U.S. patent application number 14/505258 was filed with the patent office on 2016-04-07 for performance attribution for portfolios with composite investments.
This patent application is currently assigned to AXIOMA, INC.. The applicant listed for this patent is Vishv Jeet, Vishal Shekhar. Invention is credited to Vishv Jeet, Vishal Shekhar.
Application Number | 20160098796 14/505258 |
Document ID | / |
Family ID | 55633127 |
Filed Date | 2016-04-07 |
United States Patent
Application |
20160098796 |
Kind Code |
A1 |
Jeet; Vishv ; et
al. |
April 7, 2016 |
Performance Attribution for Portfolios with Composite
Investments
Abstract
In existing performance attribution, composite investments are
resolved into simple assets, and the performance attribution
provides results only for the resolved, net investment in the
simple assets. As a result, the individual investment in the
composite investment in isolation is lost, and it is impossible to
determine if the investment in the composite investment in
isolation helped or hurt performance. Approaches are described to
determine attribution in a manner in which the attribution
hierarchy is altered so that, after reporting on the performance of
the full portfolio, a further level of attribution reports on a set
of sub-portfolios. The first sub-portfolio represents the original
investments in simple assets only while the other sub-portfolios
represent investments in each composite investment. This
composite-first performance attribution determines the individual
contribution to performance of each composite investment, resulting
in more detailed, practical, and intuitive results.
Inventors: |
Jeet; Vishv; (Marietta,
GA) ; Shekhar; Vishal; (Alpharetta, GA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Jeet; Vishv
Shekhar; Vishal |
Marietta
Alpharetta |
GA
GA |
US
US |
|
|
Assignee: |
AXIOMA, INC.
New York
NY
|
Family ID: |
55633127 |
Appl. No.: |
14/505258 |
Filed: |
October 2, 2014 |
Current U.S.
Class: |
705/36R |
Current CPC
Class: |
G06Q 40/06 20130101 |
International
Class: |
G06Q 40/06 20120101
G06Q040/06 |
Claims
1. A computer-implemented method for computing and reporting the
performance attribution of a set of portfolio holdings over time
comprising: electronically receiving and storing by a programmed
computer a set of dates defining an attribution time horizon to be
analyzed; for each date, electronically receiving and storing by
the programmed computer a set of possible investments where at
least one of the investments on at least one of the dates
represents a composite investment in two or more underlying simple
assets; for each composite investment opportunity, electronically
receiving and storing by the programmed computer a set of
underlying weights in simple assets that define each composite
investment composition; for each date, electronically receiving and
storing by the programmed computer a historical portfolio of
holdings having investment weights in the set of possible
investments where at least one of the investments on at least one
of the dates represents an investment in a composite investment;
for each date, electronically receiving and storing by the
programmed computer a set of supporting data required to compute a
performance attribution on the historical portfolios; for each
date, electronically calculating a reallocation of the historical
portfolios into sub-portfolios, that includes a sub-portfolio
representing the original investments in simple assets only and a
sub-portfolio representing each composite investment present in the
historical portfolio wherein the sum of the original investment
weights equals the sum of the investment weights across the
sub-portfolios; computing a performance attribution analysis for
each sub-portfolio; and electronically outputting the performance
attribution results using an output device.
2. The method of claim 1 in which the investment weights in the
sub-portfolio representing the original investments in simple
assets only have been altered by a function of a sum of the
original investment weights in each composite investment.
3. The method of claim 2 in which the style of performance
attribution is an asset grouping attribution and the set of
supporting data includes a benchmark, a classification of the
assets, and the asset returns.
4. The method of claim 3 further comprising: ranking the
contributions computed for each sub-portfolio.
5. The method of claim 2 in which the style of performance
attribution is a factor based attribution and the set of supporting
data includes a set of factors, factor exposures, factor returns,
and specific or asset returns.
6. The method of claim 5 further comprising: ranking the
contributions computed for each sub-portfolio.
7. The method of claim 2 in which at least one of the composite
investments is a composite of a composite.
8. A computer-implemented system for computing and reporting the
performance attribution of a set of portfolio holdings over time
comprising: a memory for storing data for a set of dates defining a
time horizon for an attribution to be performed; a processor
executing software operating to: to retrieve data defining a set of
possible investments at each date where at least one of the
investments on at least one of the dates represents a composite
investment in two or more underlying simple assets; to retrieve
data for each composite investment opportunity a set of underlying
weights in simple assets that define each composite investment
composition; to retrieve data for historical portfolios of holdings
having investment weights in the set of investible assets at each
date; to retrieve a set of supporting data required to compute a
style of performance attribution on the historical portfolios; to
compute a reallocation of the historical portfolios into
sub-portfolios comprising a sub-portfolio of investments
representing the original investment in simple assets and a
sub-portfolio representing each composite investment present in the
historical portfolio wherein the sum of the original investment
weights equals the sum of the investment weights across the
sub-portfolios; to compute a performance attribution analysis for
each sub-portfolio; and an output device electronically outputting
the performance attribution results.
9. The system of claim 8 in which the investment weights in the
sub-portfolio representing the original investments in simple
assets only have been altered by a function of a sum of the
original investment weights in each composite investment.
10. The system of claim 9 in which the style of performance
attribution is an asset grouping attribution and the set of
supporting data includes a benchmark, a classification of the
assets, and the asset returns.
11. The system of claim 10 in which the contributions computed for
the sub-portfolios are ranked by the programmed computer.
12. The system of claim 9 in which the style of performance
attribution is a factor based attribution and the set of supporting
data includes a set of factors, factor exposures, factor returns,
and specific or asset returns.
13. The system of claim 12 in which the contributions computed for
the sub-portfolios are ranked by the programmed computer.
14. The method of claim 9 in which at least one of the composite
investments is a composite of a composite.
