The Challenge of Retirement Finance in a Few Charts

I won't repeat Sharpe's over-used quote about the "retirement problem" but let's at least stipulate that retirement finance is a bit of a challenge. The intersection of uncertain returns, uncertain longevity and uncertain consumption in the absence of lifetime income creates a difficulty in how much to save, how much to spend and how to allocate resources.  The self-serving illusion articulated by much of financial services or in academic studies is that there is "a" solution -- i.e., your "number" or some other optimal calculation distilled down to a point -- where in reality there is just fuzz: fuzz now and then even more fuzz as time unfolds.  The point here is to highlight some of the idea of "fuzz" rather than solutions or points. 

While I think the real answer in retirement is more likely in some form of an adaptive operations management and control process (while keeping an eye on the usual stuff), which is another post entirely, if I want to frame this post in traditional finance terms I have to lean on standard tools related to returns, spending and actuarial expectations for the planning horizon. Let's fire up some assumptions and then blow out a chart (fig 1) that shows how long portfolios will last given a portfolio return/vol assumption for not just one but all spend rates over some interval. Then we can compare that to a distribution of likelihoods for surviving at all as a human. All of this will be projected over the next 100 years where the 100 years is a stand in for "forever" or "a really long time."  

Portfolio Assumptions 

• Portfolio: continuously compounded equivalent to a 4% return and 12% std dev (real)
• The P assumptions are arbitrary and can be pushed in many directions not shown 
• Spend rate: constant (real) spend between 1% and 12% of initial portfolio. This is framed as constant because that is easy to code. In reality: a) people adapt spending in response to changes in the environment, and b) it is stochastic with a distribution that likely has a fat upper tail.

Lifetime Assumptions

• Start Age: 63 (only affects the longevity calc not the portfolio which can sorta last forever)
• Longevity: Gompertz model conditional on age with mode 88 and dispersion 9 (avg health?)

Process Assumptions 

• The net wealth process is the formulation below in Eq1. which uses a continuous time formulation with M(j-1) representing current wealth, w is the withdrawal or "consumption," and v(tilde) is the randomized continuous compound return. The key thing to note about this process, contra asset accumulation, is that it can go to and through zero, hence the interest in portfolio longevity, the years it takes to get to zero. 

Eq 1. Net wealth process

Panel 1 (top), Figure 1. These particular assumptions above generate the scatter/heat diagram in Figure 1, top panel. The X axis is portfolio longevity in years. The Y axis is the constant spend rate which is uniformly and randomly generated in the code.  Each dot is a particular iteration amongst 25000 rendered as the number of years the portfolio lasted before depletion. Higher spend (say 12%) and we can see that most portfolios don't last very long though a few can. Lower spend (say 2-3%) and we can see that many portfolios can last forever but some don't. Note: other portfolio allocations would bend this cloud in different directions but the general effect would be vaguely similar. 

Panel 2(bottom), Figure 1. This is a continuous Gompertz model for longevity (conditional survival probability) conditioned on surviving to 63 (i.e., me) with a mode of 88 and a dispersion of 9. These parameters are a bit arbitrary but if I recall they are somewhere between a Social Security life table (average health) and a SOA individual annuitant table (healthy cohort) so here we are in a generic gender-neutral middling-life-assumption model of some kind. 

Both Panels, red dashed overlay. The horizontal red-dashed line is illustrative of a 4% spend assumption, common in the literature flawed though it may be. The vertical is an entirely arbitrary illustration of age 95 which is a common conservative fixed planning assumption (where here we have the full distribution on display). Note that this particular spend, and all spends, are constant in real terms. This "constant" assumption is unrealistic since most people adapt spending over the lifecycle in response to their environment and their variable psychology. 

Figure 1. Portfolio Longevity in Years for one portfolio
and many spend rates paired with survival probabilities for age 63

Figure 2. To add some texture to Figure 1, Figure 2 shows a cross section at the 4% horizontal line.  This is the simulated frequency distribution that illustrates some relative likelihood of PL for the given portfolio and spend assumption (4). Or, alternatively: how many of the 25,000 simulated iterations ran its portfolio to zero in that year? Note: this is clearly a defective distribution but it also shows that there are really two categories of outcome: a) the portfolios that last forever (or its proxy 100 years) or b) portfolios that have finite longevity. In this defective world it is hard to use stats like mean or median or any other moments so maybe it is just better to just visually inspect it.  It looks like this set up is a not unreasonable but also a not riskless bet. If I were to model with discrete returns it is similar but maybe a bit less sanguine, a point that is unnecessary for this post but I thought I'd throw it in. 

Figure 2. PL for given Portfolio assumptions and 4% constant spend

My Point, Such That I Have One

In the retail-facing practicing financial community -- for entirely understandable reasons of communication clarity and comprehensibility -- it is common to distill retirement "solutions," created by whatever means at hand, down to simple answers like what one might find at the intersection of the two red dashed lines above: x% likelihood of success at some age y. This answer can come by way of Monte Carlo and fail rates, deterministic formulas, advanced optimal mathematics or even some guesswork heuristics...whatever.  The reality, though, is that the answer is not really a point but an ever-shifting cloud of possibilities which is risk-goosed even more by the non-specific uncertainty around how long we will live. This moving fuzz-cloud (vs the single number point answer) is something that one has to live with and is also why imo adaptive approaches that check-in periodically with the environmental multi-method analytics are going to outmaneuver more optimal set-and-forget single-method systems (that last was a bit of a strawman. I don't think anyone really does that irl. It is also an assertion without much proof...just an opinion). 

Side Note on Kolmogorov. 

While Monte Carlo and Fail rate analytics for fixed ages are pretty well known, and often useful though flawed, there are other methods. The Kolmogorov partial differential equation for Lifetime Probability of Ruin (LPR) is another. This PDE is similar to MC only in the sense that the PDE can either be simulated in a proxy way that satisfies the construct or it can be solved via finite differences which I call a type of simulation by other means.  The advantage here with LPR is in the fact that it takes into account the full "term structure" of longevity (see bottom panel, Figure 1) rather than just "to 95."  

For example, in a custom simulation that were to satisfy the PDE: the portfolio longevity distribution illustrated in figure 2, re-rendered in relative probabilities, is multiplied by the "weighting" implied by the conditional survival probability illustrated in the bottom panel of figure 1. So when you see something like "your LPR is 5%" that is what is going on. I built a custom sim for this once I called FRET (flexible ruin estimation tool) because all I did in around 2012 or so was fret about ruin. As the word fret is generally defined it, in addition to its core meaning, generally also comes across as a little bit unnecessary or counterproductive...something which is also true about me spending too much time on retirement finance blog posts.