U.S. patent application number 13/965621 was filed with the patent office on 2013-12-12 for methodology and process for constructing factor indexes.
This patent application is currently assigned to AXIOMA, INC.. The applicant listed for this patent is Anthony A. Renshaw. Invention is credited to Anthony A. Renshaw.
Application Number | 20130332391 13/965621 |
Document ID | / |
Family ID | 49084210 |
Filed Date | 2013-12-12 |
United States Patent
Application |
20130332391 |
Kind Code |
A1 |
Renshaw; Anthony A. |
December 12, 2013 |
Methodology and Process For Constructing Factor Indexes
Abstract
Approaches to the construction of indexes are addressed wherein
a portfolio of securities such as stocks, bonds, or the like and
their associated investment weights or shares is generated. Indexes
can be used as investment tools in various ways. For instance,
indexes comprising a plurality of securities can often be bought
and sold more cheaply than buying and selling the individual
constituents of the index. This pricing differential allows
investment with reduced transaction costs. Alternatively, in
passive and enhanced indexing, investments are made with reference
to an index. Performance statistics such as return and risk are
reported with respect to the reference index. Factor indexes can
serve as active manager benchmarks or the underlyers for investable
products such as exchange traded funds and mutual funds. Computer
based systems, methods and software are addressed for constructing
indexes that replicate the returns of a quantitative factor such as
medium term momentum or value. Further, processes and methodology
are described by which the index can have the best possible
replication of the underlying factor returns as well as other
desirable characteristics. The methodology provides an approach to
determine the index even when all desirable characteristics of the
index are not simultaneously achievable.
Inventors: |
Renshaw; Anthony A.; (New
York, NY) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Renshaw; Anthony A. |
New York |
NY |
US |
|
|
Assignee: |
AXIOMA, INC.
New York
NY
|
Family ID: |
49084210 |
Appl. No.: |
13/965621 |
Filed: |
August 13, 2013 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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12958778 |
Dec 2, 2010 |
8533089 |
|
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13965621 |
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61265910 |
Dec 2, 2009 |
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Current U.S.
Class: |
705/36R |
Current CPC
Class: |
G06Q 40/06 20130101 |
Class at
Publication: |
705/36.R |
International
Class: |
G06Q 40/06 20060101
G06Q040/06 |
Claims
1. A computer based method of constructing a factor index of
portfolio weights comprising: selecting a universe of possible
investments; defining a benchmark portfolio comprising a set of
holdings in the universe; selecting a first fully specified factor
risk model defined for the universe of securities, said first
factor risk model comprising a fully specified matrix of factor
exposures, a matrix of factor covariances, and a matrix of specific
risk variances; selecting a target factor which is one of the
factors defined by the first factor risk model; constructing a
target factor portfolio for the target factor whose holdings are
fully determined by the universe, the benchmark portfolio, and the
first factor risk model, and whose exposure to the target factor is
substantially different than the exposure of the benchmark
portfolio to the target factor; selecting a second fully specified
factor risk model defined for the universe of securities, said
second factor risk model comprising a fully specified matrix of
factor exposures, a matrix of factor covariances, and a matrix of
specific risk variances; determining weights of each security for a
factor index so that the tracking error between the factor index
and the target factor portfolio as predicted by the second risk
model is less than a prescribed amount; and outputting the factor
index weights as an electronic output.
2. The computer based method of claim 1 wherein the target factor
selected represents a linear combination of one or more of the
following metrics: exchange rate sensitivity, growth, leverage,
liquidity, market sensitivity, long term momentum, medium term
momentum, short term momentum, size, value, volatility, one or more
countries, one or more industries, one or more sectors, and one or
more currencies.
3. The computer based method of claim 1 wherein the universe of
securities is selected based on a second factor.
4. The computer based method of claim 3 wherein the second factor
indicates the country, region, currency, size, value or growth of
each element in the universe.
5. The computer based method of claim 4 wherein the country factor
comprises U.S. equities.
6. The computer based method of claim 1 further comprising:
limiting the exposure of the factor index to a second factor to
insure factor neutrality to the second factor.
7. The computer based method of claim 6 wherein the second factor
is a factor defined by the first risk model but is different than
the target factor.
