- Over some pretty broad, middle of the road asset-allocation ranges and moderate spend rates, spending control looks like it trumps asset allocation as a lever for lifetime consumption utility. This is more evident as the concept of risk aversion rises. My take-away from this is that once a consumption plan has been rationalized, asset allocation -- over a broad range from 30-40% to 80-100% to a risk asset -- is of less importance than spending control.
- Risk aversion, if it is considered a valid analytical tool, has a significant effect on the results and suggests that spending control is an important tool in the presence of high risk aversion. High aversion seems to command lower spend rates.
- Random lifetime, when it is modeled as "literally random life" as opposed to just a vector of conditional survival probabilities for a given age, makes the simulation unstable except at very high iterations within the sim. My naive take-away from this is that in real life, the individual path one is dealt in terms of longevity matters a lot and may trump both spending and asset allocation as a significant factor for consumption utility over the life-cycle. In the absence of "pensionized wealth" this might be considered a vote for conservatism in spend rates given the first and second bullets. ...or maybe a vote for pensionized wealth.
- Volatility reduction, all else equal, appears to have positive effects on consumption utility and would imply support for higher (constant) spend rates. Alternatively, looking at it from another perspective, parameter estimation uncertainty in constructing efficient portfolios (e.g., estimation of parameters such as returns, vol, covariance, etc) would imply some need to think carefully about conservatism in planned spend rates. It is not shown here but it has been shown elsewhere (e.g., Claire 2017, Hoffstein 2018) that allocations to alternative risk premia and anomalies like trend following, given the effects on the volatility of portfolio returns, is accretive to retirement portfolios that are seeking higher consumption rates.
- The very lowest allocations to portfolio risk, counter-intuitively to some retirees, appear to be the least productive of lifetime consumption utility and might reasonably be avoided by those with moderate to low risk aversion. The exception, of course, might be for those in the most fearful "retirement crouch" but I have not looked at extreme risk aversion yet. When the policy for how the model deals with cases of depletion before income starts is made more assertive, high spend rates with high allocations to risk are much more heavily penalized (not shown) and would be avoided as well. That policy change may be explored in a follow-up post.
THE POINT OF THE POST
The goal of this post is to use a lifecycle utility model -- one that anticipates a wealth depletion time i.e., a time interval in the late lifecycle when non-pensionized wealth runs out and consumption snaps to available income -- to evaluate the expected discounted utility of lifetime consumption given various choices in asset allocation and (constant) spend rates. This exploration is not sufficiently rigorous to make hard conclusions about allocation optima or, for that matter, specific recommendations about spending. The exploration, rather, is intended to help me see the general shape of lifetime consumption utility in visual terms and to get some sense on how it "moves" in response to changes in risk aversion or volatility assumptions.
SOME DISCLAIMERS AND HEDGES
This post has no hard academic or professional purpose so the assumptions and output are not intended to be fully realistic or to suggest plausible courses of action. The post is intended for what I want to call "shared self-education" and to teach myself what the shape of retirement risk looks like for a small set of generic or hypothetical assumptions and 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.
My code is provisional and contains errors and omissions and questionable policy choices every time I look at it again. That last may be one of the larger flaws[3]. I'm not sure how much academic scrutiny could be withstood on close examination. Some of these policies will be revisited to see how it affects the output and conclusions.
Any optima on the surface charts can be taken with a grain of salt. Since I have not pressed yet on realistic type modeling that includes real world phenomena --like realistic forthcoming return expectations, parameter uncertainty, fat tailed distributions, sequence of returns risk, or autocorrelation -- one might (correctly) assume that the real world results would be different. It's easy to predict that spend rates would get more conservative and that high volatility, before we even get to the lumps in the tails, would not be helpful. If you happen to see constant spend rate optima of 5 or 6% (or more) below, don't get excited.
There is a ton of other parameterization not considered here. One factor of note is that annuities and annuitization are not (yet) considered here. The pensionization[2] of wealth has significant positive effects on lifetime consumption utility given random lifetime. Also note that this exploration is not very granular. For example, intra-point spend rates were not calculated in order to speed things up.
As I note below, I have -- 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 -- assumed a constant inflation adjusted spend rate. I attempt to defend this simplifying assumption here (link).
THE WDT UTILITY MODEL
The "wealth depletion time" model is explained in detail here. The schematic looks like this:
The value function looks like this:
CORE ASSUMPTIONS AND PARAMETERS
The assumptions do not reflect generalizable research objectives. They reflect, rather, my own age, curiosities, and self-learning goals.
