U.S. patent application number 12/356309 was filed with the patent office on 2009-05-21 for matched filter approach to portfolio optimization.
Invention is credited to UNNIKRISHNA SREEDHARAN PILLAI.
Application Number | 20090132433 12/356309 |
Document ID | / |
Family ID | 38862684 |
Filed Date | 2009-05-21 |
United States Patent
Application |
20090132433 |
Kind Code |
A1 |
PILLAI; UNNIKRISHNA
SREEDHARAN |
May 21, 2009 |
MATCHED FILTER APPROACH TO PORTFOLIO OPTIMIZATION
Abstract
Given a fixed amount of capital, how to invest it optimally by
distributing it among a set of stocks and securities so as to
maximize the return while minimizing the overall risk is addressed
here. Given that one has full freedom in selecting the type of
stocks, a new strategy is outlined here by maximizing the ratio of
the gain to risk--rather than minimizing the risk alone--to
determine the fraction of capital that must go to each stock. An
optimum gain versus variance plot can be used to determine the type
of stocks to be selected in addition to their relative quantity for
maximum yield over the duration of interest. By modifying the
definition of risk to include a function of the covariance matrix
of secondary stocks that are sympathetic to the primary stocks of
interest, an alternate investment strategy is also developed here.
If short selling of stocks and securities is not allowed in a
portfolio, then stock selection becomes important so as to maintain
the desired fractions to be positive. In this context, a new
iterative method that incrementally increases the diagonal loading
of the covariance matrix of the primary returns so as to achieve
positive weight factors is also developed.
Inventors: |
PILLAI; UNNIKRISHNA SREEDHARAN;
(Harrington Park, NJ) |
Correspondence
Address: |
Walter J. Tencza Jr.
Suite 210, 100 Menlo Park
Edison
NJ
08837
US
|
Family ID: |
38862684 |
Appl. No.: |
12/356309 |
Filed: |
January 20, 2009 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
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11453370 |
Jun 15, 2006 |
7502756 |
|
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12356309 |
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Current U.S.
Class: |
705/36R |
Current CPC
Class: |
G06Q 40/06 20130101;
G06Q 40/00 20130101 |
Class at
Publication: |
705/36.R |
International
Class: |
G06Q 40/00 20060101
G06Q040/00 |
Claims
1. A method for investing a given capital sum by distributing it
among a set of investments comprising determining a plurality of
weight factors, one for each investment in the set of investments
by which the capital sum will be partitioned so as to determine an
actual amount to be invested in each investment; wherein the
plurality of weight factors make up a weight factor vector; wherein
the plurality of weight factors are determined by maximizing a
ratio of an expected portfolio gain to an overall risk, wherein the
expected portfolio gain is obtained by multiplying a transpose of
the weight factor vector with a vector of mean return values, and
the overall risk is obtained by first multiplying the transpose of
the weight factor vector with a covariance matrix of a plurality of
returns corresponding to the set of investments, and multiplying
the result with the weight factor vector.
2. The method of claim 1 wherein the set of investments do not
permit short selling; and wherein each of the plurality of weight
factors is a non negative number.
3. The method of claim 2 wherein maximizing the ratio of the
expected portfolio gain to the overall risk is carried out by
minimizing a modified risk function subject to a nonnegativity
constraint on the plurality of weight factors; wherein the modified
risk function is the overall risk minus a scaled function of the
expected portfolio gain.
4. The method of claim 2 wherein maximizing the ratio of the
expected portfolio gain to the overall risk is carried out by
maximizing a modified gain function subject to a nonnegativity
constraint on the plurality of weight factors; and wherein the
modified gain function is the expected portfolio gain minus a
scaled function of the overall risk.
5. The method of claim 1 further comprising prior to distributing
the capital sum among the set of investments, selecting the set of
investments from a pool of investments by determining a plurality
of computed risks of a corresponding plurality of combinations of
investments from the pool of investments; wherein the set of
investments is one combination of the plurality of combinations of
investments; and wherein the set of investments has a computed risk
which is lowest among the plurality of combinations of
investments.
6. The method of claim 1 further comprising after the capital sum
is distributed among the set of investments, selling the set of
investments when a prior predetermined overall gain for the set of
investments is realized.
7. A method for investing a given capital sum by distributing it
among a set of investments comprising determining a plurality of
weight factors, one for each investment in the set of investments
by which the capital sum will be partitioned so as to determine an
actual amount to be invested in each investment; and wherein the
plurality of weight factors are determined by maximizing a ratio of
a square of an expected portfolio gain to an overall risk; wherein
the plurality of weight factors make up a weight factor vector;
wherein the expected portfolio gain is obtained by multiplying a
transpose of the weight factor vector with a vector of mean return
values, and the overall risk is obtained by first multiplying the
transpose of the weight factor vector with a covariance matrix of a
plurality of returns corresponding to the set of investments, and
multiplying the result with the weight factor vector.
8. The method of claim 7 wherein if short selling is not allowed,
the step of maximizing the ratio of the square of the expected
portfolio gain to the overall risk is further carried out subject
to nonnegativity constraint on the plurality of weight factors.
9. A method comprising determining a set of a plurality of return
values for each of a plurality of investments; determining a vector
of mean return values, one mean return value for each of the
plurality of investments; determining a co-variance matrix and its
inverse based on the sets of a plurality of return values, and the
vector of mean return values; determining a vector of weight
factors, one weight factor for each of the plurality of
investments, by multiplying the inverse co-variance matrix times
the vector of mean return values to obtain a vector of a plurality
of result values and normalizing each result value of the vector of
the plurality of result values by dividing each result value by the
sum of absolute values of the plurality of result values;
specifying a plurality of amounts to purchase, one amount for each
of the plurality of investments determined by multiplying a total
capital amount by a weight factor of the vector of weight factors
for each of the plurality of investments.
