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
Process 3 - Portfolio Longevity
Summary here.
"The primary determinant of retirement cost is longevity." Dirk Cotton
Process 4 - Human Mortality and Conditional Survival Probability
The first thing you need to know about this post is that I will start sandbagging myself on the title topic before the first sentence is even completely read. Actually sandbagging is not the right word since it implies "deep knowledge + deceit" used to gain some advantage. Let's say, rather that I will formally disclaim any professional facility with the topic of human mortality and conditional survival probability because I am not an actuary or a demographer. I am a retiree and an amateur-wannabe-retiree-quant. Hundreds of years, whole industries, zillions of careers, oceans of digital ink, and no small number of degrees have been dedicated to the study of longevity and actuarial science. I am clearly not one of those people and I am not working in that domain. (email me for any errors or omissions; I will correct and attribute if it makes sense)
Yes, we can safely assume that I have no role in pricing insurance products or researching population dynamics. Nor do I hedge corporate books of life-risk, either against people living too long (annuities) or people dying too soon (life insurance). But then again, "they" (the people that do that kind of thing) are not often seen rough-thumb-nailing boundaries for portfolio longevity in retirement or evaluating discounted utility of lifetime consumption across several strategies. Nor are they often seen trying to scale any one individual's estimate of lifetime ruin risk as it might apply to what might happen to that person's children if that risk were to unfold (apologies if you do this kind of thing. I'm throwing you under the bus to make a point).
So, while longevity science in it's extreme analytic professional form does not interest me much, longevity considerations, in a very general and personal sense, are of interest to me...great interest. This is because I have a lot of self interest in the long term impact of my retirement choices. One of the weirder things in blogging is to quote oneself but I will here because it is relevant to this point. In the last process-post (Process 3 - Portfolio Longevity), I said this:
...while [portfolio longevity] and human mortality are truly independent processes and it is useful to make that separation intellectually, in the end [portfolio longevity] is constrained in our imagination by the limit of our self interest which is itself imposed by how long we are allowed to live and spend. --Me.Using my own words in that way sets me up to say that this post is not intended to, didactically or otherwise, instruct anyone on anything, particularly actuarialness, which is not a word. My only goal here is to report on some few tidbits related to the broader subject of human mortality that I have found useful over the last three or four years in the retirement quant stuff I try to do. These tidbits have been quite useful to the analysis of the retirement processes that are of interest to me and on which I have written in my last several essays. They can be summarized like this:
1. It makes sense to nudge oneself off of the false certitude of point estimates for longevity and into thinking in probabilities and distributions. This is still true even if one, in the end and like me, goes back to using a fixed reference point like "30 years."
2. When thinking in distributions, it makes sense to be aware that there are different populations that can shape the probabilities of interest.
3. Life expectancy, even if we know the population and distribution, is a drifting target over time due to changes in culture and science.
4. Life expectancy, even if we happen to know the population, the distribution, and the drift, is a still a moving target because survival probabilities are conditional on the age to which one has survived.
5. There are some simple math hacks that can be used by the quantitatively inclined to model longevity if the need arises.
6. Generally speaking, it looks like the maximum expectation for human longevity does not change much...but the distribution of probability before that time does.
7. Planning for either an average longevity or a maximum longevity is probably not the wisest move when contemplating planning horizons.
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Tidbit 1 - Going From Point Estimates to Distributions and Probabilities
I've been reading a lot of Ret-Fin content for a long time now. Even many academic papers that I see from otherwise really smart people, for reasons that probably have to do with making their lives easier in doing analysis, will often assume a fixed duration of "30 years" in their financial modeling (as did Bengen, if I recall). This assumption also rolls downhill to advisor-retiree conversations, as well. I've heard it myself many times. This is probably because a common retirement age used to be 65 and "30 years" is a good covering-assumption for a conversation because it plops one right into something vaguely like the 95th percentile for survival for a 65 year old which is nicely conservative and appropriate...for a 65 year old. But I retired when I was 50 which means that 30 years would not even have gotten me to a mean life expectancy for my then-current age. Also, if I had happened to alternatively retire at 75, then "30 years" would have been a pretty fat assumption for that age. In the end, even though I am now 60 and "30 years" may work well enough (and maybe be itself a little fat given what I know about my family medical history), I think it is helpful to jump off the "30 year" ship at this point for context building. This is because longevity is, and always has been, unknown for any particular person. It is random. Random, yes, but if generalized over a large enough population, it will follow a particular shape of randomness -- or at least it does if medical innovation were to stop for a minute and the patterns of the past persist. I think it is helpful to be aware of this shape because while "average" longevity or "to 30 years" (depending on the start age) is maybe too aggressively optimistic (short) sometimes especially for early retirees, and while a maximum age of something like 120 is theoretically plausible and probably too conservative (long) almost always, neither choice is a particularly robust or helpful assumption when it comes to sussing out the nuances of retirement planning risk in the face of an unknowable lifetime. We need a more reasonable understanding of our fluid lifetime boundary.