Age: 60
Endowment: $1M
Spend Rates: 3% to 8% constant (that assumption is defended here) in 1% steps
Risk aversion: Coefficients of 1, 2, and 3 (gamma) in a CRRA (and log) utility function
Asset Allocation: 0 to 100% to a risk asset in a 2-asset portfolio in 11 steps
Utility is calculated on real spending which snaps to income at wealth depletion
Inflation, where it matters is 3% as is any discounting other than the subjective discount
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)
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)
The efficient frontier used to drive the allocation steps in the simulation, based on the parameters above, along with standard MPT math, looks like this:
 |
| Figure 1. Efficient Frontier for 2-asset portfolio |
* These return parameters are not really intended to reflect my personal expectations about forward forthcoming returns in an era when the foot of the Fed has been heavy, past returns have been weird, valuations high, and interest rates low. In general, they may be viewed as a kind of average or composite of capital market expectations recently published by JP Morgan in 2018 and so represent a type of not-totally-insane reflection of broad market expectations. For now they are a placeholder. Note that commentary in the financial press in mid-2018 argues for lower expectations almost across the board.
THE PROCESS
The assumptions and model were thrown together into an R-script and then run as follows.
1. The software, represented by the schematic and formula above, was first run for a spend rate of 3%, a risk aversion coefficient of 1, and then iteratively, given those first two parameters, for each of the 11 asset allocation "steps" along the frontier.
2. Step 1 was then repeated for spend rates of 4 to 8% in 1 point steps.
3. Steps 1 and 2 were then repeated for coefficients of risk aversion (gamma) of 2 and 3.
4. Data and output were stored and charted along the way.
5. Errors and software flaws were diagnosed and corrected and then steps 1 through 4 were repeated a few too many times. Way too many times. This appears to now be a continuous loop[3].
A way to visualize the process is to think -- for a particular spend rate (say 5%) and risk aversion coefficient (say 1) -- of applying the value function above in a simulation at each of the asset allocation steps along the EF. Then, along that EF we ask: what is the "expected discounted utility of lifetime consumption" at each interval? The original "spend and time-interval and risk-aversion" agnostic 2D efficient frontier (see above) becomes a 3D frontier with the vertical dimension being the output of the value function. Like this:
 |
| Figure 2. Efficient frontier rendered in 3d by way of a lifecycle utility simulation |
Note that the "single period" 2D arithmetic efficient frontier is transformed into its 3D shape by all of the following: (a) geometric return effects coming from volatility and multi-period time, (b) the presence of a consumption process against wealth, (c) the utility math itself, and (d) the rather blunt effect of a wealth depletion period on lifetime consumption utility. The grey 2D line can maybe also be viewed as the shadow of the blue line if the blue line were to be illuminated from above.
THE OUTPUT 1 - Baseline Surfaces
Following the process above there were three main chart deliverables that I wanted to produce: chart one 3D "allocation - spend - utility" map for each of the three levels of risk aversion (RA). For what it's worth, I picked the coefficients of 1-3 just to see incremental steps and the "movement" as RA changes. In other research I've seen lately, the researchers use something like 2 4 8 16 (where there might be a consensus around an empirically observed RA of 3 or 4??? Maybe.). But the way the math works for higher gamma values, the formula creates vanishingly small (absolute) differences in the utility outcomes. For non-economists like me this creates, in my opinion, weird and unintuitive results at high levels of risk aversion, or at least it does if we are working in utiles rather than working back into certainty equivalents which I have not attempted to do here. Note that the perspective was not held constant in order to try to give the best views of a shaped object. Also...I didn't pick the colors.
In order of rising risk aversion (gamma):
 |
| Figure 3. Utility surface, gamma=1 |
 |
| Figure 4. Utility surface, gamma=2 |
 |
| Figure 5. Utility surface, gamma=3 |
THE OUTPUT 2 - Selected Parameter Uncertainty
I have not been comprehensive in looking at sensitivity to the parameters. It might be interesting to look at uncertainty in returns and correlation along with some other inputs. In this case I just wanted to see the general shape and the movement in things for small changes in volatility of (unfortunately normally distributed) returns. To do this I did some rather blunt adjustments to the efficient frontier: I added and subtracted 1% in vol all along the EF. This is not intended to realistically represent portfolio choice or design or real world processes. It was a modeling and testing convenience. On the other hand the adjustments could perhaps be constructively viewed in two ways:
1. They, the adjustments, could provide some sense of the utility model's sensitivity to uncertainty in volatility parametrization when one is constructing efficient portfolios, especially when we know that spending, multiple time periods, and lifetime consumption utility considerations will come into play, as they do in the real lives of retirees, and
2. They could provide some intuition into what happens with the addition of "volatility reducing alternative risk premia" to a hypothetical portfolio of two assets. Of course things won't work in real life the way I did it here by creating a wholesale shift to the left. But the kind of portfolio design choice I'm thinking of here will (might), in fact, move a lot of the frontier to the left in a way similar to my blunt-force shift in this post. For an example, here is the CME slide on adding manged futures (i.e., trend following, i.e., vol reducing asset class) to a portfolio. In particular, see the chart on page 23. Also, ignore recent discussions on FinTwit of the possible decay in trend following strategies because I don't believe them. Yet.
