More Than I (Or You) Ever Wanted To Know About Spending Variance Vs. Fail Rates.

I got a really interesting result the other day when I was shaking out some of the newer spending variance features[1][2] of the simulator I recently rebuilt. But I had forgotten, for a moment, my two primary rules of financial modeling:

A. Models don't reflect some part of the infinite complexity of reality back at you, rather they reflect back a pretty thin image of something not-real that is nothing other than an artifact of one instance of one software design, a design that comes from (usually) one person's assumptions about a very, very simplified version of the world.

B. If model results seem too interesting it probably means something like: the software was broken or put together the wrong way or not tested or the underlying thing being modeled was misunderstood or misrepresented or maybe it was an accident of things combining where insufficient complexity had been designed in or the math was flawed by some degree of innumeracy or maybe something else altogether.  What it usually is not is a serendipitous discovery of something new and cool. Rarely it might be, but usually not.   

So, let's walk it through to see where I went off the trail and then maybe get back on again.

1. In My First Attempt I forgot About Auto-regression Effects[4]

I had put into the simulator a variable for both normal and skewed distributions of spending (variability other than inflation or spending policies, that is).  I did this because my own monthly spending has varied by as much as +/-50% or more over the last seven years (less now) and maybe +/- 20% annually (less now).  Constant spending in a model, even when it's inflation adjusted or bent by various "rules" doesn’t seem very realistic to me.  So I let it vary. The result I saw when I ran it was a very dramatic sensitivity of fail rates to spending variability, especially when I was using a custom, skewed, and wide distribution that roughly matches my own pattern.  I thought that was cool so I wrote it up for a post like it was something new that I hadn't seen in the lit before.  But then I realized I had been incautious and I went back to see what my software was really doing.  What it was doing was an autoregressive (AR1) process where spending this period depends on spending/variance of the prior.  That's fine and I had done that intentionally but is that how I really spend? There are maybe at least two ways to model it: 1) something dependent on prior period, or 2) a little independent variability in only the current period, like a little "in-bounds" spending shock.  Which is more like reality?  That's a trick question because it's neither. In real life spending has some randomness to it of, course, but much of the variance (in my opinion) also comes from the tug and pull of lifestyle choices, desires, and expectations which means that lifestyle is a pretty protean concept especially over time and especially when it interacts with the boundaries of financial capacity and one's sense of both long term and impending near-term risk. It can also, I think, trend.   No way to model all of that, right?  So let's at least start with process2 (independent) because process1 sends spending into autoregressive cascades that no rational person would allow to happen.  It might look like this in simplified form.  A bunch of possible paths for spending 100k auto-regressively are on the left, simple variance of a spending process is on the right. The process on the left is where I was, the one on the right is where I am going...plus a few other ideas:

2. I Also Forgot A Little Common Sense

The large fail-rate sensitivity-to-variance also failed, in retrospect, my common sense test.  While I had expected, and probably should expect, some impact on fail rates, the effect probably shouldn't have been that big.  First, spending variance is no more than a periodic re-occurrence of a too-high spend rate (and too-low for that matter).  But that, in theory, should be more or less counterbalanced by the savings from the underspends (if they are not, in fact, spent).  Second, spending variance is a logical corollary to the idea of sequence-of-returns risk, something we already know (or should know) pretty well. By that I mean that if a constant (inflation-adjusted) spending plan happens to be subjected to a bad sequence of returns and that "interaction" between spending and returns results in increased retirement fail rate risks then it has to be that a constant series of returns subjected to a bad pattern of spending variance will (or must) have a similar result. So there should be some impact but one that has a sequence-of-returns-like effect, not a nuclear bomb-like effect.  

3. But Since There Is Likely Some Effect, We Should Check It Out

In order to not let points 1 and 2 make us too complacent, let's think about it this way: what if we happened to have a bad sequence of returns and a bad series of unfortunate smaller spending events (let's call it sequence-of-spending risk) all mixed up together at the same time.  Now that would be un-fun (and, for that matter, it would look not too dissimilar to the first couple years of my own early retirement).  Add in some "out of bounds" big spending shocks and it would get really interesting.  So there should be some real impact here that can be modeled, which is the goal of the rest of the post. 

