SYSTEME DE CONSULTATION FINANCIERE

FINANCIAL ADVISORY SYSTEM

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Published without an Abstract

Revendications

CLAIMS
What is claimed is:
1. A method comprising:
generating return scenarios for each asset class of a plurality of asset
classes based
upon future scenarios of one or more economic factors;
creating a mapping from each financial product of an available set of
financial
products onto one or more asset classes of the plurality of asset classes by
determining exposures of the available set of financial products to each
asset class of the plurality of asset classes; and
simulating return scenarios for one or more portfolios including combinations
of
financial products from the available set of financial products based upon
the mapping..
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Description

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FINANCIAL ADVISORY SYSTEM
This is a continuation-in-part of application serial no. 08/982,942, filed on
December 2, 1997, that is currently pending.
COPYRIGHT NOTICE
Contained herein is material that is subject to copyright protection. The
copyright
owner has no objection to the facsimile reproduction of the patent disclosure
by any
person as it appears in the Patent and Trademark Office patent files or
records, but
otherwise reserves all rights to the copyright whatsoever.
BACKGROUND OF THE INVENTION
Field of the invention
The invention relates generally to the field of financial advisory services.
More
particularly, the invention relates to a system far advising a user regarding
feasible and
optimal portfolio allocations among a set of available financial products.
Description of the Related Art
During the l 980's, a significant trend emerged in retirement savings.
Traditional
defined beneft plan assets began shifting towards employee-directed defined
contribution
plans like 40I(k). As this trend continues, many individual investors will
ultimately
become responsible for managing their own retirement investments. However,
many
people are not well-equipped to make informed investment decisions. Further,
the
number and diversity of investment options available to individuals is rapidly
increasing,
thereby making investment decisions more complex by the day.
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Many investment software packages claim to help individuals plan for a secure
retirement, or some other intermediate goal. However, typical prior art
investment
software packages are limited in several ways. For example, some packages
provide
generic asset-allocation suggestions (typically in the form of a pie-chart)
and leave the
investor to find the actual combination of financial products that meets the
suggested
asset allocation. However, many investments available to individual investors,
such as
mutual funds, cannot easily be categorized into any one generic asset class
category.
Rather, mutual funds are typically a mix of many different asset classes. This
property of
mutual funds complicates the selection of appropriate instruments to realize a
desired
asset allocation.
Further, some prior art programs, typically referred to as "retirement
calculators,"
require the user to provide estimates of future inflation, interest rates and
the expected
return on their investments. In this type of prior art system, the user is
likely, and is in
fact encouraged, to simply increase the expected investment returns until
their desired
portfolio value is achieved. As should be appreciated, one of the problems
with this type
of program is that the user is likely to create an unattainable portfolio
based on an
unrealistic set of future economic scenarios. That is, the portfolio of
financial products
required to achieve the X% growth per year in order to meet the user's
retirement goal
may not be available to the user. Further, the idealistic future economic
conditions
assumed by the user, for example, 0% inflation and 20% interest rates, may not
be
macroeconomically consistent. Typical prior art investment packages simply
allow the
user to manipulate economic conditions until a desired result is achieved
rather than
encouraging the user to focus on his/her own decisions regarding investment
risk, savings
rate, and retirement age within the context of realistic economic assumptions.
Consequently, the so called "advice" rendered by many of the prior art
investment
software packages can be misleading and impossible to implement in practice.
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In addition, investment advice software in the prior art have various other
disadvantages which are overcome by the present invention. Notably, prior art
systems
typically do not provide realistic estimates of the investment or retirement
horizon risk-
return tradeoff given a user's specific investments and financial
circumstances. This
makes informed judgments about the appropriate level of investment risk very
difficult.
Obtaining the appropriate level of investment risk (and return) is critical to
the success of
a long-term investment plan.
In view of the foregoing, what is needed is a financial advisory system that
employs advanced financial techniques to provide financial advice to
individuals on how
to reach specific fcnancial goals. More specifically, it is desirable to
provide a system that
automatically generates future-looking realistic economic and investment
return scenarios
and allows a user to arrive at a feasible portfolio that meets both
intermediate and long-
term financial goals by a process of outcome-based risk profiling. In this
manner, the
user can focus on his/her own decisions regarding investment risk, savings,
and
retirement age while interactively observing the impact of those decisions on
the range of
possible investment outcomes. Further, it is desirable that the financial
advisory system
create a feasible optimal portfolio that maximizes the utility function of the
user by
selecting financial products that are available to the user and that provides
the highest
possible utility given the user's risk tolerance, investment horizon and
savings level. By
utility what is meant is a function that determines the relative preferences
of an individual
for different combinations of.financial products based. on one or more
characteristics of
the products (e.g., expected return, variance, etc.), and optionally one or
more parameters
specific to the individual. Moreover, it is advantageous to perform plan
monitoring on an
ongoing basis to alert the user if the likelihood of meeting their financial
goals falls below
a threshold value or if their portfolio risk level becomes inconsistent with
their risk
preferences. Finally, it is desirable to provide specific advice to the user
regarding steps
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they can take to improve their chances of meeting their financial goals while
taking into
consideration the user's personal tradeoffs among risk, savings, and
retirement age.
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BRIEF SUMMARY OF THE INVENTION
A financial advisory system is described. According to one aspect of the
present
invention, return scenarios for optimized portfolio allocations are simulated
interactively
to facilitate financial product selection. Return scenarios for each asset
class of a
plurality of asset classes are generated based upon estimated future scenarios
of one or
more economic factors. A mapping from each financial product of an available
set of
financial products onto one or more asset classes of the plurality of asset
classes is created
by determining exposures of the available set of f nancial products to each
asset class of
the plurality of asset classes. In this way, the expected returns and
correlations of a
plurality of financial products are generated and used to produce optimized
portfolios of
financial products. Return scenarios are simulated for one or more portfolios
including
combinations of financial products from the available set of financial
products based
upon the mapping.
Other features of the present invention will be apparent from the accompanying
drawings and from the detailed description which follows.
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BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS
The present invention is illustrated by way of example, and not by way of
limitation, in the figures of the accompanying drawings and in which like
reference
numerals refer to similar elements and in which:
Figure 1 illustrates a financial advisory system according to one embodiment
of
the present invention.
Figure 2 is an example of a typical computer system upon which one embodiment
of the present invention can be implemented.
Figure 3 is a block diagram illustrating various analytic modules according to
one
embodiment of the present invention.
Figure 4 is a flow diagram illustrating core asset class scenario generation
according to one embodiment of the present invention.
Figure 5 is a flow diagram illustrating factor asset class scenario generation
according to one embodiment of the present invention.
Figure 6 is a flow diagram illustrating financial product exposure
determination
according to one embodiment of the present invention.
Figure 7 is a flow diagram illustrating portfolio optimization according to
one
embodiment of the present invention.
Figure 8 is a flow diagram illustrating plan monitoring processing according
to
one embodiment of the present invention.

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DETAILED DESCRIPTION OF THE INVENTION
A financial advisory system is described. In embodiments of the present
invention, a factor model approach is laid on top of a pricing kernel model to
simulate
returns of a plurality of asset classes, and ultimately financial products,
such as securities
or portfolios of securities. The term "financial products" as used herein
refers to a legal
representation of the right (often denoted as a claim or security) to provide
or receive
prospective future benefits under certain stated conditions. In any event, the
forecasts
may then be used for purposes of providing financial advisory services to a
user. For
example, such forecasts are useful for selecting the composition of an
optimized portfolio
(based on a utility function) from a set of available financial products
conditional on
decisions and constraints provided by the user.
Briefly, fundamental economic and financial forces ase modeled using a pricing
kernel model that provides projected returns on a plurality of asset classes
(core asset
classes) conditional on a set of state variables that capture economic
conditions. The core
asset classes in combination with additional asset class estimates that are
conditioned on
the core asset classes comprise a model (hereinafter "the factor model") of a
comprehensive set of asset classes that span the universe of typical
investment products.
A factor model is a return-generating function that attributes the return on a
financial
product, such as a security, to the financial product's sensitivity to the
movements of
various common economic factors. The factor model enables the system to assess
how
financial products and portfolios will respond to changes in factors or
indices to which
financial products are exposed. The selection of asset classes may be tailored
to address a
narrow or broad range of investors. For example, asset classes may be chosen
that are
relevant only to a particular industry or asset classes may be chosen to span
the market
range of a broad set of possible investments (e.g. all available mutual funds
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equities). According to embodiments of the present invention discussed herein,
to reach
the broadest segment of individual investors, the asset classes selected as
factors for the
factor model have been chosen to span the range of investments typically
available to
individual investors in mainstream mutual funds and defined contribution
plans.
After generating future scenarios for the factor model, financial products
available
to an investor may be mapped onto the factor model. To assure that a portfolio
recommended by the system is attainable, it is preferable to generate
investment scenarios
that include only those financial products that are available to the investor.
