I had the privilege and honor this morning of being able to spend a half hour one-on-one in person with Moshe Milevsky. That was great because his book (7 Equations) more or less triggered my ret-fin explorations over the last five years as well as this blog. I had happened to email him a few weeks back to ask what he thought my joint probability approximation[1] was actually doing in the context of his paper Ruined Moments in Your Life: How Good Are the Approximations? [2003] and he very generously offered to explain it to me in person since he was giving a presentation about 10 miles away from where I live. That was today.
My goal was to come away with at least three things: 1) figure out where my ruin approximation approach fits into the scheme of things in terms of the ruin approximations [approximations to the Kolmogorov partial differential equation (PDE) for ruin] listed in his paper, 2) get more insight into my probability density function (pdf) for portfolio longevity since I had some confidence issues due to the fact that it had some weird properties and I don't have a formal background in stats, and 3) I wanted a better verbal interpretation of the third term of the Kolmogorov equation. That, and I also wanted to get a selfie with him for my twin sister who's spent 40 years in the financial services industry but I chickened out on that since at the end he was moving pretty quickly towards his next meeting.
What did I get from my chat?
1. Evidently my approach is not really like any of the approximations[2] in the paper at all. Rather it is a "brute force" expression of the ruin principle itself via simulation of at least one of the main terms, an approach that was not available (without the processing power and memory we have today) back in the 1930s when the PDE was formulated. He also said that the program I wrote was a pretty direct way of expressing the core proposition for ruin (P[T < L]) where a successful expression of P[T< L] satisfies (or should) the PDE. And keep in mind here that T is personal "lifetime probability" in some type of Gompertz time process and L is portfolio longevity over that lifetime. He said that the fact that the average difference between my approach and the finite differences solution (Crank Nicholson scheme) to the PDE is effectively zero showed that my approach describes the same process and should find the same solution. He also confirmed that ruin probability is the combination of two independent probability processes (life and portfolio) -- a combination that happened to be one of my "aha-s" in a previous post -- and evidently something that quite a few people get turned around on when looking at portfolio longevity in isolation.
2. Because I don't have a stats background and even though my solution "worked" I was having a confidence problem with the pdf for portfolio longevity. I had based it off of a frequency distribution coming out of a simulation of a net-wealth process (m*w-1) and I used that freq distribution to come up with a pdf. But it was a mis-shapen pdf, a bellish curve but with a big spike on one end, the whole of which was more or less representing the number of years a portfolio survived but no longer. The spike at the end was due to the fact that some of the sims didn't fail, they lasted forever, even when I upped a lifetime to 50,000 years. Me? I had calculated my probability density based on the frequency of the occurrences of what I was calling "end-of-portfolio-survival in some year t" over the sum of the whole mis-shapen distribution which included those sims that lasted forever. Again, this worked great and turned out to be correct in the end but I had no idea what I was doing precisely and I was not at ease with my approach. Prof Milevsky very patiently and kindly schooled me in "defective distributions" and explained that I actually had two distributions for what he formally referred to as "the longevity of the portfolio in years" rather than one. One distribution is related to P[L = ∞] (probability the portfolio will last forever; that has less of a distribution shape and more of a "column" look) and the other is the P[L < ∞ ] (probability it will last for less than forever; the shape of that one looks familiar but does not sum to 1). Integrating over the second one only, which the professor implied that some try to do, is apparently incorrect (that makes sense since it does not sum to 1) while integrating over both does work. I just happened to get lucky by doing it correctly on the first try. This is probably kind of like a broken clock getting the time right twice a day. He pointed out again that the match between the outcomes of my method and the finite differences solution to the PDE was a good sign.
3. Back when I did my first draft of the post exploring the Kolmogorov equation I was trying to figure out how to express in words what its third term meant. The coefficient of the third term (in front of the second derivative of ruin probability with respect to wealth) is (s^2w^2)/2. I had created a page-long disquisition on volatility in my post which, though more than likely correct, is a long way of trying to say something simple. Prof. Milevsky merely said it, the 3rd term, was a volatility term, a diffusion term and said it was a lot like heat diffusion in the "heat equation" (representing heat processes) to which the ruin PDE is related.
4. As a bonus, Prof Milevsky also told me that not only was the approach I had taken quite faithful to the process and equation in question, he also said that it had some distinct advantages. The example he described was in terms of creating some type of option collar around a portfolio. In that case the distribution of wealth outcomes over time becomes non-gaussian (and ruin risk can be lower). But this is a place to which the Kolmogorov equation cannot go...while a sim-based approach can. I look forward to trying that out some day.
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[1] also here, here, here, here, and here
[2] Or at least not an approximation in the sense that it is meant in the 2003 paper where people are trying to get at ruin estimates by way of proxy equations and methods that might be easier and more tractable than trying to evaluate the PDE directly.
note: If you haven't heard me say it before: I know almost nothing about differential equations.
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