4. Let's See What The Redesigned Spending Variance Looks Like v Fail Rates

I'll skip over the results of autoregressive-style spending for the reasons outlined above.  With the "independent process 2" spending that I switched to I did nothing more than add to a given sim-year a little spend randomness like this: spending this year = f(spending, inflation, spending*var) where var is a random variable created using, in R, rsnorm(mean=0,sd=x, skew=y) so think +/- 10% or 20% or something like that.  In this little test, I varied the standard deviation from 0 to .05, .10, and .20 (or approximately +/- 0%, 20%, 40%, 60-70%). Skew was kept normal.  Fail rates[3], respectively, looked like the below (using fail rates for the longest expected terminal age).

stdev        fail rate
sd=.00     20.10% black
sd=.05     20.45% purple
sd=.10     20.13% blue
sd=.20     19.57%  red

or in chart form:

So, no really big differences here.  The widest variance spend path is even the lowest risk at age 105 but we'll maybe chalk that up to some random differences that would change if we did it again.  But we are not off the hook yet.  In the chart above I tried really, really hard to be unscientific by trying to force the wider variance to be the higher risk by running it over and over until I got the result I wanted (how's that for good science, ha!). I just couldn't do it.   Which tells me something.  On the other hand when I did it with my own personal data and assumptions (maybe my retirement data is a little longer and a little lower risk vis-a-vis spending rate percents) it was completely the opposite.  I literally could not get it not to get a little riskier with wider the distributions of spending.  It looked like this below.  I stripped out fail rates because I figured out a way you can hack/back into net worth from fail rates and other data.  So I'm not doing that.

In at least one case (mine) there might be an impact of spending variability on fail rates...maybe...but it is a relatively small effect in both of the cases either way. Move along, not much to see here.[5]   

5. But What About the Skew

One generally expects spending, in a "normal" sequence of time anyway and ignoring inflation, to mostly stay roughly the same from one period to the next.  But we know it varies and we tested a little of that. But even if we expect it to stay about the same and vary a little bit, it won't really vary to the same degree up and down.  Intuition and personal experience tells me that it is much harder to cut spending than to let it fly.  So there should be some skew. And, in fact, when I look at my own monthly data, there really is a skew.  I don't have enough years of data to see it in the annual data but it's probably there, too. Here is a view of the mo/mo variance of my own data:

Skew, yes, but I probably need to be careful  here, though, because I know the big underspends come from the months after big accidental over-spends.  So, there are some "dependence" kind of things going on in here but this is all I've got for now and it does skew. Let's see what happens when we isolate skew in the model.    

In the model I gave up on my custom-skewed distribution for the moment and let R do it for me.  In this case I used rsnorm(mean=0,sd=x, skew=y) again and kept the mean at zero, the sd at .05 and let the skew variable change from  1 to 1.5 to 2.0.  I forgot to keep the actual skewness numbers but 10k random numbers plugged into rsnorm() with skew set to 2 comes up with a skewness of .78 if I installed the right packages.  Here are the fail rates for the longest longevity expectations for different skew factors:

Skew

Param      sd        Fail Rates
n/a           0         20.18% black

1             .05       20.12    purple

1.5          .05       20.17    blue

2             .05       20.08    red

Again we see almost no impact at all.  But again, note that we are not off the hook.  While in the chart above skew explains practically nothing, when I ran it on my own data, it did (longer, lower risk retirement I guess; I don't know what drives the difference yet).  Here below is me with the lowest line (black) being no skew and no variance and the highest one (red) being the highest skew:

While it looks like there is no effect in the first chart, there probably is in the second.  So maybe that means that there may, in fact, be cohorts for whom skew will matter.  Not matter much, maybe, but some.  The differences in the end-states for the second chart are trivial (maybe 30-40 bps), btw, but persistent.  Just for fun, keep in mind that the mean expectation is still "no change" in spending (other than inflation) so I find the skew effect interesting.  

6. What If Spending Trends In Ways Other Than Inflation?

Above and beyond inflation effects, which are commonly modeled by everyone, what if spending trends up or down?  In real life we see this all the time.  Lifestyle creeps up one small changed expectation at a time.  We know that from personal experience. We try to warn our kids about it.  It also creeps down.  Blanchett and others have shown that spending can often decline an avg 1-2% in retirement, independent of health cost spending.  So spending trends probably do occur over long enough cycles (I have early retirees in mind here in particular).  Think of it as a type of lifestyle inflation or deflation.  So, in addition to adding variance to a constant spend, adding a trend might make it look like this (add in inflation and it'd be even more fun):