The available
financial products may include, for example, a specific set of mutual funds
offered by an
employer sponsored 401 (k) program. In any event, this mapping of financial
products
onto the factor model is accomplished by decomposing the returns of individual
financial
products into exposures to the asset classes employed by the factor model. In
this
manner, the system learns how each of the financial products available to the
user behave
relative to the asset classes employed by the factor model. In so doing, the
system
l 5 implicitly determines the constraints on feasible exposures to different
asset classes faced
by an investor given a selected subset of financial products. Given this
relationship
between the user's available financial products and the factor model, the
system may
generate feasible forward-looking investment scenarios. The system may further
advise
the user regarding actions that may be taken (e.g., save more money, retire
later, take on
additional investment risk, seek opportunities to expand the investment set)
to achieve
certain financial goals, such as particular retirement standard of living,
accumulating a
down payment for the purchase of a house, or saving enough money to send a
child to
college.
In the following description, for the purposes of explanation, numerous
specific
details are set forth in order to provide a thorough understanding of the
present invention.
It will be apparent, however, to one skilled in the art that the present
invention may be
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practiced without some of these specific details. In other instances, well-
known
structures and devices are shown in block diagram form.
The present invention includes various steps, which will be described below.
The
steps of the present invention may be embodied in machine-executable
instructions. The
instructions can be used to cause a general-purpose or special-purpose
processor that is
programmed with the instructions to perform the steps of the present
invention.
Alternatively, the steps of the present invention may be performed by specific
hardware
components that contain hardwired logic for performing the steps, or by any
combination
of programmed computer components and custom hardware components.
The present invention may be provided as a computer program product which may
include a machine-readable medium having stored thereon instructions which may
be
used to program a computer (or other electronic devices) to perform a process
according
to the present invention. The machine-readable medium may include, but is not
limited
to, floppy diskettes, optical disks, CD-ROMs, and magneto-optical disks, ROMs,
RAMS,
EPROMs, EEPROMs, magnet or optical cards, flash memory, or other type of media
/
machine-readable medium suitable for storing electronic instructions.
Moreover, the
present invention may also be downloaded as a computer program product,
wherein the
program may be transferred from a remote computer to a requesting computer by
way of
data signals embodied in a carrier wave or other propagation medium via a
communication link (e.g., a modem or network connection).
While, embodiments of the present invention will be described with reference
to
an financial advisory system, the method and apparatus described herein are
equally
applicable to other types of asset allocation applications, financial planning
applications,
investment advisory services, financial product selection services, automated
financial
product screening tools, such as electronic personal shopping agents and the
like.
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S~rstem Overview
The present invention may be included within a client-server transaction based
financial advisory system 100 such as that illustrated in Figure 1. According
to the
embodiment depicted in Figure 1, the financial advisory system 100 includes a
financial
staging server 120, a broadcast server 11 S, a content server 117, an
AdviceServerTM 110
(AdviceServerTM is a trademark of Financial Engines, Inc., the assignee of the
present
invention), and a client 105.
The financial staging server 120 may serve as a primary staging and validation
area for the publication of financial content. In this manner, the financial
staging server
120 acts as a data warehouse. Raw source data, typically time series data, may
be refined
and processed into analytically useful data on the financial staging server
120. On a
monthly basis, or whatever the batch processing interval may be, the financial
staging
server 120 converts raw time series data obtained from data vendors from the
specif c
vendor's format into a standard format that can be used throughout the
financial advisory
system 100. Various financial engines may be run to generate data for
validation and
quality assurance of the data received from the vendors. Additional engines
may be run
to generate module inputs, model parameters, and intermediate calculations
needed by the
system based on raw data received by the vendors. Any calibrations of the
analytic data
needed by the financial engines may be performed prior to publishing the final
analytic
data to the broadcast server 115.
The broadcast server 115 is a database server. As such, it runs an instance of
a
Relational Database Management System (RDBMS), such as Microsoft SQL-ServerTM,
OracleTM or the like. The broadcast server 115 provides a single point of
access to all
fund information and analytic data. When advice servers such as AdviceServer
110 need
data, they may query information from the broadcast server database. The
broadcast
server 11 S may also populate content servers, such as content server 1 i 7,
so remote
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implementations of the AdviceServer 110 need not communicate directly. with
the
broadcast server 11 S.
The AdviceServer 110 is the primary provider of services for the client 105.
The
AdviceServer 110 also acts as a proxy between external systems, such as
external system
125, and the broadcast server 115 or the content server 117. The AdviceServer
110 is the
central database repository for holding user profile and portfolio data. In
this manner,
ongoing portfolio analysis may be performed and alerts rnay be triggered, as
described
further below.
According to the embodiment depicted, the user may interact with and receive
feedback from the financial advisory system 100 using client software which
may be
running within a browser application or as a standalone desktop application on
the user's
personal computer 105. The client software communicates with the AdviceServer
110
which acts as a HTTP server.
1 S Overview of Exemplary User Interaction with the System
During an initial session with the financial advisory system 100, according to
one
embodiment of the present invention, the user may provide information
regarding risk
preferences, savings preferences, current age, gender, income, expected income
growth,
current account balances, current financial product holdings, current savings
rate,
retirement age goal, retirement income goals, available f nancial products,
intermediate
and long-term goals; constraints on fund holdings, liabilities, expected
contributions, state
and federal tax bracket (marginal and average). The user may provide
information for
themselves and each profiled person in their household. This information may
be saved
in one or more files in the financial advisory system 100, preferably on one
of the servers
to allow ongoing plan monitoring to be performed. In other embodiments of the
present
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invention additional information may be provided by the user, for example,
estimates of
future social security benefits or anticipated inheritances.
Based on the user's current holdings the system may forecast a retirement
income
and graphically depict the current portfolio's projected growth and range of
possible
values over time.
The system may also provide the user with statistics regarding the likelihood
that
they will be able to retire when they would Like, given the projected returns
on the user's
current portfolio based upon the data input by the user, including the user's
current
savings rate, retirement age goal, and investment holdings.
Based on models and calculations that will be discussed in more detail below,
the
financial advisory system 100 may provide an initial diagnosis based upon the
user's risk
preference, savings rate, and desired risk-return tradeoffs. This diagnosis
can result in a
series of suggested actions including: (1) rebalance the portfolio, (2)
increase savings, (3}
retire Later, or (4) adjust investment risk. An iterative process may then
begin in which
the user may adjust his/her investment risk, savings rate, and/or retirement
age and have
the financial advisory system 100 evaluate the projected performance of an
optimized
portfolio given the financial products available to the user based on the
currently selected
risk tolerance, investment horizon and savings rate decisions. This process of
the
financial advisory system 100 providing advice and/or feedback and the user
adjusting
risk, savings, and retirement age parameters may continue until the user has
achieved a
desired portfolio forecast.and performance distribution. At this time, the
user may chose
to implement the optimal portfolio. The parameters and portfolio allocation
may then be
saved by the financial advisory system 100 for future user sessions.
As described further below, on an ongoing basis the financial advisory system
100
may evaluate the user's portfolio against one or more financial goals and may
notify the
user if progress towards any of the goals has changed in a material way.
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In subsequent user sessions with the financial advisory system 100, the user's
data
(e.g., the user's profile information, account holdings, plan parameters, and
tax
information) may be retrieved from memory on the AdviceServer 1 I0, for
example, and
the current forecast for the one or more goals may be presented to the user.
Additionally,
if the ongoing pian monitoring has generated any alerts, they may be presented
to the user
at this time. Alternatively, alerts may be generated proactively by the system
and
transmitted to the user via a telephone, email, fax, or standard mail
messaging system.
Based upon the alerts generated by the ongoing plan monitoring, the user may
again begin
the iterative process of adjusting the decision variables described above
(e.g., risk level,
savings rate, and retirement age) until the user is satisfied with the
likelihood of meeting
his/her goal(s). To assure accurate portfolio tracking, if the personal data
changes, the
user may simply modify the data upon which the financial advisory system's
assumptions
are based. For example, if the user's salary increases, this information
should be updated
in the user's profile. Additionally, if the user's employer adds a new mutual
fund to the
company's 401 (k) program, then the user should update the list of available
financial
products in the user profile information. This is important because the
optimal allocation
among the user's available financial products may be impacted by the addition
of a new
mutual fund, for example. In one embodiment of the present invention, the
financial
advisory system 100 may be connected to external record-keeping systems at the
user's
employer that can provide automatic updates to selected user information.
Advantageously, the user is never asked to predict the future with regard to
interest rates, inflation, expected portfolio returns, or other difficult to
estimate economic
variables and parameters. Additionally, the optimal portfolio generated by the
financial
advisory system 100 is guaranteed to be attainable. That is, the optimal
portfolio has
been determined based upon the specific financial products that are available
to the user.
