Universal portfolio algorithm

Abstract: Cover's universal portfolio has deep connections to universal data compression. In this paper, we provide a statistical view of universal portfolios in order to develop a clearer understanding of their performance on actual financial data sequences. By recasting the analysis of a universal portfolio in statistical terms - with a special emphasis on means and covariances - we are able to resolve a long standing and false perception of a disconnect between information theory and empirical finance. We first show that the universal portfolio can be characterized as a conditional expectation of a multivariate normal random variable. We then show that this implies that the universal portfolio algorithm is asymptotically approximately equal to a constrained sequential Markowitz mean-variance portfolio optimization based on estimates of the mean of a multivariate normal distribution. In light of this equivalence, we propose alternative estimation methods and conclude with some practical investment advice

Abstract: A classic problem in statistics is the estimation of the expectation of random variables from samples. This gives rise to the tightly connected problems of deriving concentration inequalities and confidence sequences, that is confidence intervals that hold uniformly over time. Jun and Orabona [COLT'19] have shown how to easily convert the regret guarantee of an online betting algorithm into a time-uniform concentration inequality. Here, we show that we can go even further: We show that the regret of a minimax betting algorithm gives rise to a new implicit empirical time-uniform concentration. In particular, we use a new data-dependent regret guarantee of the universal portfolio algorithm. We then show how to invert the new concentration in two different ways: in an exact way with a numerical algorithm and symbolically in an approximate way. Finally, we show empirically that our algorithms have state-of-the-art performance in terms of the width of the confidence sequences up to a moderately large amount of samples. In particular, our numerically obtained confidence sequences are never vacuous, even with a single sample.

Abstract: We present a new universal portfolio algorithm that achieves almost the same level of wealth as could be achieved by knowing stock prices ahead of time. Specifically the algorithm tracks the best in hindsight wealth achievable within target classes of linearly parameterized portfolio sequences. The target classes considered are more general than the standard constant rebalanced portfolio class and permit portfolio sequences to exhibit a continuous form of dependence on past prices or other side information. A primary advantage of the algorithm is that it is easily computable in a polynomial number of steps by way of simple closed-form expressions. This provides an edge over other universal algorithms that require both an exponential number of computations and numerical approximation.