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This invention claims priority to U.S. Provisional Patent Application No. 61/051,251 filed on May 7, 2008, which is incorporated herein by reference in its entirety.
This invention relates investment measurement systems and more particularly to methods and systems for estimating forward trajectory of time-varying exposure of factors in financial, econometric problems that can be solved under constraints.
Target Date Funds (TDFs) are increasingly popular investment options in retirement plans. TDFs automatically allocate an investor's wealth into a diversified set of asset classes and systematically reduce investment risk as the TDF approaches its target maturity year. According to current estimates, 80% of large companies now offer TDFs in their 401(k) plans. Furthermore, 34% of corporations automatically enroll employees into 401(k) plans, more than half of which invest in a target-date fund. The target date products industry has not previously had an objective mechanism to transparently benchmark target date investments performance.
Despite the ease of deployment and adoption for investors, TDFs are complex investing instruments. For the vast majority of investors critical analysis of TDFs is best left to the experts, in most cases the plan sponsor. Plan sponsors are being offered a growing number of TDF options including both proprietary TDF solutions and TDF methodology overlays on top of existing plan fund options. As a plan sponsor contemplates the numerous options, there are a number of factors that should be considered. To date, it has been difficult to isolate and analyze these key factors. This invention provide a method and system that will help guide plan sponsors and other interested parties as they seek to analyze and select appropriate TDFs.
Currently there is no objective way for investors to benchmark the performance of a TDF manager, to determine how particular TDFs compare to one another, or to compare a TDF asset allocation glide path and relate risk/return to an industry average for all TDF Families. An investor can not objectively determine an expected risk and return for a particular TDF or evaluate the attribution performance of a TDF manager to differences between asset allocation and other manager skills in terms of market timing and selection returns relative to a passive benchmark glide path.
A number of firms such have tried to create solutions including creating TDF benchmark index candidates. For example, one firm has created a process that generates their own hypothetical TDF indices and calculates a backward looking historical TDF risk measure based on their own derived ideal reference glide paths for all TDFs. This approach does not yield indicators of forward looking expected risk for existing investments such as TDFs that are designed to purposefully change their asset allocation over time. Unfortunately, all of the above Target Maturity benchmark index candidates are based on the proprietary methodologies and asset allocation glide path models of each of the respective index publishers and are not derived from actual investable vehicles in the market as is the case with traditionally appropriate benchmark indices for investments. None of the current TDF benchmarking attempts provide the objectivity and transparency that is truly needed to answer significant questions about Target Maturity Funds.
There are also no effective mechanisms to benchmark a target date investment managers' skills and to disaggregate the impact these skills have on performance from the investment mangers' choices of passive asset class glide path.
Starting with William Sharpe's well known Returns Based Style Analysis, (RBSA) significant advances have been made in the field of automatic analysis of investment fund behavior based on their returns histories including deriving an effective replication strategy for predictive purposes.
Returns Based Style Analysis has been widely adopted in the investment management industry as a tool to assess the performance of managers. RBSA is a statistical optimization technique that solves for a rebalance portfolio of selected indexes that most closely mimic behavior of an investment vehicle over a specified period of time. The advantage of RBSA over linear regression is that short-sale portfolio constraints can be specified in RBSA. A palette of asset class indexes that are mutually exclusive and exhaustive is preferably used in RBSA. An example of such an asset palette consists of cash, US aggregate bonds, US large cap stocks, US small/mid cap stocks and international stocks indices. The R-Squared statistic reflects how well the RBSA model explains the behavior of the investment's historical performance. RBSA assumes that the style and skill of the manager are uncorrelated and that style is constant through time. Accordingly, RBSA results can be interpreted as average style weights over the sample period. Thus, traditional RBSA cannot be used for analysis of investments like TDFs that change their asset allocation over time.
In RBSA, a “rolling window” can be used to check for shift in style over time. Rolling Window Style Analysis (RWSA) is implemented by performing an RBSA on a sub-set of the returns data that moves along with observed history. If RBSA is performed on a small window, the resultant style analysis provides an estimate of asset allocation around the mid-point time of the window used. In practice the use of small windows for RBSA is limited because measurement noise in returns distorts the optimization results and is especially bad in small samples.