8. A computer system for constructing a factor index of portfolio
weights comprising: a programmed processor for selecting a universe
of possible investments; the programmed processor selecting a
benchmark portfolio comprising a set of holdings in the universe;
the programmed processor selecting a first fully specified factor
risk model defined for the universe of securities, said first
factor risk model comprising a fully specified matrix of factor
exposures, a matrix of factor covariances, and a matrix of specific
risk variances; the programmed processor selecting a target factor
which is one of the factors defined by the first factor risk model;
the programmed processor constructing a target factor portfolio for
the target factor whose holdings are fully determined by the
universe, the benchmark portfolio, and the first factor risk model,
and whose exposure to the target factor is substantially different
than the exposure of the benchmark portfolio to the target factor;
the programmed processor selecting a second fully specified factor
risk model defined for the universe of securities, said second
factor risk model comprising a fully specified matrix of factor
exposures, a matrix of factor covariances, and a matrix of specific
risk variances; the programmed processor determining weights of
each security for a factor index so that the tracking error between
the factor index and the target factor portfolio as predicted by
the second risk model is less than a prescribed amount; and the
programmed processor outputting the factor index weights as an
electronic output.
9. The computer system of claim 8 wherein the target factor
selected represents a linear combination of one or more of the
following metrics: exchange rate sensitivity, growth, leverage,
liquidity, market sensitivity, long term momentum, medium term
momentum, short term momentum, size, value, volatility, one or more
countries, one or more industries, one or more sectors, and one or
more currencies.
10. The computer system of claim 8 wherein the set of securities is
selected by the programmed processor utilizing a second factor.
11. The computer system of claim 10 wherein the second factor
wherein the second factor indicates the country, region, currency,
size, value or growth of each element in the universe.
12. The computer system of claim 11 wherein the country factor
comprises U.S. equities.
13. The computer system of claim 8 further comprising: the
programmed processor limiting the exposure of the factor index to a
second factor to insure factor neutrality to the second factor.
Description
[0001] The present application is a divisional of U.S. application
Ser. No. 12/958,778 filed Dec. 2, 2010 and claims the benefit of
U.S. Provisional Patent Application Ser. No. 61/265,910 filed Dec.
2, 2009, which are incorporated by reference herein in their
entirety.
FIELD OF INVENTION
[0002] The present invention generally relates to the construction
of indexes wherein a portfolio of securities such as stocks, bonds,
or the like and their associated investment weights, allocations,
or shares are determined. The present invention relates more
particularly to improved computer based systems, methods, and
software for constructing factor indexes that replicate the returns
associated with a quantitative factor such as medium term momentum
or value.
BACKGROUND OF THE INVENTION
[0003] Indexes can be used as investment tools in various ways. For
instance, indexes comprising a plurality of securities can often be
bought and sold more cheaply than buying and selling the individual
constituents of the index. This allows investment in these
securities with reduced transaction costs. In passive and enhanced
indexing, investments are made with reference to an index or
benchmark portfolio. A benchmark portfolio is a portfolio intended
to represent the market in general. The holdings of a benchmark
portfolio are often proportional to the market capitalization of
each security. Performance statistics such as return and risk are
reported with respect to the reference index or benchmark
portfolio. Indexes can serve as active manager benchmarks or the
underlyers for investable products such as ETFs and mutual
funds.
[0004] Factor indexes are indexes designed to replicate the returns
associated with a selected quantitative factor. Possible factors
include those present in commercial risk models. These include
style factors, industry factors, market factors, country factors,
and currency factors. Axioma's U.S. Equity Fundamental Factor Risk
Model uses ten style factors: exchange rate sensitivity, growth,
leverage, liquidity, market sensitivity, medium term momentum,
short term momentum, size, value, and volatility.
[0005] U.S. Pat. No. 7,698,202 describes characteristics,
properties and uses of factor risk models. This patent is
incorporated by reference herein in its entirety.
[0006] Factors are defined quantitatively. For instance, the ten
factors in Axioma's U.S. Equity Fundamental Factor Risk Model are
defined as follows.
[0007] The exchange rate sensitivity factor measures the
sensitivity of a stock's historical returns to the returns of a
currency basket, referenced in U.S. dollars. The basket used is the
International Monetary Fund's Special Drawing Rights (SDR) which
contains the currencies U.S. dollar, euro, Japanese yen, and pound
sterling. A stock's exchange rate sensitivity factor exposure is
the normalized slope obtained by regressing the time series of its
past 120 day return against that of the currency basket.
