Patent Application
20090099976
The present invention relates to a method and system for determining an optimal portfolio for determining financial product to be object for purchasing among a plurality of financial products, a program therefor and a storage medium storing the program.
As a model to be employed upon determining optical portfolio, in a group of financial products to take as object for purchasing (hereinafter, object for purchasing is exemplarily assumed as universe consisted of group of stocks (two hundreds twenty-five names as whole of the First Section of Tokyo Stock Exchange), under a premise of fixing an earning rate at a predetermined value, a mean dispersion model employing a quadratic programming for minimizing secondary objective function expressed as a risk indicative of fluctuation of the earning rate, or multi-factor model are introduced in Hiroshi Konno “Chrematistics Technology I”, Nikka Giren, pp 4 to 19.
On the other hand, Japanese Patent Application Laid-Open No. 2000-293569 discloses a model according to a linear programming for maximizing a sum of expected earning rate consisted of a plurality of scenario and a period as an optimal portfolio determination method, under (1) a constraining condition by a function taking market price as parameter and (2) a constraining condition for performing control relating to possible gain and loss.
As a mathematical programming, such as quadratic programming or linear programming, an effective constraint method and so on are typically known as introduced in Toshihide Ibaragi and Masao Fukushima “FORTRAN 77 Optimization Programming” Iwanami Shoten, pp 87 to 113, and so forth, for example. In the mathematical programming, it is a typical method to repeat updating a point string from an initial point to a point where an optical solution is obtained. Upon updating the point string, a most part of process is matrix operation for deriving a direction for retrieving the point string. In the matrix appearing upon formulation into quadratic programming problem, most of factors are zero. For processing such matrix, it has been known a sparse method for implementing matrix operation with discriminating factor of zero on the program.
The sparse method is an general purpose approach as a method for matrix operation process in the quadratic programming. However, in viewpoint of application for a problem of determination of optimal portfolio, in case of a problem having several thousands of parameters, huge calculation period is required even in the sparse method for necessity of discrimination of the factors of zero on the program. While the recent computers are significantly advanced, upon practically determining portfolio, shortening of calculation period of the quadratic programming is strongly demanded for necessity of solving the quadratic programming for number of times with updating objective function or constraint function.
On the other hand, in the system disclosed in Japanese Patent Application Laid-Open No. 2000-293569, since no detail of mathematical programming has been disclosed, upon application of the mathematical programming, a system employing the sparse method is employed to encounter the similar program.
An object of the present invention to provide an optimal portfolio determining method enabling high speed determination of objective financial product which optimize availability for institutional buyer or retail investor and purchasing amount on the basis of information relating to earning rate or the like of individual name and information relating to information factors influencing for earning rate, and a system for realizing the method.
Another object of the present invention is to provide a program indicative of process procedure of the optimal portfolio determining method and a storage medium storing the program.
In order to accomplish the above-mentioned and other objects, according to the first aspect of the present invention, an optimal portfolio determining method for determining purchasing amounts of respective financial products among a plurality of financial products so as to optimize an objective function consisted of earning rate of all of a plurality of financial products and risk influencing for earning, comprises:
input step of inputting constraint parameters forming constraint condition for optimizing objective function consisted of an expected value of the earning rate of each individual financial product, individual floating factor as unique factor of each individual financial product influencing for earning, common floating factor as factor influencing for earning of overall financial products, and risk influencing for earning rate and earning of overall financial product; and
solving step of determining financial product to perchance and purchasing amount for maximizing the objective function on the basis of input data.
In the preferred construction, the optimal portfolio determining method may further comprise preliminary process step of processing of dividing a coefficient matrix appearing in the objective function into a partial matrix relating to individual floating factor of each individual financial product, and a partial matrix relating to the common floating factor, upon determining the financial product to purchase and purchasing amount.
In the alternative, the optimal portfolio determining method may further comprise preliminary process step of processing of dividing a matrix consisted of the constraint parameters into a partial matrix relating to the financial products and the common floating factor, a partial matrix relating to the common floating factor, and a partial matrix relating to the financial product and purchasing amount thereof.
