First whack at cost of stochastic inflation in decumulation

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note: some errors have been corrected since initial post
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Ok, let's sandbag right away. I don't know if I got this in the right groove but let's roll anyway.

Using a life consumption utility model, and a simple model for auto-regressive inflation based on historical data, I ran a few new models today:

1. a baseline with deterministic inflation set to the mean of the random distribution I'll use for #2

2. a simulation using stochastic inflation that is bootstrapped from history (1914-2018) with a coefficient of auto-regression over 1 period of .64. (see the link above on inflation)

3. This scenario is the same as #2 except that the initial wealth is stepped up a bit (+25%) to get the life consumption utility up back towards #1 levels (not done analytically. Pretty much just eyeballed it). This is more or less like evaluating "certainty equivalent wealth" but I'm not totally sure about that so I won't precisely make that claim.

The goal was to determine how much extra wealth might be necessary at retirement-start to make the life consumption utility roughly the same as the baseline i.e., that is, either: a) we implicitly spend less...I realize that I have not really shown that here, or b) we have some amount of "redundant" (higher) wealth (vs the baseline) that is held in reserve in a way. Either way, same thing. I might have to expand on this because I know it's a little fuzzy.


A. Core assumptions

- Model is here 
- 61 yo start age
- 1M in initial wealth for scenarios 1 and 2, 1.2 for scenario 3
- risk aversion coefficient is 2 (my guess at mine)
- lifetime is random, conditional on age 61, shaped to SOA IAM data
- asset allocation is based on a stylized but representative EF
- inflation is from inflationdata.com 1914-2018
- the mean of historical data is ~3.2% which is then used for deterministic mode
- auto regressive coefficient is .64. See link at top
- subjective utile time preference = .005
- spending choice (loop, 4-5.5%)
- asset allocation choice (loop, 0->100%)
- spending is constant for this post only
- SS is ~15k at age 70 for life, infl adjusted
- no annuities or pensions
- returns are normally distributed
- probably forgetting something

B. Model run # 1 - Baseline/scenario 1 - Deterministic Inflation

X axis is allocation to risk, 1=all bond, 11 = all equity
Y axis is expected discounted utility of lifetime consumption in utiles
Each line is a separate spend rate from 4 to 5.5% in .005 increments
W(0) = 1M

Figure 1. Baseline Scenario

C. Model run # 2 - Scenario 2 - Stochastic mean-reverting Inflation 

Same description as above except:

- grey is scenario 1
- red is scenario 2, same mean inflation but randomized with AR[1] coeff = .64
- W(0) = 1M

Figure 2. changing inflation from deterministic to stochastic

D. Model run # 3 - Scenario 3 - Scenario 2 + additional 20% in wealth to goose utility score

Same descriptions as above except:

- grey is scenario 1
- red is scenario 2, same mean inflation but randomized with AR[1] coeff = .64
- blue is scenario 3: initial wealth is increased by 20% to get utility score up a bit to scenario 1 lvls
- W(0) = 1.0M --> 1.2 M

Figure 3. Increasing initial wealth by 25%

E. Discussion

• Optimal spend rates (say 5% for scenario 1&2, 4.5% for scenario 3) look a little high here, I think. Probably because of the relatively high level of SS income in late life compared to the initial consumption expectation. I'll have to look at that.
  
  
• There is a clear consumption utility hit taken when inflation goes from deterministic to stochastic in the model. Not surprising.
  
  
• An approximate 25% bump to initial time-zero wealth (I backed into it) kinda-sorta approximates the deterministic baseline inflation scenario.  In my opinion, this can be viewed two ways: 1) spending has to be decreased by x% to implicitly reserve for future inflation risk on consumption, and/or 2) more has to be saved at retirement-start in order to deal with later inflation uncertainty. These, I think, are saying the same thing. Then, if the high inflation scenarios do not unfold I suppose we can enjoy higher consumption later or perhaps higher bequest...elegant problems to have unless it is at too late an age.  I believe that the extra capital at the start can be considered a Taleb-ian antifragility kind of thing in the sense that it is redundant capital not otherwise needed except for the uncertainty around inflation-only in this post.
  
  
• Basically think of it this way. I'd need 25% more wealth at the start of retirement (in this first-look model. In real life I have no idea, maybe it's less or more) to make my life consumption approximately as useful as in a case where inflation were to have an entirely fake "no volatility at all."
  
  
  
• Interestingly, it looks like going from static inflation to stochastic makes a higher allocation to risk more optimal (this model only). What's the intuition? Equity risk as a lotto ticket in hard times? Probably.