The point of this post is to extend the last several posts by now looking at different deferral periods for an idealized, hypothesized, and entirely fake deferred income annuity (DIA) to see what that change in deferral does to the maximum lifetime consumption utility for different spend rates across different allocations to a (idealized, hypothesized....) consumption portfolio.
I mention the "idealized, hypothesized" thing because this model is doubtful in it's fidelity to anything real that will unfold in the world we live in. Also, note that my goal is not to give myself some perfect consumption rate here. Rather I want to look at the "shapes, movement, and ranges" of spend rates and allocations given the narrow constraints of the software I built. From that I might be able to intuit how things work a little better. Maybe others have done this kind of thing before me but I need to see it for myself.
Background 1 - Previous Posts
In the series:
- Self-Evaluation of My Own Lifetime Consumption Utility
- Self-Evaluation of My Own Lifetime Consumption Utility, Part 2
- Having Some Fun with Portfolio Choice vs. Lifetime Consumption Utility
- Lifetime Consumption Utility with addition of trend following-like behavior
- An attempt at a stylized lifetime consumption utility "frontier"
- Adding a DIA to the last post that was looking at a consumption utility "frontier"
Background 2 - Efficient Frontier
In my fake model-world here I am using a vaguely stylized and illustrative-only portfolio. I have assumed out of thin air not only the return range, volatility, and co-variance in a way that is convenient to me, I have also assumed that I could successfully add a third asset that reduces volatility increasingly as we go up the EF. Like this:
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| Figure 1. EF |
| Step | R | SD1 | SD2(red) |
| 1 | 0.0350 | 0.0400 | 0.0400 |
| 2 | 0.0395 | 0.0386 | 0.0356 |
| 3 | 0.0440 | 0.0457 | 0.0397 |
| 4 | 0.0485 | 0.0583 | 0.0493 |
| 5 | 0.0530 | 0.0736 | 0.0616 |
| 6 | 0.0575 | 0.0902 | 0.0752 |
| 7 | 0.0620 | 0.1076 | 0.0896 |
| 8 | 0.0665 | 0.1254 | 0.1044 |
| 9 | 0.0710 | 0.1434 | 0.1194 |
| 10 | 0.0755 | 0.1616 | 0.1346 |
| 11 | 0.0800 | 0.1800 | 0.1500 |
I am under no illusions that anything remotely like this will have a high probability of playing out in my future. But, as I mentioned above, by seeing the shape, range and movement of things I might hope to understand a little better how all this works. In addition, note that the above was very very roughly modeled on a paper by the CME on adding managed futures to a traditional 2-asset portfolio. Their representation looked like this, which is not much to go on but enough for now:
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| Figure 2. CME EF w Managed Futures |
Background 3 - The Wealth Depletion Time Lifetime Consumption Utility Simulations
If you have the time, I have encapsulated how this works in the following page/post:
If you are in a rush, the basic idea can be sketched out in the schematic in figure 3 where the math is basically a CRRA utility function of this form
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| Eq 1. CRRA utility |
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| Figure 3. Schematic of Expected Discounted Utility of Lifetime Consumption |
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| Eq2. EDULC |
and where g[c(t)] is as above in Eq. 1, k is a subjective discount, tPx is a conditional survival probability, and S is the number of iterations. The biggest things to note here are (1) that the CRRA utility power math is very convex, so drops in consumption hurt, and (2) consumption "snaps" to available income (SS + annuity cf) when wealth depletes.
An aside: the meaning and usefulness of this whole series of posts hinges on the acceptance of economic utility as a reasonable framework for evaluating consumption and retirement strategies. I'm not sure that I myself have totally bought into this framework yet. I have no idea what my risk aversion is. I also am not particularly sanguine about the likelihood of my risk aversion being stable over time. And even if I buy either of those, is CRRA the right model and is it really additive? I am just jumping on this thing for a ride to see what happens. Recall that triangulation among several models and methods is more my thing and in that sense the current framework may not be all that un-useful.
Background 4 - Annuity Assumptions
I am pricing fake annuities using the following formula
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| Eq 3. Annuity Formula |
In this post, in addition to the SPIA I profiled in a previous post, I am pricing four different DIA's, at 70, 75, 80, and 85. If I understand these things correctly, 85 is about as late as you can go. Also note that these are nominal annuities rather than real. That means I am taking an explicit risk position on rates and inflation. In effect I think I am short inflation if that makes sense. The purchasing power will degrade over time. That's ok here because it is not real life and the presence of income, degraded or not, has a significant effect on the consumption utility.
