Ithaca, or maybe it was Lavinium, waits for thee.
"There are places where one arrives and feels: here I am meant to be, here I am whole. I had not known this place, yet it knows me, and I it, as if I had been born for its silence and its light." Rilke
Several years ago, before my recent brush with stage 3.99999 prostate cancer, I finally (finally! after six decades) was able to choose for myself a life -- lifestyle, location, focus, purpose -- that made a whole lot of sense to me. This meant a little less quant and a little more right brain and active movement than I pursued in Florida. In practice all that means right now, in Montana, is stuff like: hike, lift, read, enjoy the landscapes, feed the cat, do some photography, take some western road trips, travel a little to see kids and gf, and so forth. And...less, but not zero, quant as a byproduct.
That lifestyle has a cost but the cost, contra 12 years of my blog posts, is not really an optimized retirement spend rate "cost." It is not possible to live life prospectively and precisely to a three-significant digit spend rate plan (and it'd be a bit of a buzzkill in the attempt no doubt). I am lucky if my realized lifestyle is within even one point of the plan. That's why I watch the unfolding of the lifestyle cost and then frame what was and what is as either: rational, not rational or in the frontier in between. Course adjustments in the not rational or frontier zones are implied, however, as guided by a family balance sheet (ala Ken Steiner), an income statement, and a spend rate control process. The frontiers are neither rules nor even necessarily hard boundaries. They are fuzzy zones that are informed, but not strictly ruled, by a generalized sense of what several theoretical models (the ensemble) point to rather than by one particular optimization done years ago. Too high is often obvious especially in retrospect. Too low is less visible but still pernicious in terms of forgone opportunity costs of experiences[1]. In between are a multiplicity of choices that might be all correct in their own contexts...and note that even the frontiers are soft. Given my current left-brain-to-right-and-pursue-an-active-life-in-MT ethos, I am currently ok with all this "fuzz."
To keep this post short, below is an example of an ensemble of models[2,3] I might or might not use to frame "the frontier" and "the rational," all of which are proffered as "correct" by their advocates within their own contexts. Which one of these is right? Idk, all of them? None? I guess maybe: pick one or split the difference or float on a current inside the shores of the edges...or whatever. I think that as long as one is not willfully damaging one's future and/or bequest or living in an unnecessary and punishing penury, it's probably all ok. Keep in mind though, that this picture below is a little bit of a trick. I have been slightly tendentious and cherry-pickish with the parameters so that the look of the ensemble makes my point. I could probably blow this up a little more into a more diverse (but still coherent) picture.
Forgot the labels: x is age; y is “optimal(??)” spend rate
So, six analytic models [note 2], some quite complex, some stupidly simple, and all are saying roughly the same thing (I did this before once with a different ensemble). This was all done, fwiw, for one unspecified portfolio which doesn't matter right now; change the risk and return and aversion and longevity and you might get something wider or steeper or shallower but the underlying modeling engines are all running on similar fuel. That means there is likely a fair amount of latitude to frame a plan and to manage the process (a different topic/post).
For me, it just means that 99 days out of 100 I can ignore the details, relax, and enjoy the lifestyle that I wandered so long and far to find.
Notes -------------------------------------------
[1] Too low can be called self denial, or my as my fired advisor said "live a little bro." But in an early retirement it can be a reserve or redundancy to be able to recover from blows. Also, a point rarely made in the literature is that if one has a lifestyle that makes sense and doesn't cost very much, one is under NO obligation to spend more to make a model or an advisor more content.
[2] The models:
A. Grey dashed: Kolmogorov PDE for Life Probability of Ruin. This is the version of LPR that was solved by way of finite differences in VBA code in a spreadsheet sent to me once by Professor Milevsky. Fast, simple, and using his words it encompasses the entire constellation of asset exhaustion from now to infinity as well as considering the entire term structure of longevity. In the case above I used a solver to solve for the spend rate that resulted at each age for a ruin rate of either 85% or 98%. This dual line thing was just for framing a range of lower to higher risks that might be within a plausible policy range.