15. A computer-implemented method for determining a performance
attribution of a portfolio comprising investments in simple assets
including individual stocks and at least one composite investment
including an exchange traded fund (ETF), the method comprising:
splitting the portfolio into a first sub-portfolio for the
investments in the simple assets and an additional sub-portfolio
for each composite investment including a second sub-portfolio for
the ETF wherein the sum of the original investments equals the sum
of the investments across the sub-portfolios; determining an
attribution contribution for the first sub-portfolio; and
determining an attribution contribution for each additional
composite investment including an attribution contribution for the
second sub-portfolio.
16. The method of claim 15 further comprising: displaying the
attribution contribution for the first sub-portfolio and the
attribution contribution for the second sub-portfolio employing an
output device.
17. The method of claim 15 further comprising: electronically
receiving and storing by a programmed computer a set of dates
defining an attribution time horizon to be analyzed; for each date,
electronically receiving and storing by the programmed computer a
set of investments comprising the portfolio; for each composite
investment, electronically receiving and storing by the programmed
computer a set of underlying weights in simple assets that define
each composite investment; and for each date, electronically
receiving and storing by the programmed computer a set of
supporting data required to compute the performance attribution
contributions.
18. The method of claim 15 further comprising: utilizing a
percentage allocation to the first sub-portfolio to determine an
active weight of each simple asset of the first sub-portfolio; and
utilizing a percentage allocation to the second sub-portfolio to
determine an active weight of each simple asset of the second
sub-portfolio.
Description
FIELD OF INVENTION
[0001] The present invention relates generally to methods for
calculating performance attribution results for investment
portfolios containing composite investments such as exchange traded
funds and mutual funds. More particularly, it relates to improved
computer based systems, methods and software for calculating
performance attribution results when the portfolio of investments
contains individual investment allocations that are associated with
investments in more than one underlying investment opportunity.
BACKGROUND OF THE INVENTION
[0002] One of the goals of performance attribution is to identify
individual buys and sells for a portfolio of investments that are
likely to improve future performance. To do this, performance
attribution generally decomposes the historical performance of an
investment portfolio using a decomposition hierarchy. At the
highest level of the hierarchy, it reports the performance of the
full portfolio either in absolute terms or relative to a benchmark.
Then, in the next level of the hierarchy, it groups the performance
into groups of assets (asset grouping or Brinson style attribution)
or groups of factors (factor-based attribution). This stage of the
decomposition identifies the groups of assets or factors that were
most impactful for the historical performance. Additional levels of
decomposition can be performed until a final level is reached in
which individual asset investments are evaluated. The higher levels
of decomposition help focus the portfolio manager's attention on
key areas, but, in the end, any action must be translated into
changes of individual asset positions. The hierarchy helps identify
the most important individual positions and the desired action (buy
or sell) for each position. A portfolio manager can then increase
those individual investments that helped performance the most and
decrease those that degraded performance.
[0003] One of the problems with existing performance attribution
tools comes from the fact that modern portfolio managers frequently
invest in investments that are, in fact, portfolios of individual
investments. These include exchange traded funds (ETFs) and mutual
funds. As a result, it is quite common for a portfolio manager to
need to analyze a fund of funds; that is, an investment portfolio
in which some of the individual investment allocations are actually
investments in two or more other investments to be bought or sold
in some predetermined ratio. Any investment that involves investing
in two or more other assets is referred to herein as a composite
investment or composite asset. Any other investment is a simple
investment or simple asset. The group of simple assets includes
individual equities, bonds, and cash. The group of composite
investments includes ETFs and mutual funds.
[0004] In existing performance attribution, composite investments
are resolved into simple assets, and the performance attribution
provides results only for the net investment in the simple assets.
So, for example, if a portfolio manager invests in a large company
such as General Electric as well as an ETF of large capitalization
U.S. equities that includes General Electric, his or her position
in General Electric in the portfolio performance analysis will be
displayed as the net position from the individual investment in
General Electric plus the investment derived from the large cap
equity ETF. The performance of the net position in General
Electric, the simple asset, will be reported. However, the
performance of the investment in the composite investment and the
individual equity in isolation will not be reported. As a result,
if, say, the performance attribution suggests that the portfolio
manager should increase his or her position in General Electric,
the traditional performance attribution cannot determine if that is
best done by buying General Electric in isolation or buying the
large cap ETF. In other words, the individual investment in the
composite investment is lost in existing performance attribution
approaches.
SUMMARY OF THE INVENTION
[0005] Among its several aspects, the present invention recognizes
that in existing performance attribution, composite investments are
resolved into simple assets. Thus, the performance attribution
provides results only for the net investment in the simple assets.
As a result, the individual investment allocation in the composite
investment is lost, and it is impossible to determine if the
investment in the composite investment in isolation helped or hurt
performance.
[0006] The present invention describes a new performance
attribution approach in which the attribution hierarchy is altered
so that, after reporting on the performance of the full portfolio,
the next level of attribution reports on a set of sub-portfolios. A
first sub-portfolio represents the original investments in simple
assets while additional sub-portfolios represent the original
investments in individual composite investments. Multiple
individual composite investments may be addressed each with its
corresponding sub-portfolio. This composite-first performance
attribution determines the individual contribution to performance
of each composite investment, resulting in more detailed,
practical, and intuitive results.
[0007] The present invention overcomes problems such as those
associated with composite investments. In accordance with one
aspect of the invention, the attribution hierarchy is altered as
follows. After reporting on the performance of the full portfolio,
the next level of attribution hierarchy reports on a set of
sub-portfolios. The first sub-portfolio represents the original
investments directly in simple assets. The other sub-portfolios
represent investments in individual composite investments. Hence,
if there are J composite investments to be analyzed in the original
portfolio, the first decomposition reports results for J+1
sub-portfolios. Once these attributions have been performed, the
usual attribution results (asset grouping or Brinson for groups of
assets or factor-based for groups of factors) is performed on each
of the J+1 sub-portfolios. Once the attribution decomposition
hierarchy has been completed, the final results represent the
performance contribution of all the available investment
opportunities including composite investments. This procedure is
more detailed than the existing approach and yields more practical
and intuitive results.
