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.

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.