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United States Patent Application 
20170323385

Kind Code

A1

Sivaramakrishnan; Kartik

November 9, 2017

METHODS AND APPARATUS EMPLOYING HIERARCHICAL CONDITIONAL VARIANCE TO
MINIMIZE DOWNSIDE RISK OF A MULTIASSET CLASS PORTFOLIO AND IMPROVED
GRAPHICAL USER INTERFACE
Abstract
The traditional Markowitz meanvarianceoptimization (MVO) framework that
uses the standard deviation of the possible portfolio returns as a
measure of risk does not accurately measure the risk of multiasset class
portfolios whose return distributions are nonGaussian and asymmetric. A
scenariobased conditional valueatrisk (CVaR) approach for minimizing
the downside risk of a multiasset class portfolio is addressed that uses
MonteCarlo simulations to generate the asset return scenarios. These
return scenarios are incorporated into a modified RockafellarUryasev
based convex programming formulation to generate an optimized hedge. One
example addresses hedging in an equity portfolio with options. Testing
shows that a hierarchical CVaR approach generates portfolios with better
predicted worst case loss, downside risk, standard deviation, and skew.
Inventors: 
Sivaramakrishnan; Kartik; (Marietta, GA)

Applicant:  Name  City  State  Country  Type  Axioma, Inc.  New York  NY  US 
 
Family ID:

1000002200408

Appl. No.:

15/280144

Filed:

