Shapes and Flow

Comment from feedback: no one needs this post. It is just a weird flourish...

I never really set out to find specific answers when I embarked on my finance mission back in 2012. I was just curious about stuff...and ticked off at my grubby advisor. What I really did want was to be able to see what I call the "shape and flow" of retirement.  So, here is a fun "shape and flow" I was surprised to see, mostly because I never looked for it. It is the shape of consumption utility over time when considering the possibility of wealth depletion. Let's look at it like this in simulation mode:

• Each life (row) in the simulation has 100 periods (columns)
• the sim has 5000 lives it lives
• The spend rate (say 4%) continues until wealth fails then it drops to practically zero
• For each life, in each year (i.e., column) the spend -- either .04 (or whatever rate I was using for this illustration) or alternatively "near-zero" -- is 1) wrapped into CRRA utility [1], 2) weighted by a conditional survival probability [2], and 3) given a utile time preference discount of .005 [3]
• Thus, there are 5000 rows and 100 columns of weighted, discounted, utiles of consumption

 If I were to chart that last bullet point, it'd look like this: 

Figure 1. weighted, discounted utility of consumption

The odd shape comes mostly from a) the utility math, b) the failed paths where wealth runs out and spending drops to near zero, and c) the diminishing probability of surviving.  The latter (conditional survival probability for a 68yo(?), for certain Gompertz parameters) looks like this so you can see what happens to relative likelihood, or the "weight" at years 0->40ish:

Figure 2. Conditional Survival, Gompertz model

In Figure 1:

• The top line of the "mass" is the consumption (weighted utility of) that never gets smoked by a wealth fail.
• The "drop lines" are the ones where wealth eventually fails and spending (U) falls to near-zero
• The rising bottom line is the effect of both the weighting and discounting

If one were to take this further, which I do, one would calculate the expected discounted utility of lifetime consumption by way of the following

• sum each line
• average the sums
• and, I guess, compare to other portfolios and spend rates. Not done here

Why did I do this post? No idea really but I though the shape was rather cool. I'm guessing the shape has no real practical purpose for anyone but me. When I look at it, sure I see some kind of Retirement Finance "shape and flow," but I also see 

• A hummingbird, or maybe
• a Japanese crane

The "aesthetics of quantitative retirement finance." That kind of thing is a market of one, I think. Heh.

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

[1] [C^(1-gamma)]/(1-gamma) or ln(C) if gamma = 1. If I recall, I am using an arbitrary gamma=1.5.

[2]  {exp(exp((x-m)/b)*(1-exp(t/b)))}

[3]  1/(1+.005)^t