A good place to start is a period life table[1]. Let's take the Social Security Administration Period Life table for a male[2], for example. Starting with what I'll call unconditional survival i.e., where we look at this "from birth" rather than for someone who has conditionally survived to age 65, the period table will show the number of lives at birth of 100000 with the expected attrition in future years following year one thereafter. We can extract a probability mass for this process that looks like this (I've limited it to a viewing interval of 50-105 but it goes from birth to 119):
This is relatively decent representation of a distribution of probabilities for a lifetime estimate. It's a little better than an arbitrary "30 years." The average here is a little over 76 years, the mode is a little over 87 and the 95th percentile -- a 5% chance of living longer than that -- is around 95+. Which one is OK for use as a planning horizon? Trick question. Probably none of them except maybe by accident because neither you nor I are 1-year-old babies. But we'll get to conditional probability more appropriate for someone in their 60s in a bit.
Speaking of conditional probabilities lets flip this chart around so we can see it another way by starting to look at the survival probabilities. This can be extracted directly from the SSA table or the distribution above can be integrated to a cumulative probability of dying and then inverted by subtracting that from one to get the cumulative (unconditional here) survival probability which looks like this over the full age range:
Speaking of conditional probabilities lets flip this chart around so we can see it another way by starting to look at the survival probabilities. This can be extracted directly from the SSA table or the distribution above can be integrated to a cumulative probability of dying and then inverted by subtracting that from one to get the cumulative (unconditional here) survival probability which looks like this over the full age range:
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| Figure 2. SSA period life table survival probability at birth |
This is done for the interval that starts at birth. If this were done for someone who had survived to another age, say 65, it'd be the conditional survival probability. That CSP is useful for some retirement analysis because it can be used as a weighting or discounting factor for things like annuity estimates, cash flow evaluations, discounted utility, lifetime ruin probability, etc. In past essays when you've seen notation like tPx, that's the conditional survival probability at future time t for someone aged x which would be a point along the blue line. This is useful, but we'll get to that later, too. For now we will simply assert that a distribution of possibilities is probably better than a fixed estimate.
Tidbit 2 - Which Population Should We Look At?
The SSA table is useful but it is averaged over a large "average" population. This may or may not represent the type of risk you might think you need to evaluate based on things like family health history, wealth level, planning conservatism, etc. There seems to be some evidence, un-cited here, that affluent cohorts tend to live longer and may have different health experiences. That cohort, in addition to those that think they are going to live a long time, is more likely to buy annuities for those reasons. This kind of selection bias means that for an insurer that sells annuities to stay in business, they had better make sure they are evaluating longevity for the right population. And, in fact, if we look at the Society of Actuaries (SOA) mortality table[3] for individual annuitants (IAM) and compare it to the SSA data we start to see some differences between the average population (SSA) and the self-selected population (SOA IAM). If we chart out the SOA data over the same interval (from birth) and compare SSA and SOA in an overlay, it looks like this:
We can see that the SOA data models a longer-living population where the average expectancy (from birth) is now ~83, the mode is 90 and the 95th percentile is ~98+. Which data do we use? Well, we are still looking from the perspective of a 1-year-old so it's not a fair question. But in general for my personal planning and for quantitative modeling -- ignoring my family history and personal bad habits for now -- I will use the (conditional) SOA data because it is more conservative. If or when I have partial heart failure, that assumption may change.
Tidbit 3 - Longevity Drifts Over Time
An important thing to understand about Figure 3 is that the distribution is not static. It moves over time with changes in science and culture. Think: medical and public health advances (drift up) or maybe the opioid crisis (drift down). Or maybe think about periods of war and peace. A generally upward drift has been going on for a long time and it can make past planning or "fixed quant assumptions" obsolete in subtle ways pretty quickly. Here is one way to see it as illustrated by Nathan Yau using World Bank data from 1960+:
North and Latin America are highlighted in red and blue respectively. It is easy to see that there has been a long upward drift in life expectancy around the world. It's less easy to see that there has also been some retrenchment in the US data at the far upper right. For a little longer look-back, using selected UK data (and now it is in a survival percentage orientation), here is a chart from OurWorldinData.org:
The implications of this phenomenon are many. First, it means that one should assume a drift when modeling longevity and when one is using published tables and doing quantitative analysis. One might also assume that it is a stable drift. Current US data coming in lately might affect that last assumption but maybe we can at least assume a stable, perhaps slightly diminishing trend over time. Fortunately for me in my own planning, I don't need to be very precise with this. As a current shareholder of some insurers I have a different opinion of their need for precision and I certainly home they pay attention which I'm pretty sure they do.
