Sharpe Ratio

Bruce C. Dieffenbach ()
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Bruce C. Dieffenbach: Independent author

Chapter 62 in Conjugate Duality in Economic Analysis, 2026, pp 479-485 from Springer

Abstract: Abstract The Sharpe ratio is the maximum value of the ratio of mean to standard deviation in a moment space of random variables. The Sharpe ratio for excess returns plays an important role in optimum portfolio choice and in the capital-asset pricing model (Sharpe, W. F. (1964, September). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, XIX (3), 425–442). We show that the Sharpe ratio is attained at the mean vector. The inverse-Sharpe-ratio theorem is that the Sharpe ratio in a subspace is the inverse of the ratio in the complementary subspace. The Hansen–Jagannathan bound in a financial market is an application of this theorem (Hansen, L. P., & Jagannathan, R. (1991, April). Implications of security market data for models of dynamic economies. Journal of Political Economy, 99 (2), 225–262). Under cost and mean positivity, the maximum ratio of mean to standard deviation among all stochastic discount factors is the inverse of the Sharpe ratio for excess returns.

Date: 2026
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DOI: 10.1007/978-3-032-21396-9_62

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