10. The method of claim 9 wherein the plurality of investments do
not permit short selling; and wherein each of the plurality of
weight factors is a non negative number.
11. The method of claim 9 further comprising selecting the
plurality of investments from a pool of investments by determining
a plurality of computed risks of a corresponding plurality of
combinations of investments from the pool of investments; wherein
the plurality of investments is one combination of the plurality of
combinations of investments; and wherein the plurality of
investments has a computed risk which is lowest among the plurality
of combinations of investments.
12. The method of claim 9 further comprising purchasing the
plurality of amounts for the plurality of investments; after the
plurality of amounts for the plurality of investments have been
purchased, selling the plurality of investments when a prior
predetermined overall gain for the plurality of investments is
realized.
13. A method for investing a capital sum by distributing it among a
set of investments comprising using a first numerical filter to
un-correlate the set of investments; and using a second numerical
filter to combine the un-correlated set of investments so as to
maximize an overall gain for the set of investments.
14. A method for investing a capital sum by distributing it among a
set of primary investments in a portfolio comprising determining a
plurality of weight factors, one for each investment in the set of
primary investments by which the capital sum will be partitioned so
as to determine an actual amount to be invested in each investment;
and wherein the plurality of weight factors are determined by
maximizing the ratio of the square of the expected portfolio gain
of the set of primary investments to a modified risk; wherein the
modified risk is defined as the sum of an original risk for the
portfolio and a function of a covariance matrix of a secondary set
of stocks that are sympathetic to the set of primary
investments.
15. The method of claim 14 wherein the set of primary investments
do not permit short selling; and wherein each of the plurality of
weight factors is a non negative number.
16. The method of claim 14 wherein after the capital sum is
distributed among the set of primary investments, selling the set
of primary investments when a prior predetermined overall gain for
the set of primary investments is realized.
Description
CROSS REFERENCE TO RELATED APPLICATION(S)
[0001] The present application is a divisional of and claims the
priority of U.S. patent application Ser. No. 11/453,370, titled
"MATCHED FILTER APPROACH TO PORTFOLIO OPTIMIZATION", filed on Jun.
15, 2006.
FIELD OF THE INVENTION
[0002] This invention relates to methods and apparatus for
distributing funds among a set of investments.
BACKGROUND OF THE INVENTION
[0003] Suppose one has some capital to invest in the stock market.
How does one go about investing it? One can try picking "good"
stocks at low prices and selling them later at higher prices. That
age old strategy although quite simple conceptually, is very
difficult to implement. Stocks are inherently risky since they move
up and down in a seemingly haphazard fashion on a variety of inputs
and the common wisdom says that one should not keep "all eggs in
one basket", but rather spread out the investment so as to minimize
the risk.
[0004] In the 1950s Harry Markowitz, then a graduate student at the
University of Chicago, fine-tuned this idea, laying the foundations
of the modern portfolio theory. Markowitz's idea is easy to
understand. Let us concentrate on picking stocks. The strategy is
to pick the right mix of stocks that minimizes the overall risk in
terms of losing money that is invariantly caused by the stock
values moving below their purchased prices. Stocks move up and
down, sometimes violently, causing great volatility in term of the
total portfolio value. Markowitz's basic idea was to keep this
volatility low by picking the right mix of stocks. One would like
to keep the total portfolio value fluctuations to a minimum at all
times, i.e. no big variations, and if there are any variations they
should amount to small jitters. Actual stock variations are of
course beyond one's control, but what is controllable is which
specific stocks to add to the overall portfolio from the total
pool, and how much of each stock. The idea is to use the right mix
of right stocks to minimize the overall volatility. After all the
basic goal of a fund manager is to protect the portfolios under his
management from losing their values and hopefully increase their
return values or the overall gain. The specific stock holdings and
their relative importance within the portfolio are unimportant both
to the fund manager and to the investor, so long as the portfolio
"makes money", or performs well.
[0005] Thus two quantities play a role in portfolio selection--the
overall risk, and the overall return or gain. Obviously, the
overall risk needs to be minimized, and the overall return or gain
should be maximized at the same time. Various strategies can be
designed using these conflicting goals.
[0006] For a random variable, the variance is a good measure of the
spread of the random variable around a mean value, and hence
volatility minimization for stocks or investments, can be achieved
in terms of portfolio variance minimization.
[0007] To quantify these ideas, let P represent an overall
portfolio consisting of m stocks where s.sub.i(n) represents the
i.sup.th stock price at time index n and a.sub.i>0 the weight
factor associated with the i.sup.th stock. Note that the unit of
time can be hours, days, months or years depending on the
investment duration. Clearly
a i > 0 , i = 1 m a i = 1 ( 1 ) ##EQU00001##
and the a.sub.is are unknown to start with.
[0008] If C.sub.o represents the total capital, then C.sub.oa.sub.i
represents the capital invested in the i.sup.th stock so that
C.sub.oa.sub.i/s.sub.i(0)=k.sub.i represents the actual number of
the i.sup.th stock in the portfolio. Hence the portfolio value at
time index n equals to
i = 1 m k i s i ( n ) ##EQU00002##
and hence the portfolio return over duration (0, n) equals
P = i = 1 m k i s i ( n ) - C o = C o i = 1 m a i s i ( n ) - s i (
0 ) s i ( 0 ) = C o i = 1 m a i r i ( n ) ( 2 ) ##EQU00003##
where
r i ( n ) = s i ( n ) - s i ( 0 ) s i ( 0 ) ( 3 ) ##EQU00004##
represents the i.sup.th stock return over the duration (0, n). Thus
for portfolio return analysis, the important variable is the stock
return value r.sub.i(n) rather than the actual stock value
s.sub.i(n) itself.