September 29, 2016 
Related U.S. Patent Documents
      
 Application Number  Filing Date  Patent Number 

 62333563  May 9, 2016  

Current U.S. Class: 
1/1 
Current CPC Class: 
G06Q 40/06 20130101; G06F 2203/04803 20130101; G06F 3/0482 20130101 
International Class: 
G06Q 40/06 20120101 G06Q040/06; G06F 3/0482 20130101 G06F003/0482 
Claims
1. A computerimplemented method for interactively comparing performance
of a plurality of investment portfolios within a window of a graphical
user interface, the method comprising: electronically receiving by a
programmed computer a plurality of return distributions corresponding to
the plurality of investment portfolios wherein each return distribution
comprises pairs of return and frequency values; displaying a graphical
representation of the return distribution for each investment portfolio
within the window of the graphical user interface on a computer screen in
a display order such that each return distribution is displayed over and
potentially obscures any previously displayed return distribution;
monitoring, by a processor, a location of a user pointer indication to
detect when the user pointer indication is located and hovering within
the window of the graphical user interface displaying the return
distributions; automatically determining an indicated return value
corresponding to the return value of the displayed return distributions
wherever the user pointer indication is located and hovering within the
graphical user interface displaying the return distributions;
automatically altering, by the processor, the display order in which the
return distributions are displayed so that, at the indicated return value
corresponding to the user pointer indication, no return distribution
completely obscures any other return distribution.
2. The method of claim 1 wherein the display order in which the return
distributions are displayed is determined such that the frequency values
of the return distributions at the indicated return value are in a
decreasing order.
3. The method of claim 1 wherein the processor constantly monitors the
activation of a second user indication capable of indicating a selection
of a preferred return distribution and a corresponding preferred
investment portfolio; automatically electronically outputting the
preferred investment portfolio selection whenever the second user
indication is activated.
4. The method of claim 3 wherein each of the plurality of investment
portfolios is constructed to minimize a conditional value at risk (CVaR)
estimate at a plurality of confidence limits.
5. The method of claim 4 wherein at least one confidence interval is
between 90% and 99%.
6. The method of claim 4 wherein the CVaR minimization employs a
regularized RockafellerUryasev methodology modified to utilize returns
lying within an elliptical uncertainty set.
7. The method of claim 4 wherein a second window in the graphical user
interface displays a table of each confidence limit and the CVaR at each
confidence limit for each investment portfolio.
8. The method of claim 7 wherein the table in the second window in the
graphical user interface also displays a budget for each investment
portfolio.
9. The method of claim 8 wherein the processor constantly monitors the
table within the second window for changes in individual confidence
limits or budgets; whenever a change in the confidence limits or budget
is detected for an investment portfolio, the investment portfolio and
return distributions are recomputed using those changes, and the
displayed return distributions and CVaR values in the graphical user
interface are automatically updated.
10. The method of claim 1 wherein at least one of the plurality of
investment portfolios is an existing portfolio and at least one other
investment portfolio is a hedge to reduce a risk estimate of the existing
portfolio.
11. A computerimplemented method for interactively comparing performance
of a plurality investment portfolios within windows of a graphical user
interface on a computer screen and then selecting a preferred investment
portfolio, the method comprising: electronically receiving by a
programmed computer a plurality of investment portfolios; electronically
generating a set of simulated returns for each investment portfolio;
electronically receiving by the programmed computer an initial graphing
style and graphing order for displaying each set of simulated returns;
displaying a graphical representation of the simulated returns for each
investment portfolio within a first window of a graphical user interface
on a computer screen in the initial graphing style and graphing order
such that each set of simulated returns is graphed on top of the
previously graphed simulated returns; displaying in a second window of
the graphical user interface on the computer screen a first user
indicator that sets a modified graphing style or graphing order;
monitoring the first user indication of the modified graphing style or
graphing order; automatically redisplaying the graphical representation
of the simulated returns for each investment portfolio within the
graphical user interface on the computer screen in the graphing style and
graphing order specified by the first user indication; displaying in a
third window of the graphical user interface on the computer screen a
second user indication that selects a preferred investment portfolio from
the plurality of investment portfolios; monitoring the second user
indicator selecting a preferred investment portfolio; automatically
electronically outputting the preferred investment portfolio whenever the
second user indication is activated to select a preferred investment
portfolio.
12. The method of claim 11 wherein the graphical representations of the
returns of the plurality of investment portfolios are automatically
updated based on realtime data inputs from an electronic trading system.
13. The method of claim 12 wherein the preferred investment portfolio is
output to the electronic trading system for execution.
14. A computerimplemented method for interactively engineering and
selecting a preferred portfolio of investments within windows of a
graphical user interface on a computer screen connected to a programmed
computer, the method comprising: displaying within a first window of the
graphical user interface on the computer screen a data set that defines a
set of portfolio construction parameters needed to compute an investment
portfolio that minimizes a conditional variance of a portfolio at a set
of prescribed confidence levels, where the set of portfolio construction
parameters include: a first user indicator in the first window of the
graphical user interface of the computer screen a set of potential
investment opportunities that are eligible to be included as part of the
investment portfolio; a set of general portfolio construction
requirements that all eligible portfolios are to satisfy; a data library
for the set of investment opportunities containing data needed to model
each potential investment opportunity mathematically on the programmed
computer; a tolerance for the differences in conditional valueatrisk;
using the programmed computer and the data library to generate a sequence
of simulated returns for each potential investment opportunity;
electronically receiving by a second user indication a set of two or more
confidence levels at which a conditional value at risk is to be
minimized; sequentially calculating using the programmed computer a set
of investment portfolios that minimizes the conditional valueatrisk for
the portfolio at each confidence limit prescribed by the second user
indication such that the conditional valueatrisk for the portfolio is
within a tolerance of its minimum possible value; displaying within a
second window of a graphical user interface a distribution of returns for
the portfolio of investment; interactively changing the second user
indication to specify different sets of two or more confidence limits and
redisplaying the distribution of returns until a preferred investment
portfolio and return distribution is obtained; automatically
electronically outputting the preferred investment portfolio whenever the
second user indication is modified.
15. The method of claim 14 wherein the sequence of confidence limits
includes at least 95% and 90%.
16. A computer implemented method of estimating risks of an optimized
portfolio when assets with nonlinear and asymmetric return distributions
are included in the optimized portfolio, the computer implemented method
comprising: specifying assets that may be included in the optimized
portfolio; employing a MonteCarlo pricing engine to generate asset
return scenarios for said assets; specifying at least two confidence
levels for conditional value at risk (CVaR) estimation; specifying a CVaR
tolerance; employing a RockafellerUryasev methodology modified to
utilize return scenarios lying in an elliptical uncertainty set to
produce the optimized portfolio whose CVaR is within the CVaR tolerance
of the minimum possible CVaR at each confidence level; and outputting the
optimized portfolio and distributions of returns.
17. The computer implemented method of claim 16 wherein said assets
comprise multiasset class investments.
18. The computer implemented method of claim 16 wherein for N said assets
and S asset return scenarios, the MonteCarlo pricing engine produces an
N by S matrix of asset return scenarios.
19. The computer implemented method of claim 16 further comprising
repeating the method for additional sets of confidence levels to produce
alternative optimized portfolios and distributions of returns.
20. The computer implemented method of claim 19 wherein said outputting
step further comprises displaying at least two distributions of returns
on a computer display screen, each corresponding to a repeated
performance of the method; evaluating an order in which the distributions
of returns are displayed; and revising said order if it is determined a
revised order more clearly displays overlaid data to a user.
21. A computer implemented method of employing a hierarchical conditional
value at risk (HCVaR) comprising: specifying at least two confidence
levels for conditional value at risk (CVaR) estimation; constructing a
first portfolio that minimizes CVaR at a first confidence level
(CVaR.sub.1); storing the minimal CVaR for the first confidence level
(CVaR.sub.1); constructing a second portfolio that minimizes CVaR at a
second confidence level subject to a constraint that CVaR at the first
confidence level is less than CVaR.sub.1 times a predetermined amount.
Description
[0001] The present application claims the benefit of U.S. Provisional
Application Ser. No. 62/333,563 filed May 9, 2016 which is incorporated
by reference in its entirety.
RELATED APPLICATIONS
[0002] The present invention may advantageously be used in conjunction
with one or more of the following applications and patents: U.S. patent
application Ser. No. 11/668,294 filed Jan. 29, 2007 which issued as U.S.
Pat. No. 7,698,202; U.S. patent application Ser. No. 12/958,778 filed
Dec. 2, 2010 which issued as U.S. Pat. No. 8,533,089; U.S. patent
application Ser. No. 12/711,554 filed Feb. 24, 2010 which issued as U.S.
Pat. No. 8,315,936; U.S. patent application Ser. No. 12/827,358 filed
Jun. 30, 2010; U.S. patent application Ser. No. 13/503,696 filed Apr. 24,
2012 which issued as U.S. Pat. No. 8,533,107; U.S. patent application
Ser. No. 13/503,698 filed Apr. 24, 2012 which issued as U.S. Pat. No.
8,700,516; U.S. patent application Ser. No. 13/892,644 filed May 13,
2013; U.S. patent application Ser. No. 14/025,127 filed Sep. 12, 2013;
U.S. patent application Ser. No. 14/051,711 filed Oct. 11, 2013; U.S.
patent application Ser. No. 13/654,797 filed Oct. 18, 2012; U.S. patent
application Ser. No. 13/965,621 filed Aug. 13, 2013; U.S. patent
application Ser. No. 14/336,123 filed Jul. 21, 2014; U.S. patent
application Ser. No. 14/203,807 filed Mar. 11, 2014; U.S. patent
application Ser. No. 14/482,685 filed Sep. 10, 2014; U.S. patent
application Ser. No. 14/495,470 filed Sep. 24, 2014; U.S. patent
application Ser. No. 14/505,258 filed Oct. 2, 2014; U.S. patent
application Ser. No. 14/519,991 filed Oct. 21, 2014; all of which are
assigned to the assignee of the present application and incorporated by
reference herein in their entirety.
FIELD OF INVENTION
[0003] The present invention relates generally to methods and apparatuses
for constructing portfolios, overlay portfolios, and trade lists of
multiasset class investments that interactively manage and engineer the
conditional variance (CVaR) of a portfolio or trade list at two or more
confidence levels, as well as, to advantageous techniques for displaying
two or more return distributions utilizing a graphical user interface. By
specifying two or more confidence levels with a designated order of
importance, herein termed hierarchical CVaR, investment portfolios and
trade lists with advantageous return distributions may be engineered. An
improved graphical user interface permits a portfolio manager or trader
to readily create, display and select a preferred investment portfolio or
trade list utilizing simultaneous displays of multiple distributions of
returns to facilitate comparison with reduced impact from one return
distribution's graphical display obscuring another.
BACKGROUND OF THE INVENTION
[0004] Multiasset class portfolios are ubiquitous in finance. Asset
managers, asset owners, and hedgefunds invest in diverse asset classes
such as equities, fixedincome, commodities, foreign exchange, credit,
derivatives, and alternative investments such as real estate and private
equity. Large institutional investors such as pension funds, sovereign
wealth funds, and university endowments have moved from portfolios
composed of only equities and bonds to multiasset class portfolios.
[0005] One overall aim of multiasset class portfolio construction is to
generate a diversified portfolio with a superior risk adjusted return.
Multiasset class instruments provide a more diversified set of asset
allocation opportunities with a wide spectrum of risk and return
profiles. For example, commodities tend to be negatively correlated with
both equities and bonds. As a result, an investment portfolio holding
commodities, equities, and bonds may have less risk than a portfolio
invested in only equities and bonds due to the diversifying effect of the
commodity investment.
[0006] The tools and methods used to construct multiasset class
portfolios must overcome obstacles not found when constructing
equityonly portfolios. Many of these obstacles arise because the
distributions of possible returns for nonequity investments are often
highly asymmetric and can change substantially over time. The traditional
Markowitz meanvariance optimization (MVO) framework, that linearizes the
expected return of the portfolio and uses the standard deviation of
return as a measure of risk, does not accurately measure risk for such
portfolios. For asymmetric return distributions, downside risk, which is
the risk or likelihood of the actual or realized return being less than
the mean possible return, is often quite different than upside risk,
which is the risk or likelihood of the actual or realized return being
greater than the mean possible return. When constructing a portfolio of
investments, an investor is usually principally interested in minimizing
downside risk. The 20082009 financial crisis spurred a renewed interest
in downside risk protection for multiasset class portfolios because
credit risk, a highly nonlinear nonequity class, played a major role in
that crisis.
[0007] Consider the following investment situation that highlights some of
