Almost everything I need to set this up was already done in Lifetime Consumption Utility "Frontier" and DIAs - With Different Deferral Periods on June 23. Start there first.
The purpose here is to take the exact same set-up as in the link and now find out how much initial wealth I can annuitize (using fake in-model nominal SPIAs as proxies for access to a "risk pool") in increasing increments. I want to see if or where the optimization benefits peter out when adding lifetime income...knowing that the "theory" says you are supposed to be able annuitize all wealth for optimality. We'll see.
Theory References
There are a bazillion references on this. I'll mention only two, however, and quote a salient fragment from each.
1. Yaari, M. (1965), Uncertain Lifetime, Life Insurance, and the Theory of the Consumer
"In Case C the consumer has no bequest motive but he is constrained to meet the requirement that his transferrable assets at time of death (i.e., those assets which become a part of his estate) should be non-negative with probability one. It turns out that this case is quite simple to analyse because the consumer's assets (or liabilities) will always be held in actuarial notes [i.e., more or less "annuities," but see Prof. Milevsky's comments below] rather than in regular notes."2. Yagi & Nushigaki (1993), The Inefficiency of Private Constant Annuities
The consumption stream without any constraints [by which I assume they mean “case C” where wealth is held in actuarial notes], which Yaari derived, is considered to be the first-best optimal consumption stream in an economy where insurance is available. The consumption stream with constraints diverges from the first-best consumption stream, and this divergence can be seen as a distortion created by the constraints. In other words, the constraints imposed by the insurer and the imperfect capital markets create a loss in efficiency.Divergence from Theory
The theory reference is important here because my post diverges from the famous conclusion in #1 and I need something like #2 or other reasons to to justify my results although it is probably fairly intuitive to understand the divergence at a superficial level. We might set it up like this. If my results differ from the conclusion in #1, we might say one of the following:
a. I'm a genius and have discovered a ret-fin version of cold-fusion,
b. My software sucks and I have mis-coded due to shortcuts or oversights or error, or
c. I have mis-understood the economic theory or I am naive.
If we were in Vegas, b and c would have good odds. a? Not so much.
On its face, the divergence is easy to explain. Here are some of the first reasons I could come up with:
- I have a simplistic model that lacks the sophistication and hard math necessary to get it precisely right,
- I am using fake nominal SPIAs that, even though fabricated, are still plausibly available in the current product market where Yarri's notes aren't. Yaari, as Prof. Milevsky reminded me in private correspondence, was working with effectively real (not nominal) but also "continuous time" actuarial notes or what Prof said were "really instantaneous tontines." Going further: "Needless to say, such an insurance product doesn’t exist (and never will) and is often touted as yet another reason that full annuitization isn’t optimal in the real world."
- I have just one parameterization (one age, one fake efficient frontier, one risk aversion coefficient, one inflation rate, etc...) not many, nor am I working at the level of an entire economic system.
- As mentioned it's not all that unlikely that I have made errors and omissions.
- I'm sure there are other reasons.
Assumptions and Background
See Lifetime Consumption Utility "Frontier" and DIAs - With Different Deferral Periods. Every main assumption is already stated there. A reminder: I was using assumptions linked to me personally so that absolute assumptions on wealth or spending are redacted.
What is different here is the following:
- I am working with only SPIAs (and no DIAs) that are "purchased" before the model is run. Increments of wealth used to purchase the annuity are: 0% 10% 20% 30% 40% 50% 60% 70% and 80%. Note that we were playing with 10% and DIAs in the last post. We are still allocating, though, between these insurance products and a hypothetical three-asset-class illustrative-only portfolio with a representative (illustrative) efficient frontier.
- Spend rates in the last post were presented relative to liquid wealth remaining after the annuity purchase. That was moderately ok (but probably not really ok) when the allocations to annuities were 10% of W. Here it makes less sense to do that. So, I have norm-ed spend rates in this post to what an actuarial balance sheet would look like at plan-start. In other words, net worth should be the same before and after the annuity purchase and so the spend rate is rendered here as a % of net-worth with the present value of the annuity cash flow (effectively the price) part of the asset side of the ledger. This means that when the allocation to annuities is "80%*W," the "balance sheet" spend rate might look like 4.4% in this post but that rate against the remaining "liquid" wealth at plan initiation is more like 21%. This is expected since in the presence of a high level of annuitization, according to the theory, wealth should be depleted earlier rather than later or near the end.
The "Game"
For the nine different scenarios, we will be looking for the one that maxes out the lifetime consumption utility and/or offers the highest balance-sheet initial spend rate.
In addition we will look at the allocation increments to see what happens to the allowable risk positioning as the degree of annuitization goes up.
We, meaning me, will, however, not be annuitizing anything anytime soon in real life regardless of the conclusions. That is because I want to maintain my poster-boy-of-the-annuity-paradox status for the time being.
The Output
If you are familiar with the assumptions we can jump straight to the output.
1. Spend rate perspective for just one representative scenario.
This is for the "allocate 60% of W to an annuity" scenario only. Each line is for a different (of 11) allocation to risk and each dot is an individual spend rate assumption for that allocation along the line on which the dot lies.
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| Figure 1. |
However the lines play out for figure 2, for each spend rate the max E[V(c)] is selected. Those max values, the boundary at the top of figure 1, will become the line for this scenario in Figure 2. There were 9 of these charts which, other than this one, are not shown.
