The theory behind my custom spend calc?

The answer to the title: yeah, this is probably it...if I understood it better.  I was thinking about this because I was recently perusing a Society of Actuaries research report Value of Longevity Pooling, Feb 2018 -- and at this point it would be reasonable to think that someone somewhere might be wondering about what might be missing in my personal life that I find myself down that particular rabbit hole -- and I found some good material from Moshe Milevsky (always a good source for complex and relevant things in lifecycle finance) on "The Value of Mortality Pooling: a.k.a. Approximating Annuity Equivalent Wealth (AEW)." Down at the bottom of this particular hole I found this:

and in which formula(2) looked suspiciously familiar to me.  As a side note I even took his advice and looked at the chapter 13 material  that he mentions (here as lecture notes).  Coming out of that even deeper rabbit hole I found myself vaguely disoriented if not nauseous, not because I dislike the subject or the authors, but more likely because the very experience of: a) looking at 240 pages of nothing but ret-fin calculus, and b) me acquiring a dawning appreciation for the tragically minuscule amount that I know or even can know on this subject that I have covered for two years, left me, well, you know, it left me disoriented and nauseous.

The affinity I saw in the Milevsky slide was to my custom spend liability calc which I covered here and maybe here.  The formula in my amateur hack version, and which I laid out with some disclaimers, was this

which in other posts I rearranged and animated with randomness and simulation to contrive a stochastic present value of spending (which I finally figured out is really secretly disguised Monte Carlo Simulation by another name) for use with a household balance sheet.  I want to now try to compare my hack formula to the theory but first let's do this. Let's set a proper tone and context.

Here is the proper context for Dr. Milevsky:

 - Academically and professionally rigorous work opus

 - Formal education and training

 - Commanding use of continuous form math 

 - High theory and deep practical knowledge are combined

 - Professional stakes for being correct are high

 - Highly deserved global reach and recognition

 - Backed up by fully conceived and developed calculus-based life-cycle models

 - Status of skin in the game re retirement unknown

 
Here is the proper context for RiversHedge:

 - Layman-esque and slightly dilettantish blog posts

 - Autodidact for the most part

 - Uses discrete math if it's even correct and botches or misunderstands the continuous stuff

 - Relatively shallow and/or narrow knowledge base

 - Professional stakes for being correct or not are low

 - Reach is a few readers, one sister, one daughter, and a cat; girlfriend just rolls her eyes

 - Backed up mostly by intuition, sketchy blogging, some code and lots of internet reading

 - Clearly has skin in the game for retirement outcomes

Now let's take a look.   ≈ means I am at least in the general ball park or am given the benefit of the doubt.

1. e^-rt  ≈ my (1+hr)^-t  term

This is a discounting term for the stuff that is integrated.  In mine I have a (1+hr)^-t term which is the same thing in discrete math except I have a totally superfluous h term in there as a placeholder for hyperbolic discounting. I could probably use the continuous term e^-rt without a ton of harm in the discrete version but probably shouldn't...but I could...if no one looks too closely...but I won't since I am summing discrete (discounted) terms. 

2. tPx ≈ my tPx term

This is more or less the same in both. I call it conditional survival probability and I hope that is right. Both formulas use Gompertz-Makeham derived math (gompertz if a lambda term is zero I hear) to weight cashflows over a plausible lifetime. Here I have to be careful because there are differences between the instantaneous way of doing this and how I do it but for my hack purposes it doesn't matter. I pull my numbers from ln[p] = (1-e^t/b)e^((x-m)/b) in case you were wondering.  The point, though, is that it is the same concept either way.

3. u(c_t*) ≈ s (if you squint hard enough)

This is the term for the "optimized consumption path" as he describes it. My "s" is for what I called the custom spend which is a consumption path.  While "s" is not a rigorously optimized thing it is at least what I will call a "carefully considered spend plan" which, while not exactly an optimizing proposition, at least does not foreclose on the possibility that one of the "carefully considered" options is an optimal one.  My point is that it is an amateur hack version of the same thing.  s and c are also what differentiate the two equations from an annuity formula, which of course these really are but in disguise. In that special case s or c is 1, if I understand these things which, well....

4.  ∫

I mean really, who wants to integrate when one can sum discretely to similar effect? Let's call it a wash, though.  The continuous form integral has a "to" term of infinity but since the conditional survival probability goes to zero around age 120 my terms for summing from 0 to 120-x is effectively the same thing in practical terms.  I could have used infinity as well. 

5. U_x* ≈ my "CSL" (in the broadest of terms)

"Lifetime utility of wealth...without annuities."  That, in a nutshell, is the same game or it would be if I knew utility and calculus better, especially when the "without annuities" qualification is added. Let's just say that the CSL term on the left of equality in my equation is me trying to do the best thing that I possibly can for me and my kids over a lifetime without tipping myself into the abyss. Same game.


So I'll take it that that above is a small piece of the theory that backs up my custom spend calc and it's also the theory that teases my "intuition" with the thought that I am not always a total moron (partial but not total; ask my kids) and that I can at least get myself oriented in a generally correct direction every once in a while.  The "theory theory" stuff, the big stuff, is more to be found in the meat of the content in the chapter 13 link above, a chapter which itself depends on an awful lot of other people all the way back to at least Yarri '65 and beyond.  I like to delude myself that someday I'll be able to "see" and understand all of this kind of thing but the chances are, um, long...