[0009] Let
.mu..sub.i=E{r.sub.i(n)} (4)
represent the mean value (expected value) of the i.sup.th stock
return (see, in "Probability, Random Variables and Stochastic
Processes," Fourth Edition, A. Papoulis, and S. U. Pillai,
McGraw-Hill Companies, New York, USA, 2001). The mean value
.mu..sub.i can also be a good indicator about the future trend,
where one hopes the stock will be based on company performance and
other related parameters. One may need to predict .mu..sub.i based
on all available data. The stock return values move around their
mean values, the individual variations depending on the individual
variance and related cross-correlations among other stocks.
[0010] The expected value of the portfolio return represents the
net gain G of the portfolio. Thus the overall gain of the portfolio
in (2) is given by (C.sub.o=1)
G = E { P } = E { i = 1 m a i r i ( n ) } = i = 1 m a i .mu. i = a
_ T .mu. _ ( 5 ) ##EQU00005##
where
a=[a.sub.1, a.sub.2, a.sub.3, . . . a.sub.m].sup.T (6)
r(n)=[r.sub.1(n), r.sub.2(n), r.sub.3(n), . . . r.sub.m(n)].sup.T
(7)
and
.mu.=E{r(n)}=[.mu..sub.1, .mu..sub.2, .mu..sub.3, . . .
.mu..sub.m].sup.T. (8)
Here E{.} stands for the expected or ensemble averaging operation
as in (4). The overall risk of the portfolio is given by the
variance of the portfolio return that equals
.sigma. P 2 = E { [ P - E { P } ] 2 } = E { a _ T [ r _ ( n ) -
.mu. _ ] [ r _ ( n ) - .mu. _ ] T a _ } = a _ T E { [ r _ ( n ) -
.mu. _ ] [ r _ ( n ) - .mu. _ ] T } a _ = a _ T R a _ ( 9 )
##EQU00006##
where (see, in "Probability, Random Variables and Stochastic
Processes," Fourth Edition, A. Papoulis, and S. U. Pillai,
McGraw-Hill Companies, New York, USA, 2001).
R=E{[r(n)-.mu.][r(n)-.mu.].sup.T}>0 (10)
represents the covariance matrix (positive definite matrix) of the
stock return vector r(n). Notice that
R.sub.ii=E{(r.sub.i(n)-.mu..sub.i).sup.2}=var{r.sub.i(n)}=.sigma..sub.i.-
sup.2>0 (11)
represents the variance of the i.sup.th stock return, and
R.sub.ij=E{(r.sub.i(n)-.mu..sub.i)(r.sub.j(n)-.mu..sub.j)}=cov{r.sub.i(n-
),r.sub.j(n)}=.rho..sub.ij.sigma..sub.i.sigma..sub.j (12)
represents the covariance between returns r.sub.i(n) and
r.sub.j(n), where .rho..sub.ij is defined as the correlation
coefficient between r.sub.i(n) and r.sub.j(n).
[0011] The above equations are well known in the prior art. In
addition, the above equations have been used to formulate the
following portfolio optimization strategy, which can be called
"Prior Art: Minimize Portfolio Risk".
Prior Art: Minimize Portfolio Risk:
[0012] Find the right max of stocks that minimizes the overall
portfolio risk. Take whatever profit you get.
[0013] In the "Minimize Portfolio Risk" approach, the Portfolio
risk is minimized by minimizing the portfolio variance
.sigma..sub.P.sup.2 in equation (9) subject to the constraints in
equation (1). This gives the well-known constrained optimization
problem referred to in "Mean-Variance Analysis in Portfolio Choice
and Capital Markets", H. M. Markowitz, et. al., John Willy, New
York, 2000:
min a.sup.TRa subject to a.sup.Te=1 (13)
where e represents the "all ones" column vector
e=[1, 1, 1, . . . , . . . 1].sup.T (14)
Notice that the nonnegative constraint for a needs to be
incorporated as well. One approach of the prior art is to use the
simplex type optimization methods to incorporate the positivity
constraint for the weight vector a as referred to in "Mean-Variance
Analysis in Portfolio Choice and Capital Markets", H. M. Markowitz,
et. al. Another approach is to put additional constraints on stock
selection to realize this goal.
[0014] Eq. (13) leads to the modified Lagrangian function
min .LAMBDA.=a.sup.TRa+.lamda.(a.sup.Te-1) (15)
and its minimization yields
.differential. .LAMBDA. .differential. a _ = 2 R a _ + .lamda. e _
= 0 ( 16 ) ##EQU00007##
which gives
a _ = - .lamda. 2 R - 1 e _ ( 17 ) ##EQU00008##
a _ T e _ = 1 - .lamda. 2 = 1 e _ T R - 1 e _ or ##EQU00009##
and the normalization condition
a _ = R - 1 e _ e _ T R - 1 e _ > 0. ( 18 ) ##EQU00010##
Observe that (18) must turn out to be a positive vector. This is
clearly satisfied if R.sup.-1 is a positive (Perron) matrix as
specified in the prior art in "Matrix Algebra and Its Applications
for Statistics and Econometrics", C. R. Rao, M. B. Rao, Singapore:
World Scientific, 1998. Thus if R.sup.-1 is a positive definite
matrix, then the optimum vector a turns out to be positive since
the denominator e.sup.TR.sup.-1e>0.
