Time aggregation of the Sharpe ratio

Abstract

The Sharpe ratio (SR) is the most widely used risk-adjusted performance index. The building blocks of the SR – the expected return and the volatility – depend on the investment horizon. This raises a natural question: how does the SR vary with investment horizon? To address this question, we derive an explicit expression for the SR as a function of the investment horizon for both simple and log-returns. Assuming independent normal returns distribution, we show that for simple returns, the SR is humped shaped – it rises and then falls with the investment horizon. This finding suggests that time aggregation of the SR using the square-root-t rule will lead to significant errors in ranking portfolios. For log-returns, we show that the SR monotonically rises with the horizon and the square-root-t rule holds true. Using robust bootstrap sampling methods, we empirically corroborate our theory with annual data for a large number of important style portfolios, based on size, book-to-market, and other investment criteria. Our empirical analysis provides robust benchmark SRs for these portfolios over investment horizons that span from one to twenty-five years. Our findings have important implications for investors and portfolio managers who rely on SR for asset-allocation and performance-evaluation decisions.