the considerations in multiasset class investing. A fixed income
portfolio manager owns a large bond portfolio in early 2016. The U.S.
Federal Reserve has indicated that it will reduce its quantitative easing
program, which is likely to result in an increase in interest rates in
the near future. Although an increase in interest rates will reduce the
value of the existing bond portfolio, the manager does not want to change
the portfolio. Instead, he or she wants to hedge it against future losses
resulting from an increase in interest rates. The manager can do so by
purchasing appropriate amounts of customized overthecounter (OTC)
derivatives such as interest rate caps and payer swaptions whose returns
are strongly correlated with changes in the interest rates. These
additional investments are often called overlay portfolios or simply
overlays because they are added to an existing investment portfolio. The
portfolio construction problem facing the manager requires: (a), deciding
how to allocate an available budget towards the purchase of these
overlays; (b), effectively hedging the portfolio against an increase in
interest rates by determining the correct investment amounts in each of
the overlay investments; and (c) satisfying the general set of
constraints imposed for all the investments: namely, considerations, such
as the manager's preferences and investment insights, business
requirements, and institutional mandates.
[0008] Once a preferred overlay portfolio has been determined, the list of
trades associated with that overlay must then be transmitted to an
electronic trading system for execution.
[0009] As already indicated, one of the distinguishing features of
multiasset class portfolios, as compared with equity portfolios, is that
the return distribution for multiasset class portfolios is nonGaussian
and asymmetric with significant skew and kurtosis. For equity portfolios,
the return distribution is symmetric and approximately Gaussian. An
exemplary return distribution having these characteristics is illustrated
by chart 200 of FIG. 1, which shows a return distribution 202 for one
share in the S&P 500 Index, a welldiversified equity index. The
distribution of possible returns 202 is approximately symmetric and
Gaussian.
[0010] However, as shown in chart 204 in FIG. 2, the return distribution
206 for a covered call portfolio, which holds one share in the S&P 500
Index and is short one share (e.g., the same amount) of a call contract
on the S&P 500 Index, is highly asymmetric and nonGaussian. The
nonGaussian return profile of the covered call portfolio is due to the
call contract, a nonlinear instrument with an asymmetric payoff. The
upside of the covered call portfolio is capped by the strike price since
the call will be exercised if the share price exceeds the strike. While
the S&P 500 Index in isolation, as shown in distribution 202, has a
significant probability of having returns as high as 20%, the highest
possible return of the covered call 206 is only about 2.78%. The downside
of the covered call portfolio is considerably greater than 2.78% since
the share price of the S&P 500 Index can potentially go all the way to
zero.
[0011] FIG. 3 shows a table 208 with several common risk metrics for the
two return distributions 202 and 206 shown in FIGS. 1 and 2. One common
measure of risk is the standard deviation of the possible returns. This
metric is so ubiquitous in finance that it is often referred to as the
"risk" of the investment. The higher the standard deviation, the riskier
the investment. For the S&P 500 Index, the standard deviation of return
is 6.48%, while for the covered call portfolio, it is only 3.78%. That
is, using this metric of risk, the covered call is less risky than the
S&P 500 investment in isolation.
[0012] One criticism of the standard deviation as a risk metric is that it
fails to distinguish downside risk, the risk of a large, negative return,
as shown by the left tails of distributions 202 and 206, from upside
risk, the risk of a large, positive return, as shown by the right tails
of the distributions 202 and 206. A second risk metric is conditional
value at risk (CVaR) or expected shortfall, or left CVaR. This risk
metric is a number that determines the likelihood or probability, at a
confidence level e specified by the portfolio manager, that the realized
loss will exceed the value at risk (VaR), which is an estimate of how
much an investment might lose under typical market conditions over a
specified time period. CVaR is a measure of the downside risk of the
portfolio. As shown in table 208, for the S&P 500 Index portfolio, the
CVaR is 13.45%, while for the covered call portfolio it is 10.55%. Hence,
this measure of risk indicates that the covered call portfolio is less
risky than the S&P 500 Index portfolio in isolation.
[0013] Table 208 also indicates the right CVaR, which is the expected
excess return above the right VaR, which is the measure of the how much
an investment might gain under typical market conditions over a specified
period of time. That is, it measures the upside risk, the riskiness of
the right tail of the distribution (the large positive returns) instead
of the left tail (the large negative returns). As shown in table 208, the
right CVaR of the S&P 500 is 13.31%, a value similar to the CVaR of this
investment. These numbers are roughly the same because the overall
distribution 202 is symmetric. However, for the covered call, the right
CVaR is only 2.78%, which is substantially different and less risky than
the left CVaR of 10.55%. This difference is caused by the substantial
asymmetry of the return distribution 206 in which the range of possible
positive returns is capped by the strike price.
[0014] Prior methods for constructing a portfolio of investments with
advantageous risk and return characteristics are known. See, for example,
Markowitz, in Portfolio Selection: Efficient Diversification of
Instruments, Wiley, 1959 (Markowitz) which is incorporated by reference
herein in its entirety, developed MVO, which is a portfolio construction
approach and methodology that is widely used in equity portfolio
management.
[0015] In MVO, a portfolio is constructed that minimizes the risk of the
portfolio while achieving a minimum acceptable level of return.
Alternatively, the level of return is maximized subject to a maximum
allowable portfolio risk. In traditional mean variance portfolio
construction, risk is measured using the standard deviation of possible
returns. The family of portfolio solutions solving these optimization
problems for different values of either minimum acceptable return or
maximum allowable risk is said to form an efficient frontier, which is
often depicted graphically on a plot of risk versus return. Portfolio
construction procedures often make use of different estimates of
portfolio risk, and some make use of an estimate of portfolio return. A
crucial issue for these optimization procedures is how sensitive the
constructed portfolios are to changes in the estimates of risk and
return. Small changes in the estimates of risk and return occur when
these quantities are reestimated at different time periods. They also
occur when the raw data underlying the estimates is corrected or when the
estimation method itself is modified. Traditional MVO portfolios are
known to be sensitive to small changes in the estimated asset return,
variances, and covariances. See, for example, J. D. Jobson, and B.
Korkei, "Putting Markowitz Theory to Work", Journal of Portfolio
Management, Vol. 7, pp. 7074, 1981 and R. O. Michaud, "The Markowitz
Optimization Enigma: Is Optimized Optimal?", Financial Analyst Journal,
1989, Vol. 45, pp. 3142, 1989 and Efficient Asset Management: A
Practical Guide to Stock Portfolio Optimization and Asset Allocation,
Harvard Business School Press, 1998, (the two Michaud publications are
hence referred to collectively as "Michaud") all of the above cited
publications are incorporated by reference herein in their entirety.
[0016] One of the assumptions of traditional MVO analysis is that the
investment returns are approximately Gaussian since MVO uses the standard
deviation of the portfolio returns to measure the risk of the portfolio.
This standard deviation approach assumes that most of the asset variation
falls within three standard deviations of the mean. For symmetric,
returns distributions such as distribution 202, standard deviation is a
useful measure of the risk of the portfolio. However for asymmetric
return distributions, such as distribution 206, using the standard
deviation as a risk measure can significantly understate the volatility
or risk of the portfolio of nonlinear investments. This underestimation
of risk is one of the principal deficiencies of traditional portfolio
construction tools such as MVO when applied to portfolios with
multiasset class instruments and nonlinear return distributions.
[0017] For over three decades, commercial risk model vendors have sold
factor risk models to estimate the standard deviation risk of a portfolio
of assets. Alternatively, these same models may be used to estimate the
variance of a portfolio, since variance is the square of the standard
deviation. Such risk models can be advantageously employed to estimate
return distributions. Factor risk models provide an estimate of the
assetasset covariance matrix, Q, which estimates the future covariance
of each pair of asset returns using historical return data.
[0018] To obtain reliable variance or covariance estimates based on
historical return data, the number of historical time periods used for
estimation should be of the same order of magnitude as the number of
assets, N. Often, there may be insufficient historical time periods. For
example, new companies and bankrupt companies have abbreviated historical
price data and companies that undergo mergers or acquisitions have
nonunique historical price data. As a result, the covariances estimated
from historical data can lead to matrices that are numerically
illconditioned. Such covariance estimates are of limited value.
[0019] Factor risk models were developed, in part, to overcome these short
comings. Factor risk models represent the expected variances and
covariances of security returns using a set of M factors, where M is much
smaller than N, that are derived using statistical, fundamental, or
macroeconomic information or a combination of any of such types of
information. For each factor, every asset covered by the factor risk
model is given a score. The N by M matrix of factors scores is called the
factor exposures or factor loadings. In addition, a factor return is
estimated for each factor at each time that the model is reestimated.
Given exposures of the securities to the factors and the covariances of
factor returns, the covariances of security returns can be expressed as a
function of the factor exposures, the covariances of factor returns, and
a remainder, called the specific risk of each security. Factor risk
models typically have between 20 and 200 factors. Even with, say, 80
factors and 1000 securities, the total number of values that must be
estimated is just over 85,000, as opposed to over 500,000.
[0020] A substantial advantage of factor risk models is that since, by
construction, M is much smaller than N, factor risk models do not need as
many historical time periods to estimate the covariances of factor
returns and thus are much less susceptible to the illconditioning
problems that arise when estimating the elements of Q individually.
[0021] Over the many years that MVO and its variants have been
commercially employed, a number of practices for constructing portfolios
and trade lists using optimization have become standard. As one example,
Axioma, Inc. (Axioma) sells software for constructing portfolios and
trade lists that allows portfolio managers to construct portfolios and
trade lists that specify general rules and requirements for both the
portfolio and the trades. The portfolio can be long only, or it may be
longshort. For longshort portfolios, the ratio or leverage between the
market value of the short side can be controlled independently or as a
function of the market value of the long side. The local universe
comprising potential investment assets that may be used to construct the
portfolio or trade list can be specified. General grandfathering options
are commonly employed to allow the portfolio to hold or keep existing
asset investments if they are not in the local universe or do not satisfy
constraints that are violated by the initial holdings. In addition, the
trade list may or may not allow crossover (long positions becoming short
positions or vice versa), and may or may not use round lotting to
restrict the trade or holding sizes to multiples of a fixed numbers of
shares. The strategy may also include compliance rules that are specified
for subsets of portfolios.
[0022] The objective function, which may be minimized or maximized to
obtain the optimal portfolio, may include linear terms such as the
expected return or alpha. In MVO, the letter M refers to the mean and is
a tilt on the expected return, sometimes called alpha, which is maximized
for the optimal portfolio. The objective function may include tilts or
linear terms for the long and short holdings separately. The objective
function may include risk terms, which refer to the standard deviation of
possible returns, or variance terms, which are the square of the standard
deviations. These risk terms may be computed using the total holdings, or
they may be computed using only the active holdings relative to a
benchmark of investment holdings. In this case, the risk and variance
terms are termed active risk or active variance. In MVO, the letter V
refers to variance, either total or active, and is minimized. In many, if
not most, cases, a commercial factor risk model is used to estimate the
risk or variance of the portfolio. The objective function terms may also
include the costs of trading the portfolio. Such costs may include both
the costs charged directly as well as indirect market impact costs, such
as changes in market prices caused by the trade itself. The objective
function may also include terms designed to benefit the portfolio when
taxes are considered. Taxable losses may be maximized while taxable
gainsboth short and long term and for various ratesmay be minimized.
In modern portfolio and trade list construction software, there is great
flexibility to consider different, weighted combinations of these terms
in the objective function to compute a desired, optimal portfolio.
[0023] The portfolio construction strategy will usually include a set of
constraints that must be satisfied by the optimal portfolio or trade
list. These constraints may include maximum and/or minimum bounds on the
holdings or exposures of the holdings. For instance, the maximum and
minimum asset weights in the portfolio may be specified. Or the maximum
or minimum net exposure of assets to an industry, sector, or country may
be specified. The maximum and minimum net exposure of the portfolio or
subsets of the portfolio to general attributes such as market
capitalization or average daily traded volume may also be specified as
constraints on the portfolio or trade list. Instead of including risk or
variance in the objective function, the maximum allowable risk, active
risk, variance, or active variance may be specified as a constraint. In
addition, the marginal contribution to risk or active risk, which is the
derivative of the risk with respect to an asset's weight in the
portfolio, may also be given a maximum value. The constraints may impose
