Playing Around With Some RH40 Math for context...

I tried to come up with a reason that this post might have any real serious functional purpose.  I couldn't do it or couldn't do it very well.  This post is really just me playing around with some math to contextualize my RH40 formula by using Blanchett's simple "dynamic" formula.  There is no advice here; I'm just goofing around.


David Blanchett -- in order to make it easier for practitioners and engaged retirees to practice a more dynamic form of retirement finance, something that has been shown to improve outcomes but is difficult to do without the proper tech -- compressed a whole mass of Monte Carlo simulation into a single formula so that retirees and their advisors had access to sophisticated simulation without all the work and processing cycles. His article is Simple Formulas to Implement Complex Withdrawal Strategies which was presented in the Journal of Financial Planning. I'd cite the date and volume but I guess they don't do that at onefpa.org.  I've written about this article before. It is useful stuff.

The reason I am looking at this again is that there is so much work baked into his formula that it makes it easy for me to use it to benchmark or contextualize stuff I'm working on.  In this case I am trying to convince myself that my new formula that I built on the back of Evan Inglis's "divide by 20" rule of thumb actually has some value.  Since my RH40 formula was kinda tongue-in-cheek but also legit, what I wanted to do is to place it somewhere in a broad context of both Evan's "divide age by something between 10 and 20" and Blanchett's "simple rule" when his rule is presented as a range based on a variety of inputs.  The goal is to see if the RH40 is "reasonable" even if it is an "math artifact" kind of rule rather than a transparent economic model.   Basically, the question is can I use it for my own private purposes with any degree of confidence?

First I grabbed Blanchett's formula for retirements that are expected to be longer than 15 years (for shorter retirements, he uses an RMD style formula, which kinda makes sense. In the last post we showed that RMD works pretty well early in retirement and not so well late).  His formula looks like this:

Then I ran it for a whole bunch of different scenarios: e.g., probability of success at 85, 90, and 95%, equity percentages from 0 to 100% in 10% increments, and all of that for each start age (actually for expected years for a given start age combined with a fixed terminal expectation of 95). I realize that there are all sorts of other scenarios beyond those but these seemed like enough context for now. When I do all that, the mass of Blanchett formula data looks like this when charted:

Then I looked at the min/max of that mass and what I'll call one "representative" scenario: 60/40 allocation, end age = 95, probability of success = 90%, fees of 1%, though that last may be a little conservative. Note that we need to ignore actuarial science in the sense that if one were to estimate end age at age 50 it might be 81 or 82 but if we survived to age 90 and then estimate again it might be, I don't know, 95 or 100 or something? In any case, I didn't do that. The chart below was intended to simplify the chart above in order to isolate a few things. It looked like this; I threw in a line for 4% since everyone knows 4%. The grey lines are the min/max:

Now, to get to the fun stuff, I took that red line, the "representative" scenario, and I tried to not only compare it to my RH40 formula, I tried to see if I could hack the math and come up with a similar curve for ages 50-95 with a different equation (why? no reason, just for the hell of it. The only thing I could think of was that maybe someone might want to have a formula where the only input is age. Not sure why.  If that were really the case the RH40 formula works well enough all by itself for a lot of ages). Here is what I came up with, though.  In the chart below, I have all sorts of stuff going on but here is the gist of it.

Grey dotted lines -- since I based my formula on the Divide by 20 rule, the lower dotted line is age divided by 20. The upper dotted line is age divided by 10. 10 is the upper bound described by Inglis and 20 the lower. That was his safe-to-risky range.

Grey solid lines - these are the min/max bounds of the Blanchett formula for all of the inputs I decided to work with as described above. I could have gone lower than 85% probability of success for example, but why?

Red line - That is the Blanchett "representative example" using a 60/40 allocation, a 90% probability of success, and age 95 as a terminus.

[ formula presented above ]

Purple - My RH40 formula.

Blue - a p-hack version of the Blanchett formula in terms of age only. C in this case is .0032, x is .0369, D is 1E-13 and y is .286.

Green - green is an extension of RH40 that tries to bend it more at late ages in order to fit to the Blanchett "representative scenario" line.  Basically it is a variation on RH40 that looks like this if w is already calculated as above:

w' = w + p  

where  p = [D * e^(A*z)].  

D was 1E-13 and z  .284

These four color coded formulas plus the grey context lines look like this when all charted together:

Conclusions?

1. For most ages prior to median mortality all of the lines overlap with only trivial differences with RH40 being a little more conservative past about age 80,

2. Yes, one can always simulate stuff to death or one can maybe just lean on Blanchett instead as a speedier, simpler solution. Leaning on Blanchett is probably ok within certain boundaries and I like his formula as an alternative to simulation. Even with Blanchett and his simple formula, one could blow out billions of scenarios but it looks like one representative example can cover an awful lot of reasonable ground given how narrow the band of results can be,

3 . If you can't remember the Blanchett formula or have trouble setting it up in a spreadsheet or don't have access to a computer, simplified age-based math can take you to almost the exact same place.  The two hacked versions I came up with that use exponentials work great for mimic-ing the "representative example" but the basic RH40 works just as well for all ages up to about median mortality (for me) so why bother with exponentials? You might as well just use the Blanchett formula at that point.  Even after median mortality RH40 probably works great in that it is friendlier to the bequest motive and in real life most people have a declining spending path anyway (ex-health care),

4. The "divide by [10 or 20]" rules have awesome simplicity that I like a lot but RH40 works pretty well too for almost as much simplicity. In addition, the "divide-by" rules may be a little too simple at later ages.  RH40 maps pretty well to a robust well-tested formula so maybe it has some legs.  I have no problem using it as a rule of thumb if only for myself.