For example, a 12-month window is used to analyze the historical behavior of the fund members of TDF Family A. Sample results are displayed in FIG. 1. As expected the funds changed their asset allocation over time but there is also significant amount of statistical noise. As can be seen from the RWSA results of TDF Family A 2040 fund 102, the results indicate that the fund has around a 96% allocation to stocks in 2002 and that allocation seems to shift slightly towards bonds after 2003. While the TDF Family A 2010 fund 104 shows a much more significant and growing exposure to bonds.
One of the recent promising advances is described as Dynamic Style Analysis (DSA) for time-variant investments in U.S. Patent Application Publication No. 2004/0083152 entitled Method and system to solve dynamic multi-factor models in finance by Michael Markov, et al. RBSA and all current extensions including DSA for time-varying investment behavior provide tools to create custom benchmarks for investments with returns history. These bring transparency to the investment industry by creating passive replication strategies that serve as natural custom benchmarks for the investment managers. The alternative to RBSA is constant on-going analysis of securities based on their actual underlying investment holdings of the all managers.
However, RBSA methods' limitation is that they do not take into account the asset allocation trajectory that a reference-date dependent investment undergoes. Hence, the classifications generated with the prior art always produce backward looking estimates of how an investment behaved. Such backward looking estimates provide inaccurate a static description of trajectory of allocation mixes for reference-date dependent investments which are designed to deterministically change their future allocation mix. For example, when performing simple RBSA on a target date fund that is few years from maturity, and becoming more conservative by the year, all traditional RBSA and its advancements would inaccurately calculate historically more aggressive style allocation than at any point in time or even the true current allocation. GPSA would take into account the trajectory of changing allocation assign the appropriate allocation for each period in history or current. Similarly, performing RBSA on a target date 2040 mutual fund will result in mostly a equity exposure. However, we cannot use this static asset classification to predict the future allocation mix of this fund 30 years into future, because we know by definition of a target date fund that its allocation mix will significantly reduce its equity exposure as it approaches maturity date. Thus RBSA results in inadequate benchmarks for future performance, where as the GPSA trajectory can reveal the appropriate style allocation 30 year into the future.
New target date benchmarks published by various industry sources like TD Analytics, Dow Jones, Morningstar and Standard and Poor's also have significant problems. For example, most of these benchmarks, with the exception of Standard and Poor's, are based on idealized methodology assumptions that create indices that are not representative of the life cycle industry actual investment behavior. While attempting to benchmark target date investments, instead of producing objective, market-based passive benchmarks, these index creators impose upon their index construction their own methodological subjective opinions and constraints of how TDF should behave.
Conversely, the Standard and Poor's life cycle index is created using current holdings-based analysis of the investment options in the target date industry. The holdings are then mapped into Standard and Poor's selected asset classes which are used to interpolate a glide path. This Standard and Poor's technique is most similar to the analysis this GPSA invention performs. Such holding based analysis are inflexible for the particular investor's needs and become uneconomical to conduct in real-time.
There are major problems with all of the aforementioned industry benchmarks. One problem is that the benchmarks are not derived using a standard asset class palette. The palette used to derive such benchmarks may not reflect the choices of asset classes covered by the target date investment that is being benchmarked or the preferred palette of choice of the investor. For example, some of the benchmarks use a basic traditional set of five asset classes, while the investment being benchmarked may have been constructed and is managed using a much richer asset class palette with different, more granular asset class benchmarks and possibly including non-traditional asset classes that would make a direct traditional benchmark comparison inappropriate. Another problem is that the strategic and tactical allocation mix management execution of any particular target date fund differs significantly. For example, the allocation mix choice of equity allocation percentages for 2010 funds near maturity ranges widely from around 6% to 68% in the target date investments industry. Thus, use of a single industry benchmark is inadequate because it is impossible to attribute difference in fund performance due to pure portfolio allocation difference with respect to the industry benchmark, and the performance due to manager skill. Also some target date investment families are passively managed with low underlying investment turn-over and expenses, while other target date families are very actively managed with high underlying investment turn-over, expenses and active tactical decisions. Thus, the traditional static industry benchmarks are inappropriate measures of performance of specific managers unless the investment manager explicitly licensed a specific benchmark to manage to as in the case of Wells Fargo licensing the Dow Jones Target Date index.
