Universal portfolios generated by an inequality involving the Kullback-Leibler and chi-square divergences

An inequality involving the Kullback-Leibler and chi-square divergences is used to generate new universal portfolios for investment. The stationary vector of an objective function is determined for the purpose of deciding the next-day portfolio given the current-day portfolio and the current-day price relative vector. The two-parameter portfolio is studied empirically by running the portfolio on selected stock-price data sets from the local stock exchange. It is demonstrated that the wealth of the investor can be increased by using the proposed universal portfolio.


Introduction
Daniel Bernoulli's article about log utility and the St. Petersburg Paradox was written in 1738. The article was then translated and reprinted from Latin to English in 1954, by Dr. Louise Sommer et al. [1]. The logarithmic utility function is arguably one of the earliest ideas and contributions to risk aversion. Kelly criterion uses the log utility function in designing a gambling scheme for horse races [2]. Given credit for the idea of gambling scheme, Breiman argued that a sufficient and essential and condition for a favourable game is to consider the minimal time requirement and the magnitude condition [3]. Following the footstep of Kelly and Breiman, Thorp discussed the general mathematical theory and its application on favourable games [4]. On the other hand, the Bernoulli's subjective utility is proposed as one of the criterions for investment strategy in [5].
Markowitz's portfolio theory is the first mathematical formalization of investment diversification [6]. The mean-variance efficient set of portfolio analysis was then greatly refined by Sharpe, especially the reduction of total computing time [7]. Bell and Cover demonstrated that a game theoretically optimal portfolio is possible by maximizing the conditional expected log return [8]. An algorithm that maximizes the expected log return was also presented in [9]. The approach-exclusion theorem of Blackwell was used to define a market portfolio with universal properties in [10]. Given information of the historical data, Algoet and Cover proved that the conditional expected log return can be maximized [11]. The result in [11] is an asymptotically optimal log-optimum investment strategy. In [12], *Corresponding author: yjlee@utar.edu.my Algoet discussed the log-optimum investment in universal gambling scheme, universal prediction scheme and universal portfolio selection schemes.
There is an extensive literature regarding work on universal portfolios (see [13] for early work on universal portfolios). In [14], the asymptotic property of the universal wealth generated by the Dirichlet-weighted universal portfolios compared with the best-constantrebalanced-portfolio wealth is derived. Universal portfolios weighted by moments of probability distribution is proposed in [14]. Another method of deriving universal portfolios using the stationary vector of an objective function is initiated by Helmbold et al. [15]. This method is generalized by Tan and Lee [16] to generate universal portfolios from the Cauchy-Schwarz and Hölder inequalities. The focus of this paper is to generate a universal portfolio from an inequality involving the information divergences. For this purpose, an inequality involving the Kullback-Leibler and chi-square divergences is chosen from [17]. Universal portfolios generated from the f and Bregman divergences are discussed in Tan and Kuang [18].

Some preliminaries
Consider an investment in a market described by m stocks on the th n trading day. Let and choosing natural logarithms, it is clear from (1) Then ( ) by virtue of (5).

Main results
Consider the current day portfolio vector n b and the current price-relative vector n x , the next-day portfolio vector

Proposition 1
Let the objective function (7) where ( ) 1 || nn Q + bb is given by (6),  is the Lagrange multiplier and the parameter " 0.   Then the solution to Proof. Differentiating (7), Rewriting (9) as Remark. The Proposition is valid for any function ( ) h  different from (4) and ( )

Proposition 2
For the objective function ( ) given by (7)    Take the positive root of the equation.
Remark. Assume that The universal portfolio (13) is studied for different values of  and c in the next section. (13) is known as the Kullback-Leibler chi-square (KLCS) universal portfolio.

Empirical results
We analyse the performance of the KLCS universal portfolio (13) using the data sets from Tan and Lee [16] as given in Table 1. Table 1 gives the multiple data sets that consist of 5 Malaysian company stocks in each portfolio. The daily stock prices for each company are available from 3 rd January 2005 to 4 th September 2015, for a total of 2500 trading days. These companies are randomly selected from different industries in order to diversify the risk for long-term investment. S , remain almost constant for parameter values greater than 10. Fig. 1, Fig. 2 and Fig. 3 illustrate the wealth achieved after 2500 trading days for data sets J, K and M, respectively, for selected parameter values of  and c . The series  and c are plotted on the primary axis while the series A close study of Fig. 2(a) and Fig. 3(a) reveal that, holding the parameter  constant for the given interval, as the value of parameter c increases, shows that the accumulated wealth 2500 S also increases. However, this pattern is only applicable to data set J for 0.6 1   , as shown in Fig. 1(a).
On the other hand, holding the parameter c constant for the given interval, the accumulated wealth 2500 S decreases as the value of parameter  increases, as displayed in Fig. 2(b) and Fig. 3(b). In Fig. 1(b), it appears that the accumulated wealth   Table 2 gives the summary of the wealth obtained after 2500 trading days * 2500 S and the next-day portfolios is almost equally distributed among the companies in the portfolios K, L and N.

Conclusion
In this study, we have presented a universal portfolio generated with some information divergence, involving the Kullback-Leibler and chi-square divergences. We work under the setting such that our investment strategy depends only on the historical stock price relatives, with an assumption of no stochastic model of the stock prices. The performance of the accumulated wealth n S heavily relies on the choice parameter  and the universal portfolio (13) performs best for parameter values 01  . Other than that, the choice of a good company in the portfolio helps to maximize the investment return, as demonstrated in portfolios J and M.