Friday, February 13, 2015

Dominated Strategies

A while back, I had in mind to write a series of posts on how game theory might be useful for analyzing retirement income strategies. I wrote the first, A Tiny Bit of Game Theory, describing how game theory might be useful in deciding when to claim Social Security benefits. But, then I got sidetracked by questions from readers about bond ladders and bond funds and now, nearly two months later, I'll wander back to game theory. (This freedom to meander is a wonderful part of retirement.)

In game theory terms, strategy A is said to "dominate" strategy B if a player is always better off playing A instead of playing B, regardless of the strategies chosen by other players. This is the strong form of domination. If strategy A's payoff is never worse than B's and sometimes better, strategy A is said to weakly dominate strategy B.

Say we have two bets, A and B. A always pays $200 and B always $100, no matter what other players do. Strategy A is said to strongly dominate strategy B because the payoff is always better when playing A.

If, on the other hand, strategy B always pays $100 and strategy A always pays at least $100 but sometimes more, then strategy A weakly dominates strategy B. The difference is that with weak dominance, the strategies can sometimes have equal payoffs. With strong dominance, the dominant strategy must always have a better payoff.

Here's where identifying dominant and dominated strategies pays off: game theory tell us that a rational player should never play a dominated strategy. In fact, there are game theory operations that simply remove dominated strategies from the game and out of consideration to simplify the game's analysis.

Are there dominated retirement income strategies? If there are, we can simplify the planning process by eliminating them from consideration.

Let's consider two forms of the sustainable withdrawal rate (SWR) strategy and refer to them as SWR-Fixed (or SWR-F) and SWR-Variable (or SWR-V).

The SWR-Fixed strategy tells us to calculate some percentage of our initial wealth and to spend that fixed amount throughout retirement. Let's use 4% as a sustainable withdrawal rate and $100,000 as our portfolio value on the day we retire. The SWR-Fixed strategy tells us we can spend 4% of $100,000, or $4,000, every year for thirty years with about a 95% chance of not outliving our savings. This is the SWR strategy you read about in the popular trade press.

That 95% is the probability that you will not outlive your savings calculated on the day you retire. If your portfolio declines in value after you retire and you keep spending the same dollar amount, your probability of failure will grow beyond 95%. Possibly well beyond.

The SWR-Variable strategy is similar, except that the spending amount is recalculated at the beginning of each year as a percentage of our new portfolio value. In this example, we would also spend $4,000 the first year, but the next year's spending would be 4% of the value of our portfolio at the beginning of the second year of retirement. That portfolio value, of course, is unpredictable and could be more or less than $4,000, depending on market returns for the first year.

This raises a key issue. What do we mean by a "better payoff?" Is SWR-F better because its income is predictable? Is it better because it's simpler to implement?

Or, is SWR-Variable a better strategy because it has less SOR Risk, as I explained in Sequence of Returns Risk and Payouts and provides more income when the portfolio prospers?

Using game theory, we get to decide individually which payoffs are "better" by defining the game precisely. We might, for example, use game theory to explore strategies that provide the best payoff in terms of simplicity of implementation, though given that either strategy requires minimal work once a year, that might be a somewhat trivial objective.

We could also create a game that values predictable annual income more highly than maintaining a maximum allowable level of risk throughout retirement. While some might consider any of these objectives reasonable, I propose that the most rational game for retirees would be one that maximizes annual spending while maintaining a ceiling on the risk of outliving our savings, say, never exceeding a 90% probability of ruin throughout retirement. Let's call this the Safety First game.

To identify potential dominated strategies in the Safety First game from our set of available strategies at this point, SWR-Fixed and SWR-Variable, we would need to show that one strategy always provides higher payoffs than the other, or in the weak form, that one strategy never does worse than the other.

First, let's consider the scenario in which the retiree enjoys excellent market returns throughout retirement. His portfolio value increases every year, at least on average. In this scenario, SWR-V will always outperform SWR-F, because 4% of an ever-increasing portfolio value beginning at $100,000 will always be greater than 4% of the initial portfolio value of $100,000.

For example, let's say portfolio returns for year one are 8%. At the end of year one, the portfolio value would be $100,000 less $4,000 plus 8% of $96,000, or $103,680. SWR would still pay out $4,000 at the beginning of the second year, but SWR-V would pay out 4% of $103,680, or $4,147.

Now, let's consider the other extreme, an ever-declining portfolio value. Playing SWR-V in this situation will always provide less income than playing SWR-F, but recall that we also have an objective in the Safety First game to manage risk of ruin throughout retirement.

A retiree with a $100,000 portfolio at the beginning of 2007 planning for a 30-year life expectancy and planning to spend 4% annually had a 9.8% probability of outliving her savings, according to Moshe Milvesky's formula for probability of ruin. Had her portfolio fallen 25% by 2009 to $75,000, she had two choices. She could lower her spending to about 4% of $75,000 ($3,000), and still have a probability of ruin of about 9.8%. Alternatively, she could continue to spend $4,000, which would be a 5.33% spending rate and and would raise her probability of ruin from 9.8% to 21%.