When I add and subtract 1% vol (standard dev) from the EF and I then calculate the value function again all along the EF for a given spend rate and risk aversion coefficient, just like I did in "The Process" above, this is what I get. I have not attempted to measure exact rates of change because I was mostly interested in the confirmatory "visual" of: vol up = lower utility, vol down = higher utility. Which I got. This is the same image as above but now with the addition of the two (green) lines reflecting the change in the recalculated value function E[V(c)] for the changes in vol. The grey line (2D efficient frontier) is still the shadow of the blue line if blue were to be illuminated from above.
 |
| Figure 6. uncertainty in vol assumptions x = portf. std dev, y = portf. return, z = lifetime consumption utility |
If we wanted to re-render Figure 6 as a surface diagram for a gamma of 1 for only the "reduction-of-vol" side of utility (and for the various spend rates from 3 to 8%), it would look like this
 |
| Figure 7. U surface with reduced vol |
but this is not helpful. It's too hard to see any changes from figure 3. So let's just look at the 4, 5, and 6% spend rates at the top of the surface in Figure 3 and then see what happens to them for a 1% reduction in vol all down the EF line. That looks more like this:
CONCLUSIONS
I'll copy/paste the summary at the top of the post into the conclusions section here for completeness. But before I do that, the only other main conclusion I have at this point is that it feels like if one were to be even mildly risk averse and if one were to believe this model then the spend choice seems like it is everything and the asset allocation choice seems pretty mute by comparison. And if it is not mute, then maybe it is at least telling us to avoid extremes along the spectrum of allocation choices: stick to the middle or maybe high-middle. But I think I already knew that. The other personal take away is that if I were to happen to have low risk aversion, then it looks like I would be seriously crimping my lifetime consumption utility by staying in a fearful, low-spend, no-equity-risk crouch. But I think I knew that, too. So that begs the question: what's my risk aversion? No idea. Any chance I can get a handle on it anytime soon? Doubtful. Does it ever change? With certainty. That's what makes any definitive conclusions here hard.
From the summary above:
- Over some pretty broad, middle of the road asset-allocation ranges and moderate spend rates, spending control looks like it trumps asset allocation as a lever for lifetime consumption utility. This is more evident as the concept of risk aversion rises. My take-away from this is that once a consumption plan has been rationalized, asset allocation -- over a broad range from 30-40% to 80-100% to a risk asset -- is of less importance than spending control.
- Risk aversion, if it is considered a valid analytical tool, has a significant effect on the results and suggests that spending control is an important tool in the presence of high risk aversion. High aversion seems to command lower spend rates.
- Random lifetime, when it is modeled as "literally random life" as opposed to just a vector of conditional survival probabilities for a given age, makes the simulation unstable except at very high iterations within the sim. My naive take-away from this is that in real life, the individual path one is dealt in terms of longevity matters a lot and may trump both spending and asset allocation as a significant factor for consumption utility over the life-cycle. In the absence of "pensionized wealth" this might be considered a vote for conservatism in spend rates given the first and second bullets. ...or maybe a vote for pensionized wealth.
- Volatility reduction, all else equal, appears to have positive effects on consumption utility and would imply support for higher (constant) spend rates. Alternatively, looking at it from another perspective, parameter estimation uncertainty in constructing efficient portfolios (e.g., estimation of parameters such as returns, vol, covariance, etc) would imply some need to think carefully about conservatism in planned spend rates. It is not shown here but it has been shown elsewhere (e.g., Claire 2017, Hoffstein 2018) that allocations to alternative risk premia and anomalies like trend following, given the effects on the volatility of portfolio returns, is accretive to retirement portfolios that are seeking higher consumption rates.
