Universal portfolios generated by rational functions

The f -divergence of Csiszar is defined for a non-negative convex function on the positive axis. A pseudo f -divergence can be defined for a convex function not satisfying the usual requirements. A rational function where both the numerator and the denominator are non-integer polynomials will be used to generate universal portfolios. Five stock-price data sets from the local stock exchange are selected for the empirical study. Empirical results are obtained by running the generated portfolios on these data sets. The empirical results demonstrate that it is possible for the investors to increase their wealth by using the portfolios in investment.


Some basic theory 2.1 Definition
Investment in a stock market with m number of stocks is considered. The market is described by a sequence of price relative vector = ( ) on the n th trading day. The is the price relative of the i th stock which is defined as the ratio of the closing price of stock to its opening price on day n, for = 1,2, ⋯ , . A portfolio = ( ) is the portfolio strategy used on the n th trading day where is the weightage of the wealth at the beginning of the n th trading day invested on stock , for = 1,2, ⋯ , , 0 ≤ ≤ 1 and ∑ = 1 =1 . The initial wealth invested is assumed to be 1 unit. The wealth ( ) at the end of the n th trading day is (1) Let ( ) define a convex function on (0, ∞), is strictly convex at = 1, and (1) = 0. The Csiszar's -divergence of two probability distributions = ( ) and = ( ) is defined as The -divergence given in Equation (2) is also known as the -disparity difference between two probability distribution = ( ) and = ( ) given the ( ) is a -disparity function which fulfill the followings. A continuous function of ( ) on (0, ∞) is adisparity function if i.
(0) is determined by the continuous extension of ( ) (see [6], pg. 29) In the statistical inference literature, the -disparity function is also known as the phidisparity function. The convex function ( ) from the -divergence is an the -disparity function.
For two portfolio vectors +1 and , the -divergence is defined as

Main results
Let ( ) define a given -disparity difference. Then the Type-1 universal portfolio +1 generated by ( ) corresponding to the object function is given by where is the Lagrange multiplier and the parameter and are assumed to be constants.

Proposition 1
For 1 = 2 = > 0 and > 1, a valid version of the universal portfolio +1 generated by ( ) is given by and

Proposition 2
For = 1, a valid version of the universal portfolio +1 generated by ( ) is given by for = 1,2, ⋯ , , where is given by Equation (10) and the numerator of Equation (11) is positive for selected values of , 1 and 2 .
Simplifying the Equation (15), Solving the quadratic in and normalizing +1, , the universal portfolio (11) is obtained. Table 1 shows the list of Malaysian Companies chosen from Kuala Lumpur Stock Exchange (KLSE) for empirical study. These data are collected from Bloomberg for the period 3 rd January 2005 to 4 th September 2015. Each stock consists of 2500 trading days. Then stocks are grouped into stock-price data sets, J, K, L, M and N. Table 3 and 4 show the result obtained after the universal portfolio (9) and (11) in Proposition 3.1 is run on the five data sets J, K, L, M and N, respectively. Table 3 lists the final wealth 2500 after 2500 trading days for selected values of the parameters , and while fixing = 0.1. Table 3 also includes the final portfolio 2501 for each stock. The Table 4 gives the final wealth 2500 for selected values of parameters , , 1 and 2 while fixing 1 = 1 and 2 = −1. Table 4 also includes the final portfolio 2501 for each stock. Table 2 gives the wealth achieved by running the Type 1 Helmbold universal portfolio on stock-price data sets J, K, L, M and N. The comparisons of the performance between Type 1 Helmbold universal portfolio, universal portfolio (9) and (11) is done.

Empirical results
By comparison, the result obtained from universal portfolio (9) and (11) are closed to the result from Type 1 Helmbold universal portfolio. The stock-price data sets J, K and M are good performing portfolios while stock-price data sets L and N are poor performing portfolios. However, in-term of wealth achieved, both universal portfolio (9) and (11) perform slightly better for stock-price data sets J, K, L and M as compare to Type 1 Helmbold universal portfolio. But Type 1 Helmbold universal portfolio perform better for stock-price data set N. However, the performances between Type I Helmbold universal portfolio, universal portfolio (9) and (11) do not show any significant difference for all stock-price data sets.   Table 3. The best wealth 2500 obtained after 2500 trading days by running the universal portfolio generated by rational functions (9) Table 4. The best wealth 2500 obtained after 2500 trading days by running the universal portfolio generated by rational functions (11) over the data set J, K, L, M, N for selected value of , , 1 , 2 and the final portfolios after 2500 trading days 2501 are listed.