[0008] The present invention recognizes that it is important for
portfolio managers and other investment professionals to understand
the contribution to performance of all their investment decisions,
including investments in individual composite investments.
[0009] One goal of the present invention, then, is to describe a
methodology that enables a portfolio manager to quantitatively
assess the contribution of his or her investment in individual
composite investments.
[0010] A more complete understanding of the present invention, as
well as further features and advantages of the invention, will be
apparent from the following Detailed Description and the
accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0011] FIG. 1 shows a computer based system which may be suitably
utilized to implement the present invention;
[0012] FIG. 2 illustrates summary statistics for a backtest for a
case study demonstrating several advantages of the present
invention;
[0013] FIG. 3 illustrates the time history of tracking error for
the case study;
[0014] FIG. 4 illustrates the time history of round-trip turnover
for the case study;
[0015] FIG. 5 illustrates shows three levels of performance
attribution contributions for the case study in which the composite
investment has been resolved into simple assets;
[0016] FIG. 6 illustrates four levels of performance attribution
contributions for the case study in which the original portfolio
was reallocated into a sub-portfolio of original investments
directly in simple assets and a sub-portfolio representing the
investment in each composite investment;
[0017] FIG. 7 illustrates shows three levels of performance
attribution contributions for a modified case study in which there
are no composite investments;
[0018] FIG. 8 illustrates the underlying composition of composite
investment C in a simple example;
[0019] FIG. 9 illustrates the portfolio weights, benchmark weights,
sector assignments, and asset returns for a simple example;
[0020] FIG. 10 illustrates a performance attribution of a simple
example in which the composite investment has been resolved into
simple assets;
[0021] FIG. 11 illustrates the relative weights of each
sub-portfolio in the simple example in which the original portfolio
has been reallocated into a sub-portfolio of original simple asset
investments and a sub-portfolio representing the composite
investment;
[0022] FIG. 12 illustrates the original portfolio reallocated into
two sub-portfolios in a simple example;
[0023] FIG. 13 illustrates the performance attribution of the two
sub-portfolios in the simple example;
[0024] FIG. 14 illustrates a performance attribution across sectors
for the simple example in which the original portfolio has been
resolved into simple assets;
[0025] FIG. 15 illustrates a performance attribution across sectors
for the simple example in which the original portfolio has been
reallocated into sub-portfolios; and
[0026] FIG. 16 illustrates a flow chart of the steps of a process
in accordance with an embodiment of the present invention.
DETAILED DESCRIPTION
[0027] The present invention may be suitably implemented as a
computer based system, in computer software which is stored in a
non-transitory manner and which may suitably reside 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
solid state drive, optical storage devices, such as CD-ROM, CD-RW,
DVD, Blue Ray Disc or the like, or as methods implemented by such
systems and software. The present invention may be implemented on
personal computers, workstations, computer servers or mobile
devices such as cell phones, tablets, IPads.TM., IPods.TM. and the
like.
[0028] FIG. 1 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 or mobile device 12 including one or
more programmed processors, 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 28 or other computer through
an Internet, local area network (LAN) or wireless connection 26. In
this embodiment, both the computer or mobile device 12 and server
28 run software that when executed enables the user to input
instructions and calculations on the computer or mobile device 12,
send the input for conversion to output at the server 28, and then
display the output on a display, such as display 22, or print the
output, using a printer, such as printer 24, connected to the
computer or mobile device 12. The output could also be sent
electronically through the Internet, LAN, or wireless connection
26. In another embodiment of the invention, the entire software is
installed and runs on the computer or mobile device 12, and the
Internet connection 26 and server 28 are not needed. As shown in
FIG. 1 and described in further detail below, the system 100
includes software that is run by the central processing unit of the
computer or mobile device 12. The computer or mobile device 12 may
suitably include a number of standard input and output devices,
including a keyboard 14, a mouse 16, CD-ROM/CD-RW/DVD drive 18,
disk drive or solid state drive 20, monitor 22, and printer 24. The
computer or mobile device 12 may also have a USB connection 21
which allows external hard drives, flash drives and other devices
to be connected to the computer or mobile device 12 and used when
utilizing the invention. 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 cell phones, tablets and e-readers such as
an IPad.TM., IPhone.TM., IPod.TM., Blackberry.TM., Treo.TM., or the
like.
[0029] One embodiment of the invention has been designed for use on
a stand-alone personal computer running in Windows 7. Another
embodiment of the invention has been designed to run on a
Linux-based server system. The present invention may be coded in a
suitable programming language or programming environment such as
Java, C++, Excel, R, Matlab, Python, etc.
[0030] According to one aspect of the invention, it is contemplated
that the computer or mobile device 12 will be operated by a user in
an office, business, trading floor, classroom, or home setting.
[0031] As illustrated in FIG. 1, and as described in greater detail
below, the inputs 30 may suitably include portfolio allocations in
investible investments at one or more times where at least one
investment is a composite investment, underlying constituent
weights of the composite investments in terms of simple assets, and
supporting data required to compute a performance attribution of
the portfolios, such as asset returns, asset groupings, factor
exposures, factor returns, specific returns, benchmarks, and the
like.
[0032] As further illustrated in FIG. 1, and as described in
greater detail below, the system outputs 32 may suitably include a
reallocation of each portfolio into sub-portfolios, one
sub-portfolio representing the portfolio investments in simple
assets and the other sub-portfolios representing the investments in
each composite investment, a performance attribution reporting an
attribution for each sub-portfolio, and a ranking of the
attribution results for each sub-portfolio.
