4% spend vs. survival probabilities and a net wealth process vs. annuity boundary

This is another one of those "I wanted to see what it looks like" posts and is a partial reprise of some "spend games" I did a while back.  This time I wanted to take a closer look at: 1) a very simple visual of a basic spend calc in the context of survival probabilities that change with each year survived, and 2) the net wealth process implied by the spend calc especially as it might plausibly relate to the arc of possible annuity calcs over time...or rather as an annuity might be recalculated if it were priced at each new age for the new level of inflation adjusted consumption at that age as well as the new longevity probabilities.

Some assumptions:

• the portfolio is $1M and returns are dead-flat deterministic (4% 5% or 6% nominal), 
• there are no taxes or fees or insurance frictions or anything else inconvenient  
• inflation is 3% fixed  
• for the "inflated cash flow" annuities, survival probabilities are extracted from the SOA IAM table with G2 extension to 2018. Male.  
• for the "constant cash flow" annuities Gompertz Makeham math is used inside an annuity pricing tool I got from Milevsky
• age range for the net wealth process is 58-105. 
• age range for the annuity calcs is 58-95.  
• consumption is 4% constant

1) 4% spend vs. survival probabilities

Here in figure 1 is the path of 4% constant spending inflated at 3% and pv-discounted at 3%. Spending goes to zero when the portfolio fails.  Red is a portfolio that earns 4%, blue is 5% and green is 6%. Survival probability line 1 (dark grey, right axis) is for a 58 year old; the lighter grey line 2 is rendered for someone who has survived to 85.  The orange triangle is the mean life expectancy at age 58 while the purple triangle is the mean life expectancy if one were to survive to 85. [note: there is an error in the mean expected longevity at age 85; the mean would be ~92 not 89]

Figure 1. Constant 4% spend vs. survival probabilities for two ages

At first glance this looks pretty dumb to chart a 4% spend inflated at 3% and discounted at 3% just to get a flat line in pv terms, but: a) this came from a past post where there were other non-linear spend strategies other than a constant flat spend, b) I hadn't in the past shown the failing-end of the spend plan against a survival probability or two, and c) sometimes simple deterministic "dumb" charts can illustrate the basic dynamics of things better than all the differential equations in the world. 

Conclusion for "1"

I guess the least that figure 1 should show us is that planning a spend rate to an average life expectancy leaves an awful lot of longevity probability on the table. In addition we might say that poor returns can push your "fail" backwards enough in age terms that the conditional survival probabilities get interesting and worrisome pretty fast. There are all sorts of non-linearities in there where things intersect. I'm sure someone somewhere has the math for it.

  

2) A net wealth process vs. annuity boundaries

Here we eschew the PV math and run the 4% constant inflated spend against the $1m but with three different rates of return (4, 5, 6%). We display what happens to the residual portfolio at each age in FV terms.  Then we take a shot at calculating an annuity price for the then current cash flow for each year of the portfolio's life to compare to remaining wealth. This is an amateur boundary test; I wanted to see in a simple static model those points where one could still have enough to plausibly somehow defease an inflation-adjusted lifestyle before things come crashing down.

The annuity prices are calculated in two ways: a) as a constant cash flow with no inflation but using the cash flow to which the spend plan has inflated at each age and using adjusted longevity expectations for the new age, and b) same as "a" but here as a constant, inflated cash flow at each age.  

For "a" I ran with the ILA R-script I got from Prof. Milevsky that is part of the technical appendix to "Utility Value of Longevity Risk Pooling: Analytic Insights." It is a more formal version of what I am trying to do in "b" but it is faster and easier to use and seems to me to be trustworthy.  It is applied to a constant cash flow. The method is described in note [1].  "a" and "b" when run with similar assumptions are parallel with "b" being a little underpriced relative to "a.".  

For "b" I had to wing it with my own math and I'll try to render it like this:

where a(x) is the price calculated at each future age x, w is ~120, tCx is the spending level at each time t after we have gotten to a new age x using a 3% inflator to get to that age and in this case it is then also inflated into the future at 3% as well.  tPx is the conditional survival probabilities in each time t for the x year old for which a(x) is being calculated. d is the discount rate which I am arbitrarily forcing to 3%.  There is no loading and this method runs parallel to other ways not mentioned here that I used for a cross-check but it under-prices a bit.  The end result is a vector of a(x)s for each x between 58 and 95.

In the following chart, the red, blue, green are the residual portfolio values using the spend plan and returns as described in "1."  The upper dotted line is my 2b estimate for an annuity price at each age for the spend plan at that time if the annuity were to be inflation adjusted. The lower dotted line represents 2a. Don't forget that I have no realistic frictions in there. The chart looks like this: [note: there is an error in the mean expected longevity at age 85; the mean would be ~92 not 89]

Conclusion for "2"

The point here in "2" is that self insuring (i.e., systematic withdrawal plan) definitely looks like it works under at least some assumptions about high returns but under other, lower, assumptions, one's chance to hop off the risk-train by locking in a lifetime lifestyle by offloading risk to a third party goes away at some point, a point which might happen pretty fast.  There are really no specific or formal points here but a casual look at the chart shows that assets available to defease a lifestyle via an annuitization "rip-cord" (either inflation adjusted or constant) look like it is the most robust between 70 and early 80s with a very subjective peak around 74-78 which is consistent with other annuity optimization papers I have read.  In a past post I mentioned a paper by Patrick Collins that suggests that "boundary monitoring" is part of a wise surveillance and control system for retirees.  He is probably right.  

  

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

[1] I described his ILA tool in a past post like this:  Milevskys ILA function is an Rscript "discretized (Reimann) version of an actuarial expectation" represented by: 

where w-x is forced to an endpoint around 121 and dt is 1/52.  Gompertz Makeham math is used to represent actuarial longevity expectations similar to what's in a mortality table. The output is an annuity factor for a given age.