On page 435 of Reading 7: Statistical Concepts and Market Returns, it’s stated that, “The geometric mean return approximately equals the arithmetic return minus half the variance of return.” What is the derivation of this formula?
r_{geom} = \sqrt[n]{\prod_{i=1}^n\left(1 + r_i\right) - 1}
= \sqrt[n]{\left(1 + r_1\right)\left(1 + r_2\right)\cdots\left(1 + r_n\right) - 1}
More to come.
In the meantime, you should run some numbers in Excel. Generate 10 – 20 random returns between, say, −10% and + 10%, calculate the geometric and arithmetic means as well as the (population) variance, and compare the values. That should be enough to convince you that it works pretty well, even without the derivation.
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Yeah, the derivation is more of a curiosity than anything else. I ran some numbers in excel, and the formula works pretty well, as you said. It’s generally accurate out to two decimal places (in % form).