[0008] The growth factor gives an indication of a company's
historical rate of growth. The growth factor is calculated as the
product of the one year return on equity times one minus the
dividend payout rate. The return on equity is calculated as the
ratio of the annualized income over the last year to the common
equity value of a year ago. The dividend payout rate is calculated
as the ratio of the annualized dividends per share to the
annualized earnings per share.
[0009] The annualized income over the last year is calculated from
income before extraordinary income analogously to dividends per
share and earnings per share by going to the most recent filing in
the last 520 trading days and summing the four most recently filed
values over the previous 15 months. The common equity value of a
year ago is taken as the most recent value for common equity that
has been filed in the period between 780 and 260 trading days ago,
which is approximately three years to one year ago.
[0010] The leverage factor provides a measure of a company's
exposure to debt levels. The leverage factor is calculated as total
debt divided by market capitalization. Total debt is the sum of
long term debt and debt in current liabilities, short term debt,
taken on the most recent date over the last 520 trading days for
which both values have been filed. For market capitalization, the
20 day average is used.
[0011] The liquidity factor provides a measure of a stock's trading
activity, or lack thereof. It is defined as the natural logarithm
of the last 20 day average volume divided by the natural logarithm
of the last 20 day average market capitalization, both expressed in
currency values such as dollars.
[0012] The market sensitivity factor is a measure of a stock's
under or over performance relative to the broad market from
historical data. It is simply a stock's historical beta calculated
by regressing the time series of an asset's returns against the
market returns over the preceding 120 trading days. The beta is
then adjusted for autocorrelation in the asset returns and
asynchronous trading via the Scholes-Williams procedure with a
lag/lead of one.
[0013] The medium term momentum factor gives a measure of a stock's
past performance over the medium term time horizon. It is defined
as an asset's cumulative return over the last 250 trading days,
excluding the last 20 trading days.
[0014] The short term momentum factor gives a measure of a stock's
recent performance. It is defined as an asset's cumulative return
over the last 20 trading days.
[0015] The size factor differentiates between large and small
stocks and is defined as the natural logarithm of the market
capitalization, averaged over the last 20 trading days. Market
capitalization is computed as the product of the total shares
outstanding and closing price.
[0016] The value factor gives a measure of how fairly a stock is
priced within the market. The value factor is calculated as the
ratio of common equity to the current market capitalization (i.e.
Book-to-Price). The calculation uses the common equity value
reported on the most recent date from the last 520 trading days,
which is approximately two years. Market capitalization is taken as
shares outstanding times the closing price on the day for which the
risk model is generated.
[0017] The volatility factor gives a measure of an asset's relative
volatility over time according to its historical behavior. It is
calculated as the square root of the asset's absolute return
averaged over the last 60 days, divided by the cross-sectional
standard deviation of the risk model's estimation universe.
[0018] Axioma's U.S. Equity Statistical Factor Risk Model uses a
different set of factors to describe the universe of assets. These
factors can be used to define a factor. A factor may be defined by
one risk model, while risk and active risk, which is also known as
tracking error, is predicted with a different risk model.
[0019] There are numerous other possible factors. U.S. Pat. No.
7,620,577 lists a number of these: market price, market
capitalization, book value, sales, revenue, earnings, earnings per
share, income, income growth rate, dividends, dividends per share,
earnings before interest, tax, depreciation and amortization, and
the like. This patent is incorporated by reference herein in its
entirety.
[0020] For many factors, there are many similar methods to
quantitatively measure and define the factor score or factor
exposure of an asset to the factor. For example, value factors
commonly use several similar metrics including the book-to-price
ratio, the earnings-to-price ratio, and the earnings-per-share
ratios. In fact, Axioma's software is capable of expressing a fully
specified risk model in tennis of slightly different factors while
still giving good risk estimates. Factor indexes constructed using
similar factor scores would be similar even if the factor scores
were slightly different.
[0021] In factor indexes, the goal is to create a set of holdings
whose returns replicate the returns associated with one or more
factors. The return of a single factor can be defined in numerous
ways. It can refer to the return of a long-short, dollar neutral
portfolio that is long in those assets with the highest factor
scores and short in those assets with the lowest factors scores.