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An Exemplary Computer System
Having briefly described one embodiment of the financial advisory system 100
and
exemplary user interactions, a computer system 200 representing an exemplary
client 1(l5
or server in which features of the present invention may be implemented will
now be
S described with reference to Figure 2. Computer system 200 comprises a bus or
other
communication means 201 for communicating information, and a processing means
such as
processor 202 coupled with bus 201 for processing information. Computer system
200
further comprises a random access memory (RAM) or other dynamic storage device
204
(referred to as main memory), coupled to bus 201 for storing information and
instructions
to be executed by processor 202. Main memory 204 also may be used for storing
temporary variables or other intermediate information during execution of
instructions by
processor 202. Computer system 200 also comprises a read only memory (ROM)
and/or
other static storage device 206 coupled to bus 201 for storing static
information and
instructions for processor 202.
A data storage device 207 such as a magnetic disk or optical disc and its
corresponding drive may also be coupled to computer system 200 for storing
information
and instructions. Computer system 200 can also be coupled via bus 201 to a
display device
221, such as a cathode ray tube (CRT) or Liquid Crystal Display (LCD), for
displaying
information to a computer user. For example, graphical depictions of expected
portfolio
performance, asset allocation for an optimal portfolio, charts indicating
retirement age
probabilities, and other~.data types may be presented to the user on the
display device 221.
Typically, an alphanumeric input device 222, including alphanumeric and other
keys, may
coupled to bus 201 for communicating information and/or command selections to
proce$sor
202. Another type of user input device is cursor control 223, such as a mouse,
a trackball,
or cursor direction keys for communicating direction information and command
selections
to processor 202 and for controlling cursor movement on display 221.
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A communication device 225 is also coupled to bus 201 for accessing remote
servers, such as the AdviceServer 110, or other servers via the Internet, for
example. The
communication device 225 may include a modem, a network interface card, or
other well
known interface devices, such as those used for coupling to an Ethernet, token
ring, or
other types of networks. In any event, in this manner, the computer system 200
may be
coupled to a number of clients and/or servers via a conventional network
infrastructure,
such as a company's Intranet and/or the Internet, for example.
Exemplary Analytic Modules
Figure 3 is a simplified block diagram illustrating exemplary analytic modules
of
the financial advisory system 100 according to one embodiment of the present
invention.
According to the embodiment depicted, the following modules are provided: a
pricing
module 305, a factor module 310, a financial product mapping module 315, a tax
adjustment module 320, an annuitization module 325, a simulation processing
module
330, a portfolio optimization module 340, a user interface (U1) module 345,
and a plan
monitoring module 350. It should be appreciated that the functionality
described herein
may be implemented in more or less modules than discussed below. Additionally,
the
modules and functionality may be distributed in various configurations among a
client
system, such as client 105 and one or more server systems, such as the
financial staging
server 120, the broadcast server 115, or the AdviceServer 110. The
functionality of each
of the exemplary modules will now be briefly described.
An "econometric model" is a statistical model that provides a means of
forecasting the levels of certain variables referred to as "endogenous
variables,"
conditional on the levels of certain other variables, known as "exogenous
variables," and
in some cases previously determined values of the endogenous variables
(sometimes
referred to as lagged dependent variables). The pricing module 305 is an
equilibrium
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econometric model for forecasting prices and returns (also referred to herein
as "core
asset scenarios") for a set of core asset classes. The pricing module provides
estimates of
current levels and forecasts of economic factors (also known as.state
variables), upon
which the estimates of core asset class returns are based. According to one
embodiment
S of the present invention, the economic factors may be represented with three
exogenous
state variables, price inflation, a real short-term interest rate, and
dividend growth. The
three exogenous state variables may be fitted with autoregressive time series
models to
match historical moments of the corresponding observed economic variables, as
described further below.
In any event, the resulting core asset classes are the foundation for
portfolio
simulation and are designed to provide a coherent and internally consistent
(e.g., no
arbitrage) set of returns. By arbitrage what is meant is an opportunity to
create a
profitable trading opportunity that involves no net investment and positive
values in all
states of the world.
According to one embodiment, the core asset classes include short-term US
government bonds, long-term US government bonds, and US equities. To expand
the
core asset classes to cover the full range of possible investments that people
generally
have access to, additional asset classes may be incorporated into the pricing
module 305
or the additional asset classes may be included in the factor model 310 and be
conditioned
on the core asset classes, as discussed further below.
Based upon the core asset scenarios generatedby the pricing module 305, the
factor module 310 produces return scenarios (also referred to herein as
"factor model
asset scenarios") for a set of factor asset classes that are used for both
exposure analysis,
such as style analysis, and the simulation of portfolio returns. The
additional asset
classes, referred to as factors, represented in the factor model are
conditional upon the
core asset class return scenarios generated by the pricing module 305.
According to one
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embodiment, these additional factors may correspond to a set of asset classes
or indices
that are chosen in a manner to span the range of investments typically
available to
individual investors in mainstream mutual funds and defined contribution
plans. For
example, the factors may be divided into the following groups: cash, bonds,
equities, and
foreign equities. The equities group may further be broken down into two
different broad
classifications (1) value versus growth and (2} market capitalization. Growth
stocks are
basically stocks with relatively high prices relative to their underlying book
value (e.g.,
high price-to-book ratio). In contrast, value stocks have relatively low
prices relative to
their underlying book value. With regard to market capitalization, stocks may
be divided
into groups of large, medium, and small capitalization. An exemplary set of
factors is
listed below in Table 1.
Exemplary Set of Factors
Grou Factor
Cash: Short Term US Bonds core class
Bonds: Intermediate-term US Bonds core
class
Lon -term US Bonds core class
US Co orate Bonds
US Mort a a Backed Securities
Non-US Government Bonds
uities: Lar a Ca Stock -- Value
Lar a Ca Stock -- Growth
Mid Ca Stock -- Value
Mid Ca Stock -- Growth
Small Ca Stock -- Value
Small Ca Stock -- Growth
Forei n: International ui -- Euro a
International E ui -- Pacific
International Equity -- Emerging
Markets
1 S Table 1
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At this point it is important to point out that more, less, or a completely
different set of
factors may be employed depending upon the specific implementation. The
factors listed
in Table 1 are simply presented as an example of a set of factors that achieve
the goal of
spanning the range of investments typically available to individual investors
in
mainstream mutual funds and defined contribution plans. It will be apparent to
those of
ordinary skill in the art that alternative factors may be employed. In
particular, it is
possible to construct factors that represent functions of the underlying asset
classes for
pricing of securities that are nonlinearly related to the prices of certain
asset classes (e.g.,
derivative securities}. In other embodiments of the present invention,
additional factors
may be relevant to span a broader range of financial alternatives, such as
industry specific
equity indices.
On a periodic basis, the financial product mapping module 315 maps financial
product returns onto the factor model. In one embodiment, the process of
mapping
financial product returns onto the factor model comprises decomposing
financial product
I S returns into exposures to the factors. The mapping, in effect, indicates
how the financial
product returns behave relative to the returns of the factors. According to
one
embodiment, the financial product mapping module 31 S is located on one of the
servers
(e.g., the financial staging server I20, the broadcast server 115, or the
AdviceServer I 10}.
In alternative embodiments, the f nancial product mapping module 3 I 5 may be
located on
the client 105.
In one embodimentof the present invention, an external approach referred to as
"returns-based style analysis" is employed to determine a financial product's
exposure to
the factors. The approach is referred to as external because it does not rely
upon
information that may be available only from sources internal to the financial
product.
Rather, in this embodiment, typical exposures of the financial product to the
factors may
be established based simply upon realized returns of a financial product, as
described
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further below. For more background regarding returns-based style analysis see
Sharpe,
William F. "Determining a Fund's Effective Asset Mix," Investment Management
Review, December 1988, pp. 59-69 and Sharpe, William F. "Asset Allocation:
Management Style and Performance Measurement," The Journal of Portfolio
S Management, 18, no. 2 (Winter 1992), pp. 7-19 ("Sharpe [1992]"}.
Alternative approaches to determining a financial product's exposure to the
factors include surveying the underlying assets held in a financial product
(e.g. a mutual
fund) via information filed with regulatory bodies, categorizing exposures
based on
standard industry classification schemes (e.g. SIC codes), identifying the
factors
exposures based on analysis of the structure of the product (e.g. equity index
options, or
mortgage backed securities), and obtaining exposure information based on the
target
benchmark from the asset manager of the financial product. In each method, the
primary
function of the process is to determine the set of factor exposures that best
describes the
performance of the financial product.
The tax adjustment module 320 takes into account tax implications of the
financial products and financial circumstances of the user. For example, the
tax
adjustment module 320 may provide methods to adjust taxable income and
savings, as
well as estimates for future tax liabilities associated with early
distributions from pension
and defined contribution plans, and deferred taxes from investments in
qualified plans.
Further, the returns for financial products held in taxable investment
vehicles (e.g. a
standard brokerage account) may be adjusted to take into account expected tax
effects for
both accumulations and distributions. For example, the component of returns
attributable
to dividend income should be taxed at the user's income tax rate and the
component of
returns attributable to capital gains should be taxed at an appropriate
capital gains tax rate
depending upon the holding period.
