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:
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.