The Variance Drain and Jensen's Inequality

CAEPR Working Paper No. 2012-004

14 Pages Posted: 24 Mar 2012 Last revised: 18 Dec 2015

See all articles by Robert A. Becker

Robert A. Becker

Indiana University Bloomington - Department of Economics

Date Written: March 19, 2012


The well-known approximation of the difference between the arithmetic average and geometric average returns as one-half of the variance of the underlying returns is reexamined using Jensen's Inequality. The "defect" in Jensen's Inequality, is given an exlicit formula in terms of the variance following some ideas put forward by Holder. A new form of the AM-GM Inequality follows and is is applied to financial returns. Both exact, and approximate relations between the arithmetic average, geometric average, and variance of returns are discussed. The mathematical formulation of these relations are free of distributional assumptions governing the underlying returns process.

Keywords: AM-GM Inequality, variance, Jensen's Inequality, Holder's Defect

JEL Classification: G00, G10, C80

Suggested Citation

Becker, Robert A., The Variance Drain and Jensen's Inequality (March 19, 2012). CAEPR Working Paper No. 2012-004, Available at SSRN: or

Robert A. Becker (Contact Author)

Indiana University Bloomington - Department of Economics ( email )

Wylie Hall
Bloomington, IN 47405-6620
United States

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