This post is a continuation of several past posts where I have been using (trying to shake out) a lifecycle utility model (decumulation focus only) [1] that includes a wealth depletion framework -- i.e., where there is a time interval in the late lifecycle when non-pensionized wealth runs out and consumption snaps to available income -- to evaluate "optimal spend rates." The optimization is done by evaluating a value function (expected discounted utility of lifetime consumption) that is calculated, based on a simulation process, across different spend rates, range of ages, different risk allocations and different coefficients of risk aversion. The goal is to see what happens to optimized spend rates at different ages and how it might be influenced by asset allocation. Framed as a question it might look like this:
What kind of optimized consumption does $1M in purchasing power (age 60 baseline) buy at different ages, and for different allocations to risk, in a lifecycle/decumulation utilility model that uses a wealth depletion time framework?
As in past posts the assumptions and parameters are illustrative rather than realistic and are for self learning rather than suggesting or recommending strategies and plans of action. Also, since this uses a utility framework, you'd have to buy into that, which I'm not sure I do even though I am here doing it.
- Annuities, though part of the model, are not considered here (yet) but would have a significant (positive) impact. Social security is set to a low value and then discounted even more. This is not particularly realistic but helps me think through some issues in modeling. Earlier claim of SS might be helpful here but is unexplored (this is not a last-mile analysis). Also, note that I have a soft utility floor assumption baked in. In effect, it is the equivalent to assuming that in a worst case wealth depletion scenario with no SS income present there is income roughly equivalent to $1k available. I have reasons.
- For modeling convenience and efficiency and to support the more generalized goals of my exploration -- as opposed to me trying to give specific, realistic retirement advice -- I have assumed a constant inflation adjusted spend rate. This is not always received as a happy assumption; I attempt to defend my choice here (link). Also note that the model is still naive on things like taxes and fees not to mention autocorrelation or return regimes. To quote Gordon Irlam at aacalc.com on adaptive spending: "A scheme that does not respond to the performance of the portfolio is likely to underperform one that is responsive to portfolio performance." This is a sentiment that I agree with but do not model here. Note as well that this post has a spend rate of 2.75% as the lowest scenario tested which means it represents a minimum in the results.
- In the output, nothing is chained either forwards or backwards. The results at age 65 do not depend on what happened at 60. Also, even though I'd like to try it, this is not a backwards induction thing. I am not starting with an optimization at the last period and then walking it back to time zero. Each age-allocation-spend-risk "set" is run independently as if it is the first and only time it has ever been run.
- The model leans heavily on illustrative capital market assumptions that are not necessarily realistic and are detailed here. Taxes and fees are not modeled either and would be expected to bring all spend levels down a bit. Also note that normal distributions are assumed (but not expected). Fat tails would impinge on spend rate optima as well.
- Longevity is treated as random and conforms to the SOA IAM table with extension to 2018.
The Model
The "wealth depletion time" model is explained in detail here. The schematic looks like this:
- The model leans heavily on illustrative capital market assumptions that are not necessarily realistic and are detailed here. Taxes and fees are not modeled either and would be expected to bring all spend levels down a bit. Also note that normal distributions are assumed (but not expected). Fat tails would impinge on spend rate optima as well.
- Longevity is treated as random and conforms to the SOA IAM table with extension to 2018.
- Since the model is highly sensitive to input parameters, and since the output also depends heavily on a coefficient of risk aversion, which is subjective, this post is not a recommendation on spend rates, especially the relatively high spend rates in the low risk aversion scenarios. It is just me playing with the model as it's designed to see what it tells us and to see what it looks like for one set of parameters.
- The model is obviously not "real life." It represents no more and no less than the shape of a few lines of code I built. That means that the model is like a bottle that holds water. The shape of the water depends on the shape of the bottle. The model in this post is merely a bottle of arbitrary design; there are other bottles. The question then is whether the model has anything useful to say about reality and maybe also whether you think I'm any good at modeling and/or whether I have gone far enough. TBD.The "wealth depletion time" model is explained in detail here. The schematic looks like this:
tPx - Conditional survival probability at time t for a person age x.
k - a subjective discount on utiles
c(t) - consumption at time t; c is the consumption plan
S - number of iterations
g[c(t)] - CRRA utility function of form (c(t)^(1-gamma)-1)/(1-gamma) for gamma <> 1 else ln(c(t))
Misc. Assumptions and Parameters
Age: 60 - 80 in 5 year steps
Endowment: $1M in purchasing power pegged originally at age 60 and then equivalent-real thereafter[2]
Spend Rates: 2.75% to 12% in .25% steps; note the minimum 2.75.