[0015] In other words, to start with one may select only those
stocks to be in the portfolio for which R.sup.-1 satisfies the
Perron property (positive matrix). In that case, the minimum
volatility is given by:
( .sigma. P 2 ) m i n = a _ T R - 1 a _ = 1 _ T R - 1 e _ = 1 i j R
ij > 0. ( 19 ) ##EQU00011##
where R.sup.ij represents the (i,j)-th entry of the matrix
R.sup.-1. Also, the net gain in that case is given by
G = a _ T .mu. _ = _ T R - 1 .mu. _ _ T R - 1 e _ > 0. ( 20 )
##EQU00012##
[0016] For example, in a two-stock portfolio, the Perron property
that R.sup.-1 contain only positive entries is satisfied by any two
negatively correlated stocks since in that case
R = ( 1 - .rho. - .rho. 1 ) , 0 < .rho. < 1 and ( 21 ) R - 1
= 1 1 - .rho. 2 ( 1 .rho. .rho. 1 ) > 0 ( 22 ) ##EQU00013##
has all positive entries. Observe that equation (21) represents the
covariance matrix of two stock returns with "opposing trends" and
they are negatively correlated. Hence when one "goes up", the
tendency of the other one is to "go down" thus minimizing the risk
of loss. For large m, realizing this nonnegativity condition may be
too restrictive. From equation (18), a more relaxed condition is
that the row sums of R.sup.-1 must be all positive.
[0017] From time-to-time, the portfolio manager should recompute R
and update the portfolio mix vector a by buying/selling stocks to
keep the overall portfolio volatility low.
SUMMARY OF THE INVENTION
[0018] At least one embodiment of the present invention provides a
method and an apparatus for investing a fixed amount of capital
optimally by distributing it among a set of stocks and securities
so as to maximize the return while minimizing the overall risk. In
at least one embodiment a gain to risk ratio is maximized rather
than minimizing the risk alone, to determine the fraction of
capital that must go to each stock. If short selling stocks and
securities is not allowed in a portfolio, then stock selection
becomes important so as to maintain the desired fractions to be
positive. An optimum gain versus variance plot can be used to
determine the type of stocks to be selected in addition to their
relative quantity for maximum yield over the duration of interest.
By modifying the definition of risk to include some function of the
covariance matrix of secondary stocks that are sympathetic to the
primary stocks of interest, an alternate investment strategy is
also developed here.
[0019] At least one embodiment of the present invention includes a
method comprising determining a first and second return values for
a first stock, and determining a mean return value for the first
stock based on the first and second return values. The method
further includes determining a first and second return values for a
second stock, and determining a mean return value for the second
stock based on the first and second return values for the second
stock. The method may further include determining an inverse
co-variance matrix based on the first and second return values for
the first stock, the mean return value for the first stock, the
first and second return values for the second stock, and the mean
return value for the second stock. The method may also include
determining weighting factors for the first and second stocks,
respectively, by multiplying the inverse co-variance matrix times
the mean return values for the first and second stocks, and
normalizing the result by dividing it with their sum. The method
may also include specifying an amount of the first stock for
purchasing based on the first factor of the weight factor and
specifying an amount of the second stock for purchasing based on
the second factor of the weight factor.
[0020] The method may further be comprised of determining a vector
of a plurality of weight factors a.sub.opt including the first and
second weight factors, and determining a vector .mu. of a plurality
of mean return values including the mean return values for the
first and second stocks. The plurality of weight factors may be
given by a.sub.opt=kR.sup.-1.mu., wherein k is a normalization
constant so that the entries of the optimum vector add up to unity,
and R.sup.-1 is the inverse co-variance matrix. The quantity k may
be given by
k = 1 _ T R - 1 .mu. _ . ##EQU00014##
and e.sup.T is an all ones row vector.
[0021] The present invention, in one or more embodiments also
includes a method for investing a given capital sum by distributing
it among a set of investments. The method may include determining a
plurality of weight factors, one for each investment in the set of
investments by which the capital sum will be partitioned so as to
determine an actual amount to be invested in each investment. The
plurality of weight factors may be determined by maximizing a total
gain to overall risk ratio. Alternatively, the plurality of weight
factors may be determined by maximizing a square of a total gain to
overall risk ratio. A first numerical filter may be used to
un-correlate a primary or first set of investments and a second
numerical filter may be used to maximally combine the primary or
first set of investments.
In at least one embodiment of the present invention a vector of
weighting factors a.sub.opt for the primary set of investments is
determined from the below fraction:
a _ opt = R - 1 .mu. _ _ T R - 1 .mu. _ ( 23 ) ##EQU00015##
wherein R.sup.-1 is the inverse of the covariance matrix for the
primary set of investments, .mu. is the expected mean returns
vector for the primary set of investments, and e.sup.T is an all
ones row vector.
[0022] The present invention in one or more embodiments may include
maximizing the ratio of the square of an expected investment
portfolio gain of a primary or first set of investments to a
modified risk. The modified risk may be defined as the sum of the
portfolio risk based on the primary set of investments and a
function of a covariance matrix of a secondary set of stocks that
are sympathetic to the set of primary investments.
[0023] The present invention, in one or more embodiments may
include a method comprising selecting a primary set of investments,
and selecting a secondary set of investments, which are related to
the primary set of investments. The method may also include
determining returns for the primary set of investments, determining
returns for the secondary set of investments, and determining an
expected mean returns vector for the primary set of investments.