In the further alternative, the optimal portfolio determining method may further comprise preliminary process step of processing of dividing a matrix consisted of the constraint parameters into a partial matrix relating to the financial products and the common floating factor, a partial matrix relating to the common floating factor, a partial matrix relating to the financial product and purchasing amount thereof, and a partial matrix relating to purchasing amount of each group in the case where the financial products are grouped into a plurality of groups.
In such case, the partial matrix relating to the individual floating factor may be a diagonal matrix having elements in a portion of diagonal component corresponding to number of financial products to be selected. The partial matrix relating to the common floating factor may be a matrix taking square of the common floating factor as dimension. The partial matrix relating to the common floating factor may also be a diagonal matrix having element in a portion of diagonal component corresponding to number of the common floating factor. The partial matrix relating to constraint for purchasing amount of the financial product may be a diagonal matrix having element in a portion of diagonal component corresponding to number of the common floating factor. The partial matrix relating to the financial product and the common floating factor may be a matrix taking a product of the financial product and the common floating factor as dimension. The partial matrix relating to constraint for purchasing amount of the group, in which the financial products belong, may be a matrix taking a product of number of the groups and the financial products.
In the further preferred construction, the optimal portfolio determining method may further comprise display step outputting the risk indicative of variation of earning and earning rate consisting the objective function.
According to the second aspect of the present invention, an optimal portfolio determining system having a computer unit for determining purchasing amounts of respective financial products among a plurality of financial products so as to optimize an objective function consisted of earning rate of all of a plurality of financial products and risk influencing for earning, the computer unit comprises:
storage device storing an expected value of the earning rate of each individual financial product;
storage device storing individual floating factor as unique factor of each individual financial product influencing for earning,
storage device storing common floating factor as factor influencing for earning of overall financial products, and
storage device storing constraint parameters forming constraint condition for optimizing objective function consisted of risk influencing for earning rate and earning of overall financial product;
optimal portfolio solving device determining financial product to perchance and purchasing amount for maximizing the objective function on the basis of data stored in the storage device; and
display device outputting determined optimal portfolio.
The computer unit may comprise a server computer including respective storage devices and the optimal portfolio deriving device, and a plurality of client computers receiving information relating to the optimal portfolio calculated by the server computer for displaying, and the sever computer and the client computers are connected through a network.
According to the third aspect of the present invention, a optimal portfolio determining program being readable by a computer includes input step and solving step of the optimal portfolio determining method set forth above.
According to the fourth aspect of the present invention, a storage medium storing a program readable by a computer which stores a program executing input step and solving step of the optimal portfolio determining method set forth above.
The present invention will be understood more fully from the detailed description given hereinafter and from the accompanying drawings of the preferred embodiment of the present invention, which, however, should not be taken to be limitative to the invention, but are for explanation and understanding only.
In the drawings:
The present invention will be discussed hereinafter in detail in terms of the preferred embodiment of the present invention with reference to the accompanying drawings. In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention. It will be obvious, however, to those skilled in the art that the present invention may be practiced without these specific details. In other instance, well-known structure are not shown in detail in order to avoid unnecessary obscurity of the present invention.
The present invention will be discussed in detail in terms of a system for determining an optimal portfolio for determining an objective financial product for purchasing among a plurality of financial products and purchasing amount so as to maximize gain and to minimize risk indicative of element to fluctuate the gain by a mathematical programming, such as linear programming or non-linear programming. With the system, institutional buyer and general investor may determine the optimal portfolio using computer. The preferred embodiment of the present invention will be discussed with reference to the accompanying drawings. At first, discussion will be given for algorithm of optimal portfolio determination.
In a problem of portfolio selection taking a plurality of stocks (here, all stocks in the First Section of Tokyo Stock Exchange) as a group of financial products to be objects for purchasing, an objective function is a utility function as expressed by the following expression (1) established by a sum of an earning rate expressed by a sum of products of multiplication of expected earning rate of each stock and investing rate, and a value calculated by multiplying a measure of risk aversion and an active risk expressed by a deviation rate between bench mark ratio indicative of a rate of current value of each individual name versus total current value of overall stocks and investing rate of each individual name:
U=á
T
h
p
−ë(hp−hm)TG(hp−hm) (1)
wherein á is a vector taking expected earning rate of individual name as element, ë is measure of risk aversion held by the investor (e is set greater as giving preference for risk aversion and is set smaller as giving preference for increase of gain of entire portfolio), hp is a vector taking investment ratio of each name as element, hm is a vector taking a bench mark ratio as element, and G is a matrix taking covariance between gain rates of individual names.