The weights can cover a fixed percentage of the assets in the set
of securities, measured in terms of capitalization or equal
weighting. The weights used for each security can be equally
weighted or cap-weighted. Alternatively, the return of a factor can
be derived from a regression. This regression can be a cross
sectional regression that regresses across assets or a times series
regression that regresses over time. Numerous other methods have
been proposed for defining the return of a factor, such as the
return of a factor mimicking portfolio.
[0022] Often an additional desirable characteristic for a factor
index is to make the returns of the factor index dissimilar to the
returns associated with non-targeted factors. Dissimilar can mean a
number of different things. It can refer to uncorrelated returns,
or, it can refer to limiting or neutralizing the exposure to a
factor, for example. When a factor index is dissimilar to
non-targeted factors, it is described as being pure.
[0023] The exposure of a portfolio to a factor is often measured
with respect to a benchmark portfolio. The exposure of the
benchmark portfolio is the sum of the products of the factor scores
for each asset times the weight of that asset in the benchmark
portfolio. Since this is the exposure of the benchmark, this
exposure value is representative of the market in general. The
exposure of the factor index is the sum of the products of the
factor scores for each asset times the weight of that asset in the
factor index. The difference in these two exposures is termed the
active exposure. That is, the active exposure is the exposure of
the factor index minus the exposure of the benchmark. Small
positive or negative active exposures indicate that the factor
index is similar to the benchmark. Large positive or large negative
active exposures indicate that the factor index is dissimilar to
the benchmark. For a pure factor index, the active exposure to the
targeted factor is substantially different than zero, while the
active exposure of the factor index to non-targeted factors is
small.
[0024] Exposures and active exposures are often expressed as a Z
score, which measures the exposure in terms of standard deviations.
An active exposure Z score of 100% indicates that the factor index
exposure is one standard deviation greater than the benchmark's
exposure. Such a large active exposure would indicate that the
exposure of the factor index and the benchmark portfolio are
substantially different. A Z score of -100% indicates that the
factor index exposure is one standard deviation less than the
benchmark's exposure and would also indicate a substantial
difference between the factor index and benchmark portfolio
exposures. On the other hand, an active exposure Z score between
-5% and 5% would indicate that the difference in exposures between
the factor index and the benchmark portfolio is less than one
twentieth of one standard deviation and would indicate that the two
sets of holdings have similar exposures to the factor.
[0025] It will be recognized that what constitutes a substantial
difference in exposures between the benchmark portfolio and the
factor index is a subjective determination by each investor.
Whereas some investors may consider a Z score difference of 25% to
be small, others may believe that 25% represents a substantial
difference. The same may be true for Z score differences of 20%,
30%, 40%, or even 50%.
[0026] Factor indexes described in the prior art suffer a number of
deficiencies.
[0027] Consider first simple factor indexes. Simple factor indexes
are sets of holdings created by selecting a universe of potential
investments, defining a factor score for each element or asset in
the universe, ranking the factor scores, and then buying those
assets with the highest scores and selling those assets with the
lowest scores. Buying refers to making a long position and selling
refers to making a short positions. In some factor indexes, only
the highest scores are bought, or, alternatively, only the lowest
scores are bought instead of sold. These two alternatives produce
long only indexes. The number of assets bought and sold can be
determined in many different ways. The number can be related to the
market capitalization of the assets or it can be a fixed number.
The amount bought or sold in each asset can also be determined in a
number of ways including cap-weighting or equal weighting. The
simple factor index may or may not be dollar neutral, meaning that
the total amount of assets bought equals the total amount of assets
sold. As one particular example, consider a 35% cap-weighted,
long-short dollar neutral. This simple factor index is created by
buying the 35% of a universe measured by market capitalization with
the highest factor scores, and selling the 35% of the universe with
the lowest scores. The individual asset weights are proportional to
market capitalization.
[0028] Simple factor indexes have a number of undesirable
properties. The turnover and the number of names involved are
generally large. This can render investing in the index expensive
and impractical. In addition, the resulting portfolios are often
not neutral to other factors such as value, leverage, and even
size. The returns associated with these simple indexes are not pure
returns, but also comprise returns associated with non-vanishing
factor exposures. Investment in these simple factor indexes can
therefore easily involve unintended bets on other factors.
[0029] Another kind of factor index is a factor mimicking portfolio
constructed from a matrix of factor exposures. See for example, R.