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Additionally, the tax module 320 may forecast future components of the
financial
products total return due to dividend income versus capital gains based upon
one or more
characteristics of the f nancial products including, for example, the active
or passive
nature of the financial product's management, turnover ratio, and category of
financial
product. This allows precise calculations incorporating the specific tax
effects based on
the financial product and financial circumstances of the investor. Finally,
the tax module
320 facilitates tax efficient investing by determining optimal asset
allocation among
taxable accounts (e.g., brokerage accounts) and nontaxable accounts (e.g., an
Individual
Retirement Account (IRA), or employer sponsored 401 {k) plan). In this manner
the tax
module 320 is designed to estimate the tax impact for a particular user with
reference to
that particular user's income tax rates, capital gains rates, and available
financial
products. Ultimately, the tax module 320 produces tax-adjusted returns for
each available
financial product and tax-adjusted distributions for each available financial
product.
The portfolio optimization module 340 calculates the utility maximizing set of
1 S financial products under a set of constraints defined by the user and the
available feasible
investment set. In one embodiment, the calculation is based upon a mean-
variance
optimization of the financial products. The constraints defined by the user
may include
bounds on asset class and/or specific financial product holdings. In addition,
users can
specify intermediate goals such as buying a house or putting a child through
college, for
example, that are incorporated into the optimization. In any event,
importantly, the
optimization explicitly takes into account the impact of future contributions
and expected
withdrawals on the optimal asset allocation. Additionally, the covariance
matrix used
during optimization is calculated based upon the forecasts of expected returns
for the
factors generated by the factor module 310 over the investment time horizon.
As a result,
the portfolio optimization module 340 may explicitly take into account the
impact of
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different investment horizons, including the horizon effects impact from
intermediate
contributions and withdrawals.
The simulation processing module 330 provides additional analytics for the
processing of raw simulated return scenarios into statistics that may be
displayed to the
S user via the UI 345. In the one embodiment of the present invention, these
analytics
generate statistics such as the probability of attaining a certain goal, or
the estimated time
required to reach a certain Ievel of assets with a certain probability. The
simulation
processing module 330 also incorporates methods to adjust the simulated
scenarios for
the effects induced by sampling error in relatively small samples. The
simulation
processing module 330 provides the user with the ability to interact with the
portfolio
scenarios generated by the portfolio optimization module 340 in real-time.
The annuitization module 325 provides a meaningful way of representing the
user's portfolio value at the end of the term of the investment horizon.
Rather than
simply indicating to the user the total projected portfolio value, one
standard way of
conveying the information to the user is converting the projected portfolio
value into a
retirement income number. The projected portfolio value at retirement may be
distributed
over the length of retirement by dividing the projected portfolio value by the
length of
retirement. More sophisticated techniques may involve determining how much the
projected portfolio value will grow during retirement and additionally
consider the effects
of inflation. In either event, however, these approaches erroneously assume
the length of
the retirement period is known in advance.
It is desirable, therefore, to present the user with a retirement income
number that
is more representative of an actual standard of living that could be locked in
for the
duration of the user's retirement. According to one embodiment, this
retirement income
number represents the inflation adjusted income that would be guaranteed by a
real
annuity purchased from an insurance company or synthetically created via a
trading
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strategy involving inflation-indexed treasury bond securities. In this manner,
the
mortality risk is taken out of the picture because regardless of the length of
the retirement
period, the user would be guaranteed a specific annual real income. To
determine the
retirement income number, standard methods of annuitization employed by
insurance
S companies may be employed. Additionally, mortality probabilities for an
individual of a
given age, risk profile, and gender may be based on standard actuarial tables
used in the
insurance industry. For more information see Bowers, Newton L. Jr., et al,
"Actuarial
Mathematics," The Society of Actuaries, Itasca, Illinois, 1986, pp. 52-59 and
Society of
Actuaries Group Annuity Valuation Table Task Force, "1994 Group Annuity
Mortality
Table and 1994 Group Annuity Reserving Table," Transactions of the Society of
Actuaries, Volume XLVII, 1994, pp. 865-9I3. Calculating the value of an
inflation-
adjusted annuity value may involve estimating the approl3riate values of real
bonds of
various maturities. The pricing module 305 generates the prices of real bonds
used to
calculate the implied real annuity value of the portfoiio at the investment
horizon.
Referring now to the plan monitoring module 350, a mechanism is provided for
alerting the user of the occurrence of various predetermined conditions
involving
characteristics of the recommended portfolio. Because the data upon which the
portfolio
optimization module 340 depends is constantly changing, it is important to
reevaluate
characteristics of the recommended portfolio on a periodic basis so that the
user may be
notified in a timely manner when there is a need for him/her to take
affirmative action, for
example. According to one embodiment, the plan monitoring module 350 is
located on
the AdviceServer 1 i0. In this manner, the plan monitoring module 350 has
constant
access to the user profile and portfolio data.
In one embodiment, the occurrence of two basic conditions may cause the plan
monitoring module 350 to trigger a notification or alert to the user. The
first condition
that may trigger an alert to the user is the current probability of achieving
a goal falling
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outside of a predetermined tolerance range of the desired probability of a
achieving the
particular goal. Typically a goal is a financial goal, such as a certain
retirement income or
the accumulation of a certain amount of money to put a child though college,
for
example. Additionally, the plan monitoring module 350 may alert the user even
if the
S current probability of achieving the financial goal is within the
predetermined tolerance
range if a measure of the currently recommended portfolio's utility has fallen
below a
predetermined tolerance level. Various other conditions are contemplated that
may cause
alerts to be generated. For example, if the nature of the financial products
in the currently
recommended portfolio have changed such that the risk of the portfolio is
outside the
user's risk tolerance range, the user may receive an indication that he/she
should
rebalance the portfolio. Plan monitoring processing, exemplary real world
events that
may lead to the above-described alert conditions, and additional alert
conditions are
described further below.
The UI module 345 provides mechanisms for data input and output to provide the
user with a means of interacting with and receiving feedback from the
financial advisory
system 100, respectively. Further description of a UI that may be employed
according to
one embodiment of the present invention is disclosed in U.S. Patent Nos.
5,918,217 and
6,012,044, both entitled "USER INTERFACE FOR FINANCIAL; ADVISORY
SYSTEM," the contents of which are hereby incorporated by reference.
Other modules may be included in the financial advisory system 100 such as a
pension module and a social security module: The pension module may be
provided to
estimate pension benefits and income. The social security module may provide
estimates
of the expected social security income that an individual will receive upon
retirement.
The estimates may be based on calculations used by the Social Security
Administration
(SSA), and on probability distributions for reductions in the current level of
benefits.
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Core Asset Scenario Generation
Figure 4 is a flow diagram illustrating core asset class scenario generation
according to one embodiment of the present invention. In embodiments of the
present
invention, core assets include short-term US government bonds, long-term US
government bonds, and US equities. At step 410, parameters for one or more
functions
describing state variables are received. The state variables may include
general economic
factors, such as inflation, interest rates, dividend growth, and other
variables. Typically,
state variables are described by econometric models that are estimated based
on observed
historical data..
At step 420, these parameters are used to generate simulated values for the
state
variables. The process begins with a set of initial conditions for each of the
state
variables. Subsequent values are generated by iterating the state variable
function to
generate new values conditional on previously determined values and a randomly
drawn
innovation term. In some embodiments, the state variable functions may be
deterministic
rather than stochastic. In general, the randomly drawn innovation terms for
the state
variable functions may be correlated with a fixed or conditional covariance
matrix.
At step 430, returns for core asset classes are generated conditional on the
values
of the state variables. Returns of core asset classes may be described by a
function of a
constant, previously determined core asset class returns, previously
determined values of
the state variables, and a random innovation term. Subsequent values are
generated by
iterating a core asset class function to generate:new values conditional on
previously
determined values and a random draws of the innovation term. In some
embodiments,
the core asset class functions may be deterministic rather than stochastic. In
general, the
randomly drawn innovation terms for the core asset class functions may be
correlated
with a fixed or conditional covariance matrix.
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In alternative embodiments; steps 410 and 420 may be omitted and the core
asset
class returns may be generated directly in an unconditional manner. A simple
example of
such a model would be a function consisting of a constant and a randomly drawn
innovation term.
A preferred approach would jointly generate core asset class returns based on
a
model that incorporates a stochastic process {also referred to as a pricing
kernel) that
limits the prices on the assets and payoffs in such a way that no arbitrage is
possible. By
further integrating a dividend process with the other parameters an arbitrage
free result
can be ensured across both stocks and bonds. Further description of such a
pricing kernel
IO is disclosed in a copending U.S. Patent application entitled "PRICING
KERNEL FOR
FINAIvTCIAL ADVISORY SYSTEM," Application No. 08/982,941, filed on December 2,
1997, assigned to the assignee of the present invention, the contents of which
are hereby
incorporated by reference.
Factor Model Asset Scenario Generation
Referring now to Figure 5, factor model asset scenario generation will now be
described. A scenario in this context is a set of projected future values for
factors.
