Sharpe Ratios and Alphas in Continuous Time

Abstract

This paper proposes modified versions of the Sharpe ratio and Jensen's alpha which are appropriate in a simple continuous-time model. Both are derived from optimal portfolio selection. The modified Sharpe ratio equals the ordinary Sharpe ratio plus half of the volatility of the fund. The modified alpha also differs from the ordinary alpha by a second-moment adjustment. The modified and the ordinary Sharpe ratios may rank funds differently. In particular, if two funds have the same ordinary Sharpe ratio, then the one with the higher volatility will rank higher according to the modified Sharpe ratio. This is justified by the underlying dynamic portfolio theory. Unlike their discrete-time versions, the continuous-time performance measures take into account the fact that it is optimal for investors to change the fractions of their wealth held in the fund versus the riskless asset over time.

Keywords: Sharpe ratio, Jensen's alpha, performance evaluation, dynamic portfolio management

JEL Classification: G11