Quick and Dirty Experiment with Real Options on Future Wealth vs Spending Choice.

Goal - See the rough shape of a real option-on-future-wealth value for different spend choices

I'm dipping my toe into water here where the conditions may not favor my swimming skills.  But this is another "wanna see" post and I wanna see a net wealth process at some horizon as a real option with some strike and then to also see what happens to the value of the option as we increase or decrease the spending.  This seems pretty obvious. Spend nothing one has infinite optionality in the following periods. Spend 100% and it's all gone.  So I expect the option value to decrease with spending choice but how? What is the shape?

The shape might matter.  Spending is often "held in" by early retirees as both a hedge against residual uncertainty that is not model-able and also as a way to access future optionality on the potential to increase (and not have to decrease) lifestyle (or bequest) given the unfolding of exceptional returns. While we will say little to nothing here about either consumption utility (really low spend will have low utility; zero spend is infinitely un-useful) or risk, in terms of fail or any other metrics, it does seem worthwhile to think about optionality on its own terms.

The Models

For the net wealth process it is, as in previous posts, a recursive process that can be described by this

net wealth process

For the option model I won't be using Black Scholes or closed form math (because I can't). I'll access the intuition via simulation (because I can) by way of the sum of the probability-weighted present values of intrinsic value which I'll represent like this

R = ∑ { P(MT) * PV(max[0, MT - k]) }

real (call) option model

and where the rationale for this intuition can be found in an article by Grossman et al in "The Intuition Behind Option Valuation: A Teaching Note."  

Some spend rate ranges I've run into recently

This post is not intended to be all that rigorous or thorough. I am just recording some things I've run into recently.  I've been looking at some soft boundaries or ranges for constant spending under some rarefied and probably unrealistic assumptions into which I will not go too deeply.  Also, I will not be throwing in any major conclusions or interpretations. This is just presented so I can record my thoughts and come back to it later.

First some assertions. Some of these are what an old boss called "proof by intimidation" which means your skeptical gene can feel free to kick in.

Obscure artifacts, now with a constrained horizon

I've about beat this subject to death but let's do one more.  This time I'll take the same portfolio longevity simulation I did before that was unconstrained, or run to infinity, and here create an arbitrary horizon. I'll set the horizon H to 40 periods to perhaps reflect an early retiree or a 60 or 65 year old with long survival expectations. I'll also, as before, look for artifacts in the fake model that suggest what I was calling a "tipping point" while being reluctant to conclude whether that point has any meaning or interpretability in real life.

I'm dumber than I look

In My portfolio longevity modeling artifacts vs economic utility theory on one set of parameters I tried to cheat a little and moved too fast. So fast that I made two errors: a sin of commission and one of omission.  I won't revise the last post, I'll just keep rolling here. But I don't think I'll tweet this one. It's hard enough to admit to error, harder still to do it in public.

My portfolio longevity modeling artifacts vs economic utility theory on one set of parameters

See the correction of errors that I made in this post here.

I guess I have to confess to being a geek here.  In the last post (One more obscure artifact of Portfolio Longevity) I was looking at the second derivative of Portfolio Longevity (L) percent at L = infinity (ok 100 years) or an artifact of which I was labeling "Y." In that look-see, Y ramps up the fastest at ~3.4% spend (w) for the given, and entirely illustrative, portfolio assumptions of arithmetic input 4% return and a 10% standard deviation.

The basic math model for this was like the following where M is endowment, w is spending and V(tilde) is random, volatile returns:

Net wealth process...recursive

The universe of L and w for that parameterization and that model looked like this when run

One more obscure artifact of Portfolio Longevity

In the last post I was eyeballing the first derivative of the process that drives portfolios from finite longevity to infinite looking for a "tipping point" for the given portfolio return and vol assumptions.  Not even sure if that is legit but either way as long as I am on this road I might as well keep going.  In the last post I guessed that the max slope was around 3-4% spend.  This time I wanted to zero in on exactly where and get the min of dY/dw or where the second deriv goes to zero over an abbreviated interval.