Discussion with be given hereinafter for an example in terms of the case where the following expressions are taken as constraint expression in the utility function as expressed by the foregoing equation (1). In the following expression, e represents a vector in which all elements are 1.
eThp=1 (overall investment ratio is 1) (2)
hp 0 (constraint for inhibiting short selling)
Setting method of such utility function has been disclosed in R. C. Grinold and R. N. Kern “Active Portfolio Management” Toyo Keizai Shinbunsha, pp 81 to 87. The disclosure in this publication is herein incorporated by reference.
In a mean dispersion model, the quadratic programming is applied with taking the utility function expressed by the foregoing equation (1) as objective function. However, in the mean dispersion model, when a thousand five hundreds of names in the First Section of Tokyo Stock Exchange are taken as objective for calculation, 2250000 of values of covariance between earning rates of individual names are inherently included in the objective function. Upon solving the problem of the quadratic programming having such objective function, it is expected to take a huge amount of time. Therefore, such approach is not practical for the problem of portfolio selection with mean dispersion model.
A model to be employed for solving the shortcoming of the mean dispersion model is multi-factor model. In the multi-factor model, the earning rate of each individual name is expressed as the following equation (4) with common factor influencing for earning rate of overall names and individual factor variable depending upon factors unique to each individual name.
áj=á
j
+Óâ
jk
F
k+åj (4)
wherein âjk is a parameter representative of influence for the earning ratio of individual name j when a factor Fk of the common factor k is varied by one unit, and is referred to as factor exposure. For example, when the common factor is yen-dollar exchange rate, and if $1=¥123, 123 is assigned as Fk. On the other hand, if the earning rate is varied 0.1% when the exchange rate is varied from $1=¥123 to $1=¥124, 0.1 is assigned as âjk.
While detail has been eliminated here from so as to maintain disclosure simple enough to facilitate clear understanding of the present invention as disclosed in Hiroshi Konno, “Chrematistics I”, Nikka Giren, pp 18 to 19. The above-identified passage of the publication will be herein incorporated by reference, using a matrix B consisted of âjk, a matrix F consisted of dispersion and covariance of Fk, and a diagonal matrix A having a specific risk expressed by dispersion of åj as diagonal component, the covariance matrix G of the earning rate of each individual name is expressed by the following equation (5):
G=B
T
FB+Ä (5)
Substituting the foregoing equation (1) with the equation (5), and assuming Bhp=y, the following equation (6) is established:
In the multi-factor model, the utility function derived from the foregoing equation (6) is taken as object of the objective function to be maximized. Furthermore, in the multi-factor model, by assuming Bhp=y, new parameter y is generated. In conjunction therewith, not only the constraint expressions (2) and (3) but also the following constraint expression (7) have to be considered.
Bh
p
−y=0
It should be noted that the present invention can be embodied even in the case where the covariance matrix G is in a form other than that shown by the equation (5). Hereinafter, mode of implementation of the invention in connection with determination of optimal portfolio in multi-factor model will be discussed.
In the individual earning rate input means 101, information relating to an expected value of the earning rate of individual name is input. One example of data shown in
In the individual factor input means 102, information relating to the specific risk, in which fluctuation factor of earning rate of the individual name is discussed as factors unique for the individual name, bench mark ratio indicative of a rate of current value of each individual name versus total current value of overall stocks, are input. One example of data shown in
The common factor input means 103 inputs information relating to covariance between two common factors in among factors common to influence for earning rate of overall names (hereinafter referred to as common factor). One example of data shown in
The constraining parameter input means 104 inputs data relating to factor exposure representative that when the common factor influencing to earning rate of overall names as discussed in
On example of data shown in
One example of data shown in
In the optimal portfolio deriving means 105, objective stock to purchase and purchasing ratio are determined on the basis of information input from input means 101 to 104. In the optimal portfolio deriving means 105, measure is taken for method to determine assignment of the optimal portfolio. The measure will be discussed later.
The optimal portfolio display means 106 outputs useful information for investor or fund manager active in fund operation for capital fund deposited by customer.
