Process 3 - Portfolio Longevity [draft]

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Note: as in the previous essays, this is a draft as I hone some of this content. Also, since I view these essays as consolidating and integrating what I've learned about ret-fin so far, I will continue to add to and update this provisional latticework over time in response to new findings or errors.
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This essay is a continuation of:

• Five Retirement Processes - Introduction 
• Process 1 – Return Generation
• Process 2 - Stochastic Consumption Processes
• Summary here.

Process 3 – Portfolio Longevity   

"Risk and time are opposite sides of the same coin, for if there were no tomorrow there would be no risk. Time transforms risk, and the nature of risk is shaped by the time horizon: the future is the playing field."  Peter L. Bernstein 

The topic of portfolio longevity is not unknown, of course, in retirement finance but I think it is a little bit underappreciated in its unconditional sense[a] because we see it presented so often with some degree of prejudice or at least prejudice in its "preclusionary effect" sense.  By this I mean that when we talk about portfolio longevity we almost always see it, if we see it at all, integrated with a planning horizon such as "30 years" or random lifetime (i.e., the "conditional" sense) without first stepping back to look at it over an infinite horizon, a horizon that is the natural home of portfolio longevity as a continuous process or "a flow." We also, for that matter, seem to frequently jump very quickly over a deeper understanding of the underlying processes that are combined to create the concept of portfolio longevity in the first place.


The Two Processes That Underlie Net Wealth and Portfolio Longevity

The idea of portfolio longevity is implicit in what we'll call a "net wealth process." A net wealth process is an important but synthetic or integrated process that is made up of two or more underlying "core" processes. We have previously taken a very close look at two of the core processes in question: return generation (Process 1) and stochastic spending processes (Process 2). A close look at these is, in my opinion, a prerequisite to understanding well the net wealth process and "unconditional" portfolio longevity (i.e., viewed without prejudice with respect to finite planning horizons, random or otherwise).  Robinson and Tahani (2007) frame those core processes in general, analytic terms as two geometric brownian motion (GBM) processes along with a correlation between the two. Like this:

Eq 1.  return and consumption GBM processes with correlation

with R being the return process with GBM B, C being the consumption process with drift of alpha and a GBM engine described by Z. The third equation is the correlation in the two GBMs. This kind of thing is good enough for academic work, of course, but note that we saw in our examination of Process 1 and Process 2 that the full glory of return and consumption processes are not completely reflected by these GBM abstractions. Returns have leptokurtic features that can leave them both fat tailed and asymmetrical and they can also have perverse and counter-intuitive behavior over multi-period time. Neither of these features is directly apparent from the SDEs above. Consumption, for its part, can have shapes (discontinuities) and it also seems unlikely to exhibit true GBM over the long haul based on what we see going on in real life. The Robinson SDEs above appear to be conventions for research papers only but are at least useful to us as a baseline description for now.

The Income Process That is Missing in Net Consumption

Second, neither Robinson (or many others for that matter) nor my description of Process 2 make clear that the consumption process is maybe better represented as a "net consumption" process in this type of context.  By net I mean that there is a hidden "other" process related to income over a lifecycle --  income that would mitigate life-cycle consumption and make it "net" consumption -- that is often obscured and of which it is important to at least be aware before we go further.  Since we will not delve too deeply into income processes here and since income is highly unlikely to be a GBM process itself -- it more likely converges in late age, especially after wealth depletion, to a constant income from social sources or pension annuities -- we will propose an income process that, in my amateur fuzzy notation, might look like this:

eq 2 - income processes

where p(t) is income at time t; H(t) is monetized human capital at time t and I generally assume mine to be zero; p(ex)(t) is exogenous income, say social security or other pensions; and p(en)(t) is endogenous income, say life annuities purchased from wealth, though that last will make both wealth and income have discontinuous jumps. With respect to the function for p(en)(t): we'll say that W(t) is period wealth; α[a(x)(t)]c(t) is the percentage of remaining lifetime consumption that is annuitized from wealth at time c, the price of which has to be less than or equal to remaining wealth; and a(x)(t)) is the cost of annuitizing $1 at time t for someone aged x. Implicit in the endogenous income function is the idea that income would jump up for the remainder of lifetime if we buy an annuity.  The last equation of the three attempts to get at the idea that wealth will jump down at the annuity purchase. Note that there are other ways to model this kind of thing.  My notation here is merely a placeholder that acknowledges that the income process exists at all. [b]

Simplifying Some Assumptions

We have already spent quite a bit of time, in posts for Process 1 and 2, showing how broadly weird return generation and spending can be so I don't feel very obliged to belabor it much here in this analysis. We know we can do that later. Also, I'll note that income complexity is not all that necessary, yet, for this discussion. For those reasons we will simplify our assumptions in those areas for our own convenience in this post.  For income we will, for now, assert constant income, if any. Then, by setting the consumption drift and diffusion terms to zero (in the Robinson equations above) we will effectively assume constant consumption. There are no other consumption rules or discontinuities assumed.  We will also consider consumption to be "net consumption" after income is added back (unless income is specifically called out in some equation or income is zero).  If for some reason we need to do this, we can also consider net consumption to be 1 unit of constant net consumption which would make wealth "units of wealth."  These may be flaky assumptions, perhaps, but they are sometimes much easier to work with when doing the math.

A Net Wealth Process Represented Two Ways

When the two main processes in question (return generation and stochastic consumption...plus the hidden "other" (income) process) are combined, a net wealth process over time can be represented like this which is lifted from Habib (2017):

eq 3 - net wealth process, full

where alpha is the percentage allocated to a risk asset, (u - r) is the risk premium, W is wealth at time t, B is the geometric brownian motion driving the process, p is income and c is consumption. Initial wealth is w and wealth over time is constrained to at least zero, and if it turns out that it is at (or thru) that point, then it remains there forever.

