3D "utility surface" in a Wealth Depletion Time model: spend rates vs annuity

This is at least temporarily the end of the road for my WDT (wealth depletion time) model.  While this is still a "play" thing and not really "real" analysis yet, the pictures I derive here below were always my end-game except to the extent I might extend it to my own data some day.  The goal in this post can be stated like this:

The Goal of this Post

If we were to calculate the discounted (and weighted and deterministic) utility of lifetime consumption for a consumption plan that anticipates being reduced to available income  when wealth runs out (income that can be purchased with available wealth while it lasts), what would it look like if we did that utility calc for different combinations of initial (constant) spend rates and fair (but loaded) increments of "annuity payouts purchased" (first at age 80 and then another look is done by doing it at 75) in a rudimentary WDT model with fake-ish data?

That is a big awkward bite. So let's roll with it and see what happens.   Since this is a complex question, we should linger on the assumptions and model rather than merely point back to other posts. The concluding "utility surface" looks cool but don't jump ahead yet.


1. The Model Structure

The model is my Wealth Depletion Time Game. This is a spreadsheet lifecycle model that projects income, spending, and net-wealth paths from age 60 to 120.  It has, among other things, at least three notable features:

1. consumption is forced to available income when wealth is depleted, 
2. actuarially fair (or as fair as an amateur can do it) annuities with an insurance load can be purchased up to total wealth at any point along the way, and 
3. there is a value function that evaluates the [deterministic; no random lifetime] discounted utility of lifetime consumption using a CRRA (power, isoelastic...) utility formula. (one ring to rule them all) 

That's thin so let's look at the retirement processes in the model or what I call "zones:"

A. The Utility or Value Function Zone

The model uses a constant relative risk utility function though readers may recall that I am not a fan of utility models and math.  I do it here for: a) self education, b) because it seems to be a standard approach across the ret-fin space, and c) because the WDT math in the literature seem to go there too. Received wisdom is now officially received.

The  CRRA function will be in this form:

where c(t) is consumption at time t and gamma is the coefficient of risk aversion.  The literature is all over the place on what a realistic gamma might be but there is some centering around "3."  Or 4? It depends. Being a rebel, I use 2 as my standard...risk be damned.

The value function for discounted utility of lifetime consumption will be in this form:

Which is an amalgam from multiple mostly credible sources not listed. tPx is a survival probability vector for a 60 year old male extracted from the SOA IAM table with G2 extension applied. I hope to god I got that right.  g[c(t)] is as above.  Theta is part of the subjective discount factor. In this model I keep it at a modest .005 based on nothing other than Gordan Irlam's advice to me. But I do want to preference the model with something for the younger ages when mobility and cognitive abilities will appreciate the consumption now more than later.  t is the years rolling by.  T is the furthest possible lifetime end.  I call T = 121-60.  So: discounted (theta and tPx) utility (g) of lifetime (T by way of t) consumption (c).

Side note: there is an unshown and vestigial "certainty equivalent" calc that I do on the mean of un-discounted utility g[c(t)]. For that I still use a hard terminal end of life estimate. For this run, if I happen to use it, that T(h) value will effectively be to age 95.

see [1] Notes on WDT and utility

B. The Income Zone

Human Capital - There is no initial income or human capital in the model (yet).

Dividends - I do have a feature that splits total return into a dividend stream and a growth process ding-ed for the missing dividends but that is not used here.

Social Security - The model allows a start date and a start amount (fv) and then inflates it at some rate which, unless otherwise noted, can be assumed to be .03.  12k in income (fv) starting at age 70 was used.

Annuities - Annuities are priced in the model like this:

The notation is off re this post and the lack of brackets is confusing but it is basically a loaded annuity formula where R is the "annuity discount" and the iPx (or tPx above) is a conditional survival probability vector for an x year old extracted from the SOA table mentioned before.  For the model the annuity income can be purchased whenever but I chose age 80, then also age 75.  The probabilities are recalculated for the selected purchase/start age. The load is 10% and I may have added some uncertainty to the load but I'd have to look.  The summation is not really to infinity but to age 120. The purchase of the annuity at time t = a(x)(t) decrements "wealth at time t" = W(t) by the amount a(x)(t) subject to a(x)(t)  <= W(t).  The annuity income is inflation-adjusted subsequent to purchase. I have normed this calc to other annuity pricing methods and calculators in the world and it works pretty well as far as I can tell. A current blended treasury rate used for the annuity discount is currently a skitch over .03 so .03 is used. This is imperfect stuff but ok for illustration.

C. The Spending/Consumption Zone

Basic Spending Process - Spending starts with an initial state (at age 60) that is a parameter or variable in this post. We will vary it from 3% to 5% in .01 increments. It is inflation adjusted, so constant in real terms.   There is a feature in there to "inflect" income up to two times per lifecycle -- either up or down by x% at some age between 60 and 120.  This feature is not used in this post.

WDT Process - The whole point of the model is to cram spending down to available income when wealth is depleted.  This is what gives us the differential utility and a vague air of reality.  A lack of income during wealth depletion would blow up the model so we assume something > at least 1$.

So, to recap, spending clips along until wealth runs out and then goes to the income level that is available at wealth depletion.  At any point before depletion an annuity can be purchased that will (a) reduce wealth at that time, (b) increase income at that time, and (c) change the amount of time spent at the original spend rate and extend the amount of time at the post depletion income -- but the purchase has to cost less than available wealth.

D. The Net Wealth and Return Generation Zone

Returns -- returns are deterministic. This is an "unhappy" choice to some but I justify it here in this post (Stochastic vs Deterministic - Part 2). Return is nominal but broken into inflation and real.  If pundits say real returns will be 3% in the foreseeable future, I use 2.5% to account for some volatility effects.  I also ding the returns with a penalty for taxes, fees, and other. Don't worry about the realism because we are not making real judgements, this is still only illustrative.