That is an example of what could happen over two or three years. Most simulated SWR-Fixed strategies end with the retiree's portfolio holding about half its initial value in real dollars at the end of retirement. What happens to the 9.8% probability of ruin if a retiree's portfolio declines in value to $50,000 and she still has a 15-year life expectancy? According to Milevsky, she can continue to spend $4,000 and have a 28% probability of going broke, or lower spending to $2,600 and hold the risk steady at her original 9.8% probability of ruin. A portfolio's value can decline very quickly, or over many years.

If she played the game I mentioned above that values consistent income over maintaining acceptable risk, then SWR-F would have a higher payoff than SWR-V, but not so in the Safety First game that maximizes spending while maintaining an acceptable level of risk.

In plain English, this shows that when our portfolio declines in value and we continue to spend the same dollar amount, as with SWR-Fixed, we expose ourselves to greater risk of outliving our savings. When our portfolio value declines, we can spend less and maintain a constant probability of ruin (the SWR-V strategy), or we can spend the same amount and take on more risk of ruin (the SWR-F strategy).

The fixed-spending sustainable withdrawal strategy is mostly an invention of the financial press. Even William Bengen noted in Conserving Client Portfolios During Retirement that "the adviser should examine the projected current withdrawal rate through the entire time horizon of the clients, not just the first year of retirement."

Noted retirement experts like Michael Kitces have long suggested that SWR-Fixed is a research technique and that no one actually implements it. I hope that is true, but I have reason to doubt it. I talk to readers and clients frequently who plan to implement fixed-withdrawal SWR strategies, Money magazine recommended it for perhaps 20 years (but backed off after the huge losses of the Great Recession), and I recently received a sample Kiplinger newsletter that suggested it, so I have to think someone is doing it.

In the rising-portfolio value scenario, SWR-V always pays off more.  Risk of ruin is not a concern when portfolio values increase. In the declining-portfolio scenario, SWR-V has a better payoff (though not higher) because, although it provides less and unpredictable income, it shows the maximum amount we can spend without taking on more risk. In this game, SWR-V dominates SWR-F and, according to game theory, SWR-F should never be played.

The only retiree who should play SWR-Fixed is one who cares about the probabilities of outliving his savings the day he retires, but is unconcerned with that risk for the rest of his retirement. Sounds a bit irrational, no?

There is an important, though often overlooked point I should add. Retirees tend to spend what they need to spend. Strategies like SWR tell us how much we can safely spend, but we aren't required to spend that amount. This issue is also sometimes raised by retirees regarding Required Minimum Distributions from IRA accounts. If the withdrawal from either of these is more than you need to spend, no one is telling you that you have to spend it. We're just telling you the maximum amount of spending we think should be safe.

Is there a strategy that dominates SWR-Variable? I'll look at a candidate next time in Dominated Strategies and Dynamic Spending.


  1. A nice summary comparison Dirk. Many view these as opposing strategies. I believe they are part of the same spectrum of strategies. SWR-F may be calculated and compared to SWR-V using the dynamic updating method, i.e., annual review.

    SWR-F merely represents an amount of spending that rarely would need to be reduced (but it may if market/economic events are outliers, say 2 standard deviations or more) ... thus a close proxy for what may be called necessary spending.

    SWR-V is a prudent maximum spending amount FOR THAT YEAR given the facts at the beginning of the year with returns and expected longevity updated (as you know - but readers may not - expected longevity is an age one never reaches based on current present age).

    The difference between the two is called "discretionary spending." The safety first crowd often gets alarmed or defines failure as a need for reducing spending. Once the model incorporates a method to distinguish necessary from discretionary spending, one realizes that essentially the same goal is accomplished - with the added benefit of being able to spend a little more (discretionary) when times are good; and not when they are not. BUT, spending is still possible - spending is not gone as safety first crowd fears.

    Our research (JFP Nov 2011) shows that spending is the one variable under one's control that can make a difference on outcomes (having money when older for those necessary spending years remaining at that time - i.e., not broke). Messing with allocation is not (essentially no difference when compared to control data).

    With all this in mind - since the future is always unpredictable, a flexible strategy that can adjust for the unexpected will provide better results over one that can not, or does not, adjust. What is underappreciated and under represented in most writings on the subject, is that a flexible strategy should incorporate a lower end of spending up front so it is clear where necessary and discretionary spending are.

    The next element often understood is that we all live on the same blue marble so there is no economic system we can extract ourselves from during bad times, or insert ourselves into during good times, that isn't being experienced by all at those times too. This gets to understanding the differences between risk retention (e.g., managing the portfolio - indexed investing) versus risk transference (have an insurance company manage the portfolio - annuity of some sort). Both methods are exposed to the same systemic risks. The difference is that the single insurance company increases risk because as a single company, they could one of those companies to disappear. An indexed portfolio would not (unless all companies within the index simultaneously disappeared).

