"Arguably, for someone willing to pay careful attention to the evolving funding status of their retirement (the present value of their remaining assets divided by the present value of their remaining liabilities), asset allocation can be adjusted dynamically by matching assets to liabilities in coordination with the client’s risk capacity and risk aversion, rather than just on an arbitrary path. But for those unwilling to take such care, such as users of target date or lifecycle asset allocation funds, or for those whose goals are somewhat more complex in the balancing of current and future, income and legacy goals, what should be the default equity glidepath in the postretirement period?"
Pfau, W. and Kitces, M (2013)
"...the optimal asset allocation moves after significant market events. This means an investor needs to monitor and adjust their asset allocation after such events. They cannot simply follow a glide path."
Irlam, G (2014)
These quotes are not incompatible. I was considering this today because I was re-reading Irlam (2014) on using backward induction (BI) via stochastic dynamic programming (SDP) so that I could revisit and repair some software I built in 2017 that attempted to replicate what Irlam had done in 2014. It's a deceptively simple idea that is hard-ish to implement and even harder to figure two years later when you are trying to figure out what you did earlier (note to self: comment one's code better).
The economic problem here, in simple terms, is that if you have a horizon of 30 years and (say) 11 asset allocations to test if you are looking for an optimal path (given what you know at the time) over the plan then that means that you have 11^30 combinations to test. If you do it like Irlam, you have 101 allocations and 70 years over a full lifecycle or 101^70 combinations. It's something that he called "computationally intractable."
Better, he says (and does), to work backwards[2]. It's like deciding when to leave for the airport by working backwards from when you want to get there rather than working forward with all the possible combinations of when one could plausibly leave and then testing them all. This approach
"...treats asset allocation as a mathematical optimization problem and solves it analytically or numerically using stochastic programming or stochastic dynamic programming (SDP) techniques (Bellman (1953), Mulvey and Vladimirou(1989))"
- Irlam (2014)Irlam lays out the steps that still have a little simulation within the backward process, the same steps that I followed in a rougher, more rudimentary, more amateur fashion to try to replicate the work:
- Start with some base assumptions about lifestyle and inflation and horizon (e.g., 30)
- For, say, an horizon of 30 years, at end-time-30 in the plan (not in real life since you are doing this at time zero, run a small (small because the program takes forever) single period (we are in the last year after which the horizon has no further horizon) simulation using historical (or modeled. I used Aswath Damodaran's data at Stern for historical and a custom model I created for a model version) for all (in practicable increments) levels of wealth and all asset allocations.
- For any given level of wealth, pick the allocation that fails the least (I added a tie breaker for highest dollar outcome, legacy planning and all you know), and save that allocation and the probability of success and attach it to that year and level of wealth.
- Step back one year to year 29 and repeat the step above except this time, in the single period mini-sim, use the end wealth (rounded or bucketed) of end-of-sim-year 29 to pull from year 30 the probability of that wealth succeeding to the end of year 30, combine the mini sim probabilities for the particular allocation being evaluated, and then use the mean probability to pick the "best" allocation among the several for that year and level of wealth being evaluated.
- Walk that process just described back to year one and behold: a table of "optimal" asset allocations by year and level of wealth for the selected spend rate.
Using a 130k arbitrarily selected annual lifestyle and a 3% inflator, 11 asset allocations in 10% steps from 0% risk asset to 100%, and 100 increments of wealth from 100k to 10M, it'd look like this below in figure 1. My tool is rudimentary and quite a bit more reduced. Irlam's has more rigor and better pictures. Y axis is level of wealth at some plan year, X is plan year, colors are the "optimal" allocation with yellow being "all risk" asset and blue being no or low (0-20% or so).
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| Figure 1 |
Figure 1 was using the historical data. Figure 2 was a test run using a fabricated model/efficient frontier for some future work. The Y scale is different; cut it off at 8M
The intuition may seem obvious in retrospect. For a lifestyle of 130k real we would have a present value of 3.9M for a spend liability. If our current balance sheet says we have 3.9M net monitizable wealth now, then in theory we have a "feasible" plan and we would likely take relatively low risk[1] until late in the plan (conditional on wealth level at that time...let's assume it stays relatively high for now) at which point we'd be able to amp up the risk because we could. At 115 I'm throwing big parties. So, in other words that looks a little like an economically plausible rationale for a (conditional) glide path.
Some reasons I can see that one might deviate from that "guessed" up-glide-path are: 1) massive changes in wealth mid plan, 2) a lifestyle plan that desires an option on ratcheting up lifestyle later if possible, 3) the inability to successfully execute a weak defeasement plan over time, 4) large changes in lifestyle (creep over time), 5) and let's say legacy considerations. There may be others.
If, on the other hand, one has 1M, not 3.9 or 5M, one technically has an "infeasible" plan and the only way to win that game at the given lifestyle is to roll some dice; you can't make it with what you have now. Irlam, if I recall, calls it the lottery ticket effect. That's why there's a lot of yellow low down. Taking high risk will either crash you or save you while taking low risk will just crash you. Better to adjust the spending, I think.