Risk aversion: Coefficients of 1 and 3 (gamma) in a CRRA (and log) utility function
Asset Allocation: varies from 0/100 equity to 100/0 in 11 steps
Misc. Assumptions and Parameters
Age: 60 - 80 in 5 year steps
Endowment: $1M in purchasing power pegged originally at age 60 and then equivalent-real thereafter[2]
Spend Rates: 2.75% to 12% in .25% steps; note the minimum 2.75.
Risk aversion: Coefficients of 1 and 3 (gamma) in a CRRA (and log) utility function
Asset Allocation: varies from 0/100 equity to 100/0 in 11 steps
Asset class returns and correlation:
- Risk-high = 8% arithmetic return and 18% standard deviation
- Risk-low = 3.5% arithmetic return and 4% standard deviation
- correlation -.10 (in our real world this is highly variable)
- these are illustrative rather than reflecting my expectations and may be quite unrealistic going forward.
- Risk-high = 8% arithmetic return and 18% standard deviation
- Risk-low = 3.5% arithmetic return and 4% standard deviation
- correlation -.10 (in our real world this is highly variable)
- these are illustrative rather than reflecting my expectations and may be quite unrealistic going forward.
Utility is calculated on real spending which snaps to available income at wealth depletion
Inflation, where it matters, is 3%
Sim Iterations: 10,000
Subjective discount on utiles: .005
Longevity: from SOA IAM table with longevity extension to 2018; for age x = 60, done two ways:
a) T* is a random variable in each sim iteration and the lifetime is simulated for T*- x years
b) a vector of conditional survival properties is calculated for age x over age -> infinity (121)
Convention b is chosen here for reasons described in the model.
Social Security: 11,000 annually starting at 70 and inflated thereafter. This is low vs reality, but:
(a) it creates a stub of minimum income for the WDT module, and/or
(b) it represents a heavy risk-discounting of SS income for subjective reasons of my own
The Process
1. For age 60, a wealth level of 1M (see above), and a risk aversion coefficient of 1 the value function is estimated via simulation for each allocation step and then within each each allocation step for each spend rate between .0275 and .1200. The spend rate with the maximum expected discounted lifetime consumption utility is selected and stored for each of the 11 allocation levels.
2. Step 1 is repeated for the each age bracket (65:80) in 5 year steps. Wealth and SS assumptions are advanced nominally to keep a constant real level of wealth and income. I hope I have that right.
3. Steps 1 and 2 are repeated for a risk aversion coefficient of 3.
The Output
Spend Rate Optima for Risk Aversion Coefficient = 1
For the given assumptions and a risk aversion coefficient (gamma) = 1, spend rate optima look like this in a 3D surface format:
Or alternatively, the top down view....
Spend Rate Optima for Risk Aversion Coefficient = 3
For given assumptions and a risk aversion coefficient (gamma) = 3, spend rate optima look like this in a 3D surface format:
Or alternatively, the top down view (colors don't match)....
Or in table form...
Conclusions
- Spend rate optima (per $1M-real in this run) go up with age due to foreshortening longevity expectations. This is neither unexpected nor unknown.
- Lower risk aversion looks like it can tolerate both higher spend rates and higher allocations to risk (and seems to hate low-risk portfolios). Higher risk aversion appears to have a lower tolerance for high spend rates and also both the highest and lowest allocations to risk assets. For example, in the high (actually medium, I hear) risk aversion scenario, allocations to risk assets, before we get to alternative risk and efficient frontiers, are very middle of the road over a pretty broad interval. All of this might be mooted, however, by small shifts in capital market assumptions...which is why I am cautious at this point about making hard conclusions or recommendations (which is not the purpose of the blog, by the way).
- I was surprised that the optima surfaces were so flat or planar but I guess this is consistent with some past posts where my conclusions were that optimal consumption is relatively insensitive to allocation choices except in the extremes. The actual spend rate itself, along with age (expected longevity), appear to be very important.
- In a recent post on endowments, I made the point that I think there is very little space between the allocation and spending choices given that we are working with a net wealth process. The "insensitivity" comment above notwithstanding, I still think that return, risk and spend are closer to being "one choice" than two or three completely separate ones. They are intertwined. It's the goal of posts like this to see all three at the same time in order to see the "shape of the process."