Then, if we were to (1) think in portfolio terms rather than the allocation components, (2) consolidate consumption and income into "net consumption," and (3) work in wealth units, we might alternatively render it like this:

eq 4 - net wealth process, simple w constant net consumption in units

This way of framing it has the bless-ed advantage of making my life much easier when simulating this kind of thing in Excel for a blog post, for which I would do it like this:

W(t+1) = µW(t) -1;  µ ~ N[r,sd],  | we acknowledge the weirdness Process 1 & 2 

eq 5 - net wealth process, simple simulation

Either way we do it, this synthetic net-wealth process, when everything is put together, now becomes the genesis for the third main process in the "Five Retirement Processes:" Portfolio Longevity (PL). PL, for what it's worth, has its own properties and behaviors which are worthy of considering and understanding -- especially in the unconditional or "without prejudice" sense -- which we will look at shortly.

Net Wealth, Time, Dispersion and the Zero-Wealth Threshold

The first thing to know about a net wealth process (NWP) is that given the size of C (or 1) relative to W and the dynamics of µ, it is entirely possible (unlike the GBM process for a stock price) that wealth at some time "t" can hit or, if entirely unconstrained, go through zero.  It is this property of a NWP that causes the possibility of fail or ruin.  In most of the literature I see, fail or ruin is often described as a rate or a percentage but that is only because it is typically evaluated at some specific point time like "30 years" or "end of random lifetime."  The main point I am making here, on the other hand, is that the net-wealth "fail process" happens continuously (and increasingly) over infinite time and is not just a rate or a number at some "t" (i.e., a fail rate...see the x axis below for a given "t" value on the y axis) but is also a sorta-continuous portfolio-longevity distribution over all "t." (PL is a function related to going all the way down the y axis). These two ways of looking at it are more or less saying the same thing, they are just oriented differently.

But before we specifically get to portfolio longevity (going down the y axis) we should probably make sure we can visualize net-wealth as a diffusion process in the first place.  Since a NWP is rendered as an SDE/GBM process like we defined above, the key thing to think about here is that it will probably exhibit some degree of drift and dispersion. Let's take a look.  Below we are going to use [eq 5] to drive a NWP over 20 periods with a normally distributed return generation process with mean .04 and a standard deviation of .12. Consumption here is 1 unit into 25 wealth (w/c) units as defined at the start-time zero. When we turn on something like this, this is what happens:

Figure 1. Net Wealth Process over 20 periods

The y axis is time going from zero to twenty which advance towards us from the back. The x axis is the net wealth outcomes at some time t. Note that the x-axis is low on the right and high on the left. We have turned this 3D model around so we can see it better. z is the probability mass value which has been clipped at .40 but goes to 1 for wealth units at time zero.

The dotted line happens to be the boundary where wealth goes to, or if unconstrained goes through, zero. By looking at that dotted line we can see that time, consumption, and uncertain returns create risk --  i.e. there is an increasing likelihood that wealth depletes and that, even though some extreme wealth can occur as time passes, the distribution as a whole gets increasingly asymmetrical while the main mass of likelihood (in this case, anyway) moves towards modest or negative outcomes -- and the question for us then becomes either: (a) "what is the likelihood of hitting zero at some time t?" (i.e., x axis: fail rate at time t) or (b) "what is the likelihood of a portfolio lasting to time t?" (looking down y axis this time). The latter is, in my opinion, a more natural question in retirement but, as they say in Wisconsin, it's "a horse a piece." Milevsky has covered this kind of thing (PL) at great length (e.g., Milevsky(2012) and others). We'll focus on PL for the remainder of this post.

A Simple Portfolio Longevity Distribution...as Extracted from a Simple Simulation

Anyone who has read a lot of retirement lit might be amused by the fact that I am trying to separate mortality risk from the concept of portfolio longevity since the marriage of those two ideas feels like one of the bigger achievements of the last 60 years in retirement finance (at least in the academic lit since, oh, say 1965). But there are some benefits to separating here. First, it acknowledges that the two processes (human mortality and portfolio longevity) are independent of each other [*see the comments because in real life there are some correlations] and so each is worthy of understanding in its own right first before one attempts to integrate them. Second, it allows us to isolate and see the full scope of the synthetic portfolio longevity process on its own terms over an infinite time frame, a time frame where portfolio longevity reaches a limit. The awareness of the limit is useful only for endowments, those with strong bequest motives or maybe immortals. And third, by focusing less on a "fail rate" at some specific time t, which is really Monte Carlo territory with all its faults and foibles, and by focusing more on time in general and PL as a process, which is somewhat closer to questions that a retiree might really ask (how long will it last vs. what is my "fail rate at time t"), we are now looking at what Albrecht and Maurer (2005) called the entire "constellation of asset-exhaustion." Even though random lifetime is one of the bigger conundrums in retirement finance, and even though I have separated it out for now, this concept of "constellation" will be useful to us later when it is juxtaposed with the mortality considerations I have temporarily rejected here.

While I appear above to be a little dismissive of Monte Carlo simulation for retirement I have not entirely escaped gravity yet on simulation in general because I still need it to generate the simulated outcomes from which I can extract the shape of portfolio longevity over infinite time.  MC sims for retirement often get reasonably rapped for a variety of reasons, such as:

1. They fail to acknowledge magnitude of the fail (e.g., Kitces(2012))
2. They tend to be opaque, difficult to explain and difficult to replicate
3. Minimization of fail rates can lead to sub-optimal plan design and outcomes
4. They often ignore risk aversion and investor preferences (utility)
5. They can obscure teach-ability on the intuition about, and relationships between, key variables

But in my case, not only is simulation the only way to get what I need, I am also working at a level where some of the objections I listed don't apply or don't apply yet.  Where Monte Carlo for retirement sometimes answers questions (fail rate x at time y) that aren't really being asked and integrates processes before the processes are well understood, extracting a PL distribution from a sim is more a case of setting us up for questions that haven't been asked yet while also being faithful to understanding one process on its own terms before we move on to other things.

