Rapid method of analysis for correlation of asset return to future financial liabilities

[0002] The present invention is related generally to a method and system for selecting a portfolio of assets for achieving optimum correlation of asset return to a selected standard financial index. More particularly, the invention is related to a highly efficient, rapid method and system for choosing an asset portfolio having the optimum correlation of the asset return to a time dependent financial index, such as a financial liability, at each of a number of selectable asset return levels.

[0003] Management of portfolios of assets has historically emphasized maximizing the return on assets with the objective of at least outperforming the market. However, in some financial industries the objective, or the figure of merit, is also related to meeting future liabilities rather than just achieving high return on assets. Frequently, an institution will have a future intended use of the assets which requires their availability at some future time. If assets are performing differently than liability requirements, substantial financial hardship can result. For example, insurance companies and corporate pension plans have well defined future financial liabilities which must be met. Consequently, although return on assets is one important objective, meeting future liabilities is also important and can be even more important in many instances. In fact, many pension plan managers are now required to meet the standards set forth in FASB Statement 87 (Financial Accounting Standards Board) on pension fund accounting. Under the FASB Statement a market interest rate return on pension funds is the standard index and is to be based on A-rated ten year corporate bonds. Under this FASB Statement any deficit in corporate pension funds are now reflected on the balance sheet. Any such deficit would therefore have substantial adverse effect on the apparent net worth of the subject corporation. Consequently, this FASB Statement standard strongly encourages maintenance of a surplus for a pension fund. As an example of the importance of matching the liability requirements under the FASB Statement, consider the percentage change possible for pension plan liabilities, as measured by the accumulated benefit obligation (ABO). If, for example, interest rates increase by 1% in one year over the present rates, the present value of the ABO would decline by 10% if the ABO has a duration of 10 years. Likewise, if interest rates were to drop by 1% in one year, the present value of the ABO would increase by 10%. The potential for such dramatic fluctuations in liabilities clearly deserves careful attention by parties obligated to meet future liability streams.

[0004] In order to timely meet future financial liabilities and maintain a proper surplus fund for a pension plan, a number of methodologies have arisen including “immunization”, “cash matching”; and some preliminary efforts have even been directed to utilizing stock funds.

[0005] The “immunization” method of meeting future financial liabilities uses bonds having substantially the same duration as the liability stream. Duration is a measure of volatility expressed in years, which is similar to, but more precise than, average life. The duration is calculated as the weighted average amount of time to the receipt of the payout. There are however significant drawbacks to “immunization”, with one primary disadvantage being the relatively low excess return on assets generally achieved by the method. Additional limitations are imposed by the two major assumptions made by the strategy: The yield curve (a plot of yield to maturity on bonds versus their time to maturity) will only make parallel shifts. Consequently, regardless of maturity, when market conditions change, all bonds allegedly move exactly the same amount in yield. This clearly is not the case since there have been substantial inconsistencies in the past for the difference in rates for short-term bonds and long-term bonds. Secondly, all cash flows in excess of required annual payments can allegedly be reinvested at the yield to maturity of the portfolio. This presumption is also clearly not true since sharply declining or rising interest rate environments will make it extremely difficult to carry out reinvestment. Furthermore, this strategy does require more ongoing management of the portfolio in order to sell or buy more securities to match the actuarial schedule and maintain a proper asset/liability match.

[0006] The “cash matching” method utilizes a bond portfolio having numerous component bonds with various maturity dates and payout rates to precisely match the liability requirements of the pension plan. Such an approach has the same primary disadvantage as the “immunization” method and further requires additional effort to assemble the portfolio. Frequently, the “cash matching” method demands payment of a premium to achieve the correct mix of bond rates and maturity. Both of the first strategies (“immunization” and “cash matching”) must invest in fixed income securities to provide the assurance of receiving the necessary cash flows. In fact, they must primarily invest in U.S. treasury obligations since investments in corporate or mortgage securities increase the chance for default or for call risks which can have the effect of changing the projected cash flow.

