Correction - My post on bequests and fairness...esp wrt the delta concept

1. My mistake

In my post "Some thoughts on bequests and fairness: a mini moral-theory story about me vitiating my optimal path - draft 1" on March 17, I mis-analysed some stuff in one section.  In section "B.2.2.2 -- Swap 2: Swapping a SPIA for GLWB product - using the pooling-delta method" I misapplied Milevsky's concept of delta in the first part of that section.  When looking at his table that had the annuitization "delta" cost for wealth that is already 99% pensionized, I mistakenly applied that delta factor to what would have been original wealth forgetting I would have spent all wealth to buy a "defective" annuity like the VA/GLWB product. Figuring out delta, in the presence of prior annuitization, would instead be applied to remaining wealth which might be zero or $100 or $100,000 or something; whatever is liquid, if I have that right, which is now suspect... I had mis-understood what I was trying to do and how to read his table. I'll try to correct this as best I can.

2. The proper way is likely still the old way

The old way of using a simple annuity pricing tool for pricing the cost of the diminished payout (GLWB vs Annuity) for which I would want to be compensated still stands. That was ~128,334 and will be a proxy for my lower bound "price" for abandoning a full life annuity for a GLWB. That is the value of the incremental cost of lower expected payouts.  There is probably some additional real return risk that happened to be discussed in links in the past post which might make the switching cost higher but is not addressed here. It would be subsumed in the "range" I am proposing anyway.

3. The Delta method did not really price the move from SPIA to GLWB

Using the delta concept presented in the  paper Utility Value of Longevity Risk Pooling: Analytic Insights 2018, I don't think I know how to price the move from annuity to GLWB or I haven't figured out a good way to present it yet.  If I decided to swap 100% SPIA for 100% GLWB (if I could and if all else were equal) there would be no wealth left to annuitize either way and I think this point about "delta" would be moot. In that case, the cost of giving up something or taking on new risk might really be better analysed using the "old" method; it might be conceptually easier to calculate and understand. Whether the GLWB is a good deal in that case and whether the freeing up of the bequest possibility for the vitiation of the expected payout and increased risk would be quite subjective I think, something that has more to do with moral judgments and risk aversion than I can capture here.

4. The Delta method Is still pretty slick for evaluating compensation required to not annuitize, generally

On the other hand, we can still use delta method to evaluate the compensation required to not annuitize at all as a conceptual upper bound for: a) a situation where there is liquid wealth with no pensionized income, and b) the same but with perhaps social security or a small pension factored in. There are other scenarios but are too much for me figure out right now.  I happened to do "a)" before by hand and got a delta of .279 yielding a "price" of 501k recalling that delta means "its DELTA percent of the money you have available to annuitize at retirement. So, if you have $100,000 and decide not to purchase the longevity insurance -- and take the chance of living longer than life expectancy -- you would need to be compensated with $100,000 times."  I did not do "b" because it is really hard to do.

5. I have a new tool!

But let's look at this topic again because we now have a new tool.  "b)" can in fact be done but it requires some algorithmic genius. I don't have that so I borrowed it from the technical appendix to Milevskys "Utility...." paper.  I don't know a public link yet. In there he has R-script for: annuity formulas, wealth depletion time estimation, and an iterative algorithm for estimating delta in the presence of prior annuitization.  It get's pretty hairy but it is implementable. [See note1].

When I implement the code and plug in the parameters from the old post (age=58, mode =89.95, b = 8.4, w = 1,798,528, r=.0301, gamma = 2) and then add a new parameter that I could not before (let's call it pre-existing pensionized income = 30,000 as a proxy for Soc Security and/or a pension) and I run this more formal code using methods that, while discretized, are closer to continuous than I'll ever get...I get the following answer:

[1] "Analytic Delta Zero"
[1] 0.2685505
[1] "Pensionized (Psi)"
[1] 0.2294185
[1] "Delta for given Psi:"
[1] 0.233

"Analytic Delta Zero" (26.9%) is the delta as if there were no pension income. This is the baseline definition of delta presented in #4 above and the definition used in the last post. This answer here is a whisker or two away from my hand drawn .279 in the last post so I'm feeling pretty good about that.

"Pensionized (Psi)" (22.9%) is the fraction of total (balance sheet) wealth that is initially pensionized. That "balance sheet" part of the phrase is important because the pensionized cash flow estimate is capitalized on to the balance sheet along with liquid wealth to come up with that psi number.

"Delta for given Psi:"  (23.3%) is more or less the same as Analytic Delta Zero except here a complex algorithmic testing of things like wealth depletion time and utility values goes on to come up with what is a more realistic "price" to compensate me for not annuitizing in the presence of pre-existing pensionized income:  .233 x 1,798,528 = 419,057.

Final corrected conclusion

So my corrected conclusion might be that the price for abandoning my SPIA quest might on the high side be 419k-501k if I am going from an SPIA-quest to a systematic withdrawal plan (SWP) with maybe a little pension income on the side. On the low side, using the old method and assuming we might abandon the SPIA for a GLWB product, the "price" might be 128k.  Not sure I totally have this down but that is probably a more correct way to set it up than I did in my previous post.





----------------------------------------------------

[1] Excerpts from the Milevsky and Huang Technical Appendix. This is not comprehensive just a flavor of the complexity of the enterprise... I'll provide a link when I can find a public one...

...

...