- The very lowest allocations to portfolio risk, counter-intuitively to some retirees, appear to be the least productive of lifetime consumption utility and might reasonably be avoided by those with moderate to low risk aversion. The exception, of course, might be for those in the most fearful "retirement crouch" but I have not looked at extreme risk aversion yet. When the policy for how the model deals with cases of depletion before income starts is made more assertive, high spend rates with high allocations to risk are much more heavily penalized (not shown) and would be avoided as well. That policy change may be explored in a follow-up post.
SELECTED REFERENCES
Habib, Huang, Milevsky, Approximate Solutions to Retirement Spending Problems and the Optimality of Ruin, 2017
Hoffstein, C. Leverage and Trend Following, 2018. https://blog.thinknewfound.com/2018/05/leverage-and-trend-following/
Hoffstein, C. The New Glide Path, 2018, https://blog.thinknewfound.com/2018/07/the-new-glide-path/
Lachance, M. (2012), Optimal onset and exhaustion of retirement savings in a life-cycle model, Journal of Pension Economics and Finance, Vol. 11(1), pp. 21-52.
Leung, S. F. (2002), The dynamic effects of social security on individual consumption, wealth and welfare, Journal of Public Economic Theory, Vol. 4(4), pg. 581-612.
Leung, S. F (2007), The existence, uniqueness and optimality of the terminal wealth depletion time in life-cycle models of saving under certain lifetime and borrowing constraint, Journal of Economic Theory, Vol. 134, pp. 470-493.
Leung Notes and Comments - Uncertain Lifetime, the theory of the consumer, and the lifecycle Hypothesis, 1994
Milevsky and Huang, The Utility Value of Longevity Risk Pooling: Analytic Insights, and the Technical Appendix, 2018
Yaari, Uncertain Lifetime, Life Insurance, and the Theory of the Consumer. 1965 .
NOTES
[1] I'll also mention that V(c) is still a random variable here because of the randomizing effects of returns and volatility on the start time of WDT. Random T* is expressed via the conditional survival probability vector tPx . So rather than WDT being random both in beginning and "hard" end it is now random at beginning and then trails to infinity but is then also weighted by a probability of longevity for being age x. Hence V(c) is still a random variable and hence the need for E[V(c)]. If I have that right...
[2] I borrow the semantic concept of "pensionization" from Prof. Milevsky. I do this here because it has been my experience that the word annuitization has been a trigger word for certain financial professionals in the past. These professionals correctly, in my opinion, clarify that (some) products exist that give, with varying degrees of efficiency and effectiveness and consistent availability, access to the concept of pooled risk (and lifetime income) or at least a simulacrum of pooled risk, but do so without necessarily requiring a complete and total assignment of existing capital in early years of an arrangement. So, for my purposes, I refer to pensionization here as the general idea of gaining, with private wealth, access to "a risk pool" and therefore "pool subsidized" lifetime income by whatever means the profession and academics consider "worthy" these days. This worthiness varies with the era (going back 1000 years, say) but let's throw in stuff like income annuities, deferred annuities, GLWB riders, tontines, private socialized swap agreements, social or religious organizations, or even my own idea for a fixed-horizon-tontineized-interval-fund, whatever (maybe DB pensions too but I'm thinking here of private wealth vs that tied up in labor arrangements)... Risk and capital go into a pool, income comes out, and sometimes capital might even still be available for a bit of time if bad things happen. Just don't say annuitization in the wrong crowd.
[3] For the skeptical on stuff like this, i.e., if you are like me and have doubts about utility, note that because running out of money has infinite disutility that means that even one sim iteration that has a "life that runs out of money before income kicks in" makes it hard to code around the issue or rather it requires some policy decisions on modeling. This means we need to make some simplifying assumptions and policy choices just to make it work. These distort the results, though, depending on the choices. I have provisional assumptions in there for dealing with this kind of thing but they are rudimentary. If I do this again the results will look different and may, in fact, undermine what I have already done. I'll have to deal with that when the time comes. Here is one example: the higher spend rates and allocations are not nearly penalized as much as they should be (opinion-wise). On the other hand, to fix this might make the summary points above even more emphatic so I'm good for now. Since this post is what I call "shared self-education," I guess the reader is along for the journey.
Let's chart out this future-post in teaser form. I have Goldilocks choices. I can have a hard approach where I allow infinite disutility to occur, except that I have a hard time coding and analyzing that (may say more about my coding ability than it does about utility math or design approaches). Or, I can take a soft approach, like I did above where I ignore infinite disutility and assume some institutional or societal support for wealth depletion states. Or, I can take a middle approach and have some kind of firm floor on the utility function that is not (-)infinite. If I assume a nominal dollar of income to which consumption snaps, which is not exactly how I did it but I at least floor-ed utility to that level, then we have an intermediate policy choice. It looks like this if charted out.
