Buying some extra "certainty equivalent constant" spending given a wealth depletion framework.

See one of my last posts on stochastic vs deterministic models because this post is deterministic so it has some serious weaknesses but is not 100% without insight.  This post leans on the WDT game I built here and from which I inherit some structural model assumptions as well as parameters, the biggest of which we should note is that consumption is coerced to whatever pensionized income is available when wealth is depleted. It might be hard to understand this post without reading the prior work and even then this is pretty opaque.  The goal in this post can be stated like this: 

Starting with some assumptions around $1M in wealth and a baseline 4% spend rate for a particular level of risk aversion (say the coefficient is 2 or 3) for a 60 year old and a particular (fixed not stochastic) real return of say 2%, if I want to then consider an increase in spending in .5% increments, but I also want to keep my certainty equivalent consumption constant (with respect to what it was at 4%) each time I ponder a spending step up increment and I want to consider all of this over a full (fixed not stochastic) life-cycle interval (as defined by [60,T=100]), then: how much of a insured income payout, if any, do I have to buy at age 80 for each spending step-up to make that constant CE thing happen?

Whew...that may be the weirdest and longest framing of a question that I have done here at RH. I'm not even sure I know what I am asking myself (aren't blogs great) nor do I know if this is question is worth asking. Also there are so many variables to play with. But let's roll with what we have. This is in a class of what I want to call "holy grail" questions because it is a part of a quest to manage lifestyle up without blowing out risk at late ages when there is so much uncertainty about the future.  And don't forget: no bequest to the kids...yet. 

Assumptions 

Initial wealth = $1M, age = 60, utility planning horizon = age 100, initial spending is 4% (varied for the illustration) and inflated at 3%, there are no spending "inflections," real returns are 2% with a 1% penalty for taxes fees and vol etc, 15k (FV) SS starts at age 70 inflated at .03, a fair but loaded annuity is purchased at age 80 in amounts necessary to keep life-cycle certainty equivalent spending constant for changes in initial spending by way of tuning the payout to make that happen, and the CRRA risk aversion coefficient is 2 (and then 3 just for fun).

Results

The blue line represents how much of an annuity payout I'd have to purchase at each level of spending I am contemplating in order to keep the CE constant as compared to what it was at a 4% spend rate assumption (given all the assumptions and constraints above). 

The red line is for a risk aversion coefficient of 3 rather than 2 for the blue. The CE income is different for each RA value so there should be no real interpretation of the spread, just the shape of the curve.

Note that I could not go to 6% spend rate because the annuity purchase required to keep a constant CE was not feasible. We did not have enough wealth at the end of age 79 to make it happen. 

Conclusions

Based on the tiny world of this model I'd say that yes, in fact, you can buy your way into a higher lifestyle without totally shooting your self in the foot in terms of lifecycle risk*.  Right now I buy my own certainty by lower spending. I may have to reconsider that someday.  Also the ability to pull off this trick seems to vanish pretty fast for higher spend rates.  I mean really, I can't spend 10% with no risk and no magic? What's wrong with the world anyway? Outside of the "tiny world" represented here all bets are off. 


Post Post Notes.....
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This is the model for RA=2 and 50k in spending with  a purchase of an annuity payout of 21.5k at age80.  This shows the trickery of CE analysis. The CE was held constant vs the 4% spend but the timeframe over which consumption is diminished is now pretty high.  For the 4% spend the consumption (blue) was high for longer but also much lower than the green in this chart shows thereafter.  This makes it clear that it is at some point all about judgments about longevity probability. Something to consider...

That last comment was confusing so here is the 4% spend with no annuity purchase...

*So, I'd modify my conclusion like this: Screw Utility theory. I'll take my chances that I am not going to survive that long and that for the age range 84-97 I'll take back my higher consumption thank you very much. Who came up with this utility stuff anyway.  Or maybe I just need to figure out how to probability-weight Utiles.  That is on my list.