Dominated Strategies, Illogical Strategies, Problematic Strategies and Strategies That Just Make Me Queasy

In Pure and Mixed Strategies, I noted that we can make life simpler for retirees by winnowing out dominated strategies and strategies that are logically unsound. I showed in Dominated Strategies and Dynamic Spending that dynamic updating of sustainable spending amounts dominates the SWR-Fixed and SWR-Variable strategies, so I cross those off my personal list of reasonable alternatives for retirement income plans.

I believe that there are strategies that are flawed, some logically sound strategies with problems, and others that just give me a queasy feeling. So, in this post, I will share my feelings on those three categories of retirement income strategies.

Dominated Strategies: There are strategies that appear to be dominated by better strategies and, according to game theory at least, should not be played. In previous posts, I noted that SWR-Fixed and SWR-Variable strategies are dominated by a dynamic updating strategy. Are there other dominated strategies that are often proposed? 

The RMD Strategy, in which the retiree bases her spending on the required minimum distribution amounts the IRS mandates for IRA’s is also a dominated strategy. (You can read about RMDs here.) Dynamically updating all important variables of sustainable spending will always perform better than a strategy of updating only the retiree’s age and new portfolio balance, as the RMD strategy does.

Logically Unsound Strategies: There are also retirement income strategies that simply aren't logically sound, like the "Spend Dividends Only" strategy. This strategy seems to be based on a misunderstanding of how dividends work.

I won’t devote a lot of explanation here, because the topic has been thoroughly vetted by others. I recently recommended this explanation by Canadian Couch Potato.  But, in short, when a company pays you a dividend on a stock worth $10.00, they pay you perhaps 30 cents in cash and the market immediately reduces your stock’s value to $9.70. You are no better off and no worse off. (You may actually be a tad worse off if you hold the stock in a taxable account because dividend payments can be a taxable event.) Furthermore, investing primarily in dividend-paying stocks will limit your portfolio’s diversification.

I cross Spend Dividends Only off the list because it is illogical and, as Wade Pfau recently put it, simply isn’t a valuable strategy. 

SWR is another logically unsound strategy when applied literally. It ignores conditional probabilities of failure. A common counter-argument is that no one implements SWR literally. I hope that is correct, but I doubt it.

Problematic Strategies: Now, let’s look at an example of a strategy that has not been shown to be dominated and that is logically sound, yet still problematic. The Bond Ladder and Longevity Insurance (BLLI) strategy proposed by Professor S. Gowri Shankar, for example, is logically sound but suffers from two problems.

(If the BLLI strategy is unfamiliar, there is a nice summary of it and several other strategies in this paper by Wade Pfau and Jeremy Cooper.)

The BLLI strategy proposes building a 20-year TIPS Bond Ladder and funding later years (beginning age 85 for someone retiring at 65) with Deferred Life Annuities (DLAs). The idea is that the retiree won’t have to give up control of his capital for the first 20 years – a common complaint with annuities – and that deferred life annuities are cheaper than immediate life annuities. (See Wade Pfau's Why Retirees Should Choose DIAs Over SPIAs.)

The BLLI strategy, however, has an inflation problem. Because DIAs will provide income well into the future, insurance companies typically won’t offer them with inflation protection and, of course, "well into the future" is when inflation takes its biggest toll. When inflation protection is offered, it does not cover the period from purchase to the first payout, in this example 20 years.

This is different than funding half of your retirement income with a TIPS bond ladder and the other half of the income with annuities. BLLI suggests exclusively funding the first half (or so) of retirement with a TIPS bond ladder and the second half exclusively with annuities.

The idea of maintaining control of capital and liquidity is also somewhat problematic, since the retiree will need the money in her bond ladder for living expenses to age 85 and can’t really spend it in an emergency. Funding the entire 30 years with a TIPS bond ladder and prematurely spending the most distant rungs is problematic enough, but it is possible that she won't live the full 30 years and may not need to spend that money ever. Diverting funds meant to meet living expenses during the first 20 years of retirement is significantly riskier.

Nonetheless, it is neither flawed nor dominated and stays on the list.

Strategies that Make Me Queasy: Some strategies can be logical and not dominated and perhaps not problematic to some, but still make me feel uncomfortable. To wit, the Floor-leverage Rule and Zvi Bodie's floor-and-upside strategy of 90% TIPS bonds and call options (LEAPS). Both are known as "barbell strategies" because they invest in extremely safe and extremely risky assets with nothing in between.