Littennan, Modern Investment Management: An Equilibrium Approach,
John Wiley and Sons, Inc., Hoboken, N.J., 2003 (Litterman), which
gives detailed descriptions of factor mimicking portfolios and
which is incorporated by reference herein in its entirety. Often,
the matrix of factor exposures is associated with a factor risk
model, but that need not be the case. Factor mimicking portfolios
are designed so that they have exposure to one and only one factor.
The exposure to all other factors in the risk model or matrix of
factor exposures is zero by construction. Factor mimicking
portfolios are pure.
[0030] Unfortunately, factor mimicking portfolios are expensive to
buy and trade. Furthermore, there is no explicit control over the
returns associated with these portfolios. In practice, the returns
of a factor mimicking portfolio may be quite different than the
returns associated with a factor.
[0031] A number of new products have been introduced that are
similar to simple factor indexes. These new factor index products
all explicitly control implementation costs such as turnover, the
number of names held, and the like. However, none of these new
products use a tracking error constraint or minimize tracking error
when constructing the products. Instead, they control or maximize
the exposure to the targeted factor.
[0032] Factor index products that construct factor indexes by
controlling the factor exposure of the target exposure are much
more likely to underperform compared to those products that
explicitly constrain the tracking error of the factor index. That
is, the returns of factor indexes using exposure control are likely
to differ from the true factor returns, defined here as the returns
of a target factor portfolio.
SUMMARY OF THE INVENTION
[0033] Among its several aspects, the invention described herein is
designed to overcome various shortcomings associated with known
factor index construction methodologies. Expertise in optimization
and portfolio construction enables the construction of indexes that
are practical to implement, for example, having low turnover,
controlled transaction costs, and limited number of names held, and
which are neutral to non-target factors. During times of high
volatility, it is often impossible to track factor returns well and
simultaneously manage index turnover and maintain factor neutrality
to other factors. The invention described here is designed so that
an advantageous tradeoff between these competing portfolio
characteristics is achieved whenever possible. Research indicates
that an advantageous tradeoff between competing portfolio
characteristics is generally not obtained simply by softening
constraints in a prescribed order, as has been proposed in the
prior art. Instead, a sequence of different construction
optimizations must be used to determine the most advantageous
combination of portfolio characteristics.
[0034] A further aspect of the present invention relates to
improved computer based systems, methods, and software for
constructing indexes that replicate the returns of a quantitative
factor such as momentum or value.
[0035] A further aspect of the present invention relates to
improved computer based systems, methods, and software for
constructing indexes that replicate the returns of more than one
factor simultaneously. That is, a universe of potential investments
is constructed that has a large exposure or tilt towards a first
factor. For example, there are well known universes which consider
stocks with known exposures to size, value or growth. Within this
investment universe, a factor index can be constructed that targets
a second factor. This produces a factor index associated with more
than one factor index.
[0036] Another aspect of the invention addresses processes and
methodologies by which the index can substantially replicate true
factor returns as well as other desirable characteristics. An
approach is described to determine the index even when all
desirable characteristics of the index are not simultaneously
achievable as described further below.
[0037] The description that follows describes the construction
methodology for one particular, prototypical factor, a momentum
factor, for both a large cap universe and a small cap universe.
These factor indexes therefore incorporate known characteristics
related to two distinct factors: momentum and size. It should be
understood that this particular factor index has been presented by
way of example only, and not limitation. The methodology,
processes, and techniques described would apply equally well to
other isolated factors and combined factors for the construction of
other factor indexes.
BRIEF DESCRIPTION OF THE DRAWINGS
[0038] FIGS. 1A and 1B illustrate monthly, one-way, single-side
turnover for a simple 35% cap-weighted, long-short momentum
index;
[0039] FIGS. 2A and 2B illustrate price to book ratios for a simple
35% cap-weighted, long-short momentum portfolio;
[0040] FIG. 3 shows a computer based system which may be suitably
utilized to implement the present invention.
DETAILED DESCRIPTION
[0041] Non-optimized factor indexes normally hold a large number of
names and experience large turnover. FIG. 1A shows a graph 102 of
the monthly, one-way, single-side turnover for a simple 35%
cap-weighted, long-short momentum index based on a large cap
universe from January 1995 through June 2009. FIG. 1B shows a
similar graph 150 for a small cap universe. The lines 110 and 160
indicate the turnover of the long side, and the lines 120 and 170
show the turnover of the short side. On average, the monthly
turnover is 20% for each side. However, the small cap results have
annual spikes in turnover associated with an annual reconstitution
of the small cap universe itself. This level of turnover is
undesirable both for long term and short teen investment
purposes.
