Method and system for portfolio optimization from ordering information

• US 20060190371A1
• Filed: 02/24/2005
• Published: 08/24/2006
• Est. Priority Date: 02/24/2005
• Status: Active Grant

• Current Assignee: Everything Optimal LLC
• Original Assignee: Everything Optimal LLC
• Inventors: Almgren, Robert, Chriss, Neil
• Granted Patent: US 7,630,930 B2
• Time in Patent Office: Days
• US Class Current: 705/35
• CPC Class Codes: G06Q 40/00 Finance; Insurance; Tax strategies; Processing of corporate or income taxes
• CPC Class Codes: G06Q 40/06 Asset management; Financial planning or analysis

Abstract

A method of optimizing a portfolio includes selecting an investment universe with a finite number of assets, forming a belief matrix based on one or more homogeneous inequality relationships among the expected returns of assets in the universe, selecting those asset returns that are consistent with the belief matrix to form a consistent set of return vectors, selecting a set of allowable weight vectors for the assets in the universe, determining a centroid vector of the consistent set of return vectors with respect to a probability measure, and finding an optimal portfolio by finding a weight vector on a boundary of the set of allowable weight vectors that maximizes an inner product with the centroid vector.

Claims

1. A program storage device readable by a computer, tangibly embodying a program of instructions executable by the computer to perform the method steps for optimizing a portfolio, said method comprising the steps of:

providing an investment universe with a finite number of assets;

forming a belief matrix from one or more relationships among the expected returns of assets in the universe;

selecting those asset returns that are consistent with the belief matrix to form a consistent set of return vectors;

determining a set of allowable weight vectors for the assets in the universe;

determining a centroid vector of the consistent set of return vectors with respect to a probability measure; and

finding an optimal portfolio by finding a weight vector in the set of allowable weight vectors that maximizes an inner product with the centroid vector.

2. The computer readable program storage device of claim 1, wherein the set of allowable weight vectors is determined based on one or more constraints on the universe.

3. The computer readable program storage device of claim 2, wherein said constraints include a risk limit, a total investment limit, a liquidity restriction, a risk constraint with market neutrality, or a transaction cost limit.

4. The computer readable program storage device of claim 1, wherein the probability measure is spherically symmetric and decreases sufficiently rapidly with increasing radius so that a total probability measure is finite.

5. The computer readable program storage device of claim 4, wherein the probability measure h is defined by

h(ρ

)=Cρ

<sup>n−

1</sup>exp(−

ρ

²/2R²),

where ρ

is a magnitude in the set of return vectors, n is the dimensionality of the vectors, R is a cutoff magnitude, and C is a normalization constant chosen so that the total probability measure is equal to one.

6. The computer readable program storage device of claim 4, wherein the probability measure is a distribution uniform on a sphere of radius R.

7. The method of claim 1, wherein the one or more beliefs are in the form of one or more homogeneous inequality relationships among the expected returns, wherein the homogeneous inequality relationships comprise an ordering of the assets in the universe.

8. The method of claim 7, wherein a plurality of sorts are performed on the assets of the universe, a consistent set of return vectors is formed associated with each sort criteria, and the centroid is determined under a probability measure that gives equal weight to each consistent set.

9. The computer readable program storage device of claim 7, wherein the assets of the universe are divided into sectors, and the assets within each sector are ordered according to a separate criteria.

10. The computer readable program storage device of claim 9, wherein if the order of assets in a sector has no significance, the method further comprises the step of computing a superposition of centroids of the remaining sectors of assets for which the expected returns have significance, and averaging the centroid components within the sector where the ordering has no significance.

11. The computer readable program storage device of claim 1, wherein the step of determining the centroid comprises a Monte Carlo computation.

12. The computer readable program storage device of claim 9, wherein the centroid is calculated by performing a Monte Carlo calculation in each sector, and forming a superposition of the centroid of each sector.