According to this embodiment, the factors may be mapped onto the core asset
factors by
the following equation:
r;, = a; +~3,,ST _ Bonds, +,1~2, LT Bonds +,03; US Stocks, + s, {EQ # 1 )
where
r;, represents the return for a factor, i, at time t
~3~; represent slope coefficients or the sensitivity of the factor i to core
asset class j
ST _ Bonds, is a core asset class representing the returns estimated by the
pricing
module 305 for short-term US government bonds at time t
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LT-Bonds, is a core asset class representing the returns estimated by the
pricing
module 305 for long-term US government bonds at time t.
US Stocks, is a core asset class representing the returns estimated by the
pricing
module 305 for US stocks at time t.
a ; is a constant representing the average returns of factor asset class i
relative to
the core asset class exposures {"factor alpha").
~, is a residual random variable representing the returns of factor asset
class i that
are not explained by the core asset class exposures ("residual variance").
At step 510, the beta coefficients (also referred to as the loadings or slope
i 0 coeff cients) for each of the core asset classes are determined. According
to one
embodiment, a regression is run to estimate the values of the beta
coefficients. The
regression methodology may or may not include restrictions on the sign or
magnitudes of
the estimated beta coefficients. In particular, in one embodiment of the
present invention,
the coeff cients may be restricted to sum to one. However, in other
embodiments, there
may be no restrictions placed on the estimated beta coefficients.
Importantly, the alpha estimated by the regression is not used for generating
the
factor model asset scenarios. Estimates of alpha based on historical data are
extremely
noisy because the variance of the expected returns process is quite high
relative to the
mean. Based on limited sample data, the estimated alphas are poor predictors
of future
expected returns. At any rate, according to one embodiment, a novel way of
estimating
the alpha coefficients that reduces the probability of statistical error is
used in the
calibration of the factor model. This process imposes macroconsistency on the
factor
model by estimating the alpha coefficients relative to a known efficient
portfolio, namely
the Market Portfolio. Macroconsistency is the property that expected returns
for the
factor asset classes are consistent with an observed market equilibrium, that
is estimated
returns will result in markets clearing under reasonable assumptions. The
Market
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Portfolio is the portfolio defined by the aggregate holdings of all asset
classes. It is a
portfolio consisting of a value-weighted investment in all factor asset
classes. Therefore,
in the present example, macroconsistency may be achieved by setting the
proportion
invested in each factor equal to the percentage of the total market
capitalization
S represented by the particular factor asset class.
At step 520, a reverse optimization may be performed to determine the implied
factor alpha for each factor based upon the holdings in the Market Portfolio.
This
procedure determines a set of factor alphas that guarantee consistency with
the observed
market equilibrium. In a standard portfolio optimization, Quadratic
Programming (QP) is
employed to maximize the following utility function:
X rC(r ~' T.
E(r) X- ,u X =1 (EQ #2)
T
where,
E(r) represents expected returns for the asset classes,
C(r) represents the covariance matrix for the asset class returns,
T, Tau, represents a risk tolerance value,
X is a matrix representing the proportionate holdings of each asset class of
an
optimal portfolio comprising the asset classes, and
a is a vector of all ones.
C(r) may be estimated from historical returns data or more advantageously rnay
be
estimated from projected returns generated by a pricing kernel model.
inputs to a standard portfolio optimization problem include E(r), C(r), and
Tau
and QP is used to determine X. However, in this case, X is given by the Market
Portfolio, as described above, and a reverse optimization solves for E(r) by
simply
backing out the expected returns that yield X equal to the proportions of the
Market
Portfolio.
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Quadratic Programming (QP) is a technique for solving an optimization problem
involving a quadratic (squared terms) objective function with linear equality
and/or
inequality constraints. A number of different QP techniques exist, each with
different
properties. For example, some are better for suited for small problems, while
others are
better suited for large problems. Some are better for problems with very few
constraints
and some are better for problems with a large number of constraints. According
to one
embodiment of the present invention, when QP is called for, an approach
referred to as an
"active set" method is employed herein. The active set method is explained in
Gill,
Murray, and Wright, "Practical Optimization," Academic Press, 1981, Chapter 5.
The first order conditions for the optimization of Equation #2 are:
E(r) = 2C(r) X + Ku (EQ #3)
r
where K is a Lagrange multiplier; hence, knowing the Market Portfolio and any
two values of E(r) (for example, the risk free rate and the return on US
equities) the full
set of expected returns that are consistent with the Market Portfolio can be
derived. The
two values of E(r) required for the reverse optimization follow from the
expected returns
of the core assets.
At step 530, factor returns may be generated based upon the estimated alphas
from
step 520 and the estimated beta coefficients from step 510. As many factor
model asset
scenarios as are desired may be generated using Equation #1 and random draws
for the
innovation value s, . A random value fors, is selected for each evaluation of
Equation # 1.
According to one embodiments, is distributed as a standard normal variate. In
other
words E, is drawn from a standard normal distribution with a mean of 0 and a
standard
deviation of 1.
Advantageously, in this manner, a means of simulating future economic
scenarios
and determining the interrelation of asset classes is provided.
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Financial Product Exposure Determination
As discussed above, one method of determining how a financial product behaves
relative to a set of factor asset classes is to perform returns-based style
analysis.
According to one embodiment, returns for a given financial product may be
estimated as
a function of returns in terms of one or more of the factor asset classes
described above
based on the following equation:
r f - a~ + 'fhri r +.SrZr2 r'+'.. .-1-S~!'nr -~ 6r (EQ ~)
where,
a,r is the mean of the left over residual risk ("selection variance") of the
financial
product return that cannot be explained in terms of the factor loadings.
rg is the return for financial product f at time t,
rnt is the return for factor n at time t, and
s, is the residual at time t that is unexplained by movements in the factor
returns.
The financial product exposure determination module 3 l 5 computes the factor
asset class exposures for a particular fund via a nonlinear estimation
procedure. The
exposure estimates, Sfn, are called style coefficients, and are generally
restricted to the
range [0,1 ] and to sum to one. In other embodiments, these restrictions may
be relaxed
(for example, with financial products that may involve short positions, the
coefficients
could be negative). Alpha may be thought of as a measure of the relative under
or over
performance of a particular fund relative to its passive style benchmark.
At this point in the process, the goal is to take any individual group of
assets that
people might hold, such as a group of mutual funds, and map those assets onto
the factor
model, thus allowing portfolios to be simulated forward in time. According to
one
embodiment, this mapping is achieved with what is referred to as "returns-
based style
analysis" as described in Sharpe [1992], which is hereby incorporated by
reference.
Generally, the term "style analysis" refers to determining a financial
product's exposure
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to changes in the returns of a set of major asset classes using Quadratic
Programming or
similar techniques.
Figure 6 is a flow diagram illustrating a method of determining a financial
product's exposures to factor asset class returns according to one embodiment
of the
present invention. At step 610, the historical returns for one or more
financial products to
be analyzed are received. According to one embodiment, the financial product
exposure
module 315 may reside on a server device and periodically retrieve the
historical return
data from a historical database stored in another portion of the same computer
system,
such as RAM, a hard disk, an optical disc, or other storage device.
Alternatively, the
financial product exposure module 325 may reside on a client system and
receive the
historical return data from a server device as needed. At step 620, factor
asset class
returns are received.
At step 630, QP techniques or the like are employed to determine estimated
exposures (the S coefficients) to the factor asset class returns.
At step 640, for each financial product, expected future alpha is determined
for
each subperiod of the desired scenario period. With regards to mutual funds or
related
financial products, for example, historical alpha alone is not a good estimate
of future
alpha. That is, a given mutual fund or related f nancial product will not
continue to
outperform/under perform its peers indefinitely into the future. Rather,
empirical
evidence suggests that over performance may partially persist over one to two
years while
under performance. may persist somewhat longer (see for example, Carhart, Mark
M. "On
Persistence in Mutual Fund Performance." Journal of Finance, March 1997,
Volume 52
No. 1, pp.57-82).
For example, future alpha may depend upon a number of factors, such as
turnover,
expense ratio, and historical alpha. Importantly, one or more of these factors
may be
more or less important for particular types of funds. For example, it is much
more costly
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to buy and sell in emerging markets as compared to the market for large
capitalization US
equities. In contrast, bond turnover can be achieved at a much lower cost,
therefore,
turnover has much less affect on the future alpha of a bond fund than an
equity fund.
Consequently, the penalty for turnover may be higher for emerging market funds
compared to large cap U.S. equities and bond funds. Improved results may be
achieved
by taking into account additional characteristics of the fund, such as the
fact that the fund
is an index fund and the size of the fund as measured by total net assets, for
example.
