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Sharpe Ratios and Alphas in Continuous Time

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  • Nielsen, Lars Tyge
  • Vassalou, Maria

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 that it is optimal for investors to change the fractions of their wealth held in the fund vs. the riskless asset over time.

Suggested Citation

  • Nielsen, Lars Tyge & Vassalou, Maria, 2004. "Sharpe Ratios and Alphas in Continuous Time," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 39(1), pages 103-114, March.
    • Handle: RePEc:cup:jfinqa:v:39:y:2004:i:01:p:103-114_00
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    Cited by:

    1. Eric Benhamou & Beatrice Guez, 2018. "Incremental Sharpe and other performance ratios," Journal of Statistical and Econometric Methods, SCIENPRESS Ltd, vol. 7(4), pages 1-2.
    2. Bermin, Hans-Peter & Holm, Magnus, 2021. "Leverage and risk relativity: how to beat an index," Knut Wicksell Working Paper Series 2021/1, Lund University, Knut Wicksell Centre for Financial Studies.
    3. Edward J. Lusk & Michael Halperin & Atanas Tetikov & Niya Stefanova, 2010. "Forecasting Financial Market Annual Performance Measures: Further Evidence +," American Journal of Economics and Business Administration, Science Publications, vol. 2(3), pages 300-306, September.
    4. Hans-Peter Bermin & Magnus Holm, 2024. "The geometry of risk adjustments," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 47(1), pages 83-120, June.
    5. Guo, Ming & Ou-Yang, Hui, 2021. "Alpha decay and Sharpe ratio: Two measures of investor performance," Economic Modelling, Elsevier, vol. 104(C).
    6. Ramiro Losada López, 2016. "Managerial ability, risk preferences and the incentives for active management," CNMV Working Papers CNMV Working Papers no. 6, CNMV- Spanish Securities Markets Commission - Research and Statistics Department.
    7. Morten Mosegaard Christensen & Eckhard Platen, 2007. "Sharpe Ratio Maximization And Expected Utility When Asset Prices Have Jumps," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(08), pages 1339-1364.
    8. Mahesh K.C & Arnab Kumar Laha, 2021. "A Robust Sharpe Ratio," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 444-465, November.
    9. Eric Benhamou & Beatrice Guez, 2021. "Computation of the marginal contribution of Sharpe ratio and other performance ratios," Working Papers hal-03189299, HAL.
    10. Akuzawa, Toshinao & Nishiyama, Yoshihiko, 2013. "Implied Sharpe ratios of portfolios with options: Application to Nikkei futures and listed options," The North American Journal of Economics and Finance, Elsevier, vol. 25(C), pages 335-357.
    11. Hans‐Peter Bermin & Magnus Holm, 2021. "Kelly trading and option pricing," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(7), pages 987-1006, July.

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