Sequence risk fudge - I added a rudimentary non-random sequence risk factor.  For the the "hard life end = 95" we split that period in two and then in half-1 we decrement returns by a factor and in half-2 we increment them. I think I had .005 or .001 in there at the time I ran it.  The geometric return is unaffected over the long run. This is another conservatising factor that may be redundant with the penalty for volatilty but that's ok for illustration.

Initial Wealth State - We start with $1M

Net Wealth Process - Net wealth (NW) in some period t is = [NW from the prior period + (income from all sources - consumption - annuity purchases)] * (1+(modified(r)). Note that the income and consumption occurs at the beginning.


2. Model Assumptions for the Illustration

Have I mentioned that this is illustrative only? Good. Some of the following is dodgy but I just wanted to get to the end charts.  The inputs for this post are as follows, and may have been mentioned above. These are mostly accurate except in those cases where I lost track of what I used and accidentally saved a file over what I had done. 

Age. start = 60, end = 120
Wealth. start = $1M
Spend rate. variable: .03 to .05 in .01 increments, growth at .03., forced to income at WDT
Spend inflections. not used
Returns. .045 nominal (.025 real), .01 penalty added. Simple seq. risk modeled to age 95 +/- .005
Social Security. 12k (fv) starts at 70
Annuities. Priced for age [75, 80] in 10k increments from 0 to available wealth. Start = [75,80]
Annuity Load. 1.10 + x
Risk Aversion. as needed = [2,8]
Subjective utility discount. .005
Lifetime. Probability weighted. If a hard date is used it is = 95. Not stochastic here.
Inflation. .03


3. The Output  

When all of that exhausting stuff above is run through its paces we get the following charts. First up is the variations of spend rates and the annuity purchases that are made at age 80; the second run is for annuity purchases at age 75.

If I got any of the above wrong through coding errors or deep naivete, then all of the following will be terribly embarrassing. But even it is fatally flawed to the roots, I thought that the aesthetics of the 3d charts I came up with were still compelling and when, really, is the last time you heard someone talk about the aesthetics of retirement finance? That's right. Never. Until now.

Annuity Purchases at 80 - RA = 2

Annuity Purchases at 75

Annuity Purchases at 80 - RA = 1/2

Some disclaimers

- If I haven't sandbagged before, I'll do it here again.  I think I got the math and the mechanics and the model and the implementation right or right enough. But I'm not totally sure. I'd need an economist and a 15 year old coder in my back yard and I don't have either.

- This is scale sensitive and also I do not have confidence in utility.  I really have no idea how big of a real difference there is in increments of .00005 utiles.  In real life, I think none. Maybe.  In fact if the risk aversion is raised even a little bit the chart literally goes flat.  That means either I messed up the math, don't understand utility, made a coding error, or something else and or all of the above.

- The model has a ton of short cuts and over-simplifications. I doubt this is useful for real applications to real retiree problems. (yet)

- the output does not reflect anything real. It describes, rather, the shape and boundaries of what is in some rows and columns of spreadsheet. That limits how far this can be taken seriously.

- No bequest utility so that's probably a flaw.

- Other.  Always other.

Conclusions, such that they are...

1. Just step back and look at the pretty colors.

2. This corroborates an approximate 4% spending "range" give or take. The peak in the first chart is at  4.2% spend rate to age 80 and then a 50k payout purchase at 80 that depletes wealth.  That was about a -10% shift in lifestyle. The peak in the 2nd chart is at 4.6% and then a 40k payout purchase at 75 that depletes wealth. That was about a 30% drop in lifestyle; but then our spending when young was decently high. The peak, when I ran a lower risk aversion (.5) is high spend-low annuity which makes sense I guess.

3. Annuitization seems to buy utility as the theory would suggest. Even with lower risk aversion.

4. I have no real bead on real maxima.  I'm sure there are recursive bellman dynamic programming methods for that kind of thing. I don't have that. yet.

5. Spending more than 4% means the plan is way more sensitive to the need for buying income although that gets muted at low risk aversion.

6. The calculus is interesting. Walking through the cube (for the first two charts only) and starting at a spend rate of 5% with no annuity and then going down in spending (to the left): at 5%, the velocity of utility for increments of annuitization is high  (U' is steep) and the acceleration decreasing (U'' <0). At 4-ish percent the velocity is moderate and the acceleration is stable and slightly diminishing.  In the 3% range, velocity of U in terms of annuitization is relatively low (U' is flat) but is also accelerating (U'' >0). Not sure how all that interprets yet in real-world.

7. In the absence of annuitization the optimum is < 4% except for the lower risk aversion chart.

8. The higher spend rates clip their own wings in terms of how much annuitization can be bought as can be seen where the surface ends as the annuity purchase increments go up but is a viable part of the game since high utility can be bought.  And the "high spend rates without annuitization" appears to pauper itself because it guarantees a long wealth depletion time with nothing but social security.

9. Low spend rates have lower utility especially in the young important (to me) years but can dramatically upgrade themselves in utility terms (acceleration) when the income is bought and for which there is a decent reserve at late ages due to the precautionary savings.

10. I'm not sure how I'd do it but adding a utility of bequest would be interesting. It would certainly have an impact on the low-spend-high-precaution scenarios.


Other Stuff

This is the process chart for the first surface chart above

----------------------------------------
[1] Notes on WDT and utility

This is from "Approximate Solutions..."

This is from "The Utility Value of Longevity Risk Pooling..." Page 8. Note the split utility.

Which reduces further to...

Which makes Milevsky's case that knowing T(d) is both subtle and interesting in the context of utility math as well as bolstering my interest in my own self-challenge.