The SSA tables are published relatively often with the current being dated 2015 so my guess is that they pick up the drift in the process of periodic updating. The SOA IAM table I use, on the other hand, is from 2012 so it doesn't pick up any drift since then. But the SOA also provides tables (2583 and 2584) and a methodology [4] for taking 2012 table data and projecting the drift into the future. Using the probability of death format we can see the drift in Figure 6 by applying the projection scale from 2012 to 2019 and then comparing that with the original table, like this:
The projection makes the table a little older in tiny increments.
Tidbit 4 - Life expectancy, even if we know the right population, the distribution, and the overall drift, is a still a moving target because survival probabilities are conditional on the age to which one has survived.
This topic of conditional probability is obvious and well known...to some people. It just that it took me personally a while to figure it out. This section is a "tidbit" because this particular insight has been uniquely useful to me in modeling financial processes that deal with longevity. The concept of "moving longevity targets influenced by age attained" can be inferred directly from the tables, whichever one you want to use. The main takeaway is that (I'm using the SOA data here) if the average longevity for a male at birth is 83, it's not 83 if you survive to age 60. It's closer to 86 at that point. And higher still at an attained age of 80. Humans don't live past 120 so there is a limit to how much the expectation can move but the movement is there. as an example, here is the conditional survival probability distributions for "at birth," at 60, and at 80 using the SOA data:
This kind of thing is important because, as I mentioned in Tidbit-1, the conditional survival probability vector tPx Ɐ(t) [the green or blue or yellow lines above] is defined as the probability at future times "t" that a person, aged x at the time when the analysis is done, will still be alive. This "tidbit" alone was worth the price of entry for me in terms of me spending my time to understand how human mortality works mathematically. It was worth it because it allows me to impose from outside a well-reasoned and probabilistic "kill time" on what would otherwise be independent, infinite (at least from a modeling perspective) processes such as portfolio longevity (whence lifetime probability of ruin, or LPR, back in Process 3), lifetime income (this is how annuities can be priced), and spending programs (it's one of two ways to weight or discount lifetime utility of consumption). I say well-reasoned here because we are not using assumptions like "30 years" or to age 95. We are, rather, taking into account "the full term structure" of mortality probability which is a paraphrase of a language I borrowed from Milevsky (2011).
One final view of this concept is to take the conditional survival probability vector for each age survived-to and, by summation, collapse the vector to get the mean longevity expectation. Then when we chain those means together we can see the movement in mean of expected longevity as each subsequent age is attained. This is why planning is hard:
Tables of mortality data are useful and purport represent some real statistics of how people live and die. As long as nothing radical happens in human longevity science or there are no specific, known issues with any one individual's personal health then these tables are worthy sources of insight. For those of us that automate things and run models and simulate processes, however: sometimes the tables are cumbersome. Either it is a hassle to load the tables and structure the data sets for processing or the table data does not shape itself well to something we are trying to test or hypothesize. In those cases it is sometimes easier to use a mathematical model to get closer to what you are trying to do or to simplify one's programming life. For that purpose, the most common equation that I've seen in the literature -- and as a non-actuary non-demographer I have to assume there is a vast literature on this kind of thing beyond my paper-thin amateur view -- is a Gompertz Equation. I first ran into it in Milevsky (2012). In his book it looks like this below. I have also run into over and over again in this and related forms in the ret-fin literature (see note [5] for some other examples):
So, pretty close. The neat thing about this is not just that we have a simple model that more or less can mimic some given actuarial table it's also that we now can play with it. If we wanted to hypothesize a jump in longevity science we could move the mode out. If we wanted to model lower expectations due to health issues we could move the mode in. If we want more or less uncertainty or dispersion, we change b.
