Uncertain Longevity Teased Out in a Couple Different Ways

 I thought I'd try this kind of thing again just for "fun."  Fun! ;-)  Heh. 

I've used a number of different evaluative frameworks over the years including fail rates/Monte Carlo, Lifetime Probability of Ruin, Perfect Withdrawal Rates (PWR), consumption utility, formulas, etc. A while back I took PWR (Suarez 2015) and feathered in random life time by way of Gompertz math that I used to take a random draw on terminal age consistent with a distribution tuned to actuarial tables. That was interesting and while I don't have my link handy I recall that I got some counter-intuitive results (misc measures of spend rates went up) due to the inclusion of the possibility of very very short lifetimes vs the typical "30 years" combined with the fact that long lifetimes will still be circumspect with respect to spending. 

This time I wanted to shake up the "PWR + random life" thing a little more. This was spurred by reading Huang et al 2011 which presented a way of randomizing force of mortality via diffusion process. Just out of curiosity I wanted to see what happened in a PWR context when doing that. (recall that PWR is the consumption rate that with perfect foreknowledge of returns would allow one to spend to zero over a given horizon.). After messing around with this a little too much for some very minor new info I had these various ways of shaking things up:

1. No shakeup, just PWR and 30 year horizon

2. PWR with a simple parameterized Gompertz derived distribution.

3. PWR with parameter uncertainty

4. PWR with parameter uncertainty - bias to the dispersion param

5. PWR with parameter uncertainty - bias to the mode param

6. PWR with a faked pseudo-chaotic nudge

7. PWR with stochastic force of mortality - lower sigma

8. PWR with stochastic force of mortality - higher sigma 

Assumptions 

• 20,000 iterations
• r = .04 normal w .12 vol
• start age = 65
• max age = 120
• M = 90 (mode)
• b = 8.5 (dispersion)
• T = random lifetime, drawn from a probability mass
• key metrics = 5th and 20th percentile of PWR withdrawal distribution plus mean and median

Perfect Withdrawal Rate

I don't want to dwell on this too much just write it up so you'll have to go look up Suarez or search past posts.  The form of the PWR math, though, is this

Fig 1. Perfect Withdrawal Rate

and running it 20,000 times with random r gives you 20,000 w in a distribution that can be examined and from which some inferences about what can be spent can be more or less inferred.

Longevity Math

When the horizon is not just "30 years" fixed I am using variations of Gompertz math from Milevsky (2012) and Huang et al (2003, 2011) etc. I'll be using these and the various relationships along the way. Without comment, the various expressions look like this:

Fig 2. Misc Longevity related math

Notes on Scenarios 1 and 2

I ran PWR with a fixed horizon (black) and with a simple Gompertz distribution (blue). Some more detailed results are tabled below. More on that later. The thing to note is what I mentioned above which is that the randomized horizon means that very short lives have high spend rates and nudge the PWR upward. Very long horizons will have PWR results similar to the 30 year fixed and so that does not influence the PWR much.

Fig. 3

Notes on Scenarios 3 - 5 - PWR with Longevity Parameter Uncertainty

 In the PWR algo I am doing a random draw from the PDF (EQ 4 or it's equivalents, usually by way of EQ3) to get an age of death and thus the number of periods.  Just for the hell of it I decided to see what happens if I make "m" and "b" random variables. For better or worse, and this is probably easily criticized (but I am just playing a game, my rationale and methods are obscure), I shook up "m" and "b" like this in R code: 

• M.r <-  M*(1-rnorm(1,0,Mv)*M.amp)
• b.r  <-   b*(1-rnorm(1,0,bv)*b.amp)

The exact measures of vol are not presented because it was pretty arbitrary. Basically I was just trying to move them around a bit on the theory that perhaps the original fit of the model/parameters was wrong or in some unknown zone and I wanted to see what would happen. Visualized in PDF mortality probability terms (ie fx(t)) it went like this, noting that the X axis is years from age 65 not age and Y is the measure of probability. Each curve sums to 1: 

• left panel is generalized uncertainty
• center is biased towards "b" vol
• right is biased towards "m" vol

Figure 4

Ok, now when I applied this kind of parameter volatility, basically all I got was the red line in figure 3 above for each of the PDFs in figure 4. Didn't move the needle much on PWR. Scratched my head for a bit and decided that over 20,000 iterations it kinda evens out, you win some you lose some and it all washes out. Figure 4 shows only 10 or 20 iterations; figure 5 shows many many 1000s of iterations (grey) overlaid with the deterministic PDF using the untouched Gompertz math. I thought I'd get more from doing this exercise but it was a "meh."