[0033] The output information may appear on a display screen of the
monitor 22 or may also be printed out at the printer 24. The output
information may also be electronically sent to an intermediary for
interpretation. For example, the performance attribution results
for many portfolios can be aggregated for multiple portfolio
reporting. Other devices and techniques may be used to provide
outputs, as desired.
[0034] With this background in mind, a detailed discussion of the
invention and its context follows. Consider a universe of
investible assets including J composite investments with the
following:
[0035] S=the set of simple assets,
[0036] C=the set of composite investments,
[0037] w.sub.i=the portfolio weight of the i-th asset,
[0038] b.sub.i=the benchmark weight of the i-th asset,
[0039] .lamda..sub.k.sup.j=the weight of the k-th simple asset in
the j-th composite investment,
[0040] r.sub.k=the period return of the k-th simple asset,
[0041] B.sub.km=the exposure of the k-th asset to the m-th factor,
and
[0042] f.sub.m=the period factor return of the m-th factor.
It is assumed that the composition of each composite investment in
terms of underlying weights in simple assets sums to one
k .di-elect cons. S .lamda. k j = 1 for each j .di-elect cons. C
##EQU00001##
If there are composites of composites, where the underlying weights
include weights on other composite investments, the weights
.lamda..sub.k.sup.j represent the weights once each composite
investment allocation has been decomposed into simple assets. The
weights of the portfolio and benchmark may or may not sum to one,
depending on whether or not they are fully invested. If there is no
benchmark, then all the benchmark weights can be taken as zero,
e.g., b.sub.i=0. However, some forms of performance attribution,
such as Brinson attribution, explicitly require a benchmark against
which the performance of the portfolio is compared.
[0043] In traditional attribution approaches, when there are
composite investments present, either in the portfolio or
benchmark, the composite investments are resolved into net
investments in simple assets only:
w ~ i = w i + j .di-elect cons. C w j .lamda. i j ##EQU00002## b ~
i = b i + j .di-elect cons. C b j .lamda. i j ##EQU00002.2##
Then, attribution is performed in the usual manner using
allocations {tilde over (w)}.sub.i and {tilde over (b)}.sub.i
instead of w.sub.i and b.sub.i. However, since there are no
investments in composite investments in the resolved allocations
{tilde over (w)}.sub.i and {tilde over (b)}.sub.i, attribution on
these allocations gives no sense for the performance of the
individual investments in composite investments, which have been
entirely lost.
[0044] In the approach proposed here, the active holdings of the
original portfolios are first decomposed or reallocated into a set
of J+1 sub-portfolios. To do this, a mathematically consistent
procedure is employed to assign the correct investment weights to
the J+1 sub-portfolios. Such a formula is derived by considering
the sum of the active weights and rearranging it as follows.
i .di-elect cons. S C ( w i - b i ) = ( i .di-elect cons. S w i + j
.di-elect cons. C w j ) - ( i .di-elect cons. S b i + j .di-elect
cons. C b j ) = ( i .di-elect cons. S w i + j .di-elect cons. C w j
( k .di-elect cons. S .lamda. k j + b k - b k ) ) - ( i .di-elect
cons. S b i + j .di-elect cons. C b j ) = ( i .di-elect cons. S w i
+ j .di-elect cons. C w j ( i .di-elect cons. S .lamda. i j - b i )
) - ( i .di-elect cons. S b i + j .di-elect cons. C b j - j
.di-elect cons. C w j i .di-elect cons. S b i ) = ( i .di-elect
cons. S w i + j .di-elect cons. C w j i .di-elect cons. S .lamda. i
j - j .di-elect cons. C w j i .di-elect cons. S b i ) - ( i
.di-elect cons. S b i + j .di-elect cons. C b j - j .di-elect cons.
C w j i .di-elect cons. S b i ) = i .di-elect cons. S w i - i
.di-elect cons. S b i + j .di-elect cons. C i .di-elect cons. S w j
b i + j .di-elect cons. C i .di-elect cons. S w j .lamda. i j - i
.di-elect cons. S b i - j .di-elect cons. C i .di-elect cons. S w j
b i = i .di-elect cons. S ( w i - ( 1 - j .di-elect cons. C w j ) b
i ) + j .di-elect cons. C i .di-elect cons. S ( w j .lamda. i j - b
j .lamda. i j - w j b i ) ##EQU00003##
[0045] The final line can be interpreted as a sum of the "revised
active weights" of J+1 sub-portfolios, where, by construction, the
sum of the "revised active weights" equals the sum of the original
active weights. In the case where there is no benchmark, e.g.,
b.sub.i=0, then the sum of the original weights equals the sum of
the "revised weights".
[0046] The first sub-portfolio is derived from the original simple
asset holdings or investments. The "revised active holdings" for
this portfolio are the original simple asset holdings minus the
benchmark weights reduced by the amount
( 1 - j .di-elect cons. C w j ) . ##EQU00004##
[0047] The other J sub-portfolios are derived from the original
composite investment holdings, with the j-th portfolio having
"revised active holdings" of
(w.sub.j-b.sub.j).lamda..sub.i.sup.j-w.sub.jb.sub.i.
[0048] Since each of these J+1 sub-portfolios only has "revised
active holdings" in only the simple assets, any traditional form of
performance attribution can be unambiguously performed on each of
these portfolios. However, unlike direct resolution to simple
assets only, in this case, results are obtained for the allocation
to each of the J composite investment sub-portfolios. As a result,
the resulting attribution results can be advantageously employed to
judge the effectiveness of each composite investment in isolation
as taught herein.
[0049] Although the above decomposition is employed in a presently
preferred embodiment for the present invention, it will be clear to
those skilled in the art that alternative decompositions or
reallocations of the original portfolio into different
sub-portfolios are possible. For example, in the above formula, the
weights of the composite investments in the benchmark, b.sub.j,
could be resolved in each formula, so that only the composite
investment allocations in the original portfolio, w.sub.j, remain.