I wasn't sure how to model trends in a sim so I did it like this.  I multiplied variable spending in any given year by (inflation and...) a "trend inflator" (t) compounded to that year: spending = f(spending, inflation, spendvar, trend=(1+t)^n). Who knows if this is realistic or meaningful (or right)?  A "t" of zero means no trend, .01 is 1% etc.  I didn't know what values were realistic and even if I did, if we happened to be looking at my own personal data, I already have two spending deflections in my model that work out to be about a 1.5-2% deflator over the long haul (that means a trend on a trend; not very nice).  In this case I just threw the following mindless trending inflators at the generic un-deflected model: -.002, -.001, 0, .001, .002.  This is what I came up with for fail rates for the longest longevity expectations:

Trend               Fail

-.002                7.39% black

-.001                13.07  purple

0.000               19.71  blue

+.001               29.19  red

+.002               37.70  green

That seems like some pretty intense sensitivity...so much so that I went back and looked at the lowest level data inside the loop logic, line by line, loop by loop.  It looks like it works the way I intended.  I'm 99.8% sure I got it right but maybe not.  I'll look at this again at some point. This one worries me a bit. I know the principle is right, it's just the execution I want to make sure of.  It matches the results using my own data in terms of degree.

7. And What About Spending Shocks?

When it comes to retirement finance, one of the bigger under-explored or under-examined things out there is that retirements are (sometimes) subject less to stochastic processes and the normal types of probabilistic randomness than they are to chaos principles, spending shocks and various positive-feedback style cascades that lead to early bankruptcy. The real risk is not necessarily sequence of returns risk (from normal real or simulated stochastic/random return series) and then slowly running out of money (which you can see coming) in year x. The real risk is getting a medical bill for $1M and being thrown into bankruptcy next year because everything hits the fan all at once…and you didn't see it coming. According to Dirk Cotton (Dirk has some good stuff on this at theretirementcafe.com), the prediction horizon for retirement, whether in or out of a simulator, is probably good out about a year, maybe two. After that there are too many both normal-random and chaotic-random things for a sim to be of any real value. But let's try this anyway.

Since I can barely do plebe statistics at my age and I have zero mandelbrotian/chaos-theory capabilities, I wasn't sure how to model this. The best I could come up with was to throw a fixed dollar amount at any given year's spending with some probability x that it will happen. So, that means that in any given sim-year I have an x% chance that an additional $y "need" will show up and be spent...and then I let the sim run as it will. For a "y" I picked 4x the generic initial spend [3] which happens to be vaguely under (maybe as under as a half) an NPV for longterm care estimates for someone my age. So this is a sorta-kinda arbitrary number...but not totally unhinged either. I also inflate "y" at z% (3% in this example) to the years in which it hits. In the same format as before the chart looks like this with the low lines to higher lines being 0%, 1%, 5%, and 10% probabilities.

Prob      Fail

0%       19.71% black

1%       22.46% purple

5%       31.34% blue

10%     42.47% red

That chart also shows a really big sensitivity effect but in this case I actually think it is smaller than I was expecting in a way.  It matches the analysis using my own data in degree but the absolute numbers are quite a bit smaller in mine. Again, I went back to look in the detail of the loop logic and it looks like it is working the way I intended.  And again, I will go back and look at this at some point, just not today.  For the moment, we'll let simple intuition stand on its own two feet: a higher chance of a spend shock is almost always bad news. 

8. What Do I Think About All Of This?

For now, I'll leave it at this:

a. Random spending variance does happen and it has at least some negative and often un-modeled or un-considered effects on retirement risk for at least some retirement cohorts.  For those cohorts, wide and skewed distributions matter more than narrow and none.  The cohorts might include longer retirements and, counter-intuitively, lower risk retirements but that is TBD.  In general the effect looks pretty muted as long as the underspends are reinvested.

b. "Normal" spending variance and skew have a much smaller impact than I expected and can probably be mostly ignored unless they become extreme or retirements get long. Skew, in particular, I thought would have a bigger effect but I guess I'm glad that it doesn't.  Unless skew somehow turns itself into an upward "trend," that is, in which case it can be deadly.  

c. Spending trends and spending shocks look like they have pretty big effects (but we kinda knew that) and they really have to be anticipated and defended against.  Spending trends, on the one hand, just need to be managed assertively while acknowledging that they can come and go. The trick is catching them early and redirecting oneself back to the plan before too much damage is done.  Spending shocks, on the other, can't be managed very well at all by definition.  But one can at least accrue a liability for this risk on an actuarial balance sheet. (Ken Steiner at howmuchcaniaffordtospendinretirement.com has a lot of good resources on this).  Often this accrual is funded by working longer where possible but it may also require some seriously hard reallocation choices between things like: current lifestyle, future spending, risk reserves, and legacy goals.  Any way you cut it though, ignoring the risk doesn't make it go away. But unrealized risk is what? Maybe a legacy fund...?

