My annuity game extended: now vs RMD with 3 different return expectations

This is an extension of a previous spend game, the description and rules of which are here (Early retirement: inflation adjusted annuity vs. 2 SWPs). The game is the basically the same but can be summarized like this:

• For a $1M endowment at age 58: accept the lifestyle implied by an inflation adjusted annuity with a fair price + 10% (which I gather to be somewhere around an initial payout of 33k, but I might be off) 

• Either buy the annuity or replicate the annuity by spending the RMD or the annuity spend rate, whichever is lower between age 58 and 120, 

• Then see how much lifestyle has to be forsaken (in probability weighted PV terms) in the SWP (systematic withdrawal plan) in order to not run out of money at late ages by using the rules of the SWP.

The Extension

The extension here is to use at least three different return assumptions for the RMD plan and note the effects of the different growth rates on the game.  See the prev link for constraints and caveats...like: there is no randomness or vol in the model, the RMD for age 58-70 is an extrapolation, static inflation of 3%, discount of 3%, etc. none of which are all that realistic or maybe even reasonable. Hence the framing of this as a "game" rather than reality.


PV of Lifestyle

The lifestyle for age 58+ in PV terms looks like figure 1.

Figure 1. Lifestyle paths in PV terms

Red is the infl adj annuity cash flow
orange is the capped RMD with 4% growth rate
green is the capped RMD with 5% growth rate

blue is the capped RMD with 6% growth rate
grey are the survival probabilities for age 58 and 85 (right y axis)
triangles are the mean life expectancies  at 58 and 85

In this chart, constant returns of 6% look like they would sustain the lifestyle over the full cycle.  One could then maybe consider ditching the rule on the front end but one would only know in retrospect if it worked out so we have to stick with the rule. Even 5% growth looks reasonable here vis-a-vis the longevity probabilities.  4% constant returns appear to commit us to an abridged lifestyle.

The "Cost" of the SWP

If we take the difference between the cash flows (annuity vs SWP) and probability weight it and sum across the lifecycle we get what I want to call the cost of the SWP.  If I do that with the game data and chart it out it looks like figure 2.

Figure 2. Cost of SWP in relative, discounted, weighted terms

For all level of returns, especially the higher ones, the "cost" is heavily weighted towards the early years due to the effects of discounting and probability weighting and the rule.  In the higher returns there is no difference in the cash flows in later years so the cost is all in the front end. The rules in the early years depend on my extrapolation of the RMD to young ages which you may not buy but it is not too dissimilar to my RH40 rule or the PMT rule in Waring and Seigel's ARVA or the Inglis 'divide by 20' rule so I am not too far out on the lake-ice I think.

What Happens to Residual Wealth Along the Way?

Since the game has players/rules that opt to not irreversibly commit capital to an annuity, we should see what happens to wealth along the path for the set-up above.  The effects look like this in figure 3.  This is converted into PV terms.

Figure 3. Residual Wealth Effects

red - RMD with 4% growth (left y axis)
blue - RMD with 5% growth
green - RMD with 6% growth
dotted - annuitization boundaries: upper = infl adjusted, lower = constant, for each age
grey - the survival probabilities for a 58 year old with no additional lines for later ages (right axis)
triangles - mean longevity expectations for age 58 and 85

For what it's worth, the 4% rule would have seen these lines drop to zero at some point so the SWP we have here does a good job, lifestyle support notwithstanding in low return environments, of not crashing over a reasonable interval of the cycle.  Also, if my math is right (and that is a big if; I've made plenty of embarrassing mistakes before), we have not forsaken the ability to opt out into the annuity along the way.  I have to double check this because it looks like the RMD with a 4% growth would be able to increase lifestyle by opting into the annuity for most of the lifecyle which seems counter-intuitive to me. That's TBD I guess, but if we go with it the chart tells me that by not irreversibly committing capital we continue to be able to hold an option on annuitization.  Add some stochasticity to the anlaysis and we have the "real option of not annuitizing is generally positive for a while" which Milevsky covered formally and I tried to cover casually here.

Conclusions

Accepting a lifestyle constrained to roughly what an annuity purchase could deliver and then using an actuarially sound rule while praying for anything but really low real returns looks like it can favor a SWP, which I already knew.  It's the "praying but getting a bad market experience, sequence or otherwise" that can torpedo one's ability to fund very late age realized longevity risk. That is the attraction of: 1) throwing some or all of that longevity risk onto a third party sooner or later, and 2) keeping a weather eye on the boundary where that option starts to be taken off the table by the circumstances that are only revealed by time.