Try #2 on a rough validation of a 3.6% break-even inflation rate for a real annuity

Before my vacation to MT, I briefly proffered a post that tried to validate Joe Tomlinson’s guess that the current market (ok, it’s only one company and product) for inflation adjusted annuities suggests that there is a “breakeven rate” of around 3.6% inflation when considering those products (Which Annuities Offer the Best Inflation Protection, June 2019). I happened to retract my post because I thought it looked and sounded innumerate and even if it wasn’t innumerate it was at least mixing quantitative apples and oranges and that, at a minimum, felt embarrassing. I also used the wrong name for the insurance company that offers the inflation adjusted annuity. But, I *am* still interested in things like inflation, lifetime income, and consumption utility over a full lifecycle, especially in the decumulation stage. 

A. The Basic Setup and Assumptions
Now, let’s take another look at this thing, and this time I’ll try to keep the same "fruit" in the same basket. And this time we’ll do several things to be consistent:

1. use a standard formula for pricing annuities*

2. Price, using market observables, some simple SPIAs**

3. Price, using market observables, one CPI-adjusted annuity**

4. Assume, without evidence, that the simple SPIA has a load of 5%

5. Solve, using the formula in #1 and price from #2, for a discount rate (R[n=nominal])

6. Solve, using the formula in #1 and price from #3, for a discount rate (R[r=real])

7. Estimate one insurer’s inflation estimate by way of {R[n] – R[r]} for different {R[r]} loads*

* The annuity formula I’m using looks like this:

where the process here is to solve for different "R" (implicit discount rate) given A(t,x) (the annuity price), cf (cash flow), and tPx (conditional survival probability) and also given different levels of “l” (insurance company load). The goal is to gauge the spread between R[n=nominal] and R[r=real] as some type of indicator of insurance company expectations for inflation...depending on one’s own expectations for insurance loads.

** the assumptions for the SPIA and the CPI-adjusted product are, in general: age = 61, an arbitrary 4000/mo in income starting immediately (7/20/19), nominal cash flow for the SPIA and real for the CPI adjusted, state of FL for residence, a toxic male annuitant, etc.

B. Market Observables
1. The SPIAs, given the assumptions above in the assumptions and using immediateannuities.com ranged from ~806k to 866k with an average of 829k. I will used the low end of that range for this post on the possibly misguided assumption that the load at that level will be the lowest and closest to my own assumption of 5% for the nominal SPIA. Who knows? TBD...

2. The CPI-adjusted annuity, using the same general assumptions came from from Principal (not Provident). It was extracted from immediateannuities.com as well. The price quoted, given the assumptions, was $1,243,285 as of late July 2019.

3. The conditional survival probabilities for a life conditioned on achieving age 61 (and that is an achievement...) and male, for individual annuitants, was derived by way of using an SOA table available on the SOA site.

Loads are either asserted or variable in my test. Cash flow is flat out asserted. 

C. The Output
When we use assumption #1 and the formula in #1’s note, and when we look at loads for R[n] of 5% (which I’ll assert for convenience) and loads for R[r] of 5, 10, and 15%, then we get, for the given market observables in B, a solved-for R[n]-R[r] of the following:

Load R[r]         R[n]-R[r]

5%                   3.61%

10%                 3.27%

15%                 2.93%

Discussion
I don’t know if this confirms Joe’s 3.6% expectation or not but at least we are in the ball park. For at least one of the CPI-adj load assumptions (5%) we are at something close to 3.6%. Whether they, the insurance companies, price it exactly like that or not is unknown. I’ll call it close enough for now. The basic idea for Joe's article, and for this post, is that right now, in mid 2019, if one were to buy a CPI-adjusted annuity (from at least one company), then an inflation expectation, over the long haul, of more than 3.6%, for at least one parameterization, looks like it makes the CPI annuity a better deal than not. Caveat emptor, though, I guess we should say.