d. Early retirees, in my opinion, have a special obligation to be tuned into the possibility of these spending variability effects. If you are 70 with a lot of money and have had three heart attacks it probably doesn't matter all that much. If you are 50 and on a razor's edge of long retirement risk and plan to live to 105, then it does.  My opinion is that there is a whole giant pile of separate rules necessary for early retirees.  

e. Don't trust models all that much, especially this one. Also, a sharp eye will notice that I wasn’t very careful in defining and being consistent with how I was using and applying the word variance.  Someday I'll go back and tighten that up a bit. Don't forget, while we're at it, my two primary rules of financial modeling back at the beginning.  Anyway you cut it, take all of this, the whole post, with a big fat grain of salt, except for the next point…

f. Treating normal-variable-spending as a process and using process control techniques (on which I have written before) is probably a pretty good idea.  Keeping the running spend-variance low and in-bounds (whatever those bounds may be) in an ongoing "process" has some modest benefits like:

               - risk from over-spends and "sequence of spending risk" goes down a little

               - predictability for planning and budgeting is increased
               - trends get nipped in the bud in their infancy before they do much harm

               - it's more efficient: my opinion is that there is less personal, 

                 social, and environmental waste from unnecessary consumption. 

Notes

 --------------------------------------------------------

[1] For example, added to the model: spending variance (no variance, normally distributed, skewed, custom, with and without autoregression, etc.), spending shock (x% chance of a y * initial spend in any given sim-year), spending "trends" that are independent of inflation and that can be positive or negative.

[2] Main Simulator Variables

- Random longevity (distributed by way of 2013 SS Life Table)

- Random stock and bond returns (distributed by way of Stern school data)

- Random inflation (distributed by way of inflationdata.com history)

- Option for 2 streams of future income, e.g., SS at age 70, and/or annuity

- Random spending shocks of x% probability and $y magnitude per sim-year

- 2 +/- spending "deflections" over plan range

- Spending variance: none, normal, or skewed [AR(0)] (other version tbd for AR(1))

- Spending trends beyond just inflation effects (e.g., lifestyle creep)

- Standard parameters for

            start age

            endowment

            stock/bond alloc. and tbill/tbond allocation

            option to suppress returns by x% first y years

            rudimentary fees

            rudimentary taxes

            cap historical returns by .xx

            cap random longevity to age xxx

            other...

[3] Assumptions in the sims that are reflected in the charts:

            age:                      60

            endowment:        1M

            initial spend:       40k

            alloc:                   60/40

            deflections:         none for generic model

            fees:                   .6%

            tax effect:           10%

            return suppression:     -2% over first 10 years

            longevity cap:             120

            income stream1:         500/mo at age 70 as a partial SS provision

            income stream2:         0

            spend shocks:             as modeled above

            spend var/skew          as modeled above

            spend trends               as modeled above
            runs                            10k, except when busy or bored

[4] I'm retired, not a statistician or a researcher or a practitioner.  If I have the terminology wrong or imprecisely applied, let me know.

[5] I've come to understand that the grownups in the room (academic researchers and very serious and smart practitioners) use concepts like utility functions and certainty equivalents when interacting with the issue of spending variability.  I haven't really wrapped my head around the math and its application to this problem but I could probably figure it out if I tried.  The problem I have is that in 30 or 40 years of dealing with my own personal financial planning -- and at the end of  that path I don't feel like a total slouch...partial, maybe, but not total -- I have never once myself or in conversation with any financial planner (from the lowest capability to CFA'd portfolio managers) used the word utility or certainty equivalent.  That means the utility (pun, yes) of digging into this is low.  Me? I'd rather have my model just throw in some variability at whatever level and let it fly because that's what happens in real life; there are no certainty equivalents and there are, for that matter, no 10,000 retries on a retirement.  But, in the end, it looks like it (the variance) doesn't matter so much, but only if the under-spend value sticks around long enough and compounds in order to save me from the over-spends.  But I get the spend-variance-risk-aversion issue.  It's a little like mean variance analysis and planning: a) high variance sometimes, not always, means higher return but the "feel" of the high variance sucks so we seek lower variance stuff for similar return and try to find our optimal efficient frontier a task at which we will prospectively never succeed, and b) we often forget that variance (in portfolio or spending) is not even real risk, it's just a "feeling" thing. Real risk is the permanent loss of capital and thus lifestyle.