What is needed is a method that extends the flexibility of RBSA to provide custom benchmarks for TDF based on a specific asset class or industry palette to project the target date investment family's strategy upon. This would allow disaggregation of target date investment returns and performance attribution between target date families into their underlying components some examples of which are the differences due to allocation mixes and differences due to manager skill.
A system and method of automatically benchmarking and evaluating individual reference-date dependent investments and families of such investments using their historic returns performance is provided. A Glide Path Style Analysis creates a custom replication strategy that is a reference date dependent trajectory of portfolio allocations for any family of target date investments using reference benchmarks for the behavior attribution. The GPSA uses periodic RBSA to create estimate history of appropriately dated portfolio style allocations for each member from a family of TDF. The historical centered windowed periodic RBSA style allocations for the family are aligned by distance from the reference date to reveal. This style allocation estimate is then results of a fund family are processed through an using an algorithm that exponentially weights the historic time-windows and that mathematically functionally fits a glide path allocation mixcalibrates a mathematically functions to the portfolio trajectory through time. This calibrated function is then used to construct a portfolio style glide path for user supplied set of dates with respect to reference date.
Embodiments of the invention for the first time creates a custom benchmark for individual or family of reference date investments. This allows comparison of same reference dated funds between various TDF families, by disaggregating the sources of fund returns into differences between their asset and industry allocations, manager specific behavior such as market timing and investment selection skill. The glide paths style allocations can be projected to any custom palette of asset classes, industry sectors and dynamic factors. These glide paths of various families can then be averaged to create a natural industry benchmark index, as well as custom peer benchmarks. The invention provides output that can be coupled with co-patent, that also allows investors to access forward looking metrics of risk/return and investment value at projected performance at various confidence for number of specified periods into the future. In summary, the invention solve some of the most widely published problems with TDF funds.
Illustrative embodiments of the present invention provide a method and system for custom benchmarking and evaluating performance of investments that are designed to change their allocation mix deterministically over time with respect to a reference date through simultaneous analysis of one or more members from a family of such investments. A Glide Path Style Analysis (GPSA) method according to an illustrative embodiment of the invention extends RBSA to provide custom benchmarks for reference-date dependent investments and family of such investments.
Interlinking of TDF within a target date fund family is used to derive assumptions regarding an investment manager's strategic trajectory portfolio allocation with respect to reference date. For example, TDF from the same investment manager and investment family most likely follow some coherent common glide path, such that the 2040 fund will most probably have a allocation mix similar to what the family's 2030 fund's allocation mix will be ten years into the future. Thus, the near-maturity dated TDF returns provide hints of how the far-from-maturity dated funds will evolve their allocation mix.
The GPSA technique performs periodic RBSA on a set of historical investment returns of each investment fund. This technique can be used with number of RBSA extensions, including rolling window RBSA, exponential weighted RBSA, and even with Dynamic RBSA, for example.
It leverages an insightful interpretation of the RBSA results namely that the RBSA calculated style allocation mix is the average allocation mix for the historical returns window period. For the centered window RBSA, the technique equally weighs each point, thus the weighted period is the mid-period of the window. For exponentially weighted RBSA, the exponential weighted time-lag is the assigned period of the calculated style allocation.
Traditionally time-lag of RBSA has been considered a weakness of the technique and primary driver of various advances including exponentially weighted RBSA and Dynamic RBSA, for example. However, GPSA leverages this property in a unique way. The historical periodic style allocations provides an estimate of the allocation mix at the weighted-time period of the RBSA during which a target date investment is slowly changing its allocation mix over. When these periodic style allocations for an entire target date fund family are aligned with respect to their distance from the target maturity date it reveals a statistically noisy view of the allocation mix transitions over time of the target date family with respect to target maturity dates. GPSA then takes this statistically noisy allocation mix and fits parsimonious mathematical functions that represent a trajectory of the portfolio with respect to a reference date. This fitted calibration can then be used to derive an estimate of the allocation mix transition for a user range of specified periods to produce a allocation glide path.