[0007] Pension plan liabilities or other future liabilities, such as are present in the insurance industry, are long term in nature. Therefore, a future liability stream can greatly benefit from the compounding effect of investment in higher returning assets, such as common stocks. However, attempts to characterize stocks in terms of a time duration parameter or otherwise have not been successful. In the last few years many unsuccessful attempts have been made to develop a system whereby a portfolio of equities is linearly optimized relative to a liability stream. There have been attempts to parallel the “cash matching” techniques with the use of stocks, instead of bonds. This approach has involved matching the expected dividend flow of the portfolio to the liability stream. Unfortunately, stock dividend yields are unpredictable, particularly beyond 3 years in the future. Another major effort in equities has been directed to an “immunization” type treatment. In this effort an attempt was made to calculate the duration of stocks on an individual basis, as well as on a portfolio basis; but these attempts also have been unsuccessful, primarily due to the long term unpredictability of stock dividends.

[0008] In a related patent application, incorporated by reference herein and having U.S. Ser. No. 281,560 and filed Dec. 8, 1988, now abandoned an improved method and system were set forth directed to correlating return on assets to a financial objective over time. In performing the analysis to determine the optimum assets of a portfolio to track the financial objective, the machine time and efficiency of the evaluation process can limit the number of assets considered in constructing the portfolio. Such limitations on the number of assets which are considered for inclusion in the optimum portfolio can also limit the performance of the selected portfolio.

[0009] It is therefore an object of the invention to provide an improved method and system for determining the optimum portfolio of assets for tracking a financial index.

[0010] It is another object of the invention to provide a new method and system of efficiently selecting the optimum portfolio of assets for tracking a financial index.

[0011] It is an additional object of the invention to provide an improved method and system of rapidly analyzing a large number of potential assets to select the optimum portfolio of assets to track a financial index.

[0012] It is a further object to provide a new method and system of enlarging the number of potential assets under consideration for inclusion in a portfolio of assets, while reducing the time required to select the portfolio of assets which best track the behavior of a financial index.

[0013] It is another object of the invention to provide a rapid, more efficient method and system of selecting the weighted values for assets selected from a universe of possible assets for a portfolio designed to track a financial index.

[0014] It is an additional object of the invention to provide an improved method and system for reinvesting cash flow from a portfolio starting with that current portfolio of assets.

[0015] Further objects and advantages of the present invention, together with the organization and manner of operation thereof, will become apparent from the following description when taken in conjunction with the accompanying drawings described hereinbelow.

[0016] FIG. 1A is a functional flow chart illustrating operation of one method of portfolio construction and

[0017] FIG. 1B is a flow chart illustrating operation of a particular method of the invention;

[0018] FIG. 2A illustrates simulation results of annual total liability returns of a pension plan (dashed) and annual total asset returns (solid line) for a preferred form of the invention and

[0019] FIG. 2B shows annual total liability returns of the simulated pension plan (dashed) and annual total asset returns (solid line) for the Standard & Poors 500;

[0020] FIG. 3A is a bar graph of simulation results for funded status returns over time for a preferred method of the invention and

[0021] FIG. 3B is a bar graph of funded status returns over time for the Standard & Poors 500;

[0022] FIG. 4 a comparative plot of cumulative funding status for simulation results over time for a pension plan liabilities (dashed), a portfolio derived by a preferred method of the invention (solid line) and the Standard & Poors 500 index (dashed and dotted);

[0023] FIG. 5A shows the correlation between asset return and liability return for a preferred method of the invention and

[0024] FIG. 5B illustrates correlation between asset return and liability return for the Standard & Poors 500 index; and

[0025] FIG. 6 illustrates the boundary line of minimum risk for various future asset return levels.

