UNIVERSAL RISK BUDGETING

I juxtapose Cover’s vaunted universal portfolio selection algorithm ([Cover, TM (1991). Universal portfolios. Mathematical Finance, 1, 1–29]) with the modern representation of a portfolio as a certain allocation of risk among the available assets, rather than a mere allocation of capital. Thus, I define a Universal Risk Budgeting scheme that weights each risk budget, instead of each capital budget, by its historical performance record, á la Cover. I prove that my scheme is mathematically equivalent to a novel type of [Cover, TM and E Ordentlich (1996). Universal portfolios with side information. IEEE Transactions on Information Theory, 42, 348–363] universal portfolio that uses a new family of prior densities that have hitherto not appeared in the literature on universal portfolio theory. I argue that my universal risk budget, so-defined, is a potentially more perspicuous and flexible type of universal portfolio; it allows the algorithmic trader to incorporate, with advantage, his prior knowledge or beliefs about the particular covariance structure of instantaneous asset returns. Say, if there is some dispersion in the volatilities of the available assets, then the uniform or Dirichlet priors that are standard in the literature will generate a dangerously lopsided prior distribution over the possible risk budgets. In the author’s opinion, the proposed “Garivaltis prior” makes for a nice improvement on Cover’s timeless expert system, that is properly agnostic and open to different risk budgets from the very get-go. Inspired by [Jamshidian, F (1992). Asymptotically optimal portfolios. Mathematical Finance, 2, 131–150], the universal risk budget is formulated as a new kind of exotic option in the continuous time Black–Scholes market, with all the pleasure, elegance, and convenience that entails.

A Note on Universal Bilinear Portfolios

This note provides a neat and enjoyable expansion and application of the magnificent Ordentlich-Cover theory of “universal portfolios”. I generalize Cover’s benchmark of the best constant-rebalanced portfolio (or 1-linear trading strategy) in hindsight by considering the best bilinear trading strategy determined in hindsight for the realized sequence of asset prices. A bilinear trading strategy is a mini two-period active strategy whose final capital growth factor is linear separately in each period’s gross return vector for the asset market. I apply Thomas Cover’s ingenious performance-weighted averaging technique to construct a universal bilinear portfolio that is guaranteed (uniformly for all possible market behavior) to compound its money at the same asymptotic rate as the best bilinear trading strategy in hindsight. Thus, the universal bilinear portfolio asymptotically dominates the original (1-linear) universal portfolio in the same technical sense that Cover’s universal portfolios asymptotically dominate all constant-rebalanced portfolios and all buy-and-hold strategies. In fact, like so many Russian dolls, one can get carried away and use these ideas to construct an endless hierarchy of ever more dominant H-linear universal portfolios.

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.

Universal portfolio analysis of Malaysia’s stocks for different durations

In this research, four proposed finite order universal portfolios were used to study Malaysia’s stock market comprehensively and the constant rebalanced portfolio (CRP) was used as a benchmark for comparison. The empirical performance of the four universal portfolio strategies was analysed experimentally concerning 95 stocks from different categories in Kuala Lumpur Stock Exchange (KLSE) from 1 January 2000 to 31 December 2015. Combinations of three stocks data from the selected 95 stocks are used for study for short-term (1-year duration), middle-term (4-years and 8-years durations) and long-term (12-years and 16-years durations). The empirical results showed that the performances of the proposed universal strategies are outperform CRP in 1 year and 4 years durations, but did poorly in 8-years, 12-years and 16-years durations. Therefore, these four UP strategies are empirically considered to be good investment strategies in the short-term.

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.