Or another lesson for me anyway. I went back today to the article by Richard
Michaud "A Practical Framework For Portfolio Choice" Journal of
Investment Management 2003. There was
another example he used (vs another post I did) to illustrate geometric returns that I wanted to figure
out because I wasn't quite sure where the numbers came from. He can roll over this stuff...he has a PhD. in
math. Me, I gotta go slow. Here is his
example on page 5 (if you are a quant this is probably like doing coloring
books in pre-school but me, I need to know):
Suppose an asset with two equally probable outcomes in each investment period: 100% or −50%. What is the expected geometric mean return for investing in this asset over the investment horizon? In general it is not 0%. A correct answer requires more information. Suppose we plan to invest in this asset for only one period. The expected return of the investment is 25% not 0%. Suppose you are considering investing in the asset for two or three investment periods. The expected geometric mean return is 12.5% over two periods and 8.26% over three periods. For any finite horizon, the investment has a variance as well as an expected return. It is only at the limit, when the number of investment periods is very large, that the expected growth rate of investing in this asset is 0%.
I got the 25% thing he's talking about because it's easy. I got the 0% thing because I
had used the formula for the limit of the geometric mean before. It was the
12.5% and 8.26% numbers that I didn't quite get...or rather that I could not quite duplicate. But let's back up a bit. There are two ways
to go about this kind of thing: 1) the dumb way, and 2) the
I-won't-even-call-it-smart-because-it's-obvious-in-retrospect way. #1 is to go back and try to use a decision
tree used in a prior project that was probably flawed in conception and
execution and then screw around with it for an hour trying to understand why
you couldn't replicate the results. # 2 is to take the paragraph literally at
face value and use simple math to calculate the result. Guess which path I went down? uh, yeah, that's right.
If I had been listening attentively I would have realized
it's as easy as using that formula above over N periods for each scenario. Sounds obvious right? I won't show you all
the wrong ways I did this. That would just be self-flagellation. But here is what
it looks like when you do it right (and confirming the 12.5% and the 8.26%
values, btw):
I threw in that N = 4 section as a bonus.
And then, if one were to be even a little more anal about it than normal, like someone we might know, one might want
to chart it out like this…
And again, just for fun, here is the limit-math from the
Michaud article just to show that the limit of geometric return, using only his example, as N gets big is, in
fact, zero…
And so there you have it, Michaud decoded. But why am I
doing this? Three reasons:
1. As I've mentioned before, I am not trying to be an
encyclopedia with "answers." I am just trying to report on my own journey
on how and when I come across ret-fin stuff that I run into every week that I
find interesting and that I want to figure out,
2. By writing stuff down online as I figure it out, it helps
me consolidate my understanding and learning in a formal way that I can
remember and go back to,
3. I hope that in some way that I am helping someone
somewhere sometime. On the other hand let's be realistic about this just for fun. Let's
say there are 3/7Billion people that might have the same interest, 5000/7B
people that read the article, and 3/7B people that read my blog and want to
know. That might make the cumulative probability that I am
helping anyone anywhere somewhere around 1.3e-25. But, hey, at least it is non-zero…
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