Illustrative embodiments of the invention automatically create a custom passive replication benchmarks for any reference-date dependent family of investments on any asset class or industry palette. These benchmark glide paths allow the performance difference between two TDF to be attributed to, among other things, the difference of their respective style allocation mix choices.
In other embodiments, this invention can serve a TDF manager's performance attribution purpose. The benchmark glide paths also can be used to create synthetic historic returns of a passive replication strategy to evaluate the skills of the investment manager relative to the custom benchmark. The excess risk/return performance attributes of the fund manager can be measured against this passive glide path benchmark, for particular fund or the entire family.
In still other embodiments, the GPSA derived custom benchmark glide paths for a number of different fund families from different investment managers can be averaged or weighted-averaged to create a target date industry benchmark index. The aggregate target date allocation mix benchmark indexes then serve as neutral reference points for the target date industry which can be projected onto any allocation mix palette(s) such as various asset class or industry/sector palettes.
Furthermore, a benchmark may be created for a sub-set of target date families that are the closest peers based on investment behavior and choice of asset class palette. For example, a plan sponsor may choose to use an embodiment of the present invention to construct a specialized peer benchmark in which peer managers are included in an index only if they have significant exposure to specific asset classes such as Real Estate Investment Trust (REIT) or Treasury Inflation Protected Securities (TIPS), for example. This power to create custom peer target date benchmark indexes provides a very powerful tool to construct appropriate relative measures of behavior and evaluation.
Illustrative embodiments provide estimates of the future behavior of the target date family which are not provided by RBSA. In an illustrative embodiment of the invention, when GPSA is performed on daily returns, the tactical changes to a manager glide path allocation style can be detected quickly. This can provide timely information on the change in behavior of a manager relative to their stated allocation mix and/or provide information on over-concentration in specific industry sectors and thus serve as an early warning system for investors/plan sponsors to further investigate the behavior of their target date manager.
Glide paths derived according to embodiments of the present invention can then be used by an appropriate simulation forecasting analytic engine to periodically rebalance investments according to the target date glide path and to evaluate future distribution of future wealth outcomes based on a specific investor profile with an initial wealth and an initial savings profile. These forward simulations can also be used to provide forward looking statistics on risk/return. Simulations may also provide estimates for distribution of value of hypothetical performance of a current investment in a specific dated fund a number of years into the future with at various confidence percentiles. Such forward looking performance estimates of each TDF based on its own particular style allocation provide a significant improvement compared to traditional backward looking risk measures used to measure current TDF performance.
An illustrative embodiment of the invention provides a computer implemented method for Generating a Glide Path Style Analysis. The computer implemented method includes the steps of performing a periodic Returns Based Style Analysis (RBSA) over history of returns for a family of target maturity funds, interpreting each results from said RBSA as average allocation for specific period that is time-weighted based on RBSA characteristic, assigning to each specific period style allocation, a time distance from reference date, chaining resulting said periodic style allocations together ordered by time distance from reference date to reveal a noisy estimation of changing asset allocation as a function of temporal proximity to maturity, smoothing said noisy period estimates by algorithm that fit appropriate mathematical functions to create a parsimonious representation of the portfolio trajectory over time based on reference date; and supplying a set of time periods with respect to reference date, to obtain set of corresponding portfolio allocations for each period. In the illustrative embodiment, periodic RBSA could be of the whole length of the fund, rolling window RBSA, exponential weighted RBSA, or even Dynamic Style Analysis. For a corresponding RBSA technique, a weighted-time period needs to attribute the style allocation. For RBSA and Window RBSA, this period is the center of the time period used for the entire sample or window. For exponentially weighted RBSA, it is the exponential weighted time lag period.
The foregoing and other features and advantages of the present invention will be more fully understood from the following detailed description of illustrative embodiments, taken in conjunction with the accompanying drawings in which:
FIG. 1 is a rolling window style analysis sample results according to the PRIOR ART;
FIG. 2 is a system block diagram showing an overview of a life cycle family GPSA according to an illustrative of the present invention;
FIG. 3 is a flow diagram showing the steps of a GPSA according to an illustrative embodiment of the present invention;
FIG. 4 is a flow diagram showing a fit allocation used in generating a allocation glide path according to an illustrative embodiment of the present invention; and
FIG. 5 is an aggregate rolling window style analysis chart according to an illustrative embodiment of the present invention.