As I suggested in Hope and Your Retirement Portfolio, most retirees won't be comfortable with the possibility of losing all or most of their upside "hope" even with a comfortable floor in place. Retirees who purchase call options will often see those options expire "out of the money", in other words, worthless. The financial argument will be that the options served their intended purpose, even the ones that expired worthless, and that is correct. Most of the retirees I know, however, will find that cold comfort when they see a few ten thousand-dollar call positions disappear in a poof of smoke. Actually, it isn't even that dramatic. It's just not there, anymore.

Zvi Bodie's friend, Jeremy Siegel, seems to agree (download PDF). On multiple occasions Siegel has said, "You know, I find it a little strange — Zvi says he’s giving conservative investment advice, and then advising all your clients to buy call options."

The Floor-leverage Rule makes the LEAPS approach look tame in comparison. The idea for its upside portfolio is to employ triple leverage at the equity end of the barbell. The typical investor can't purchase a stock portfolio with triple leverage. It's illegal. But, as Sharp and Watson point out, they can purchase shares of a 3x leveraged ETF like UPRO.

I'm crossing Floor-leverage off the list of reasonable strategies not because it makes me queasy to apply huge leverage to my entire upside portfolio (it does), but because the upside strategy doesn't work. The problem is that these ETFs are not the same as a triple-leveraged stock index fund. In fact, they aren't portfolios of stocks, at all. They're portfolios of derivatives that only track stock indexes for short periods of time. They're best suited to short term investments, which shouldn't be part of a retirement income plan.

Bodie's options strategy stays on the list. It's riskier than I can accept and I don't think it will fit the temperament of most retirees, but it isn't flawed. It will outperform other strategies in some scenarios, so it isn't dominated.

Some of the strategies I've crossed off the "sound" list may fit your individual financial situation and may be bets you're willing to take. Some of them will even be recommended by advisers. I would pare down the list of reasonable retirement income strategies by at least a third, as shown in the following table. This is my personal perspective and not everyone would agree.

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Glide Paths

Retirement researchers Michael Kitces and Wade Pfau published papers on glide paths in the Journal of Financial Planning in 2014 and 2015, including Retirement Risk, Rising Equity Glide Paths, and Valuation-Based Asset Allocation, suggesting a rising glide path for retirees beginning with a low equity allocation early in retirement and rising throughout. Researcher David Blanchett has published contradicting studies that show better performance from declining equity glide paths throughout retirement, which has been the traditional recommendation, in Revisiting the Optimal Distribution Glide Path.

As you can imagine, these conflicting recommendations have caused more than a little anxiety among advisers and do-it-yourselfers and experts like William Bernstein have even entered the discussion.

Ultimately, the research has left some retirees asking whether they should follow a rising glide path or a declining glide path.

The correct answer is "none of the above" – they should follow their personal financial situation as retirement progresses. (See Dominated Strategies and Dynamic Spending.)

Imagine two 90-year old retirees. Rising glide paths would suggest that they both hold large equity positions, while declining glide paths would suggest they both hold large bond positions. Now imagine that one still has tons of savings left, more floor income than she needs and is, as William Bernstein puts it, investing for her heirs. The other has barely enough wealth left to cover another five years of expenses. Should both invest with the same asset allocation because they are the same age? A similar argument can be made at any age and the correct answer is that we need to consider more than age or retirement date when determining our asset allocation.

This research aims to isolate one factor, the retiree's planned retirement date (or, roughly speaking, the retiree's age), and to determine its effect on the retirement investment problem. The question it attempts to answer could be framed in a number of ways, like:

• What would be the best asset allocation pattern for a retiree if we only knew that retiree's planned retirement date and nothing more?


• What allocation should we recommend to a retiree who wants to do little or no investment planning? (Note that this person is unlikely to be a reader of this blog.)


• What glide path should be implemented by a target-date mutual fund that serves a broad target market of retirees who happen to share the same planned retirement date?

I feel certain that all of the authors of these papers would agree that a customized, individual retirement plan will always be preferable to one based entirely on the retirees age/retirement date. (One of the authors actually told me this, so it isn't entirely a guess.)