B. Black Line: Hazard rate weighted joke heuristic. This was my RH40 formula that was based on an Evan Inglis article that said spend rates, with some reasonable justification, can be estimated by age/20. I took that reasonableness and made an absolutely mathematically unnecessary and arbitrary variation and then for the hell of it added a hazard rate kicker. This was all a bit of a lark but it works over the interval.
Age/[40-Age/3]/100 * (1+λx+t ). Don't judge me ;-)
C. Blue Lines: Hamilton Jacobi Bellman Dynamic Spend Optimization by ChatGPT. Except it isn't really HJB but a proxy because the nature of the problem is too hard for AI or it'll time out in the attempt. Initially AI gave me a prompt that it said was ready to run and then when I said "run it" AI said "sorry bro, no can do, here is a proxy." But it was a reasonable attempt I think. One blue line was for risk aversion coefficient of 2 and the other is for 4. Wealth was normalized to 1 at time zero.
See Note [3]
D. Orangeish/redish line: 1/e ; e is expected remaining lifetime | age. Expected remaining is a policy choice where I pulled years from the Soc Sec table at the 95th percentile at each age. 1/e was in a note by Gordon Irlam on easy-to-use rules of thumb.
E. Green Line. Excel PMT function (=PMT(rate, nper, pv, fv. type). In this case nper is years in the SS table at the 98th percentile but could be fixed horizon too for simplicity. This was offered as a rule of thumb in a paper by Waring and Seigel: Annually Recalculated Virtual Annuity (ARVA) (The Only Spending Rule Article You Will Ever Need M. Barton Waring and Laurence B. Siegel, 2015
F. Peachy/Pink Line: PWR (Perfect Withdrawal Rate). PWR is the spend rate that ends a horizon's wealth at zero with perfect foreknowledge of the sequence of rates. In this case the rates are randomized based on some arbitrary portfolio and the horizon is stochastic lifetime using a Gompertz function tuned, I think, to a mode of 88 and a dispersion of 9. That process generates a distribution of withdrawal rates and so as a policy choice, the 5th percentile withdrawal was selected. The percentile is conceptually kinda sorta a type of fail rate and behaves similarly. See Suarez Suarez & Waltz, "The Perfect Withdrawal Amount, A Methodology, 2015, Trinity University
[3] Here is what ChatGPT says its dynamic spend optimization both is and is not. I am inclined to be sympathetic to its approach if for no other reason that it is fast-ish and close enough:
- It’s not continuous calculus — it’s a discrete Bellman loop.
In theory, the HJB equation is a continuous-time PDE. In practice, we solve it as a discrete dynamic-programming recursion over age and wealth grids. That turns the math into a finite-difference or simulation problem, not an analytic PDE solution. - We’re maximizing expected utility, not solving for a closed-form control.
The algorithm searches for the consumption rate c^*(W,x) that gives the highest expected lifetime utility at each grid point, given returns and mortality. It’s the same logic as HJB but done numerically by iterating backward, not by deriving an explicit formula. - Mortality enters as an extra discount rate, not a state variable.
We fold survival probability into the discount factor \rho + h(x). That’s an economic shortcut — true HJB with stochastic lifetime would treat death as a stopping condition with its own state boundary. - Portfolio returns are treated statistically, not as stochastic processes.
Instead of solving the diffusion term \frac{1}{2}\sigma^2 W^2 V_{WW} directly, we approximate it using Monte Carlo or quadrature over lognormal returns. That’s a numerical integration step, not a differential operator. - It’s really an expected-value policy search guided by HJB logic.
You’re applying the structure of HJB — the idea that optimal control satisfies a local first-order condition — but computing it through discrete grids, interpolation, and simulation. So what you’re “really doing” is an HJB-inspired dynamic program, not a formal PDE solution.
No comments:
Post a Comment