[0026] Broadly stated, a method and system are described for selecting a portfolio of assets and correlating a future asset return of the portfolio to a financial index, such as, a liability index, an inflation index, or any other accepted index and mixtures thereof. Specific examples of indices are liability indices, such as, individual corporate pension plan liabilities and insurance company liabilities. The consumer price index and wage growth index are examples of an inflation index, and other indices can include accepted stock price indices and futures markets indices. The method includes selecting asset portfolios which optimally correlate portfolio returns to the future desired payouts or payments needed over time to fulfill the desired financial objective. In the general case the user selects a standard index to which optimum correlation is desired for the selected portfolio having a future asset return over time. The process of selecting the standard index can involve obtaining input (such as actuarial) in terms of the characteristics of future cash payments discounted to present value based on a range of discount rate and wage (inflation) values. This information can be used to construct a functional behavior for the present value of the liability. A decision is then made, such as by a company pension fund manager, that certain discount rates and inflation assumptions should be made. On this basis the current liabilities are projected back in time using these assumptions and a plurality of assets are examined to determine their sensitivity to the past behavior of the liability returns. In the most general sense if one can determine an index to which a portfolio of assets has a strong correlation, this sensitivity can be used to select a set of assets which will match the behavior of the index as it changes over time. As a particular example an actuary can provide specific ranges of present value liability for a range of discount rates and inflation rates. The change over time of the liability from month to month over a twenty four month period can yield a liability return. The analysis to be described in more detail hereinafter determines which selected ones of a plurality of assets track the liability returns with best correlation. The resulting weighted set of assets form the portfolio to follow the future liability returns. An analysis using the selected standard index can be performed on a plurality of assets, such as, for example, at least one of the following categories of assets: stock securities, real estate investments, futures contracts, options, commodities, currencies and precious metals. The analysis allows the identification of the combination of weight percentages of selected ones of the plurality of assets yielding the optimum correlation of the future asset return to the standard index. Optimum correlation is thus achieved by calculating a minimum standard deviation or a variance for the difference between the return of the portfolio of assets and the selected standard index return. This method and system are particularly applicable for, but not limited to, the insurance industry and management of pension fund liabilities.

[0027] FIGS. 1A and 1B illustrate in functional flow charts the procedures followed in carrying out two forms of the invention. In the first method shown in FIG. 1A (and described previously in pending application having U.S. Ser. No. 281,560, now abandoned) the correlation of the expected asset return of a portfolio to a standard index one is initiated by input of various basic information. This information includes, for example, establishing the fundamental statistical characteristics of liability returns, and future payment schedules for matching a desired index, such as the future stream of financial liabilities of a pension plan. As described hereinbefore, the future payment schedule for a pension plan can be determined by using actuarial data. These future liabilities can be characterized in terms of an accumulated benefit obligation (ABO), that is, the price you would have to pay if the liabilities were sold at a selected time. The total outlay required to pay retirement wages for the pension plan are discounted back to the present value at the market rate interest (currently 10%). Other related characterizations can be used, such as a projected benefit obligation (PBO), by accounting for inflation in the growth of wages at retirement. This amount is converted to a percentage and an expected salary at retirement, discounted to present value. Therefore, although the ABO is affected primarily by interest rates, additional standard measures, such as the PBO, account for inflation. Therefore, the method is also generally effective for calculating the convolution of complex effects with one another. The method only requires optimizing correlation of the time behavioral performance of future asset return relative to the particular standard index, which includes any conceivable selected characteristic which assets are found to be sensitive to.

[0028] In the manner illustrated in step 1, box 21, of FIG. 1A, various input files are therefore created to begin the analysis. These input files can include, for example, asset return information for the universe or plurality of assets to be sampled in the analysis. Also established as data files are the data representative of the standard asset return over time, such as target returns for a future liability stream of a pension plan or an insurance company. The future liability stream can depend on interest rates and/or inflation rates and other variables which can affect the liability stream. For example, as described hereinbefore, a surface can be generated which describes the behavior of liability return as a function of both interest rates and inflation rates. Other information in the data files can be identification information for the plurality of assets, current price and market capitalization of the assets, as well as the characteristic weight percentages of assets in a previously selected portfolio. Weight percentages, for example, from a prior period would be used in the most preferred embodiment.