Obscure Artifacts

In my last two posts on portfolio longevity (see content and assumptions here) I ran a simulation that randomized both returns and withdrawal rates (uniform) to create a 3D picture of portfolio longevity with respect to withdrawal rates.  I was interested in one of the shapes in that chart. In particular I was interested in the nonlinear change in the percent of portfolios that lasted to infinity in that simulation (right side of figure 1).  The number of portfolios that do that start slow and then ramp up.  It seemed to confirm, in a way, in my own head anyway, my contention that 4% (given those assumptions only) looks like a "tipping point," rather than a "rule."

Here is another way to look at it. First here is the original chart with the region of interest pointed out by the red arrows...

Figure 1. 3D PL vs W

Now let's focus on this red curve exclusively. Let's also shift here from raw frequencies to the percent, for each withdrawal rate in .10 chunks, of portfolios that survive to infinity (or 100 years in this case) which, for convenience, we'll call Y though it happens to be the z axis in figure 1 if it were to be re-conceived as a probability mass along each w.  Then, in addition, let's now also look at the first derivative of Y with respect to w and see for fun where the rate is highest. Humor me by imagining for a moment that the curve is actually smoothly continuous.

The wall of probability implied by what happens at PL = 100 in figure 1 looks like this below in figure 2:

Figure 2. Percent of portfolios in sim that go infinite at diff withdrawals

Now, I'm guessing that this has more to do with the real growth rate being around the withdrawal rate (I'd have to check other parameterizations) but I thought that it was interesting that the rate at which these things go infinite reaches a peak or inflection point when changing from 3% to 5% or around 4.  This is another example of why in my 5-process posts I said I thought second derivatives were as insightful as anything else in retirement finance.  Note also that the inflection point is roughly where the median number of portfolios go into orbit.  hmmm.  Is this useful info? No idea. Just fun to see.


random other thoughts
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- I've never really considered 4% "safe" for anyone younger than 60. Even here in this post, where age is not part of the game, somewhere around 4%, half of this parametization has failed before infinity and half not. That begs the question on horizon which is typically discussed as "30 years" or "random human life." I recall a paper I read that showed that there is a direct link between consumption rates and mortality or hazard rates which dominate the spending choice. Probably true. I'll try to find the link.

- You can see in figure 1 that an opinion about whether a portfolio will last is, in fact, sensitive to choice about horizon. You can also see why spend rates can go up with age as the horizon comes in and why for early retirees, discretion might be the order of the day.

- The 4% rule, a lucky outcome from what might be called a lucky back-test looks here like one of an infinite array of possible rules.  In trading terms one might say the 4% rule was curve-fit to the data (and a fixed lifetime) and might suffer under an out of sample test (and random lifetime).

- My portfolio choice of .04/.10 was illustrative only and maybe arbitrary but, as in a prev post, it's not too far off from a blended portfolio after taxes and fees depending on what era the data model is based.

Where's Waldo? (or where is the 4% rule)

I recently tweeted out my last blog post about my 3D visualization of portfolio longevity vs. withdrawal rates. I followed up that first tweet with another tongue-in-cheek second tweet about finding the 4% rule within that image...like a where's Waldo game.

I took up my own challenge on that and now I thought I'd throw it out there. Here, based on all the flawed assumptions inherited from the last post, is the visualization of where the 4% rule shows up on my chart, all else being equal. Please admire me people for not writing "ceteris paribus:"

I haven't bothered to sync up my own portfolio (return and vol) assumptions with Bengen but they are close enough for my point.  And the point is that "4%" always existed within a broader probabilistic guessing game and while it worked out for that one study (alone) it still seems a little thinly thought out to me.  

