Kelly Dynamic Portfolio Optimization With Risk Constraints

For a portfolio manager, there are two aspects to improving a trading strategy. The first and most important goal is to achieve a positive expected return. Once this has been achieved, the portfolio manager needs to know what percentage of his capital to risk on each trade. Furthermore, he must concerns the balance between asset allocation and growth. Despite explosive development over multi decades and extensive application to the construction of equity portfolios, traditional portfolio theory rely heavily on assumption of return distribution, real world stochastic processes. At the other extreme,continuous time model is somewhat intractable in a world of risk, transaction costs and other constraints. We believe that Kelly model is essential and has a great importance to real world trading.We investigate the problem of dynamic optimal capital growth of a portfolio under risk and transaction costs constrained. A general framework that the wealth process never falls below a fixed fraction of its maximum-to-date, and one strives to maximize the long term growth rate of its expected log utility was developed. However, when applying to portfolio management with many assets, optimization algorithms such as quadratic programming run into difficulties. In our research, we get the fraction for a portfolio in continuous time by combining law of large numbers and the additivity of the logarithm utility functions. The original model has a major flaw in practice due to the lack of consideration of risk constrains. To meet this gap, we introduced the financial risk measurement model into Kelly portfolio.The results show that the investor’s wealth will exceed the initial value when the fraction is chosen less than critical value. If the asset allocation equals the critical value,the portfolio return equals risk free interest. But, if larger than the value, ruin is almost sure. Growth versus security is a function of the allocate size. Empirical results with real financial data show the feasible allocation set, the higher of the fraction, the faster the growth(return) with the bigger drawdown(risk). We also find that if an investor is willing to accept a larger drawdown, then he will enjoy a higher growth. Since the growth starts to decline while the accepted drawdown exceeds the certain value, we should keep the fraction under certain threshold.To pick an appropriate drawdown level continuously in time so that wealth stay above a desired wealth growth path. Based on this, tailored asset management packages can be designed in different risk level for investors with various risk tolerance abilities.Simultaneous independent risky investment with multi-period, mean variance model has found little use to deal with the growth. Continuous time approximation failed when portfolio with risk and transaction costs constraints. If the wealth and the utility function is positively correlated, this Kelly dynamical portfolio model can be applied. However,we shall argue that it is most risky utility function one should ever consider using and it is most dangerous, this means, a strong ability of risk tolerance as well as the corresponding psychological bearing are required in practice.