[0029] As illustrated in the second box 22 of FIG. 1A, the number and type of the plurality of assets should be specified, the time period for matching the standard, the number of returns used to calculate a statistical correlation and the number of asset returns used to calculate averages for the plurality of securities to be analyzed to select the optimum set of weighted assets.

[0030] The method in FIG. 1A then, in box 23, initializes information preparatory to analyzing the plurality of assets, such as, establishing names of securities, associated identifier information, industry codes, prices of securities, market capitalization, weight and percent of the previously calculated prior portfolio, the period for the asset return, the name of the target or standard for measuring a standard asset return and the time period for the standard returns.

[0031] In the next box 24 in FIG. 1A the correlation between the future asset return and the standard index is optimized by first generating a covariance array. While other nonlinear statistical analyses are possible, this method being described is a preferred method of carrying out the analysis. For example, another useful statistical method of analysis is correlation parameterization which is embodied in the computer software program Appendix III. As shown in the next step, box 25, in FIG. 1A, the average return of each security is calculated followed by imposing certain constraints on the calculation, box 26, such as setting a range of weight percentages to be tried. The calculation is then implemented to a solution by a standard computer program quadratic technique (see Appendix I), boxes 27 and 28. This step is then followed by determination, box 29, of various statistical parameters, such as a and B, standard error, portfolio returns over various time periods and for selected weights. The analysis is then completed, box 30, by printing output see attachment to Appendix I) such as asset weights, sensitivity factors for selected assets of the portfolio, statistical parameters, sorted buy and sell orders and sector weights.

[0032] A simple example of utilizing the preferred statistical method is illustrated for a portfolio containing three stocks (designated 1, 2, and 3). In order to find the optimum weight percent for each of the three stocks in the portfolio, the minimum standard deviation (square root of variance) is calculated for the differences between the assets of the portfolio and the future liabilities as represented by the standard asset return over time. The risk is therefore defined as the standard deviation of differences: $\mathrm{Risk}=\frac{{\left[\sum _{i=1}^n\left(R_{\mathrm{pi}}{R}_{T}i\right)-n{R}_p{R}_t\right]}^{1/2}}{{\left(n-i\right)}^{1/2}}=\sigma \left[R_p-R_t\right]={\mathrm{variance}}^{1/2}$

[0033] where:

[0034] R_pi=total return on the portfolio during period i;

[0035] R_Ti=total return on target or standard in period i,

[0036] σ=standard deviation;

[0037] R_p=average return on portfolio, i=1,n; and

[0038] R_T=average return on target or standard portfolio of assets,

[0039] i=1, . . . , n . . .

[0040] The portfolio return equals percentage weight for each stock times the return on that stock: $R_{\mathrm{pi}}=\sum _{j=1}^3{x}_j{R}_{\mathrm{sj},i}$

[0041] x_j=the weight in the portfolio of the stock j

[0042] R_sj,i=the return on stock j in periods i

[0043] Now in the definition of risk, as set forth above, we can substitute the following: $R_{\mathrm{pi}}=\sum _{j=1}^3{x}_j{R}_{\mathrm{sj},i}$

[0044] Making this substitution, a determination of risk in the manner set forth above results in the calculation of the covariance of the stock with each of the other stocks in the portfolio after subtracting the return of the target, or standard index, from the future asset return of each stock.

[0045] The covariance of stock 1 with stock 2 is therefore: $\left(R_1,R_2\right)=\sum _{i=1}^n\left(R_{\mathrm{s1i}}-R_{\mathrm{Ti}}\right)\left(R_{\mathrm{s2i}}-R_{\mathrm{Ti}}\right)-n\left({\hat{R}}_1-{\hat{R}}_T\right)\left({\hat{R}}_2-{\hat{R}}_T\right)$

[0046] We calculate all the spanning covariances and put them in a matrix form: $\left[\begin{array}{cc}\mathrm{Cov}\left[\left(R_1-R_T\right),\left(R_1-R_T\right)\right]& \mathrm{Cov}\left[\left(R_1-R_T\right),\left(R_s-R_T\right)\right]\\ \mathrm{Cov}\left[\left(R_3-R_T\right),\left(R_1-R_T\right)\right]& \mathrm{Cov}\left[\left(R_3-R_T\right),\left(R_3-R_T\right)\right]\end{array}\right]$