The chart above shows that 4% over 30 years is just "one thing" among a probabilistic "infinity of things."  It might actually still be a good "bet" for a 65 year old with short longevity expectations.  But it is a bet.  And the thing about bets, especially if you've ever done track betting, is that there are not only probabilities (my chart) but also odds or what I'll call consequences. Payoffs I guess they call it.  The consequences of the world to the left of the red line that extends up and to the right from "30 years" can be severe. Then, when you consider that that 30 number is not even knowable, and also when you consider that my return and vol assumptions in the sim may be a little sanguine (i.e., it'd move the mass a little bit to the left), then they, the consequences, feel like they get even dicier, to continue the gambling metaphor.  The chart above represents only the probabilities, btw. The odds, on the other hand, are ours alone, and for each of us in our own way. 

You, my friend, can stick your head in the sand and just go with a fixed 4% rule.  Or, you can even wake up and also look at the probabilities like so many ret-fin papers suggest we do. Me? Uh, well, I'll do that as well but I will also be keeping a weather eye on the shifting numbers on the odds-board because that is what will hurt me if I don't watch out. ...or help my kids if I do.   

3D visualization of portfolio longevity vs withdrawal rates

This, hopefully for you, is (almost) the last in a series of me trying to visualize portfolio longevity. There are probably some readers who may be wondering by now, and muttering under their breath, "why does he do this?" Well, just to see it.  I like to know the shape of retirement math in my gut and this kind of thing helps.

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.

Withdrawal Rates, Stochastic Portfolio Longevity, and Trend Following

Here's another look at portfolio longevity with some stochastic assumptions about returns (normally distributed for now) and withdrawal rates (uniformly distributed to generate the chart) inserted into a net wealth process run in an unconstrained manner over infinity (using 150 years as a proxy). The key difference here from the previous posts is that I am now imposing a "stylized" and very extreme version of what might be called trend following phenomena to see what happens to PL compared to a base case.

Volatility and Stochastic Portfolio Longevity

This is an extension of a prior post illustration of Withdrawal Rates vs. Portfolio Longevity.  Without (too much) commentary, this is another heat scatter, this time for "input" standard deviation vs. portfolio longevity, where

- returns are randomized normally with a real return of .04
- standard dev "input" is uniformly distributed between of .05 and .40
- withdrawal rates are either .03 or .07 with an addition for .12
- a mini sim of the number of years a portfolio lasts is run 500,000 times
- "150 years" stands in here for infinity

The dotted boxes are not analytical but rather aesthetic. They are intended to draw the eye to what I'm thinking about. The conclusions, if I can even make them, might include:

- I'm left vaguely wondering if I mis-coded this...
- lower spend rates last longer, obviously
- lower spend rates seem to insist on lower vol strategies
- higher spend rates look more horizon-dependent.
      - short horizons want lower vol
      - immortals, early retirees, and endowments might need the lottery ticket of higher vol.
- I probably need to think about this more...

My interview that showed up in Kiplinger's Retirement Report

This linked article has some quotes from me that actually have a reasonable fidelity with what I said...  And the message is not too far from the various themes in this blog.

https://www.kiplinger.com/article/retirement/T037-C000-S004-how-to-draw-a-steady-portfolio-paycheck-retirement.html

A "tipping point" and the 4% rule?

The 4% rule gets flogged so often, the poor thing.  It more or less worked for Bengen due to its getting "close enough" over a "hard coded 30 years" using historical data that happened to be pretty good but which may never replicate. Or maybe it will.   My last illustration of withdrawal rates vs portfolio longevity has an implied comment on the 4% rule: "it's just one thing among an infinity of things." There is zero guarantee that it'll work.  On the other hand, it looks to me that, even though that comment is true, the 4% spend level seems to be a gateway to what I'll call a tipping point.

By tipping point I mean that a 4% spend, given a balanced portfolio allocation with its associated realized geometric return expectations for the future, seems to be about where portfolio longevity "tips" from more finite space (PL in years is < infinity) to more infinite space (PL lasts forever).  See here below in a revision of the chart I used in the last post. Assume that 4% return and 12% vol is more or less near where Bengen's 50% US equity 50% 5 year Treasury would land, which I think might be true. Then somewhere between a 4% and 3% spend (let's assume 2% withdrawal on the other side is, or can be, effectively a perpetual process) portfolio longevity "tips" from finite world to infinite.  Take a look and tell me what you think...

also, just for the hell of it, I amped up the iterations 10x to 500k to see how it fills out. There was no reason for this except curiosity...

illustration of Withdrawal Rates vs. Portfolio Longevity

Without comment, this is a heat scatter of constant withdrawal rates vs. portfolio longevity, where

- returns are randomized normally with a real return of .04 and a standard dev of .12
- withdrawal rates are spread out uniformly between 2 and 12 %
- a mini sim of the number of years a portfolio lasts is run 50,000 times
- "150 years" stands in here for infinity

No purpose other than to see what it looks like...

