At Play in the Fields of the Sim

This post has no real analytical meaning or purpose. Certainly there is no pedegogy intended.  I am just messing around with a simulator for the hell of it.  Since I have been reading and thinking about geometric returns and utility functions as well as trying to fine-tune my understanding of return variance over the last few weeks I thought I'd fold some of what I've absorbed into the software.  So I added, for each sim-life (in this particular case age 60-90), a synopsis, at the end of the sim-life, of the realized geometric return (product of (1+Rt) for each t=sim year) and the standard deviation of the vector of returns (= stddev[ln(1+r)] for r in each sim-year)  over the life.  Then, of course, I wanted to see what it looked like. Assuming I did not fall flat on my face programming-wise, here below are some snapshots from my sim travels. It's meaning and purpose and application will come some other day...


For some generic assumptions[1] I ran a few sim cycles to see what the charts would look like.  Here is present value of terminal wealth (y axis in chart 1) and the CRRA utility (y axis in chart 2 with a zoom-in of the same thing in chart 3) by way of the geometric return (x axis).  Whether this is correct analysis or correct econ thinking is beside the point.  I say code the code, save the data, then solve the meaning later.  I zoomed into the CRRA utility (W^(1-gamma)/(1-gamma)) in chart 3 because it is hard to see what is really going on when this stuff blows up exponentially.  The highest  U is for highest return; no surprise there...

Then, just to round out my curiosity, here were the same charts with the x-axis now being the standard deviation of sim-life returns.  Again the rightmost chart is just a zoom-in on the middle chart. While there is an itch for a 3D chart somewhere in here let's not get the cart before the horse.

Well, that was fun. I promised no analytical content but really, is looking for high returns with low vol so un-obvious?  No, not really.  Here is what is interesting, though.  If you believe someone like Thomas Piketty, (and I might...but only a tiny tiny bit; and it seems that I am one of a small club that actually read the whole book) the rate of return on invested capital over the last several thousand years has been running, with the exception of the recent century or so, at around 5%[2]... at least before the years when there was inflation and capital taxation and so forth.  That means when I go back and look at stuff like this, while its fun to think I might earn 10 or 12 or 20 % in some years at some super cool level of vol, it's not so fun to assess the probabilities and the risk of that coming to pass. Or rather it is perhaps reassuring, to look at it another way, that lower risk approaches might have a little bit of lift over a lifetime.






---------------notes---------------------------

[1] Assumptions are standard stuff: $1M endowment, 4% spend, age 60-90, no spend flux, 50/50 allocation, no return suppression, relative risk aversion coefficient =4, no SS, fixed longevity, 10000 runs, etc.

[2] Mostly, it seems, government bonds or proxies and real estate or proxies.