limits on the kinds and size of trades employed. That is, some assets may
not be allowed to trade, while other asset positions may be entirely
liquidated. The total transaction cost of trades may be constrained to be
less than a maximum allowable amount. The total number of names held or
traded may also be constrained. The taxable gains and liabilities for the
investment holdings may be constrained.
[0024] Of course, with more sophisticated software, the number and variety
of possible objective terms and constraints increases.
[0025] The various combinations of the above listed objective terms and
constraints comprise the typical requirements of a portfolio or trade
list construction strategy. These are the general preferences and
investment insights, business requirements, and institutional mandates
that must be satisfied by a portfolio manager when constructing a
portfolio, overlay, or trade list.
[0026] The commercial importance and expertise required to build high
quality factor risk models, as well as, high quality portfolios and trade
lists has led to many patented innovations for factor risk models. These
include U.S. Pat. Nos. 7,698,202, 8,315,936, 8,533,089, 8,533,107, and
8,700,516, all of which are assigned to the assignee of the present
invention and are incorporated by reference herein in their entirety.
[0027] There are numerous, well known, variations of MVO that are used for
portfolio construction. These variations include methods based on utility
functions and the Sharpe ratio.
[0028] Having recognized that standard deviation can be a poor measure of
risk for multiasset class portfolios, there has been considerable
interest in downside risk measures that reflect the financial risk
associated with losses. There have been two primary downside risk
measures that have been proposed and studied: value at risk (VaR), and
conditional value at risk (CVaR) which is also known as expected
shortfall or left CVaR.
[0029] Value at risk (VaR) estimates how much a set of investments might
lose under typical market conditions over a fixed time period such as a
day. VaR is calculated as a threshold loss value such that the
probability that the loss on the portfolio over the given time horizon
exceeds a given confidence limit or probability. So, for example, a
portfolio VaR of ten million dollars at a 95% confidence level over a ten
day period indicates that there is 95% confidence that the portfolio will
not suffer losses greater than ten million dollars over a ten day period.
[0030] VaR played a prominent role in the Basel regulatory framework. See,
for example, A. J. McNeil, R. Frey, and P. Embrechts, Quantitative Risk
Management: Concepts, Techniques, and Tools, Revised Edition, Princeton
University Press, 2015, (McNeil), which is incorporated by reference
herein in its entirety. VaR is the most widely used risk measure for
multiasset class portfolios.
[0031] The conditional value at risk (CVaR) at a confidence level e is the
expected value of the loss exceeding VaR. CVaR is an alternative to VaR
that is more sensitive to the shape of the loss distribution in the tail
of the distribution. CVaR was introduced to overcome the shortcomings of
VaR. In certain contexts, VaR has poor mathematical properties. It is not
coherent in the framework of P. Artzner, F. Delbaen, J. M. Eber, and D.
Heath, "Coherent Measures of Risk," Mathematical Finance, 9(3), pp.
203228, which is incorporated by reference herein in its entirety. In
particular, it is not subadditive. In other words, the VaR of a
portfolio can be larger than the sum of the VaR of the portfolio
constituents. So, if VaR is used to set risk limits, it can lead to
concentrated portfolios. CVaR, on the other hand, is a coherent risk
measure encouraging diversification.
[0032] VaR does not measure the left tail of the portfolio loss
distribution. Consequently, the worst case loss can be much larger than
VaR. CVaR, on the other hand, is a tail statistic that incorporates the
losses that occur in the left tail of the loss distribution.
[0033] VaR for a nonlinear portfolio is difficult to optimize in practice
as it requires the solution to a nonconvex optimization problem. CVaR,
on the other hand, can be optimized via a scenario based linear program.
This follows from the work of R. T. Rockafellar and S. Uryasev,
"Optimization of ConditionalValueatRisk," Journal of Risk, 2(2000),
pp. 493517 and R. T. Rockafellar and S. Uryasev, "Conditional
ValueatRisk for General Loss Distributions," Journal of Banking &
Finance, 26(2002), pp. 14431471, (these two RockafellarUryasev
publications are hence referred to collectively as
"RockafellarUryasev"), both of which are incorporated by reference
herein in their entirety.
[0034] Although this prior art portfolio and trade list construction
approach can be solved using linear programming, it has important
limitations. The method is particularly sensitive to estimation errors in
the scenarios or Monte Carlo simulations. When employing this prior art
approach, it is unclear whether the optimization procedure is minimizing
CVaR or the estimation error embedded in the CVaR estimate or a
combination of both (Michaud).
[0035] In addition, in practice, many portfolio and trade list
construction problems that minimize CVaR result in an objective function
to be minimized with relatively flat gradients. Hence, although a global
optimal solution may exist that minimizes CVaR, there may be alternative
portfolios or trade lists that have nearly the same CVaR. Existing
approaches that minimize CVaR do not take advantage of the fact that
there may exist alternative portfolios or trade lists with nearly optimal
CVaR risk estimates as well as other advantages.
[0036] Other downside risk measures have also been proposed, but these
have significant deficiencies compared with CVaR. There has been prior
research to improve the risk estimates and avoid the underestimation
caused by using the standard deviation by incorporating higher moments
such as skew and kurtosis in the portfolio construction. Mathematically,
the standard deviation is the second moment of the distribution of
returns, while skew and kurtosis are the third and fourth moments of the
distribution of returns. See E. Jondeau and M. Rockinger, "Optimal
Portfolio Allocation Under Higher Moments," European Financial
Management, 12(1), 2006, pp. 2955, which is incorporated by reference
herein in its entirety.
[0037] However, approaches that incorporate higher order moments in their
measure of risk have a number of deficiencies. First, these approaches
are limited in the size of the portfolios that they can handle. Second,
these approaches require a long return time series to estimate the third
and the fourth moments. This data may not be available; even if the data
is available, the estimates often include significant estimation errors.
Third, the portfolio construction problem is a nonconvex polynomial
optimization problem that cannot be solved efficiently in practice.
[0038] Semivariance and lower partial moments have also been proposed as
measures of downside risk. However, there is no practical approach to
minimize either semivariance or lower partial moments when constructing a
portfolio, overlay or trade list.
SUMMARY OF THE INVENTION
[0039] Among its several aspects, the present invention recognizes that
existing approaches for strategically constructing portfolios and trade
lists comprising multiasset class investments suffer from important
limitations as addressed in detail above and further below.
[0040] One general problem considered by the present invention is how to
more effectively allocate a budget towards purchase of overlays, hedge
the portfolio and satisfy the general constraints imposed on the
portfolio. Particular attention is paid to hedging market risk and credit
risk; that is, reducing the risk of a portfolio associated with market
risk or credit risk. Market risk is the risk of a change in the value of
a portfolio due to changes in the value of its holdings, for example,
changes in the market value of an existing portfolio. Credit risk is the
risk of not receiving promised payments due to default of the
counterparty in investments such as credit default swaps.
[0041] A further significant aspect of the present invention concerns a
graphical user interface that can easily and automatically alter the
manner in which return distributions such as distributions 202 and 206
are displayed. In several aspects of the present invention, improved ease
of interaction between a portfolio manager and a graphical user interface
is addressed. In some embodiments of the invention, an iterative
interaction occurs between the portfolio manager and a graphical user
interface in order to construct and select a portfolio whose distribution
of potential returns has desirable properties. Those desirable properties
are advantageously displayed on the graphical user interface as addressed
further herein.
[0042] The present invention recognizes that traditional portfolio
construction tools that use standard deviation as a risk measure can
significantly underestimate risk when assets with nonlinear and
asymmetric returns distributions are included in the portfolio. Such
underestimation is commonly the case for multiasset class portfolios,
and it is an important limitation of traditional portfolio construction
tools. When portfolios, hedges, and trade lists are constructed for
multiasset class securities, the likelihood of underestimating the risk
of the portfolio is high. In other words, a hedge produced using standard
deviation as the risk metric may not reduce risk as much as the standard
deviation metric suggests
[0043] The present invention recognizes that portfolio construction and
trade list construction approaches that minimize VaR have significant
limitations. These include a lack of mathematical coherence as well as
mathematical and practical difficulties in determining portfolios or
trade lists that minimize VaR.
[0044] The present invention recognizes that among the different measures
of downside risk that have been proposed, CVaR has many advantages. These
advantages include mathematical coherence, its focus on the left tail of
the distribution of returns describing losing returns, and practical
solution techniques.
[0045] Although portfolio construction techniques that minimize CVaR exist
in the prior art, there are a number of limitations to these existing
approaches. These include the fact that the traditional linear
programming solution is sensitive to estimation error and the fact that
the objective function for many practical problems that minimize CVaR are
nearly flat objective functions.
[0046] The present invention recognizes that existing efforts to improve
portfolio construction tools for portfolios with nonlinear and asymmetric
return distributions by including higher order moments such as skew and
kurtosis are often not satisfactory since they lead to nonconvex
polynomial optimization problems that cannot be efficiently solved in
practice.
[0047] The present invention also recognizes the limitations in how two or
more distributions of returns are displayed. When two or more
distributions are plotted simultaneously, each distribution must be
displayed either on top of or behind each of the other distributions. As
a result, the distributions plotted on top of other distributions
potentially hide or obscure the distribution results for those
distributions plotted behind them. Such obscuration may make it difficult
to easily compare the distributions or decide if one has better
properties than the other.
[0048] One goal of the present invention, then, is to provide a
methodology that enables a portfolio manager to effectively construct a
portfolio with superior downside risk properties. In particular, the
portfolio is constructed so that the CVaR is minimized for two or more
specified confidence levels.
[0049] Another goal is to provide improved tools for displaying and
comparing the predicted performance for alternative portfolios.
[0050] Another goal is to provide an improved, interactive tool in a
graphical user interface for use by a portfolio manager who wishes to
manage and minimize the CVaR of his or her portfolio at several
confidence levels.
[0051] Another goal is to provide an improved, interactive tool embedded
in an electronic trading system so that a portfolio manager who wishes to
manage and minimize the CVaR of his or her portfolio may efficiently and
interactively construct a desirable hedge and transmit and trade that
hedge on the electronic trading platform.
[0052] According to one aspect of the present invention, downside risk is
measured using CVaR. Multiasset class portfolios are constructed by
minimizing the CVaR of the portfolio using an improved methodology based
on the RockafellarUryasev approach modified as taught herein and
employed in conjunction with a MonteCarlo framework to generate the
asset return scenarios for the multiasset class investment
opportunities.
[0053] In one aspect of the present invention, improved tools for
simplifying and improving interactions between a portfolio manager and a
graphical user interface are provided. In some embodiments of the
invention, the invention embodies an interactive tool embedded within a
window of a graphical user interface which is used by the investment
manager to interactively alter various test portfolios in order to
construct and select a preferred portfolio whose distribution of
potential returns has desirable properties.
[0054] In other embodiments of the invention, the invention embodies an
interactive tool embedded within an electronic trading platform which is
used by the investment manager to interactively construct a set of trades
which alters the distribution of potential returns so that they have
desirable properties.
[0055] The present invention describes new and improved methods for: (a),
modeling the possible performance of multiasset portfolios, overlays,
and trade lists; (b), constructing portfolios and trade lists that
minimize the downside risk of the portfolio, overlay, or trade list; (c)
iteratively and efficiently interacting with a graphical user interface
so that the differences in the performance of different portfolios,
overlays, and trade lists can be quickly and accurately compared and a
preferred portfolio, overlay or trade list can be engineered and
identified; and (d), efficiently transmitting a preferred portfolio,
overlay or trade list to a portfolio database or electronic trading
system.
[0056] To such ends, a computerimplemented method for interactively
comparing performance of a plurality of investment portfolios within a
window of a graphical user interface is provided. The method may suitably
comprise: electronically receiving by a programmed computer a plurality
of return distributions corresponding to the plurality of investment
portfolios wherein each return distribution comprises pairs of return and
frequency values; displaying a graphical representation of the return
distribution for each investment portfolio within a first window of the
graphical user interface on a computer screen in a display order such
that a second return distribution is displayed over and potentially
obscures a first return distribution; monitoring, by a processor, a
location of a user pointer to detect when the user pointer is located and
hovering within the window of the graphical user interface displaying the
first and the second distributions; automatically determining an
indicated return value corresponding to the return value of the first and
second return distributions whenever the user pointer is located and
