Let's look at the response of a particular metric "portfolio longevity in years" (unbounded by human life scales, btw) to asset allocation along an arbitrary but not totally unrealistic efficient frontier...but: in the presence of high spend rates.
High spend rates here could be interpreted as either: 1) a literal high spend, or 2) as an underfunded retirement. Same thing. There are probably some pension finance corollaries here too while we are at it. I don't remember but I might have done this before here or I might have at least done some pieces of this before but whatever, let's dive in yet again to see how it looks. That (how things look) is my perennial question that got me from 2012 to 2022 on this blog and its precursors.
1. The Portfolio
This portfolio will be entirely made up. Also, I think for this post we can assume the return parameter to be rendered in "real" terms but it doesn't really matter since I have no client-deliverables at stake nor am I am shooting for tenure. Heh. Real.
I will assume a 2 asset portfolio (2A). I could have gone with 3 or 5 or 10 but: a) those get mathematically and practically intractable for me as an amateur dude, and b) it doesn't matter for reasons in the last paragraph and plus we can imagine the multi-asset equivalent of the "2A" just nudged up and left a bit while anchored to the low risk point (if that makes sense). So, I think we could basically sorta kinda replicate a 5 asset portfolio with a different parameterization with a nudge in the mid-upper risk range. Hard to explain. I will also assume a normal distribution of returns (not left-tailed) and stuff like iid because, well, because it is easier. Plus after doing this for a decade the impact on the margin of pushing those assumptions is sometimes too minor to make me want to try something else. These basic assumptions are close enough. I mean, I am cutting a lawn with a lawn mower, here, not cutting with scissors so we are just trying to get something done.
The assets:
- Low risk: r = .035, sd = .04
- High risk: r = .08, sd = .25
(coeff of corr = -.20)
These are debatable parameters, of course, but are just placeholders for illustration. The correlation coeff in particular, whether retrospectively or prospectively, is fun to debate but we won't. I could modulate these assumptions, but won't. yet.
The portfolio EF return will be defined like this as per Markowitz and Bray 2014 but otherwise generally well known:
R(g) = R(a) - V/2
which is rather silly since that formula is: a) at an infinite horizon which makes no sense for humans, and b) is one of at least 6 estimators and is apparently the least of them. But it is a good framework because it is easy to use and remember and provides a plausible outer boundary for our task. Plus the lawnmower metaphor and all... For the rest of this post we will ignore the geometric return and its frontier.
Combine all that gobbledygook and this is what comes out:
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| Figure 1 |
The black dots will represent our 11 different asset allocations. I could have done 101 or more but I don't need the complexity yet.
2. Portfolio Longevity (PL)
Here we will, along the EF, for each of the 11 black dots, evaluate "portfolio longevity in years" for some arbitrary spend rate(8%). I will use an algo I inherited from Professor Milevsky and later reproduced in his book "Retirement Recipes in R." I will use what I consider a high rate of spend = 8%. This is entirely arbitrary but in my experience 8 is either "high" or, as mentioned, it can be an artifact of an underfunded retirement but it is the same (imo) either way: risky. Note that the "PL in years" algo produces a distribution not a number. The other params here include:
- Scope of PL year playing field = 100 where 100 is a proxy for "a very long time" or even infinity.
- Iterations: let's go big with 100k
- Portfolio: as defined above
- Spend: as mentioned above but 8 in this post
The result....
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| Figure 2 |
3. Evaluation or Does it Make Sense?
Ok so I have no idea how to evaluate this. Obviously as one takes on more risk the dispersion of what happens in terms of PL spreads out. This is common sense. But is there anything we can say for these params? Maybe. But even then we have to consider the objectives which will drive the meaning. For example, here are some plausible objectives or questions:
1. On some central-tendency basis (let's say median since we are into a weird, defective, not even really lognormal type distribution; mean will suck at this game) does any allocation "win" in terms of how long it lasts? Here we are ignoring human lifetime btw, just PL solo.
