In addition to other methods we’ve discussed, a third type of variable spending model uses dynamic programming methods. These methods rely on complex computing power and mathematical equations to integrate spending and asset allocation decisions more completely over the life cycle.

Dynamic programming provides a road map at each point in time for optimal spending and asset allocation, which have been determined by first considering optimal future behavior stemming from today’s decisions.

Because the complexity of solving for what is optimal today based on how today’s decisions will affect what happens tomorrow (and tomorrow’s tomorrow, and so on), mathematical simplifications have been developed which basically involve working backward through time.

Optimal end of life decisions are determined, which then feed into making optimal decisions at each younger age.

Due to their mathematical complexity, dynamic programming methods are mostly discussed in the realm of academia and have not yet become a common part of the toolkit for individual retirees.

Gordon Irlam’s AACalc and Laurence Kotlikoff’s E$Planner are two software programs based on dynamic programming that are available for consumer use.

Gordon Irlam and Joseph Tomlinson’s 2014 article “Retirement Income Research: What Can We Learn from Economics?” tries to make dynamic programming methods more accessible to those outside of academia.

It is hard to generalize about the solutions for dynamic programming methods, but Irlam and Tomlinson provide an example based on a case study in their article. In terms of understanding their findings, we can think about dynamic programming as a different way to choose an evaluation criteria.

I have been discussing the choice of the PAY Rule™ and a visual comparison of the results as a way for individuals to decide on a retirement income strategy.

For dynamic programming, individuals would not make these decisions directly. Instead, a utility function would be determined, which provides a link to the satisfaction the individual derives from different amounts of spending and legacy.

This utility function would build in a degree of spending flexibility about how willing the individual is to spend more today with the tradeoff that this could require less spending in the future. Given the mathematics of this tradeoff, the remainder of the strategy will follow from the dynamic programming methodology about how much to spend each year and how to adjust asset allocation each year as circumstances evolve.

The results are meaningful to the extent that the mathematical function expressing the tradeoffs between spending certainty and the average level of spending—as well as between spending and legacy—adequately reflects the preferences of the individual using the model.

Irlam and Tomlinson build a case study for a conservative individual with $500,000 of financial assets, $20,000 of Social Security benefits, and no particular desire to leave a legacy.

They compare outcomes for the solution offered by the dynamic programming methodology to the outcomes for various decision rule and actuarial methods and asset allocation strategies.

For spending, they consider constant inflation-adjusted spending strategies, a fixed percentage strategy, and a strategy similar to the RMD rule spending. For asset allocation, they consider fixed allocation strategies, age in bonds, age minus 10 in bonds, and target date funds.

Among the other strategies, they find that the closest match to their optimal dynamic programming solution is the RMD-styled strategy with a fixed 90% stock allocation. Next is a 6.8% fixed percentage strategy with a 90% stock allocation. In third is a constant inflation-adjusted spending strategy with a 60% stock allocation.

All of the other asset allocation strategies result in worse outcomes because they lack sufficient aggressiveness. It is interesting that the variable spending strategies are accompanied by higher stock allocations, the reason being that varying withdrawals provide a safety value for stock volatility.

With constant inflation-adjusted spending, a lower stock allocation to reduce volatility is the only avenue for alleviating sequence risk.

For readers who are interested in further exploring the mathematics of dynamic programming, Samuel Pittman, Yuan An Fan, and Steve Murray also provide a simplified example in their 2013 article, “Optimizing Retirement Income: An Adaptive Approach Based on Assets and Liabilities.”

In it, they use dynamic programming to create a retirement income strategy for a simplified world with three time periods and two investment outcomes.

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