Stochastic vs Deterministic - Part 2

Subtitle: How I might "wing it" with a deterministic model (knowing something about how return and net wealth processes evolve over time) when lacking a simulation that might add a little more "realistic" randomness.

Let's say we are contemplating a net wealth process over 35 periods (age 60 to 95 maybe) and we have a spreadsheet model that has arithmetic real return expectation of .03 that is modeled deterministically. Standard deviation of return if we had a stochastic model would be maybe .10.  Wealth units are 25 and the spend unit is 1 (4% spend rate).  How could I work the deterministic model to tease out insights that might only be found by randomizing in a simulation?


The net wealth process in "deterministic mode" might look like this (there are probably better ways to write this but it works ok in a spreadsheet):

and for r = .03, by my calc, I have 9.88 wealth units after 35 periods.  But, you say, "in what world is a series of exactly .03 returns one after the other going to happen.  That's so unrealistic, you should simulate. It's really more robust."  And it is but we can still use the spreadsheet model.

We'll do it like this. First let's iterate the equation above 4000 times in a simulated version (this is in excel otherwise I'd do more; unfortunately the small nbr of iterations may distract from my point) by randomizing r with mean .03 and sd .10 (normal; no gaussian mix today). Then we'll illustrate the evolving distribution of net wealth over 35 periods. Then we will look at some summary stats at t=35 and compare them to the deterministic model.  When we do all that, we get something like this where the wealth unit unit axis is in the front (notice that the image is reversed so negative on the right), the periods go from 1 to 35 coming towards you back-to-front.  The center mass at t=1 is at or near the starting wealth units = 25 and it is vaguely normally distributed at that point...but not so much later:

So here you can see that the net wealth estimate using the deterministic .03 ends up at about the mean (black bar) of the simulated distribution at t=35 which I guess is what one might expect.  But since this net wealth process looks more lognormally distributed at that point, which we would also expect, the median (blue bar) is of much more interest to us as a summary statistic since we are working with a geometric process and the median of terminal wealth = (1+G(m))^N (ignoring consumption).  So let's go back and re estimate "r" as, instead, an n-period geometric return in the presence of vol.  We can borrow one of the simpler formulas that looks like this:

Expected N-per Geo return = r - (1-1/N)/sigma^2/2

For N=35 and sigma = .10 we'd get an r =~ .025.  Plug that into the deterministic model and we get a terminal wealth result of around ~4.4.  So not perfectly synced with the sim but this is a cruddy simulation. But we are at least a little closer to having a better link to the idea of a simulated distribution and the lognormal processes for terminal wealth. For starters...

Conclusion

I guess my point here is that if you are (a) aware of the imperfections of deterministic models, (b) know how random return generation works, (c) know how multiplicative processes work over non-infinite time frames, and (d) know how net wealth processes work and evolve...then you can roughly, by holding your thumb up to the horizon and swallowing your quantitative pride, point a dumb deterministic model at an expected distribution -- where the expected distribution is still only in your imagination -- and still get kinda, vaguely close to a meaningful point of interest (say roughly a median expectation).

By the way, if you look closely at the figure above, you may notice what is not there.  There are extreme wealth-unit outcomes not shown; they go all the way up to ~250.  These are what pull the mean up pretty far. These are also surprisingly uninteresting to us.   If we become bazillionaires in 35 years, who cares? Everything worked out and we buy a lear jet.  What is of interest for planning purposes, is the other side of the distribution (right side in this case because we reversed the image).  That is where all the stress comes from. And there, we can still use a deterministic model to play with the far side of the median (idk, maybe think about sequence risk or retiring in the wrong era...). In this case we would play around interactively, perhaps testing lower and lower r just to see what happens knowing we are probably going to be at least somewhere in the distribution. The only problem is that we just don't know the likelihood or the extent of the boundaries.  Which we never would anyway because even the simulation is just a model and not real life.  So, personally, I think that deterministic can work just fine in a pinch as long as you can see the processes and distributions in the mind's eye and you know what you are getting yourself into.



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On Stochastic vs. Deterministic Models - Part 1