The method may further include determining a covariance matrix for
the primary set of investments, and modifying the covariance matrix
for the primary set of investments by adding a diagonal matrix
generated from a covariance matrix for the secondary set of
investments. The method may also include determining weighting
factors a for investing in the primary set of investments
iteratively from the below fraction:
a _ k + 1 = ( R + .sigma. 0 2 a _ k 2 I ) - 1 .mu. _ _ T ( R +
.sigma. 0 2 a _ k 2 I ) - 1 .mu. _ . ( 24 ) ##EQU00016##
wherein R is the covariance matrix for the primary set of
investments, .mu. is the expected mean returns vector for the
primary set of investments, e.sup.T is an all ones row vector;
.sigma..sub.o.sup.2 is the sum of the variances of the secondary
stocks that influence the primary stocks that is obtained by
summing the diagonal entries of the covariance matrix of the
secondary set of investments. Here
.parallel.a.sub.k.parallel..sup.2=a.sub.k.sup.Ta.sub.k>0
represents the square of the norm of the vector a.sub.k, and I
represents the m.times.m identity matrix (with ones along the main
diagonal and zeros elsewhere). In at least one embodiment of the
present invention, the constant term .sigma..sub.o.sup.2 above may
also be treated as a free positive variable and increased in
numerical value until the vector of weighting factors for the
primary set of investments turns out to be positive.
[0024] The primary or first set of investments may include any
investment, such as a stock, security, mutual fund, hedge fund, or
index following fund, and they may be selected so that the above
weight factor vector turns out to be positive.
BRIEF DESCRIPTION OF THE DRAWINGS
[0025] FIG. 1 shows a flow chart of a method in accordance with an
embodiment of the present invention;
[0026] FIG. 2 shows a diagram of a whitening filter followed by a
matched filter;
[0027] FIG. 3 is a chart showing gain versus square root of
portfolio variance (standard deviation); and
[0028] FIG. 4 shows a flow chart of another method in accordance
with another embodiment of the present invention.
DETAILED DESCRIPTION OF THE DRAWINGS
[0029] One embodiment of the present invention makes use of matched
filtering concepts for the purpose of picking stocks. These matched
filtering concepts were previously used in the field of electrical
engineering but were not previously used for the purpose of picking
the right mix of stocks in a portfolio.
[0030] At least one embodiment of the present invention provides a
method for picking stocks, which maximizes gain and simultaneously
minimizes risk. This is achieved by maximizing the ratio of gain to
risk. The right mix of stocks are selected so as to maximize the
gain G while simultaneously minimizing the overall risk
.sigma..sub.P.sup.2. In at least one embodiment the following ratio
is maximized:
G .sigma. P 2 . ( 25 ) ##EQU00017##
Equation (25) represents the gain over the portfolio risk. However,
in one embodiment of the present invention, instead of maximizing
(25), the following ratio is maximized:
G 2 .sigma. P 2 = a _ T .mu. _ 2 a _ T R a _ ( 26 )
##EQU00018##
subject to the normalization constraint a.sup.Te=1. Strategy-1 (In
Accordance with an Embodiment of the Present Invention):
[0031] Clearly, equation (26) represents a more aggressive strategy
in term of maximizing gain, but more interestingly, the ratio in
equation (26) is the same as the familiar SNR (Signal to Noise
Ratio) maximization strategy used in classical receiver design in
Communication theory, in Electrical Engineering, where a signal
corrupted by interference and noise is presented to a receiver to
minimize the effect of output interference plus noise while
maximizing the output signal component at the decision instant as
referred to in "Signals Analysis", A. Papoulis, McGraw-Hill
Companies, New York, USA, 1977, and also "Digital Communications",
Fourth edition, J. Proakis, McGraw-Hill Companies, New York, USA,
2001. The solution to the SNR maximization problem leads to well
known matched filter (MF) solution as referred to in "Signals
Analysis", A. Papoulis.
From ( 26 ) , with S N R = G 2 .sigma. p 2 , we get S N R m ax = (
G 2 .sigma. P 2 ) m ax = max a _ a _ T .mu. _ 2 a _ T R a _ < _
.mu. _ T R - 1 .mu. _ ( 27 ) ##EQU00019##
since by Schwarz's inequality
|a.sup.T.mu.|.sup.2=|(R.sup.1/2a).sup.T(R.sup.-1/2.mu.)|.sup.2.ltoreq.(a-
.sup.TRa)(.mu..sup.TR.sup.-1.mu.), (28)
With equality if
a.sub.opt=kR.sup.-1.mu.. (29)
This gives
SNR m ax = ( G 2 .sigma. P 2 ) m ax = .mu. _ T R - 1 .mu. _ . ( 30
) ##EQU00020##
[0032] In general the entries of the optimum portfolio mix vector
shown in equation (29) can be both positive or negative. Negative
entries indicate that the corresponding stock is to be shorted. If
short sale strategies are prohibited, for example, as in the case
of most of mutual funds, then one needs to maintain a>0 and in
that case one can perform a constrained optimization strategy of
maximizing equation (26) subject to the non-negativity constraint
of a. This leads to a suboptimum solution with all positive or
non-negative entries for the vector a that requires no short
selling. This strategy can be applied to any given set of stocks
and securities that the investor has a-priori selected. In that
case the capital will be partitioned according to the entries of
the suboptimum vector so obtained and invested in the corresponding
stocks.
[0033] An alternate strategy is to keep the pool of the desired
stocks and securities to be selected as potentially open, and
select them from a larger pool of stocks and securities in such a
way that the inverse of their covariance matrix R.sup.-1 turns out
to be a positive matrix. If this condition turns out to be too
restrictive or severe especially for a portfolio containing a large
number of stocks, one can also settle for the less restrictive new
condition
R.sup.-1.mu.>0 (31)
by
[0034] (i) the judicious selection of stocks that go into the
portfolio and by
[0035] (ii) the choice of .mu. vector in (29) that represent the
expected average return.