This allocative approach could be made using the formula for {tilde
over (b)}.sub.i previously described. For long-short portfolios,
the decomposition could also involve distinguishing long and short
holdings so that there are sub-portfolios representing the long
holdings and sub-portfolios representing the short holdings. In the
inventors' experience, the decomposition detailed above is the most
practical decomposition.
[0050] Having decomposed or reallocated the original portfolio into
sub-portfolios, a performance attribution is performed on each
sub-portfolio. There are several common and important forms of
performance attribution. In R. Litterman, Modern Investment
Management: An Equilibrium Approach, John Wiley and Sons, Inc.,
Hoboken, N.J., 2003 (Litterman), which is incorporated by reference
herein in its entirety, two major categories of performance
attribution are described. See, for example, Litterman, Chapter 19,
"Return Attribution", pages 297-333, which gives a detailed
technical summary of both of these categories of performance
attribution. The first is referred to as asset grouping or Brinson
Style attribution. In this approach, the assets in the portfolio
are grouped into different groups such as sectors or countries. The
second major category of performance attribution is called
factor-based attribution. In this approach, the return is explained
using a set of factors, factor exposures, and factor returns. In
most cases, the factors, factor exposures, and factor returns are
taken from a commercial factor risk model. Axioma sells a range of
country and regional factor risk models using fundamental factors,
statistical factors, and macroeconomic factors. Factor-based
attribution has been used for several decades and is described in
Litterman as well as R. C. Grinold, and R. N. Kahn, Active
Portfolio Management: A Quantitative Approach for Providing
Superior Returns and Controlling Risk, Second Edition, McGraw-Hill,
New York, 2000, which is incorporated by reference herein in its
entirety.
[0051] For asset grouping or Brinson style attribution, assets are
reallocated into different groups (sectors, countries, etc.), and
then the "allocation" and "selection" effects are computed, where
the allocation effect for the n-th group of assets is
A n = ( i .di-elect cons. n w i - i .di-elect cons. n b i ) ( i
.di-elect cons. n w i r i i .di-elect cons. n w i - i .di-elect
cons. S C b i r i ) ##EQU00005##
and the selection effect of the n-th group of assets:
S n = ( i .di-elect cons. n w i ) ( i .di-elect cons. n w i r i i
.di-elect cons. n w i - i .di-elect cons. n b i r i i .di-elect
cons. n b i ) ##EQU00006##
In the above formulas, whenever a denominator is zero, the ratio of
the sums is taken to be zero.
[0052] A large number of alternative formulas have been proposed
for use in place of the above formulas. See Litterman at pages
306-310 and 331-333 which gives an asset grouping approach that
includes an interaction term. Other examples are provided in G. P.
Brinson, and N. Fachler, "Measuring Non-U.S. Equity Portfolio
Performance," Journal of Portfolio Management, Spring 1985, pp.
73-76, which is incorporated by reference herein in its entirety.
See also G. Bacon, "Performance Attribution", Chapter 5 in
Practical Portfolio Performance Measurement and Attribution, Wiley,
2011, which is incorporated by reference herein in its entirety.
Such alternatives may also suitably be used in conjunction with the
present invention.
[0053] In the original Brinson models approach, the selection
effect is split into selection and interaction effects. Some
practitioners prefer to split that term that way, as they see value
in the two separate numbers. Others, however, prefer to keep them
together as that better mirrors their investment process. In a
hierarchical investment process, budget allocation decisions
(across different sectors or groupings) precede stock selection
decisions. Thus, to compute the correct stock selection effect, the
sector weights fixed by the allocation decisions are used instead
of using sector weights in the benchmark. This order of decisions
(allocation and then selection) eliminates the interaction between
them, thus leading to a cleaner attribution. In the original
Brinson models approach, interaction effect is an artifact of the
implicit assumption that allocation and stock selection decisions
do not have an order of precedence.
[0054] In factor based attribution, the factor contribution of the
m-th factor is
F m = i .di-elect cons. S C ( w i - b i ) B im f m ##EQU00007##
and the factor contribution from the n-th group of assets for the
m-th factor is
F nm = i .di-elect cons. n ( w i - b i ) B im f m ##EQU00008##
The specific contribution is the residual formed by subtracting all
the factor contributions from the total contribution
S = i ( w i - b i ) r i - i m ( w i - b i ) B im f m
##EQU00009##
Most risk models do not explicitly define the exposures of
composite investments, so the exposures in the formulas above,
B.sub.im, may not be defined immediately by the factor risk model
for the composite investments. However, they can be derived from
the underlying weights in simple assets of the composite
investment, .lamda..sub.i.sup.j. Alternatively, one can resolve the
composite investments into simple assets, but then the identity of
the composite investments is lost.
[0055] A factor risk model comprises an asset return model
r=Bf+.epsilon.
and a corresponding factor risk model
Q=B.SIGMA.B.sup.T+.DELTA.
where
[0056] r is an N dimensional vector of asset excess returns (return
above the risk free rate)
[0057] B is an N by M matrix of factor exposures (also called
factor loadings)
[0058] f is an M dimensional vector of factor returns
[0059] .epsilon. is an N dimensional vector of asset specific
returns (also called residual returns)
[0060] Q is an N by N matrix of asset covariances=Cov(r, r)
[0061] .SIGMA. is an M by M matrix of factor covariances=Cov(f,
f)
[0062] .DELTA. is an N by N matrix of security specific
covariances=Cov(.epsilon., .epsilon.); often, .DELTA. is taken to
be a diagonal matrix of security specific variances. In other
words, the off-diagonal elements of .DELTA. are often neglected,
e.g., assumed to be vanishingly small and therefore not explicitly
computed or used. U.S. Patent Application Publication No.
2004/0078319 A1 by Madhavan et al. also describes aspects of factor
risk model estimation and is incorporated by reference herein in
its entirety. In general, the number of factors, M, is much less
than the number of securities or assets, N.