The other way to engage with the imperative for "simplification and automation" might be to use a quick and dirty exponential random lifetime approach where we would assume, incorrectly, that human life conforms to a constant force of mortality. I don't use this simplifier (yet) but I've seen it in the lit. In Milevsky (2005), for example, he frames it like this where T is remaining lifetime, t is the number of years in the future from the current age, and lambda is force of mortality in discrete lifetime given the "conditional" age achieved:
The expected value of the (exponential) terminal lifetime in this world would be 1/λ. That means that if one were to be 60 years old and the mortality rate were to be ~.024, then the remaining lifetime is ~42 years so terminal age is ~102. The median value is = ln(2)/λ which also means that λ can be inferred from the actuarial tables by extracting the median age, converting it to years and dividing it into ln(2). The purpose here is not to be deeply expository, it is merely to have a couple "close enough" tools in the tool belt, relative to the actuarial tables, for approximating longevity and survival probabilities if we need them. Here is Milevsky (2005):
Tidbit 6 - Generally speaking, it looks like the maximum expectation for human longevity does not change much...but the distribution of probability before that time does.
The oldest person ever was Jeanne Calment (1875–1997) of France, who supposedly died at the age of 122 years. I say supposedly because I just read something that suggested that she had assumed her sister's identity and wasn't really 122 when she died. The oldest man was ~116. The point here is that this kind of age range (116 to 122) seems to generally be used as a type of un-changing outer bound for longevity. And 120 or 122, the far end of the outer bound, may be over-generous given the questions on Jeanne's identity. Barring radical advances in medicine, what this has meant, and is likely to continue to mean, is that while average longevity advances a bit each year, terminal longevity doesn't. And for us? I think it means that we can generally stop worrying about retirement planning horizons in the 105 to 120 year old range, a range that I just made up but seems reasonable for "extreme" since the likelihoods are so low and unlikely to change. Note that the concept of a "pegged outer-bound with moving average expectancy over time" can be seen back in Figure 5. Alternatively, if we were to model it using our math-hack and then rendered it in probability of death terms while shifting the mode out an arbitrary three years and shrinking the dispersion by an arbitrary -.50 (those shifts are designed to reflect some hypothesis about changes in longevity statistics), it might look like this in grey:
Tidbit 2 - Which Population Should We Look At?
The SSA table is useful but it is averaged over a large "average" population. This may or may not represent the type of risk you might think you need to evaluate based on things like family health history, wealth level, planning conservatism, etc. There seems to be some evidence, un-cited here, that affluent cohorts tend to live longer and may have different health experiences. That cohort, in addition to those that think they are going to live a long time, is more likely to buy annuities for those reasons. This kind of selection bias means that for an insurer that sells annuities to stay in business, they had better make sure they are evaluating longevity for the right population. And, in fact, if we look at the Society of Actuaries (SOA) mortality table[3] for individual annuitants (IAM) and compare it to the SSA data we start to see some differences between the average population (SSA) and the self-selected population (SOA IAM). If we chart out the SOA data over the same interval (from birth) and compare SSA and SOA in an overlay, it looks like this:
 |
| Figure 3. Different populations implied in SSA and SOA data |
We can see that the SOA data models a longer-living population where the average expectancy (from birth) is now ~83, the mode is 90 and the 95th percentile is ~98+. Which data do we use? Well, we are still looking from the perspective of a 1-year-old so it's not a fair question. But in general for my personal planning and for quantitative modeling -- ignoring my family history and personal bad habits for now -- I will use the (conditional) SOA data because it is more conservative. If or when I have partial heart failure, that assumption may change.
Tidbit 3 - Longevity Drifts Over Time
An important thing to understand about Figure 3 is that the distribution is not static. It moves over time with changes in science and culture. Think: medical and public health advances (drift up) or maybe the opioid crisis (drift down). Or maybe think about periods of war and peace. A generally upward drift has been going on for a long time and it can make past planning or "fixed quant assumptions" obsolete in subtle ways pretty quickly. Here is one way to see it as illustrated by Nathan Yau using World Bank data from 1960+:
 |
| Figure 4. Life Expectancy Drift |
 |
| Figure 5. Live Expectancy Drift England and Wales from 1851 |
The implications of this phenomenon are many. First, it means that one should assume a drift when modeling longevity and when one is using published tables and doing quantitative analysis. One might also assume that it is a stable drift. Current US data coming in lately might affect that last assumption but maybe we can at least assume a stable, perhaps slightly diminishing trend over time. Fortunately for me in my own planning, I don't need to be very precise with this. As a current shareholder of some insurers I have a different opinion of their need for precision and I certainly home they pay attention which I'm pretty sure they do.