Illustrative embodiments of the present invention provide a system and method of automatically benchmarking and evaluating individual reference-date dependent investments and families of such investments using just their historic returns performance. These investment products include but are not limited to TDF for retirement wealth accumulation, 529 plan target date investments, as well as certain target date retirement income funds. The innovation extends William Sharpe's Returns Based Style Analysis (RBSA) technique EQUATION 1, and its recent advances and modifications. The innovation allows for simultaneous analysis of a family of instruments that deterministically change their investment allocation mix with respect to a reference date over time, and produces a passive replication strategy that is a trajectory of style allocations with respect to the reference date. The reference benchmarks for the behavior attribution can be of any granularity of based on any benchmark palette of asset classes, industrial sectors or dynamic factors.
In an illustrative embodiment, the GPSA uses the historical center of time-windowed RBSA results using the history of returns for each member from a family of TDF (e.g., 2010, 2015, 2020, 2025, etc.). The historical centered windowed RBSA results of a fund family are processed using an algorithm that exponentially weights the historic time-windows and mathematically functionally fits a glide path allocation mix that conforms to budget constraints and non-negativity constraints.
GPSA provides appropriate passive benchmarks for investment managers that change allocation based on a reference date. The inventive method can simultaneously align windows RBSAs of all funds in a family based on years to maturity. It then fits a multivariate generalized logistic functional form fit to each of the asset classes based on time dependent trends. It then uses the calibrated function to calculate portfolio allocation for user specified time periods with respect to a reference date, such that each portfolio follows constraints of budget and non-negativity. This provides a passive glide path strategy that most closely replicates the historical performance of the TDF investment.
In the illustrative embodiment, GPSA is applied to reference-date dependent investments like Target date, TDF and Retirement Income Funds by using a multivariate generalized logistic functional form to estimate maturity date based allocation mix. GPSA performs rolling window RBSA on history of one of more members of a reference date dependent investment, and simultaneously fit portfolio allocation trajectory with respect to the reference date, such that it recovers glide path is subject to non-negativity and budget constraints. The algorithm first assigns the centered windows RBSA allocation mix as the mid-date estimate for the portfolio allocation. RBSA algorithm enforces budget and non-negativity constraints. These mid-dates are then transformed to distance from reference date. Once each window is given a distance, all the windows for entire fund family can be sorted based on this distance from reference date to reveal a noisy glide path of portfolio allocation. A smooth trajectory of style allocation needs to be fitted with an algorithm.
The fit algorithm can have multiple embodiments that fit both marginal as well as cumulative transitions of the portfolio allocations with respect to reference date using a number of mathematical function families, piece-wise polynomial or linear fits.
In the illustrative embodiment, the cumulative transition portfolio allocations is used. The time periods are segmented into four equal quarters. The average portfolio allocations at first quarter and the last quarter of time are compared to determine the changes in allocations. The benchmark indices order in the portfolio are sorted based on differences. The algorithm works if sorting is ascending or descending. This ordering of benchmarks is applied to all the aligned RBSA windows with respect to reference date. This transformation looks like an “S-Curve” type transitions of cumulative portfolio style allocation over time. Accordingly “S-Curve” generalized logistic functional is chosen to naturally fit the cumulative transition of each benchmark over time. EQUATION 3a is the modified logistic function equation for cumulative asset class transition from lower to higher values as it approaches reference date, while EQUATION 3b is fits the reverse transition pattern. This choice of function is easy to interpret and has parsimonious form. In other embodiments, other mathematical functions can be chosen including piece-wise linear, polynomial forms to fit either cumulative or marginal portfolio transitions.