For example, since extracting a PL distribution gets to the root of fail "magnitude" by looking at the full scope of all of the "constellation of asset-exhaustion possibilities" over all time frames possible, objection-one does not really obtain. We are looking at magnitude directly in its raw form.  Also since I am using eq5 to do the simulation (while keeping knowledge of the shape of processes 1 and 2 in mind) it is as simple and replicable and can be.  Also since the output of PL simulation is an input into other analysis and other conversations, objections 3-5 are more or less mooted, too. For now.

So, if I run Eq 5 with the same assumptions I used above (4% real return, 1 unit spend on 25 of wealth[4%], 12% vol) over, say, 200 years and I do it many thousands of times I get a simulated net-wealth process that unfolds like figure 1 (where I happened to stop at 20 years for the sake of illustration). 200 years is an arbitrary choice and will stand in for infinity because it is generally long enough to see how planning choices would play out since we have no prejudice (yet) with respect to any finite planning horizons. 200 years also makes my life easier numbers-wise. From that long-game process I can now extract the shape of PL by looking at the relative frequency[c] of the first occurrence of a depletion event in every iteration and over all years. For a spreadsheet sim where I have a mere 4000 iterations (a little thin) the relative frequencies rendered as a probability mass would look like this:

Figure 2. PL probability mass

If you think about it, this distribution answers the question "how long will my money last?" under some relatively restrictive assumptions (e.g., constant spend, stable independent returns and vol, etc) but it is a weird looking distribution. Since I am not formally educated in math, statistics, or economics, Prof. Milevsky, when I met him, very gently schooled me on defective distributions (which this is) and reminded me that it really represents two distributions: (1) those portfolios that fail in finite time and (2) those that make it to infinity (i..e, succeed forever). Integrating both will add to P=1.  He also, like almost every professor ever, asked me five times until I could name the distribution correctly ("what is the name of this distribution...try again...try again...again..."  I mean, really, I'm 60 and I felt like a first year idiot). The correct answer: "portfolio longevity in years."

Integrating this pmf/pdf or, alternatively extracting an empirical cumulative distribution from the sim data directly, gives us the same thing in a different form:

Figure 3. PL cumulative distribution

Here it's easier to see that there are no failures in the first 12 years, fails pick up speed thereafter, and limit out at around 70% starting somewhere around 100 years.  ~30% of the portfolios happen to succeed forever.  This is why I have not imposed any conditions like: evaluate to "30 years" or "random lifetime" yet. We get to see how this process unfolds as a fully expressed process over eternity.  We can now see where the "arbitrary point-in-time fail rates" come from and how they can misrepresent the process and answer questions that might or might not have even been asked.  To me, this also helps show me how the Bengen 4% rule worked. His specific parameters (e.g., 30 years, historical returns from "recent" history) just happened to work out for a process that might actually have been a true 50 or 60 or 70% fail when seen over the very very long run.  The only reason 30 years happened to work is because the interval was lucky to end just early enough. In other words: the "30" threshold is pretty arbitrary and not very reassuring, especially for early retirees.

How a Simple Portfolio Longevity Distribution Behaves in Response to Changed Parameters

This is just a model of course and not reality but we should at least see how it behaves with changes in the few parameters we are using.  The neat thing about the simplicity of modeling consumption as we did above, as "1 unit," means that there is no real difference here between a reduced endowment and increased spending. The relationship is clear. There is no difference either between increased income and lower spending or between higher fees and taxes and higher spending.  That leaves three things to play with: returns, endowment/spend, and volatility. Let's take a look. This will not be rigorous or exhaustive. We are just shaking it up a little bit to see how the changes look visually.

0. Base case, as above: real return = .04, sd = .12, W/c = 25/1 [blue]
1. Increase real return assumption by one point from .04 to .05 [red]
2. Decrease endowment (increase spending) by 20% from 25 units (.04) to 20 (.05) [green]
3. Reduce volatility by an arbitrary 2 points from .12 to .10 [grey]

The changes, charted, look like this. A reminder that this is illustrative only:

Figure 4. Rudimentary sensitivity of PL to parameters

Table 1. fail estimates using PL at different "t"

I have not been exhaustive with parametization changes and sensitivity since we are just taking a quick peek, nor have I tried to create equal magnitude changes across the altered parameters. It would be interesting to see, however, if other combinations have the same effect I see here where the domain over which the changes have an impact does not appear strictly uniform. For example, if we look at the first 30-60 years, the impact of volatility is high relative to its impact later on after the effect loses "speed" especially if we compare it (grey) to scenario 1 (red).  This might be easier to see if we: (1) look only at the fails that happen in finite time (< 200 years here), (2) we zoom in on t = 1 to 60, and (3) render it as a density. The same color coding scheme applies:

Figure 5. PL density for finite-time fails only t=1:60

While the vol reduction (see our caveats above) fails to convince us on its overwhelming effectiveness over infinite time in this particular example, it certainly does convince us over an interval closer to a realistic lifetime. According to Corey Hoffstein at Newfound Research: "this is intuitive, because if portfolio hits ‘escape velocity’, slightly increased vol won’t matter much." If this kind of effect holds over other parameterizations, and I'm guessing that it will, this is where there would probably be a good case for active engagement with portfolio design, hedging and risk management programs. In particular I am thinking about trend following or other alt risk diversifications that seek efficiency through vol and downside risk suppression.  There are other means to do this no doubt but the point here is that with a little bit of smart, evidence-based portfolio design, you've bought yourself something really important: time. This is an unexplored topic here but my Fin-twit friends at Newfound and ReSolve Asset Management tend to cover this kind of thing in depth and with integrity.

Other thoughts on this without diving too deep for now... The effect of the endowment reduction we did here, equivalent to a market draw-down, would also be expected to fluctuate with markets and would be hard for me personally to watch. Also, since we are working in units (which conflate endowment and spending), I'll note that a reduction in spending would be the equivalent in this model of an increase in the endowment with a significant impact on portfolio longevity occurring in the opposite direction from what we saw with the move in the green line in scenario 2 when we clipped the endowment by 20%. This spend change would have the added benefit that that course of action would be expected to be more permanent and less volatile than changes in the endowment itself due to market flux. I'll look at this more closely in another post. 