The SSA tables are published relatively often with the current being dated 2015 so my guess is that they pick up the drift in the process of periodic updating. The SOA IAM table I use, on the other hand, is from 2012 so it doesn't pick up any drift since then. But the SOA also provides tables (2583 and 2584) and a methodology [4] for taking 2012 table data and projecting the drift into the future. Using the probability of death format we can see the drift in Figure 6 by applying the projection scale from 2012 to 2019 and then comparing that with the original table, like this:
 |
| Figure 6. Using a projection scale to counter drifting longevity |
The projection makes the table a little older in tiny increments.
Tidbit 4 - Life expectancy, even if we know the right population, the distribution, and the overall drift, is a still a moving target because survival probabilities are conditional on the age to which one has survived.
This topic of conditional probability is obvious and well known...to some people. It just that it took me personally a while to figure it out. This section is a "tidbit" because this particular insight has been uniquely useful to me in modeling financial processes that deal with longevity. The concept of "moving longevity targets influenced by age attained" can be inferred directly from the tables, whichever one you want to use. The main takeaway is that (I'm using the SOA data here) if the average longevity for a male at birth is 83, it's not 83 if you survive to age 60. It's closer to 86 at that point. And higher still at an attained age of 80. Humans don't live past 120 so there is a limit to how much the expectation can move but the movement is there. as an example, here is the conditional survival probability distributions for "at birth," at 60, and at 80 using the SOA data:
 |
| Figure 7. conditional survival at birth, 60, and 80. |
One final view of this concept is to take the conditional survival probability vector for each age survived-to and, by summation, collapse the vector to get the mean longevity expectation. Then when we chain those means together we can see the movement in mean of expected longevity as each subsequent age is attained. This is why planning is hard:
 |
| Figure 8. mean conditional longevity for different survival ages |
Tidbit 5 - There are simple math hacks that can be used to model longevity if the need arises.
Tables of mortality data are useful and purport represent some real statistics of how people live and die. As long as nothing radical happens in human longevity science or there are no specific, known issues with any one individual's personal health then these tables are worthy sources of insight. For those of us that automate things and run models and simulate processes, however: sometimes the tables are cumbersome. Either it is a hassle to load the tables and structure the data sets for processing or the table data does not shape itself well to something we are trying to test or hypothesize. In those cases it is sometimes easier to use a mathematical model to get closer to what you are trying to do or to simplify one's programming life. For that purpose, the most common equation that I've seen in the literature -- and as a non-actuary non-demographer I have to assume there is a vast literature on this kind of thing beyond my paper-thin amateur view -- is a Gompertz Equation. I first ran into it in Milevsky (2012). In his book it looks like this below. I have also run into over and over again in this and related forms in the ret-fin literature (see note [5] for some other examples):
 |
| Eq 1. Gompertz approximation to survival probabilities |
where p is the conditional survival probability, x is the age attained, t is the future year, m is the mode of the expected distribution, and b is a measure of dispersion around the mode. If we recast it as a PDF of mortality, then the PDF curves (SOA and Gompertz conditioned on age 60; mode = 90, b=8.5) look like this:
 |
| Figure 9. Gompertz approximation fit to SOA IAM table for 60yo male |
So, pretty close. The neat thing about this is not just that we have a simple model that more or less can mimic some given actuarial table it's also that we now can play with it. If we wanted to hypothesize a jump in longevity science we could move the mode out. If we wanted to model lower expectations due to health issues we could move the mode in. If we want more or less uncertainty or dispersion, we change b.
The other way to engage with the imperative for "simplification and automation" might be to use a quick and dirty exponential random lifetime approach where we would assume, incorrectly, that human life conforms to a constant force of mortality. I don't use this simplifier (yet) but I've seen it in the lit. In Milevsky (2005), for example, he frames it like this where T is remaining lifetime, t is the number of years in the future from the current age, and lambda is force of mortality in discrete lifetime given the "conditional" age achieved:
prob(T>t) = e^-λt (eq. 2)
The expected value of the (exponential) terminal lifetime in this world would be 1/λ. That means that if one were to be 60 years old and the mortality rate were to be ~.024, then the remaining lifetime is ~42 years so terminal age is ~102. The median value is = ln(2)/λ which also means that λ can be inferred from the actuarial tables by extracting the median age, converting it to years and dividing it into ln(2). The purpose here is not to be deeply expository, it is merely to have a couple "close enough" tools in the tool belt, relative to the actuarial tables, for approximating longevity and survival probabilities if we need them. Here is Milevsky (2005):
"Although human aging does not conform to an exponential or constant force of mortality assumption...for the purposes of estimating a sustainable spending rate [in this case he's using it in the context of his simulation-free reciprocal gamma approach to spend rates], it does a remarkably good job when properly calibrated."