[0042] Furthermore, non-optimized indexes often also exhibit strong
exposure tilts in other important systemic factors. FIGS. 2A and 2B
show graphs 200 and 250 for the price-to-book ratio, a common
metric of value, for a simple 35% cap-weighted, long-short momentum
index for a large cap and a small cap universe, respectively, from
January 1995 through June 2009. The exposure of the long side is
shown by the lines 210 and 260, and the absolute exposure of the
short side is shown by the lines 230 and 280. The exposure of the
benchmark is shown by the lines 220 and 270.
[0043] For the large cap universe, there is a strong correlation
between momentum and value during 1999-2001. The value exposure of
a simple momentum index is long line 210 and short line 230. From
1999-2001, the long line 210 is significantly higher than the
benchmark while and the short line 230 is significantly less than
the benchmark. In other words, the momentum portfolio has a strong
value tilt during this time period. The situation is even more
complicated for the small cap universe, which has several periods
of strong momentum-value correlation.
[0044] Non-optimized factor indexes may be adequate for investment
purposes if implementation costs and non-momentum neutrality are
not important characteristics. However, when these are important
considerations, optimization techniques can be used to embed these
characteristics into a factor index while still capturing the
return performance associated with the factor.
[0045] The construction of a momentum index for two different
universes describing large cap and small cap equities is described
in detail herein. Each index is intended to be one of a family of
factor indexes focusing on factors such as momentum, volatility,
liquidity, and leverage. The illustrative indexes track momentum
returns closely while simultaneously possessing a number of
desirable, practical features related to index implementation and
factor neutrality. The optimization methodology used to construct
these indexes has been tested extensively with historical backtests
to ensure that the realized index characteristics and returns are
as intended.
[0046] The illustrative indexes are designed to overcome the
implementation and trading issues associated with simple factor
index construction.
[0047] The index construction methodology is designed so that the
factor index possesses the best possible combination of portfolio
characteristics, or, in the event that it is not possible to obtain
all the desirable characteristics, an advantageous tradeoff among
these competing portfolio characteristics is achieved.
[0048] The momentum indexes are constructed using Axioma's
state-of-the-art optimization software and U.S. Equity Fundamental
Factor Risk Model. Different universes, different optimization
tools, and different factor risk models could also be used. The
optimization procedure has been backtested historically since
January 1995 to ensure that the resulting indexes track momentum
returns closely and possess the other desired implementation and
neutrality characteristics.
[0049] Optimized factor indexes replicate the return sequence of
simple, non-optimized indexes while simultaneously reducing the
index implementation costs and improving the momentum signal. The
optimized factor indexes replicate the return sequence of simple,
non-optimized indexes by imposing a tracking error constraint on
the active risk between the optimized factor index and the simple,
non-optimized factor index.
[0050] The index construction methodology uses state of the art
optimization technology to specify exactly how the index is
constructed. In this methodology, a series of optimization problems
are solved to produce the factor indexes. After solving the first
optimization problem, if the characteristics of the portfolio
obtained are acceptable, it is used as the actual factor index. If
the characteristics of the portfolio obtained are not acceptable,
then a second optimization problem is considered. This sequence of
optimization problems results in an advantageous tradeoff among
competing portfolio characteristics.
[0051] The specific quantitative targets of this optimization are
the following.
[0052] The target factor portfolio is defined to be a simple 35%
cap-weighted, long-short, dollar neutral momentum portfolio. This
portfolio defines true factor returns. Other factor portfolios
could also be used.
[0053] The tracking error between the momentum index and a 35%
cap-weight long-short momentum target factor portfolio is minimized
whenever possible and never greater than 5%. Momentum is defined
using the medium term momentum exposure in Axioma's U.S. Equity
Fundamental Factor Risk Model.
[0054] Whenever possible, the one-way turnover of each side of the
momentum index is less than 7.5%. At times, it is impossible to
maintain the other index characteristics with this low level of
turnover. When this occurs, the turnover is allowed to increase up
to 15%. The turnover is also allowed to be greater for the small
cap momentum index whenever the small cap universe is
reconstituted.