[0063] The covariance and variance estimates in the matrix of
factor-factor covariances, .SIGMA., and the (possibly) diagonal
matrix of security specific covariances, .DELTA., are estimated
using a set of historical estimates of factor returns and asset
specific returns.
[0064] Both the covariance and variance computations may utilize
techniques to improve the estimates. For example, it is common to
use exponential weighting when computing the covariance and
variance. This weighting is described in Litterman as well as
Grinold and Kahn.
[0065] The covariance and variance estimates may also incorporate
corrections to account for the different times at which assets are
traded across the globe. For example, U.S. Pat. No. 8,533,107
describes a returns-timing correction for factor and specific
returns and is incorporated by reference herein in its
entirety.
[0066] The covariance and variance estimates may also incorporate
corrections to make the estimates more responsive and accurate. For
example, U.S. Pat. No. 8,700,516 describes a dynamic volatility
correction for computing covariances and variances, and is
incorporated by reference herein in its entirety.
[0067] Traditionally, commercial factor risk models come in three
varieties: fundamental factor risk models, statistical factor risk
models, and macroeconomic factor risk models.
[0068] In fundamental factor risk models, the factor exposures are
defined using explicit market and security information. Typically,
fundamental factor risk models include style factors which measure
the exposure or loading of each security to factors such as value,
growth, leverage, size, momentum, volatility, and so on. The
exposures are often given as Z scores, in which the raw
measurements of these metrics have been normalized by subtracting
the cap-weighted mean value and dividing the result of the
subtraction by the equal-weighted standard deviation of the
original measurements. See Litterman pages 353 and 354 for further
details. By performing this resealing, a factor such as size
(measured as market cap, with values such as billions of dollars)
can be effectively compared to a factor such as volatility
(measured in terms of annual volatility, which is a number less
than one). Fundamental factor risk models also include categorical
factors such as industries, countries, market, and currency
factors. In binary models, such as those sold by Axioma, the
exposure of any simple security is non-zero and equal to one for
only one industry, one country and one currency. Other commercial
factor risk model vendors sometimes spread out the exposure of an
individual security across more than one categorical factor in each
of these categories, with the restriction that the total exposure
across each category adds up to 100%. So, for instance, General
Electric may have non-zero exposure to both health and finance
industries.
[0069] Other categorical assignments can be used as well. For
instance, the global industry classification standard (GICS)
taxonomy developed by MSCI and Standard & Poor's has four
classification levels: industry sub-groups; industries; industry
groups, and sectors. Countries can be grouped by region (Americas,
Europe, or Asia) or by economy (developed or emerging).
[0070] Once the factor exposures have been defined, the factor
returns for a fundamental factor risk model are estimated using a
cross-sectional regression across the security returns at any point
in time.
[0071] In statistical factor risk models, the matrix of security
returns across the universe of securities and back through time is
analyzed to determine factors that best represent the volatility of
returns. Often, principal components analysis is used to determine
these factors. See Litterman pages 345-348. By construction,
statistical factors represent the risk of the assets well. However,
since the exposures are determined mathematically, it is often
difficult to develop intuition about what each statistical factor
may mean in terms of traditional metrics such as size and value.
Furthermore, since the factors can change from day to day, any
intuition developed on one day for a particular model may not be
applicable on another day.
[0072] In macroeconomic factor risk models, the factors are chosen
to represent the correlation or beta of each security to a time
series of macroeconomic data such as GDP, interest rates, corporate
spreads, and the like.
[0073] In the present invention, any factor risk model may be used
for factor-based performance attribution.
[0074] In traditional performance attribution, period contributions
are compounded and linked together so that their contributions sum
to the total active return of the portfolio. See Litterman on pages
311-319 for details of several methods for compounding and linking
contributions including the methodology proposed by the Frank
Russell Company and the methodology proposed by Mirabelli. See
also, D. R. Carino, "Combining Attribution Effects Over Time,"
Journal of Performance Measurement, Summer 1999, pp. 5-14, which is
incorporated by reference herein in its entirety.
[0075] As a particular example, in the method proposed by the Frank
Russell Company, the portfolio return and one-period sources of
return are computed in terms of percent returns. Then, each
one-period percent return is multiplied by the ratio of the
portfolio log-return to the percent return for that period. Then,
the resulting returns are converted a second time back into percent
returns by multiplying by the ratio of the full period percent
return to the full period log return. This approach achieves the
important attribution characteristic of having multi-period sources
of return that are additive. These transformations perturb the
realized risk of the contributions since the original period
contributions are perturbed. In general, the modifications derived
from linking for both contributions and risk contributions are
small.
[0076] Next, aspects of the present invention are illustrated by
constructing a performance attribution of a historical backtest
case study. The historical portfolios are rebalanced monthly from
Jun. 30, 2005 to Dec. 31, 2013 (103 monthly rebalances). At each
rebalance, the portfolio only holds constituents of the Russell
2000 index of small cap U.S. equities, which are the simple assets
for this example, and SPY, the SPDR S&P 500 ETF Trust, which is
the composite investment. SPY is an ETF that tracks the S&P
500, an index of large cap stocks. In this example, there is only
one composite investment, so J=1.
[0077] Axioma's medium horizon, fundamental factor U.S. equity
factor risk model (AXUS3-MH) is used as the risk model. This factor
risk model has both style factors and industry factors, which are
used to explain the returns and predict risk of portfolios of U.S.
equities.
[0078] An alpha signal of expected asset returns is constructed by
averaging the medium-term momentum and value factors from the risk
model.
[0079] At each rebalance, the expected return (alpha) is maximized.