Evaluating and Using Portfolio Longevity

Now that we have had a short introduction into PL, let's look at some different ways to evaluate or use the concept assuming we have successfully extracted a PL distribution from a modeled net wealth process...or if we at least have the basic input assumptions for endowment, real growth expectations, and consumption.

PL Evaluation and Use - Simple Formulas (no or solved-for horizon)

Not all evaluations of portfolio longevity require calculus or advanced simulation techniques. There are back-of-the-napkin or simple deterministic formulas to use first.  The simplest is dividing consumption into wealth.  This gives a wrong answer but at least starts the conversation on "how long." If I spend 40k of $1M start-value I have 25 wealth units to spend over time. In the absence of growth I have 25 periods to go.  This misconceives the problem a bit because what we really have, if we are more careful, is a "present value" problem. i.e., "is the present value of the 40k spend (or alternatively 1 unit) more or less than $1M (25WU) using some discount rate that reflects growth expectations net of inflation, taxes and fees?"

Milevsky (2012) gets at this PV idea[d] by presenting a formula he attributes to Fibonacci though he points out that Fibonacci did not write in equations like this and anyway he predates the invention of logarithms.

Eq 6. Milevsky's Fibonacci eq for PL

r is the real growth rate, c is consumption, W is the initial wealth. Milevsky points out that Fibonacci's method, represented here by this equation he didn't use, was the first instance of using present value analysis if you look closely at how he articulated the problem being solved. And here we are 800 years later still blogging on retirement finance!

Michael Zwecher (2010), without me explaining his terms, offers this version of "annuity number of periods" in his end-notes (chapter 3 note 3 p 252) and points out that this is usably invertable for working with loan payments outbound in addition to annuity payments inbound in relation to the number of periods:

Eq 7. Zwecher's Annuity number of periods

But if we are going to use Eq. 6 or 7, both of which have an implicit present value perspective, why not just recognize we are doing present value analysis in the first place and go to page 1 of any finance text or to any financial calculator or to Excel for that matter and come at this directly. If we stipulate that PV math looks like this below and that 1 unit of consumption means PV = wealth units = W/c = $1M/40k = 25,

Eq 8. Basic Present Value

then we just solve for "T." Excel solver makes it particularly easy and has the added virtue of not (always) choking on situations where limits are involved. It'll probably give a wrong-ish answer in that case but it will at least answer the question in human terms and allow exploration of the limit. This tool is simple, fast and hopefully a little more intuitive, depending on the audience.  Any one of these is useful enough, though, for quick work for most retail retirees.  It allows a conversation by separating portfolio longevity from a planning horizon. PL and horizon can be brought together later for a series of what ifs on the core spending and return assumptions. That approach also keeps us pretty far away from technical analysis, Monte Carlo simulation, fail rates and ruin risk, a distance which is not always a bad thing for a first round conversation.

PL Evaluation and Use - Baseline Simulation  (either no or solved-for horizon)

Simulation seems to sometimes be a dirty word for many of the reasons I listed above. It also is sometimes seen as an inelegant, brutish method by academics. But academics often write for tenure or other academics, rather than retail retirees (or even advanced practitioners) and so they have other incentives at work. Sometimes simulation is just easier to do and provides insights that might otherwise not be accessible to mortals. Most simulation I have seen, though, is not set up to work with portfolio longevity per se. It is designed to evaluate fail rates (and sometimes magnitudes) at specific points in time. Sometimes (rarely) it is done for random lifetime but the results can be hard to interpret.  To get a portfolio longevity distribution and the full "constellation of asset-exhaustion" one would have to run a MC sim for each horizon separately, collect the cumulative fail rates at that horizon, differentiate the results to get the incremental changes in PL and then, importantly, not forget to add back the implied results for those iterations that survived to infinity (maybe you could say "failed" at infinity but that's kind of a wrong way to say it).

The alternative might be to spend 10 minutes to spreadsheet it out and access PL directly.  The output would then be usable for the same conversation I mentioned in the previous section, except now we can talk in "distributions and probability" which is a useful shift in the conversation.  I'll give you an example.  Take the assumption of a 4% spend on $1M (or $1M/40k = 25 wealth units and a 1 unit spend) along with a 4% real return. If we plug that into Milevsky or Zwecher we get a choke because we are at a limit and the interpretation is that the portfolio lasts to infinity. Highly doubtful. Doing it like W/C = 25 periods is wrong too because we have growth. If we simulate it, on the other hand, we get the distribution we illustrated above in figure 2 and 3. This is a better tool for the given assumptions.

So what else can we say at this point? We could try to articulate some summary statistics but this is a defective distribution and summary stats can be more or less meaningless in that case. The mean is not useful at all. Median gets at it better but I have to ask: the median of both distributions combined or just the fails in finite time? The mode has the same problem but may be more interesting because it will give two very different answers for the combined vs single distribution. Portfolio longevity, when looking at the two modes, is sort of like light in that PL has "dual nature": one interpretation of one mode is that "PL = infinity" (kinda true and the answer from Eq 6), the other interpretation is that "PL ~29 years" which is also kinda true and useful for a conversation when a planning horizon or longevity expectation (which we'll get to in Process 4) is in hand. Here are the summary stats for Figure 2:

Table 2. Summary stats from Figure 2

This kind of conversation around a distribution and its flawed summary stats may require some jedi-level skills for a given audience but it seems to me to be a more honest way to come at it than using "simulated fail rates at a specific arbitrary horizon are x%" or saying "it lasts forever according to a formula."  Figure 3 as a tool would be useful to this conversation as well since more moments of the distribution are visible and the threshold level chosen (which is arbitrary or a policy choice) has direct access to knowledge of the cumulative probability of the scenarios chosen. Also, I think that very aggressive, high-risk plans would likely have a useful discussion around the speed (slope) of PL as it moves towards the limit as well as the number of years that are at stake since the CDF would probably start to approach the limit on the scale of a human life. My guess, without doing the analysis, is also that "PL years gained by a strategy change" would be way more sensitive to parameter changes (e.g., spending reductions) in a high risk plan than low. That's another post.