2. If we are nervous nellies, maybe we focus on: "what is the likelihood of the portfolio, along the various allocations, crashing and burning in less than, say, 10 years?"
3. If we have high testosterone perhaps we might ask: "what is the chance that with a given allocation the portfolio will last more than 50 years?"
I am sure there are other metrics and other questions but let's stick with these for this post. And note that the answers and the judgements will all change with different parameters and boundaries.
3a. Median - PL in years
This is the median of the distributions in figure 2 along each one of the 11 asset allocations. Note that this is for this parameterization only. This changes in weird ways if we shift the assumptions around. Maybe I'll tackle that in another post to describe how it works but not today.
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| Figure 3. Median PL for 11 allocations |
X is each of 11 allocations. Y is median PL in years. Note that I did this in excel before I did it in R and got TOTALLY different results. That weirds me out. Idk if I mis-coded or if maybe doing 100k iterations shook some shit out. No idea. I'll just go with what I have. I've been coding R for a decade so I have some degree of confidence that I have what I have and since this is a fake world, not real world data, it kind of doesn't matter unless I am waaaay wrong in which case I'll apologize in a later post. TBD.
Analysis? A high allocation to risk under these params leads to high median PL.
3b. Left Tail - PL in years
Let's select some arbitrary number of years below which we might freak out if a portfolio were to fail. This frontier is totally fake and maybe there are better ways to select it. I mean, think about it. In this post I use "10" but if my spend rate were to be, say, 15%, 10 might be higher than the median so "not left tail." I will use 10. For fun.
Now, for each of the 11 allocation steps, and for the resulting probability mass, let's pretend it is continuous and integrate from 0 to 10 to figure out the chance that the P will crash before 10. If my notation doesn't suck (I wish I knew the math better) I think this might be the function:
Give me shit in the comments if I have this notation wrong. In any case, if I do what I think I am trying to say this is what we get:
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| Figure 4. The left Tail eval at PL < 10 |
Analysis? A high allocation to risk under these params creates a crappy outcome. Stay "safe" if your horizon is under 10 years. All this might change for different portfolios and spend rates.
3c. Right Tail - PL in years
Let's select some arbitrary number of years ABOVE where we might want to stretch if we want to max out PL with some allocation. This is the "high testosterone approach" and while I am relatively high T (now, finally) I do not recommend this mindset in Retirement finance. Here I will pick "50." That, again, is totally arbitrary. I picked 50 because: a) it seemed to fit figure 2 pretty well for "right tail," defective distribution notwithstanding, and b) when I retired I had a plausible 50 year horizon. As mentioned, change the params and this assumption gets corrupted. So: 50.
As before, if I get the math right, this is what we are doing recognizing that the integration to infinity picks up that defective right tail where some portfolios last forever or at least > 100. I might do another post on that later but not here. I just remember Prof Milevsky drilling me on that right tail like I was a first year (I mean I was 60!) when we had coffee. Heh. It is important to understand. Here:
Comment-ding me if I am off. Always eager to learn. The output for the given params if I have it right:
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| Figure 5. The right tail. P[PL>50] |
Analysis? A high allocation to risk under these params creates a great outcome if we ignore the left tail in figure 4 that says high risk can smoke you if you have a short horizon. Another thought is that, while we have a max PL at the max allocation to risk, the marginal rate at which we gain PL for PL >50 maxes at the mid allocations. That would be a really good policy conversation between the high-T and the circumspect consultants, something I won't hazard an opinion on yet but I know what I think.
The other comment is that I recall a paper by Gordon Irlam on backward induction, stochastic dynamic programming, and asset allocation. In cases of low wealth, he said, sometimes a high allocation to risk is like a lottery ticket. While we are almost certainly destined to fail in that scenario, the high risk might be the only way out, hence it it is sometimes taken as an option. Ha. Dark, but true. Don't want to be there, though.
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