Tidbit 6 - Generally speaking, it looks like the maximum expectation for human longevity does not change much...but the distribution of probability before that time does.
The oldest person ever was Jeanne Calment (1875–1997) of France, who supposedly died at the age of 122 years. I say supposedly because I just read something that suggested that she had assumed her sister's identity and wasn't really 122 when she died. The oldest man was ~116. The point here is that this kind of age range (116 to 122) seems to generally be used as a type of un-changing outer bound for longevity. And 120 or 122, the far end of the outer bound, may be over-generous given the questions on Jeanne's identity. Barring radical advances in medicine, what this has meant, and is likely to continue to mean, is that while average longevity advances a bit each year, terminal longevity doesn't. And for us? I think it means that we can generally stop worrying about retirement planning horizons in the 105 to 120 year old range, a range that I just made up but seems reasonable for "extreme" since the likelihoods are so low and unlikely to change. Note that the concept of a "pegged outer-bound with moving average expectancy over time" can be seen back in Figure 5. Alternatively, if we were to model it using our math-hack and then rendered it in probability of death terms while shifting the mode out an arbitrary three years and shrinking the dispersion by an arbitrary -.50 (those shifts are designed to reflect some hypothesis about changes in longevity statistics), it might look like this in grey:
 |
| Figure 10. Playing with a hypothetical change in the longevity mode |
This happens to look suspiciously like Figure 6 where the SOA had baked a projection assumption about drifting longevity into its IAM table. The resemblance here in Figure 10 is by design and it sorta resembles a pegged outer bound...but only for small changes in m. Larger changes in m get weird but are not un-useful in terms of modeling more radical departures from the current status quo. For example, using our math-hack, as m ® ¥ the conditional survival probabilities would ® 1, i.e., it would be the endowment model which might be useful if you are immortal. Back in the real world, we generally assume human mortality is a stable predictable process because it has been watched and described for a long time by smarter people than me. But it probably isn't really stable and predictable or it won't be forever. At least we have some tools to play around with to see what it means if that were to happen.
I came into this essay asserting that it is common to hear "30 years" as a planning horizon. That is anecdote because I have no survey to prove it. But I read a lot. And I have been in conference rooms with a lot of advisors over the years. "30 years" comes up a lot. One of my main points here has been that using 30 years as a boundary for planning is flawed and misconceives longevity because it does not treat it like a random, independent process or something that can be described by a probability distribution. You are probably now thinking I am going to offer some complex solution to this "30 year problem" or maybe recommend using to-age 120 or 105 just to be hyper-conservative.
Nope. Actually I like "30 years" for me. I like it now because, while it was mind-numbingly wrong for me at age 50 and will be a joke again at 75 or 80, it is not too bad right now at 60. Plus it's easy. In fact, I have to also say here that this whole topic has forced me to take a very, very close look at what assumptions I do or should use for myself, which was a "close look" I wasn't expecting when I started this post. Over the last 10 years I have been almost absurdly conservative out of an abundance of caution as an early retiree, a hazy understanding of how retirement processes work, and an early retirement crouch-of-fear that came from a move to a new crazy state, a divorce, and the global financial crisis. Now at 60, however -- when I carefully consider that the men in my family often have heart attacks in their late 40s and 50s (I'm 60) and the men and women that survive the heart problems seem to lose their minds in their mid 80s -- I have to be pretty clear with myself that 30 years might actually be a little too conservative now.
Planning for a terminal age 76 (that's the average risk based on the SSA table, but from birth) is clearly a little dumb because I am not a baby. Better to look at the conditional probabilities for a 60 year old where the average is closer to mid-80s. Or maybe better to look at the 95th percentile at around age 95 and after which the probabilities get a little silly to consider and before which one might plausibly still have a reasonable life. On the other hand, and if self-annuitizing, planning for age 120 would clearly be a disservice to me and my family in terms of me funding (spending on) important life-building "experiences" while I still can (and should) between now and my mid-80s, or at the latest 95. Anyway, much of that really late-age superannuation risk -- from, say, age 85 or 95, the probabilities of which it is very important to at least be aware if not exactly to act on -- can either be hedged out with life income, deferred or otherwise, or maybe conceived of as some type of minimal threshold existence that is unlikely to occur...or both. Me? I'll do both.[6] That means that for me, planning for something like modal (or maybe 90-95th percentile or maybe even less...) terminal longevity is probably not totally moronic at this point. That's somewhere around age 90 (to as much as the late 90s) for me. That'd put me right on the cusp of a fixed "30 year" horizon which is why that bad assumption no longer offends me as much as it used to. As long as I know that I am dealing with a probability rather than a "point-estimate certainty" and that things might all radically change next year (if I'm still breathing...and monitoring) no matter what I assume, I should be OK.
