This idea of being aware of the two pdfs and then combining them is probably revelatory to no one and obvious to all that have gone down this path. But to me (hey, I'm an amateur) I really did have one of those aha! moments when, after thinking about a Corey Hoffstein post on the "Lie of Averages," I first plotted a distribution (defective distribution, btw) for portfolio longevity in years and then for fun overlaid a pdf for risk of death over the same "years." I was looking at the mean of each pdf thinking there was nothing that logically connected them at all, one could be early, the other late, and vice-versa - who knows. Then I was thinking if the portfolio longevity happened to fall in the 5th percentile and the death year in the 95th percentile, you'd be screwed, kind of a supersized ruin scenario.
I realize this may be a little obscure but finally, after staring at the two pdfs, I realized that (the aha was here) if I switched the mortality pdf to a 1-CDF I'd get a conditional survival probability (CSP) for each year of the portfolio longevity distribution. That meant I could use the CSP to weight the portfolio longevity pdf with the probability of being alive in any of those years. Then I figured if I added up the result of that weighting to somewhere around the point where the CSP is zero (let's call it 120 or so), that should be a lifetime ruin estimate. On my first try I choked with absurd results and almost dropped the idea but I saw an error and did it again and then it looked ok. I checked the answer against the VBA version of the K-equation I got from Prof Milevsky and got about the same answer. Then, with a few steps in between, I came up with ~100 other scenarios and still got the same answers as the other equation every time. I thought that was pretty cool, first because I could finally see how the process worked in my head, and second because I was getting the same answer as a famous and brilliant equation from the 1930s with something I "amateur-hacked" on my own.
I'm not sure what exactly I can say about hindsight here. Maybe it's that I wish I hadn't spent so many years modeling fixed duration scenarios, 30 years or whatever. That approach can be helpful (and simple and fast) but in probability terms it now seems more or less meaningless. I also wish that I had been able to see the "two separate and independent processes" overlaid on one another earlier. I might have figured out some things I've been thinking about sooner. I might have also spent less time over-designing and overbuilding a longevity-varying and much slower (though more flexible) simulator. The real hindsight though, I guess, is that I am glad that I slogged away on the problem for so long and got to a place that made sense because my goal at the beginning of the blog and my research efforts was to be able to see this ret-fin world a little better than when I first wandered blind into early retirement -- and I think this helped.
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Here, from a previous post by the way, was my amateur notation attempt (I need a math guy) at lifetime probability of ruin (LPR) where the gw(t) term is portfolio longevity in years and tPx is the probability that an x year old will be alive in t years. n could be infinity but I use 120:
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