Using approximations to intuit the output of simple non-spend MC simulation

My first foray into retirement finance was in 2012 with Milevsky's "7 Equations..." book. Those seven equations took me pretty far and I still think that simple deterministic equations (ok, well maybe not the Komogorov equation in that book, but even that can be handled by amateurs) embedded in an adaptive triangulation methodology along with some common sense can go a really long way into teasing out some intuition about the consequences of the portfolio and spend choices that one needs to make over a lifetime.

A fantasy of exogeneity

The Setup

Past this sentence there is no modeling of real financial phenomena. This is just playing around with an idea just to see what it looks like.  This is also the second whack at an idea about modeling "critical states" like forest fires, sand pile avalanches and earthquakes.  Here is the idea: most research papers I read perseverate on returns and return distributions.  The normal distribution is the flawed baseline but usually close enough. There are others. T-distributions have usable fat tails but need to be fit. Gaussian mixes (GM) are often very usable but also need to be fit. I like GM since there is a "high note" of a relatively regular, probabilistic, narrow variance return process and a "low note" of much lower and/or very wide variance returns. This is easy to model but conceiving of the low note as a stochastic process might be "fittable" in the end but also wrong. What if the world had darker forces -- sometimes related to returns -- that are not a regular random process and not always a function of returns.  What if the earthquakes that hit us financially come from things other than returns (or regular spending).  Here we can take a stab at some ideas for what I mean:

Estimating geometric returns and wealth over time vs a simulated path

No grand goals here, just looking at an estimator for geometric returns over time (what we really earn) as well as it's correlate - wealth accumulation - and then compare to one very arbitrary simulated path.  Just for fun and to get the formulas into a spreadsheet.  For the estimator I am using R Michaud's estimator for the Nth period geo return and it's variance. Like this:

On Snow

“In any man who dies there dies with him, his first snow and kiss and fight. Not people die but worlds die in them.”   Yevgeny Yevtushenko quotes (Russian Poet, b.1933)

"Maybe it's wrong when we remember breakthroughs to our own being as something that occurs in discrete, extraordinary moments. Maybe falling in love, the piercing knowledge that we ourselves will someday die, and the love of snow are in reality not some sudden events; maybe they were always present. Maybe they never completely vanish, either.”  Peter Høeg, Smilla's Sense of Snow

On May 1st, 20__, I happened to walk out of my house directly into snow. Snow seemed, on May 1st, as improbable as a rain of frogs, but in Minnesota, in May, that might be a slight exaggeration. The improbability caught my attention, however, because it reminded me that whenever I go into my head and pull the name "X" from the catalogues of memory, as I had done just moments before I walked out the front door, the image of snow always comes with it. I am always amazed at how the brain works like that. I’ve been told, or read somewhere, that unrelated neurons can be triggered just by recalling the memory of another thing held in closely adjoining brain space. In this case I could almost feel the neighborly neurons firing and bringing up their dual images, the name and the image, X and snow. 

Playing with Gaussian Mixes and "jumps" again

There is no real hard science or rigor past this sentence so begone if you need that sort of thing.  

The reason to fling blog crud today was that I was coerced into reading a paper[1] on jump-diffusion processes by the inimitable David Cantor. I know nothing, really, of those processes but I came to the conclusion that my amateur attempts at doing a Gaussian mix in the past was kinda close. I mean, my vol was not really stochastic but both the return and vol "jump" within a random process and so we can, over time, with either jump modeling or mixes - close enough, right? - more or less mimic the fat tails of real world distributions, which in modeling-for-retirement terms is desirable. I think. 

Merton and a special case of optimal consumption

I think I've done this before but what the heck? If a retirement blog can't drool and repeat itself every once in a while, then is it really a retirement blog?[1] The occasion here is some thoughts coming out of reading Merton's 1970 paper on "Optimum Consumption and Portfolio Rules in a Continuous Time Model."  Either a hat tip or an accusatory finger pointed at David Cantor for this outrage  

In Memoriam - Dirk Cotton 195? - 2021

Dirk Cotton, one of the great retirement bloggers of recent years, was not really a close friend or an accomplice in Ret-fin crimes. But I did know him.  Here are some of my memories.