus-function limitation or step-plus-function limitation is defined by the following conditions for a specific claim limitation: a) “means for ) "Or" step for "is explicitly stated, b) the corresponding function is clearly stated, c) the structure, material, or the act of supporting that structure is stated It is only used if all of the conditions that are not specified exist in the limitation.

320 computers
322 Central processing unit (CPU)
324 input / output (I / O) devices
326 bus
328 memory
330 Market and accounting data
338 software
340 modules
341 modules

Claims (39)

A method for optimizing a plurality of portfolios, wherein each portfolio includes one or more interests in one or more tradeable assets, the method comprising:
a) receiving asset data defining a plurality of said portfolios;
b) receiving one or more individual portfolio optimization decision variables corresponding to one or more of the plurality of portfolios;
c) receiving one or more global optimization decision variables;
d) optimizing the asset data for each of the plurality of portfolios based on a corresponding one or more of the individual optimization decision variables;
e) aggregating the optimized asset data to generate aggregate optimized asset data;
f) determining whether the aggregate optimized asset data satisfies the one or more global optimization decision variables;
g) outputting the optimized asset data only if the one or more global optimization decision variables are satisfied in step f. In step b, the individual portfolio optimization decision variable is
The method of claim 1, comprising one or more individual optimization constraints and one or more individual optimization objective functions. In step c, the global portfolio optimization decision variable is
The method of claim 2, comprising one or more global optimization constraints and one or more global optimization objective functions. h) if the one or more of the global optimization constraints are not satisfied, based on each of the optimized asset data and the aggregated optimized asset data, the one or more individual optimization objective functions 4. The method of claim 3, further comprising the step of adjusting each. 5. The method of claim 4, further comprising: i) reoptimizing the asset data based on the adjusted decision variable.   5. The method of claim 4, wherein step h adjusts the one or more individually optimized objective functions using Lagrange relaxation and dual problem techniques. The method of claim 3, further comprising: determining whether the one or more global optimization objective functions are satisfied if the one or more global optimization constraints are satisfied. If the one or more global optimization objective functions are not met, the one or more individual optimization objective functions are determined based on each of the optimized portfolio data and the aggregated optimization asset data. 8. The method of claim 7, further comprising the step of adjusting. Step a is
At least one of the name of the asset, the asset symbol, the market price of the asset, the average price at which the asset was purchased, the number of asset interests in one of the portfolios, and the number of asset interests in the portfolios The method of claim 1, further comprising: receiving one.   The step of receiving one or more individual optimization constraints includes the maximum number of shares that can be bought and sold for the portfolios, whether “latecomers” are allowed, whether “crossings” are allowed, “fairness” 3. The method of claim 2, further comprising receiving at least one constraint defining whether or not is considered, or a total maximum transaction cost for all portfolios, a maximum level of acceptable risk.   The method of claim 2, wherein receiving the one or more individually optimized objective functions further comprises receiving at least one objective related to at least one of risk, rate of return, or buying and selling costs. Method.   4. The method of claim 3, wherein the one or more global objectives include at least one of minimizing a total individual portfolio deviation or minimizing a worst individual portfolio deviation. A computer readable storage medium having stored therein computer executable program code comprising the following operations:
a) receiving asset data defining a plurality of said portfolios;
b) receiving one or more individual portfolio optimization decision variables corresponding to one or more of the plurality of portfolios;
c) an operation that receives one or more global optimization decision variables;
d) optimizing the asset data for each of the plurality of portfolios based on a corresponding one or more of the individual optimization decision variables;
e) an operation of aggregating the optimized asset data to generate aggregate optimized asset data;
f) an operation of determining whether the aggregated optimization asset data satisfies the one or more global optimization decision variables;
g) for optimizing a plurality of portfolios by performing the operation of outputting the optimized asset data only if the one or more global optimization decision variables are satisfied in operation f A computer readable storage medium having stored therein computer executable program code. In operation b, the individual portfolio optimization decision variable is
14. The computer readable storage medium of claim 13, comprising one or more individual optimization constraints and one or more individual optimization objective functions. In operation c, the global portfolio optimization decision variable is
The computer readable storage medium of claim 14, comprising one or more global optimization constraints and one or more global optimization objective functions. h) if the one or more of the global optimization constraints are not satisfied, based on each of the optimized asset data and the aggregated optimized asset data, the one or more individual optimization objective functions 16. The computer readable storage medium of claim 15, wherein further instructions for performing operations to adjust each are stored therein. 17. The computer readable storage medium of claim 16, wherein further instructions for performing an operation to re-optimize the asset data based on the adjusted decision variable are stored therein.   17. The computer readable storage medium of claim 16, wherein operation h adjusts the one or more individual optimization objective functions using Lagrange relaxation and dual problem techniques. If the one or more global optimization constraints are satisfied, further instructions are stored therein for performing operations to determine whether the one or more global optimization objective functions are satisfied; The computer-readable storage medium according to claim 15. If the one or more global optimization objective functions are not met, the one or more individual optimization objective functions are determined based on each of the optimized portfolio data and the aggregated optimization asset data. 20. The computer readable storage medium of claim 19, wherein further instructions for performing the adjusting operation are stored therein. The operation a is
