Generalizing the Sharpe Ratio and Infinite Divisibility in Financial Markets

Abstract

Investments with high Sharpe ratios tend to be fixed income investments, particularly short-term bonds. Such securities dominate income investment, but not growth investment. We generalize the Sharpe ratio to interpolate smoothly between risk-averse and risk-seeking behaviors, with the traditional Sharpe ratio as a special case. The generalized ratios may be used to optimize portfolios within a lifestyle strategy or a lifecycle strategy. We demonstrate that the Sharpe ratio has an interpretation connected to value at risk (VaR), if returns are normally distributed. We study broad market indices and indicators for US stocks, US corporate bonds, gold, and Japanese stocks. Using monthly data obtained from FRED, we show that these datasets are neither normal, nor log-normal, nor stable. We argue that none of these distributions is appropriate for market data. We argue instead for using infinitely divisible distributions, and fit Pearson distributions to the data.