13. The computer readable program storage device of claim 1, wherein the step of determining the centroid comprises calculating the expression

$\begin{array}{c}{c}_{j,n}=\end{array}$

n ! ( j - 1 ) ! ⁢

( n - j ) ! ⁢

∫

∞

∞

⁢

w ⁢

 

⁢

F ⁡

( w ) n - j ⁢

( 1 - F ⁡

( w ) ) j - 1 ⁢

f ⁡

( w ) ⁢

ⅆ

w = ⁢

n ! ( j - 1 ) ! ⁢

( n - j ) ! ⁢

∫

0 1 ⁢

 

⁢

F - 1 ⁡

( z ) ⁢

z n - j ⁡

( 1 - z ) j - 1 ⁢

ⅆ

z ,  

wherein c_j,n is the jth component of an n-dimensional centroid vector, f(w) is the probability density, and F(w) is a cumulative distribution of f(w), and w is defined by

$\mathrm{Prob}<br><br>\left\{\begin{array}{c}w<<br><br>y_{j,n}<<br><br>\\ w+d<br><br>\text{\hspace{1em}<br><br>}<br><br>w\end{array}\right\}=\frac{n!}{\left(j-1\right)!<br><br>\left(n-j\right)!}<br><br>{F<br><br>\left(w\right)}^{n-j}<br><br>{\left(1-F<br><br>\left(w\right)\right)}^{j-1}<br><br>f<br><br>\left(w\right)<br><br>d<br><br>\text{\hspace{1em}<br><br>}<br><br>w,$

wherein y_j,n is a jth component of a vector comprising the asset universe sorted into decreasing order.

14. The computer readable program storage device of claim 13, wherein if both j and n are large, approximating the integral by either F⁻

1(z_mean), wherein

$z_{\mathrm{mean}}=\frac{n-j+1}{n+1},\mathrm{or}<br><br>\text{\hspace{1em}<br><br>}<br><br>F^{-1}<br><br>\left(z_{\max }\right)<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{wherein}<br><br>\text{\hspace{1em}<br><br>}<br><br>z_{\max }=\frac{n-j}{n-1}.$

15. The computer readable program storage device of claim 13, wherein if the probability distribution is a normal distribution, approximating the integral by the Blom approximation

$c_{j,n}\approx$

F - 1 ⁡

( n + 1 - j - α

n - 2 ⁢

α

+ 1 ) ,

wherein α

is a parameter in the range from about 0 to about 1.

16. The computer readable program storage device of claim 15, wherein α

is defined by the formula α

=A−

Bj<sup>−

1</sup>, wherein A, B, and β

are determined by comparing the Blom approximation to a numerical integration of the expression for c_j,n.

17. The computer readable program storage device of claim 15, wherein α

is defined by the formula α

=A−

Bj^β

, wherein A is about 0.4424, B is about 0.1185, and β

is about 0.21.

18. A method for optimizing a portfolio, said method comprising the steps of:

receiving a criteria for providing an investment universe with a finite number of assets, and selecting said investment universe;

receiving a set of beliefs concerning the expected returns of assets in the universe and forming a belief matrix from said beliefs;

selecting those asset returns that are consistent with the belief matrix to form a consistent set of return vectors;

determining a set of allowable weight vectors for the assets in the universe;

determining a centroid vector of the consistent set of return vectors with respect to a probability measure; and

finding an optimal portfolio by finding a weight vector in the set of allowable weight vectors that maximizes an inner product with the centroid vector.

19. The method of claim 18, wherein the criteria for selecting the investment universe includes one or more constraints on the universe, and wherein the set of allowable weight vectors is determined based on the one or more constraints.

20. The method of claim 19, wherein said constraints include a risk limit, a total investment limit, a liquidity restriction, a risk constraint with market neutrality, or a transaction cost limit.

21. The method of claim 18, wherein the probability measure is spherically symmetric and decreases sufficiently rapidly with increasing radius so that a total probability measure is finite.

22. The method of claim 21, wherein the probability measure h is defined by

h(ρ

)=Cρ

<sup>n−

1</sup>exp(−

ρ

²/2R²),

where ρ

is a magnitude in the set of return vectors, n is the dimensionality of the vectors, R is a cutoff magnitude, and C is a normalization constant chosen so that the total probability measure is equal to one.

23. The method of claim 21, wherein the probability measure is a distribution uniform on a sphere of radius R.

24. The method of claim 18, wherein the set of beliefs are in the form of one or more homogeneous inequality relationships among the expected returns, wherein the homogeneous inequality relationships comprise an ordering of the assets in the universe.

