Fractional Growth Portfolio Investment

We review some fundamental concepts of investment from a mathematical perspective, concentrating specifically on fractional-Kelly portfolios, which allocate a fraction of wealth to a growth-optimal portfolio, with the remaining fraction collecting (or paying) interest at a risk-free rate. Using stochastic differential equations, we lay out a coherent continuous-parameter time-series framework for analysis of these portfolios, explaining relationships between Sharpe ratios, growth rates, and leverage. We see how Kelly’s criterion prescribes the same leverage as Markowitz mean–variance optimization. Furthermore, for fractional Kelly portfolios, we state a simple distributional relationship between portfolio Sharpe ratio, the fractional coefficient, and portfolio log-returns. These results provide critical insight into realistic expectations of growth for different classes of investors, from individuals to quantitative trading operations. We then illustrate application of the results by analyzing performance of various bond and equity mixes for an investor. We also demonstrate how the relationships can be exploited by a simple method-of-moments calculation to estimate portfolio Sharpe ratios and levels of risk deployment, given a fund’s reported returns.