[0055] The exposure of the momentum index to all non-momentum,
style risk factors in Axioma's U.S. Equity Fundamental Factor Risk
Model is less than 25%; in other words, one quarter of one standard
deviation of the universe Z scores. The non-momentum exposure of
each side of the momentum index (i.e. the long and the short side)
is similarly constrained to fall within a quarter of a standard
deviation of the corresponding non-momentum exposure of the base
universe (i.e. large cap or small cap). These nine style factors
are exchange rate sensitivity, growth, leverage, liquidity, market
sensitivity, short term momentum, size, value, and volatility.
[0056] The index is long-short and is dollar neutral. The index
holds only assets that are constituents of the base universe. The
index does not hold any assets on a restricted list of assets. The
restricted list identifies hard-to-trade assets whose holdings
could substantially impact the implementation of the index. The
index does not short any assets on a "Do Not Short" list of assets.
The "Do Not Short" list identifies hard-to-short assets and assets
that cannot be shorted.
[0057] For the large cap index, the maximum number of long names is
200 and the maximum number of short names is 200. For the small cap
index, the maximum number of long names is 300 and the maximum
number of short names is 300. The maximum absolute weight (long or
short) in any asset is limited to the smaller of either 2%, or, 10%
of the 20 day average daily volume traded based on the long side of
the index having a value of ten million U.S. dollars. For example,
shares held could not exceed 5,000 in the index if the 20 day
average daily trading volume of is 50,000 shares.
[0058] In addition to these specifications, additional constraints
restricting the holdings, trades, exposures or risk of the index
could also be imposed if desired.
[0059] The index is rebalanced on a monthly basis.
[0060] An important distinction of this illustrative momentum index
compared with those described in the literature is that the
momentum exposure is not explicitly managed. Instead, tracking
error to the 35% cap-weighted long-short momentum target portfolio
is substantially reduced and is not more than 5%. Since the goal is
to have the momentum index returns be as close as possible to true
momentum returns as defined by the returns of the simple momentum
index, the appropriate quantitative metric is tracking error, not
exposure. Arbitrary exposure constraints do not produce true
momentum returns regardless of how large the exposures are.
[0061] Turning to the exemplary, detailed methodology, each monthly
index construction consists of a sequence of up to three different
optimizations performed in a specified order.
[0062] The momentum indexes are constructed with the following
priorities: low tracking error, low turnover, and non-momentum
factor neutrality. Tracking error is the most important property,
followed by turnover, and then factor neutrality. These
characteristics are achieved using the following sequence of
optimizations.
[0063] In a first optimization scenario, the objective is to
minimize the tracking error between the momentum index and the
target factor portfolio subject to the constraints already
detailed.
[0064] If the momentum index returned as the solution of the above
optimization problem has a tracking error of less than 5%
(annualized tracking error) and a one-way, one-sided turnover less
than 7.5%, this solution is accepted. Otherwise, a second
optimization is performed.
[0065] In a second optimization scenario, the objective is to
minimize turnover subject to a maximum allowable tracking error of
5%.
[0066] If the momentum index solution has a one-way turnover for
each side of less than 15%, this solution is accepted. Otherwise, a
third optimization is performed.
[0067] In a third optimization scenario, no objective is
considered. Instead, the problem is driven by the following
constraints and Axioma's Constraint Hierarchy.
[0068] Limit the tracking error between the momentum index and the
target factor portfolio to at most 5% annual volatility. This
constraint is placed in Axioma's Constraint Hierarchy but is the
last constraint to be softened.
[0069] The one-way turnover of each side of the momentum index is
less than 15%. This constraint is also placed in Axioma's
Constraint Hierarchy and, if necessary, will be softened before the
tracking error constraint.
[0070] The exposure of the momentum index (long minus short) and
each side of the index to the base index benchmark to all
non-momentum, style risk factors in Axioma's U.S. Equity
Fundamental Factor Risk Model is less than 25%. These constraints
are placed in Axioma's Constraint Hierarchy and are the first
constraints to be softened in the event of infeasibility.
[0071] This third solution is always accepted, regardless of any
constraints that are softened.
[0072] If the turnover of the index universe itself is particularly
high in any given month, then the optimization sequence above is
modified. The second optimization problem is skipped and the
turnover constraint in the other two optimizations scenarios is
eliminated.
