But before I go down that path I thought I'd look at simulation and the utility of terminal wealth distributions one more time. It doesn't hurt to assimilate some of this stuff as I move through the math of retirement finance. Today's game is simulate different asset allocations for a 4% and then a 3% spend rate and look at the utility of terminal wealth for different coefficients of risk aversion. I'm pretty sure this has been done before but I'm also pretty sure that I haven't done it myself so off we go.
I used some generic sim assumptions[1] and ran the software thru it's paces for each spend level. The runs were shorter than I wanted (6000) because I was under pressure to get ready for my kids and I wanted to speed things up. Note that I focused in the charts on coefficients of risk aversion > 1 because "1" seems to recommend 100% risk allocation every time. I probably need to check my software to make sure I didn't screw something up but it seems to be working. Here is what I came up with for a 4% spend for different allocations. All this, of course, is quite debatable but let's take it as it is:
And here is the same thing with a 3% spend rate...
And then for a 3% spend...
Does any of this mean anything? Maybe, but first some caveats:
- Don't forget that I am accepting, if not embracing, the idea of simulated negative wealth. This is not shared by everyone.
- Relatedly, I am using two utility functions, one for wealth greater than 0 (for RA =1) and another for wealth > $1 (for RA >1). There is no rationale for this but it works for an amateur just winging it.
- You can probably ignore retirement "Omega." This is my bailiwick. The calc happens to be the area of the CDF above zero returns divided by the area of the CDF below. I am playing with this to see if it might be meaningful.
- Note that the coefficient of relative risk aversion when it is 1 means the Utility is LOG(Wt) which is different from the CRRA function for RA > 1 which is W^(1-gamma)/(1-gamma). That means something to someone somewhere but it screws up my charting so I am focusing on RA >1.
- I'd much rather be focused on consumption utility than terminal wealth utility but I have not gotten there yet.
- The sim runs were pretty short because I was lazy and under the gun re family.
- I used an over simplified 2-asset portfolio (risk and non-risk) but what the heck
- I seem to have been making mistakes lately. For example, I accidentally messed up my own econ math last week. I had wanted to use a U(of negative wealth) = -1/(gamma-1) that I put up on my blog posts accidentally as 1/(1-gamma). I fixed it and it had no impact but you have to realize I am sometimes prone to errors.
- Beware of the scale of the Utility charts. Each U line means something different. In isolation each one scaled to itself makes sense but putting all 4 in one chart, while efficient, may not be as meaningful.
- The "yellow zones" are judgement not math.
- I worry that I have been incautious in my programming or math but that is the advantage of having only two or three readers, the impact is not life or death.
- Markowitz in his recent book warns of the risk of ad-hoc optimality analysis, something in which I indulge frequently, like here; beware.
- I chose 3 % spending and 4% spending kind of arbitrarily. Past analysis tells me that higher than 4 gets into the peril of overspending and fail risk. Alternatively, past analysis tells me that under 3% tends to be a success area if not an area where one gets into some serious self-denial. Beware of that range as my personal bias range. The range might actually be wider in real life, I just picked it as a convenience to me.
So given all that, are there any conclusions I can make? Sure, why not. Let's take a run at this based only on the assumptions we have made and the caveats I have tendered. I'd say it looks like this:
1. For higher spend rates it looks like one must take more risk and the cost of that risk is in much higher magnitudes of retirement fails in terms of the duration of the fail.
2. Lower spending means one can be, if not should be, more cautious or conservative in terms of asset allocation. But this is a risky conclusion. This is where I want to look more at the utility of consumption because the analysis of wealth alone may lead to some serious underspending.
3. The answers to asset allocation variation (in utility terms) seem, with the exception of a RA coefficient of 1, to be pretty consistent across different risk aversion levels for each of the two spend rates.
4. The spend rate, as mentioned in my past posts, seems to dominate everything.
5. Lower spending, as I mentioned earlier in this post, is deceiving. It boosts Utility here but the point of retirement is to spend as much as makes sense without taking on too much risk. That fades into the background in this analysis.
6. For higher spend rates, like the 4% in the example charts, there appears to be a pretty big zone where asset allocation does not matter much, say 40-100% where for lower spend rates (like 3%) the range does matter and is much lower, say 20-50% allocated to risk.
7. For higher spend rates it looks like traditional simulation analysis (fail rates and maybe fail duration) might lead to more conservative and more useful conclusions than utility of terminal wealth. Looking at fail rates while also trying to mitigate fail "magnitude" looks like it leads to a lower and more useful allocation to risk...while looking only at utility (or my omega thing) would lead to high allocation to risk that might increase the magnitude of retirement fails. Which is better? TBD...
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[1] age 65, $1M, terminal age 90, no spend shocks or variance or trends but a constant inflated spend, some tax and fee effects, no return suppression, allocations vary per analysis, 2 asset world, spending varies per analysis, historical returns using Stern data, no SS, no annuity, etc.
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