The optimal portfolio deriving means is constituted of step of generating optimization problem on the basis of information input through respective databases (input means) 101 to 104, and step of solving the optimization problem. As a solution for the optimization problem, mode of implementation according to an interior solution, in which number of times of updating of point string becomes small even for large scale problem and demonstrate superior performance, will be discussed. Mode of implementation of the invention may also employ simplex method in linear programming problem or active set method in quadratic programming problem.
The optimization problem is normally formulated into standard type of quadratic programming problem as expressed by the following expressions (8) and (9).
Minimization: cTx+xTQx/2+d (8)
Constraining Expression: Ax=b x≧0 (9)
wherein c is N-dimension vector, Q is N-dimension quadratic matrix, A is M×N matrix, b is M-dimension vector.
In the objective function of
All elements in remaining part are zero. Namely, when number of names and number of common factors are 1432 and 13, respectively, among about 2,080,000 of overall elements containing, elements containing other than 0 are 1600 which is less than 0.1% of overall elements. By considering of nature of secondary coefficient matrix, speeding up of optimizing operation becomes possible. It should be noted that, in a vector c indicative of primary coefficient of the objective function, most of elements are other than 0. However, comparing with the secondary coefficient matrix, no significant problem will be arisen for much lesser number of elements.
In the constraint expression of
Next, upon focusing parameters hp and y appearing in the objective function of the optimization problem shown in
Y=y+s*e (10)
After such conversion, namely after modification by substituting with y=Y−s*e, the structure of the quadratic programming problem becomes as shown in
Next, discussion will be given for optimal portfolio deriving means. At first, the interior solution as solution of the optimization problem will be briefly discussed with reference to the drawings.
Basically, the retrieving direction is derived by the Newton's method to make a different of objective functions of the primal problem (original problem) and dual problem (quadratic programming problem derived from the primal problem) as small as possible, for updating point string. By repeating point string as set forth above, when the difference of the objective functions becomes 0, the optimal solution can be obtained.
On the other hand, in the interior solution, when the optimal solution of the quadratic programming problem is assumed as x* and appropriately selecting y* and z* corresponding to equation constraint and inequality constraint (non-negative constraint of x), (x, y, z)=(x*, y*, z*) satisfies the following non-linear equation. The theoretical background has been disclosed in Hidetoshi Ibaragi and Masao Fukushima “FPRTRAN77 Optimal Programming” Iwanami Shoten, pp 453 to 457. The disclosure of the above-identified publication is herein incorporated by reference, and detailed discussion is eliminated for keeping the disclosure simple enough to facilitate clear understanding of the present invention. The constraining condition of the primal problem is expressed by the following expression (11), the constraining condition of the dual problem is expressed by the following equation (12), and complementary condition is expressed by the following expression (13).
Ax=b (11)
A
T
y−Qx+z=c (12)
xTz=0, x≧0, z≧0 (13)
The solution of the quadratic programming problem may be attained by solving the foregoing non-linear equation. In the interior solution, modifying the non-linear equation by using positive real number and modifying the complementary condition as the following expression (14):
xTz={grave over (l)}, x>0, z>0 (14)
Particularly, {grave over (l)} is set at positive number which is great in some extent, approximately solving the non-linear equation, the point string (xk, yK, xK) (K=0, 1, 2, 3, 4 . . . ) is updated sequentially with making smaller value to 0, the optimal solution for the quadratic programming problem can be derived.
In actual programming operation, {grave over (l)} is set in a*XKTZK/n so that retrieving direction is controlled in such a manner that the retrieving direction is shifted to be closer to the value 1 when the solution is out of the constraining region, and to be closer to the value 0 when the solution falls within the constraining region, and the Newton's equation shown by the following expressions (15) to (17) is solved.
Deriving the retrieving direction by solving the Newton's equation, and reducing the violation amount of the constraining condition and complementary condition set forth above, a step width satisfying x>0 and z>0 is calculated for updating the point string. It should be noted that in the foregoing expression (17), Xk and Zk are diagonal matrix taking the vector at respective repetition point as diagonal element, and e is vector where all elements are 1.