Observe that
a _ T e _ = i a i = 1 ##EQU00021##
can be easily maintained with the constant k in equation (29)
chosen to be
k = 1 _ T R - 1 .mu. _ . ( 32 ) ##EQU00022##
This gives the desired portfolio mixing vector to be
a _ opt = R - 1 .mu. _ _ T R - 1 .mu. _ ( 33 ) ##EQU00023##
that maximizes the gain and minimizes volatility. In this case,
G opt = a _ T u _ = .mu. _ T R - 1 .mu. _ _ T R - 1 .mu. _ > 0
and ( 34 ) ( .sigma. P 2 ) m i n = .mu. _ T R - 1 .mu. _ ( _ T R -
1 .mu. _ ) 2 > 0. ( 35 ) ##EQU00024##
[0036] FIG. 1 shows a flow chart 10 of a method in accordance with
an embodiment of the present invention. At step 12, a stock returns
vector, such as r(n) calculated by equation (3) and (7), is
determined for m stocks. Next a means of returns vector, such as
.mu. calculated by equation (4) and (8), is determined at step 14.
At step 16, an inverse matrix of m stocks, such as R.sup.-1 is
determined. At step 18 weighting factors, such as a are determined
based on the inverse co-variance matrix times the means for returns
vectors divided by the sum of the vector so obtained, such as by
the equation (29) or (33).
[0037] Equation (33) can be given the whitening followed by matched
filtering interpretation as well, as will be shown with reference
to FIG. 2. FIG. 2 shows a diagram 100 of a whitening filter 102
followed by a matching filter 104. This technique was previously
used in classical receiver design in Communication theory in
Electrical Engineering, but not for the purpose of picking the
right mix of stocks in a portfolio. In this example, the input to
the whitening filter 102 is stock returns vector r(n). The
whitening filter 102 reduces the noise or volatility in the stock
returns vector r(n) and produces the filter output shown below:
x _ ( n ) = [ x 1 ( n ) x 2 ( n ) x m ( n ) ] = R - 1 / 2 r _ ( n )
( 36 ) ##EQU00025##
[0038] The filter output above is uncorrelated and has unit
variance since its covariance matrix equals
R x = E { ( x _ ( n ) - E { x _ ( n ) } ) ( x _ ( n ) - E { x _ ( n
) } ) T } = R - 1 / 2 E { ( r _ ( n ) - E { r _ ( n ) } ) ( r _ ( n
) - E { r _ ( n ) } ) T } R - 1 / 2 = R - 1 / 2 RR - 1 / 2 = I ( 37
) ##EQU00026##
and to maximally combine these outputs, the coefficients {b.sub.i}
in FIG. 2 must be selected so as to maximize the average portfolio
gain
G = E { P } = E { i b i x i ( n ) } = b _ T .mu. _ x ( 38 )
##EQU00027##
where
.mu..sub.x=E{x(n)}=R.sup.-1/2.mu.. (39)
[0039] From Schwarz's inequality (see (28)), Eq. (38) is maximized
if
b=k.mu..sub.x=kR.sup.-1/2.mu.. (40)
[0040] Thus b in (40) is a maximal combiner with respect to
.mu..sub.x. Hence,
G=b.sup.T.mu..sub.x=k.mu..sup.TR.sup.-1/2.mu..sub.x=k.mu..sup.TR.sup.-1/-
2R.sup.-1/2.mu.=a.sup.T.mu. (41)
or
a=kR.sup.-1.mu. (42)
as in equation (29).
[0041] Interestingly, Equations (33)-(35) can be used to generate a
gain-risk plot by varying over all sustainable .mu.s. Following
equation (33), an arbitrary .mu. is said to be sustainable if
R.sup.-1.mu. is a positive vector. Using a sustainable .mu., one
can compute the optimum gain and .sigma..sub.P using equations
(34)-(35).
[0042] Notice that although scaling .mu. does not affect the
variance in equation (35), it does affect the gain in equation
(34). Hence to avoid duplication by simple scaling, the first entry
.mu..sub.1 in a sustainable .mu. may be normalized to unity. As an
example, FIG. 3 shows a diagram 200 of an optimum average portfolio
gain G as in (34) versus square root of the risk in (32) (standard
deviation) plot using arbitrary sustainable normalized mean vectors
for various sets of portfolios containing different numbers of
actual stocks. The stocks in each portfolio are selected for
illustrative purposes only. Table 1 lists the actual stocks used
for FIG. 3. Observe that FIG. 3 shows cases for portfolios where
the number of stocks equals m=12, 8 and 6. The results for cases
m=12, m=8, and m=6 is shown as A, B, and C, respectively in FIG. 3.
Covariance matrices in each case have been calculated using sample
data collected for the period of January 2001 to December 2004 with
weekly duration representing a time unit.
TABLE-US-00001 TABLE 1 Stock symbols used in FIG. 3. m = 12 m = 8 m
= 6 `SLB` `TWX` `NOC` `BK` `COST` `BAC` `GD` `SBUX` `SBUX` `SBUX`
`MER` `AAPL` `TWX` `NOC` `GE` `Dell` `AAPL` `GD` `NOC` `AFL` `CFC`
`DST` `BA` `GIS` `EBAY` `MHP`
[0043] From FIG. 3, as the number of stocks in a portfolio
increases, the risk in terms of overall variance decreases.
Interestingly, for the strategy of stock weight picking shown by
equations (29) or (33), for a given set of stocks, the risk is more
concentrated compared to the spread in gain. Each point in FIG. 3
corresponds to a positive weight vector that is optimum for the
corresponding normalized mean vector and the given stocks. The
desired gain and risk tolerance of the investor will dictate the
actual point of interest that will be selected for investment.
[0044] Interestingly, other variations of the stock picking
strategy shown by equations (29) and (33) also lead to the same
result.