2. Spend rate maxima for all scenarios: 0-80% allocation to lifetime income.
Again, remember that all the spend rates have been norm-ed to the initial actuarial balance sheet. Each line is a scenario of an initial allocation to an SPIA. X axis is the norm-ed spend rate. Y is the expected value of the value functions, i.e., the maxima from the last chart. The top dark red line in Figure 2 is the 60% allocation to annuity income. The peak for that line is around a 4.6% lifestyle for this parameterization. Higher spend may still be a decent bet. See the discussion below.
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| Figure 2 |
3. The Overall Allocation-Step perspective.
If figure 2 were a 3D chart seen from one side, another side is the allocation to risk, so if we rotate figure 1 90 degrees to left or right we'd see the max lines with respect to risk. It's a similar process. If figure 1 were "allocation steps," rather than spend rates, with different lines for spend rates, we would select the maxima for each increment of risk to render the lines in Figure 3.
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| Figure 3. |
4. Inside the "60% scenario" - Consumption in real terms.
Figure 4a shows the (first 50) iterations of the sim in terms of the real consumption path (over the first 34 periods). One can see that spending is a constant something (4.6% here or 21% of liquid wealth) and when an iteration depletes wealth, consumption "snaps" to available income or in this case the current real value of the annuity stream which is low -- when finally needed due to effects of in-model inflation -- and it's getting lower.
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| Figure 4a |
Figure 4b is the same thing but with more granularity. This is for all iterations for the 60% scenario and for all periods that the sim considered. Here we can see that consumption can crash anywhere along the periods with some degree of probability (not estimated). The very earliest of crashes happen after the SPIA (which starts at t=0) but before SS kicks in. The degradation of the SPIA income, due to inflation, can be seen. On the other hand, from an economic utility perspective (highly convex), anything is better than zero which has infinite (or is it undefined) disutility. The Y axis is stretched here a bit to anticipate a wealth line being drawn in at some point.
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| Figure 4b |
5. Inside the "60% scenario" - median nominal wealth, median real consumption and the consumption-snap-to-income at wealth depletion.
This is a little bit of a mixed grab bag but I didn't know how else to illustrate it. Also the many crash lines above in 4b are distracting. So, in this Figure (5) we are working with medians rather than dispersions and distributions and we are working from the simulation output rather than being strictly stylized.
We see median wealth in nominal terms in the black dotted line. The consumption path is blue and the crash line represents a median expectation. The lifestyle safely allowed without insurance (before the annuity) is in red though without any sense of a fail date here though it certainly exists. The snap of consumption to income can be seen around period 36 or so. The question is then what is the expected terminal horizon date? 30 years? 50? More?
The advantage here for utility-style analysis (other than that we are looking at consumption rather than terminal wealth) is that the utiles are both discounted subjectively a bit to reflect a current-vs-future bias as well as weighted for survival probabilities. So, the "distant" future-crash does not bite as much as a near one and not nearly so severely as if we had let consumption go to zero without insurance. A real annuity might be better in this context but only one company in the US offers one precisely like that. There are similar products, though (COLAs, VAs with GLWBs etc). Joe Tomlinson has written on this at Advisor Perspectives
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| Figure 5 |
Discussion
1. It's clear that the optimal annuitization is not going to be 100% here. In this run the highest lifetime consumption utility comes from the 60% allocation to an immediate nominal annuity after which the higher allocations suffer a bit by comparison. 50% is a close second. That (60%) also enables the highest spend rate, optimally speaking. The uninsured path now looks like a hard sell.
2. The difference between the theory and the outcome is no doubt partly due to the design of the software and my biases or naivete. More likely it is due to the fact that these are nominal not real and discrete rather than the instantaneous load-less tontines or annuities that can be bought and sold freely in theory-world.
3. Consistent with papers I've read before, not only are spend rates and consumption of wealth "high and early" with the higher degrees of annuitization (I can't recall but I think this was in some papers by LaChance, Milevsky, and maybe Leung) but we can also see the other point that others have made (see figure 3) that well insured lifetime income can favor much higher allocations to risk for the liquid wealth portion of the estate. For example, we see the "uninsured" allocation nestled at around a 40% risk posture while the higher allocations to annuities are looking at up to 100% risk.
4. In figure 2 we can see that even though the optimal point (using the balance sheet approach) is in the mid fours, the tolerance of an insured income for higher-than-optimal spend rates is much more forgiving than the un-insured approach. Using the 60% annuity line we can see that the lifetime utility at 6% consumption is still higher than the most optimal uninsured line.
5. As in other posts on this blog on other unrelated topics, this (figure 3) reinforces a point (ex the unisured path, anyway) that I've made before that except for the extremes of low and high risk, the analysis needle is often not moved very far by the risk allocation. My other point that I made in the very last post, as I will also make here, is that I often marvel at portfolio engineers that ignore spending and the risk pool and instead perseverate on minute differences between "factor" returns and methods. That's fine in a no-spend and infinite or single-period context. But not here in retirement-land.
6. When someone like Dirk Cotton or even myself cautions us about the formal concept of "ruin risk" in papers or posts there are many reasons. A post like this is probably one of them. One is rarely really "ruined" as such. Spending is cut, part time work is sought, family and institutions and social benefits step in. And, if it is not clear, one can always buy one's way out of ruin risk if one has the foresight to do so while there are still sufficient resources to execute that choice.
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