Algorithm of the quadratic programming designed in consideration of the foregoing matters is constituted with steps 1201 to 1210 as shown in
At step 1201, data for quadratic programming problem are input. Data to be input here are data relating to an expected value of the earning rate of each of the individual names shown in
At step 1202, number of constraint expressions and number of parameters in the quadratic programming problem are set. Assuming that the common factor input at step 1201, business category group to be considered as constraint (when not considered as constraint, 0 is set), and number of individual names as K, S and N respectively, numbers of the constraint expression and parameters are respectively expressed by (K+1+S) and (K+N).
At step 1203, Newton's equations (15) and (16) and a normed value of right side vector of right side vector indicating violation amount of the constraint condition and a value of xTz of left side of the constraint condition (13) are calculated.
The right side vector of Newton's equation (15) implements calculation by blocking as shown in
On the other hand, the right side vector of the Newton's equation (16) implements calculation by blocking as shown in
<Step 1204: Checking Whether Complementary Condition and Violation Amount of Constraint Condition is Less than or Equal to Predetermined Value>
At step 1204, at currently obtained repetition point, judgment is made whether the violation amount of the constraint condition and the complementary condition fall within allowable error range or not. In practice, judgment is made whether the constraint conditions (11) and (12) and the complementary condition (13) are satisfied or not. In practical arithmetic operation on the computer, judgment is made whether the conditions (11), (12) and (13) are approximately satisfied or not. The complementary condition (13) is expressed as the following expression (13′).
∥xTz∥<å (13′)
An inequality having a sufficiently close to zero (e.g. 10−10 and so forth) is used for judgment of optimality.
<Step 1205: Calculation of Value of {grave over (l)}>
At step 1205, the value of {grave over (l)} relating to the Newton's equation (14) is calculated. In practice, (â*xkTzk/n) shown in the foregoing equation (17) is set as the value of {grave over (l)}. It should be noted that the current repetition point does not satisfy the constraint condition (11), in order to retrieve the repetition condition satisfying the constraint condition (11), the value of â is set at a value close to one (e.g. 0.99). On the other hand, when the current repetition point satisfies the constraint condition (11), the value of â is set at a value close to 0 (e.g. 0.01) for retrieving the optimal solution. Such setting method of a respectively correspond to the processes at steps 1102 and 1103 as shown in
At step 1206, calculation of the right side vector of Newton's equation (17) is performed.
At step 1207, the Newton's equations (15), (16) and (17) are solved to derive a retrieving direction (dx, dy, dz) of the current repetition point. Upon solving the simultaneous equations, with the following equations (18) to (20), the solutions of dy, dx, dz are derived in order of (18), (19), (20). In the following equations, g(x), g(y) and g(z) respectively correspond to −(b−Axk), −(ATyk−QXk+zk−c), −{Xkzk−(âxkTzk/n)}.
A(Q+X−1Z)−1ATdy=−g(x)−A(Q+X−1Z)−1(g(y)−X−1g(z)) (18)
(Q+X−1Z)−1dx=−g(y)+X−1g(z)−ATdy (19)
dz=X
−1
g(z)−X−1Zdx (20)
In the equations (18), (19) and (20), X and Z are respectively diagonal matrixes having x and z in diagonal element.
For solving the equation (18), process is performed by blocking the matrix. However, since the contents of the process is complicate, discussion will be given with reference to
Upon solving the equation (18), at first, it becomes necessary to derive inverse matrix of Q+X−1Z. Number of dimensions of the matrix Q+X−1Z becomes (N+K) wherein the individual name and number of common factor are respectively N and K. Accordingly, in the example from
As shown in
Concerning the left upper portion of
After deriving (Q+X−1Z)−1 as set forth above, a product of the matrix A and (Q+X−1Z)−1 is derived. The element structure of respective matrix in the Newton's equation (18) is as shown in
Namely, when the constraint of the business category group is not considered, as a preliminary process for solving the problem of the optimal portfolio, the matrix A consisted of constraint parameters is divided into a partial matrix relating to financial products and common floating factor, a partial matrix relating to common floating factor and a partial matrix relating to the financial product and the purchasing amount thereof. On the other hand, when the constraint of the business category group is considered, as a preliminary process for solving the problem of the optimal portfolio, the matrix A consisted of constraint parameters is divided into a partial matrix relating to financial products and common floating factor, a partial matrix relating to common floating factor, a partial matrix relating to the financial product and the purchasing amount thereof, and a partial matrix relating to the purchasing amount of each group when the financial products are grouped into a plurality of groups.