Background 5 - Other Assumptions
These are the same as the last post or two but we can repeat here:
- The models and questions and data are roughly tuned to my personal situation so I can practice my blog on myself for a change. Note that all of this is illustrative and very fake. This is not an empirical study just a mind-game,
- The post/model has a lifetime consumption utility focus with the utility model described formally here,
- Age = 61,
- Risk aversion coefficients = 2; see the links for why “2,”
- Social Security that is realized with full probability over the entire life interval starts at age 70,
- Available assets are net of liabilities and include only what is reasonably monetizable over the life-cycle,
- High-risk asset nominal arithmetic return is .08 with standard deviation of .18,
- Low-risk asset is nominal arithmetic .035 with standard dev of .04,
- Correlation coefficient for standard EF is -.10,
- Portfolio is combined in 11 steps from 0% risk asset to 100%
- Addition of a “stylized” (i.e., fake) third asset class is considered. In this case it just means the vol of the portfolio is reduced between 0 and 3 points between 0 and 100% risk asset with the addition. The precise allocation to the third is implicit and not known. EF with the third asset roughly mimics CME representations for addition of of managed futures to a portfolio,
- Inflation = 3% where needed; discount rates, if needed are 3%, subjective discount on real consumption utiles in the utility model is .005. Annuity discount rates might be different,
- Spending is constant with all the flaws of that assumption and snaps to available income (ss+annuity cf) when wealth depletes,
- Every parameter is (ahem) assumed stable over time with no shocks,
- There were 10,000 iterations per data-point,
- Survival probabilities (conditioned on my age of 61) were extracted from an actuarial table for annuitants (SOA),
- Initial wealth is redacted but in “unit terms” would range from ~17 to 50 depending on spend,
- I abbreviated runs of the sim to reduce manual effort
- I’m probably missing a few others…
Background 6 - Catching up on the charts from prior posts
1. Spend rate perspective. Each line represents a "scenario." Each scenario was run for spend rates between 2 and 7% and then for each of those spend rate across 11 allocation steps up the EF in figure 1. For each scenario, the max value function is calculated for each spend rate (if looking at spend rates [see note 1 for how this works]) across all allocation steps, or the max value is calculated for each allocation step (if looking at allocation steps) across all spend rates. This is better as a 3D surface but this two dimension approach is easier to see. This is where we are so far:
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| Figure 4 |
B - is the max E[V(c)] for each spend rate for the red EF (with trend) above with no annuitization.
C - is the max E[V(c)] for each spend rate for the red EF above with an annuity cash flow purchased with 10% of initial wealth
D - is the max E[V(c)] for each spend rate for the red EF above with an annuity cash flow purchased with 20% of initial wealth
2. Allocation Step Perspective. Same as above for the most part, just a different perspective. A-D are the same except we are looking at the max at each step across all spend rates.
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| Figure 5 |
At this point forward we will be working with "C" plus some of the DIA additions.
Background 7 - Some "Whys?"
1. Why 10% of wealth?
This is arbitrary. But it allowed some uniformity of something across scenarios. Also, this is about what, at age 61, I'd be willing to wade into for right now when it comes to annuities (I haven't yet). It's a material amount but not over-committed. In the model it is just enough for me to see the shape, regions, and movement of things. Maybe another post goes higher. We know, from Yaari '65, that with some constraining assumptions (such as no bequest among other things) that it would be optimal, consumption wise, to annuitize 100% of wealth. I'm not personally going to go that far probably ever. Let's start with 10% and see how it goes.
2. Why nominal annuities?
Mostly this is for my convenience. They are easier to model and they are much easier to find in real product markets. The problem is that they are an explicit bet on rates and inflation and risk some serious degradation of lifestyle in high inflation environments. I came of age in the late 70s so this is not a trivial oversight, it's a big one. See this post and paper by Bodie and Cotton: Bodie and Cotton on Real Annuities
The Output
Here we are going to start with scenario C, the SPIA scenario paired with a three asset contrived portfolio. Then I am going to add DIAs that start payout at age 70, 75, 80, and 85. Any more granular would have been too much work for this old guy. The cash flow is based on what 10% of initial wealth will buy at each deferral age. Effectively, it's higher for each deferral increment. The price is determined as described above. In the first chart, we see the effects on lifetime consumption utility of deferral where the SPIA is more or less a deferral of zero or age 61.
1. Spend rate perspective.
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| Figure 6 |
A - is the SPIA or the scenario C above
B - Defer to age 70
C - Defer to age 75
D - Defer to age 80
E - Defer to age 85
2. Allocation Step perspective.
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| Figure 7 |
A - is the SPIA or the scenario C above
B - Defer to age 70
C - Defer to age 75
D - Defer to age 80
E - Defer to age 85
Discussion
- In the narrow confines of this model (not necessarily in real life) there is a case to be made for using a deferred annuity over an immediate. The cash flow purchased covers a shorter less certain interval and the cash flow is basically a little higher...even though it is not keeping up with inflation.
- All cases here are better in a utility context with annuitization than without as illustrated in the background above.
- In the narrow confines of this model, deferring to age 75 or 80 shows the highest lifetime consumption utility gains (expected value anyway) in terms of spend rates allowed and risk positioning that can be taken. This seems consistent with other papers I've seen on annuitization but I can't put my finger on them.
- Deferring to 85 takes on more wealth depletion risk prior to age 85 than the model seems to like given the particular parameterization I used. This was illustrated (figure 8) in a past post with the first 50 consumption paths generated by the sim. It shows wealth depletion caused consumption crashes (the "snap" to income) before the DIA kicks in. I hypothesized at the time that an earlier deferral start would scoop up enough of these crashes to be worth a look and it appears so.
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| Figure 8 - consumption crashes before DIA at 85 due to wealth depletion events |
- As in the past post, I still marvel at portfolio engineers that debate tiny differences of return factors while perseverating over minute differences in trend models and yet ignore (a) spending, and (b) the beneficence of the risk pool. These two things have dramatic impact compared to splitting factor hairs. This myopia is not necessarily of service to decumulators. It would be, perhaps, if we lived forever and never spent ... but that is not our world.
---------- Notes --------------------------------------------------------
[1] Figure 9 below is a scenario. This one happens to be for the SPIA scenario or the black line in Figures 4, 5, 6, and 7. That black line is created by taking the max of E[V(c)] from each spend rate across the allocation steps. In this figure below, the allocation steps are the colored lines and the spend rates along the x axis are the dots up the y along each line. So, that's one scenario in figure 9, and each of the other scenarios -- or each solid line in figures 4-7 -- are created the same way, Each dot is created by 10,000 iterations per run. There were a minimum of 11 x 11 runs for each "allocation x spend" step, sometimes more, so let's say at least 1.2M iterations per scenario, sometimes up to 2 or 3M if I needed more detail. My computer is hot now.
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| Figure 9. a scenario, this one for the SPIA |
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