The WDT late-lifecycle retirement game now has these features:
Spending
- The spending process inflates and, optionally, trends
- The spending process is forced to the level of any pensionized income when wealth is depleted
- Spending can be "inflected," typically (but not necessarily) down, at two future ages
- A fair (loaded) mostly actuarially correct[1] annuity purchase[2] is allowed at age x. This decrements the net wealth process at x and is priced for age x using conditional probabilities
- Social income starts at selected age and then inflates
- No other starting income or human capital is assumed
- Net wealth process grows with a deterministic return and changes with (income - spend)
- The return generation process is not stochastic. This is typically a problem but (a) you can ding the return generation with a penalty for taxes/fees/vol and (b) you can change the assumptions to see what happens. Hint: it's not a linear effect.
- Note that this attempts, without thinking too deeply, to be a no-bequest model
- Age 60 is currently a hard coded start age
- Mean expectation for longevity is shown (by collapsing an annuity factor with r set = 0)
- For model simplicity the conditional survival probability = 1 [60:T] and then 0 [T:infinity]
- Annuity pricing uses the conditional probabilities from an SOA annuitant life table (G2-2018)
- For a utility horizon the model uses an arbitrary age 100 for T
- Maybe T should be stochastic but I wasn't in the mood to move towards simulation yet
- This is a hatchet job here but CRRA utility is used in this form: (c^(1-g))/(1-g) [see note 3]
- The certainty equivalent comes from mean horizon utility and then we back into CE via a transformation of the CRRA function. The validity of this approach is pending...
- For model simplicity (for now) the utility planning horizon is [60:T] all with prob = 1
- I have yet to figure out how to discount or weight CRRA. Any economists in the house?
- Discount rates have been split into inflation and subjective components
- Wealth Depletion Age is estimated by start age + WDT
- Visualized display of consumption, wealth depletion, and income from SS and annuity purchase...which was the whole point.
The Test-Run Setup
(note: all of this is illustrative rather than realistic)
- Start age: 60
- Initial Wealth: 1M
- Spending: initially 40k (4%), then inflates at 3%, no trend, inflects -10% at 80, and then finally forced to (annuity +SS) income at WDT
- Return generation: 3% inflation + 1% real - 1% penalty
- Social Sec: 15k (FV) at age 70, inflated at 3% thereafter
- Annuity: SPIA is priced/purchased at age 80 using age 80 probabilities, a 3% annuity discount factor, and a load=1.10. The purchase amt will be varied (in payout terms) from 0 to 50k in 5k increments (until no wealth left to purchase)
- Utility: aversion coefficient "gamma" = 2, planning horizon is to a fixed age = 100
- Longevity: using the assumption that for [60:T] the conditional survival prob. =1 and for [T:infinity] P=0; an "unhappy" assumption, as Yaari might say
The Results
For the trial run, I'll show only three of the steps:
- annuitization at 80 = 0
- annuitization at 80 = 25k payout (fv),
- annuitization at 80 = 50k payout (fv), and then
1. Step = 0 annuity
Consumption and income is on the left axis, net wealth on the right. Blue columns are spending, green is (SS+annuity) income. The dark blue line is the net wealth process. The red bar is mean longevity; the black bar is the planning horizon. A mortality PDF for age 60 based on SOA IAM table with G2 extension to 2018 is provided for context. Note: (a) the blue spending inflection at 80, (b) the start of green SS and the related bend in the net wealth process at 70, (c) that there is no annuity income in this step, and (d) that wealth is depleted at 92 where spending drops to Soc Sec.
2. Step = a 25k (FV) annuity payout is purchased
Note: (a) wealth depletion is now 93, (b) the green bars reflect (i) a step up in income from the annuitization of some of the wealth and (ii) there is higher income than before in the WDT interval, and (c) the net wealth process has a steep inflection down from the annuity purchase after which there is a new slope in the line (which confirms my original hypothesis weeks ago).
3. Step = a 50k (FV) annuity payout is purchased
Note that: (a) wealth depletion is now age 80 (it's not shown but between payout = 0 and payout = 45k, WDT rose from age 92 to an infinite WDT but in this latter case wealth was "an infinity of tiny wealth"), (b) it's hard to see it but if you look closely you can see that income and spending now exceed the original (post-age-80 spending) "plan" that had been a 10% cut from the original 40k.
4. Summary = certainty equivalent consumption over the horizon for misc annuity purchases [3]
Note: (a) the rising utility for insuring consumption (ok, insuring income...) which seems to show the theory of lifetime utility vis-a-vis annuities is correct (or at least it is in this amateur post and model) and (b) the utility pops up at the right. For (b) I have the urge to claim some economic phenomenon is at play here and certainly we could maybe point to the gain in consumption (income) over and above "the plan" by way of the full annuitization, but I have to also acknowledge the coding errors and shortcuts that, on close inspection, might have a big role to play as well.
Conclusions
It's hard to make conclusions on a trial run of a prototype like this but I'd maybe say that if I had a triad of the following:
1. A good grasp of the analytic expressions of WDT to guide my understanding and interpretation
2. A fully simulated version of this to breath some life and robustness into the model, and
3. Something like this spreadsheet -- just to play around with visually
...then I'd have a pretty good planning toolset to add to the RH tool belt. I already have #2...except for the utility calcs, the fair annuity purchase, and the coercion of spending towards income. But that wouldn't be that hard to fix.
Notes
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[1] Disclaimers and admissions...
There are errors and omissions, of course. I am neither an academic nor a practitioner
I haven't explored vehicles other than SPIA (e.g., DIA, VA with GLWB, SWPs...)
This is pretty deterministic and so misleads
I haven't figured out discounting and weighting of utiles yet
This is a no-bequest model. Yaari adds a "weighted" utility of bequest to the value function
The annuity pricing is "fair" and pretty close to actuarially correct but not perfect
I'm sure there are other issues TBD
[2] model assumes inflation adj annuity
[3] This is my attempt at some of the notation.
The value function is this, if I got it right
Links
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Coming soon -- The Wealth Depletion Time Game?
- Wealth Depletion Time - Commentary
- Is this an easy "Wealth Depletion Time" analysis method by proxy?
- Wealth Depletion Time - Selected Excerpts
- Wealth Depletion Time - Adding again to my pile with Leung '94
- Wealth Depletion Time - an Hypothesis and a Self-Challenge
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