On the other hand, when the constraint of the business category group is considered, the structure of the matrix A is characterized in that the partial matrix relating to the financial products and the common floating factor is the matrix taking the product of the financial products and the common floating factor as number of dimensions, and the partial matrix relating to the common floating matrix is the diagonal matrix having the elements in the portion of the diagonal product corresponding to number of the common floating factors, and the partial matrix relating to the constraint of the purchasing amount of the financial products is the partial matrix having the element in the portion of the diagonal component corresponding to number of the financial products. On the other hand, when the constraint of the business category group is considered, relative to the case not considering, the partial matrix relating to the constraint of the purchasing amount of the group, in which the financial product belongs, is the matrix taking the product of the number of groups and the financial products as number of dimensions.
The matrix (Q+X−1Z) is subject to the preliminary process to be divided in the similar method as the coefficient matrix Q appearing in the objective function. On the other hand, since A×(Q+X−1Z)−1 appears in left side and right side
Furthermore, after calculating AדQ+X−1Z)−1×AT, the element structure of the matrix becomes as shown in
Through a sequence of matrix process in
At step 1208, the step width indicative of degree of updating at the current repetition point is calculated. Calculation method of the step width is as follow.
á
p=min(−xk/dx), taking all elements of dx where dx<0 is established (21)
ád=min(−zk/dz), taking all elements of dz where dz<0 is established (22)
As shown in the foregoing expressions (21) and (22), upon execution of interior point method, the point string is updated so that the values of parameters xk and zk to be object of non-negative constraint become positive.
At step 1209, the current repetition point is updated on the basis of the retrieving direction (dx, dy, dz) and the step width (áp, ád) respectively calculated at steps 1207 and 1208. Updating is performed with the following equations.
x
k+1
=x
k
+á
p
dx (23)
y
k+1
=y
k
+á
d
dy (24)
z
k+1
=z
k
+á
d
dz (25)
At step 1210, since it is known that the repetition point after updating satisfies the optimal conditions (11), (12) and (13), this repetition point is set at the optimal solution. These information relating to the repetition point is displayed in the optimal portfolio display means.
Discussion will be given for the embodiment for outputting the information relating to the optimal resource derived in the optimal portfolio deriving means 105 in the optimal portfolio output means 106.
It should be noted that, in
In
In a central processing unit, an application software for performing calculation of the optimal portfolio and a program for displaying a result of simulation to the user are installed for executing simulation for calculating the optimal portfolio based on data input from the four database. Data relating to the optimal portfolio calculated by the central processing unit is transferred to a client computer on the side of the customers via the computer network.
The client computer on the side of the customer receives the information relating to the optimal portfolio calculated by the computer on the side of the server to display the optimal portfolio. Also, in the client computer, an application program for displaying the optimal portfolio and application program for inputting data relating to optimization indicia for the customer have to be installed.
Thus, by establishing the system construction of the optimal portfolio determination system according to the present invention, it becomes possible to determine optimal portfolio.
With the portfolio determining method and system, the fund manager or the like investing to the stock and so forth being deposited capital fund by the customers may efficiently determine the financial product, such as stock of the individual name as purchasing object and purchasing amount for optimizing utility of the investor consisted of the risk and return. It should be noted that, in determination of the purchasing object, the parameter indicating of the earning ability or the like of the individual investing object has to be predicted by executing statistical process, such as regression analysis are predicted for a plurality of times and the mathematical programming problem formulated by solving quadratic programming problem has to be solved for many times. The present invention is significantly effective in shortening the period for calculating the optimal portfolio.
Although the present invention has been illustrated and described with respect to exemplary embodiment thereof, it should be understood by those skilled in the art that the foregoing and various other changes, omission and additions may be made therein and thereto, without departing from the spirit and scope of the present invention. Therefore, the present invention should not be understood as limited to the specific embodiment set out above but to include all possible embodiments which can be embodied within a scope encompassed and equivalent thereof with respect to the feature set out in the appended claims.
| Number | Date | Country | Kind |
|---|---|---|---|
| 2001-330506 | Oct 2001 | JP | national |
| Number | Date | Country | |
|---|---|---|---|
| Parent | 10091033 | Mar 2002 | US |
| Child | 12240903 | US |