hovering within the graphical user interface displaying the first and
second return distributions; and automatically altering, by the
processor, the order in which the first and second return distributions
are displayed so that, at the indicated return value corresponding to the
user pointer location, no return distribution completely obscures any
other return distribution.
[0057] According to a further aspect of the invention, each of the
plurality of investment portfolios are constructed to minimize a
conditional value at risk (CVaR) at more than one confidence limit.
[0058] 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
[0059] FIG. 1 shows a possible return distribution for one share of the
S&P 500 Index to illustrate an aspect of the Background of the Invention;
[0060] FIG. 2 shows a possible return distribution for a covered call
portfolio composed of one share of the S&P 500 Index and a short of one
share of a call on the S&P 500 Index to illustrate an aspect of the
Background of the Invention;
[0061] FIG. 3 shows a table of different risk metrics for the S&P 500
Index and the covered call portfolio to illustrate an aspect of the
Background of the Invention;
[0062] FIG. 4 shows two different charts simultaneously displaying the two
return distributions shown in FIGS. 1 and 2;
[0063] FIG. 5A shows an illustrative screenshot produced by portfolio
construction graphical interface software including a table of portfolio
statistics which may be suitably employed as a source of investment
portfolios and the like for use in conjunction with the present
invention;
[0064] FIG. 5B shows an illustrative screenshot produced by portfolio
performance attribution software including bar and line charts adapted
for use in conjunction with the present invention;
[0065] FIG. 6 shows an overall system embodying aspects of the present
application;
[0066] FIG. 7 shows a user computer system which may be suitably utilized
in conjunction with the present invention and suitable for use in the
overall system of FIG. 6;
[0067] FIG. 8 shows a table of insample and outofsample statistics
confirming the stability of scenariobased results determined utilizing
aspects of the present invention;
[0068] FIG. 9 shows return distributions for an original portfolio and
three different hedges with a 1% budget;
[0069] FIG. 10 shows return distributions for an original portfolio and
three different hedges with a 5% budget;
[0070] FIG. 11 shows return distributions for a CVaR and a hierarchical
CVaR hedge with a 1% budget;
[0071] FIG. 12 shows return distributions for a CVaR and a hierarchical
CVaR hedge with a 5% budget;
[0072] FIG. 13 shows a table of statistical results for an original
portfolio and four different hedges with a 1% budget;
[0073] FIG. 14 shows a table of statistical results for an original
portfolio and four different hedges with a 5% budget;
[0074] FIG. 15 shows a representative portfolio construction graphical
user interface employing the invention;
[0075] FIG. 16 shows a representative flow diagram of how the invention
may suitably be utilized within an interactive graphical user interface;
[0076] FIG. 17 shows an exploded view of portions of the distributions
illustrated in FIG. 10 and illustrates aspects of a display tool in
accordance with the present invention; and
[0077] FIG. 18 shows a programmed microprocessor embodiment illustrating
aspects of the present invention.
DETAILED DESCRIPTION
[0078] The present invention may be suitably implemented as a networked
computer based system, such as networked system 600 of FIG. 6, in
computer software which is stored in a nontransitory 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 CDROM, CDRW, 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, as well as networked connections of such devices which are
preferably optimized to rapidly perform the large number of computations
on very large data sets employed in various contexts in which the present
invention appears most likely to be employed.
[0079] The charts 210 and 218 in FIG. 4 show two possible display formats
for simultaneously displaying return distributions, such as the return
distributions 202 and 206 of FIGS. 1 and 2. In upper chart 210, the
distributions are shown using filledin, colored bars. The legend 212
indicates that light gray is used to represent the return distribution of
the S&P 500 Index. The return distribution 216 in FIG. 4 is identical to
distribution 202 in FIG. 1. The legend 212 indicates that dark gray is
used to represent the return distribution of the covered call composed of
the S&P 500 Index and the short call. The return distribution 214 in FIG.
4 is identical to distribution 206 in FIG. 2.
[0080] As can be seen in chart 210, one of the advantages of displaying
more than one return distribution on the same chart is that the scale of
the returns can be more easily compared. In this case, it is quite
evident that the most likely return of the covered call 214 is 2.78%
since such a return occurs more than half the time. However, one of the
disadvantages of this approach is that one return distribution hides or
obscures the other. In the chart 210, since distribution 214 is drawn on
top of distribution 216, it potentially obscures the differences between
the two return distributions. For example, in chart 210, it is hard to
know what the return distribution 216 does for small positive returns
because its light gray bars are hidden by the dark gray bars of
distribution 214.
[0081] Note that in chart 210, the order in which the two return
distributions have been plotted matters. As shown in chart 210, with
distribution 214 plotted in front of distribution 216, most of both
distributions are visible. If, however, the order had been reversed and
distribution 216 had been plotted in front of distribution 214, then
virtually the entire left part of distribution 214 would have been hidden
by distribution 216. Such a plotting order would be notably inferior to
the order shown in chart 210. As addressed further herein, one aspect of
the invention allows the user to readily select and change this order.
According to another aspect, automatic processes and apparatus are
provided to insure an advantageous order of plotting.
[0082] One potential solution to having one distribution hide a second
distribution is shown in lower chart 218 of FIG. 4. In this
representation, only the tops of the bars of chart 214 are shown. The
legend 220 in 218 uses the same colors as in legend 212, namely light
gray for the S&P 500 Index investment, dark gray for the covered call.
The outline of the S&P 500 return distribution 224 is shown by the light
gray line. The outline of the covered call return distribution 222 is
shown by the dark gray line. This representation greatly reduces the
possibility of not being able to see a distribution. However, as can be
seen, when the two lines overlap, as they do for small positive returns,
there is still the possibility that the thickness of the line may hide
the details of the distribution.
[0083] When there are more than two distributions being shown, the visual
ability to see and distinguish the distributions is potentially further
reduced, as there is more chance of one overlapping and hiding the other.
Also, there are simply too many lines on the chart to facilitate easy
user comparison of the differences in the returns.
[0084] FIG. 5A shows a screenshot 226 produced by a portfolio construction
graphical user interface software, such as the Axioma Portfolio
Optimizer.TM. software, modified to operate in conjunction with the
present invention. The graphical user interface includes buttons or user
indicators 228 and 230 that may be selected by the user using a mouse,
keyboard or other user control device. When the button 228 is selected
with a mouse click, keyboard keystroke, or a touch of a touchscreen icon,
the graphical user interface displays the "Data" perspective of the
program. When the tab 230 is selected, the graphical user interface
brings this tabbed window to the front of all windows in the display. In
screenshot 226, only one window is shown for clarity, but users may open
dozens of windows simultaneously and toggle back and forth between them
with appropriate use of the graphical user interface tools.
[0085] FIG. 5A includes two menu selectors, 232 and 233. When menu
selector 232 is selected, the graphical user interface displays a list of
portfolios that can be displayed in the main window. By clicking on
selector 232, a drop down menu of portfolios is displayed. Then, the user
can choose which portfolio or portfolios are under analysis using the
selection menu. When selector 233 is selected, the user is presented with
a drop down menu of different confidence levels to select from. Once a
portfolio is selected with selector 232 and one or more confidence levels
are selected using selector 233, an optimized hedge is found by the
software.
[0086] Screenshot 226 includes selectors 232 and 233 to provide a user
selection tool to select portfolios or hedges from a drop down menu
consistent with the devices addressed in FIGS. 815 and to specify
confidence levels .epsilon..sub.1, .epsilon..sub.2, . . .
.epsilon..sub.n, respectively. The data in screenshot 226 is in tabular
form. In the window 230, the properties of a portfolio of investments are
shown including columns displaying the Asset, Description, and Value.
While such graphical user interfaces are common, enhanced interaction
between the user and the interface is an important aspect of the present
invention. In some embodiments of the invention, the invention is
embodied as an interactive tool embedded within a window of a graphical
user interface which is used by the investment manager to interactively
alter various portfolios in order to construct and select a preferred
portfolio whose distribution of potential returns has desirable
properties as addressed further below.
[0087] FIG. 5B shows a second screenshot 234 produced by Axioma's
performance attribution graphical user interface software, Axioma
Portfolio Analytics.TM., adapted to add improved functionality as
discussed further below. In the chart, there are two sequences of colored
bars 238 and 240, respectively, each sequence representing time series of
exposures and returns associated with an investment portfolio.
Alternatively, the cumulative return is represented by a graphical line
242 extending across the window.
[0088] In short, then, the graphical user interface associated with
financial investment portfolios commonly employ user indicators that
control the manner in which both tabular and graphical data are displayed
in the graphical user interface.
[0089] Customized hardware and software to improve the interaction between
a portfolio manager and an electronic trading platform is one aspect of
the present invention. As further illustrated in one embodiment of the
present invention shown in FIG. 6, further customized hardware and
software are addressed herein. FIG. 6 shows a networked system 600 in
which an electronic trading platform 610 communicates utilizing a network
communication system 640 with a portfolio optimization and management
system 650 and a user system 670 used by a portfolio manager or the like.
The electronic trading platform 610 may receive bids and asks directly
from a trader utilizing user system 670 or through the system 650.
Individual investors and traders buy or sell securities, foreign
exchange, and other financial derivative products over the electronic
trading platform 610. Exemplary electronic trading platforms, such as
NASDAQ, NYSE Arca, Globex, London Stock Exchange, BATS, ChiX Europe,
TradeWeb, ICAP, and Chicago's Board of Trade, provide virtual
marketplaces comprising an information technology infrastructure for
buyers and sellers to bid for and sell financial instruments. A trader
submits a bid to the electronic trading platform via an electronic
terminal such as the computer 671 having a graphical user interface as
described further herein. The electronic trading platform 610 maintains
databases 612 and 614 of real time ask and bid information and also
transmits realtime asking and bidding information that reflects pricing
information of a financial instrument via the communication network 640.
While a single user computer 671 with a graphical user interface is shown
for ease of illustration, it will be recognized that an electronic
trading platform such as the platform 610 typically communicates with a
large number of computer terminals of a large plurality of different
trading entities. The user system 670 also comprises a mouse 673 as well
as a touch screen 675 displaying user selectors, such as selectors 232
and 233 addressed above in connection with FIG. 5A.
[0090] The market data transmitted by the electronic trading platform may
include quotations, last trade feeds, and/or other market information.
The electronic trading platform 610 may also suitably communicate with
any kind of exchange, market data publisher, alternative trading system,
electronic communication network (ECN), dark pool, and/or the like. The
electronic trading platform 610 may comprise a data exchange 616 that may
execute a trading order. The electronic trading platform may further
comprise a matching engine 618 and a smart router 620 that suitably
operate to match, route and/or reroute any orders to one or more data
exchanges, which may be affiliated with the affiliated with the
electronic trading platform, or located at another electronic trading
platform.
[0091] Portfolio managers of all types may participate in electronic
trading of investment positions over the electronic trading platform 610.
For example, high frequency trading (HFT) participants may take advantage
of the present invention to execute preferred hedges. Other portfolio
managers may include any broker, individual investor, or other trading
entity, who enjoy data transmission capability at an electronic trading
platform.
[0092] An interaction between an electronic trading platform 610 and a
portfolio manager user system 670 may be similar to the interaction that
may occur between a portfolio manager and a graphical user interface. A
number of different portfolios may be evaluated, and strategies for
employing quantitative metrics that describe the advantages of each
portfolio must be compared. The quantitative metrics may change as
realtime updates of price information and the like are obtained from the
electronic trading system. When more than one portfolio or trade list is
considered, a decision must be made identifying a final decision on the
portfolio or trade list to use. Then, the preferred trade list must be
transmitted to the electronic trading platform to be executed.
Alternatively, once a preferred portfolio or trade list is identified, it
may be transmitted to a database 672 for storage.
[0093] As one example of how a portfolio manager may suitably develop a
portfolio or trade set, the user system 670 is used to communicate
through the communication network 640 with a portfolio optimization and
management system 650. System 650 comprises plural high speed servers
652.sub.1, 652.sub.2, . . . , 652.sub.n, a pricing database 654, a
dataset database 656, a factor risk model module 658, an optimizer module
660 and software 662 to construct portfolios and trade lists from inputs
provided by the portfolio manager. In the present invention, the software
662 operates in conjunction with a Monte Carlo pricing engine 663 to
compute an optimized portfolio or trade list that minimizes CVaR and to
solve secondorder cone (SOCP) problems utilizing a modified
RockafellarUryasev solution engine 664 and method. While various modules
