Increasing the machine's interval of interest to age 60-->95

Reason for this Post

See the integrated cover of what I'm doing here:

• Machine Spending  

In the last post

• Adding some lifetime income to the machine learning model

I wondered what would happen if I looked at the interval not just from 60-80 but from 60-95 because I wondered if the machine could make itself converge, when presented with the beneficence of lifetime income, towards a "shape" of spending that looks more or less like optimal consumption in a formal economics LCM (life cycle model) context.  The mental frame-of-reference that I have for "the shape" -- though there are other sources for this -- is from a 2010 paper by Marie-Eve LaChance titled 

Optimal onset and exhaustion of retirement savings in a life-cycle model; Cambridge U Press. 2010 

On page 35 she sketches it out like this:

Figure 1. Shape of optimal spend in LCM, LaChance 2010

Since I am not dealing with the accumulation phase of the life-cycle, I was expecting to see, with income present, the right side of the upper chart: a declining spend that goes to the floor set by available income (the right side of the dashed line at the top). I had in the past only used the segment from 60-80 because: a) the machine is really slow, and b) I am way more interested in the years before 80 in my personal plan. For what it's worth: there is no necessary reason for the shapes of our consumption paths not to be similar since we are using a similar consumption utility function and life-cycle approach. 

The Setup

I set up the machine to go from age 60 to age 95 which seemed long enough without having the PC take forever and bore me to tears. Probably should have run it to 105 but maybe later.  Then I (re)ran 2 scenarios 

• Scenario 1 - No income, similar to a prev post, see those for assumptions and parameters
• Scenario 2 - add 15k of life income starting at 70. As before that's new-found wealth[1]

Then: 

• Train the machine long enough to get reasonable results without getting too bored.  
• Look at the recommended spend policy for each scenario from the machine, and
• Look at the 1st quartile real spend over the last 1000+ sim-lives over the selected interval[2]

This is not precision analysis, btw, just a "shape check." I ran 6000 iterations for #1 and about 15000 for #2. Tiny but ok for this.  

The Output

1. Spend rate policy emerging for scenario 1 (red). Note that the Y scale will be different for Figure2.

2. Spend rate policy emerging for scenario 2 (red). Shown against PMT function (grey) and RH40 (blue).[3] Past posts in the link at the top will describe in more detail the benchmarks I tend to use and why. In this figure, the PMT function, which has been proffered as a spend rule, has no real insight into stochastic longevity which may explain its optimism towards spending at late ages. At least the convexity has a kinship. The RH40 benchmark mostly got lucky here but part of me want's to claim unearned genius at the same time. Yeah, no. The machine is seriously playing the right game though. I'll back that up later.

Figure 3

3. 1st quartile real spend[4], Scenario 2. I'll show scenario 1 next but I'll do it in context with scenario 2, which is something I should have done with figure 2 and 3 but didn't. I have focused on Figure 4 here separately because it shows a declining spend with at least 1 data point (at 95) falling to ambient income (15k). This, in my mind, is very very close to the the confirmation of shape I was looking for re figure 1.  Squint. You'll see it. Maybe I just got lucky...

Figure 4

4. 1st quartile real spend, scenario 1(no income, red) vs scenario 2 (income at 70, blue). 

Figure 5

To my untrained eye this looks to me like classic economics.  The presence of life income allows one to have earlier and higher (more optimal) consumption and -- counter-intuitively to some -- much earlier wealth depletion than running it all the way to the the expectation for terminal lifetime. This is because there will be no ruin at late ages like 95 or 100 or 105 or 117 due to the presence of lifetime income that saves us from total ruin. i.e., spend your money early and then ride the income stream later.

For the "no income" scenario, there must be a "reserve" created by reduced spending early. That reserve is then held and "transferred" over time in order to be available for spending at the old ages that may never come to pass (think 118).  At the same time, the portfolio over that walk to 118 always bears the risk of falling to zero before the real end of life. So that sets up a serious problem: a big bequest left on the table or...big ruin. This is why SS and annuities tend to be important.

Frankly this imagery turned out a little better than I expected relative to what I know is the theory but then again I wasn't sure what to expect from a machine that is teaching itself.  

------------- Notes ----------------------------

[1] weighted PV is ~226k but it is in fact a flow, one that lasts forever.

[2] Due to some reason (under training?) some of the late ages had massive outliers in real spending which affected the mean a bit.  Hoping it's not too corrupt or tendentious of me to do so, I used the first quartile of the spending distribution at each age as a proxy. This is arbitrary but at least it shows a similar tendency as the mean without the weird results as well as conforming to figure1 well. 

[3] RH40 is a rule I cooked up using Evan Ingliss' divide-by-20 rule with some adjustments. It get's it's strength by making some implicit and hard to explain assumptions about rates and longevity. It has in the past, in a no-income scenario, stacked up well against a Life Ruin calc pegged to a 95% success rate. I don't expect anyone but me to know or care. I use it for its shorthand convenience and past utility.

[4] By 1st quartile, if ur not a stats geek, I mean that at some age x, say 75, the machine over 10s of 1000s of iterations will conclude that some real spend or other is the one to choose.  That means that it is a distribution. I pick the 25th percentile for reasons in a note above. The distribution of the spend in the many iterations looks like this in scenario 2 for age 75. At late ages it bifurcates into a two headed distribution because spending tends towards low income constrained spending and the high spend rates from super high luck wealth outcome. Gotta look at that later.