CAEPR Working Paper No. 2012-004
14 Pages Posted: 24 Mar 2012 Last revised: 18 Dec 2015
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: Suggested Citation
Becker, Robert A., The Variance Drain and Jensen's Inequality (March 19, 2012). CAEPR Working Paper No. 2012-004. Available at SSRN: https://ssrn.com/abstract=2027471 or http://dx.doi.org/10.2139/ssrn.2027471