and engines discussed above may be implemented in software operating on a
processor or server, it will be recognized that they may be implemented
as a combination of software and hardware or principally as hardware,
such as an array of field programmable arrays (FPGAs) or application
specific integrated circuits (ASICs).
[0094] FIG. 7 shows a block diagram of a computer system 100 which may be
suitably employed as one implementation of the user system 670 of FIG. 6
and used to implement aspects of the present invention. System 100 is
implemented as a desktop computer or a mobile computing 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 that
connects to a server, database, or an electronic trading system 28 like
electronic trading platform 610, as well as other user computers through
an Internet, local area network (LAN) or wireless connection 26. The
server, database, or electronic trading system 28 or LAN 26 may also
connect to a portfolio optimization and management system, such as system
650 of FIG. 6, or a database that stores and manages investment
portfolios. In this embodiment, both the computer or mobile device 12 and
server, database, or electronic trading system 28 run software that when
executed enables the user to input instructions, user indications, and
calculations in accordance with the present invention as described
further herein to be performed by the computer or mobile device 12, send
the input for conversion to output at the server, database, or electronic
trading system 28, and then display the output on a graphical user
interface 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, database, or
electronic trading system 28 are not needed.
[0095] As shown in FIG. 7 and described in further detail below, the
system 100 includes software that is run by the central processing unit
of the computer or mobile computing device 12. In one embodiment, system
100 employed as the user system 670 communicates with an optimization and
management system, such as system 650, to license and download
application software to perform the processes and analyses described
further below. A file transfer protocol (FTP) high speed download
transfer site is established and the software is downloaded from the
system 650. This software customizes the system 100 and transforms the
system into a special purpose computer providing unique functionality as
addressed further herein. The computer or mobile device 12 may suitably
include a number of input and output devices, including a keyboard 14, a
mouse 16, CDROM/CDRW/DVD drive 18, disk drive or solid state drive 20,
monitor 22 which may be a touchscreen, and printer 24.
[0096] The mouse 16 and keyboard 14 can be used to provide user
indications to be displayed on and selectors to be acted upon utilizing
the graphical user interface display 22 and monitored by the computer or
mobile device 12 as addressed further below in connection with the
discussion of FIG. 17. For mobile devices and other suitable devices, the
user indications or selectors may be input using a touchscreen display.
In addition, the server, database, or electronic trading system 28 or LAN
26 or electronic trading system or portfolio database may also monitor
the interaction with the graphical user interface 22, respond to user
indications from the mouse 16 or keyboard 14, touchscreen, and so on.
[0097] The computer or mobile device 12 may also have a USB connector 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 so long as the transformative aspects of the
present invention are employed therein. For example, the system 100 may
be implemented in a network configuration with individual workstations
connected to a server as discussed further above in connection with FIG.
6. 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 ereaders such as an IPad.TM., IPhone.TM.,
IPod.TM., Blackberry.TM., Treo.TM., or the like.
[0098] One embodiment of the invention has been designed for use on a
standalone personal computer running Windows 7. Another embodiment of
the invention has been designed to run on a Linuxbased 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.
[0099] 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.
[0100] As illustrated in FIG. 7, and as described in greater detail below,
the inputs 30 may suitably include a data set of investable assets from
which a portfolio of investments or a hedge is to be constructed; a set
of requirements the portfolio of investments must meet; a model for
generating simulated returns of all potential investments; one or more
confidence levels; one or more investment portfolios or hedges; as well
as one more user indications for a graphing order or graph type to be
used in the graphical user interface or the preferred portfolio.
[0101] As further illustrated in FIG. 7, and as described in greater
detail below, the system outputs 32 may suitably include a set of graphs
displaying the portfolio return distributions in a specified order, and a
preferred portfolio, hedge, or trade list.
[0102] The output information may appear on the graphical user interface
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
electronic trading platform. The output information may also be
electronically sent to an intermediary for interpretation. Other devices
and techniques may be used to provide outputs, as desired.
[0103] Next, a CVaR based framework in accordance with the present
invention is described for use in analyzing and improving portfolios.
There are two main parts to this framework: (a), a MonteCarlo pricing
engine 663 shown in FIG. 6 that generates return scenarios for all the
potential multiasset class assets that may be included in the portfolio
or the set of trades to be determined; and, (b), a modified and improved
calculation engine 664 and method based on a modified version of the
existing RockafellarUryasev mathematical formulation to determine a
portfolio, overlay, or trade list that minimizes CVaR at a sequence of
confidence limits, as described further herein.
[0104] A presently preferred MonteCarlo framework that is used to
generate the instrument return scenarios or simulations is first
described. General information on MonteCarlo techniques can be found in
P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer,
2003, (Glasserman), which is incorporated by reference herein in its
entirety. The MonteCarlo pricing engine generates a matrix of asset
return scenarios. If there are N assets and S MonteCarlo simulations or
scenarios, then the matrix of asset return scenarios will be an N by S
matrix. The pricing engine will usually model each individual asset as
being driven by a set of multivariate Gaussian market factors. These
factors may include traditional equity factors such as industries,
countries, currencies, and style factors such as size, value, growth, and
momentum. Details of these equity factors are described and utilized in
Axioma's suite of equity risk models. In addition, this set of market
factors may include option pricing factors such as implied volatility and
risk free rates such as LIBOR. The profit and loss (P/L) of each asset in
each scenario is computed by numerically simulating the driving factors
appropriately scaled over the time horizon and then computing the
aggregate profit or loss of each asset according to the valuation
characteristics of the asset. The valuation may use market prices or
notional prices, depending on the kind of asset involved. This process
produces a matrix of asset profit and loss (P/L) scenarios which is then
converted to a matrix of asset returns by normalizing each asset's profit
and loss by its initial market or notional value. Because of the large
amount of data and the complexity of the calculations, the Monte Carlo
pricing engine is preferably embodied in a network system of high speed
servers which communicate with large databases of pricing data and models
and one or more electronic trading platforms as shown in FIG. 6, for
example, to achieve the desired speed of operation and to utilize the
most up to date data as desired in the expected context of the present
invention.
[0105] The portfolio manager may employ flexibility and discretion in
choosing the pricing factors he or she deems best. For example, consider
a portfolio that consists of all the assets in the S&P 500 Index and put
options on each of the individual assets. Pricing all the puts requires
the prices of all the S&P 500 underlying constituents as pricing factors.
One can think of each of the equity prices as pricing factors. This
approach is referred to as the granular risk resolution approach.
Alternatively, a fundamental factor model, such as one provided by
Axioma, can be used to map the individual equity prices to a set of
fundamental factors (industry and styles factors such as value, momentum,
etc.). For example, in the Axioma U.S. fundamental factor risk model,
there are 78 fundamental factors that represent a parsimonious set of
pricing factors. The equity prices are mapped to the fundamental factors
via their exposures in the fundamental risk model. Alternatively,
Axioma's U.S. statistical factor risk model may be employed to map equity
prices to statistical factors via their exposures in the statistical risk
model.
[0106] For most factors, including factors such as equity prices and
volatility, the factor return is defined to be the relative change in the
factor value. For fixedincome factors such as interest rates and credit
spreads, the factor return is defined to be the absolute change in the
factor value. The factor return distribution can also be parameterized by
a student t distribution that has longer tails than the Gaussian
distribution. Copulas can also be used to parameterize the joint
distribution of the factor returns where the marginal distributions
follow Gaussian or student t distributions. See McNeil and/or Glasserman,
for example.
[0107] The asset scenarios are incorporated into an improved methodology
based in part on an improvement upon the RockafellarUryasev methodology.
In the RockafellarUryasev formulation, w=(w_1, w_2, . . . , w_N)
represents a vector of the market or notional value dollar holdings in
the N different assets in the portfolio. It is desired to find a solution
for each element of w that minimizes CVaR at confidence level .epsilon.
among all possible w that satisfy the constraints representing the
portfolio manager's preferences and institutional mandates. The
confidence level is a probability between zero and one hundred percent.
Typically, it takes values close to 100%. For example, .epsilon. could be
95% or 90%. Alternatively, instead of minimizing the CVaR of the
portfolio, the same problem can be solved by imposing a maximum upper
bound on the CVaR of the portfolio.
[0108] RockafellarUryasev showed that the portfolio w that minimizes CVaR
subject to linear constraints on the portfolio can be formulated as a
linear programming problem. However, the solution to the linear
programming approach is, in practice, sensitive to the estimation error
in the scenarios.
[0109] In the improvement disclosed here, this approach is extended and
improved upon by assuming that the scenarios are not point estimates but
rather that the ith scenario lies in an ellipsoidal uncertainty set.
Technically, this ellipsoidal uncertainty set is defined in terms of an
assetasset covariance matrix Q and a constant .kappa.. If m_i is the
mean return vector for the N assets in the ith scenario, and r_i is the
simulated return vector for the N assets in the ith scenario, then the
uncertainty set for r_i is defined by
(r_im_i).sup.T Q.sup.1(r_im_i)<.kappa..sup.2 (1)
To immunize the optimal solution against estimation errors, a robust CVaR
optimization problem is solved that finds the w among all w that satisfy
the general constraints on the portfolio holdings and where r_i belongs
to the ellipsoid set described by equation (1) that minimizes CVaR at
confidence .epsilon.. This problem can be formulated as the solution of
the scenario based secondorder cone (SOCP) problem:
min w , .alpha. , u .alpha. + .kappa. w T Qw +
1 S ( 1  ) i = 1 S u_i ( 2 )
##EQU00001##
subject to the three sets of constraints. The first set of constraints
relates w, .alpha. and u_i:
m_i.sup.T w+.alpha.+u_i.gtoreq.0, i=1, . . . , S (3)
The second set of constraint makes the auxiliary variables u_i all
positive:
u_i.gtoreq.0, i=1, . . . , S (4)
The third set of constraints represents the traditional constraints on
the portfolio. These can include limits on the maximum asset holdings,
the maximum risk or tracking error of the portfolio, and the like. In
this problem, S is the number of scenarios or simulations; r_i denotes
the ith column of the assetscenario return matrix; .alpha. is a dummy
variable that provides an estimate of the VaR of the portfolio, and is
determined as part of the solution process. The auxiliary variables u_i
are determined as part of the solution, and their number is the same as
the number of scenarios, S. The auxiliary variable u_i measures the
excess loss in the ith scenario over VaR. This variable is zero if the
loss in the ith scenario is less than VaR, implying that only scenarios
with positive u_i actually contribute to the CVaR of the portfolio. Q is
the assetasset covariance risk model in linear form. Hence, {square
root over (w.sup.T Qw)} represents the standard deviation risk of the
portfolio, and .kappa. is a weighting factor to be chosen by the
portfolio manager. Note that the addition of this risk term is not
described in RockafellarUryasev, and it advantageously converts
RockafellarUryasev's linear programming problem into an SOCP problem.
The calculation of .alpha. is advantageous since it provides an estimate
of VaR as part of the solution.
[0110] Axioma has been commercially selling portfolio and trade list
construction software for over 15 years that efficiently solves SOCP
problems including problems that include elliptical uncertainty regions
such as the minimum CVaR optimization problem just described. The
optimization engine of this product includes a SOCP solver, and the
software has been designed to easily handle a wide range of commonly
occurring portfolio and trade list constraints that might be imposed.
This previously existing software has been adapted by the present
invention to compute the results presented herein.
[0111] Alternatively, specialized firstorder and decomposition approaches
are available to solve this SOCP approximately and quickly when the
number of samples is large. See, for instance, G. Iyengar and A. K. C.
Ma, "Fast Gradient Descent Rule for MeanCVaR Optimization," Annals of
Operations Research, 205(2013), pp. 203212, and A. KunziBay and J.
Mayer, "Computational Aspects of Minimizing Conditional ValueatRisk,"
Computational Management Science, 3(2006), pp. 327, both of which are
incorporated by reference herein in their entirety.
[0112] When solving this portfolio construction problem, more scenarios
are needed when the confidence level .epsilon. increases towards 100% and
techniques such as importance sampling (Glasserman) may be used to
selectively generate more scenarios in the tail of the distribution.
[0113] In general, it is desirable to test the number of scenarios used in
the optimization for both insample stability and outofsample
stability. For insample stability, one tests if the optimal CVaR values
exhibit small enough variation across scenarios of the same size. For
outofsample stability, one tests if the sequence of optimized
portfolios constructed using scenarios of the same size exhibit small
variance in CVaR when computed on a much larger master set of scenarios.
[0114] One test sequence for testing both insample and outofsample
