Early retirement: inflation adjusted annuity vs. 2 SWPs

This is another quick and dirty spend game...

Assumptions

If we were to be willing to accept these assumptions (and I'm not yet):

1. we are 58 and we have $1M
2. we are potentially willing to commit all capital to a SPIA and cut lifestyle from, say, a hypothetical 4% spend plan to match the annuity cash flow, 
3. the annuity baseline is inflation adjusted and has a 10% load... 
4. our spend rates will be based on either the annuity or the players below
5. the portfolio return is 5% nominal
6. inflation is 3% where it comes into play 
7. the survival probabilities will be extracted from the SOA IAM 2012 table with extension
8. nothing is stochastic here
9. for some reason we trust my annuity pricing, and 
10. we refuse to do any formal utility analysis today...

Constraints

...and then we want to compare the annuity above to a systematic withdrawal plan where we are willing to accept these additional constraints: 

1. we never spend more than the annuity cf
2. we accept the logic of the SWP and will spend less than the annuity when the rules say so
3. we accept an extrapolation of RMD rate logic to before age 70

The Question

...then we can compare an annuity to a SWP and ask this question: 

"in incremental, present value, probability weighted terms, how much does it cost us to use the SWP vs. the annuity over a full life-cycle?"  

The Players

...where the players are:

1. ANNUITY -- An inflation adjusted annuity that is approximated by a(x) = sum (t=1:120)[c(t)*tPx*(1+d)^-t] where x is current age of 58, tPx is a conditional survival prob. extracted from the SOA IAM table for that age, d is an arbitrary but not totally unrealistic weighted average discount rate pulled from the current yield curve, and c(t) is the inflated cash flow "inside" the annuity formula.

2. RMD -- spend rate using the IRS table except that before age 70 there is a linear interpolation backwards to age 58 from 70 using the 70-80 "rate of change."  

3. RH40 -- is a spend rate I made up based on posts and a rationale you can look up. It is age and risk adjusted and the basic formula is: withdrawal = [age / (40-age/3)]/100.  I am parti pris...

The Chart of the game

I know I am abbreviating this game and post but it looks like this:

Red line = annuity cash flow

Green line = RMD plan with some backward extrapolation 70->58

dotted line = my RH40 spending rule

grey lines = conditional survival probabilites for a 58 and 85 year old (right axis)

orange triangle = mean life expectancy for a 58 year old

purple triangle = mean life expectancy for an 85 year old

black arrow is where the 4% rule would have put us...it fails somewhere between age 86 and 100

Some Considerations

The basic idea is that the 4% rule is not really sustainable over the long haul but the annuity is forever...but pays out less. Then the idea is that we maybe feel like we "hate" annuities and also feel like we want to replicate its foreverness by being conservative and spend like the annuity but accept some lower spending to make it work.  Here is a quote from Milevsky and Huang in  Utility Value of Longevity Risk Pooling: Analytic Insights:  

For example, when gamma = 1 and utility is logarithmic, the optimal consumption function c*(t) in equation (12) collapses to the hypothetical annuity consumption w/a(x) times the survival probability (tpx), which is clearly less than what a true annuity would have provided. The individual who converts all liquid wealth w into the annuity would consume c*(t) = w/a(x) for ever, but the non-annuitizer must continue to reduce consumption in proportion to their survival probability as a precautionary measure.[emphasis added]

So in the chart above what we want to know is the sum of the difference between the RMD (or RH40) spending and the annuity in present value terms weighted by the probability of surviving to the age in question.  We are not updating probabilities here but lock them in for the 58 year old. Also recall that we are accepting the annuity lifestyle as an upper limit and that our alternative spend plans are not exactly proportional to survival probability over the full lifecycle though the desire to be is in there somewhere.  

The Results  

I don't provide the data* or worksheets but the final results I came up with if we do it like this:

game result = sum[t=1:infinity (or 120-age(x))]( PV(annuity cash flow - SWP) * tPx ) where the positive value is the cost of the SWP vis-a-vis the annuity...

look like this:

cost of RMD = 54,180

cost of RH40 = 24,410

Conclusions and Caveats

1. This is a sloppy post --  first of all this is seriously deterministic not to mention casual and fast. Also the results would be hugely impacted but the stochastisity of returns.  This was just a "quick and dirty" post.   

2. RMD is popular in the press these days as a SWP method but that is mostly because it is: a) actuarially more or less sound, and b) it is well known and the language of RMD is not "threatening" or confusing to current retirees. But then again, as implied above, there are other ways to rope a horse...

3. The 4% rule, which is not obsessed upon here, is a an active risk accepting plan if I have not made that point before

4. The "cost" of the SWPlans is way less than I thought.  Before I probability weighted it the PV was coming up as something like 500k which was 1/2 the grubstake.  That was outlandish and confused me.  That was too much of a reserve for the future superannuation that may never happen. Weighting it made more sense.  The weighted result of 54k for RMD, on the other hand, at around 5% of capital makes more sense as a reserve for really old age. I can buy that.

5. The "cost" of the SWP is almost ALL made up of the up front cost of spending less now for later risk in weighted PV terms.  By that I mean that the total NPV of the difference between the annuity and the SWP in weighted terms is almost entirely made up of the conservatism of the first 9-15 years.  This is not a new point and has been made by smarter people than me more or less forever. 

6. Recall that we are not comparing anything against the 4% plan.

7. Would it be rude of me to point out that my RH40 rule yet again (in a cherry-picked scenario, perhaps) shows itself as a viable and efficient spend method...and that it is not all that bad at balancing both early lifestyle need vs. late age risk?

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* I have ONE reader (maybe 2) that cares about this stuff.  So, since this is currently a "self expression" blog, if you question the analytics in an abbreviated post like this...then email me and I will treasure the dialogue and provide the amateur rationale for what I am trying to do.