Figure 5

Notes on Scenario 6 - A Fake, Pseudo-Chaotic Nudge 

By nudge I mean this: I assumed that the goal was for me to realize that an initial modeled assumption about T (terminal age) is A) unlikely to be 30, B) unlikely to even be as modeled even if standard random, and C) that the risk is for any given T plucked from somewhere that it might be influenced to the upside by changes in med tech or maybe just some unexplainable chaos. My proxy for chaos here is to say: "what if for any T I pull in the code to establish a horizon, there is a 15% chance that that T will be goosed to the upside by a factor of 1.3.  I made those up btw. This means that if T is 100, it would be nudged sometimes to 130. Not likely but this is a "what if?" so I can do it. 

Figure 6 is not PDFs or PWR distributions, it is the density of the occurrences of T (horizon) under standard Gompertz (black) and "nudged" T as above. Bumps the right tail of age a bit and does have a small impact on the PWRs tabled below. I thought this would be a bigger deal. 

Figure 6. Density of terminal age T, two methods

Notes on Scenarios 7 and 8 - Stochastic Force of Mortality in Play

You'll have to read Huang et al 2011 to see what they are doing. But they were trying to turn EQ2,3 into EQ5 (figure 2) i.e., force of mortality with drift and diffusion. Spoiler alert: I didn't really figure out what they were doing so I had to hack my way around this. That means I don't know if these scenarios mean anything. Took a shot tho. 

First thing to note was that I hacked a diffusion in a certain way that doesn't look like their paper. In noticed that in the deterministic Gompertz that the ratio between lambda in period x and period x-1 was constant. So, I made the force in any time t drift with that constant and some variation around it. In Figure 7, red is the original force calculated using the non-randomized formula and the grey lines are the diffusing using my hack EQ6 in figure 2. 

Fig 7. Force of Mort. diffusion

If I amp up the diffusion a bit and look at the mean and median and the first 20 iterations it looks like this where red is deterministic, grey is the mean and blue is the median:  

Figure 8

Ignoring whether I mis-modeled for now, at least the mean of my diffusion is on the right line. The 2011 paper explains in the appendix why Fig 8 might happen and what it means but I have not spent enough time on it and probably won't. 

Assuming I can still run with this then, in order to pull a random T under these last 2 scenarios then I need a PDF. In order to do this I can back into it by way of EQ3 in Fig 2 to construct EQ4 with lambda as a diffusion process. I think I have that right and if I don't then I am wrong BIG. But whatever. 

When I run EQ4 inside the simulator I get a family of iterated PDFs that look like this and not radically different ( a little diff) than the left panel in Fig 4 which means nothing much I think, just noting it:

Figure 9. PDFs constructed from a randomized lambda process

Table of PWR results

Discussion

That was a hell of a lot of autistic diddling with Excel and R code and formulas for pretty much nothing. The main conclusion was in the opening paragraph and thus already known. Me personally? I literally do not think or worry about spend rates at that level of precision in my daily life. Some uncertainty around parameters or the unknowability maybe would make me a little more circumspect but not by much according to what I see here. The chaos nudge might be something interesting to explore. 

Note that I am not pricing annuities or doing liability hedging of an insurer's book and using ONLY the PWR eval framework so maybe it's just not that big a deal for a retiree the way I am looking at it. I don't know. Probably won't tackle this again. Got better things to do in my new Montana home. 

References -----------------------------------

Huang, Milevsky, Salisbury. 2011. Yaari's Lifecycle Model in the 21st Century: Consumption Under a Stochastic Force of Mortality, SSRN 

Huang, Milevsky,Wang. 2003. Ruined Moments in your Life: How good are the approximations, SSRN

Milevsky, Moshe2. 2012. The 7 Most Important Questions for your Retirement, J Wiley canada

Suarez Suarez & Waltz, 2015. The Perfect Withdrawal Amount, A Methodology… Trinity University