Just to recap, we are working not in what are called "fail rates" as commonly understood but in "portfolio longevity" because it gives us the full constellation of asset exhaustion possibilities over infinity rather than, say, over a fixed 30 years, or even "a lifetime." The basic concept is that an endowment M evolves with both volatile returns and consumption over time. This joint-dynamic means we are subject to both realized multi-period geometric returns -- which are not arithmetic single-period, or what goes into mean-variance optimization -- and consumption. Consumption is constant in this post but doesn't have to be; see my 5-processes paper. These two joint factors give a net wealth process its pungent uniqueness, a uniqueness that gets lost, well it gets lost almost always, except in rare hands of advisors that know what they are doing.
Borrowing from Milevsky's lecture notes, we can say that, generally, M evolves like this in a recursive process:
 |
| Eq 1. Net wealth process |
(where M is the endowment and v is the real growth and w is the withdrawal). But note that what we might really want to look at is what happens when M(j) <=0 which will be our definition of portfolio longevity: L=j when M hits zero. For each iteration in a sim we get a new L. Over all iterations if we integrate L as a probability mass over a particular w we get a cumulative fail rate over 0 to infinity for that w. That is way more interesting than a Monte Carlo simulation and a fail rate for some arbitrary fake time t. But then again I sometimes geek out on this kind of stuff.
Now let's visualize this in all its expansive glory. Let's pick one portfolio with normally distributed returns (.04/.10) and then run Eq 1 for all w between 2 and 12 % to get L. Then we'll consolidate frequencies of L and render it in a 3D map -- L, w, and freq as x, y, and z.
The assumptions are these:
r = .04 real
sd = .10
w = uniform 2-12% grouped in 10ths of percents
L is capped at 100 which will be a proxy for infinity
100,000 iterations even though the chart says 50k
Continuous re-balance
Note that this is not reality, just a simple fake model.
The output, from two different perspectives, looks like this:
and this
Conclusions
Conclusions? We don't need no stinkin' conclusions. Or, rather, we can now just observe the true nature of L as revealed by the 3D charts. Lower spend rates mean (a) an increasing likelihood of portfolio fails that happen later into infinity (we are ignoring planning horizons here but could consider them later) and (b) there is an increasing incidence of portfolios that seem to last forever. 4%, coincidentally, looks to my untrained eye to be roughly about where this really starts to tip over but that is just an artifact of this sim and does not mean that 4 is a magic number. This would all be a little bit more hairy with higher vol or lower r, by the way. 10% vol was maybe a little thin for the illustration[1].
Also, just for one tiny moment think like a 50-year-old retiree. That 50-year-old might have up to a 50 year horizon (or more). Now re-interpret the charts in that early-retiree light and tell me what you see. I was 50 when I retired. I literally didn't see this until today but I did start to intuit it back in about 2012. Hence the blog...
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[1] on the other hand, a 60/40 Vanguard fund I looked at had around 8% arithmetic return and something like an annualized 11% standard dev looking at the data from the late 20th and early 21st centuries. 8% rendered as real after inflation, taxes and fees would be pretty low. 4% here seems pretty generous. Geo "realized" returns, of course, would be lower still at t = T* or end of life.
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