Looking Back at the Last Couple of Posts
Let's recap. Given the assumptions in note 1 and 33 different simulation runs for the various combinations of allocations and spend rates (see What About a "Retirement Omega?") , we got this for fail rates and median fail duration
where we had to infer what an optimal allocation and spend rate might be, though you can kind of see it if you look closely at both charts at the same time. Then we took a look at an Omega-ratio type calculation using a ratio of partial moments of the terminal wealth distribution. That gave an interesting answer for lower spend rates that looked like:
and pointed towards some preliminary conclusions about spending and asset allocation. Then I switched it up and used the expected value of Log(Wt) (Utility function for a risk-taking decision maker with an adjustment for negative wealth) for each of 22 sims (I skipped the 5% spend runs) to see what that looked like. It looked like this:
Adding Risk Aversion to the Mix
Now here I am going to use a Constant Relative Risk Aversion (CRRA) utility function for thinking about a more risk-averse decision maker. CRRA is evidently one of a large family of utility functions that I know almost nothing about but it seems to be commonly used in personal finance applications. It looks like this (W^(1-g))/(1-g) where W is either wealth or consumption and g or gamma is a factor for risk aversion. "Normal" is evidently somewhere around 2, give or take. If this were a wiki page I'd get a "citation needed" flag here. The function is concave so that lower wealth outcomes feel bad fast and higher wealth outcomes matter less and less as it goes up. In applying it I'll make some bold and maybe unwarranted, if not completely wrong, assumptions:
1. Letting the simulator crash through zero wealth into negative territory is "ok" and actually means something, neither of which is necessarily true in real life. I reason that keeping the full shape of the final simulated wealth distribution is useful as long as we are both aware and skeptical.
2. Since the dis-utility of running out of money is infinite and the logs and some power functions of negative numbers are not quite feasible, I say let's have two U functions, in this case one for all terminal wealth above $1 and another for equal to or below $1. The above-function is CRRA while the below, for me, for this post, without any credible reason, will be a constant set equal to -(1/(g-1)) which I'll consider to be the minimum utility of any particular game. Just to see what happens.
3. We'll suspend any questions about economic legitimacy while we play this.
Given that, when I do the math for 22 runs using a split utility (CRRA + a constant minimum for any terminal wealth less than $1) and a risk aversion factor of 4 it comes out like this:
I'm going to do this for other risk aversion levels soon...
Conclusions?
I am hard pressed to say much of anything different than I did in the last couple of posts especially since I am playing with things I don't fully know and I have not been comprehensive in my testing. My conclusion is still that within a central range, asset allocation does not move the needle nearly as much as changes in spending behavior. Also, higher spend rates force one to take more risk as the lottery type effects of the risk-allocation can sometimes dig you out of a hole...with the cost being bigger and earlier and longer flame-outs. For some new conclusions I can only come up with these:
1. It does feel like there is a "range" for asset allocation where inside the range things are generally ok and outside of which is trouble. It depends on the risk aversion and the spend rate of course but I still think (from other posts) that a range from 40-70% and maybe up to 90-100% risk (remember this is a two-asset 25 period fake sim world, not real life) is the outer boundary within which other factors matter more[3] than allocation. Solve those first and then come back to allocation.
2. The omega-ratio-style probability-weighted approach did a pretty good first-cut job (at the lower spend rate, anyway) at finding the center of what the utility analysis seems to conclude. The problem there is that it is not easy to calculate or explain though most people don't talk in utility terms either.
3. The higher spend rates always (always?) means a lower utility of terminal wealth, which sounds bad but there is a case for that in a no-legacy assumption world. Not sure how to calculate that though.
notes ---------------------------------
[1] $1M age 65, 25 periods, tax effects, fee .006, historical returns, returns suppressed 1st 10 years, no SS or annuity, no spend shocks or random variance, 10000 sim runs, etc.
[2] Formally, gamma is the "coefficient of relative risk aversion." I read somewhere, in different accounts, that observed "normal" risk aversion can range from 1 to 4 depending on the source I was reading. Without knowing or researching or asking I'll use 1 to 4 as a range of normal risk aversion parameters.
[3] The list might look like this for things to have for retirement success:
1. Spending control and a decent endowment
2. Luck
3. A good marriage (see Darrow Kirkpatrick's recent post)
4. tbd
5. tbd
...
8. optimal asset alloction
...
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