Arithmetic Means, Geometric Means, Accumulation Functions, and Present Value Functions

Skoog, Gary R. and James E. Ciecka. 2008. “Arithmetic Means, Geometric Means, Accumulation Function, and Present Value Functions.” The Earnings Analyst 10: 1-29.


There are “well known” relations between the arithmetic mean of a random variable and the mean. Part of this paper is offered with the belief that perhaps some of these relations are not as well-known and transparent as is sometimes assumed. In addition to asserting several relations, we provide proofs of propositions with the thought that forensic economists engaged in personal injury and business valuation litigation will find it useful to have a single and convenient reference for concepts and propositions involving the arithmetic and geometric mean. This paper also deals with a proposition and an often-cited example that has been proffered to show that the arithmetic mean of returns on an investment, when compounded for multiple time periods, gives the expected value of wealth. We examine this proposition and show sufficient conditions under which it is correct. Finally, and perhaps of most interest to forensic economists, this paper examines the expected present value function when discounting with the arithmetic mean and geometric mean.


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