According to one embodiment of the present invention, a more sophisticated
model is employed for determining future alpha for each fund:
I ~ at = aboae + P 'ahiaorirnl abort
(EQ #S)
where,
a,,Q,e is the baseline prediction for future Alpha of the fund
p , Rho, governs the speed of decay from a,,;sio,;~r t0 able
ahictorical is Alpha estimated in Equation #4
According to one embodiment,
able = C+~3,Exper~.se_Ratio+~QZTurnover+~33Fund Size (EQ #b}
where the parameters are estimated separately for each of four different
classes of
funds: US equity, foreign equity, taxable bond, nontaxable bond. These
parameters may
be estimated using conventional econometric techniques, such as ordinary least
squares
(OLS}. According to one embodiment, Rho is estimated by first calculating
historical
deviations from ab~.e ("residual alpha") and then estimating Rho as the f rst
order serial
correlation of the residual alpha series.
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Portfolio Optimization
Portfolio optimization is the process of determining a set of financial
products that
maximizes the utility function of a user. According to one embodiment,
portfolio
optimization processing assumes that users have a mean-variant utility
function, namely,
that people like having more wealth and dislike volatility of wealth. Based on
this
assumption and given a user's risk tolerance, the portfolio optimization
module 340
calculates the mean-variance efficient portfolio from the set of f nancial
products
available to the user. As described above, constraints defined by the user may
also be
taken into consideration by the optimization process. For example, the user
may indicate
a desire to have a certain percentage of his/her portfolio allocated to a
particular financial
product. In this example, the optimization module 340 determines the
allocation among
the unconstrained financial products such that the recommended portfolio as a
whole
accommodates the user's constraints) and is optimal for the user's level of
risk tolerance.
Prior art mean-variant portfolio optimization traditionally treats the problem
as a
I S single period optimization. Importantly, in the embodiments described
herein, the
portfolio optimization problem is structured in such as way that it may
explicitly take into
account the impact of different investment horizons and the impact of
intermediate
contributions and withdrawals. Further the problem is set up so that it may be
solved
with QP methods.
Referring now to Figure 7, a method of portfolio optimization according to one
embodiment of the present invention will now be described. At step 710,
information
regarding expected withdrawals is received. This information may include the
dollar
amount and timing of the expected withdrawal. At step 720, information
regarding
expected future contributions is received. According to one embodiment, this
information may be in the form of a savings rate expressed as a percentage of
the user's
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gross income or alternatively a constant or variable dollar value may be
specified by the
user.
At step 730, information regarding the relevant investment time horizon is
received. In an implementation designed for retirement planning, for example,
the time
horizon might represent the user's desired retirement age.
At step 740, information regarding the user's risk tolerance, Tau, is
received.
At step 750, the mean-variance efficient portfolio is determined. According to
one embodiment, wealth in real dollars at time T is optimized by maximizing
the
following mean-variance utility function by determining portfolio proportions
(Xi):
i0 U= E(WT)_ Var(W,.) (EQ #7)
T
where for a given scenario,
E( WT ) is the expected value of wealth at a time T
Var(WT)is the variance of wealth at time T
z is the user's risk tolerance
T-I T T-I T
WT-XILr'-I11(1+Rjl)+...+Xn~CI ~(1+Rjn)+g #8
tap j.t+1 t=0 j=t+!
where,
Xi represents the recommended constant proportion of each net contribution
that
should be allocated to financial product i.
Ct represents the net contribution at time t,
Rji represents the expected returns for financial product i in year j,
n is the number of financial products that are available for optimization,
g is the value of constrained assets for a given scenario,
The product of gross returns represents the compounding of values from year 1
to
the horizon. Initial wealth in the portfolio is represented by contribution
Co.
Importantly, the f nanciaI product returns need not represent fixed
allocations of a
single financial product. Within the context of the optimization problem, any
individual
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asset return may be composed of a static or dynamic strategy involving one or
more
financial products. For example, one of the assets may itself represent a
constant re-
balanced strategy over a group of financial products. Moreover, any dynamic
strategy
that can be formulated as an algorithm may be incorporated into the portfolio
optimization. For example, an algorithm which specifies risk tolerance which
decreases
with the age of the user could be implemented. It is also possible to
incorporate path
dependent algorithms (e.g., portfolio insurance).
According to Equation #8, contributions are made from the current year to the
year prior to retirement. Typically, a contribution made at time t will be
invested from
time t until retirement. An exception to this would be if a user specifies a
withdrawal, in
which case a portion of the contribution may only be held until the expected
withdrawal
date.
An alternative to the buy and hold investment strategy assumed above would be
to
implement a "constant mix" investment strategy or re-balancing strategy. For
purposes of
this example, it is assumed that the recommended fixed target asset-mix will
be held in an
account for each year in the future. Therefore, each year, assets will be
bought and/or sold
to achieve the target. Let f be the fraction of account wealth targeted for
the i-th asset,
then the sum of the fractions must equal one.
In the following "evolution" equations, nominal wealth aggregation is modeled
for
a single taxable account from the current time t = 0 to the time horizon t =
T. It is
assumed that "N" assets are in the account, labeled by the set of subscripts
{i=1, ..., N}.
The superscripts minus and plus are used to distinguish between the values of
a variable
just before, and just after, "settlement". The settlement "event" includes
paying taxes on
distributions and capital gains, investing new contributions, buying and
selling assets to
achieve the constant mix, and paying load fees. For example, VV~(t) is the
total wealth
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invested in all assets just after settlement at time "t". The evolution
equations for the pre-
and post-settlement values, the "dollars" actually invested in each asset,
are:
Wf (0), t - 0,
(19a} W;-(t} _
(1 + R; (t)] ' W, {t -1) - Ilk; {t)~I, 0 < t <_ T,
(19b) W;+ {t) _ ~' ' W + {t ), 0 <- t < T,
0, t=T.
In the above equation, the double-bar operator ~~ ~~ is equal to either its
argument or zero,
whichever is greater. From Eq.( 19a), we see that the pre-settlement value at
any time
(after the initial time) is j ust the gross return on the post-settlement
value of the previous
time less the "positive-part" of any distribution, i.e. the "dividend". Here,
k~(t) is the
portion of the return of the i-th asset that is distributed, and R~(t) is the
total nominal
return on the i-th asset in the one-year period (t-1, t]. We also assume that
an initial, pre-
settlement value is given for each asset. Eq.(19b) defines the post-settlement
value in
terms of the asset's constant mix and the total account value after
settlement. Since we
"cash-out" the portfolio at the time horizon, the final amount in each asset
at t = T is zero.
The pre- and post-settlement, total values are governed by the pair of
equations:
N
(I9c) W-(t) _ ~13;-{t), 0 _< t ST,
(19d ) W + (t) = W -{t) + C(t) + D(t) - L(t) - S(t), 0 <_ t <_ T .
In Eq.(19d), C(t) is the nominal contribution to the account at time "t", D(t)
is the total of
all distributed "dividends", L(t) is the "leakage", the total amount paid in
loads to both
rebalance and to invest additional contributions, and S(t) is the "shrinkage",
the total
amount paid in taxes on distributions and capital gains. We note that W+(T} is
the final
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horizon wealth after all taxes have been paid. The value of D(t), the total of
all
distributed dividends, is the sum of the positive distributions:
N
(19e} D{t) _ ~I~k; (t)~I, 0 5 t < T.
«,
Similarly, the "leakage" L(t) is the total amount of dollars paid in loads,
and L;(t) is the
number of dollars paid in loads on just the i-th asset. These individual loads
depend on I;,
the front-end load fee (a rate) on the i-th asset.
(19.f ) Lr (t) _ [li ~(1- la )J ' 1l W + (t) - ~~kr (t)~~ - ~', (t)'~°
0 5 t S T.
N
{l 9g) L(t) _ ~ L; (t), 0 <- t 5 T.
;_,
If there is a short-term loss (negative distribution), the load fee paid on an
asset's purchase
is just a fixed fraction of the purchase price.' When there is a short-term
gain (positive
distribution), we can re-invest any part of it without load fees, and pay fees
only on
purchases in excess of the gain. Note that at the horizon, we "cash-out", and
don't pay any
load fees.
The equation for the "shrinkage" S(t}, the total amount paid in taxes, has two
terms. The
first term is the tax on distributions and is multiplied by the marginal tax-
rate; the second
term is the tax on capital gains and is multiplied by the capital gains tax-
rate.
N N
(19h) S(t)=r",.~k,(t)+z~g-~[l-B;(t-1)lW;-(t)J~I~W,-(t)-W,.+{t)JI, 0<-tST.
r=t
In Eq.( 19h), the capital gains tax depends on the basis B;(t), the total of
all a$er-tax
nominal-dollars that have been invested in the i-th asset up to time "t". Note
that there can
' The dollar amount of a load fee is proportional to the ratio l / ( 1- !).
That's because our wealth variables
are all measured as "net" loads. To see this, suppose we make a contribution
c. After loads, we are left with
W = ( I- n c. In terms of W, the amount we paid in loads is L = ! c = ( ! / (I-
!) ] W.
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be either a capital gain or loss. The double-bar operator ensures that capital
gains are
triggered only when there is a sale of assets. At the horizon, we sell all
assets, and
automatically pay all taxes. The basis B~(t), evolves according to the
following recursion
equation:
Br (0)~ t = 0,
(19i) B;(t)= B;(t-1)+I~W,+(t)-W,.-(t)I~+L; (t)
-~Br(t-1)~W (t)~'~~~% (t)-~i+(t)II' 0<t<T.