Notes
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[1] There are differences between period and cohort tables that I won't broach here. It's probably worth knowing so use google and "difference between period and cohort tables" or go to the main SSA page for life tables where there is a good definition and differentiation. For us, since we are not doing a moon shot, we can probably go with any table for the most part and probably not worry about that difference or, for that matter, even worry about gender differences. The mere fact of going from an average or a static number to a probability distribution takes us pretty far down the road we need to go...which may be enough.
[2] no, not male because of any toxic masculinity or because of cisheteropatriarchy considerations...it's just because I am retired and solo and have a fiduciary and loving responsibility to three children. I go with male tables for now. Fwiw, women live longer on average...for now.
[3] SOA IAM 2012 basic with G2 projection scale to 2019.
[4] The mortality rate for a person age x in year (2012 + n) is calculated as follows: qx2012+n = qx2012(1 – G2x)n The resulting qx2012+n shall be rounded to three decimal places per 1,000, e.g., 0.741 deaths per 1,000. Also, the rounding shall occur according to the formula above, starting at the 2012 period table rate.
[5] Here are some alternative mortality-related formulations. These all evaluate to the same thing or at least when they are instantiated in a spreadsheet they shape the same PDF curve. I have not done the algebra. Note that most of the time that I see this kind of thing, the authors tend to refer back to some publication of M Milevsky. For example Babbel and Merrill (2006) have a formula I hadn't seen but it was referenced to a 2001 paper by Prof M that I had not read.
Common assumptions seen:
m = mode of the future lifetime
b = a scale parameter of the future lifetime random variable
x = current age
t = time, future year
λ = force due to accident, often = 0
λ(x+t) = force of mortality or hazard function for age
From Huang, Milevsky and Wang, 2003:
Hazard Rate function:
Conditional Survival function:
From Babbel and Merrill (2006) who attribute to Milevsky (2001) which I don't have:
Hazard Rate:
Conditional Survival:
[6] I found an echo of this sentiment in Habib (2017): "Stated bluntly, if there is only a 5% chance of reaching the age of 100, it is quite rational to (i.) assume that you won’t and (ii.) reduce your consumption to the minimal pension level, if you do."
References
------------------------------------------------------------
2015 Social Security Administration - Life Tables for the United States Social Security Area 1900-2100. ACTUARIAL STUDY NO. 120 https://www.ssa.gov/oact/NOTES/as120/LifeTables_Body.html
2012 Society of Actuaries - Individual Annuity Reserving Report & Table https://www.soa.org/experience-studies/2011/2012-ind-annuity-reserving-rpt/
Nope. Actually I like "30 years" for me. I like it now because, while it was mind-numbingly wrong for me at age 50 and will be a joke again at 75 or 80, it is not too bad right now at 60. Plus it's easy. In fact, I have to also say here that this whole topic has forced me to take a very, very close look at what assumptions I do or should use for myself, which was a "close look" I wasn't expecting when I started this post. Over the last 10 years I have been almost absurdly conservative out of an abundance of caution as an early retiree, a hazy understanding of how retirement processes work, and an early retirement crouch-of-fear that came from a move to a new crazy state, a divorce, and the global financial crisis. Now at 60, however -- when I carefully consider that the men in my family often have heart attacks in their late 40s and 50s (I'm 60) and the men and women that survive the heart problems seem to lose their minds in their mid 80s -- I have to be pretty clear with myself that 30 years might actually be a little too conservative now.
Planning for a terminal age 76 (that's the average risk based on the SSA table, but from birth) is clearly a little dumb because I am not a baby. Better to look at the conditional probabilities for a 60 year old where the average is closer to mid-80s. Or maybe better to look at the 95th percentile at around age 95 and after which the probabilities get a little silly to consider and before which one might plausibly still have a reasonable life. On the other hand, and if self-annuitizing, planning for age 120 would clearly be a disservice to me and my family in terms of me funding (spending on) important life-building "experiences" while I still can (and should) between now and my mid-80s, or at the latest 95. Anyway, much of that really late-age superannuation risk -- from, say, age 85 or 95, the probabilities of which it is very important to at least be aware if not exactly to act on -- can either be hedged out with life income, deferred or otherwise, or maybe conceived of as some type of minimal threshold existence that is unlikely to occur...or both. Me? I'll do both.[6] That means that for me, planning for something like modal (or maybe 90-95th percentile or maybe even less...) terminal longevity is probably not totally moronic at this point. That's somewhere around age 90 (to as much as the late 90s) for me. That'd put me right on the cusp of a fixed "30 year" horizon which is why that bad assumption no longer offends me as much as it used to. As long as I know that I am dealing with a probability rather than a "point-estimate certainty" and that things might all radically change next year (if I'm still breathing...and monitoring) no matter what I assume, I should be OK.
