Statistical inference for Kelly portfolios and its applications

Abstract

This thesis focuses on statistical inference for the optimal Kelly portfolio and its relevant applications.

Statistical inferences for Kelly criterion are conventionally made via the asymptotic and the likelihood based methods, which perform poorly when the sample sizes are not large enough. As an alternative, we propose making accurate inferences based on the higher order asymptotic theory, through which a more accurate statistic about the optimal Kelly is derived. Extensive simulation studies indicate that the higher order asymptotic statistic indeed improves the inference accuracies compared to those conventional approaches. Besides, we provide statistical evidences to explain the popularity of the half Kelly strategy via extensive simulation studies. Our studies suggest that the half Kelly seldom surpasses the true full Kelly, and is with high probabilities to fall into confidence intervals of the full Kelly at certain significance levels. We further propose a confidence-interval-based (CI-based) Kelly strategy, which utilizes the lower confidence limit of the optimal Kelly at certain confidence levels to make practical investments. Compared to the full Kelly and the half Kelly strategies, our CI-based Kelly strategy can provide good trade-off between investment growth and risk. Several scenarios of simulation studies are presented to demonstrate such advantages. The conclusions are additionally confirmed by empirical applications to two real stock market indices.

Although Kelly portfolio is theoretically optimal in maximizing the long-term log growth rate, in practice it is not always so. In the second part, we first show that the plug-in estimator of the Kelly portfolio weights is actually biased, and then we propose an unbiased estimator alternatively. We further derive a shrinkage estimator under the objective of minimizing the expected growth loss of the actual growth from the true growth. Explicit formula for the shrinkage coefficient is established. Statistical properties for the shrinkage coefficient are studied through extensive Monte Carlo simulations, and conditions for obtaining accurate estimates for the shrinkage coefficient are also discussed. Effectiveness of the proposed unbiased and shrinkage Kelly portfolios in reducing the expected growth loss are validated by various simulation studies. It is found that our proposed shrinkage Kelly portfolio has superior performances in growth loss reduction, followed by the unbiased Kelly portfolio, and the sample plug-in Kelly portfolio. The advantage of our proposed unbiased and shrinkage Kelly portfolios in long-term investments are additionally confirmed by stock investments in the U.S. market.

In the last part, we make accurate inferences for the Kelly portfolio weights through the Skovgaard method, the Bartlett correction method, and the generalized pivotal quantity (GPQ) approach, due to the unsatisfactory performances of the likelihood ratio statistic. Analytical formulas for computing the Skovgaard statistics are established. A parametric bootstrapping procedure for the Bartlett correction method and an algorithm for the GPQ approach are also proposed. More accurate performances of these three methods than those of the likelihood ratio statistic are demonstrated through simulation studies. A real portfolio example is also used to illustrate the differences among those considered methods on inferring the Kelly portfolio weights.