Shrinkage estimation of Kelly portfolios

Although the Kelly portfolio is theoretically optimal in maximizing the long-term log-growth rate, in practice this is not always so. In this paper, we first show that the sample plug-in estimator of the Kelly portfolio weights is actually biased, and we then propose an unbiased estimator as an alternative. We further derive a shrinkage estimator under the objective of minimizing the expected growth loss of the actual growth relative to the true growth. An 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. The 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 advantages of our proposed unbiased and shrinkage Kelly portfolios for long-term investments are additionally confirmed by stock investment in the U.S. market.