[0045] One may use a less aggressive strategy in terms of returns
while maintaining low volatility. Then we may maximize:
G .sigma. P 2 = a _ T .mu. _ a _ T R a _ ( 43 ) ##EQU00028##
instead of equation (26). This leads to
max a _ a _ T R 1 / 2 R - 1 / 2 .mu. _ a _ T R a _ .ltoreq. .mu. _
T R - 1 .mu. _ a _ T R a _ . ( 44 ) ##EQU00029##
Equality is achieved by solution given by equation (33) and in that
case
( G .sigma. P 2 ) max = e _ T R - 1 .mu. _ 2 . ( 45 )
##EQU00030##
[0046] On the other hand, one can use a much more aggressive
strategy such as maximizing
max a _ G .sigma. P 2 = a _ T .mu. _ 4 a _ T R a _ ( 46 )
##EQU00031##
subject to equation (1).
[0047] Notice that equation (46) is weighted more towards higher
gains.
max a _ G .sigma. P 2 .ltoreq. ( a _ T R a _ ) ( .mu. _ T R - 1
.mu. _ ) 2 a _ T R a _ = a _ T R a _ ( .mu. _ T R - 1 .mu. _ ) 2 .
( 47 ) ##EQU00032##
Once again, equality is obtained in equation (47) by solution given
by equation (33).
[0048] In summary, for a variety of optimization strategies, the
new portfolio mixing vector in equation (33) represents the optimum
strategy for building a portfolio. If short selling stocks is
allowed, the above strategy can be applied to any set of stocks; if
short selling is not permitted, then the selection of stocks and
their number that goes into the actual portfolio becomes important
and it must be accomplished so as to maintain the desired portfolio
mixing vector to be positive, while maintaining a high yield (gain)
with minimum fluctuations (risk).
Strategy-2 (In Accordance with Another Embodiment of the Present
Invention):
[0049] In another embodiment of the present invention, a variation
of the maximization of the gain to the risk strategy, to be
described below, leads to a somewhat different result in terms of
the desired portfolio vector. In the previous method, the risk is
defined as the variance of the portfolio under consideration as in
equation (9). In a method in accordance with an alternative
embodiment of the present invention, this definition is extended as
follows:
[0050] The actual stocks and securities that go into a portfolio as
the primary stocks are identified, and equation (7) represents
their returns. In that case the risk defined as in equation (9)
represents the variance of the exact combination of the returns of
these primary stocks that make up the portfolio. The method next
includes identifying through market research and other means
another set of stocks as secondary or sympathetic stocks that are
correlated to these primary stocks in equation (7). Let R.sub.0
represent the covariance matrix of the returns of these secondary
stocks that is also defined similar to equation (10). Since the
secondary stocks have some influence on the behavior of the primary
stocks, the argument here is that their covariance matrix R.sub.0
must also contribute to the overall risk of the portfolio. Thus in
this approach, a scalar function of R.sub.0 is added to the primary
risk factor in equation (9). In our case, the trace of R.sub.0 (sum
of the diagonal entries of the covariance matrix R.sub.0) of the
secondary returns is used as the scalar function. This gives the
modified risk of the portfolio to be
.sigma..sub.P.sup.2=a.sup.TRa+.sigma..sub.o.sup.2 (48)
where
.sigma..sub.o.sup.2=tr(R.sub.0) (49)
represents the trace of R.sub.0. In this case the optimization
problem in equation (26) gets modified as
G .sigma. P 2 = a _ T .mu. _ 2 a _ T R a _ + .sigma. 0 2 . ( 50 )
##EQU00033##
[0051] Let the vector a.sub.0 represent the optimum nonnegative
vector (constrained optimization using the non-negativity
condition) that maximizes the ratio in equation (50) and whose
elements add up to unity. In this approach, the capital will be
partitioned according to the entries of this new vector a.sub.0 and
invested in the primary stocks.
[0052] The nonnegative vector a.sub.0 above represents a suboptimum
solution, and as in equations (26)-(33) there exists an
unconstrained (without the nonnegative condition) optimum vector
b.sub.opt that maximizes equation (50), and once again the capital
can be partitioned according to the entries of the vector b.sub.opt
and invested in the primary stocks. In this case the strategy can
include short sales as well.
The globally optimum vector b.sub.opt may be solved by noticing
that (50) can be rewritten as
G .sigma. P 2 = a _ T .mu. _ 2 a _ T ( R + .sigma. 0 2 a _ 2 I ) a
_ , ( 51 ) ##EQU00034##
where I represents the m.times.m identity matrix (with ones along
the main diagonal and zeros elsewhere), and
.parallel.a.parallel..sup.2=a.sup.Ta>0 (52)
represents the norm (square of the length) of the vector a. In
(51), proceeding as in (27) through (29), we obtain the following
solution
a _ = c ( R + .sigma. 0 2 a _ 2 I ) - 1 .mu. _ ( 53 )
##EQU00035##
(where c is a normalization constant) that suggest the
iteration
a _ k + 1 = c ( R + .sigma. 0 2 a _ k 2 I ) - 1 .mu. _ . ( 54 )
##EQU00036##
After normalizing (54) as in equations (32)-(33) so that its
entries add up to unity, we obtain the desired iteration to be
a _ k + 1 = ( R + .sigma. 0 2 a _ k 2 I ) - 1 .mu. _ e _ T ( R +
.sigma. 0 2 a _ k 2 I ) - 1 .mu. _ . ( 55 ) ##EQU00037##
that can be used to solve for the above optimum vector b.sub.opt,
The above iteration is seen to converge in a variety of
situations.
