# reconciling arithmetic and geometric compounding

I have just been through 4 papers that make all sorts of clever claims about the 'alternate universes' of arithmetic returns and geometric returns, how thr twain shall never meet, and how they are inconsistent inherently. I don't see this. Can someone clarify, and correct me if I am wrong?

Let's assume you are looking at 1-year stock market returns. One approach is the 'geometric' one, where we say $r_g=ln(\frac{p_{365}}{p_0})$ The other is arithmetic, where we say $r_a=\frac{p_{365}}{P_0}-1$. If we started at 100 dollars, and end at 110, the latter gives 10.0%, while the former gives 9.53% due to continuous compounding.

Now, many an author claims these can't be reconciled; you need to pick arithmetic compounding vs geometric compounding; etc.

But unless I am being a dunce, after 1 year the arithmetic approach gives $100\cdot(1.1)=110$, while the geometric approach gives $100\cdot e^{.0953}=110$ as well. You can actually go work it out on a spreadsheet, but by induction it is obvious year 2 will give you 121, either by again multiplying by $(1+.10)$ or $e^{.0953}$ - i.e., year two's value is either $100\cdot(1+r_a)^2$ or $100\cdot e^{2r_g}$. In fact for every year t, it is $100\cdot(1+r_a)t$ or $100\cdot e^{rt}$, which is identical.

Is that not correct, or is there some subtely (that I don't appreciate, or perhaps doesn't matter to me) that will cause non-convergence -if say, you looked at one-tenth of a year or daily trading? Though even then, it seems to me the answer is still the same, but we just have ,.1 (or 1/365) in the exponent for t.

BTW, I know it will get a bit 'messier' if we go stochastic. But when we do, isn't that, itself, just an artifact of the larger asymmetry that arises because $(1+r_a+\delta)$ has less additional return that $e^{(r+\delta)t}$ for some random (upside) perturbation of sized $\delta$ (and vice versa on the downside)?

• I don't know where you read that, but I think the authors you read mean that "10.0%$\neq$9.53%" is the irreconcilability. Which is a false problem in my view. There's nothing interesting going on here. – Raskolnikov May 23 '18 at 23:08
• That's why I asked. I saw this paper and found it mighty confusing; it seems to me the reason returns becomes $\sigma$-dependent is because, when you use the 'geometric' approach you end up, in the limit, getting log-normal price distributions, and the nature of that distribution is its stochastic evolution is that it depends both on $\mu$ and $\sigma$. Also, I think people blindly plug in $\mu$ from measured annual returns into the underlying parameters for the log-normal. It all started from this page: quantivity.wordpress.com/2011/02/21/why-log-returns – eSurfsnake May 23 '18 at 23:34