- Long ago I proffered, as a type of joke, a custom age-based spend rule I called RH40 which was supposed to be conservative, easy to calculate and remember, and roughly consistent with maintaining a constant level of risk at each age. I was surprised by how well it held up here, at least for the higher level of risk aversion. The rule was that a spending rate can be estimated on the fly by: Age/(40-age/3). For age 60-65-70-75-80 the outcomes from that calc are 3, 3.5, 4.2, 5 and 6% respectively...this is not totally out of line with the second set of results though maybe a little conservative on the front end which is not necessarily a bad thing.
- It looks like allocations to risk assets that are tuned to the appropriate risk aversion level (good luck on that risk aversion determination, by the way) allow one to bring consumption rates that are more appropriate for the future forward (or is it backward?) into the present.
- Unexplored is the impact of lifetime income from higher SS assumptions, pensions, and income purchased via annuities. They would be expected to have a significant positive impact on spend rates at, I presume, all ages. A past post indirectly illustrates this by showing spend rates over different ages at different levels of wealth that imply different proximities to lifetime income.
-----------------------------------
[1] Saying "lifecycle" is probably not completely fair since this only looks at the decumulation stage but that seems to be the convention for this type of analysis. Send me a better name.
[2] This reveals a small bias. Since I do a lot of modeling with my own perspective in mind that bias creeps into assumptions. For example, in this case the model reflects a choice of SS at age 70. That is a policy choice. Then, since the SS inflates in the model thereafter, the initial endowment has to inflate from my "anchor point" of 60 (my age) otherwise the utility math would distort the results since the pre-depletion income would be relatively closer to SS than it would be without the endowment inflation. The effects of this alternative modeling were seen in a previous post. There are other ways to do this kind of thing, of course, but since there are many I chose one that is more reflective of my point of view. Bias. Maybe.
Inflation, where it matters, is 3%
Sim Iterations: 10,000
Subjective discount on utiles: .005
Longevity: from SOA IAM table with longevity extension to 2018; for age x = 60, done two ways:
a) T* is a random variable in each sim iteration and the lifetime is simulated for T*- x years
b) a vector of conditional survival properties is calculated for age x over age -> infinity (121)
Convention b is chosen here for reasons described in the model.
Social Security: 11,000 annually starting at 70 and inflated thereafter. This is low vs reality, but:
(a) it creates a stub of minimum income for the WDT module, and/or
(b) it represents a heavy risk-discounting of SS income for subjective reasons of my own
The Process
1. For age 60, a wealth level of 1M (see above), and a risk aversion coefficient of 1 the value function is estimated via simulation for each allocation step and then within each each allocation step for each spend rate between .0275 and .1200. The spend rate with the maximum expected discounted lifetime consumption utility is selected and stored for each of the 11 allocation levels.
2. Step 1 is repeated for the each age bracket (65:80) in 5 year steps. Wealth and SS assumptions are advanced nominally to keep a constant real level of wealth and income. I hope I have that right.
3. Steps 1 and 2 are repeated for a risk aversion coefficient of 3.
The Output
Spend Rate Optima for Risk Aversion Coefficient = 1
For the given assumptions and a risk aversion coefficient (gamma) = 1, spend rate optima look like this in a 3D surface format:
Or in table form...
| Age | |||||
| allocation | 60 | 65 | 70 | 75 | 80 |
| 0% | 0.0425 | 0.0475 | 0.0575 | 0.0675 | 0.0800 |
| 10% | 0.0450 | 0.0500 | 0.0600 | 0.0675 | 0.0825 |
| 20% | 0.0450 | 0.0525 | 0.0600 | 0.0700 | 0.0850 |
| 30% | 0.0475 | 0.0550 | 0.0625 | 0.0700 | 0.0850 |
| 40% | 0.0500 | 0.0550 | 0.0650 | 0.0750 | 0.0875 |
| 50% | 0.0525 | 0.0575 | 0.0675 | 0.0750 | 0.0875 |
| 60% | 0.0550 | 0.0600 | 0.0675 | 0.0750 | 0.0875 |
| 70% | 0.0525 | 0.0625 | 0.0700 | 0.0800 | 0.0900 |
| 80% | 0.0550 | 0.0650 | 0.0700 | 0.0800 | 0.0950 |
| 90% | 0.0550 | 0.0650 | 0.0700 | 0.0800 | 0.0950 |
| 100% | 0.0550 | 0.0650 | 0.0700 | 0.0800 | 0.0950 |
For given assumptions and a risk aversion coefficient (gamma) = 3, spend rate optima look like this in a 3D surface format:
Or alternatively, the top down view (colors don't match)....