stability is the following. A master set is defined using 50,000
scenarios. Next, 1,000 subsamples of a fixed size are created from this
master set. For example, the fixed size may contain 1,000 or 5,000 or
10,000 of the master scenarios, each randomly chosen. For each
subsample, the optimized portfolio that minimizes the CVaR is computed.
[0115] Insample stability can be assessed by comparing the statistics for
each of these different portfolios. This comparison is shown by table 244
in FIG. 8. Here, the mean CVaR and standard deviation of CVaR are
computed for subsamples of size 1,000, 5,000, and 10,000. As can be
seen, the metrics are all quite similar, especially for the 5,000 and
10,000 subsample sizes. This comparison establishes insample stability
for the 5,000 and 10,000 subsample sizes.
[0116] Outofsample stability is assessed by comparing the statistics on
each of the optimal portfolios using the full master set of 50,000
scenarios. Table 246 from FIG. 8 shows these metrics. Once again, there
is little variation for 1,000, 5,000, and 10,000 subsample sizes. Hence,
we can be confident that we have used sufficient sample sizes to obtain
accurate results.
[0117] One of the important advantages of the present invention is derived
from the recognition that, as a practical matter, when minimizing CVaR
for investment portfolios, the objective function does not have a steep
gradient. As a result, even though there may be a unique portfolio that
minimizes CVaR over all possible portfolios satisfying the constraints
imposed, there may be different portfolios that have advantageous
properties. One of the disadvantages of traditional CVaR minimization is
that the answer obtained is only optimal for the confidence level e
prescribed. For different confidence levels, the portfolio obtained may
be different. In other words, minimizing CVaR at the 95% confidence level
only minimizes the average of the 5% worst losses in the return
distribution. Minimizing only the CVaR value at the 95% confidence level
may give a portfolio that has an undesirably high CVaR at, say, the 90%
confidence level. Since the CVaR objective function is not steep, it may
be that nearoptimal solutions may do a better job of minimizing CVaR at
several confidence levels at once.
[0118] One of the novel and important parts of the invention disclosed
herein is the concept of hierarchical CVaR. Hierarchical CVaR is a method
for constructing a portfolio or trade list that simultaneously minimizes
or nearly minimizes the CVaR of the portfolio or trade list at more than
one confidence level. Such portfolios have many advantageous properties
as discussed further herein.
[0119] One general procedure for utilizing hierarchical CVaR is the
following. Using a graphical user interface and a selector within the
graphical user interface, a portfolio manager chooses a set of two or
more confidence levels, .epsilon..sub.1, . . . .epsilon..sub.Z, where the
confidence levels are listed in descending order, and where Z, the number
of confidence levels, is two or larger.
[0120] For example, the portfolio manager may specify two confidence
levels, .epsilon..sub.1=95% and .epsilon..sub.2=90%. Then, a sequence of
portfolios are constructed. In the first portfolio construction, a
portfolio is constructed that minimizes CVaR at confidence
.epsilon..sub.1 for all the constraints imposed on the portfolio. The
final numerical value of CVaR obtained from this optimization is saved as
CVaR.sub.1.
[0121] Next, a second portfolio construction problem is solved in which
the CVaR is minimized at confidence .epsilon..sub.2 subject to the
original portfolio constraints plus the additional constraint that the
CVaR at confidence .epsilon..sub.1 is less than CVaR.sub.1 times one plus
.DELTA.. That is, for this second portfolio, the .epsilon..sub.2 CVaR is
minimized and the .epsilon..sub.1 CVaR is no worse than CVaR.sub.1 times
(1+.DELTA.). Typically, .DELTA. is on the order of 5%.
[0122] This procedure can be continued for multiple confidence levels,
with each new solution ensuring that the CVaR for each of the previously
obtained confidence levels is no worse than a factor of one plus .DELTA.
from the globally optimal CVaR at the confidence level. As a practical
matter, the scenarios used for each confidence level should be
independent, so for Z confidence levels, a total of Z times S total
scenarios are needed.
[0123] As will be illustrated below, the return distributions obtained for
portfolios and trade lists constructed using hierarchical CVaR are highly
advantageous. Hierarchical CVaR allows the user to adjust the left tail
of the return distribution in a direct and advantageous manner. In fact,
by integrating the user specification of the confidence levels with the
display of the resulting return distribution within a graphical user
interface, an advantageous, powerful portfolio construction tool is
obtained.
[0124] Next, the advantages of portfolio and trade list construction using
CVaR and then hierarchical CVaR are illustrated using several
representative examples.
[0125] In the first example, an equity portfolio is hedged with equity
index options. Assume that an investment portfolio of longonly equities
is owned as of Dec. 19, 2014. The equity assets may include any asset in
the S&P 500 index. It is desired to hedge the risk of this portfolio over
a three month horizon with European index puts on the S&P 500 Index with
different strikes and three months to expiration. The universe or set of
investments held comprises all equities in the S&P 500 index as of Dec.
19, 2014, and the universe or set of potential hedging investments is the
set of puts on the S&P 500.
[0126] The risk of the equity assets is modeled using Axioma's WorldWide,
Fundamental Factor, Equity Risk Model.TM., which includes style, country,
industry, market, and currency factors.
[0127] For this first example, three different portfolio construction
approaches are considered in order to illustrate the differences obtained
by each approach.
[0128] In the first portfolio construction approach, the CVaR of the
combined portfolio comprising the original equities held plus the hedging
puts is minimized at the 95% confidence level over a three month hedging
horizon. A budget is defined for the investment in the puts, which
defines an upper limit to the cash value that can be invested in the
puts. This budget is specified as a percentage of the value of the
original equity portfolio value. Budget values of 1% and 5% of the equity
investment value are employed. These two values represent a small budget
and a large budget. Note that for this first example, hierarchical CVaR
is not employed. CVaR is simply minimized with one confidence level in
order to highlight its advantages over other portfolio construction
approaches.
[0129] In the second portfolio construction approach, termed the "MVO
deltarhovega" approach, the risk derived from the puts is modeled by
linearizing their return characteristics about their current prices and
the standard deviation of the risk derived from this model is also
minimized. The three parameters in the name of this approachdelta, rho,
and vegaare parameters in the BlackScholes pricing model, which is
described in J. C. Hull, Options, Futures, And Other Derivatives, 7th
edition, Prentice Hall, 2008, (Hull) which is incorporated herein in its
entirety. For the short, three month hedging horizon of this example, the
option delta dominates the risk obtained in this approach, and the risk
modeling mimics the delta hedging approach commonly used by traders. Note
also that since the longonly equity portfolio is hedged with several
possible index puts, there is no unique global optimal solution. That is,
a solution can be found using any individual put in isolation.
[0130] In the third portfolio construction approach, termed the "MVO
deltagamma" approach, the risk model of the MVO deltarhovega approach
is employed by adding the risk associated with option gamma (Hull).
Unlike MVO deltarhovega approach, the MVO deltagamma approach has a
unique solution because it selects the portfolio that minimizes both the
delta risk and the portfolio gamma.
[0131] The CVaR, MVO deltarhovega, and MVO deltagamma approaches are
first compared when the optimizer 650 is only allowed to purchase options
with a budget of 1%. 50,000 scenarios are utilized to obtain the results
shown in FIG. 9 which includes two charts 250 and 260 representing the
same four return distributions. The return distribution for the original
portfolio is plotted with the darkest color and is labelled "No Hedge".
This distribution is return distribution 258 in chart 250 and return
distribution 268 in chart 260. The return distribution for the
MVOdeltagamma hedged portfolio is plotted with the next darkest color
and is labelled "DeltaGamma". This is return distribution 256 in chart
250 and return distribution 266 in chart 260. The return distribution for
the MVOdeltarhovega hedged portfolio is plotted with the next darkest
color and is labelled "DeltaRhoVega". This is return distribution 254
in chart 250 and return distribution 264 in chart 260. Finally, the
return distribution for the CVaR hedged portfolio at 95% confidence is
plotted with the lightest color and is labelled "CVaR". This is return
distribution 252 in chart 250 and return distribution 262 in chart 260.
The charts 250 and 260 may be displayed on a display, such as display 22
of FIG. 7, with the tools described in conjunction with the discussion of
FIG. 17.
[0132] A number of features are evident in chart 250. First, and foremost,
when filling the bars on the return distribution, many salient aspects of
the different return distributions are hidden. In particular, the left
tail of the no hedge distribution 258 completely obscures the left tails
of the three hedges 252, 254, and 256, so it is impossible from this
chart to determine what the left tail looks like for the three hedges.
This obscuration is a significant disadvantage.
[0133] For charts such as chart 250, it would be highly advantageous to be
able to alter the order in which the return distributions are graphed
interactively so that a better comparison can be made. It would also be
advantageous to switch back and forth from the representation in chart
250 to the representation in chart 260, as discussed further below in
conjunction with FIGS. 17 and 18.
[0134] In chart 260, each shape of each of the four return distributions
is visible. Notice that the left tail of the three hedges are similar for
returns less than about 12%, and that all three of the hedges are
effective at reducing the likelihood of a large, negative return when
compared to the unhedged original portfolio. Also, notice that the right
tails of the three hedges are virtually indistinguishable for returns
greater than zero. Chart 260 indicates that the principal difference
between the three hedges is the shape of the return distribution between
approximately 12% and 0%. These differences are clearly visible in chart
260. The ability to effectively and interactively construct portfolios
that advantageously shape the return distributions in a region, such as
this narrow region of returns and effectively display them are important
aspects of the present invention.
[0135] Next, the CVaR, MVO deltarhovega, and MVO deltagamma approaches
are compared when the optimizer 650 is only allowed to purchase options
with a larger budget of 5%. The larger budget gives flexibility to the
portfolio construction problem, and it is anticipated that the
differences between the three different hedges will be more pronounced.
As before, 50,000 scenarios are used to obtain the results. The results
are shown in FIG. 10 which includes two charts 272 and 282, representing
the same four return distributions. The return distribution for the
original portfolio is plotted with the darkest color and is labelled "No
Hedge". This is return distribution 280 in chart 272 and return
distribution 290 shown as a dotted line in chart 282. The return
distribution for the MVOdeltagamma hedged portfolio is plotted with the
next darkest color and is labelled "DeltaGamma". This is return
distribution 278 in chart 272 and return distribution 288 shown as a
dashed line in chart 282. The return distribution for the
MVOdeltarhovega hedged portfolio is plotted with the next darkest
color and is labelled "DeltaRhoVega". This is return distribution 276
in chart 272 and return distribution 286 in chart 282. Finally, the
return distribution for the CVaR hedged portfolio at 95% confidence is
plotted with the lightest color and is labelled "CVaR". This is return
distribution 274 in chart 272 and return distribution 284 in chart 282.
[0136] A number of features are evident in chart 272. First, and foremost,
when filling the bars on the return distribution, many salient aspects of
the different return distributions are hidden. In particular, the left
tail of the no hedge distribution 280 completely obscures the left tails
of the three hedges 274, 276, and 278, so it is impossible from this
chart to determine what the left tail looks like for the three hedges.
[0137] As with chart 250, it would be advantageous in chart 272 to be able
to alter the order in which the return distributions are graphed
interactively so that a better comparison can be made. It would also be
advantageous to be able to switch back and forth between the display in
chart 272 and that in chart 282, as discussed further below in
conjunction with FIGS. 17 and 18.
[0138] In chart 282, the shape of each of the four return distributions is
visible. With the larger 5% budget, the three hedges are less similar
than they were for the 1% budget case. The CVaR hedged portfolio return
284 has the best downside risk in that it is extremely unlikely that a
return below 8% will occur. The deltagamma return distribution has the
next best downside risk in that it is unlikely that a return less than
12% will occur. Finally, the deltarhovega distribution has the next
best downside risk in that it rarely has returns less than 16%. All
three of these hedged portfolios possess improved downside risk as
compared against the return distribution of the, no hedge, original
portfolio 290.
[0139] FIG. 10 also includes a user supplied indication line 1002
indicating a return value (6.8%) at which the user may want to carefully
control the display order of the distributions in 282. As will be
explained further herein, the ability to place the indicator 1702 at
different places within the return distribution and advantageously
display the return distribution in a specific order for that location is
an advantageous aspect of the present invention as discussed further
below in connection with FIGS. 17 and 18 below.
[0140] Next, the same two examples are takenhedging a portfolio with S&P
500 puts with budgets of 1% and 5%and the CVaR solution is compared
with a hierarchical CVaR solution. For the hierarchical CVaR solution,
CVaR is minimized at the 95%, 90%, 85%, and 80% confidence levels, in
that order.
[0141] Charts 294 and 302 in FIG. 11 show the distribution of returns for
the CVaR and hierarchical CVaR solutions with a budget of 1%. In chart
294, the CVaR distribution 296 is shown with the light gray bars 296
while the hierarchical CVaR distribution 300 is shown with the dark gray
bars 300. In chart 302, the CVaR solution is shown with the light gray
bars 304 while the hierarchical CVaR distribution is shown with the dark
gray bars 306.
[0142] In chart 294, as before, it is seen that with return distributions
plotted on top of each other, the details of the CVaR distribution 296 in
the left tail are hidden by the hierarchical CVaR distribution 300. This