But Table 2 was a little bit of a trick. I chose the parameters intentionally to sabotage the analysis and force the "infinity" so that we could focus on the distribution without being distracted too much by the deterministic formulas.  Now lets back up a bit and lower the return expectation to force an interpretable deterministic depletion result from the formulas and so that we can try to see if we can norm the distribution those results.  Everything is the same here, then, except that the real return assumption goes from .04 to .03.  When we do this the revised table looks like this:

Table 3. PL stats with real return now = .03 vs .04

and the deterministic PL results from the simple formulas look like this:

 - Milevsky's Fibonacci:   46.21 years
 - Zwecher ANP:              46.90
 - Excel PV with solver:   46.90

Plotting the latter three against the new, updated probability mass:

Figure 6. PL with .04 spend, .03 rr, .12 sd

What is the real PL here? Hard to say. 11% succeed forever. The mode of the first distribution is 25. The mean is 41 and the median is 33.  The formulas say 46.  The truth, in the end, is the distribution itself -- and even that is fake.  All of this is more or less meaningless until you throw a threshold barrier in there for age at death or a planning horizon. And then we are back to a conversation and we would probably ask for a re-run with different parameters to flesh the conversation out a bit more.  If I saw this for my own planning I guess I'd be inclined to pay particular attention to the mode or median but that is just me and a highly personal and subjective choice.   

PL Evaluation and Use - Life-Cycle Utility  (horizon dependent)

The presence of the concept of portfolio longevity combined with the idea of random lifetime instantly creates the potential for depletion before life ends with or without the availability of lifetime income. Either way, this is called "ruin" or "fail" or wealth depletion time and has been studied in economics for quite a while.  I was going to ignore horizons or mortality in this post but it is hard to do when we discuss PL so I'll broach it here because economic utility is useful for comparing strategies where the comparison is done on consumption over a lifetime rather than the portfolio or its longevity.

The formal way to do this, as I have come to understand as an amateur, is through additive, separable, life cycle utility evaluation.  The paper I think of the most is Yaari (1965) but there have been many since (see here).  The way it is often set up is via a value function like this which is my amateur deterministic version of what might be seen in Yaari's case A (before he gets to bequest) or most grad macro textbooks:

Eq. 9. Value Function: Expected discounted utility of lifetime consumption

where tPx is a conditional survival probability (we won't see this until Process 4) but could be absent if the sum were to run to random life or to some discrete planning horizon, theta is a subjective discount and g[] may be a CRRA utility function (but could be something else) that can look like this where gamma is the coefficient of risk aversion and c is consumption:

Eq 10. CRRA Utility

The context for using this kind of thing is typically one where

- wealth potentially depletes before life ends
- income is present and consumption snaps to it at wealth depletion
- lifetime is random so the depletion date and duration are both random
- preferences are stable and independent over time
- risk aversion is non-linear and known
- utility analysis is believed to be able to find an optimal consumption strategy

The strawman model that I have used to visualize this in the past looks like this

Figure 7. Schematic for Wealth Depletion Utility Model

When I have done this kind of evaluation in the past, I have used simulation to generate the random end to the portfolio and to life as well. This randomness is what creates the "expected" value in E[V(c)]. But we could also do a simple deterministic calc to demonstrate the idea or to propose an heuristic for faster easier analysis.

Let's assume everything is deterministic, the subjective discount is .005, the conditional survival probability is ignored in favor of a very long fixed planning horizon of 45 years (to age 105, say, for a 60 year old), income available at wealth depletion is 1/4 of the base constant spend strategy (.04/4) and is the level to which spending "snaps," assumptions for real returns are .03 which are net of inflation taxes and fees, the spending rate is either .04 real constant (where PL is asserted to be a fixed 33 years occurring at the beginning of year 33) or, alternatively .035 (where PL is asserted to be 40 years), and the risk aversion coefficient is an entirely arbitrary 1.  The illustration could then be proposed to look like this:

Figure 8. EDULC examples with different PL

Spending less is less fun and evaluates here to lower utility in each period but portfolio longevity impinges on the higher spend eventually to create a longer wealth depletion time that over a lifetime evaluates to lower discounted lifetime consumption utility than the lower spend, lower period-utility strategy.  Knowing the number of PL years can either be obscured in a complex black box simulation or it can be explicitly transparent in a more simple deterministic analysis. Either way, it matters and life-cycle utility can be a useful tool to see the impact.


PL Evaluation and Use - Coverage Ratio  (horizon dependent)

It is rare to see papers in ret-fin step out of fail-rate analysis so I noticed a recent paper by Estrada and Kritzman (2018) that proposed an analytic framework based on a type of portfolio longevity.  It does not work with a distribution as such but however one comes by the number of years, they propose to use it to evaluate what they call a coverage ratio.  This they define as something that captures the number of years of withdrawals supported by a strategy relative to the length of the retirement period considered. Again this is horizon dependent but a useful addition. They suggest that it addresses a couple of concerns with fail-rate approaches: "First, it [MC] fails to distinguish failures that occur near the beginning of retirement from those that occur near the end. Second, it fails to account for surpluses that could be left as a bequest."

If Yt is the number of years inflation-adjusted withdrawals are sustained by a strategy and L is the length of the period under review then the coverage ratio Ct they propose is

Ct = Yt/L ;      Eq. 11 

Since this does not capture either the diminishing returns to bequest utility or the full force of high magnitude shortfalls they also propose a utility overlay where the function is kinked at a ratio of 1 where portfolio would have covered withdrawals to precisely the terminal date. Their kinked function looks like this:

Eq. 12 - Estrada and Kritzman (2018) kinked utility

For this post I have reserved judgement for now on whether this "coverage ratio" approach adds anything to either the direct examination of the PL distribution (above) or the conventional life-cycle utility value function that is typically used and that has at least 60 years of historical weight behind it (above). My guess is "no" but I think the ratio itself has a ton of communication value above and beyond any technical analytic insights that might be revealed. But then again we would have already known that from either the simple formulas or the mass of likelihood in the PL distribution. TBD.


