Portfolio longevity, defective distributions and some small thoughts on fail magnitude

It's possible that I am skating on thin ice here but I wouldn't even be on this part of the lake if Prof Milevsky hadn't encouraged me a bit.  In particular, when we were talking about portfolio longevity he schooled me on defective distributions and the effects of things like returns and vol on that type of distribution.  I mean I'm in my 60th year and he pressed and quizzed me like a sophomore in the second row of an intro finance class and it was just the two of us in a hotel coffee shop.  "What is the name of this distribution?" [mumble] "try again" [mumble] "try again" [pause] "the answer is that it is portfolio longevity in years...now what happens to the shape if we increase vol?" [uh, it shifts left?] "ok but what happens to the defective right side where we count the number of occurrences of infinite longevity?" [uh, it goes down?] "yes, now again, let's look at...." I didn't think I was going to sweat but it was like being 19 again in the crosshairs of the seminar prof. Whew.



So now let's take a little look at portfolio longevity in the terms that he and I were talking about.  I had taken the coefficient of the 2nd term of Kolmogorov equation which is nothing more than a simple brownian motion thing for wealth, returns, and spending.  I had animated returns by randomizing them.  That creates a simple net wealth process (sorry professor, portfolio longevity in years) that more often than not has a probability of failure greater than zero.  Combine that with mortality probability and it becomes relatively simple to calculate a lifetime probability of ruin that is roughly the same as the Kolmogorov equation.  But today in this post, even though I have no formal training in stats, we'll look at the first part of this process (portfolio longevity) just to shake it out.

Let's do this.  Let's start with a hyper-conservative scenario and look at the components.  Let's take a real rate of return of 1%, std dev of 10% and a spend rate of 4%.  We'll assume a 60 year old but that'll become irrelevant soon enough[1]. Here is the lifetime probability of ruin using the joint probability math I've used before

Lifetime probability of ruin (LPR) is ~47% partly due to longevity expectations and partly due to portfolio longevity expectations.  Let's look at the latter.  In the prior chart I am displaying the cumulative prob of portfolio death but that is just for communication purposes. In reality I am using the pdf of portf. longevity in years to calculate the ruin estimate.  Let's take a closer look at that portfolio longevity pdf.  Since we are frequentists in this post, here is the frequency distribution of when portfolios go insolvent.  The ones that look like they go "insolvent" at the edge of my parameter (200 periods run) aren't really insolvent at all. They are in fact the ones that last forever. Hold that thought.   Here is the portfolio longevity histogram with the x axis expressed in terms of years that the portfolio survives:

So, portfolio longevity is a distribution (and there are few that survive forever at the far right).  The mean longevity is 30.4 years, the probability weighted version is 29.3 years and the analytic version using Milevsky's portfolio longevity formula[2] is 28.8 years.  These all seem close enough and it makes sense to me.

Now let's start increasing return expectations and see what happens.  Here we up the net real returns to 3% and hold all else equal.

LPR is ~23% but again, let's look at portfolio longevity and here it gets more interesting.

Now we start to see what Prof M called a defective distribution (I wish I knew more stats).  10.3% of the portfolios last forever; the other 89.7% don't.  The mean longevity is ~59 years or, rather, let's call it 44 years if we exclude the portfolios that last forever and just look at the ones that don't.  The probability weighted version (excluding the forevers) is ~43 years. Using the formula in note 2 we get something like 46.2 years.  I think this still makes sense.

Now let's take it a step further in returns.  Here we will up the net real returns to 6% and hold all else equal.  Let's see what happens. When we do this, the LPR is now .024 but let's look at portfolio longevity again.

Here, ~82% of the portfolios last forever but there are still a few that don't (18%).  The mean longevity if we ignore the forevers is ~50 years and the prob weighted version is 49.4.  The analytic formula comes up with "nothing" because it can't evaluate it because the ln(negative numbers) doesn't work.  But there are still some portfolios that will fail so I'm thinking the analytic "equation" version distorts our view a little bit at this point.

Now let's work backwards a bit from that last scenario by adding some more vol. and see what happens (note this is back to the lecture I got from Prof M). Let's keep everything ceterus paribus again except we'll crank up vol to 25% from 10%.  When we do this the LPR is now ~.35 but let's look at portfolio longevity again.

~26% of the portfolios last forever but the rest don't. Mean longevity for them is 37 years with the probability weighted version = 36.2.  Again the analytic version can't help because it chokes on the math. You can see that if the vol were to be increased yet again the % that last forever would go down and the distribution of "the others" would shift left and the duration (the central tendency anyway) would get a little shorter.

That's a lot to chew on but the main point here is the same point that Prof M has made in a few articles and papers and the same point he made to me in person. Talking to a client (or anyone) about portfolio longevity in the context of a lifetime mortality expectation can be pretty useful.  If you think you are going to live to 90 or 95, it might make sense to gauge how long you think the portfolio might last.  This is often done these days with simulation but there are other ways to do it, too.  It can be done analytically via formula but I think I just showed that that particular path can be deceptive or confounding when we break the math.  But it is a good conversation to have one way or another.  This method I have used may be one more way.

This brings me to my last point.  It is de rigueur these days to fuss over fail magnitudes in simulation and this is a valid point to make.  Failing in the last half of the last year is different than failing with 15 years to go.  But the reality is that we are still talking about a probabilistic process so there are no hard and fast results to discuss in the end when it comes to magnitude.  Also one can't really discuss "magnitude" unless one has an absolute reference point for a death estimate which is not realistic.  If I pick 90 or 95 or 85 I can do simulations or I can do something like the above and guess when the portfolio will fail and then gauge the difference between the fail and the death estimate to come up with magnitude but in the end this is still kind of a fake.  First, the death estimate is unknown; it is stochastic, a random variable.  Second, the portfolio longevity estimate is not a number it is a distribution based on some pretty sketchy assumptions.  We can pick some kind of central tendency even if we like the assumptions but that still seems a little reductive.

What's the answer? Well, there is none I gather.  I think the best we can do is assess levels of risk in general given what we know today and take action on that today. And then start over again tomorrow.  The lifetime probability of ruin as a summary metric is probably a good "general" tool.  Then, looking at the mismatches between our planning horizon and portfolio longevity estimates is probably another rich vein to tap whether via simulation or other tools.  I think maybe just the knowledge that there are two processes that are uncertain (longevity and portfolio...and this is probably blindingly obvious to everyone) is helpful in the first place.  Knowing how they work is worth the effort and I hope that something like the above might be helpful (to at least me).





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[1] other assumptions: longevity mode =90, 5000 iterations, 200 periods, max age ~121...

[2] Milevsky's formula for portfolio longevity from his 7-Equations book.