A quick peek at consumption utility with a deferred annuity

SUMMARY 

• [note -- this is a casual and mostly non-rigorous post]  

• It's been well known before I ever came along, that risk pooling can enhance consumption utility by hedging out longevity risk and shifting it to the pool or insurer.  This post is doing a quick drive-by to see how much spending might be nudged under some narrow, artificial and simplified assumptions using my lifecycle model with a deferred annuity (DIA).  

• Last year I estimated, in another casual post, that if one were to try to keep risk the same (i.e., holding ruin risk constant in a ruin-risk-based simulator at that time) when hedging out a minimal level of age 85+ consumption with a DIA, one could increase spending ~14% (the way I did it then). This time, using the lifecycle utility model I recently built, and hedging out some of age 80+ consumption with a nominal DIA representing a "consumption floor," one can, under some generic, arbitrary, and very simplified assumptions, perhaps increase spending between 10-20% (or more) while keeping "discounted lifetime consumption utility" the same or higher. This is apples to oranges, of course, and probably optimistic, but the general magnitude still makes sense and is consistent with the results from last year.  

THE POINT OF THE POST

This is a drive-by look-see to see what happens in a lifecycle utility model when some portion of wealth is allocated up front to a DIA intended to support very late age (if any) consumption.  I had done something similar last year with a standard Monte Carlos sim and this question, reposed to myself this year, was a chance for me to add some code to my WDT-utility model to be used for working with nominal streams of income in addition to what I already had. In this case I was thinking about DIAs or pensions.  In this post, I'm taking one tiny set of shoot-from-the-hip parameters and taking a quick shot to see what happens.  I realize that this kind of DIA analysis has been done before quite often. I just wanted to shake out part of my model for some new features and to do a superficial validation of the spend increase benefits of hedging longevity that I had tried to do last year.



ASSUMPTIONS AND BACKGROUND

The life-cycle model is here. To understand what I am doing in the simulation and how I am running the value function, you'd have to click through to take a look.

The assumptions are mostly the same as this post. I'll try to be clear if I do something different.

The basics: age 60, $1M endowment, SS of 11k at age 70, spend rates are varied in the test but are otherwise in a constant process (defended here). Risk aversion varies. I'm working with a 60/40 allocation model for now with an 8%/3.5% return profile that may be unrealistic in 2018. Returns are normally distributed in this run.  No annuities other than the DIA.

Consumption utility in the model is evaluated in terms of real spending but spending is run out in the sim in nominal terms first so the DIA is sized to match a nominal "policy floor" at age 80. For today I'll call the policy floor 1/2 of the nominal spend rate at age 80 (arbitrary for the look-see) for a 4% constant plan (even as we vary spend rates). For a $1M portfolio that would be a 20k (real) payout starting in year 20. Since the DIA is constant I realize it loses purchasing power over time but so be it.

The annuity is priced three ways, once by me with math in the links above and then also with immediateannuities,com and with aacalc.com.  All three gave the same answer within tolerances that were ok for this wing-it run.

The DIA price for the given setup was almost 14% of the endowment at time-zero which was decremented for those scenarios that purchased a DIA.

There are a ton of elisions in this drive-by: taxes, fees, unexamined parameters, and so forth. Be wary.

THE PROCESS

1. The lifecycle model was run for each spend rate from 3% up in .25 increments.  The coefficient of risk aversion was set to 1.  The output is expected discounted utility of lifetime consumption. That result was left in discounted utiles and not converted to a certainty equivalent.

2. The model is then adjusted for the purchase, at T0 of a DIA that will pay constant dollars at T20. Inital wealth is decremented for the estimated purchase price. The coefficient of risk aversion is set to 3 and then Step1 is repeated except for the risk aversion.

THE OUTPUT

Here are the two charts I ran, one for a risk aversion coefficient of 1 (let's call it low) and one for a coeff of 3 (let's call it high but I hear that some consider it moderate).  The blue line is before the DIA purchase. The red line is after.

Figure 1. WDT with DIA, gamma=1

Figure 2. WDT with DIA, gamma=3

Point A is the original "optimal" (grain of salt...) spend rate for the given risk aversion. Point B is a local optimum for the value function with the DIA. Point C is the level at which the value function was maxed without the DIA in the original base case but now projected onto the new with-DIA line.

For gamma = 1

 - base spending was best at ~5% (not a recommendation by the way)
 - with-DIA, optimum is ~5.5%
 - keeping base utility the same spending could go as high as 6% (given the casual assumptions)
 - spending could go up 10-20%, keeping the value function constant or higher than the base case

For gamma = 3

 - base spending was best at ~3.5% (this has been a budgeting level for me)
 - with-DIA, optimum is ~4%
 - keeping base utility the same spending could go as high as 4.75% (given the casual assumptions)
 - spending could go up 14-34%, keeping the value function constant or higher than the base case

CONCLUSION

The results here are not surprising. I have seen a few papers show the same (or similar) thing. I also had vaguely similar results last year.  My guess is that I am over-estimating the benefits a bit here, while also ignoring a lot of assumptions that need to be explored. But in terms of a first run shake-out, it was interesting to see it work in a way that I had been expecting. The biggest question coming out of an analysis like this is why I do not allocate to a DIA.  I've been thinking about it for years and it seems self-evident every time I take a look.  My guess is that it is a combination of behavioral issues (ok, not issues, let's call it obstacles) and a perhaps misguided view of the "option value" of waiting to annuitize.  That concept my not apply to DIAs but I'd have to read up on that for a bit.  I guess I am an annuity-paradox poster child.  Also I feel like I am the risk-aversion poster child as well.  Gamma=3 seems to be me.