A *very* prelim look at nominal vs real annuities and Lifetime consumption utility

I was updating my software for simulating (expected discounted utility of lifetime consumption) and I thought I might as well make it work for a really basic comparison of nominal vs real annuities. I thought it was in the SW but the original amateurism appalled even me.  The mods were really really annoying, though.  I am totally convinced that I should never ever modify software again but do EVERYTHING from scratch.  It's like remodeling houses.

Now, here, I won't give too much detail cuz I don't want to show my own hand on personal data. Mostly this kind of thing uses personal details, though.  Let's at least assume some of the following


- Age 61 and SOA annuitant mortality assumptions apply
- I used a nominal annuity that kinda defeases 25% of spending, costs maybe 16% of wealth in yr 1
- 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.
- Inflation is NOT stochastic in this model, a flaw, and I'm working on that... set at 3% for now
- 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 (the different lines in figure 2)
- Asset allocation varies between 0% risk to 100% risk along an EF (illustrative only[1]),
   Y axis and figure 2.
- 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 is
      lower payout at the start date (~30 % lower) but then again it inflates with model-CPI
- Real annuity was fully priced using immediateannuities.com and the Principal ins co.
- 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

I know I am leaving some explanation on the table but...


Y axis here in Fig 1 is the cumulative additive lifetime consumption utility for the max spend rates across all allocations to simple, illustrative risk (contrived frontier) for different spend rates...

Figure 1. EDULC for real vs nominal annuities for "my assumptions"

Y axis is the same as figure 1 but here we see the EDULC for different allocations to simplified risk for different spend rates (the different red lines) for a fake real annuity (red) compared to different spend rates (the different grey lines) for a fake nominal annuity.  1 thru 11 are allocations to risk where 1 is 0% and 11 is 100% allocation to risk.

Figure 2. EDULC for diff spend rates and diff allocations to risk, real vs Nom annuitty...

I'll look at this more later, esp re stochastic inflation, I but wanted to get something down on paper for now...

Discussion

no real discussion here. But if I were to say anything, I'd at least say that a real annuity "wins" here even thought the initial cash flow is less...





[1] like this, for now...
r = .0350, .0395, .0440, .0485, .0530, .0575, .0620, .0665, .0710, .0755, .0800
sdv = .04, 0.0386, 0.0457, 0.0583, 0.0736, 0.0902, 0.1076, 0.1254, 0.1434, 0.1616, 0.18