PL Evaluation and Use - Stochastic Present Value (horizon dependent)

Some days all finance topics feel like they boil down to nothing other than present value analysis. The evaluation of PL can't escape it either.  Papers in the ret-fin space will demonstrate how a net wealth process projected into the future, one that creates the potential for the constellation of fails we can call PL (or fail rates at specific terminal points), can also be transformed and re-conceived as a stochastic present value where the same evaluations are done except that here we discount (stochastic discount) future consumption into the present to compare it to current wealth. We saw this kind of thing in our review of Process 2 or here is Milevsky (2005):

In the language of stochastic calculus, the probability that a diffusion process that starts at a value of w will hit zero prior to an independent “killing time” can be represented as the probability that a suitably defined SPV is greater than the same w. 

A simple expression of SPV [Milevsky (2005)] would be like this where T(tilde) is random lifetime, R(bar) is the stochastic discount and prob() is conditional survival in the continuous analog (bottom) of the discrete version (top):

Eq. 13 - SPV in discrete and continuous form

Again, this kind of analysis has to lean on a planning horizon which I was going to avoid but I think we should touch on it in this essay anyway. In this case we'll think of T as a fixed horizon for a moment rather than random just to keep things simple.

The stochastically discounted present value of consumption would become a distribution and something in that distribution would be compared -- based on some policy choice about what metric is to be used -- to initial wealth.  If the policy metric is, say, that the 95th percentile of SPV(spending) has to be less than initial wealth we could visualize it like this:

If we stick here with the idea that the horizon is fixed, then we could get at the portfolio longevity concept indirectly with SPV. My guess is that no one ever has done or would do this but you could: dynamically evaluate T using .xx prob as a threshold to find the T that satisfies that threshold such that SPV[P=.xx)] <= w. There are probably easier ways to get at this (i.e., what is an estimate for PL?) and I'd start with the Excel PV solver or direct examination of the simulated PL distribution before I did this kind of thing. The only reason I added it here is that (1) it works in present time vs the future and evaluates against a present observable in the form of initial wealth which gives it some credibility or at least intuitiveness and (2) I'm pretty sure I can perform this analysis without simulation if I am reading Milevsky (2005) right on his reciprocal gamma distribution approximation[e], something I did not do in the example below. If I were to do it by directly simulating (and after doing it this way for this example I'm pretty sure I won't do it again), it would look like this. Hopefully I'd have a more automated dynamic optimizing machine to do it for me next time.

Figure 10. SPV for different terminal years

Here we have a nest egg of 2.5M, an initial spend rate of 4%, a discount in real terms of 4%/12% r/sd, and a chosen threshold probability of SPV fail of .20. A rudimentary spreadsheet sim with a too-small number of iterations(3000) was used to generate the dispersion.  Starting with a fixed spend horizon of 35 years (grey), the area under the solid grey line to the right of the grey dotted is a 20% chance of the spend being larger than an SPV value = x which in the grey case starts at ~3.1M. Going to T=30(red), the same area starts at ~2.7M. At T=25 (blue) it's ~2.4M.  So for that particular threshold level and that spend, PL dynamically evaluates to a little over 25 years. I have not attempted to compare the process I did here with the other formulas and methods above nor have I carefully examined and rationalized the sim output for coherence or error, all of which I may do in another post. We'll call it illustrative for now.[e]

See Milevsky (2005) for how to evaluate SPV without resorting to simulation. See Robinson (2007; Appendix) for a concise proof on equivalency between a GBM net wealth process (in which PL is embedded) and SPV analysis.


PL Evaluation and Use - Lifetime Ruin Probability (horizon dependent)

I've tried my best in this essay to stay focused on PL in its unconditional sense. But it's hard to hold that kind of gaze for too long because however interesting it is to regard PL as an eternally continuous process, anything past about sixty years for a sixty year old is of no real practical interest.  So, while PL and human mortality are truly independent processes and it is useful to make that separation intellectually, in the end PL is constrained in our imagination by the limits of our self interest which are themselves imposed by how long we are allowed to live and spend.   The impulse to integrate independent retirement processes (returns, spending, life) into one thing in the name of self interest has a long history. Milevsky dates one example to the 1200s and Fibonacci but there are echoes of similar thinking in the Old and New Testaments if not earlier sources than that. I have not read the Rig Veda yet.

My own interest in "integrating" PL and lifetime first came from me trying to build my own simulators to make an end run on a fee that a banker was trying to charge me back in 2010. I also was dazzled later on by an inscrutable equation in Milevsky (2012) describing Kolmogorov's partial differential equation for Lifetime Probability of Ruin (LPR).  That equation dated from the 1930s and a Soviet era mathematician. It seemed to magically encapsulate all of these independent processes we've discussed in one tidy visible mathematical relationship (how did he do that!?).  In Milevsky's 2012 book, he does the notation like this:

Eq. 14 - Kolmogorov's PDE for Lifetime Probability of Ruin

This PDE shares similarities with the heat equation in physics but here describes LPR.  At first, since I don't know differential equations, this was totally unreadable nonsense to me. So I taped it to my refrigerator for a while to let it sink in. Now, you might recognize the (µw-1) coefficient from what we did above but it took me about nine months of staring at it, coffee in hand, to finally go "oh...yeah, now I get it, it's just the drift term of the net wealth process." And until I started working in unit terms that "1" was just a number that I didn't get or see for what it is: the consumption term.  The rest of the terms to the right just describe the diffusion process while the lambda term on the left is what brings in the mortality hazard rate (see, it's independent).  It's easier to see the relationships now but it took me quite a while.