[0053] FIG. 4 shows a flow chart 300 of a method in accordance with
an embodiment of the present invention. At step 302, a stock
returns vector, such as r(n) calculated by equation (3) and (7), is
determined for m primary stocks and at step 304 another stock
returns vector is calculated for a certain number of secondary
stocks in a similar manner. Next, at step 306 a means of returns
vector, such as .mu. is calculated by equation (4) and (8), and a
covariance matrix R is computed as in equation (10) for the primary
stocks, and at step 308 a covariance matrix R.sub.0 for the
secondary stocks is computed similar to equation (10). Equation
(49) is used to determine the trace of the secondary returns
.sigma..sub.o.sup.2 at step 310. Next at step 312, an initial
starting vector is determined, which becomes the old vector a.sub.k
at step 312 to start the iteration, and determine the norm of
.parallel.a.sub.k.parallel. as in equation (52). At step 310, a
modified inverse covariance matrix such as
( R + .sigma. 0 2 a _ k 2 I ) - 1 ##EQU00038##
is determined. At step 310 weighting factors, such as the new
vector a.sub.k+1 are determined based on the above modified inverse
covariance matrix times the means for primary returns vectors
divided by the sum of the new vector entries so obtained, such as
by equations (54) or (55). At step 314, the difference of the new
vector a.sub.k+1 and old vector a.sub.k is defined as the error
vector. At step 316 the error norm is computed as in equation (52)
for the error vector, and it is compared with a predetermined
threshold value, such as for example 0.001 etc. If the error norm
is less than the preset threshold value, the new vector obtained at
step 310 is taken as the desired weighing factors at step 318.
Otherwise, the old vector is replaced with the contents of the new
vector at step 320 and it is fed back to step 310, where the entire
cycle is repeated till the desired accuracy is achieved.
[0054] Optimum Nonnegative Solution: Interestingly, it is possible
to guarantee the solution given by equations (54)-(55) to be
nonnegative by treating .sigma..sub.o.sup.2 in equations (54)-(55)
as a free parameter. Recall that .sigma..sub.o.sup.2 represents a
measure of the effect of the correlation of the secondary stocks on
the primary stocks, and for a given value of .sigma..sub.o.sup.2,
the optimum vector in (54) can have both positive and negative
entries. In such situations, by increasing the value of
.sigma..sub.o.sup.2 the optimum vector can be made nonnegative
there by avoiding short sales. In fact, for any given covariance
matrix R and nonnegative vector .mu., there exists a minimum
positive value for the constant .sigma..sub.o.sup.2 in the
equation
a=(R+.sigma..sub.0.sup.2I).sup.-1.mu. (56)
for which the vector a becomes nonnegative. The proof follows by
expanding equation (56) and noticing that as the constant
.sigma..sub.o.sup.2 becomes large, the perturbation terms to the
first term .mu. in the expansion become of decreasing importance,
and hence the vector a becomes nonnegative. Using this approach in
equations (54)-(55), it follows that there exists a minimum
threshold value for the sympathetic stocks' variance term
.sigma..sub.o.sup.2, above which the optimum vector b.sub.opt
remains non-negative. Using any value above this threshold value
for .sigma..sub.o.sup.2 in equations (54)-(55) avoids short sales
for the optimum portfolio mixing strategy. To determine this
threshold value, one may proceed using the iterative steps in FIG.
4 where the term .sigma..sub.o.sup.2 is treated as a free
parameter. For a preset value of .sigma..sub.o.sup.2 if the final
weight factor vector at stage step 318 in FIG. 4 turns out to have
negative entries, the whole process is repeated with a larger value
for the preset term .sigma..sub.o.sup.2 until all entries of the
weight vector factor at step 318 turns out to be positive.
[0055] As an example, consider a portfolio containing three stocks
whose 3.times.3 covariance matrix is give by
R = ( 1 .0 0.40 - 0 .25 .40 0.80 - .30 - 0 .25 - 0.30 2 .0 ) ( 57 )
##EQU00039##
and let .mu.=(0.20 0.50 0.40).sup.T represent their the expected
return values vector. In that case the solution in equation (33)
that maximizes the overall gain to risk ratio is given by
R.sup.-1.mu.=(-0.237 0.7531 0.310).sup.T and it has one negative
entry and hence it involves short sales. However using
.sigma..sub.o.sup.2=0.034995, the solution in (56) after
normalization turns out to be a=(0.0000025 0.683533
0.293664).sup.T. Since the new solution has all nonnegative
entries, it avoids short sales. Using any other value above this
threshold for .sigma..sub.o.sup.2 results in all positive values
for the solution and it avoids short sales in the optimum
portfolio.
[0056] In summary, methods in accordance with embodiments of the
present invention for determining the optimization strategies for
building a new portfolio mixing vector are disclosed. In at least
most if not all of these cases, the ratio of the overall portfolio
gain function to the portfolio risk is maximized, where the
definition of the portfolio risk is extended in one case to include
the influence of stocks that are sympathetic to the primary stocks
of interest. If short selling stocks is allowed, the above
strategies can be applied to any set of stocks; if short selling is
not permitted, then the selection of stocks and their number that
goes into the actual portfolio becomes important and it must be
accomplished so as to maintain the desired portfolio mixing vector
to be positive, while maintaining a high yield (gain) with minimum
fluctuations (risk). This can also be accomplished by extending the
definition of risk to include a free variable term that denotes the
effect of a secondary set of stocks, and by increasing this term
the desired portfolio mixing vector can be made positive through an
iterative procedure.
[0057] Although the invention has been described by reference to
particular illustrative embodiments thereof, many changes and
modifications of the invention may become apparent to those skilled
in the art without departing from the spirit and scope of the
invention. It is therefore intended to include within this patent
all such changes and modifications as may reasonably and properly
be included within the scope of the present invention's
contribution to the art.
* * * * *