display is disadvantageous. Improved display techniques are addressed
further below in connection with FIGS. 17 and 18.
[0143] In chart 302, the return distribution of both solutions, CVaR 304
and hierarchical CVaR 306, shown as a dotted line, are visible. The
hierarchical CVaR approach 306 shifts the return distribution to the
right when compared with traditional CVaR. The right tails of both
distributions are essentially indistinguishable, indicating that there is
little difference in the upside risk of both approaches. The CVaR value
at the 95% confidence level for the hierarchical CVaR approach 306 is
7.7%, as compared with CVaR value for the CVaR approach 304 which is
7.5%. That is, CVaR in the hierarchical CVaR approach is substantially
the same as it is in the CVaR approach 304.
[0144] Notice as well that the differences in the two return distributions
304 and 306 fall primarily in the range of returns between 10% and 0%.
Utilizing the tools addressed in FIGS. 17 and 18 below, portions of the
display where differences exceed a predetermined percentage can be
identified and highlighted. It is a range, such as the 10% and 0%, for
this example, that a portfolio manager has the most discretion to alter,
engineer, and improve the return distribution to achieve his or her
objectives.
[0145] Charts 308 and 314 in FIG. 12 show the distribution of returns for
the CVaR and hierarchical CVaR solutions with a budget of 5%. In chart
308, the CVaR solution is shown with the light gray bars 310 while the
hierarchical CVaR solution is shown with the dark gray bars 312. In chart
314, the CVaR solution is shown with the light gray line 316 while the
hierarchical CVaR solution is shown with the dark gray line 318.
[0146] In chart 308, as before, it is seen that with return distributions
plotted on top of each other, the details of the CVaR distribution 310 in
the left tail are hidden by the hierarchical CVaR distribution 312.
Again, this display is disadvantageous. Improved display techniques are
described below in conjunction with FIGS. 17 and 18.
[0147] In chart 314, the CVaR 316 and hierarchical CVaR 318, distributions
are both visible. The hierarchical CVaR approach 318 shifts the return
distribution to the right when compared with traditional CVaR. The right
tails of both distributions are essentially indistinguishable, indicating
that there is little difference in the upside risk of both approaches.
[0148] To give some a perspective on the solutions described in detail in
FIGS. 9 through 12, the S&P 500 was trading at $2,070.65 on Dec. 19,
2014. When the budget is 1%, the CVaR approach spends its entire budget
purchasing several put contracts with strikes of 2000. The hierarchical
CVaR approach uses its entire budget to purchase put contracts with
strikes of 2060. These latter contracts are more expensive but offer
better downside risk protection. The MVO deltarhovega approach uses its
entire budget to purchase cheaper put contracts with strikes of 1925. The
MVO deltagamma approach uses part of its budget to purchase cheaper puts
with strikes of 1925 and the rest in purchasing expensive puts with
strikes of 2075.
[0149] When the budget is 5%, the CVaR and the hierarchal CVaR approaches
both purchase the expensive put contracts with strikes of 2075. In the 5%
budget case, neither of the approaches uses its entire budgetthe
hierarchal CVaR approach spends a little more of its budget than the CVaR
approach. The MVO deltarhovega approach continues to purchase the less
expensive put contracts with strikes of 1925 and does not use up its
entire budget. The MVO deltagamma approach is the only approach that
uses up its entire budget. It spends about 60% of the budget in
purchasing the less expensive put contracts with strikes of 1925 and the
rest of the budget in purchasing the most expensive contracts with
strikes of 2075.
[0150] FIG. 13 shows a table 330 that reports the statistics associated
with each of the five return distributions addressed above: the unhedged
portfolio, the deltagamma hedge, the deltarhovega hedge, the CVaR
hedge, and the hierarchical CVaR hedge. The statistics reported include
CVaR (the expected loss at the 95% confidence level), the mean return,
the standard deviation of returns, MVO (which is the objective function
of the MVO deltarhovega model), the worst return, the best return, VaR,
and right CVaR (the expected gain at the 95% confidence level). The table
330 may be displayed on a display, such as display 22 of FIG. 7 in
conjunction with a chart like FIG. 9 adapted as discussed in connection
with FIGS. 17 and 18 to allow a user to readily customize the display to
more clearly and easily focus on the differences.
[0151] When the budget is small, as it is in this example, the four hedges
give similar results for these statistics. Of course, the portfolio with
the smallest CVaR is the CVaR portfolio, with a CVaR of 10.06%. The
hierarchical CVaR is only slightly larger at 10.43%, and the CVaR of the
other two hedges are similar as well at 10.50% and 10.26%, respectively,
and both are substantially less than the value for the unhedged portfolio
at 13.99%. The lowest MVO value is obtained for the MVOdeltarhovega
approach, but, again, the values for the four hedges are relatively close
and all are substantially better, in other words, less risky, than the
value for the unhedged portfolio.
[0152] FIG. 14 shows a table 332 that reports risk statistics associated
with each of the five return distributions: the unhedged portfolio, the
deltagamma hedge, the deltarhovega hedge, the CVaR hedge, and the
hierarchical CVaR hedge. The statistics for the unhedged portfolio are
the same as in table 330. The CVaR hedged portfolio return 284 and the
hierarchical CVaR portfolio return 318 have the best downside risk
statistics. More particularly, the worst returns, and standard deviations
for CVaR and hierarchal CVaR are smaller than those of the MVO
deltarhovega 286 and MVO deltagamma 288, respectively. Moreover, the
right CVaRs for the CVaR and hierarchical CVaR portfolios are very close
and slightly smaller than the right CVaR for the MVO deltarhovega and
MVO deltagamma portfolios, showing that the CVaR hedged portfolio and
the hierarchical CVaR portfolio have the largest skew among these
portfolios. In other words, the CVaR and hierarchical CVaR approaches
reduce the portfolio downside without taking much away from the upside.
This reduced downside with slightly reduced upside is one of the
advantages of the CVaR and hierarchical CVaR approaches when compared
with the traditional MVO approaches.
[0153] Although, the MVO deltarhovega objective values for the MVO
deltarhovega and MVO deltagamma portfolios are much smaller than the
corresponding values for the CVaR hedged and hierarchical CVaR
portfolios, the other distribution statistics are worse. That the MVO
portfolios are better for the values which they minimize is not
surprising. The poor performance of the MVO approaches can be attributed
to the linear model for the puts, which does not capture the asymmetric
payouts of these instruments.
[0154] FIG. 15 presents an illustrative graphical user interface or
display that may be employed to take advantage of various aspects of the
present invention. The graphical user interface is comprised of upper and
lower portions 346 and 334, respectively. In the lower portion 334, an
interactive table and user indicators are used to interactively construct
portfolios. In portion 334, there is a main design table 336 in which
different portfolios or hedges are specified. In the illustrative upper
portion 334, there are four portfolios or hedges listed, numbered 1, 2,
3, and 4. These four portfolios or hedges may be selected from a drop
down menu on a display screen such as display screen 226 of FIG. 5A or
display screen 234 of FIG. 5B modified to include a drop down menu 233 or
235, respectively. A second column in the table includes checkboxes which
can turn portfolios from active to inactive portfolios. Active portfolios
are analyzed and displayed. Inactive portfolios are not. In this
instance, the design parameters include the budget size, the CVaR
tolerance to use for hierarchical CVaR optimizations, and a series of up
to four confidence limits. Each portfolio has an "Export" user indicator
338 which, when indicated, will export the portfolio to either a database
or electronic trading system. Finally, the user may list the plotting
order in the boxes 340. When the return distribution is displayed, the
return distributions will be plotted in the order indicated in box 340.
[0155] The interactive hedge design environment includes an "ADD" button
342. Depressing or activating this button adds another row to the table
so that additional portfolios may be designed. Finally, there is an
"UPDATE' button 344. Depressing or activating this button causes the
return distribution to be updated and redisplayed, in the order listed in
box 340.
[0156] The upper portion of this user interface shows the three active
portfolio return distributions in chart 346. In other embodiments, a
table of statistics may also be included as part of this display.
[0157] The purpose of the multiasset class hedging environment shown in
FIG. 15 is to show that portfolio managers may interactively craft
different hedges and investment solutions. By adjusting the various
design parameters in table 334, a portfolio manager may obtain and select
a preferred hedging portfolio, and then export it either to a database or
electronic trading system.
[0158] The flow diagram 350 shown in FIG. 16 illustrates how the present
invention may suitably be utilized within an interactive graphical user
interface. In a first step 352, the portfolio manager or other user
provides hedging inputs to the system. These may include hedge
construction parameters for one or more portfolios such as confidence
limits, budgets, and tolerances. These inputs would be similar to those
shown in table 334. In addition to the hedging parameters, the portfolio
manager or other user could input directions for how the return
distributions would be graphed such as a style of graphing or an order in
which to draw the graphs. As an alternative to manual input of graphing
order, an automatic approach as illustrated in FIGS. 17 and 18 may be
employed.
[0159] In a second step 354, the graphical user interface operates to
ensure that all the portfolio manager's inputs were correctly input into
the system, compute all the necessary calculations for determining the
simulated return scenarios, return distributions, and the hedge or
portfolios which optimally satisfied the constraints imposed on it by the
portfolio manager. Once these computations are performed, the graphical
user interface and its related software and hardware then coordinate the
display of the results. This display may suitably include graphical
representations of the return distributions for each portfolio, displayed
in the format and order requested, as well as other relevant statistics
used to evaluate the results. These statistics could include CVaR and VaR
for each confidence limit analyzed, the mean return, the standard
deviation of returns, the best and worst returns, and the right CVaR. In
other words, the statistics would be similar to charts 330 and 332.
[0160] In a third step 356, the portfolio manager would interactively make
decisions. One decision would be to alter the inputs and recompute the
results. In this case, the flow chart would return to the first step,
352, and repeat the sequence of steps 352, 354 and 356. Alternatively,
the portfolio manager may decide that one or more hedges or portfolios
were final. Note that in some cases, the decision on what portfolio or
hedge is final would be automated and be determined automatically. In
other cases, the portfolio manager may manually make that decision and
interact with the graphical user interface accordingly.
[0161] In a fourth step 358, when hedges or portfolios are deemed final,
these portfolios are electronically output to files, printers, other
graphical user interface windows, other software analysis packages,
databases, or electronic trading systems for execution.
[0162] FIG. 17 shows an exploded view of portions of the distributions
284, 286, 288 and 290 of FIG. 10 located within a window 1700 surrounding
a line 1702 like the line 1002 shown in FIG. 10, but moved slightly to
the right. A cursor 1710 is shown hovering in FIG. 17. The x coordinate,
5.8%, of this hovering is detected as addressed further below and line
1702 is generated and displayed on a display, such as the display 22 of
FIG. 7. The window 1700 has a predetermined width, such as 2%, as shown,
and is also preferably displayed. Intersection points, A, B, C and D of
line 1702 and distributions 288, 286, 290 and 284, respectively, are also
shown. The y coordinates of each of these distributions at the x
coordinate, 5.8%, are also determined. Knowing the y coordinates or the
portions of the curves 288, 286, 290 and 284 of interest, these curves
can be automatically displayed in the order shown so that overlapping
does not obscure the data illustrated in the window 1700 of interest. It
will be recognized that in another window of interest, such as one around
an xcoordinate of 3%, for example, a different ordering of
distributions 284, 288, 286 and 290 will be advantageously, automatically
generated to prevent obscuring the relevant data in the window of
interest. Alternatively, as noted above, a user may manually select an
order of display or the user may check on a particular distribution
bumping it to the top or showing it in isolation as desired by the user.
[0163] As such, the present invention provides a useful tool for
displaying multiple distributions simultaneously while allowing a user to
readily focus on portions of the distribution of most interest. As noted
above, once an area of interest is located by the user, tables of
statistical data can be readily generated and displayed to further flesh
out pros and cons of one distribution vis a vis a second, or a third or
so on.
[0164] FIG. 18 shows a programmed microprocessor 1800 which receives
location data from a mouse device 1810. This location data is utilized by
microprocessor 1800 to control location of a cursor displayed on a
display 1814 by controlling a display driver 1812. Memory 1820, including
RAM 1822 and ROM 1824, stores software which is provided to
microprocessor 1800, and receives and provides data to microprocessor
1800. Clock 1830 provides clock signals to microprocessor 1800 which
utilizes such signals and software to detect if the mouse 1810 is
hovering by remaining in place for a predetermined time in which case a
line such as line 1702 of FIG. 17 is displayed on the display 1814. A
distribution database 1840 stores data for distributions, such as
distributions 284, 286, 288 and 290 of FIGS. 10 and 17, for example. Such
data may be obtained from an optimizer system 1860 or optimizer 650 of
FIG. 6 which the microprocessor 1800 communicates with using a
communication network 1850.
[0165] Further user input may be provided by a user utilizing the mouse
1810 to click on an icon or selector displayed on display 1814 or
utilizing a keyboard 1870. Examples of further user inputs include
multiple confidence levels, .epsilon..sub.1, .epsilon..sub.2, and
.epsilon..sub.3, such as 95%, 90% and 85%, for example, and selections of
a hedging objective function to be minimized such as CVaR, hierarchal
CVaR, deltarhovega and deltagamma, for example.
[0166] 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.
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