The portfolio is only allowed to hold at most 50 names, where SPY
counts as one name. The individual investment weights in the simple
assets must be either 0% or between 0.01% and the benchmark weight
in the Russell 2000 plus 5%. The individual investment in SPY can
be between 0% and 50%, excluding weights greater than 0% and less
then 0.01%. The maximum tracking error of the 50-name portfolio to
the Russell 2000 is 1% and the maximum round-trip turnover is 40%
at each rebalance. However, both the tracking error and turnover
constraints are placed in Axioma's constraint hierarchy, so they
may be violated at any rebalance if necessary to avoid an
infeasible portfolio construction problem. In addition, while an
investment of 0% is allowed, the portfolio has a minimum investment
size of 0.01% in all assets including the composite
[0080] Table 202 of FIG. 2 shows a summary of the backtest. The
active return of the portfolio is -0.94%, leading to an information
ratio of -0.209. The average turnover is 67.8% and the average
predicted tracking error is 5.0%. The constraint on turnover was
violated in some periods.
[0081] Table 204 of FIG. 3 shows the time history of tracking error
206 for the optimized portfolio. The tracking error 206 never
satisfies the 1% tracking error constraint, and has its worst
violations during the great recession of 2008-2009.
[0082] Table 208 of FIG. 4 shows the time history of turnover 210
for the optimized portfolio. Despite the turnover constraint, the
turnover is never less than 40% and sometimes is greater than
100%.
[0083] In order to better understand the historical performance, a
factor-based attribution using the resolved portfolio is performed.
The investment in the ETF (SPY) is converted to or resolved into
investments in the underlying simple assets. The results of this
decomposition are shown in summary form in table 212, FIG. 5. The
results in table 212 show three levels of the attribution
decomposition hierarchy. Level 1 is at full portfolio level and
reports the total, benchmark, and active contributions. Level 2
decomposes the active contribution into factor and specific
contributions. Level 3 then decomposes the factor contribution into
style and industry contributions. Although not shown in table 212,
at Level 4, the style contributions would be broken down into
individual style factors, and then the industry contribution would
be broken down by individual industry factor contribution. Finally,
at Level 5 of the hierarchy, the individual industry and style
factor contributions would be broken down by simple assets.
[0084] The active return or contribution of -0.94% comes from a
factor contribution of 6.55% and a specific contribution of -7.02%.
Hence, the factor bets paid off historically, but the particular
stocks chosen did not help, as indicated by the negative specific
contribution. The active factor contribution of 6.55% can be
decomposed to a deeper level of granularity by dividing it into a
style contribution of 7.02% and an industry contribution of
-0.47%.
[0085] However, since the analysis illustrated in table 212 is
constructed using the resolved portfolio, all information
concerning the investment in SPY has been lost in this analysis. It
cannot be determined if SPY has helped or hurt this particular
historical performance.
[0086] To assess the impact of SPY, an alternative factor-based
performance attribution is calculated in which the active portfolio
is first decomposed or reallocated into a sub-portfolio of original
simple asset investments and a sub-portfolio representing the
investment in SPY. The high level results of that analysis are
shown in tables 214 and 216 of FIG. 6.
[0087] In this new analysis, four levels of decomposition are
shown. The highest level, Level 1, which reports the total,
benchmark and active contributions of the full portfolio of
holdings, remains the same. However, in Level 2 of the hierarchy,
the full portfolio is split into the sub-portfolio of simple asset
holdings and the SPY sub-portfolio. By making this decomposition or
reallocation immediately with Level 2, vital information about the
effect of holding SPY is retained. Then, having split the
portfolios, Level 3 decomposes the contributions into specific and
factor contributions, and Level 4 then decomposes them into style
and industry contributions. So, in the modified approach, there is
one more level to the attribution decomposition hierarchy.
[0088] Upon examining the actual contributions in tables 214 and
216, it is seen that holding SPY led to a small, negative
contribution to the overall performance of the portfolio of -0.38%,
while the investments in just the simple assets of the Russell 2000
led to a contribution of -0.56%. Neither investment beat the
benchmark, but the loss from holding SPY was smaller than the loss
from the simple assets.
[0089] A portfolio manager who examined these results would be
tempted to believe that eliminating SPY from the universe or set of
possible investments while still keeping only 50 names might do
worse than the results shown in tables 214 and 216, since the
investment in individual simple assets performed worse than the
investment in SPY, even though both under-performed the
benchmark.
[0090] In fact, for this particular example, eliminating SPY from
the set or universe of investment opportunities does lead to a
significantly worse performance. Table 218 in FIG. 7 shows a
performance attribution summary of the backtest eliminating SPY.
The active return was -2.36%, almost two and a half times worse.
So, not only does the composite-first hierarchy retain information
about the performance of the investments in the composites, that
information is intuitive, practical, and useful.
[0091] Next, the present invention is illustrated with a simple,
explicit model. Consider the following simple example involving a
single portfolio and benchmark at a single point in time. The
attribution contributions are the sum of the asset active weights
times the asset returns. However, since one of the assets is a
composite investment of the other simple assets, there are
alternative approaches to computing the contributions.
[0092] A portfolio contains five assets, named A, B, C, D, and E.
Assets A, B, D, and E are simple assets while asset C is a
composite investment. Table 302 in FIG. 8 gives the underlying
composition of composite investment C. It is composed of a 30%
allocation to A, a 30% allocation to B, and a 40% allocation to
D.
[0093] Table 304 in FIG. 9 lists the asset types, sector
assignments, portfolio weights, benchmark weights, and returns for
the universe of five assets. Assets A and B are assigned to a
"value" sector, while assets D and E are assigned to a "growth"
sector. The sector assignment of the composite investment C is not
shown as it contains assets from both value and growth. In this
simple example, the benchmark does not hold the composite
investment C.
[0094] First, the original portfolio is resolved into holdings of
only simple assets by applying the underlying weights 302 to the
portfolio allocation 304. The attribution results of the resolved
approach are shown in table 306 of FIG. 10.