S
Note that all new purchases are made with after-tax dollars, and add to the
basis; all sales
decrease the basis. Further, any load paid to purchase an asset adds to the
basis. We
assume that the initial basis B~(0) of an asset is either given, or defaults
to the initial, pre-
settlement value so that the average basis is initially equal to one.
A "constitutive" equation for k~(t) is needed to complete our system of
equations. Short-
term distributions depend on the "type" of asset; here we model mutual funds:
k;(0), t=0,
(20a) k; (t ) _
K; ~R;(t)~W;+(t-1), 0<t<T.
Often, we set the initial distribution to zero, and assume that the asset's
initial pre-
settlement value has already accounted for any non-zero, initial value. We
note that the
distribution is proportional to the amount of wealth at "stake" during the
prior-period. For
mutual funds, we assume that the distribution is a fraction K~ of the prior-
period's total
return, and therefore is also proportional to Rj(t). Note that the
distribution in Eq.(20a)
can be a gain {positive) or a loss (negative). In contrast, the constitutive
equation for
stocks takes the form:
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t=0,
(20b} k;(t)= x~,~l+R,(t)]-W;+(t-i}, 0<t<_T.
For stocks, the proportionality constant x; models a constant dividend
"yield", and the
distribution is always a gain (non-negative}. For stocks (mutual funds), the
distribution is
proportional to the gross (simple) return.
Before we leave this section, a word on 401 (k) plans and IRA's (with no load
funds). For
these accounts, the loads and taxes are ignored, and there is no basis in the
asset. At
"settlement", the user just re-balances their account. The evolution equations
for these
accounts is trivial in comparison to the equations for a general taxable
account:
(21a) W,' (t) = f, ~ W+(t), 0 <_ t 5 T,
W+(0), t = 0,
(21b) W+(t) _ ~'
1+~j,~R,(t) -W+(t-i)+C(t), 0<tST.
,_~
At the time horizon T, the total wealth in a non-taxable account is just
W+(T). This is a
pre-withdrawal total value. When retirement withdrawals are made from a tax-
free
account, they are taxed at the client's average tax-rate, za. Therefore, the
"after-tax"
equivalent value is equal to "pre-tax" wealth W+(T) times the tax factor (1 -
Ta).
How do we aggregate taxable and non-taxable accounts to get total portfolio
wealth? We
choose non-taxable accounts as a baseline. If all the funds in a non-taxable
account were
converted to an annuity, and the annuity payments were taken as withdrawals,
they the
withdrawals would mimic a salary subject to income taxes. This is precisely
the client's
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pre-retirement situation. Before aggregating a taxable account, we scale its
"after=tax"
value to this baseline using the formula:
(22) WbQ«,;~~ - WoJ7er-rar ~(1- Ta )~
Essentially, the baseline equivalent is obtained by grossing up values using
the average
S tax-rate.
The evolution equation variables appear "implicitly" in the recursion
relations. Hence, we
need to "iterate" at each time step to solve for "explicit" variable values."
We illustrate
this process with an example. Consider the simple case where there are no
distributions,
contributions, or taxes; just loads, and a constant-mix strategy. Here, the
evolution
equations simplify to a single equation for the total, after-settlement wealth
W+(t}:
N N
(23) W+(t)=W'(t-I)~~f,~~l+R;(t)~-~f,'[l;~(1-Ir)J'~~W+(t)-~1+R;(t)~'W+(t-1~~'
;-t ;=t
Note, we only know W+(t) as an implicit function of W+{t-1), but given a guess
for it's
value, we can refine the guess by substituting it into the right-side of
Eq.(23).
It's instructive to re-write Eq.(23) as the pair of equations in terms of an
"effective" return
Re(t):
(24a) W+(t)=[1+Re{t)J'W+{t-1),
N N
(24b) Re (t ) _ ~ f, ~ R; {t) - ~, .f ' [l~ ~(1- h )J ' ~ R~ (t) - R; {t)~~~
;--t ;=t
Eq.(24a} is the evolution equation for a single asset with the effective
return. Eq.(24b) is
an implicit equation for the effective return Re(t) in terms of the asset
returns R~{t): i~Ve
" In practice a robust root-finding algorithm may be used rather than
iteration.
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solve for the effective return using iteration. When the loads are equal to
zero, as
expected, the effective return is just a weighted-average of the asset
returns. Even when
the loads are not zero, this average return is a good initial guess for the
iteration
procedure. In fact, using the average return as the initial guess and
iterating once yields
S the following explicit approximation for the effective return:
N
(25a) RWg, (t) _ ~ f, ~ R~ (t)~
N
(25b) R~ (t) ~ RWg, (t} - ~ f, ' l; ' IIRWgr (~) - R. (r)II~
r=E
Eq.(25b} is consistent with our intuition, and agrees well with higher order
iterates.
To determine the mutual fund input moments we must first calculate the kernel
moments. This procedure calculates successive annual kernel moments and
averages the
result. The resulting mean and covariance matrix is then utilized by the
reverse
optimization procedure and also as an input into the optimization procedure.
To calculate analytic core moments, first we must describe the wealth for each
core asset for an arbitrary holding period. For each of the core assets, the
resulting wealth
from an arbitrary investment horizon can be written as: [Note, this is an
approximation
for equities]
r _i
W,.,. - exp ~ a + bX ~+, + c II ~+, + d d~+, + eX f + fTI ~ + g 8 J
~=r
Where:
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a, b, c, d, e, f, g - Constants
X~ - Real rate in year j
1'h - Inflation rate in year j
8~ - Dividend growth rate in year j
The expectation of wealth for any of the core assets given information at time
zero is
then:
TE~eX;+bX;.i TE~IT~;+cll;.i rE~Ba;+d6;.i
E Ojl,'T - a o~r_,>E oe ;_,
I O Since X, II, and b are independent, we can deal with each of these
expectations
separately. For example, consider the contribution in the above equation from
inflation.
The summation can be rewritten as:
Eo exp ~~ ffI J + cII ;+~ = Eo exp f'II r + ( ~~ (.f + c)II ;) + cII ,.
j=r j=r+1
Next, we need to use iterated expectations to determine this expectation. We
can
write the expectation at time zero as the repeated expectation over the
various
innovations. For example, the equation for inflation can be rewritten as:
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r-1
E o exp .~ r + ~ ~ ~ f + c ) n j ) + c II T
r,1
T-1
_ Es~E~=...E'6reXp .~n~~-~~ ~.t+c)nj)+cIiT
j=rfl
T-l
E6I~.~= ...Esr-~ eXp ~r + y, ~.f + C)1Zj ) E6r LgonrJ
f=.+1
Assuming inflation follows a modified square root process:
II, = fl~ + pn II,-1 + U
S
Where ~~ ~~ denotes the Heaviside function
0 if II , _< 0
Ii , if II , > 0
Now we recursively start taking the expectations over epsilon starting at the
end and
working backward. So:
EEr recur ~ _Esr ~ecl~rfcPrnr-i+cox ~~nr-n Er
c(Yr+C~.nr-~ +72ca"nr-1)
~e
Where the approximation is due to the Heaviside function.
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Combining this with the above equation yields:
! ... E~T_ exp ~ fII , + { ~~ {f + c)n ~)~E'r ~e ~nT~
j=t+I
= E6E6=... EET_=exp fII , + (~~ {f + c)n j) E6T_ ~e°~r+~~Pr
+/~iax+~+J)nT_!)
j.!+~
In general for any time period t, an exponential linear function of II has the
following expectation:
Le .1;+Bln, ~- E.6t Le Al+Bl{Nr+Prne-i+or~~n!-I ~~sn
_ a Al+B/Pr +Blnr_ 1 (Pr+ lay'\BI
Al+B/pr+(Bl (pr+~iBlllni-t
a
a ~,_,+Bl_,n,_,
The critical feature is that an exponential linear function of I-I remains
exponential
linear after taking the expectation. This invariance allows for the backward
recursion
calculation. Only the constant (A) and the slope (B) are changing with
repeated
application of the expectation operator. The evolution of A and B can be
summarized as
A.r + a '~' f~ ~r Br + t
~,, L z
BJ - BJ+I ~~r + 2 6a B!+1
In addition, the BJ coefficient has to be increased by (c + f) to account for
the additional
IZ~ term in the summation. To implement this recursive algorithm to solve for
expected
wealth, first define the following indicator variable:
1 if t, 5 j 5 tz
I{t » t z ) -
0 Otherwise
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Next, the following algorithm may be employed:
InitialConditions J = T, AT - 0, BT - c
(1) J = J - 1
(2) AJ - AJ +W' f~ ~ BJ + i
BJ - BJ +, lP x + i ~ ~ BJ +, J + c ~ I(t + I , T - 1 ) + f ~ I(t, T - 1 )
(3) if J = 0, End
E W = a A'+B,n °
( r ,T )
(4) Go To ( I )
The same technique applies to X since it is also a square root process. A
similar
technique can be used to create a recursive algorithm for the b component. The
only
difference is that 8 is an AR( 1 ) process instead of a square root process.
