Visualizing Simulated Spend Crashes

Here in Figure 1 [note 1] is a quick visualization of a couple retirement process: 

1. in grey, different spend paths from a portfolio without any "life constraint" over time, and 
2. in red dashed, the "life constraint" in terms of a survival probability conditional on achieving (here) age 68 and parameterized to look a little like a Social Security life table.  

On the left-Y axis we have the spend (using .045) i.e, #1. Fwiw, in this case the spend is an amount not a rate, even though it looks like a rate, because initial wealth is = 1.  On the right-Y axis we have the conditional survival probability, red dashed line, for a 68yo. i.e., #2. The X axis is an arbitrary 50 years, long enough to capture early retirements but not too absurdly long.

We can see that:

1) an infinite net wealth process, untethered to human mortality, can have a spending crashes (when wealth runs out) either before or after a reasonable interval for life. We know with certainty neither the portfolio longevity (and hence the upper level of spend in grey in Fig 1) nor the lifespan (red dashed in probability terms here rather than a random draw). 

2) were there to be an income stream for life (pension, annuity etc) -- covering some or all of the lifestyle supported by a portfolio before it runs out i.e., the upper grey lines -- there would be a small(er) and maybe even trivial difference between the spending before and after a "crash" (the stepdown to the lower grey lines) forever thereafter.

3) The interval after the blue line (that is an example of one life in the sim) steps down and before the survival probability trails off to zeroish should be of great interest to us. The "magnitude" of the risk can either be: a) measured in years (i.e., when academics talk about fail rates and "fail magnitude" in years) or b) evaluated in terms of additive consumption utility, with a risk aversion parameter, over a whole life. The convexity of the latter construct is what would punish the occurrence of a fail state that falls "too far and for too long" and/or that is without some reasonable level of life income. This is why, even though people don't seem to buy them, annuities show so well in an economic context. 

Figure 1. Spend Crashes vs Longevity Probability

[1] This has been done before by others and myself. Nothing too inspirational or new here, just test-driving a new piece of software.