The other thing that made it easier for me to see how this LPR thing works was what happened when I was once messing around -- in separate, unrelated projects in 2017 -- with (1) simulating portfolio longevity and (2) modeling human mortality and conditional survival probabilities with Gompertz math. That last I also got from Milevsky's book.  After an inspirational nudge from something I read in a Newfound Research article (The Lie of Averages, Sep. 2017), I decided on a whim to put PL and conditional survival probability together and "aha!" I now had a recipe that almost precisely satisfies the PDE above while explaining LPR to myself in an intuitive way that I finally understood.

Visually it looks like this. Take a PL distribution like the one we created above with say 4% return, 12% vol, and 4% consumption. Run a ton of iterations this time (100k in this case) to get a reliably smooth probability mass. Limit the view to the interval of interest (stop at age 120 for a 60 year old so 60 periods). Chart it.

Figure 11 - P[Portfolio longevity in years]

Then, in an independent, unrelated process not shown here[f] calculate the conditional survival probability -- for example, here it's done for a person that has survived to age 60 and now has a modal life expectation of ~90 -- for the interval in question.  Overlay that on the chart above (I've scaled it separately, right and left, to make it easier to see). The added red line is the likelihood of surviving to each year in the interval. Chart it:

Figure 12. overlay of PL in years and CSP for 60 year old

Then, and this was the big "aha" for me, multiply the two.  And there you have Lifetime Probability of Ruin.  LPR can be viewed in this sense as the probability of portfolio longevity in any given future year weighted or discounted by the likelihood of surviving to that year.  In this case, for the given parameters, it turns out to be around 18%. For clarity I will often chart it as two cumulative distributions because I think that makes the conversation clearer and also puts it on the same scale. It also provides some intuition to me on why the 4% rule, when it used historical data and a "fixed interval," might or might not have been a lucky fluke. Like this

Figure 13. Cumulative PL prob and conditional survival

This approach also satisfies the Kolmogorov equation more or less precisely. I figured that out by taking a finite differences solution (which is simulation by another name in the way that it discretizes a continuity) to the PDE that Prof. Milevsky sent me. In the face of hundreds of permutations of the parameters as test conditions, both approaches gave the same answers with a little variance from my approach due to the simulation required to get PL. So it satisfies the equation and it satisfied me.

While the PDE makes the relationships involved explicit and transparent, its implementation and use and interpretation are difficult and opaque which is why the simulation approach works better for me. First let's describe the simulated PL approach more formally and then mention some pros and cons.  In it's discrete version we'd describe it like this:

Where tPx is a conditional survival probability for a given age year in subsequent year t (the red line...but this not covered until Process 4), the summation is formally to infinity but practically to t = 120-age since the survival probabilities are effectively zero at that point and beyond, and g(w)(t) is the function that produces the vector of portfolio longevity probabilities out of a drifting and diffusing net wealth process that we did above (the blue bars).  This type of approach to LPR has several advantages.

1. It's eminently implementable. The code to do this is about a page and its fast in processing time. In a tongue-in-cheek moment I called it my FRET tool for "flexible ruin estimation tool" but also because fret is what I catch myself doing when I spend too much time thinking about ruin and not enough time living my life.

2. It clearly shows in the notation the separation of survival risk and portfolio longevity as two separate independent processes.

3. It is intellectually and economically well grounded in the sense that it satisfies a well known PDE that describes the problem in a concise, transparent and rigorous way and that has been part of the literature of risk for at least 90 years.

4. Contra Monte Carlo simulation, where we typically speak of a fail rate at a fixed point in time, here we have both a full constellation of asset exhaustion over an infinite interval as well as a full constellation of longevity probabilities over the same interval rather than an arbitrary "30 years" an interval which may or may not have anything to do with longevity risk for any given retiree. As someone who retired at 50 I can say that the 30 year "convention" was meaningless for me at that time.

5. The results will be very consistent with most honest MC simulations instantiated with reasonable parameters and internal design.  The problem here, as with most simulation, is that the internal mechanism of MC is invisible to both retirees and, often, their advisors and can have hidden assumptions and biases that can nudge the results in ways that are not transparent. 

6. As is the advantage of all simulation, the FRET approach is customizable.  The PDE assumes normal return distributions, a GBM diffusion process, and constant spend. FRET, on the other hand, can take on any distribution shape desired and designable. It can also nudge the other underlying processes involved, too.  This can take us in a black-boxy direction but at least it is there if we need it.

7. The relationships are maybe not as self evident and transparent as they are in the PDE but that equation is pretty obscure anyway. One advantage here is that the mortality process is clearly seen to be independent and invariant when it comes to the key levers of retirement: returns, volatility, and spending.  In simplifying the core net wealth process, from which PL is derived, to (µw-1) the topic of volatility gets a little lost but at least the impact of return is clear and intuitive. And now spending (which seems to disappear in most chats I've had with advisors) emerges as a dominating consideration.  The vol effects can be illustrated in iterative demos and might require some knowledge of what we discussed in Process 1 about return generation in multi-period time.

8. The magnitude of "fail" does not get lost or vitiated since we are illustrating both the full scale of life and of portfolio longevity to eternity. As opposed to a discussion -- and this is a frequent critique of what is missing in MC sim -- of an objective but fake measure of fail magnitude of "x years" here we have a potentially more useful discussion of the full scope of the more subjective probabilities involved.

The disadvantages of FRET are the same as all MC simulation:

• It requires some degree of opening a door into the black box of simulation to understand what is going on.   

• Also, to the extent that the person interacting with this tool views ruin as a prediction of the future rather than merely another dashboard metric like fuel level or speed, then this will be less helpful than it could be.  As a side note: while I think the rate of ruin as a first derivative concept is less useful than commonly thought, I do think that the second derivative, "changes in the rate", probably is a useful indicator to which it makes sense to pay attention. 

• It ignores preferences and risk aversion.  

• Excessive focus on ruin minimization can sometimes lead to sub-optimal outcomes and plan design.