First derivative of deterministic portfolio longevity

Using a deterministic formula (see figure 1) that I borrowed from Milevsky's book 7 Equations as well as from his lecture notes, provides some insight into the effect of withdrawal rates on portfolio longevity (L): High w = shorter, low w = longer. That is trivially easy and intuitively accessible without the math.

Figure 1. Portfolio Longevity

A little bit more obscure but perhaps of more interest is the derivative of L, also lifted from the lecture notes and seen in figure 2, here presented over different w . 

Simple reality check vs. complex SPV

This is a follow-up to the "Impact of Longevity Expectations on the Stochastic PV of Spending" post and inherits its assumptions and worldview.

My Personal Planning Components

Contrary to what one might come to believe based on a quick scan of the content on this blog, I do not personally do very complex financial planning.  The bulk of my work revolves around keeping family financial statements like an income statement and a "flow-based" or actuarial balance sheet.  I then manage an infrequent monitoring operation that focuses primarily on:

- periodic feasibility (net monitizable assets > than the PV of consumption) tests
- periodic, simple, and very skeptical review of "sustainability" via lifetime ruin risk, and
- watching for trends or acceleration in effective spend rates within very broad, moving control boundaries
- plus maybe a couple other things

That means I do not personally perseverate on things like stochastic present value analysis. In the development of the balance sheet, however I do have to have a proxy for the SPV of spending. Typically I recommend following a program like that recommended by Ken Steiner at howmuchcaniaffordtospendinretirement.blogspot.com. For my purposes, though, I often do just a "reality check" that both simplifies this PV and adds a complicator which I'll describe.

Impact of Longevity Expectations on the Stochastic PV of Spending

The stochastic present value (SPV) of consumption -- from what I can tell an inverted way to do Monte Carlo as a feasibility check -- typically depends on a "horizon" (i.e., fixed to, say, 30 years) to do its thing. I thought I'd instead throw some random longevity at SPV and see what happens to a couple different summary thresholds(median and 95th percentile, with a nod to the absolute value for the starting endowment) for evaluating the distribution.

The Geography of Retirement

There is a particular geography of  retirement that only the retired have seen, and not even all of them. It exists somewhere on the frontier between reasonably judicious and self-aware caution on the one hand and un-reasoned paranoia on the other. Panic and fear do not dwell there but they might be said to live somewhere over the ridge-line that marks the horizon to the south. Academics (often with tenure, W2 income, or pensions) have certainly read about it in books and can, of course, render a map of the place with a fair degree of fidelity. Young professionals -- with all their focused ambition, un-examined proprietary tools and compensated service -- can only in their fevered dreams truly imagine the place, having heard tell mostly from strangers and old men passing on the road ("there be monsters").  The retired that have walked that frontier know the place.  This is why I trust them.

Coming in July: RH consulting

All that curiosity about quantitative retirement finance and 4 1/2 years of reading all that academic dreck has to go somewhere, right? So, for those that might need it and because no one scores goals for which they didn't take a shot, here is a consulting shingle set out to swing in the wind:

RiversHedge Consulting

Pricing for customer one and two is a pretty good deal. I'd jump on it. This'll be effective July 14 which will also happen to be my 61st birthday. And my twin's, too. Pricing probably says less about my value to you than it does the following: (a) shows some willingness on my part to invest in early adopters, and (b) the marginal value of my liberty and time can tend to be pretty high. 