[0047] In order to calculate the risk, we add up all the Cov terms times the weights in each stock: $\mathrm{Risk}=\sigma \left(R_p-R_T\right)=\{{\left[\sum _{i=1}^3\sum _{j=1}^3{X}_i{X}_j\mathrm{Cov}\left[\left(R_i-R_T\right),\left(R_j-R_T\right)\right]\right\}}^{1/2}$

[0048] In order to minimize this “risk” function, we determine the combination of weight percentages for stocks 1 thru 3 which produces the smallest statistical risk. The above described risk can readily be calculated by various means, such as, by a computer program (which is included in Appendix I). The output (see attachment to Appendix I) of the calculation includes the weight percent of each stock and the associated overall risk level. This calculation can be repeated for a range of expected asset return levels and results in generating a nonlinear type “bullet” shape defining the limits of minimum risk over a range of asset return levels for associated standard deviations of funding level (see FIG. 6). The method uses historical returns for the plurality of stocks analyzed in order to calculate the resulting covariance between the standard liability returns and the future returns of the potential portfolio of assets. Appendix II illustrates an example of a computer program for calculating typical liability return data. The method of analysis results in choosing a selected set of assets for the portfolio with a strong inclination of the selected set of assets to respond in a manner such as the standard asset returns over time, which alone can be valuable output. As mentioned hereinbefore, in other embodiments, the nonlinear analysis of a plurality of assets can involve other methods, such as, index correlation parametrization for matching the performance of a target index return (see Appendix III).

[0049] In one embodiment of the invention illustrated in FIG. 1B, the method is a simplification of the more formal procedure of FIG. 1A. The method of FIG. 1B accomplishes, however, the same result as in FIG. 1A but with much greater efficiency and speed.

[0050] The method of FIG. 1B is broken into eleven steps, and the first three steps, boxes 21, 22 and 23, are substantially the same as the method of FIG. 1A. The eleven steps and details of each each step are described below:

[0051] Read in Returns of Selected Securities.

[0052] Various input files are created to begin the analysis, box 21. Large numbers (thousands of securities) can be utilized in this procedure. An input file contains the returns for each security in the prior periods, such as the previous 24 months. Various data is included, such as, security name, industry groups code, market capitalization, trading volume, recent prices, specific identifiers and estimated bid/ask price spreads. The effect of transaction costs associated with the spread in the bid/ask price can be included in the performance analysis. In box 22 the number and type of the plurality of assets are specified. Then, in box 23, the method initializes information preparatory to analyzing the assets.

[0053] Read in Returns of Target Index to Track

[0054] If a portfolio is to be constructed for tracking a specific financial target index, the returns to that target for the relevant period are read into memory arrays. Additional identifying information is also read in from the target data file.

[0055] Set-up Constraints on Upper and Lower Bounds in Terms of Percent of Portfolio for each Security and each Sector or Industry.

[0056] In box 43, the method defines variance as the sum of squared difference between portfolio returns and the target and generate return series for the largest in terms of all assets in the portfolio. Constraints are generated as security type sector type individual security rights. A maximum and minimum percentage weight of the portfolio for each security can be specified to constrain the portfolio. This can be used to insure portfolio diversification and to control costs associated with trading. In addition, sectors of the universe, for example, utility stock, can be constrained by maximum and minimum boundaries. If one has a single target, one can “short” a stock and take a negative minimum position.

[0057] Calculate the Covariance of Each Security with the Target Index, box 35.

[0058] If a target is used, the covariance is calculated for each security and stored in an array. If no target index is used, a zero value for each security is stored in the array.

[0059] Create an Initial Portfolio by Selecting Highest Covariance Securities and Weighting Them at their Upper Limits as Defined by Selected Constraints.

[0060] This step, box 36, creates an initial feasible solution to the problem by filling the vector of portfolio weights according to the constraints and in order of highest covariance.