Of course I am not the first to come up with LPR by way of simulation. This and related concepts are common in the literature. It's just that I did not run into before I came to it by way of my personal epiphany which was probably a helpful sequence in terms of my own understanding of the concept.  The concept is quite visible in the much of the work of Prof Milevsky and I believe it is part of his core curriculum at York.  The most explicit concurrence with the approach I have here that I've seen is in the work of Albrecht and Maurer (2001) which I did not see until last year. I'll give them some space here because I think it is usefully confirmatory:

At the centre of interest is the possibility Tx [life remaining] > τR, i.e. the wealth-exhaustion has been realized at a point in time when the observed person is still alive. In particular, the probability POR(R) of this event is of relevance, whereby 

PoR(R) := P(Tx > τR )

[...] If one, for the present purposes, excludes the case R[consumption] > C [initial capital], i.e. τR = 0 , then in total the following results:  

(B3 in appendix B)

The advantage in the representation (B3) consists in the separation of the mortality law on the one hand, and the constellation of asset-exhaustion at a fixed point in time on the other. The mortality law is given in the DAV-mortality table 1994 R. The probabilities P(τR = t) however, must be determined by Monte-Carlo-Simulation. 

tPx here is as we described it above. The P() term to the right is the same as my g(w)(t).  This then, is precisely FRET in both spirit and notation and it confirms my intuition.

So, whether it has advantages or not, is useful or prone to misuse, I do think that FRET, for many of the reasons described in this essay on PL which is essential to its use and understanding, is a felicitous integration of the five core processes I see in retirement: 1. return generation, 2. spending, 3. portfolio longevity as a combination of 1 and 2, 4. human mortality and conditional survival, and 5. continuous management and monitoring processes.  I will continue to keep it in my tool box while being aware of its limitations. And, if for no other reason, it shows that at least a rudimentary understanding of portfolio longevity has its uses.

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Notes

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[a] Milevsky (2000) makes a distinction between the conditional state (portfolio longevity delimited by a planning horizon and/or random lifetime) and unconditional (let it run to infinity). The latter he describes in a footnote as something where he is "perhaps abusing conventions."

[b] I typically opt to model my human capital at any point in my prospective, forthcoming retirement plan (H(t)) at zero. It may not exactly be zero but it is more or less negligible unless I start to make some heroic effort at some point to revivify it; any takers? For those that don't, Horneff (2005), following Cocco, Gomes, and Maenhout (2005), models labor income like this:

Y(t) = exp[f(t)]P(t)U(t)

P(t) = P(t-1)N(t)

where "f(t) is a deterministic function of age to recover the hump shape of income stream. Pt is a permanent component with innovation Nt and Ut is a transitory shock. The logarithms of Nt and Ut are normally distributed with means zero and with volatilities σN , σU , respectively. The shocks are assumed to be uncorrelated."

[c] There are several ways to do this.  If the output is a grid with years (t) going horizontally and sim iterations vertically, the PL can be extracted from (a) (looking vertically) the frequency of the first occurrence of depletion in years, (b) (looking vertically) the empirical cdf of all first events in years and from which the pdf (pmf) can be inferred, or (c) (looking horizontally) from the cumulative fail rate over time from 0:200+ from which the pdf (pmf) can be inferred.  (c) will miss the "succeeds to infinity" required to integrate the defective distributions to sum(P)=1. 

[d] Seven Equations... 2012, Chapter 2, p7. This page and this equation, for what it's worth, was the one that started me down the path to this blog and this essay. A five year circle.

[e] I did try to double check this later with Milevsky's SPV using reciprocal gamma approximation for age 60, median death age of 90 (lets call that similar to a fixed age range of 30 years???), 4% real rate with .12 vol and a spend of 100k. The 20% area under the curve starts for those assumptions around 2.65M which is really really close to the result of the equivalent red scenario above. And it took about a minute. If I understand what he is doing then this method is pretty slick and worth another look.

[f] Cumulative variation of Gompertz equation in this form P = e^(1-exp(t/b)exp((x-m)/b) if I have typed it right. x is age at calc, m is modal life and b is the dispersion. t is the year.

References

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Albrecht, Peter & Maurer, Raimond (2001) Self Annuitizatio, Ruin Risk in Retirement and Asset Allocation: The Annuity Benchmark, Working Paper - actuaries.org
Habib, F., Huang, H., Milevsky, M. (2017) Approximate Solutions to Retirement Spending Problems and the Optimality of Ruin

Estrada, J., Kritzman, M. (2018) Toward Determining the Optimal Investment Strategy for Retirement. IESE Business School and Windham Capital Management.

Habib, Huang, Milevsky (2017), Approximate Solutions to Retirement Spending Problems and the Optimality of Ruin.

Harlow, W., Brown, K. (2014) Market Risk, Mortality Risk, and Sustainable Retirement Asset Allocation: A downside risk perspective. Putnam and U of Tx.

Horneff, W., Maurer, R., Stamos, M. (2005) Life-cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good deal?

Kitces, M. (2012) Is the Retirement Plan with the lowest "risk of failure" really the best choice? Kitces.com

Milevsky, M., Robinson, C. (2000) Is Your Standard of Living Sustainable During Retirement? Ruin Probabilities. Asian Options, and Life Annuities. SOA Retirement Needs Framework.

Huang, H., Milevsky M., and Wang J. (2003), Ruined Moments in Your Life: How Good are the Approximations? York University

Milevsky, M. and Robinson, C. (2005), A Sustainable Spending Rate without Simulation FAJ Vol. 61 No. 6 CFA Institute. 

Milevsky, M and Huang H (2011), Spending Retirement on Planet Vulcan: The Impact of Longevity Risk Aversion on Optimal Withdrawal Rates.

Milevsky, M. (2012) The Seven Most Important Equations for Your Retirement. J Wiley

Robinson, C, Tahani, N. (2007) Sustainable Retirement Income for the Socialite, the Gardener and the Uninsured. York U.

Vertes, Druce (2013) Save withdrawal rates, optimal retirement portfolios, and certainty equivalent spending, SSRN

Yaari, M. (1965), Uncertain Lifetime, Life Insurance, and the Theory of the Consumer

Zwecher, M. (2010), Retirement Portfolios - Theory, Construction, and Management. J Wiley