I like to drink rye ... or ... Fitness after 60

I was recently flattered by having a guy in the UK use my before-and-after pic to promote a fitness program.  I mean, really, I'm turning 61, wtf? The "game" seems almost over at this point. But it's not really.  The whole point of RiversHedge is not to sweat pennies and do simulation, it is to live a life.  To live a "life" needs healthy finances, of course, but that is a dead-end without health.  Two years ago I did NOT have health, now I do. More on that later because that is the point of this post.

In case anyone might find it useful, in this post I will only describe what I do. I will not promote or sell a program or cite science or evidence. That is easy enough to find and everyone seems to have their own evidence these days, anyway.  I will just describe what I did should that be useful.

Five Retirement Processes, now in PDF form

I went to the trouble, just for you, of integrating, editing, making coherent, and saving as a PDF the "Five Retirement Processes" posts that I recently wrote here on RH. 

The link to Google Docs is here:  Five Retirement Processes

I reserve all rights.

The title page:

Darrow Kirkpatrick on Tracking Spending in Retirement

Darrow Kirkpatrick has a new post out called Budgeting or Expense Tracking: How Much is Enough?   This will be a short post. All I have to say is that while I do not do it exactly like Darrow and while I'd tweak it a bit, and while I also tend to go a little more custom via Excel, Darrow is 100% correct in both his method and his sensibility.  Track expenses and net-worth -- these are the core family financial statements: income statement and balance sheet, preferably an actuarial balance sheet. Do this consistently but not obsessively and do it at the right level of granularity ... and much of the retirement-finance universe seems to fall into place.

Gaze Heuristic in Retirement

I was reading a tidy little blog post the other day by Dr Cameron K. Murray on the Gaze heuristic (Improving 'Neoclassical man' with a gaze heuristic Thursday, February 12, 2015) and thought it quite relevant to retirement finance. With a proviso, of course, that I want to modify it a bit. 

Here's Cameron quoting Gigerenzer on how a ball might thought to be caught:

The gaze heuristic is the simplest one and works if the ball is already high up in the air: Fix your gaze on the ball, start running, and adjust your running speed so that the angle of gaze remains constant. A player who relies on the gaze heuristic can ignore all causal variables necessary to compute the trajectory of the ball––the initial distance, velocity, angle, air resistance, speed and direction of wind, and spin, among others. By paying attention to only one variable, the player will end up where the ball comes down without computing the exact spot. [emphasis added] 

Clare et al on Trend, Sustainable Withdrawal and Glidepaths, March 2019

This recent paper (Absolute Momentum, Sustainable Withdrawal Rates and Glidepath Investing in US Retirement Portfolios from 1925, Clare, Seaton, Smith, Thomas.  March 2019) consolidates a few of the things that I've been thinking about this spring with respect to retirement outcomes and trend following.  Andrew Clare and his collaborators have worked in the past on explicating perfect withdrawal rates (PWR) and the constructive contributions of trend following to traditional portfolios. Here, that previous work is extended to evaluating the addition of trend strategies in both overlay and allocation form to traditional portfolios as well as comparing them to other risk management techniques such as target date portfolios with "glidepaths."  Rather than summarize, I will extract, and the extract bullets should, I think, speak for themselves:

AQR on Trend Not Being Dead, Yet

In this AQR document (You Can’t Always Trend When You Want, AQR, Spring 2019) they come to the conclusion that despite trend under-performance since 2009, something I have perseverated on here before, that their best guess is that the anomaly premium is not gone yet.  Now I have to admit they do have a pony or two in this trend circus but I trust them since they have been a forthright and deeply quantitative crew for a long time. Full disclosure: I own some AQR trend related assets.

Cameron Murray's Thumbnail Summary of Ole Peters on Non-ergodicity

This article (Revisiting the mathematics of economic expectations) by Cameron Murray, a self-bio-ed economist is a good thumbnail summary of the difference between time and ensemble averages and does a good job doing a cover on what Ole Peters discusses often on Twitter and at his blog at ergodicityeconomics where Ole can sometimes be more opaque (to me) than not.