[0095] For this simple example, the portfolio return is 7.73% while
the benchmark return is 5.30%. This gives an active return of
2.43%. In the resolved attribution, the 2.43% active return is
decomposed into four asset contributions which sum to 2.43%: asset
A contributes 1.88%, asset B contributes -0.15%, asset D
contributes 0.90%, and asset E contributes -0.20%. The active asset
contribution in each case is the sum of the active weight times the
asset return. While this analysis gives a sense for the relative
contributions of assets A, B, D, and E, it provides no information
about the 25% allocation to the composite investment C. It is not
known, for instance, if the 25% allocation to asset C resulted in a
positive or negative contribution to the performance of the
portfolio.
[0096] To remedy this, the active holdings of the portfolio are
decomposed into two sub-portfolios: the holdings in C and the
holdings in the portfolio of simple assets, which are denoted as
portfolio H. As shown in table 308 of FIG. 11, 25% of the portfolio
is allocated to sub-portfolio C while 75% of the portfolio is
allocated to portfolio H.
[0097] Table 310 of FIG. 12 shows the composition of C and H in
terms of simple assets. The allocation to portfolio C is 25%, as
indicated by the sum of the simple asset allocations. The
allocation to the H is 75%, as indicated by the sum of the simple
asset allocations.
[0098] Using the weights of these two sub-portfolios, the active
weights are computed. In table 310 of FIG. 13, the benchmark is
resealed for portfolio C to sum to 25%, the active weights of the
composite investment C across the simple assets A, B, D, and E are
2.5%, 1.3%, 3.8%, and -7.5%. The sum of the active weights is zero.
This leads to contributions (active weight times asset return) of
0.375%, 0.025%, 0.225%, and -0.075%, for a combined active
contribution of 0.55%.
[0099] As shown in table 312 of FIG. 13, once the benchmark has
been resealed to sum to 75%, the active weights of portfolio H are
10% in A, -8.8% in B, 11.3% in D, and -12.5% in E. The sum of the
active weights is zero. The contributions are 1.5%, -0.175%,
0.675%, and 0.125%, leading to a total contribution from H of
1.88%. Adding the contributions of C and H together gives a total
contribution of 2.43%, as shown in table 314.
[0100] Hence, as required, the total performance contribution is
the same regardless of whether the assets are resolved, as they
were in table 306, or if they are reallocated into sub-portfolio,
as they were in tables 310 and 312. However, the advantage of
reallocating them into sub-portfolios, as they were in tables 310
and 312, is that performance results are obtained for the 25%
allocation to the composite investment C. In this case, the
allocation to C contributed 0.55% to the total performance of
2.43%. In other words, the allocation to C contributed about one
fifth of the total, positive performance. This insight is absent
from the resolved attribution.
[0101] Next, a second attribution is considered using the data
shown in tables 302 and 304 but in this case, the holdings are
allocated into two sectors, the value sector and the growth sector,
as indicated in table 304.
[0102] First, an asset grouping (Brinson style) attribution is
performed using the resolved holdings as shown in table 306. The
Brinson analysis splits the contribution of each sector into an
allocation contribution and a selection contribution and is shown
in table 316 of FIG. 14.
[0103] Using the resolved holdings, the 2.43% contribution is
derived from a 1.73% contribution from value and 0.70% contribution
from growth. For both value and growth, the selection effect is
large and substantial (1.34% for value and 0.86% for growth), while
the allocation effect is small (0.39% for value and -0.16% for
growth). In general, a large, positive selection effect indicates
that the portfolio manager has identified the best performing
investments within a group, while a large positive allocation
effect indicates that the portfolio manager has done a good job at
budgeting his allocation among the groups (sectors in this case)
available. Of course, with our simple two-group, five-asset
example, these effects may be more intuitive than they would for a
larger, real-world example with a large number of groups and
potentially thousands of assets.
[0104] In table 318 of FIG. 15, the attribution is done for the
sub-portfolio of simple asset holdings H, while the attribution for
the sub-portfolio C is shown in table 320. As seen in tables 310
and 312, the total contributions from H and C are 1.88% and 0.55%,
respectively. Those totals remain the same in tables 318 and 320.
However, in table 320, the allocation and selection effects are
seen for the composite investment allocation C, across both
sectors. This result was absent in table 316, making it hard to
judge the impact of the 25% allocation to C.
[0105] FIG. 16 shows a flow diagram illustrating the steps of a
process 1600 embodying the present invention as applied to
performance attribution. In step 1602, a set of dates is defined
over which the performance attribution will be performed.
[0106] In step 1604, at each date, data is obtained for the
historical portfolios of holdings of investible assets where at
least one of the portfolio holdings on at least one of the dates
represents an investment in a composite investment.
[0107] In step 1605, at each date, data is obtained for the
underlying weights of each composite investment expressed in terms
of simple assets only.
[0108] In step 1606, at each date, supporting data required to
calculate a performance attribution for the historical portfolios
is obtained. This data may include asset returns, asset groupings,
factors, factor exposures, factor returns, specific returns,
benchmarks, and so forth.
[0109] In step 1608, at each date, a reallocation of each
historical portfolio into sub-portfolios is performed, resulting in
one sub-portfolio representing the original investments in simple
assets and the other sub-portfolios representing each original
composite investment.
[0110] In step 1610, performance attribution for each sub-portfolio
is performed.
[0111] Finally, in step 1612, a comparison of the relative
contributions and performance of each sub-portfolio is performed.
For example, in table 216 we compared the active contribution of
the investments in simple assets to the active contribution in SPY,
and determined that SPY out-performed the investment in simple
assets. This comparison enables portfolio managers to determine the
relative effectiveness of their individual investments. So, for the
results in table 216, a portfolio manager may decide to increase
his or her investment in SPY based on the analysis.
[0112] 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 suitable applied to
other environments consistent with the claims which follow.
* * * * *