Since I was on a roll with this spending-choice thread (here and here, both of which were using my own assumptions rather than generic), and since risk-aversion has such a big impact in risk aversion math (surprise, that, eh?) I thought that I'd take the work I was doing recently with trying to optimize spend rates for a given set of assumptions (those are in the original post) using a consumption utility simulator...and now push it further. After the last run I was wondering:
"what would all of the spending optima would look like if they were strung, like pearls, along an axis of risk aversion (RA, or alternatively "gamma") from low RA to high (abbreviated interval, in this case)?"
Method
That thought required of me an iterative process, at each increment of RA, that searched for the optimal spend rate, again predicated solely on my personalized assumptions in the prior posts, at that value of RA. So for RA-s from .05 to 4 (I stopped at 5 where I had a computing limit on small numbers and digit cut-offs), I searched, more or less methodically but manually using a utility simulator that anticipates a wealth depletion time (formally, here), for each spend rate optimum at those RAs. I then selected the spend rate with maximum "expected discounted utility of lifetime consumption," or as close as I could get to a max given variations in the sims.
Then, all you have to do is chart it. But here I wanted to throw in some amateur interpretation for fun. High risk aversion seems to me like some sort of odd psychopathology where the compassionate among us would probably offer some help to those suffering from their paralytic fear, as they no doubt also would to the over-confident that are standing too close to the edge of a cliff without their wingsuit. I wanted, in this case to draw some lines, some boundaries of fear past which I think risk-aversion-modeled spending seems somewhere between silly and pathological. Since our modern educational institutions seem to be so enamored with safe spaces, there might also be a case, in the chart/map, to look for one of those, too.
Output
Here is my "psychology of spending" masterpiece. Maybe I can license this...
The Y axis is the optimal spend rate as a function of X. This was a mostly methodical but manual effort to find the maxima at each x. I only did it for RA = .05, .5, 1, 2, 3, and 4. 5 and beyond are not explored. The value funciton V(c) and the simulator that gives us E[V(c)] are described here. A reminder that this chart is conditioned on using precisely the same parameters I used for myself, a perhaps unrealistic expectation.
Green is my "opinion" for the zone of under spending while yellow is my opinion for overspend (in this chart and post, anyway). Blue is my "safe space" as they say in college. Too bad there are none in real life.
To the right of "RA=3" is my guess on when you probably need some help with your fear. An advisor might work, but so would a psychotherapist. Look at it this way, a 2% spend is roughly around where we might start seeing perpetuities. You are not one of those, or shouldn't be ex-excessive-bequest fetishes or living like an endowment, so if you spend like one or even spend less than that, you are likely irrational and need help.
Likewise, to the left, past an arbitrary "RA = 1", these are the people that in other domains of life would be the (now deceased) lion tamers, the (now flattened) wing-suit wearers, and maybe the (now bankrupt) owners of OptionSellers.com. I celebrate their confidence and enthusiasm but at some point it's over-confidence and their family would have thanked me for pulling them back from the edge. I draw the fictional, amateur line at "1" for no real strong reason I can come up with for this post. Maybe because it is because it is Log utility at that point but that's a meaningless artifact.
It amuses me that a 4% constant spend ends up right about in the center of the green-line I called rational.
Case Study: Coefficient of Risk Aversion: < 1
Case Study: Coefficient of Risk Aversion: > 3
Related Links
- A WDT Model
- Five Retirement Processes
- Self-Evaluation of My Own Lifetime Consumption Utility
- Self-Evaluation of My Own Lifetime Consumption Utility, Part 2
Related Reading
- Lachance, M. (2012), Optimal onset and exhaustion of retirement savings in a life-cycle model, Journal of Pension Economics and Finance, Vol. 11(1), pp. 21-52.
- Leung, S. F. (2002), The dynamic effects of social security on individual consumption, wealth and welfare, Journal of Public Economic Theory, Vol. 4(4), pg. 581-612.
- Leung, S. F (2007), The existence, uniqueness and optimality of the terminal wealth depletion time in life-cycle models of saving under certain lifetime and borrowing constraint, Journal of Economic Theory, Vol. 134, pp. 470-493.
- The Utility Value of Longevity Risk Pooling: Analytic Insights, and the Technical Appendix, Milevsky and Huang 2018
- Notes and Comments - Uncertain Lifetime, the theory of the consumer, and the lifecycle Hypothesis, Leung 1994
- Approximate Solutions to Retirement Spending Problems and the Optimality of Ruin, Habibm, Huang, Milevsky 2017
No comments:
Post a Comment