Lastly, if one has an initial 10M, then a 130k lifestyle represents a mere initial 1.3% spend rate and the chances that you'll succeed no matter how you allocate is relatively high. It looks like it's optimal, given only this framework -- which I don't recommend -- to take a lot of risk.
Note that the feasible plan (say one with $4M in assets) that is positioned in low risk early may not necessarily be an allocation -- as in this hypothetical simple 2-asset model with 11 allocation steps -- to bonds. More likely a professional would recommend early on that one might want to start to construct some type of income floor via TIPS ladders or annuities (esp in a world where the horizon is uncertain). The remainder can tolerate high risk though.
The point of the post here, as I revisit this topic, is that dynamic allocation and fixed glide paths are not incompatible. The BI provides at least some economic rationale for the existence of a glide path. While any given retiree or their advisor may not have the will or energy or courage to be fully dynamic, they should at least know that "best" asset allocation is probably more likely to be dynamic than static, but dynamic conditional on information (e.g., level of wealth) that can only unfold in the future. A static glide path up is probably an helpful heuristic that can guide discussion for most practical people that are not, like me, coding BI engines for a blog. Irlam usefully points out that this dynamism and its conditionality that is seen in much of the literature is usually just dynamic in the horizontal (time) but often misses the vertical component (wealth at time t). His paper has a good cover of the history and literature of the methods and tools that have been in play related to this topic over time (including Markowitz, Trinity Study, Bellman, Shiller, and Basu and Drew among others).
Spending, by the way, I should mention, is no doubt a better tool than allocation, in my opinion, for the dynamic changes necessary to nudge a plan in the right direction over time, especially in the context of evaluating consumption utility. Or at least it's a tool of "first resort." Also, I should mention that forward projection of risk combined with feasibility tests is more useful, by a long shot, in dynamic mode than a backward induction allocation chart. But these kind of considerations are for another post or three.
Some reasons I can see that one might deviate from that "guessed" up-glide-path are: 1) massive changes in wealth mid plan, 2) a lifestyle plan that desires an option on ratcheting up lifestyle later if possible, 3) the inability to successfully execute a weak defeasement plan over time, 4) large changes in lifestyle (creep over time), 5) and let's say legacy considerations. There may be others.
If, on the other hand, one has 1M, not 3.9 or 5M, one technically has an "infeasible" plan and the only way to win that game at the given lifestyle is to roll some dice; you can't make it with what you have now. Irlam, if I recall, calls it the lottery ticket effect. That's why there's a lot of yellow low down. Taking high risk will either crash you or save you while taking low risk will just crash you. Better to adjust the spending, I think.
Lastly, if one has an initial 10M, then a 130k lifestyle represents a mere initial 1.3% spend rate and the chances that you'll succeed no matter how you allocate is relatively high. It looks like it's optimal, given only this framework -- which I don't recommend -- to take a lot of risk.
Note that the feasible plan (say one with $4M in assets) that is positioned in low risk early may not necessarily be an allocation -- as in this hypothetical simple 2-asset model with 11 allocation steps -- to bonds. More likely a professional would recommend early on that one might want to start to construct some type of income floor via TIPS ladders or annuities (esp in a world where the horizon is uncertain). The remainder can tolerate high risk though.
The point of the post here, as I revisit this topic, is that dynamic allocation and fixed glide paths are not incompatible. The BI provides at least some economic rationale for the existence of a glide path. While any given retiree or their advisor may not have the will or energy or courage to be fully dynamic, they should at least know that "best" asset allocation is probably more likely to be dynamic than static, but dynamic conditional on information (e.g., level of wealth) that can only unfold in the future. A static glide path up is probably an helpful heuristic that can guide discussion for most practical people that are not, like me, coding BI engines for a blog. Irlam usefully points out that this dynamism and its conditionality that is seen in much of the literature is usually just dynamic in the horizontal (time) but often misses the vertical component (wealth at time t). His paper has a good cover of the history and literature of the methods and tools that have been in play related to this topic over time (including Markowitz, Trinity Study, Bellman, Shiller, and Basu and Drew among others).
Spending, by the way, I should mention, is no doubt a better tool than allocation, in my opinion, for the dynamic changes necessary to nudge a plan in the right direction over time, especially in the context of evaluating consumption utility. Or at least it's a tool of "first resort." Also, I should mention that forward projection of risk combined with feasibility tests is more useful, by a long shot, in dynamic mode than a backward induction allocation chart. But these kind of considerations are for another post or three.
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[2] Other frameworks include single period MPT (Markowitz '52, Black-Litterman '92, etc) and classic multi-period (Monte Carlo) simulation using static allocations and fixed horizons.
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Pfau, Wade D. and Kitces, Michael, Reducing Retirement Risk with a Rising Equity Glide-Path (September 12, 2013). Available at SSRN: https://ssrn.com/abstract=2324930 or http://dx.doi.org/10.2139/ssrn.2324930
Irlam, Gordon (2014), Journal of Personal Finance;2014, Vol. 13 Issue 2, p9
On Hacking Out a (Really) Rough Asset Allocation Optimizer by Using Backward Induction and Dynamic Stochastic Programming. Jan 26 2017 RiversHedge
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