Why Volatility Squared Appears in the Geometric Return Correction
Summary
The document raises a teaching question about the apparent mismatch in subtracting half the squared standard deviation from a rate when estimating geometric growth. It notes the familiar approximation that geometric mean return equals arithmetic mean return minus half the variance, and points to the derivation through Itô’s lemma and the lognormal distribution. The requested explanation is intended for learners with little statistical background.
The source does not provide a resolved explanation; it asks for one and rejects a hand-waving appeal to the correction being numerically small. Thus its value is mainly identifying the conceptual issue: rates and volatility need consistent time units, and variance is a rate-squared quantity whose time scaling is essential in continuous-time models. The document includes no worked numerical example or derivation, so it cannot on its own establish a complete dimensional analysis or clarify the approximation’s assumptions.
Key ideas
- The geometric growth correction subtracts half the variance from the arithmetic growth rate in the stated approximation.
- The question concerns how a squared volatility term can be reconciled with the units of a rate.
- The cited background connects the correction to Itô’s lemma and lognormal returns.
- The document asks for an explanation but does not itself supply a complete derivation or example.
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Full text
# How do the units compare inside the (rate - 0.5*sigma-squared) correction? # How do the units compare inside the (rate - 0.5*sigma-squared) correction? Usually, I find the units of the mean and the standard deviation of a distribution to be (quite obviously) the same. Can anyone come up with a really simple explanation (for MBA students, some of whom are essentially “poets”, taught the absolute minimum of statistics), of the seeming paradox specifically as regards the units involved, of the subtraction of half the square of SD in calculating the geometric rate of return: (r-0.5*σ^2) I should add that I have already checked the origin of this expression, via the application of Ito’s Lemma, and its relationship to the difference between the arithmetic and geometric mean in relation to the lognormal distribution, including a variety of Wikipedia entries – and even asked a couple of experts to explain, but none has been able to make clear the answer to this apparently simple question about the units involved. Answer What are the units of the variables appearing in a standard stochastic differential equation for a Wiener process? comes close, but doesn't quite answer it sufficiently for my target audience. The best source on this seem to be http://www.timworrall.com/eco-30004/bscholes.pdf in which Tim Worrall makes clear on p. 17 that this correction factor is in fact an approximation ``` geometric mean ≈ arithmetic mean – 0.5 variance ``` But I'd rather not give the students an unsatisfying “hand-waving” answer that "It's a small number and the seeming difference in the units doesn’t really matter at the end of the day." Help greatly appreciated.
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