25. The method of claim 24, wherein a plurality of sorts are performed on the assets of the universe, a consistent set of return vectors is formed associated with each sort criteria, and the centroid is determined under a probability measure that gives equal weight to each consistent set.

26. The method of claim 24, wherein the assets of the universe are divided into sectors, and the assets within each sector are ordered according to a separate criteria.

27. The method of claim 26, wherein if the order of assets in a sector has no significance, the method further comprises the step of computing a superposition of centroid of the remaining sectors of assets for which the expected returns have significance, and averaging the centroid components within the sector where the ordering has no significance.

28. The method of claim 18, wherein the step of determining the centroid comprises a Monte Carlo computation.

29. The method of claim 26, wherein if the assets are ordered into a plurality of sectors, the centroid is calculated by performing a Monte Carlo calculation in each sector, and forming a superposition of the centroid of each sector.

30. The method of claim 18, wherein the step of determining the centroid comprises calculating the expression

$\begin{array}{c}{c}_{j,n}=\end{array}$

n ! ( j - 1 ) ! ⁢

( n - j ) ! ⁢

∫

∞

∞

⁢

w ⁢

 

⁢

F ⁡

( w ) n - j ⁢

( 1 - F ⁡

( w ) ) j - 1 ⁢

f ⁡

( w ) ⁢

ⅆ

w = ⁢

n ! ( j - 1 ) ! ⁢

( n - j ) ! ⁢

∫

0 1 ⁢

F - 1 ⁡

( z ) ⁢

z n - j ⁡

( 1 - z ) j - 1 ⁢

ⅆ

z ,  

wherein c_j,n is the jth component of an n-dimensional centroid vector, f(w) is the probability density, and F(w) is a cumulative distribution of f(w), and w is defined by

$\mathrm{Prob}<br><br>\left\{\begin{array}{c}w<<br><br>y_{j,n}<<br><br>\\ w+d<br><br>\text{\hspace{1em}<br><br>}<br><br>w\end{array}\right\}=\frac{n!}{\left(j-1\right)!<br><br>\left(n-j\right)!}<br><br>{F<br><br>\left(w\right)}^{n-j}<br><br>{\left(1-F<br><br>\left(w\right)\right)}^{j-1}<br><br>f<br><br>\left(w\right)<br><br>d<br><br>\text{\hspace{1em}<br><br>}<br><br>w,$

wherein y_j,n is a jth component of a vector comprising the asset universe sorted into decreasing order.

31. The method of claim 30, wherein if both j and n are large, approximating the integral by either F⁻

1(z_mean) wherein

$z_{\mathrm{mean}}=\frac{n-j+1}{n+1},\mathrm{or}<br><br>\text{\hspace{1em}<br><br>}<br><br>F^{-1}<br><br>\left(z_{\max }\right)$

wherein

$z_{\max }=\frac{n-j}{n-1}.$

32. The method of claim 30, wherein if the probability distribution is a normal distribution, approximating the integral by the Blom approximation

$c_{j,n}\approx$

F - 1 ⁡

( n + 1 - j - α

n - 2 ⁢

 

⁢

α

+ 1 ) ,

wherein α

is a parameter in the range from about 0 to about 1.

33. The method of claim 32, wherein α

is defined by the formula α

=A−

Bj<sup>−

β</sup>

, wherein A, B, and β

are determined by comparing the Blom approximation to a numerical integration of the expression for c_j,n.

34. The method of claim 32, wherein α

is defined by the formula α

=A−

Bj<sup>−

β</sup>

, wherein A is about 0.4424, B is about 0.1185, and β

is about 0.21.

35. A program storage device readable by a computer, tangibly embodying a program of instructions executable by the computer to perform the method steps for optimizing a portfolio, said method comprising the steps of:

providing an investment universe with a finite number of assets;

forming a belief matrix from one or more homogeneous inequality relationships among the expected returns of assets in the universe;

selecting those asset returns that are consistent with the belief matrix to form a consistent set of return vectors; and

determining a centroid vector of the consistent set of return vectors with respect to a probability measure.

36. The computer readable program storage device of claim 35, wherein the method further comprises determining a set of allowable weight vectors for the assets in the universe, wherein the set of allowable weight vectors is determined based on one or more constraints on the universe, and finding the optimal portfolio by finding a weight vector in the allowable set that maximizes an inner product with the centroid vector.