Question: What do geometric returns look like when exposed to time, spending and volatility?
I went back to this topic, one that I covered before a few times, for a couple reasons the most important of which was that I had some code errors (resolved) and some design trade-offs (not really perfectly resolved) in presenting the analysis via my R-script that I was concerned about. I'm a little more comfortable now but not 100%. It's a good thing I can't get fired. I did this analysis in the past, and here, using the formula picked up at ERN:
where right of the arrow is the "perfect withdrawal rate" version of the formula presented here in the past month or so and left of the arrow is the "FV of $1" version based on compound returns and spending (C1 is the compound total return before spending, w is consumption, Ct is the backward products that are summed, w[sumCt] is an opportunity cost concept, and s is a scaling factor not used here, pt is pensionized income which I typically ignore for simplification).
There were some other reasons, too, now that some time has passed, that I decided to revisit this. One example is that, when put into simulation mode, I'm pretty sure that this is the same "net wealth engine with volatility" that represents the second two terms of the Kolmogorov equation profiled a few weeks back. It is the same process, it's just a little more tractable in this form since only return volatility is isolated in a way I can manage without high order math. A Monte Carlo simulator does the same thing it is just a little more opaque and a little less tractable. For another example, I think it's always good to remind oneself that returns are not the only game in town and that there are cross-over points where lower vol strategies can win over higher vol, efficient frontiers notwithstanding, especially in the presence of spending, something that mean-variance analysis more or less ignores.
Let's run this again just for fun. Two strategies A and B. The parameters are kinda arbitrary, but maybe I cherry-picked a little bit. I'll note that higher returns with same or lower vol are probably going to dominate no matter how I present it. At least it's worth checking. Strategy A here has a mean return expectation of .08 and a std dev of .15. B has mean .07 and sd of .05. Consumption is a constant 4% though I probably have to be careful here; I'll double check before the next post. We'll use 30 periods and 10000 iterations.
In the first illustration on the top I can see the effects of vol on annualized (expected) compound returns over time. Note that the two strategies more or less converge in the out-years. This is the nature of geometric returns in the presence of volatility. I didn't estimate it but at long horizons an approximator for geometric returns (say: mean return - sd^2/2) would be ok but definitely not over shorter horizons which some of us might have. Spending (bottom lines) exacerbates it even more and can turn a higher return strategy upside down...and sooner... compared to a lower more placid strategy (in retirement terms this might equate to sequence of returns risk).[1]
Since the annualization I did above has some trade-offs in both the analysis and presentation, this is where I went back to the code and added a view of this process without annualization. In this case I took the FV in the formula above and let the 10000 iterations rip. Then I took the median path of $1 of wealth as well as the minimum and the maximum. I did this for each strategy A and B (but only for the spending case which is equivalent to the lower two lines in the chart above). The results are going to look a little like Monte Carlo Simulation because that is more or less what this is except that this is a little easier to work with. When I plot it out it looks like this:
and here I clipped the upper and lower 5% in terms of quantiles just to get rid of some extremes...
We can see the nature of the two strategies pretty clearly here in these 2 charts. The lower vol strategy avoids ruin unambiguously and monotonically while it also has a (slightly) higher median outcome over most years. There is a diminution of the upside but that, of course, is part of the game we are playing in order to survive as retirees. Volatility combined with time and spending (and vaguely similar returns) is not our friend. Efficiency, depending on how you define it, is. If one were to have a choice[2] it's pretty obvious where one would go.
There is more to say here, I think, but this is one of those post-hurricane posts where I have hungry children that also want to be driven to their friend's house. When choosing between thorough and accurate retirement-quant stuff and un-volatile kids, the choice is clear....
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[1] this is another place I have to be careful. Because I have rendered this in annualized terms the math gets a little wanky when the net-wealth engine (when spending is present) goes negative. I made some design choices that might overstate the annualized expected returns net of spending in the later years 15 or 20+. Another thing to remember is that these are expected value lines. Individual paths, just like the spaghetti models in weather and Monte Carlo reports can be all over the place.
[2] Strategy B is more or less my alternative systematic risk strategy. A is a strawman that stands in for a 60/40 portfolio that has more vol than history would imply in real life so far... but you never know.
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