Have I mentioned that I do not have much formal training in statistics or calculus? I will sandbag in that area again, not as a lead-in to a humble-brag, but because my innocence in such things can lead to misleading errors or embarrassment. The latter I can handle but the former makes me uncomfortable and I think I engaged in the former by calling my recently built ruin estimation tool a "joint probability approximation" of lifetime ruin risk. I don't think that is quite right because I am combining a pdf with a weighting factor (1 - cdf) rather than strictly another pdf. I don't know if that is a "joint" thing or not and will leave it to others that know the math better.
I had a rough idea of what I was doing when I posted the formal definitions before (when I linked to Milevsky's paper on ruin calculus[1]). In that paper, on a recent re-read, he was pretty clear on what I was trying to do: "One can heuristically think of the integral as ‘adding up’ the probability of [portfolio] ruin at t, weighted by the probability the individual will survive to this time." ...or the way Milevsky characterized it in the paper: "the quantity...is precisely the probability of lifetime ruin." When I spoke to him in person, I didn't ask about the joint probability thing but he did confirm that my approach was consistent with the above and was a direct expression of the P[T≥L] ruin concept which also satisfies the PDE. So, not a "joint probability approximation" I gather. It's something else. Time for a new name.
Given the opportunity to rename the tool, I first came up with the folksy sounding FRED (flexible ruin estimation device). But that didn't hit me quite right so I shifted the last letter to a T. That gave me FRET for "flexible ruin estimation tool." That had the virtue of being both an accurate description of the software and also being an accurate description of what I do when I spend too much time thinking about lifetime ruin risk. So FRET it is.
My Guess at a Tweaked, Amateur Version of Notation for FRET
As in the comments above, I already had the formal notation from the same paper I mentioned, I just hadn't connected it very precisely in my head to what I was doing. I wanted to pin it down here a little better so that if I needed to explain to anyone what I was doing I could be more formal or accurate even though I kinda stink at reading or writing math notation. Since I am weighting a PDF with a conditional survival factor it looks like I can go with equation 20 in the linked paper which looks like this (you can read the paper to see where this comes from):
I'm not sure how exactly correct that is notation-wise but it does have some minor adjustments for readability and also for what I am doing in the tool. From left to right... First, LPR or "lifetime probability of ruin" reads better than the formal math left of the equality in equation 20. Then, on the right side, I am integrating to "n" rather than the ∞ that is in the original notation. In the tool I stop integrating (or, rather, summing discrete values) at a lifetime age of something like 120 or 150 or so (I'd have to go look). At that point the conditional survival probability is and continues to be zero so it doesn't really matter how much farther one goes. I suppose I could have left ∞ in there but "n" is what I actually do. The tPx term was suggested by Milevsky in this comment "Note also that in some literature the symbol (tPx) is used to represent 1 − Fx(t), which is the conditional survival probability." I think that reads better since the phrase "the probability an x-year-old will survive t years" for tPx rolls off the mental tongue a little better than anything I could do for 1 - Fx(t).
Deconstructing the Two Terms in the Notation a Bit...In Case You're Curious
For the first term gw(t) I am doing a mini simulation of (mw - 1) where w is a wealth unit term (think: 1 / a spend rate), m is the return variable that is randomized in the simulation, and 1 is the spend. The random return variable m can be sampled from a normal distribution programmatically or it can be sampled with replacement from other empirical or custom distributions (which is where it gets interesting). The pdf gw(t) is derived from a process that counts the frequency of the simulated number of years that the portfolio w is still alive after the start year. (mw - 1) is the coefficient of the second term of the Kolmogorov PDE by the way.
For the second term (tPx) in FRET I use Gompertz-Makeham math based on
In particular, I am using a function I lifted from Milevsky's "7 Equations" book, chapter 2
where you can also see the direct link to the 1 - Fx(t) or tPx term in FRET notation above.
Multiply the two terms and sum over a range and one is able to FRET all day long if one wishes to do so. ...and you get the same answer that comes from solutions to the Kolmogorov PDE.
Other Considerations: Real Men and Women Don't Simulate?!
In re-reading the linked Huang Milevsky Wang paper[1]. I came across this
"With today’s advanced computing power – and the intellectual simplicity of simulation – it is quite easy to fall-back on Monte Carlo techniques to derive all forms of lifetime ruin probabilities. This is especially common amongst practitioners who are interested in quick and-dirty heuristic approximations. In this paper we have shown how to formulate and then numerically solve the PDE representation of the lifetime ruin probability; a quantity which has been investigated by numerous authors in the finance and insurance literature""intellectual simplicity," "quite easy to fall back on," "quick and dirty heuristic." I get the sense here that if one can't derive something analytically without the help of brute force (intellectually simple fall-back) simulation, it's not really legit. I guess I am ok with my illegitimacy because the chance that I am going to analytically derive anything in a closed form tour-de-force is exactly zero. Plus the mini-sim I do gives me the flexibility to push the math around a little bit in experimental and interesting ways...hence the F in FRET.
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[1] Ruined Moments in Your Life: How Good Are the Approximations? H. Huang, M. Milevsky and J. Wang York University1, Toronto, Canada 1 October 2003
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