The mathematical fits for each benchmark allocation creates series of calibrated parameters for functional representation of the glide path. This functional representation is used to calculate estimates of portfolio allocation for user specified periods with respect to the reference date as shown in EQUATION 4. Each of the dated portfolios are then processes through constraints on budget, non-negativity, rounding. In the TDF embodiment, these dates could be for monthly portfolio allocations starting from 50 years from retirement to 10 years past retirement resulting in a glide path of style allocations that covers broad range of accumulation users that transition into retirement mode, for example. A reverse look-up is then performed from the sorted list benchmark to delivers the glide path portfolio allocation in the original order of benchmark.
A glide path function algorithm using multivariate generalized logistic functions can fit any time-dependent allocation mix in cumulative transition form and performs very well for a palette with a limited number of benchmarks. However, this technique degrades in performance in a highly granular palette, with 10 or more benchmarks, for example. In the illustrative embodiment, the glide path function algorithm uses a Hierarchical GPSA algorithm. This algorithm makes sure that the most important salient characteristics of the style allocation trajectory are captured first, and then iteratively zooms into refine the details. It performs a hierarchical iterative “depth-first” glide path fit using an asset class hierarchy. The hierarchical algorithm performs the higher tiers benchmark allocation fit first, and constrains the children benchmark to lower and upper bounds provided by the benchmark's parent at each point in time. This first fits the highest tier stock/bonds/cash, and then fits the US/International stock split, etc. This creates highly stable allocation mix glide paths that make reasonable sense at each level of granularity.
In an illustrative embodiment, an allocation mix glide path strategy is extracted by transforming RBSA data to fit a parsimonious multivariate logistic function which defines an allocation mix glide path. The RBSA results centered windows' allocation mixes are exponentially weighted and a decay rate of the windows can be solved for to maximize the out-of-sample forecasting stability of the resultant glide path. For example, in applying the algorithm to a passive target date fund that simply rebalances to a static allocation mix glide path, all the historical windows would be equally weighted. In contrast, applying the algorithm to an aggressively active target date fund would have a decay weighting that takes into account more recent allocation mix management. Budget constraints and non-negativity constraints are enforced to provide allocation mixes that conform to the norms a current mutual fund investment management industry practices.
Optionally, a glide path can be created with a most simplified asset class palette based on evaluation criterion. In the illustrative embodiment, the algorithm uses a step-wise GPSA that can start with a single asset class and recursively add benchmarks that are significant, or conversely it can perform the initial GPSA and recursively drop benchmarks that are insignificant. The demonstrated step-wise drop-simplification first creates the glide path with all asset classes, and then iteratively drops the least relevant asset classes from the glide path based on evaluation criterion. In an illustrative embodiment, the highest adjusted R-Squared values are produced. The entire process is then repeated recursively, to continue dropping the next benchmark with the smallest maximum weight in the glide path until further simplification of GPSA model can no longer increase adjusted R-Squared. This automatically generates a glide path with the fewest simplest subset of assets to achieve the highest explanation of variance of returns of the reference-date dependent investment. In another embodiment, other model evaluation criteria such as Bayesian Information Criterion are used, for example.
An overview of the various embodiments of the present invention is described with reference to FIG. 2. Historical investment returns for a family of life cycle funds 202 including one or more life cycle fund returns 204 is analyzed according to a life cycle family GPSA 206. Certain configuration parameters 208 are received for inclusion in the GPSA 206. The life cycle family GPSA 206 generates an allocation mix glide path 210 for the family of life cycle funds. The allocation mix glide path can be used for benchmarking or may be aggregated to for use as an index, for example.
The GPSA of TDF family according to an illustrative embodiment of the invention is described with reference to FIG. 3. First a centered window return based style analysis (RBSA) is 302 is performed on a family of life cycle funds. Next, all of the style allocation windows are dated with time from reference date 304. This creates a style allocation transition with respect to reference date. The style allocation transition data is then fitted 306 to a generalized logistic functions. This fitted function is then used to create the style allocation trajectory glide path for user supplied time periods relative to the reference date. Portfolio rules constraints are enforced 308 on the calculated portfolios to generate an allocation glide path 310.