In particular,
Sr - f~a + G~ Sr-1 + Ua Er
For this AR(1) process, the expectation is of the following form.
E'' Le AJ+BJW ~= E6r /e ~l+BI~Ps+Pss~-~+asE~)~
AJ+BJpd+ =a" BJ+BJp68~_ i
a .I J_~+B J_~B~_~
The evolution of A and B is thus summarized as:
AJ - '4J+1 + BJ+l C~a + 2 ~d J
BJ - BJ+1 Ps

CA 02399046 2002-07-31
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The recursive relationship for 8 is then:
InitiaiConditions J = T, AT - 0, BT - .d
(1) J = J - I
(2) y - AJ+I + Br+I ~f~s + i 6s
B~ - B~+, pa + d ~ I(t + l, T - 1) + g ~ I(t, T - 1)
(3) if J = 0, End
E W a ~~+s,ao
( ~ ,r ) _
{4) Go To (1)
This framework for calculating expected wealth can also be used to calculate
the variance
of wealth for an arbitrary holding period . From the definition of variance,
we have:
Vo(Wr,r ) = Eo(~ir )- Eo(wr.r )
but
r _. 2
W,.T - exp ~ a + bX J+, + c II ~+, + d S~+, + eX ~ + fTI ; + g ~l
,_~
T-I
- exp ~ 2~a + bXJ+, + cII;+, + d~~+I + eX~ + frl~ + g~i~
So the same technique can be used with a simple redefinition of the constants
to be twice
their original values. Similarly, the covariance between any two core assets
can be
calculated by simply adding corresponding constants and repeating the same
technique.
For the current parameter values, the constants for Bills, Bonds, and Equities
are:
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a b c d a F g
Bills 0.0077 0 -1 0 1 0.7731 0
Bonds 0.0642 -2.5725 -3.8523 0 2.5846 2.9031 0
Equities 0.0331 -2.4062 -3.7069 4.4431 2.48 2.79 -3.5487
Above, a methodology was described for calculating core asset analytic moments
for arbitrary horizons. This section describes how these moments are
translated into
annualized moments. The procedure described in this section essentially
calculates
successive annual moments for a twenty (20) year horizon and computes the
arithmetic
average of these moments. These 'effective' annual moments may then be used as
inputs
into the reverse optimization procedure and the individual optimization
problem.
For this calculation, first make the following definitions:
M; = Expected return for j'° asset over the period t, t + 1
Cove' = Covariance of returns on asset i with asset j over the period t, t + 1
These expected returns and covariance are calculated using the formulas
described above
The effective annual expected return for asset j is then calculated as:
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T
M' _ ~~ M~
r=,
Similarly, the effective annual covariance between returns on asset l and
returns on asset j
are calculated as: (Note, the weights, co,, are between zero and one, and sum
to one.)
r
Cov'~' _ ~ r,~, Cov'
. _,
In one embodiment, this annualizing technique could be personalized for a
given
user's situation. For example, the user's horizon could specify T, and their
level of
current wealth and future contributions could specify the relevant weights.
However for
purposes of illustration, the relevant 'effective' moments for optimization
and simulation
are computed assuming a horizon of 20 years (T=20), and equal weights (i.e.
1!T).
The techniques described in this section allow for the calculation of the
following
effective annual moments:
Output Description Units
parameter name
M' Bills: expected return Return per year
MZ Bonds: expected return Return per year
M3 Equity: expected return Return per year
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Covlv Bills: variance of returns (Return per year)z
Cov2~ Bonds: variance of returns (Return per year)Z
Cov3~ Equity: variance of returns (Return per year)2
Covt~2 Bills and Bonds: covariance {Return per year)2
Covt~3 Bills and Equity: covariance (Return per year)2
Covz3 Bonds and Equity: covariance (Return per year)2
Plan Monitorine
Exemplary conditions which may trigger an alert of some sort from the plan
monitoring module 350 were described above. At this point, some of the real
world
events that may lead to those alert conditions will now be described. The real
world
events include the following: ( I ) a financial product's style exposure
changes, (2) the
market value of the user's assets have changed in a significant way, (3} new
financial
products become available to the user, (4) the risk characteristics of the
user's portfolio
have deviated from the desired risk exposure, or {5) the currently recommended
portfolio
no longer has the highest expected return for the current level of portfolio
risk (e.g., the
portfolio is no longer on the mean-variance efficient frontier). An efficient
frontier is the
sets of assets (portfolios) that provide the highest level of return over
different levels of
risk. At each point on the efficient frontier, there is no portfolio that
provides a higher
expected return for. the same or lower level of risk.
When a financial product's exposures change it may pull the user's portfolio
off
of the e~cient frontier. That is, due to a shift in the investment style of a
particular
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financial product, the portfolio as a whole may no longer have the highest
expected return
for the current level of risk. According to one embodiment of the present
invention, if the
inefficiency is greater than a predetermined tolerance or if the inefficiency
will
substantially impact one of the user's financial goals, such as his/her
retirement income
S goal, then the user is notified that he/she should rebalance the portfolio.
However, if the
inefficiency is within the predefined tolerance then the plan monitoring
module 350 may
not alert the user. In one embodiment, the predefined tolerance depends upon
the impact
of the inefficiency on expected wealth. In addition, the tolerance could
depend upon
relevant transaction costs.
A significant change in the market value of the user's assets may affect one
or
both of the probability of achieving a financial goal and the current risk
associated with
the portfolio. In the case that the user's portfolio has experienced a large
loss, the
portfolio may no longer be within a predetermined probability tolerance of
achieving one
or more financial goals. Further, as is typical in such situations, the risk
associated with
the portfolio may also have changed significantly. Either of these conditions
may cause
the user to be notified that changes are required in the portfolio allocation
or decision
variables to compensate for the reduction in market value of the portfolio. A
large
increase in the value of the user's portfolio, on the other hand, could
trigger an alert due
to the increase in the probability of achieving one or more financial goals or
due to the
altered risk associated with the newly inflated portfolio.
When one or more new financial products become available to the user, the user
may be alerted by the plan monitoring module 350 if, for example, a higher
expected
return may be possible at lower risk as a result of diversifying the current
portfolio to
include one or more of the newly available financial products.
Having explained the potential effects of some real world events that may
trigger
alerts, exemplary plan monitoring processing will now be described with
respect to
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Figure 8. At step 810, the data needed for reevaluating the current portfolio
and for
determining a current optimal portfolio is retrieved, such as the user profile
and portfolio
data which may be stored on the AdviceServer 110, for example. Importantly,
the user
profile may include investment plan profile information stored during a
previous session,
such as the probability of reaching one or more f nancial goals, the risk of
the portfolio,
and the like. As described above, selected user information on the
AdviceServer 110 may
be kept up to date automatically if the financial advisory system 100 has
access to the
record-keeping systems of the user's employer. Alternatively, selected user
information
may be updated manually by the user.
At step 820, a current optimal portfolio is determined, as described above.
Importantly, changes to the user database and/or portfolio data are taken into
consideration. For example, if one or more new financial products have become
available
to the user, portfolios including the one or more new financial products are
evaluated.
At step 830, the current portfolio is evaluated in a number of different
dimensions
to determine if any trigger conditions are satisfied. For example, if the
increase in
expected wealth, or the increase in the probability of reaching one' or more
investment
goals resulting from a reallocation to the current optimal portfolio is above
a
predetermined tolerance, then processing will continue with step 840.
Additionally, if
the risk of the current portfolio is substantially different from the
investment plan profile
or if the probability of achieving one or more fnancial goals is substantially
different
from the investment plan profile, then processing continues with step 840.
At step 840, advice processing is performed. According to one embodiment of
the
present invention, based upon the user's preference among the decision
variables, the
system may offer advice regarding which decision variable should be modified
to bring
the portfolio back on track to reach the one or more financial goals with the
desired
probability. In addition, the system may recommend a reallocation to improve
efficiency
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of the portfolio. An alert may be generated to notify the user of the advice
andlor need
for affirmative action on his/her part. As described above, the alert may be
displayed
during a subsequent user session with the f nancial advisory system 100 and/or
the alerts
may be transmitted immediately to the user by telephone, fax, email, pager,
fax, or similar
messaging system.
Advantageously, the plan monitoring module 350 performs ongoing portfolio
evaluation to deai~with the constantly changing data that may ultimately
affect the
exposure determination process and the portfolio optimization process. In this
manner,
the user may receive timely advice instructing him/her how to most efficiently
achieve
one or more financial goals and/or maintain one or more portfolio
characteristics based
upon the available set of financial products.
In the foregoing specification, the invention has been described with
reference to
specific embodiments thereof. It will, however, be evident that various
modifications and
1 S changes may be made thereto without departing from the broader spirit and
scope of the
invention. The specification and drawings are, accordingly, to be regarded in
an
illustrative rather than a restrictive sense.
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