| age | |||||
| Alloc. | 60 | 65 | 70 | 75 | 80 |
| 0% | 0.0325 | 0.0375 | 0.0425 | 0.0475 | 0.0525 |
| 10% | 0.0350 | 0.0400 | 0.0450 | 0.0500 | 0.0550 |
| 20% | 0.0350 | 0.0400 | 0.0450 | 0.0500 | 0.0550 |
| 30% | 0.0375 | 0.0400 | 0.0450 | 0.0500 | 0.0575 |
| 40% | 0.0375 | 0.0400 | 0.0450 | 0.0500 | 0.0575 |
| 50% | 0.0350 | 0.0400 | 0.0450 | 0.0500 | 0.0550 |
| 60% | 0.0350 | 0.0400 | 0.0450 | 0.0500 | 0.0550 |
| 70% | 0.0350 | 0.0400 | 0.0425 | 0.0500 | 0.0550 |
| 80% | 0.0350 | 0.0375 | 0.0425 | 0.0475 | 0.0550 |
| 90% | 0.0325 | 0.0375 | 0.0425 | 0.0450 | 0.0525 |
| 100% | 0.0275 | 0.0350 | 0.0400 | 0.0450 | 0.0500 |
Conclusions
- Spend rate optima (per $1M-real in this run) go up with age due to foreshortening longevity expectations. This is neither unexpected nor unknown.
- Lower risk aversion looks like it can tolerate both higher spend rates and higher allocations to risk (and seems to hate low-risk portfolios). Higher risk aversion appears to have a lower tolerance for high spend rates and also both the highest and lowest allocations to risk assets. For example, in the high (actually medium, I hear) risk aversion scenario, allocations to risk assets, before we get to alternative risk and efficient frontiers, are very middle of the road over a pretty broad interval. All of this might be mooted, however, by small shifts in capital market assumptions...which is why I am cautious at this point about making hard conclusions or recommendations (which is not the purpose of the blog, by the way).
- I was surprised that the optima surfaces were so flat or planar but I guess this is consistent with some past posts where my conclusions were that optimal consumption is relatively insensitive to allocation choices except in the extremes. The actual spend rate itself, along with age (expected longevity), appear to be very important.
- In a recent post on endowments, I made the point that I think there is very little space between the allocation and spending choices given that we are working with a net wealth process. The "insensitivity" comment above notwithstanding, I still think that return, risk and spend are closer to being "one choice" than two or three completely separate ones. They are intertwined. It's the goal of posts like this to see all three at the same time in order to see the "shape of the process."
- Long ago I proffered, as a type of joke, a custom age-based spend rule I called RH40 which was supposed to be conservative, easy to calculate and remember, and roughly consistent with maintaining a constant level of risk at each age. I was surprised by how well it held up here, at least for the higher level of risk aversion. The rule was that a spending rate can be estimated on the fly by: Age/(40-age/3). For age 60-65-70-75-80 the outcomes from that calc are 3, 3.5, 4.2, 5 and 6% respectively...this is not totally out of line with the second set of results though maybe a little conservative on the front end which is not necessarily a bad thing.
- It looks like allocations to risk assets that are tuned to the appropriate risk aversion level (good luck on that risk aversion determination, by the way) allow one to bring consumption rates that are more appropriate for the future forward (or is it backward?) into the present.
- Unexplored is the impact of lifetime income from higher SS assumptions, pensions, and income purchased via annuities. They would be expected to have a significant positive impact on spend rates at, I presume, all ages. A past post indirectly illustrates this by showing spend rates over different ages at different levels of wealth that imply different proximities to lifetime income.
-----------------------------------
[1] Saying "lifecycle" is probably not completely fair since this only looks at the decumulation stage but that seems to be the convention for this type of analysis. Send me a better name.
[2] This reveals a small bias. Since I do a lot of modeling with my own perspective in mind that bias creeps into assumptions. For example, in this case the model reflects a choice of SS at age 70. That is a policy choice. Then, since the SS inflates in the model thereafter, the initial endowment has to inflate from my "anchor point" of 60 (my age) otherwise the utility math would distort the results since the pre-depletion income would be relatively closer to SS than it would be without the endowment inflation. The effects of this alternative modeling were seen in a previous post. There are other ways to do this kind of thing, of course, but since there are many I chose one that is more reflective of my point of view. Bias. Maybe.
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