Revisiting Perfect Withdrawal Rates But with Variable Duration

Perfect Withdrawal rates

Over the last 3 or 4 years a number of different sources[1] have discussed a way of viewing retirement consumption analysis by way of "perfect" or "maximum" withdrawal rates (PWR).  Quoting Suarez 2015: PWRs are the "constant amount that will draw the account down to the desired final balance if the investment account provides...any...particular sequence of annual returns." i.e., it is the rate in any of some number of parallel simulated universes that draws an account down to zero (or a bequest level) at the end of the duration of the analysis.  If one were to simulate 10000 parallel universes, one would thus have a distribution of 10000 constant "perfect" withdrawal rates for each parallel world.  Constant is a debatable proposition but for first pass examination of retirement processes (not last mile real planning) it is ok enough.

The distribution, then, can be examined by retirees and planners for making a joint "allocation and spending" choice.  It is, in effect, no more and no less than inside-out monte carlo simulation except that the fail rate is set to zero over the interval of interest and spend rates are allowed to vary to make it work.  Looking at the left side of the distribution (the lower spend rates, at or below which were the rates that worked in really bad situations) is interesting. Looking at the right is not or rather is an elegant problem to have.   The cumulative percentile (say 5% or 10%) of spend rates that worked in the worst of situations is roughly equivalent to a fail rate analysis except that it is, in my opinion, a wee bit more transparent and conversation-worthy.


The Nitty Gritty Plus Sequence Risk

The math for PWRs looks like this (from Suarez 2015)

where K(s) is the starting balance, K(E) is the ending balance, r(i) is the return in some period i, and j is the process for calculating the opportunity cost of consumption and the contribution of sequence risk. That last point is a little opaque so I'd recommend earlyretirementnow.com to explain it for me. In simple terms the denominator is the sum of the products of returns (r1*r2*r3*...rn) + (r2*r3*...rn) + (r3*...rn) +.........  The presence of more later return terms in the sum means that good returns late and crappy early affect the denominator and are the mathematical interpretation of sequence risk. This is useful to know. Check it out in the link. 

The Point of the Post

My interest today is not so much PWR as it is on the effects of "duration" on the PWR results.  I've touched on this kind of thing before.  I once even added stochastic longevity to the code for doing the PWR but I now think that that kind of thing is not constructive. Random draws on life, actuarially correct or not, makes a hash of the analysis and buries important detail in the distribution.  Better, I think now, to keep it (PWR) the way it is and then look at different discrete durations over which PWR can be analyzed.  This duration thing can be thought of in two ways:  (1) we can have a variable start age (50, 60 70...) and a fixed end age (say 95) for planning discussions, or (2) we can have a fixed start age (say 60) and variable end dates (75 80 85...). Given that framework we can look at PWRs for different durations, some given return assumptions, and see what happens over those different intervals and then see whether it informs the discussion.

Assumptions

This is a quick illustrative look-see post. 4% real returns, 10% standard dev, normal returns, no taxes, etc.  Durations will be for 15 - 45 years in 5 year steps.  10,000 iterations.  To the extent we need to think in terms of begin and end ages, see the table below.

Output

Charting the density (easier to see than a bunch of frequency distributions) for all of the seven intervals when we calculate distributions of PWRs, we get this:

Is this meaningful? Maybe.  As mentioned above, examining the cumulative percentile of the curves on the left side is the most interesting and works more or less like a MC sim evaluating fail rates except in this case the cum% starting from the left are the rates that worked for the worst series of returns.  0 is not really realistic so I looked at the 10th percentile (which pops up now and then as a threshold for simulation success).  In tabular form:

For context I also added the spend rates from: (a) a recent post on optimal spend rates by age using (borrowing someone else's) backward induction to optimize allocations, and (b) from my recent efforts to use decumulation consumption utility with a wealth depletion time framework to evaluate optimal spend rates.  Note here that when doing the PWR analysis, if we were to be really honest about advancing age, we would note that the terminal date would not be fixed it would be moving due to dynamism in conditional survival probabilities. That means the table above over estimates the 10th %tile at later "ages."

Conclusions

I think that PWR analysis is useful and usefully transparent but I also think that they are of a piece with other retirement analysis when the duration dimension is thrown in. It is just another way to look at it and it confirms in my head, as Newfound Research often states, "risk can't be destroyed, only transformed" (but it sure as hell can be created! and by transformed I really mean here that a new way of looking at things e.g., PWRs, won't change the fact that retirement, especially early ones, are risky and need some circumspection on the planning choices. All methods of analysis say more or less the same thing. No methodology will give me a free pass to spend 50% in retirement at age 60[2]). I also see here that with foreshortening longevity and decreasing planning horizons that spend rates go up.  This is common sense. It is also consistent with other approaches for evaluating risk and spend rates that I have looked at before, as is seen in the last three columns above.




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[1] Without all the links and by memory I'll point to: Suarez and Suarez 2015, Blanchett, Earlyretirementnow.com (ERN) (technical appendix #8), Andrew Claire et. al on trend following in retirement (at ssrn; references Suarez), and Javier Estrada on Maximum Withdrawal Rates (on his personal page if I recall).  The most accessible, for me...in my opinion, is ERN at https://earlyretirementnow.com/2017/02/01/the-ultimate-guide-to-safe-withdrawal-rates-part-8-technical-appendix/   I have worked this topic before and can be searched using "PWR" or "pefect withdrawal."

[2] The wall street journal had an article today that showed, via survey, that 3% of retirees think that a 50% withdrawal rate will work!  wtf?