The Model
The model has been discussed before but I am using an animated value function based on a survival weighted, subjectively discounted, CRRA utility of consumption over a lifetime where consumption snaps to income when wealth runs out before end of life. The model was discussed here so I won't belabor it in this post.
Core Assumptions
The assumptions in this post are perfunctory and kind of shoot-from-the-hip but will do for now. They include the following, noting that personal wealth info is redacted: age = 61, risk aversion coefficients* of 1 and 2 are used in separate runs, max Social Security that is realized with full probability over the entire life interval starts at age 70, available assets are net of liabilities and include only what is reasonably monetizable over the life-cycle, nominal return of .05 with standard deviation of .11, inflation = 3%, spending is constant with all the flaws of that assumption, every parameter is stable over time with no shocks, and there were 10,000 iterations per data-point. Survival probabilities (conditioned on my age of 61) were extracted from an actuarial table for annuitants.
You'll note, by the way, that the real return implied is 2%. Seems low maybe but most papers written in 2018 and 2019 look like they are selling the idea that we should be thinking conservatively like this now.
*A Note on my Risk Aversion Assumptions
I know nothing about the real economics of risk aversion. I have heard by rumor that some think that a value of gamma of around 3 is sometimes observed in the field. I have no idea. And that's only for people that believe in utility functions. Others might reject this kind of thing out of hand.
Me? I'll suspend disbelief for a second and will go with utility theory. I'll assert, based on nothing, that a coefficient of "3" might actually be a risk-aversion frontier past which we'd find "pathological fear" more than we'd find "aversion." That is territory that is more likely to respond to counseling than it would respond to simulation and financial modeling so I don't feel compelled to go beyond that boundary.
A coefficient of 1 happens to mean log utility which is a useful baseline. Below 1 doesn't interest me. I therefore pick, in advance and with minimal thought, a coefficient of 2 as my number for no other reason than it's between 1 and 3 and seems about right based on what little I know. So that means that I will first run a "low risk aversion" baseline with RA = "1" and then run a second set using "my" fake number of 2. We already know that if we used RA = 8 or 16 or whatever, spending would go really really low because that kind of utility math represents people hunched over in a debilitating crouch-of-fear. That's not me...not yet, anyway, so I don't feel like modeling that. Oh, and I forgot: I also assume that risk aversion is stable over time, which is a bogus assumption.
The Results
For selected spend rates (x axis) these (figure 1) are the expected values of the value function or the "expected discounted utility of lifetime consumption (EDULC) (y axis)." Note, if I haven't mentioned it, that spending snaps to income when wealth depletes. This has a big impact on the power utility function summed over life . Also note that the only income in these runs is social security. Life income (annuities) could have been allocated from wealth with positive effects in EDULC but wasn't. Maybe next time.
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| Figure 1. EDULC for RiversHedge |
Maxima:
Gamma = 2 3.3%
Gamma = 1 4.5%
Discussion
The lowest risk-aversion I'll model (RA=1) means a spend rate optimum occurs around 4.5%, though there is a fairly wide flat region around that point from, say, 4 to 5%. The highest risk-aversion I'll go (RA=2) means an optimum around 3.3%, again with a flat region from 3 to 3.7. That gives me an overall range which is perhaps useful for managing spending control. Also, I'll note that 3.3 just happens to be the current "aspirational goal" for the center of my real life spend control program. So, that was one hell of a lucky guess, I guess. You know what else was a lucky guess? Remember (anyone?) when I was pitching my RH40 age-based spend rule? (reminder, it was this: a very conservative spend rate, based on age, could be approximated by [Age/(40 - Age/3) + x] where x is a subjective nudge where I use up to 1/2 percent). For a 61 year old, that'd be 3.1% to 3.6% for x = 0 to .5%, which is scarily close to the optimum for gamma = 2.
Either way I guess I'm comfortable, based on this and some recent other posts on portfolio longevity calculus, that spend rates below around 3.3. or 3.4 % are self denying, spend rates of 2% are toying with perpetuality, and spend rates over 4.5 to 4.8% (for THIS age, anyway) are skating a little close to the edge as well as getting into a zone where the future optionality of a net-wealth process (past post) starts to dwindle.
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post script -- I got curious after I did this on what RA = 3 might look like for the given assumptions. It looks like this below. The optimum here is around a 2.7% spend rate. That seems a little fussy to me and I'll reject it, or at least I will until the next global financial crisis.
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