A second look at a joint probability approximation for ruin risk

In my last post I wrote that it had finally dawned on me that ruin risk can be approximated (within the narrow bounds that I've actually tested it, anyway...and this may not be new to others by the way, just to me) by a joint probability of: a) the chance still being alive in x years, and b) the chance of a failure of the net-wealth process in x years. Just to make sure it was not a fluke I thought I'd take a second look. To un-fluke, the second look attempts to put the approximation in a context (against the kolmogorov equation and two monte carlo simulators one of which is internally structured at least three different ways) over a narrow range of at least one variable that I might care about (3 to 6 % spend rate beyond which boundaries I am not yet interested)[1]. When I run all of them with the various assumptions in note 1 we get this:

What does this tell me? Mostly it tells me that they are all generally playing the same game and that the approximation and the K-equation in particular are very closely playing the same game. All of them attempt to describe the same real-world process. While there is increasing dispersion as the spending intensity rises, it does not surprise or bother me. Since something like a 10% spend rate at age 60 does not interest me much, the dispersion past 6% I can leave alone for now. The approximation being in the center mass of the various models is reassuring in a way -- though that may just be a personal reaction rather than something that is necessarily meaningful analytically.

Then, for good measure, if we chart out the joint probability like I did in the last post but now with four different spend rates it looks like this:

I guess one lesson, if we were to traffic in lessons, is to watch one's spending and not let it get away from you. Another could be that there might be some generalized explanatory utility in charting out the elements of ruin risk like this. But maybe that's just me...










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[1] Some other assumptions embedded here -- I won't tediously detail all of it -- are real rate of 4% (so net of inflation taxes fees etc) longevity mode of 90 for the approximation, and age 60. The underlying distribution for portfolio longevity is working in wealth units so the "endowment" is really in wealth units determined by the spend rate. The MC sims include one from the internet that is relatively well known and the other is mine but configured several ways: fixed age 85 and 90 and variable longevity using Gompertz mode90 math, one run adds a little soc. security, and one uses 60/40 vs 50/50 allocation. The point was not necessarily to detail the assumptions but to do a change up on MC methods in order to contextualize the approximation.  None of this is particularly scientific if you look too closely.

The probability of still being alive is empirically derived from the SOA annuitant mortality table for a 60 year old but without the delta-longevity nudge (I don't remember what they call it and I didn't understand it at the time) they throw in if you look at their literature.  The probability of a net-wealth-failure is empirically derived from a rudimentary simulation matrix (200 periods x 4000 iterations) of a net-wealth process using mW-1, the coefficient of the second term in the kolmogorov equation, with a return assumption of 4% a spend rate of x% and volatility of 10%.