At least one of the name of the asset, the asset symbol, the market price of the asset, the average price at which the asset was purchased, the number of asset interests in one of the portfolios, and the number of asset interests in the portfolios 14. The computer readable storage medium of claim 13, further comprising an operation of receiving one.   The operation a, which receives one or more individual optimization constraints, is the maximum number of shares that can be bought and sold for the portfolios, whether “latecomers” are allowed, whether “crossings” are allowed, “fair” 15. The computer readable storage medium of claim 14, further comprising receiving at least one constraint that defines whether "sex" is a consideration, or a total maximum transaction cost for all portfolios, a maximum level of acceptable risk.   15.The operation of receiving one or more individually optimized objective functions further comprises receiving at least one objective related to at least one of risk, rate of return, or buying and selling costs. Computer-readable storage medium.   The computer-readable storage medium of claim 15, wherein the one or more global objectives include at least one of minimizing a sum of individual portfolio deviations or minimizing a worst individual portfolio deviation. A system for optimizing multiple portfolios of assets,
Receive asset data defining a plurality of the portfolios, receive one or more individual portfolio optimization decision variables corresponding to one or more of the portfolios, and receive one or more global optimization decision variables Using the asset data and a corresponding one or more of the individual optimization decision variables to optimize each portfolio of the plurality of portfolios to generate aggregate optimized asset data Aggregated asset data, determine whether the aggregated optimized asset data satisfies the one or more global optimization decision variables, and the one or more global optimization decision variables are satisfied Limited to include a client interface configured to output the optimized asset data. System.   26. The system of claim 25, wherein the received individual portfolio optimization decision variables include one or more individual optimization constraints and one or more individual optimization objective functions.   27. The system of claim 26, wherein the received global portfolio optimization decision variables include one or more global optimization constraints and one or more global optimization objective functions.   If the one or more of the global optimization constraints are not satisfied, each of the one or more individual optimization objective functions is based on each of the optimized asset data and the aggregated optimization asset data. 28. The system of claim 27, wherein the client interface is further configured such that the client interface coordinates.   30. The system of claim 28, wherein the client interface is further configured to reoptimize the asset data based on the adjusted decision variable.   30. The system of claim 28, wherein the client interface is further configured to adjust the one or more individual optimization objective functions using Lagrangian relaxation and dual problem techniques.   The client interface is further configured such that, if the one or more global optimization constraints are satisfied, the client interface determines whether the one or more global optimization objective functions are satisfied 28. The system of claim 27.   If the one or more global optimization objective functions are not met, the one or more individual optimization objective functions are determined based on each of the optimized portfolio data and the aggregated optimization asset data. 32. The system of claim 31, wherein the client interface is further configured such that the client interface coordinates.   At least one of the name of the asset, the asset symbol, the market price of the asset, the average price at which the asset was purchased, the number of asset interests in one of the portfolios, and the number of asset interests in the portfolios 26. The system of claim 25, wherein the client interface is further configured to receive one.   Maximum number of interests that can be traded for the multiple portfolios, whether “latecomers” are allowed, whether “crossings” are allowed, whether “fairness” is taken into account, or the total maximum for all portfolios 27. The system of claim 26, wherein the client interface is further configured to receive at least one constraint that defines a transaction cost, a maximum level of acceptable risk.   27. The system of claim 26, wherein the client interface is further configured to receive at least one purpose related to at least one of risk, rate of return, or buying and selling costs.   The client interface is further configured to receive the one or more global objectives including at least one of minimizing a total individual portfolio deviation or minimizing a worst individual portfolio deviation. 27. The system according to 27.   28. The system of claim 27, further comprising a user display device for displaying the output optimized asset data.   28. The system of claim 27, wherein the output optimized asset data takes the form of a trade list for transmission to a trade system. A method for applying a “fairness” principle to optimization of a plurality of portfolios, wherein each portfolio includes one or more interests in one or more tradeable assets, the method comprising:
a) receiving asset data defining a plurality of said portfolios;
b) receiving one or more individual portfolio optimization decision variables corresponding to one or more of the plurality of portfolios;
c) optimizing the asset data for each of the plurality of portfolios based on a corresponding one or more of the individual optimization decision variables;
d) determining whether any of the plurality of portfolios is adversely affected by an optimization solution of any of the other plurality of portfolios;
e) adjusting the one or more individual portfolio optimization decision variables to compensate for the adverse effects;
f) reoptimizing the asset data based on the adjusted one or more of the individual optimization decision variables;
g) outputting the optimized data.