A prelim look at a real annuity and lifetime consumption utility - with a stochastic inflation model

Reason for the Post

This is almost the exact same post as the last one (A *very* prelim look at nominal vs real annuities and Lifetime consumption utility) except that I added the ability to make inflation (drawn from history using a table from inflationdata.com) stochastic and, in addition, the ability to stylistically model auto-regressive inflation using an approach offered by Brown, Mitchel and Poterba [2001] in The Role of Real Annuities and Indexed Bonds in an Individual Accounts Retirement Program. The stylized AR(1) process is described in the note in this post (Model Sensitivity: inflation vs lifestyle).

This post is not really research as such. Since modifying existing software is like updating the electrical in a seven bedroom 1911 house not touched for decades (I know whereof I speak and can tell you how long it took and how much it cost) we can consider this post a "software shakeout" rather than anything conclusive. How's that for sandbagging?


Base Assumptions

The assumptions are the same as before except that we have added randomized and partially autoregressive modeled inflation.  To save you the wear and tear of going back one post I will copy paste, with some adjustments, the assumptions that are partially redacted to protect the personal info I used for the shakeout:

- Age 61 and SOA annuitant mortality assumptions apply
- I used a nominal annuity that kinda defeases ~25% of spend. Costs ~16% of wealth in yr1
- Price of a real ann that is fit to nominal$ defeases 17% of (init) spending -- still 16% of wealth
- SS is in the mix and is, let's say, around 14% of initial spend. Infl-adjusted cash flow, though
- This is a model and looks nothing like real life... But you knew that, right?
- Ignore entirely that there is only one company in US with a real annuity
- The coefficient of risk aversion, for better or worse, of 2, more on that later...
- Spend rates are varied
- Asset allocation varies between 0% risk to 100% risk along an EF
- Note that the spend rates are a "% of wealth" after the annuity purchase
- Utility is lifetime additive CRRA and is ever so slightly discounted over time
- The nominal SPIA is calibrated to immediateannuities.com. and is ~16% of initial wealth
- Real annuity is calibrated to the SPIA/Principal price-wise but has

   quite a bit of a lower payout at the start date but then again it inflates with model-CPI
- 10k iterations per spend-allocation choice so close to 1M iterations to get through this
- The model for doing this, in more detail, is here and reflects the concept of a wealth depletion time
- Our stylized efficient frontier uses

     o a lower risk asset with r/sd = .035/04

     o a higher risk asset with r/sd = .08/.18

     o a correlation coeff, if I recall, of -.10

plus now...

- Coefficient of autoregression in the model is .64 based on Brown et al.
- Inflation data comes from inflationdata.com over the 1914-2018 timeframe
- Historical model has a mean of 3.2% and median of 2.7. Sd is ~.048

Output

I only have chart here since I have other things to do this afternoon. I might come back and add more. In this chart, visualize a table of the results of a simulated value function over a random lifetime, where the columns are the steps along an efficient frontier in 11 steps from 0% risk asset to 100% risk asset, and the rows are different (constant...sorry) spend rates. The lines in the chart represent the max{E[V(c)]} along each row for each scenario.  

Discussion

Pass.