Courage and the Annuity Boundary

The recent sell-off hasn't rattled me...yet...but it got me to thinking about something I wrote about a while back.  Managing a retirement is, to me, a "process methodology" rather than a one-time fixed event. That "process" involves management and monitoring concepts such as, among other things, keeping track of an "annuity boundary" above which (in balance-sheet wealth terms) we might not annuitize quite yet -- because the option value of not doing so might still be such that we can potentially ratchet up lifestyle more than we might have had we annuitized all wealth earlier -- and below which we might be at risk of permanently foregoing the lifestyle we might have maintained forever, given Methuselah-like DNA, had we just annuitized our consumption when we could have done so with the wealth we had when we had it.

That boundary idea above is a pretty neat concept.  But the sell off reminded me that, like other wealth management concepts, pulling the trigger on a strategy like the annuity boundary is not as simple in real life as it is on paper.  In this case, I was reminded that the speed of change in the environment we live in might be such that there could be psychological barriers to executing what we need to execute when push comes to shove in extreme circumstances.  Me? I seriously doubt that I could quickly and decisively and efficiently sell a large percentage of my net worth on short notice -- if the speed of the movement of my balance sheet towards the boundary is high -- to fund an insurance product that might or might not be the right thing to do at that exact moment.  What if I balked, paralyzed by fear and over-thinking?  Am I screwed or might I still be ok?

Purpose Of The Post 

My main goal here is not really rigorous analysis, it is to take a simple look at what would happen if my net worth were to sail through and below the annuity boundary and I then took my sweet time to make a decision ... and in the end I were to annuitize only some of my wealth with what I had left rather than me fully immunizing consumption -- with perfect timing and anticipation -- at the exact boundary when I might have looked like a genius if I had done so.  This will be a deterministic and shoot-from-the-hip analysis without full comprehensiveness because it's too much work to do otherwise and anyway I just wanted to see what the basic shape of what I am asking about looks like.


Some Assumptions and Setup - Scenarios

While the cognoscenti and academics will approach this kind of task with differential equations, I'm going to run through this with five different scenarios across two dimensions in an excel spreadsheet because no normal retiree uses differential equations.  Dimension-wise it'll be set up like this:

1. Returns: I'll run this with returns at: (a) a deterministic 5% nominal return and then I'll do a second run (b) with the first five years in a bad "regime" with 0% return and then 5% thereafter. Realistic? No, but I'm trying to make a point.

2. Posture on Annuitization:   (a) ignore it and let wealth run out as it runs out, (b) annuitize 100% of consumption at or just before wealth hits the boundary with perfect timing and anticipation, and (c) annuitize 1/2 of lifestyle sometime after wealth falls through the boundary but while there is still enough wealth to buy at least 1/2 of lifestyle.

The table of scenarios might look like this:

Some Assumptions and Setup - Key Assumptions

Inflation where applicable - 4%
Discounting where applicable - 4%, this includes annuity pricing
Annuity Loads - 5% which might be wishful thinking
Allocated returns - 5% nominal, 0% for first 5 years in "bad regime" scenarios
Coefficient of risk aversion - 1, which is debatable
utile discount - 0.5% which is also debatable
Initial Endowment - 1M
Initial Spend - 40k, inflated thereafter...
Consumption - Snaps to available income (SS and annuity cash flow) when wealth depletes
SS income - 12K inflated at age 70.  I can't remember where I got this assumption, but...
Age range - 60-95.  Annuity pricing and other longevity weighting things go to age 120

Some Assumptions and Setup - the math

1. I price annuities like this:

This effort will not get close to pricing it like the current market does. My goal is only to be internally consistent in my fake modeling world which I think I've done but who knows...  I use the SOA IAM table for mortality but extended to 2018. Note that I end up overpricing vs the market but that's ok inside this world because my math is the market.

2. Net Wealth Process

W(t) = [W(t-1) - consumption + SS + annuity CF - annuity purchase] * (1+ return)

3. Utility and Discounted Utility

I use CRRA utility but only for a risk aversion coefficient of 1 which means I only use log utility:

U[c(t)] = ln[c(t)] 

This assumption is thin...

4. Expected discounted utility of lifetime consumption

The "expected" part is moot because we are in deterministic mode but I calculate lifetime consumption utility like this in this post:

where tPx is the conditional survival probability, theta is the utility discount, and g is the CRRA function which in this case is ln[c(t)] for gamma = 1 as mentioned above.

Some Assumptions and Setup - the meat of the post

The evaluation of the cases in this post focuses on the value function in #4 above.  Consumption utility is the main goal here and can be observed, indirectly, in the green lines below.

Some Charts, by Case

Blue is the net wealth process in nominal terms - left axis
Red dotted is the annuity boundary in FV, age adjusted terms[1] - left axis
Green is consumption - right axis
Note: wealth drops when annuity is purchased
Note: spending snaps to income when wealth depletes


Case 1 - Normal returns, let wealth run out with no annuitization

Case 2 - Normal returns, annuitize at (or before) boundary

Case 3 - bad initial returns, let wealth run out and spending snap to income, no annutization

Case 4 - bad initial returns, annuitize at (or before) boundary

Case 5 - bad initial returns, annuitize < 100% (50%) of CF below boundary in some fuzzy zone...

Summary Table of Utility by Case Scenario

When we tabulate the sum of the discounted utility of the green (consumption) line we get this:

Conclusions

It's hard to make super-firm conclusions here but I guess we can say the following:

1. Annuitizing some wealth at the boundary in this fake world has higher utility than not.
2. Annuitizing wealth in this fake world leads to less jarring lifestyle drops at wealth depletion time
3. Bad early return regimes might force pretty early annuitization at the boundary
4. Missing the boundary is bad but not life threatening when we only annuitize 1/2 cash flow
    and it's still better than not annuitizing at all.
5. I still think I would have a hard time anticipating the boundary and pulling the trigger at the
   exact right time.  Also, I think that, even though this post doesn't really prove it, missing
   the boat but still annuitizing at least some wealth when under the gun is probably still
   going to be ok.



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[1] The boundary is based on the idea of buying an SPIA in year (t) that annuitizes spending in that year that has been inflated since year 0 and then the annuity CF is held constant (not inflated) thereafter.  This is flawed but simplifying... The goal is not really immunizing the consumption path but hedging out some portion of longevity and particularly that portion of lifetime when wealth depletes and one is left with whatever income is available. It would have been perhaps ideal to have an inflation adjusted income lifetime product but what little I know of the market is that it is more or less incomplete. But this post is abstract and stylized anyway so I (hope I) can ignore perfect products and complete markets for a minute.