Model Sensitivity: inflation vs lifestyle

In his recent Advisor Perspectives article (Which Annuities Offer the Best Inflation Protection June 2019), Joe Tomlinson, one of the more generous and helpful retirement writers now living, writes:

It turns out by pure coincidence that the 10th percentile for the COLA SPIA almost exactly matches the inflation-adjusted indexed SPIA. So there’s a 90% chance that purchasing the COLA SPIA will generate more real income than the inflation-indexed SPIA and a 10% chance it will do worse. This result is consistent with Blanchett’s findings. The upward-sloping median result for the COLA SPIA indicates that it beats inflation on average; this would be expected given its 3.6% annual increases in payments versus inflation averaging 2.3% overall. The 90th percentile beats inflation by a bigger margin.  

These results look very favorable for the COLA SPIA, but it’s worth cautioning that they are a direct reflection of my particular inflation model, which was built with considerable subjectivity. Higher future average inflation would reduce the advantage of the COLA SPIA and higher inflation volatility would increase the spread between 10th and 90th percentile results. The inflation-indexed SPIA has the unique property of providing a pure hedge against very high inflation, which is a material risk that every retiree must acknowledge and consider. [emphasis added]  

That last sentence is important so the investigations and the conclusions are important as well. Also, if we acknowledge that the inflation-indexed annuity form is the only decent way these days for a retail investor to get access to both a longevity risk pool paired with a pure inflation hedge, it is necessary, imo, to be at least aware of the idea of these products.


Models of Inflation and Purpose of the Post

I disagree with none of the conclusions in the article (yet) but I thought I'd at least take a look at the first sentence in the second paragraph about "a direct reflection of my particular inflation model."  All modeling is sensitive to the the form of the model.  To the extent that I understand how Joe modeled inflation, and he described it like this:

The modeling involves generating 10,000 25-year inflation scenarios. Each of the 10,000 scenarios involves random selection of average future inflation based on the above probability distribution [not pictured but see assumptions]. Then, based on inflation volatility since the early 1980s, I generate random year-by-year inflation assuming a 1.3% standard deviation.

then I'll propose three models for inflation for the purposes of this post and a threshold comparison:

A. T-model - 48k is inflated based on the above to the extent that I understand it correctly,

B. Historical Random - this takes historical inflation between 1914 and 2018 from inflationdata.com where we do a random draw from this table for each year within each iteration to inflate a "lifestyle" starting at 48k. Lifestyle times inflation is chained recursively over a circumscribed lifetime.

C. Historical Random with an AR(1) autoregressive process - Same as B but here we use a stylizing trick described in Brown et al. [1] to connect inflation over time. This has the modeling effect of broadening the dispersion of lifestyle (realized multi-period inflation) over time.

Comparison: 48k is inflated at a constant rate of 3.6%.

General Assumptions  

1. Inflation values for the T-model are .01,.02,.03,.04 with probabilities of  .10,.60,.20,.10. Period variance when needed within an iteration is 3%
2. "Historical model B" is from inflationdata.com with mean of 3.2% and median of 2.7
3. Iterations = 50k cycles
4. Longevity is 25 years to match the linked article
5. Lifestyle is 4000 per month or 48k per year, and would persist forever in this model 
6. Cost of living growth embedded in "comparison" lifestyle is 1.036 to match to linked article
7. Autoregression coefficient, where needed, is .64 based on Brown [1]  

Notes on the general assumptions:

- Spending and income are usually not the same thing. Or at least they are not necessarily the same, especially before wealth depletion, after which income would become a constraint.

- Wealth depletion would change spending by forcing it to income. This means that an evaluation of lifetime consumption utility would be useful here and would heavily influence product, allocation and spending choices depending on things like: age, risk aversion, wealth, lifestyle choice, longevity expectations, other income, etc.

- I am not really working with products such as annuities here. I am agnostic on products and I am speaking here only in terms of an hypothetical "comparison lifestyle" that starts at 48k and grows by 1.036.  Whether or not that benchmark can be fully or partly defeased over a full lifecycle depends on wealth available at time of a product purchase, the availability of more or less complete product markets to make an appropriate purchase, and resources and diversified income available from all sources thereafter especially in the late life-cycle, among other things. 

- To keep things simple I am working entirely in nominal terms here. "Real" effects can be inferred. I am temporarily avoiding real due to subjectivity that I sometimes find in the methods for discounting cash flows. 

The Output

Running the assumptions through the three inflation models and then: (a) illustrating the distribution of the lifestyle as driven by models A B and C at year 25, and then also (b) comparing it to the 3.6% constant-growth lifestyle, we get:

It's hard to see the blue line as blue but it's the middle one between the multi-modal T-model and the red AR(1). The red dashed vertical line is the value, at t=25, of the 48k inflated at a constant 3.6%. This red line might be compared, under certain relatively restrictive assumptions, to the "boundary income" produced by a COLA-adjusted annuity at t=25 with constant high inflation present over the years.  I think.

Discussion

- We can see that this basically corroborates Joe's comment "there’s a 90% chance that purchasing the COLA SPIA will generate more real income than the inflation-indexed SPIA and a 10% chance it will do worse" given only his inflation model assumptions and his 3.6% COLA construct while also using only a non-real and non-product-based model for the corroboration process here,

- We can also see that this basically also corroborates Joe's other comment "...results look very favorable for the COLA SPIA, but it’s worth cautioning that they are a direct reflection of my particular inflation model, which was built with considerable subjectivity."

- A real annuity would eliminate a lot of "risk" related to inflation relative to the COLA product but it would not eliminate all residual uncertainty. One would perhaps wonder about the insurer's corporate-balance-sheet-level hedging and their prospects for long term solvency under conditions that reach, or merely approach at a distance, hyper-inflation.

- The custom-blended 3.6% COLA product described in the article is probably "ok," depending on one's subjective outlook today about inflation...but it is not really a hedge for assertive inflation scenarios nor is it necessarily appropriate for a "less bounded" interpretation of how inflation might play out (i.e., the model problem).

- Just for reference, the constant rate COLA adjustment, using the AR(1) model for lifestyle dispersion, that would be at the 90th percentile of that dispersion is ~1.056.  The difference, maybe coincidentally(?), between the 90th percentile of historical inflation table (.0889) and mean inflation in the same table (.0325) also happens to be around 5.6%.


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[1] (The Role of Real Annuities and Indexed Bonds in an Individual Accounts Retirement Program [2001] Jeffrey R. Brown, Olivia S. Mitchell,and James M. Poterba). Their stylized AR process is described like this:

We consider two cases for the inflation process, corresponding to different assumptions about the degree of inflation persistence over time. The first case treats each annual inflation rate as an independent draw from our six-point distribution. This approach to modeling inflation tends to understate the long-run variance of the real value of fixed nominal payments and thus serves as a lower bound on the effect of inflation. Ourempirical findings in the last section demonstrate clearly that inflation is a highly persistent process.  

In the second case, we incorporate persistence by allowing inflation to follow a stylized AR(1) process. In the first period, inflation is drawn from the same six-point distribution as in the i.i.d. scenario. In later periods, however, there is a probability γ that πt+1 will be equal to πt and a probability 1- γ of taking a new draw from the six-point distribution. An attractive feature of this approach is that γ is equal to the AR(1) coefficient in a regression of inflation on its one-period lagged value, and thus can be parameterized using historical inflation data. Using U.S. historical data from the period 1926–97, the AR(1) coefficient for inflation is equal to 0.64, and this is the value of γ that we use in modeling a persistent inflation process.