An Analytical Confidence Interval for the Treynor Index: Formula, Conditions and Properties

The Treynor index, a well-known, widely-used measure of portfolio performance, is the ratio of the mean excess rate of return of the portfolio to the portfolio's beta. We derive an analytical formula that is designed to yield rigorous confidence intervals on the index. Necessary and sufficient conditions for the Treynor index to be statistically different from zero, are provided. We illustrate our approach with detailed examples and include simulations to help analysts choose the number of periods to study. Finally, some interesting properties of the interval and some sensitivity results are provided.