Ahhhhh...Now I get it! Ruin risk is a joint probability of separate distributions

In my last post "a riff on Hoffstein's "Lie of Averages" it was all there right in front of my face and I didn't even see it (ok it was maybe an hour later that I saw it).  All the components of the Kolmogorov equation (and many Monte Carlo simulators) were there (age, force of mortality or longevity expectations, spend rate, wealth units, real return, and volatility) and I had even simulated wealth depletion paths using Kolmogorovs coefficient  mw - 1  to create a distribution of "portfolio lives" in that last post.   All I had to do now was realize that I had the two probability distributions[1] I needed to finish the job (job = decode or hack ruin risk without using complex simulators or PDEs): one for the probability of still being alive at some future age and one for the probability that net wealth (or wealth units here) will burn out at any given remove from the present for the returns and spending assumed.  That's all it was and all I had to do was put it together by joining the probabilities which is no more than asking "what is the probability I will run out of money given that I am still alive i.e., ruin risk appears to be:  P(future wealth crash) * P(still alive) [2].

First let's start with a little calibration.  For three ginned-up scenarios (that's all I'm doing this afternoon) the question is: what does the excel-VBA version of the K-equation that I got from Dr Milevsky say for fail/ruin rates.  Then we'll see if I can get close with my amateur hack. This is the set-up, with the numbers in the big box being the scenario ruin rates calculated with the Kolmogorov math:

The first step was to use the SOA annuitant mortality table data (for my age), pull a probability distribution out of it, and then create a CDF.  Taking 1-CDF I'll have the probability of being alive at any age after 59 (I think...pretty sure anyway). So right there we have a rough proxy for the K-equation "force of mortality" at my age though it is not exact since that force is a moving target as one ages.  

Then in step 2 I need the probability distribution (empirically derived from a rudimentary simulation) for the portfolio longevity.  And remember here I had been simulating portfolio longevity outcomes (see linked post ) using the net-wealth-process coefficient from the second term from the K-equation ( mw - 1 ) with some randomized volatility thrown in (remember volatility is in the third term of the K-equation).  The key inputs, then, are spend rate (which gives the wealth units), return, and volatility. The output is a distribution of portfolio life durations. That is converted into a rough probability distribution that represents the probability of a wealth fail at any given future year/age

And in theory that is all the raw material one needs to figure out (roughly) the likelihood that one will run out of money while still living.  Chart it out and it looks kinda like this: 

Beautiful.  I wish an advisor had shown me something like this 10 years ago. It would have made perfect sense right away and maybe even changed some clueless behaviors at the time. And in fact this probably is a pretty useful way to display it for someone that knows a little stats.  Once this is up on the wall, and this is where I had my little epiphany this afternoon, all one has to do is join two distributions to figure out "what is the chance that I will run out of money over the period that I am still likely to be alive? This is the same question that the K-equation and simulators ask, of course, but now it's really two questions rather than one: 1) what's the probability of being alive over time? and 2) what's the probability of running out of money over time? Separate; all we need to do is put them together.  Simulators and complex equations can sometimes distract or confuse or obscure this I think. Here, when I finally saw it presented this way, it looked like it was wide open and (relatively) simple.  I'll throw out some numerical results for what I came up with, but first let's look at the other two scenarios on a chart because I think the shape of the curves is instructive.

Same spending as above, but now higher returns with predictable results...

Same return as the last chart, but now much higher spending and again predictable results...

So how does it work in real number terms when we then join them together? Pretty darn well I think from my amateur hack perspective (I don't really have a very robust background in stats and have only rudimentary calculus so I'll take what I can get). Lets take the same setup table from above and now fill it in with my hacked approximations (in yellow) based on the joint probabilities I roughed out: 

Ok, so it is not exact-exact but given that I took a bunch of simplifying short cuts and given that I am winging it without partial differential calculations or a big Monte Carlo simulator and since I only did one rudimentary and preliminary excel pseudo-sim that took maybe 2 seconds to calculate, I'm feeling pretty good about these results so far. Either way, though, I'm 100% sure that unless I find that I did something stupid, which is not entirely out of the question, I'll add this kind of thing to my toolbox if for no other reason than I think the visualizations are pretty useful.  

Postscript

You know, over the last couple of years I committed myself, for whatever reason, to digging into retirement finance mostly so I could "see" it or have a generally intuitive sense for how things worked. This was done so that I could manage my own process and be 100% sustainable for my remaining lifetime without having to trust strangers that are selling things I don't understand -- and 9 out of 10 times unworthy of my trust anyway no matter what they are selling.  If I got this right here (knock on wood) then I think it takes me a  few steps further towards my goal. I'm hoping my first efforts were not some random artifact of luck. 

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[1] empirical from simulation

[2] I probably still need a stats prof to walk me through what I'm doing here. Some of my terminology may be off, too.  I'm sure there are giant piles of books with page after page of notation that shows this kind of thing the way it's supposed to really be done.  I just don't have access to any of it and probably could not decipher it anyway.  This was just for fun to see if I could do it.