The Cost of Retirement Certainty

This post is a little bit of a reprise of a post I did a couple years ago. That one, as is this one, is dependent on past conversations and correspondence with others. First, this post here is a riff on an article by Gordon Irlam on the "cost of safety,"  of which this is derivative...although he was working in utility terms and I am working in Life Probability of Ruin terms (LPR). Second, this post is the result of thinking about some conversations with David Cantor on the topic of hedging tail risk (retirement portfolios, not necessarily accumulation portfolios) by way of either technique (options hedging, trend following, risk parity, etc.) or redundancy (surplus capital, more on which later in another post). Fwiw, David has been almost my only interlocutor over the last 5 years and has been a very productive influence. The vast majority of the papers I read now come from his enthusiastic referrals.


The Setup

The basic idea here is to gauge how much capital is required, under one illustrative parameterization only, to get to some level of certainty where certainty is denominated in "Lifetime Probability of Ruin" -- as well as how much capital is required per step for the incremental halving of risk which is a subset question.

LPR sounds like Monte Carlo simulation but isn't quite the same.  For instance, the approach here considers the full constellation of asset exhaustion possibilities (Albrecht and Maurer, not referenced) and the full term structure of longevity expectations (Milevsky, not referenced). Most MC sims just stop at 30 years and kick out a number. Also, the LPR approach happens to satisfy the 1930s era  Kolmogorov PDE for LPR which means only that we are on some solid ground here analytically. The following probably makes it harder than it looks but it is also as simple as it looks, if that makes sense:

Lifetime Probability of Ruin

...where tPx is a conditional survival probability for the 65 yo (x) in each year forward (t) and gw(t) is a distribution of portfolio longevity assumptions (to infinity by way of future year t) given return, vol and spending assumptions.  So: probability of "portfolio longevity in years" at various future ages weighted by the likelihood of living to those ages.

Recall from past posts that ruin can be a flawed metric. We are using it here because it was easy and fast and relatively rigorous (I had the software ready to roll).

The Inversion

Typically one would make wealth the dependent variable and LPR the independent. Here I'll flip it and make LPR dependent and Wealth independent. We'll also be working in wealth units (WU). In this setup, a wealth unit of 1 means that one is spending of 1 unit of capital. That means that 25 units of wealth is a 4% spend rate. So wealth units here means one can infer the spend rate by way of 1/WU.  20 wealth units is a 5% spend rate and so forth.

Core Assumptions

I weary of detailing assumptions these days but detail we must. The basics are as follows:

Wealth Units (vary; the independent variable)
LPR (dependent var): 100% to 0% in steps (I'll explain the steps later)
Iterations per run: 30k
Longevity uses a Gompertz model:
 - mode = 90 years
 - dispersion = 8.5 years. These are kinda arbitrary for now but are consistent w life tables
Age = 65 (let's call this a unisex thing)
Return = 4% real
Std Dev = 12%, this is roughly akin to a balanced 60/40 portfolio depending on one's beliefs
Spend rate depends directly on wealth units. Invert WU to infer spend rates.

The Output

To cut to the chase this is the LPR v Wealth chart. The main idea is that for increasing degrees of certainty, denominated in "% Lifetime Probability of Ruin" as defined above, how much wealth in wealth units is required to secure that ruin rate given a particular, singular parameterization? What happens as we seek to have more certainty about the future for the 65yo? Ignore that we might be able to do this in economic utility terms or maybe some other method.

Figure 1. Cost of Retirement Certainty

Discussion

Ok, this should be pretty obvious. The relationship is not very linear. As one seeks a lower life ruin rate, the capital required goes stratospheric. I mean, I guess that makes sense, right? 100% certainty is no doubt an infinite cost (in 2012 it took me a while to understand this idea).  Recall that: 1) in the chart that 25 wealth units is a 4% spend rate, and 2) this is merely one particular parameterization.  The curve, however, will no doubt be visible in all parameterizations though I have not proven it.

The other interesting thing, something I did not get directly from this data (I fit a curve to make WU =~ 7.12*ln(ruin)+9.34 which was pretty close but not shown; it sits right on the line above), was that for each halving of risk (80-40% LPR, 40-20%, 20-10%, 10-5%, etc) we are required to have just about 5 more wealth units.  The interpretation of that is for another post but it is no doubt in the nature of the power laws that it looks like we have in play here. Not sure if it'd show up in every parameterization but I'd guess it would.

The take away from this limited peek? It looks like it cost's a ton to be more and more certain of not running out of money and more the more certain we want to be.  This analysis is not totally out of synch with what one might read in some N Taleb book on anti-fragility.  The "level" of comfort, btw, is clearly a policy choice not some secret crisp answer just waiting to be found.