Notes
-----------------------------------------------------------
[1] There are differences between period and cohort tables that I won't broach here. It's probably worth knowing so use google and "difference between period and cohort tables" or go to the main SSA page for life tables where there is a good definition and differentiation. For us, since we are not doing a moon shot, we can probably go with any table for the most part and probably not worry about that difference or, for that matter, even worry about gender differences. The mere fact of going from an average or a static number to a probability distribution takes us pretty far down the road we need to go...which may be enough.
[2] no, not male because of any toxic masculinity or because of cisheteropatriarchy considerations...it's just because I am retired and solo and have a fiduciary and loving responsibility to three children. I go with male tables for now. Fwiw, women live longer on average...for now.
[3] SOA IAM 2012 basic with G2 projection scale to 2019.
[4] The mortality rate for a person age x in year (2012 + n) is calculated as follows: qx2012+n = qx2012(1 – G2x)n The resulting qx2012+n shall be rounded to three decimal places per 1,000, e.g., 0.741 deaths per 1,000. Also, the rounding shall occur according to the formula above, starting at the 2012 period table rate.
[5] Here are some alternative mortality-related formulations. These all evaluate to the same thing or at least when they are instantiated in a spreadsheet they shape the same PDF curve. I have not done the algebra. Note that most of the time that I see this kind of thing, the authors tend to refer back to some publication of M Milevsky. For example Babbel and Merrill (2006) have a formula I hadn't seen but it was referenced to a 2001 paper by Prof M that I had not read.
Common assumptions seen:
m = mode of the future lifetime
b = a scale parameter of the future lifetime random variable
x = current age
t = time, future year
λ = force due to accident, often = 0
λ(x+t) = force of mortality or hazard function for age
From Huang, Milevsky and Wang, 2003:
Hazard Rate function:
From Babbel and Merrill (2006) who attribute to Milevsky (2001) which I don't have:
Hazard Rate:
[6] I found an echo of this sentiment in Habib (2017): "Stated bluntly, if there is only a 5% chance of reaching the age of 100, it is quite rational to (i.) assume that you won’t and (ii.) reduce your consumption to the minimal pension level, if you do."
References
------------------------------------------------------------
2015 Social Security Administration - Life Tables for the United States Social Security Area 1900-2100. ACTUARIAL STUDY NO. 120 https://www.ssa.gov/oact/NOTES/as120/LifeTables_Body.html
2012 Society of Actuaries - Individual Annuity Reserving Report & Table https://www.soa.org/experience-studies/2011/2012-ind-annuity-reserving-rpt/
Albrecht, Peter & Maurer, Raimond (2001) Self Annuitization, Ruin Risk in Retirement and Asset Allocation: The Annuity Benchmark, Working Paper - actuaries.org
Babbel, D., Merrill, C. (2006) Rational Decumulation, Wharton School
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?
Huang, H., Milevsky M., and Wang J. (2003), Ruined Moments in Your Life: How Good are the Approximations? York University
Jones, D. R., (2013) Mathematical Analysis of Mortality Dynamics.
Huang, H., Milevsky M., and Wang J. (2003), Ruined Moments in Your Life: How Good are the Approximations? York University
Jones, D. R., (2013) Mathematical Analysis of Mortality Dynamics.
Milevsky, M., Robinson, C. (2000) Is Your Standard of Living Sustainable During Retirement? Ruin Probabilities. Asian Options, and Life Annuities. SOA Retirement Needs Framework.
Milevsky, M. (2001) Optimal Annuitization Policies: analysis of the options. North American Actuarial Journal Vol 5:1 57-69.
Milevsky, M. and Robinson, C. (2005), A Sustainable Spending Rate without Simulation FAJ Vol. 61 No. 6 CFA Institute.
Milevsky, M. (2001) Optimal Annuitization Policies: analysis of the options. North American Actuarial Journal Vol 5:1 57-69.
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
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