FIG. 4 provides a more detailed description of fitting the allocation mix transition data to a functions according to an illustrative embodiment of the invention. The allocation mix transition data is optionally sorted 402 by maximum or minimum transition. Cumulative transition allocation is then fitted 404 using generalized logistics, beta, piecewise linear, piecewise polynomial or piecewise cubic spline techniques for example. Such fitting is repeated 406 for each allocation. The fitted allocation data is optionally unsorted 408 and output as an allocation glide path 410.
An example of output from a glide path style analysis is described with reference to FIG. 5. It should be noted that the analysis in this example is performed on the same TDF family A funds which provided the Rolling Window Style Analysis shown in FIG. 1. GPSA transforms these rolling window results into a common time based measure by subtracting the midpoint of the RBSA from the target date of the fund. Thus, the center of each time window provides an estimate of the fund manager's allocation based on length of time from target date. This common unit allows GPSA to consolidate all the rolling window style analysis for all TDF from a particular family. The aggregate results 502 for TDF family A as shown in FIG. 5 provides new insight to the fund's behavior. For example, in this particular family of funds, it is apparent that their style exposure behavior becomes more conservative as they approach their target dates. Additionally, for this particular fund family, their glide path allocations continue to become more conservative even after fund maturity dates.
To address the Rolling Window Style Analysis noise, the GPSA method according to the various embodiments of the invention fits a generalized logistic function for each that represents style allocation transition over time. In performing this functional fit, GPSA makes assumptions that the TDF asset allocation decisions are deterministic simply based on time from maturity date and are consistent across all funds within a particular TDF family. The GPSA method solves for the model parameters that make each glide path's annual asset allocation glide point model returns as close as possible to the historical returns of the style asset allocation of the TDF Family. The GPSA method covers and allows for both linear and non-linear mathematic function families, as well as single and piece-wise fits for different time horizons.
While the present invention is described generally herein in terms of a benchmarking system which may include certain algorithms and software processes it should be understood that illustrative embodiments of the invention include computer system components such as processors, memory and communication devices that are particularly configured and or programmed to perform the various processes and implement the various systems and data transformations described herein.
While the invention has been described with reference to illustrative embodiments, it should be understood by those skilled in the art that various other changes, omissions, and/or additions may be made and substantial equivalents may be substituted for elements thereof without departing from the spirit and scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teaching of the invention without departing from the scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed for carrying out this invention, but that the invention will include all embodiments, falling within the scope of the appended claims. Moreover, unless specifically stated any use of the terms first, second, etc., do not denote any order of importance, but rather the terms first, second, etc. are used to distinguish one element from another. Certain equations that can be used in various illustrative embodiments of the invention are presented in the Appendix filed herewith which is incorporated by reference in this specification.
William Sharpe's RBSA method uses quadratic optimization to minimize the variance of the excess return of the manager over a linear combination of the style benchmarks.
A nonlinear model is defined as an equation that is nonlinear in coefficients, or a combination of linear and nonlinear in the coefficients. The nonlinear in matrix form is:
y=ƒ(X,β)+ε EQUATION 2
y is an n-by-1 vector of responses.
ƒ is a function of β and X.
β is a m-by-1 vector of coefficients.
X is the n-by-m design matrix for the model.
ε is an n-by-1 vector of errors.
The fitting process starts with initial estimate for each coefficient. In our GPSA fit model, we have transformed the style transition S-Curve such that parameters have values that can be interpreted to have intuitive meaning.
For benchmark allocation transition from low to high
For benchmark allocation transition from high to low
a: the lower asymptote;
c: the upper asymptote (100% weight) minus A;
m: the time of maximum growth, around starting value;
b: the growth rate;
t: affects near which asymptote maximum growth occurs.
0≦t≦2 (most of the transition happens near the target date for accumulation TDF funds)
For the starting values of the curve fit we use mid-points of trust range. First, step is to produce the fitted curve for the current set of coefficients. The fitted response value is given by
ŷ=ƒ(X,β) EQUATION 4
And then we calculate the Jacobian of ƒ(X,β), the matrix of partial derivatives taken with respect to the coefficients.
Next step is to adjust the coefficients and determine whether the fit improves. The fitting algorithms determine the direction and magnitude of the adjustment depend. We use the trust-region algorithm, because it allows for coefficient constraints.