    A great post once again Dirk!

  2. Thanks, Larry. You add some important additional points for consideration.

    I believe it was your referenced research that shows that the important variables of probability of ruin are spending rate, expected rate of return, life expectancy and asset allocation, in that order. I agree, tweaking the allocation is a second-order consideration.

    In addition to risk retention versus transferring that risk it to an insurance company, we also have to consider the risk-pooling benefit of annuities that cannot be duplicated by an individual retiree. In one case you have to worry about the risk of a single company and, in the other, the risk of a single retiree. A helping of each might be in order.

    You make great points. Thanks, again, for contributing!

  3. Indeed Dirk. That comparison was made in Wade and my paper (JFP Apr 2014) "Lifetime Expected Income Breakeven Comparison between SPIAs and Managed Portfolios,"

    Properly managed, portfolios can outlive people too. The red herring has always come from using set end dates instead of using rolling expected longevity ages. If there are always future years to be funded, there needs to always be future portfolio balances to do so. Unless the retiree spends too much - but that is the whole point of measurement and monitoring isn't it - along with meaningful Decision Rules about what to do and when to do it.

    Our Apr 2014 paper suggests waiting until older ages before committing to risk transfer. Purchasing power of the dollars between now and elder age is factored in the paper. We also "punished" portfolio cash flows by taking the portfolio fee out of the cash flow (thus reducing it by that amount), versus taking the portfolio fee out of the portfolio balance first before making the cash flow calculation. A subtle yet important distinction for those who feel the paper is tilted towards anti-SPIA.

    Social Security is the first helping in the risk-pooling category.

    Your welcome ... and great conversation Dirk!

    PS. Links to read any of our published JFP research papers are at for those who don't have login credentials to read JFP papers otherwise.

  4. Note to my readers: The referenced paper can be found here.

  5. Hi again Dirk, your combined posts for the 6th and 13th of Feb got me thinking about a couple of other points:

    1) The problem I see with trying to be conservative with estimates, one about risk return parameters (point 2 below) and the second about longevity … and my main point here … is what happens with overly conservative longevity estimates, say age 95 for example as a suggested end age for all … is that the many are under spending just in case they’re the few who may outlive age 95 (a low percentage, depending a present age, but more so when most SWR perspectives look at 30 years implying the marker is only age 65).

    This is why myself and others like Ken Stein, and at times you, suggest using life expectancy tables to mark time periods – and then update that time period each year (what I call Dynamic Updating) along with updated risk & returns data. Such an approach shifts spending into the more likely years one may live as well as when they’re also more likely to be the go-go years (why save spending now for when you’re in the no-go years or not here at all?). Now I know the marble is rolling around – but herein comes the point where customization based on individual desires comes into play. The 4% rule has nothing to do with most people’s portfolio characteristics nor their current age.

    A comparison between bottom and top spending may be found here:

    2) asset allocation should be targeted towards a much forgotten purpose – to shift the standard deviation curve away from the bad, left-tail, events as described succinctly by Larry Swedroe in his book “Reducing the Risk of Black Swans.” Basically the purpose of proper allocation is to first target one’s risk parameter and then to both tighten the standard deviation as well as shift the curve to the right side. More on that here:

    In summary, 1) people should understand that conservative time frame estimates may result in their under spending relative to time periods closer to expectation from any actuarial table. 2) people should look at their portfolio's specific risk return characteristics instead of applying a rule of thumb not representative of their specific holdings. That’s enough to ponder – in short – for the moment: Rules of thumb are often not representative of specific people.

    Great blog posts both Dirk.

    1. Larry, I agree – perhaps more than you think.

      "and at times you, suggest using life expectancy tables to mark time periods"

      I don't think at times that retirees should consider life expectancy when spending from a volatile portfolio, I always think they should. Sorry if I haven't made that clear enough. The post I am working on for this week on game theory should make that abundantly clear.

      Second, I don't suggest that conservative estimates of safe spending are a good answer, but only that they are a safer guideline if you are going to play SWR than a more aggressive spend rate would be. If you look at the amount of savings you would need in order to spend 3.5% versus what you would need to spend 4.5%, you can see that conservative estimates are extremely costly. My point is not that SWR is fine with conservative estimates, but that SWR is a very risky strategy and that is shown by our inability to accurately know a safe spending rate. Hence, the range.

      I'm not a fan of SWR unless the retiree has a lot of money compared to her spending needs. (If you have a 1% spend rate, just about anything works.) I prefer a sound floor that takes the pressure off spending from the upside portfolio.

      Regarding asset allocation, I'm with William Bernstein:

      [Step] 1. Determine your basic allocation between stocks and bonds. First, answer the question, "What is the biggest annual portfolio loss I am willing to tolerate in order to get the highest returns?"

      That will be easier for some readers to understand than shifting the standard deviation curve away from Black Swans, but it means the same thing.

      Thanks for the additional explanation, Larry. I try to make my posts readable for a broad audience, but I think the discussion section is great for those "for extra credit" kind of guys!