[0073] The present invention may be suitably implemented as a
computer based system, in computer software which resides on
computer readable media, such as solid state storage devices, such
as RAM, ROM, or the like, magnetic storage devices such as a hard
disk or floppy disk media, optical storage devices, such as CD-ROM
or the like, or as methods implemented by such systems and
software.
[0074] FIG. 3 shows a block diagram of a computer system 100 which
may be suitably used to implement the present invention. System 100
is implemented as a computer 12, such as a personal computer,
workstation, or server. One likely scenario is that the system of
the invention will be implemented as a personal computer or
workstation which connects to a server or other computer running
software to implement the processes of the present invention either
through a network, Internet or other connection 26 or via a web
hosting connection 28. As shown in FIG. 3 and described in further
detail below, the system 100 includes software that is run by the
central processing unit of the computer 12. The computer 12 may
suitably include a number of standard input and output devices,
including a keyboard 14, a mouse 16, CD-ROM drive 18, disk drive
20, monitor 22, and printer 24. In addition, the computer 12 may
suitably include an Internet or network connection 26 for
downloading software, data and updates or for providing outputs to
remote system users. It may also use a web hosting server 28. It
will be appreciated, in light of the present description of the
invention, that the present invention may be practiced in any of a
number of different computing environments without departing from
the spirit of the invention. For example, the system 100 may be
implemented in a network configuration with individual workstations
connected to a server. Also, other input and output devices may be
used, as desired. For example, a remote user could access the
server with a desktop computer, a laptop utilizing the Internet or
with a wireless handheld device such as a Blackberry.TM., Treo.TM.,
or the like.
[0075] One embodiment of the invention has been designed for use on
a standalone personal computer or workstation.
[0076] According to one aspect of the invention, it is contemplated
that the computer 12 will be operated by a user, such as a factor
index construction employee working in an office setting. However,
if desired, it would also be possible to practice the invention
with the user using an off-site computer and either loading the
below-described software onto the off-site computer or connecting
to a server computer running the software.
[0077] As illustrated in FIG. 3, and as described in greater detail
below, additional inputs 30 may suitably include databases of
historical data for back testing and the like, data sources for
assets which may be included in portfolios, such as the asset
symbols, tickers, or identification number, the current prices of
stocks, bonds, commodities, currencies, options, other investment
vehicles, and the like, data, such as current factors, risk models
and return data, and the like. This data may also include
historical information on macroeconomic variables, such as
inflation and the rates for United States Treasury bonds of various
maturities, for example. It will be recognized that a wide variety
of additional inputs may be provided including without limitation
other complementary or supplementary software, such as portfolio
optimization modeling software, for example.
[0078] As illustrated in FIG. 3, and as described in greater detail
below, the system inputs 32 may suitably include the index
universe, which defines a set of securities over which to define
the factor index; a targeted factor, which defines a numerical
value for each security in the universe; non-targeted factors,
which define numerical values for other factors for the securities
in the universe; risk models, which can be used to compute tracking
errors; and data for the securities in the universe such as average
daily trade volume, price, benchmark weight, and the like.
[0079] As further illustrated in FIG. 3, and as described in
greater detail below, the system outputs 34 may suitably include
the factor index portfolio, defined in terms of shares, weights or
currency. It may also include quantitative statistics for the
factor index such as the number of names held and the predicted
tracking error, or the like. The output information may appear on
the monitor 22 or may also be printed out at the printer 24. The
output information may also be electronically sent to an index
distributor or some other intermediary. Other devices and
techniques may be used to provide outputs, as desired.
[0080] In one embodiment of the invention, software is utilized to
generate a number of computer display screens for receiving inputs
from, and providing outputs to, a user of the system.
[0081] It is anticipated that the models of the present disclosure
will be implemented in software. The software may be stored in any
appropriate computer readable medium, such as RAM. The software may
be executed on any appropriate computer system, such as the system
12 as shown in FIG. 3.
[0082] The optimization problem used to define the index
construction can have various alternative formulations. For
example, a constraint can be utilized that limits the total risk of
the factor index to be a fraction of a benchmark risk value. This
and various other alternatives can be used to define different
indexes with different properties.
[0083] While the present invention has been disclosed in the
context of various aspects of presently preferred embodiments, it
will be recognized that the invention may be suitably applied to
other environments consistent with the claims which follow. For
example, while a large number of exemplary factors have been
discussed herein, it will be recognized that other factors can be
defined. The invention is not limited to the kind of factors
considered herein.
* * * * *