[0061] Calculate the Objective Function Value at Initial Portfolio Weightings, box 37.

[0062] The objective function can be defined in a number of ways. The computer program allows monthly or moving quarterly returns to be used for optimization. Transaction costs can be considered and their importance magnified or reduced relative to other objectives. With minor changes, other such goals can be incorporated into the objective function. The key is that once the objective function is specified, partial derivatives can be used to guide the search for an optimal portfolio. Any example of a preference that can be created is an increased weighting for a stock with a likely dividend versus one with no dividend.

[0063] Calculate Partial Derivatives For Each Security at Initial Portfolio Weightings.

[0064] A partial derivative is calculated, box 37, for each variable (in this case each security), and a direction can be determined in which to move the individual security weights in order to obtain an improved portfolio solution. The partial derivatives are also used to determine if the weights are optimal. In the prior art, the solution techniques require storage of a full covariance matrix array. This storage requirement has limited the practical number of securities which could be considered at one time in the past methodologies. Indeed, the storage requirements, and to a large degree the processing time, varies as the square of the number of securities in the portfolio under consideration. See the example discussed hereinafter in which the previous methodology is compared to the invention.

[0065] In addition, the methods of solution for these problems in the prior art were slow and cumbersome and subject to failure when the full covariance array was sensitive or a nonunique set of solutions were achievable. This current method requires much less storage, uses a rapid solution technique and allows control of the tolerance used for optimality. Consequently, the improved methods will select a portfolio when several combinations are equally desirable.

[0066] Not only is the amount of necessary memory reduced and the computer calculational time greatly reduced, there is substantial flexibility in defining the objective, assurance of the solution is enhanced and simultaneous considerations of large number of securities allows substantial improvement in optimizing the expected return of the portfolio compared to the target index.

[0067] Change of Portfolio Weightings by Moving in Direction of Improvement is Indicated by Partial Derivatives, box 39.

[0068] Adjustment of the portfolio weights is achieved by a search technique which moves along the constraints and changes in a proper direction of improvement of the objective function. The objective function is calculated at the new weights and a test of improvement is made.

[0069] Test for Convergence to an Optimal Solution Defined by the Kuhn-Tucker Conditions.

[0070] If the objective function is not improving or if the step size used to adjust the portfolio weights becomes extremely small, the search is terminated. This solution is normally a Kuhn-Tucker point (conventional method of establishing optimality conditions) or extremely close thereto within an acceptable epsilon to such a point.

[0071] Recycling Conditions.

[0072] If the termination conditions are not satisfied, then, in box 40, one re-calculates the objective function value, re-calculates the partial derivatives, makes changes in the portfolio weightings to achieve an improved solution and test for convergence.

[0073] The process is by nature iterative and continues cycling until a solution is reached in which diminishing returns are achieved by further cycling.

[0074] Output of Information.

[0075] Relevant portfolio information is output, box 41, with security weightings, objective function values, purchases and sales necessary to achieve the optimum portfolio and industry weightings.

[0076] The detailed output is written into a computer file which then can be examined for relevant information. Order to buy and sell securities can be developed from the information in the output file.

[0077] Included in Appendix IV is an exemplary computer software (source output) program illustrating critical steps of the method of FIG. 1B. Table XI shows exemplary results for a program simulation wherein the target index is the Standard and Poors 500 stock index. Appendix VI illustrates significant distinctions from the optimizer methodology used in the copending patent application having U.S. Ser. No. 281,560, now abandoned.

[0078] Quantifying Magnitude of Reduced Computation Time

[0079] A test was performed on an IBM compatible PC to compare the solution speeds of two portfolio optimization systems. In a prior system, the problem of handling large numbers of securities in a portfolio selection process increased in proportion to the number of securities squared. Thus, a problem involving one hundred securities would take approximately one hundred times as many calculations to solve as a problem with ten securities.

[0080] In the current system the solution difficulty increases by a factor of less than one times the number of securities. In addition, the computing memory required to solve the problem is proportional to the number of securities rather than, as in the prior system, that number squared.

