Component Analysis #3 - Mortality

“….One need hardly be reminded that a consumer who makes plans for the future must in one way or another take account of the fact that he does not know how long he will live…” - Menahem Yaari 1965

Up until maybe 5 or 10 years ago, most papers I've read in RetFin did not take random lifetime seriously (might be wrong, just an amateur observation) which is not a great place to start a conversation with people that don't live exactly 30 years after a retirement, the start date of which is itself uncertain for some. 

In chapter 8 of Moshe Milevsky's Retirement Recipes in R (RinR), he gives a slick tool, divorced from formal "life tables" and based on "the Gompertz Law of Mortality," that can model conditional - and continuous - survival probabilities (CSP) with some nice malleable parameters. Probably won't land a rocket on Mars but very effective for RetFin models. 

Talebian Redundancy - redux

Like the last post, don't take this post too seriously. This is playtime with Taleb and one of his anti-fragility ideas now expanded a bit.  As in the last post, it is more or less like this with some changes highlighted in blue: 

IF

- robust systems are identifiable by the reduction of single points of failure and redundancy of resources at critical points, and

- we assume a $1M portfolio P1 for a 60-95yo, spending an age adjusted spend[1], and

- of that 40k (in t(0) only) in spend, 20k (real, all periods) is a life-or-death floor forever, and

- we use SS-like life table to assess the probability of spending anything at a future time but now conditional on advancing age, and

- the PV at t(0) of the probability-weighted cashflow of the floor is variable by age , and

- we simply and blindly double that part of the portfolio (.43) that defeases the floor at t(0), and then we also, as age advances:

- recalculate the spend as the "heuristic rule spend amt" divided into the "total" capital, where the total now includes the extra redundancy

A Talebian Spend Rate

First of all, don't take this post too seriously. This is playtime with Taleb and one of his anti-fragility ideas. I think I have done this before but I wanted to mentally run through it again. 

IF

- robust systems are identifiable by the reduction of single points of failure and redundancy of resources at critical points, and

- we assume a $1M portfolio P1 for a 60yo, spending a deterministic (real) 40k, and

- of that 40k in spend, 20k is a life-or-death floor, and

- we use SS-like life table to assess the probability of spending anything at a future time, and

- the PV at t(0) of the probability-weighted cashflow of the floor is .43 of the initial P1, and

- we simply and blindly double that part of the portfolio (.43) that defeases the floor at t(0),

THEN

- the initial portfolio P1 is now 1.43 x P1 = P2

- The spend rate of 40k of P1 is now .028 of P2 at age 60

- The need for redundancy will decline towards zero at later ages. I haven't gotten into that.

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.

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:

A 500 year portfolio and min-max spend rates

What is the spend rate that maximizes median terminal wealth at a horizon of 500 years (why 500? idk, just messin' around and anyway that's the game we played in the last post)? Obviously the proper spend rate for doing that is zero. That's why it is sometimes hard to talk about portfolio growth optimality in a retirement spending context, granting that the 500 years here is absurd. Better to discuss portfolio longevity + mortality, and/or (gasp) life consumption utility. 

Ok, what is the threshold spend rate that minimizes median terminal wealth at a 500 year horizon? idk, let's look.

Some Random Thoughts on Growth Optimal Portfolios

“Theorem: If Harry repeatedly invests in a portfolio whose E log(1+R) is greater than that of Paul [i.e., the growth-max proposition], then -- with probability 1.0 — there will come a time (T(0)) when Harry’s wealth exceeds Paul’s and remains so forever thereafter.” Harry Markowitz in a 2016 book poking fun at Paul Samuelson on their past argument about growth optimal investing criteria.

"Mean log of wealth then bores those of us with tastes for risk not real near to one odd (thin!) point on the line of all tastes for risk -- and this holds for each N with N as big as you like...For N as large as one likes, your growth rate can well (and at times must) turn out to be less than mine -- and turn out so much less that my tastes for risk will force me to shun your mode of play. To make N large will not (say it again, not) make me change my mind so as to tempt me to your mode of play. QED"  Paul Samuelson (1979)

Intro

There seems to have been an uptick in the last few years of interest in the growth optimality consideration for portfolios. Ergodicity Economics feels like the new kid on the block, but this topic has a pretty long history, and the general interest bubbling up I find useful because it is an interesting and worthy topic. I won’t recapitulate all the math or notation in this post since it is tedious blogging [7] and one can read it comprehensibly, for the most part, and quite usefully, in the following bulleted references. Or check out the recommended reading list at the end and in particular pay attention to the references inside the various papers: 

A Random List of 50 Things I've Learned over a Decade of Blogging Quant Retirement Finance

[recent updates in brackets, see item 50]9/20/21

1. The planning interval of keen interest in retirement, often pegged at “30 years” by many advisors, academics and retirees, is actually a random variable by way of the dynamics occurring at both ends. The end, and in an underappreciated way the beginning, of retirement are uncertain. This uncertainty has to be acknowledged somehow in the planning process with some method whether it is scenarios, ranges, distributions for key parameters and output variables, probability weighting, random draws, provision for life income, estate planning etc.
   
   
2. Fail rates are a bogus metric. Dirk Cotton, now gone, wrote on this better than I can. No one in the modern world does a real mathematical fail where there is a continuous dive into the ground. People adapt, spending changes, annuity ripcords are pulled if possible, family steps in, institutions – to the extent we even trust institutions anymore in 2021 – of government and association help out. Stuff happens. I mean bankruptcy is possible but that’s a different problem. Also most savvy commentators will mention that magnitude is ignored. Failing by a dollar under some goal on the last day of life is different than running out of money at 72 with another 30 years to go.