I don't think this undermines my points yet. Moderate spend rates and moderate allocations still prevail if they are not accentuated even more here. Changing the policy even more (lim->"0 consumption") might make it a little weird but I think this is as far as I'll go for this post. The conclusions from a more aggressive policy, which I tried to summarize above, would be to avoid super high spend rates and excessive risk allocations. I don't think that thought would be all that controversial. The main point where I think spending control is a big deal compared to allocation has not been contradicted yet. But I'm also willing to listen.
POST SCRIPT
When, on a long road trip with a daughter, and she wants to know what I am thinking about as we are driving along, I am skeptical that telling her that "it is everything above this sentence" is either helpful or reassuring. Especially in heavy traffic.
Yaari, Uncertain Lifetime, Life Insurance, and the Theory of the Consumer. 1965 .
NOTES
[1] I'll also mention that V(c) is still a random variable here because of the randomizing effects of returns and volatility on the start time of WDT. Random T* is expressed via the conditional survival probability vector tPx . So rather than WDT being random both in beginning and "hard" end it is now random at beginning and then trails to infinity but is then also weighted by a probability of longevity for being age x. Hence V(c) is still a random variable and hence the need for E[V(c)]. If I have that right...
[2] I borrow the semantic concept of "pensionization" from Prof. Milevsky. I do this here because it has been my experience that the word annuitization has been a trigger word for certain financial professionals in the past. These professionals correctly, in my opinion, clarify that (some) products exist that give, with varying degrees of efficiency and effectiveness and consistent availability, access to the concept of pooled risk (and lifetime income) or at least a simulacrum of pooled risk, but do so without necessarily requiring a complete and total assignment of existing capital in early years of an arrangement. So, for my purposes, I refer to pensionization here as the general idea of gaining, with private wealth, access to "a risk pool" and therefore "pool subsidized" lifetime income by whatever means the profession and academics consider "worthy" these days. This worthiness varies with the era (going back 1000 years, say) but let's throw in stuff like income annuities, deferred annuities, GLWB riders, tontines, private socialized swap agreements, social or religious organizations, or even my own idea for a fixed-horizon-tontineized-interval-fund, whatever (maybe DB pensions too but I'm thinking here of private wealth vs that tied up in labor arrangements)... Risk and capital go into a pool, income comes out, and sometimes capital might even still be available for a bit of time if bad things happen. Just don't say annuitization in the wrong crowd.
[3] For the skeptical on stuff like this, i.e., if you are like me and have doubts about utility, note that because running out of money has infinite disutility that means that even one sim iteration that has a "life that runs out of money before income kicks in" makes it hard to code around the issue or rather it requires some policy decisions on modeling. This means we need to make some simplifying assumptions and policy choices just to make it work. These distort the results, though, depending on the choices. I have provisional assumptions in there for dealing with this kind of thing but they are rudimentary. If I do this again the results will look different and may, in fact, undermine what I have already done. I'll have to deal with that when the time comes. Here is one example: the higher spend rates and allocations are not nearly penalized as much as they should be (opinion-wise). On the other hand, to fix this might make the summary points above even more emphatic so I'm good for now. Since this post is what I call "shared self-education," I guess the reader is along for the journey.
Let's chart out this future-post in teaser form. I have Goldilocks choices. I can have a hard approach where I allow infinite disutility to occur, except that I have a hard time coding and analyzing that (may say more about my coding ability than it does about utility math or design approaches). Or, I can take a soft approach, like I did above where I ignore infinite disutility and assume some institutional or societal support for wealth depletion states. Or, I can take a middle approach and have some kind of firm floor on the utility function that is not (-)infinite. If I assume a nominal dollar of income to which consumption snaps, which is not exactly how I did it but I at least floor-ed utility to that level, then we have an intermediate policy choice. It looks like this if charted out.
I don't think this undermines my points yet. Moderate spend rates and moderate allocations still prevail if they are not accentuated even more here. Changing the policy even more (lim->"0 consumption") might make it a little weird but I think this is as far as I'll go for this post. The conclusions from a more aggressive policy, which I tried to summarize above, would be to avoid super high spend rates and excessive risk allocations. I don't think that thought would be all that controversial. The main point where I think spending control is a big deal compared to allocation has not been contradicted yet. But I'm also willing to listen.
POST SCRIPT
When, on a long road trip with a daughter, and she wants to know what I am thinking about as we are driving along, I am skeptical that telling her that "it is everything above this sentence" is either helpful or reassuring. Especially in heavy traffic.
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