[0081] Solution Time Comparison

[0082] Hardware: IMB compatible PC, 386-20 Mhz CPU, 80387 co-processor

[0083] Problem Size: 100 securities

[0084] 3% portfolio weight upper bound on each security

[0085] 0% portfolio weight lower bound on each security

[0086] Time to solution:

[0087] Prior system—9 minutes 34 seconds

[0088] Current system—52 seconds

[0089] Overall speed-up: 11.0×prior system

[0090] This speed increase is actually of a larger magnitude since approximately 40% of the 52 seconds to solution with the current system is spent reading information from external files and writing information to other files. The solution time speed-up, allowing for reading and writing files, is approximately 17.3× prior system. This speed-up ratio increases in proporation to the number of securities considered squared.

[0091] The speed increase and the reduced computer memory requirements by the current system allow large problems to be solved in a short time, requires limited computer memory, and uses computer hardware which is relatively inexpensive.

[0092] Circumstances arise regularly in the investment field which rapidly change the prospects for securities. The impact of these sudden changes must be incorporated into the security valuation system so that rational alternations in the investment portfolios may be made.

[0093] Examples of sudden changes include: a company is presented with a buy-out offer by another firm; a disaster occurs, such as an oil spill, which may impact a firm's stock price; monetary or fiscal policy changes are implemented by the government. It is important for an investment system to be flexible and fast enough to evaluate the impact these changes may have on a security portfolio.

[0094] The current system allows for estimates of partial monthly returns to be calculated on any day of the month, for these returns to be used in the optimization process, and for the results of the analysis to be completed within a few minutes.

[0095] One advantage of the current system is that analysis of the current investment opportunities can be completed rapidly and recommendations for buying and selling securities can be quickly generated. This allows investment decisions to be made and implemented quickly with confidence.

[0096] Further illustrations of the invention are exemplified by various historical simulations shown in FIGS. 2A, 2B, 3A, 3B and 4 Tables I-III which are taken over the time period of 1975 to 1987. As listed in Table I and in FIGS. 2A, 2B, 3A and 3B the liability stream for a selected pension plan can undergo substantial variation with time. A portfolio of assets has been analyzed in accordance with the preferred statistical method described hereinbefore, and details of the selected portfolio are set forth in Tables IV-X. Over the 1975-1987 time period, the resulting portfolio of assets shows substantially better correlation to the liability stream as compared to the Standard & Poors 500 return. Moreover, as seen in Table II and FIG. 4, the overall cumulative return for the portfolio of assets selected by the preferred method is far better than the Standard & Poors 500. The greatly enhanced stability and good

[0097] statistical correlation with the liability return is further evident in Table III and FIGS. 5A and 5B, wherein detailed comparisons are made between the selected portfolio of assets and the standard liability return.

[0098] In FIG. 6 a range of simulation funding returns for the portfolio of assets are compared with a typical pension fund a mixture of stocks, bonds, real estate and treasury bills. Clearly, the risk is much higher for the typical pension fund; and dramatic improvement in the return, or reduction of risk, results when only 35% of the standard pension fund is modified using the method of the invention.

[0099] In another embodiment, a portfolio of assets can be constructed by selecting a portion of a total portfolio with assets having optimal correlation of asset return to a liability or financial index. The remainder of the portfolio comprises futures contracts which are combined with the correlated portfolio portion to achieve in effect an optimum correlation for the entire portfolio of assets. Further details are set forth in Appendix IV.

[0100] In another aspect of one embodiment, control can be exerted over pension plan surplus by adjusting the level of risk selected for a portfolio of assets. As illustrated in FIG. 6, the expected return can be selected at various levels with the degree of risk, or standard deviation of the funding level, generally increasing as one moves from a position of minimum risk at the top of the “bullet” to higher future returns. Control over a pension plan surplus, or for that matter any plan for which you wish to respond dynamically to control risk/return in concert, can be accomplished over a wide range of risk and return values. Such an approach can be used to manage return under variable risk positions and minimize insurance costs for protecting against underfunding of a plan, such as falling below a predetermined minimum floor. Consequently, as the funding level approaches 100% a minimum risk portfolio of assets should be constructed using the methods described hereinbefore. As the surplus accumulates, the acceptable risk level can be increased for the portfolio of assets by dynamic modification of the portfolio asset components. One can utilize futures contracts as an overlay for an underlying portfolio of assets, having been selected by the basic invention described previously, to create in effect an optimum statistical correlation for the entire portfolio, including the futures contracts. As the surplus approaches 10%-20% excess, a portfolio of assets can be constructed resulting in a much higher level of future return. For example, in FIG. 6, the change in future return from minimum risk to the highest return data point is about a 35% greater return but with an accompanying 70%-80% increase in standard deviation compared to the minimum risk point.

[0101] In a further embodiment of the invention one can overcome problems associated with an indefinite covariant matrix. The current system and method does not use the full covariance matrix to extract a partial derivative to guide the search process, and thus it is not subject to one of the failures that results using standard quadratic programming.

[0102] To us standard quadratic programming algorithums the covariance matrix must be positive semi-definite, or positive definite. This means technically that no row of the matrix can be replicated by a linear combination of other rows.

[0103] However, this condition occurs when there are fewer returns than the number of securities under consideration. This is a significant shortcoming of the standard methodlogy. For example, to consider five hundred securities simultaneously, the user must supply at least five hundred and one returns for each security being considered. If monthly data is being used for the return series, at least forty-one years of data must be available for each security under consideration. Most securities have not been in existence for this period of time.

[0104] If not enough returns are available or if the matrix is indefinite, the standard quadratic solution techniques will fail to find a solution point that is optimal.

[0105] The current system allows a solution, which is at least as good as any other feasible solution, to be achieved without regard to the sensitivity of the covariance matrix.

[0106] Problems that otherwise could not be solved by standard technologies are solved by the current system. This allows practical portfolios to be selected even though there is limited available return information.

[0107] In another aspect of the invention involving practical applications of the current methodology to managing security portfolios, the ability to invest dividend income and other cash flows efficiently is also an important element in effective management. The current system allows a portfolio manager to invest available cash in the most efficient securities while considering the current portfolio holdings. Effectively this allows the manager to invest in securities which best enhance the current portfolio position without selling any of the current holdings. In a practical portfolio management system the ability to reinvest cash flows efficiently is always an important consideration. The current system provides this ability. An illustration of a reinvestment solution is provided in Appendix V which lists data used and results obtained in performing the reinvestment method.

[0108] In addition to finding optimal portfolios for tracking financial targets, the current system allows other objectives to be considered and incorporated in the solution. Examples include supplementing the basic objective function with an income objective, tax impact objective, or a company cash flow objective. The system is flexible enough to allow the objective function to be customized for particular applications.

[0109] As an example, an investor who has a preference for securities with high dividend yields will specify an objective function which explicitly states the trade off between portfolio tracking and dividend income. The computer routine to optimize a portfolio (the optimizer) will extract the partial derivatives of this objective function and proceed to select an optimal portfolio which exactly incorporates the specified trade-off between dividend income and tracking. This investor then has a custom solution which addresses his particular concerns and requirements.

[0110] The consideration of transaction costs associated with buying and selling securities is incorporated into the portfolio optimization system to control expenses due to trading. The trade off between tracking accuracy and transaction costs can be specified by the investor. This allows for a customized objective function, with regard to expected transaction costs, for each client.

[0111] Investors may also have tax effects to consider when trading from one security to another. The taxable gains and losses and their impact on expected return can be specified by the investor so as to control these costs.

[0112] Another preference which can be incorporated into the objective function and handled explicitly by the optimization system is the consideration of cash flows. An investor may have a preference for investing in companies which have large and positive cash flows. This objective can be incorporated into the system and resulting portfolios will reflect this investment goal.

[0113] While preferred embodiments of the present invention have been illustrated and described, it will be understood that changes and modifications can be made therein without departing from the invention to its broader aspects. Various features of the invention are defined in the following claims.

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