As explained in Wade's post of March 2^nd on RetirementResearcher.com, To Rise or Not to Rise, the authors are currently comparing notes to understand why their results differ. Initial thoughts are that the differences are explained by different capital market expectations. Both sets of expectations, or assumptions about future markets, are imperfect predictions and either (or neither) might turn out to be right.

As Wade explains, "… The choice of glide path is not always fixed, as it does vary with the stock market valuation level at the time of retirement."

So, which glide path is best if we ignore everything about a retiree except the planned retirement date? We're not sure, but the research does seem to show that it depends on stock market valuations and interest rate expectations at the time of retirement (no surprise).

How does this information help the typical retiree invest her savings if she doesn't really care how the mechanisms of retirement funding work? It doesn't help much, in my opinion, at least not at this point. As Blanchett suggests, more research is needed.

More importantly, this information won't apply to most advisers or do-it-yourselfers because they will be willing to do more work than to simply write down a planned retirement date. They can find a much better fit.

An individual household's optimum asset allocation should be estimated by considering several more factors, including how well their retirement is funded and the maximum loss they could stomach in a bad market crash. By continually managing asset allocation based on current conditions of the retiree's finances, at the end of retirement that retiree's glide path will have followed some curve that will probably look like one of the half dozen or so currently recommended glide paths simply because they pretty much cover the entire range of possibilities. (The 2014 Investor's Guide issue of Money magazine identified six glide path recommendations that had little in common. Pozen's has a tongue sticking out the end. What's up with that???)

We will not know in advance which glide path that retiree will end up following. It will be determined by a random walk through market returns, how long we live, our expenses, the decisions we make and many unpredictable factors.

It's important to understand that research of any kind is new information, or a new argument, that we should consider in light of what we already believed to be true. It doesn't automatically replace what we thought we knew. (See Think Like a Bayesian Pig.)

When this evidence conflicts, it gives us a chance to learn more. This research doesn't tell us that one glide path is better or worse — both sides make good arguments — but it begins to tell us under what conditions one glide path might perform better. That doesn't necessarily mean one is right and the other wrong or even that a well-planned retirement needs one. Please keep that in mind when you read academic papers. They are usually new evidence or new arguments and not universal truths that dispel old beliefs.

What should a retiree do with this information about glide paths? Probably nothing. You should customize your asset allocation instead of following a glide path. There are a couple of interesting points you can use, however. First, it turns out that a 60% equity portfolio throughout retirement isn't a bad "couch potato" allocation.

Second, as Wade points out, you might give more consideration to cash and short bonds than to intermediate bonds. "Another important point in that summary is that we did look at both short-term (6 month or 1 year) bonds as well as 10-year bonds in our analysis, and we found that the shorter-term bonds were of much greater help than longer-term bonds. When the focus is on protecting from downside risks, the additional volatility caused by the 10-year bonds hurt retirement outcomes by more than could be compensated by their higher average yields. This ties into David’s article as well, since his bond allocation was 75% to (I believe) 10-year bonds, and 25% to cash."

This is an important point to remember when you think about taking on more risk to increase the return of the bond portion your retirement portfolio. Bonds and cash are there to protect against downside risk, not to maximize portfolio return.

You can also just follow the ongoing research. I find the discussion quite enjoyable. It's a work in progress.

Pure and Mixed Strategies

In a few recent posts, I suggested how game theory might be used to gain a different perspective on the Social Security claiming decision (Game Theory and Social Security Benefits) and why updating your sustainable spending amount periodically (Dominated Strategies and Dynamic Spending) will always perform better than spending a fixed amount based on your initial portfolio balance in retirement (SWR-Fixed), or by spending a fixed percentage of remaining portfolio balance each year but ignoring other determinants of portfolio survival like decreasing life expectancy (SWR-Variable).

The benefit of the spending strategy analysis it that is allows us to winnow out inferior strategies when we choose our retirement income plan. SWR-Fixed and SWR-Variable are dominated strategies. Game theory tells us never to play a dominated strategy, which only makes common sense.

I admit two motives for these posts. The first is that I am fascinated by game theory and believe it provides valuable perspective on the retirement planning problem and the second is that I'm convinced we can simplify retirement planning. 

How does this simplify the retirement income strategy choice? By eliminating dominated strategies as game theory recommends, and eliminating other strategies that aren't logically sound, we can winnow a dozen or more proposed strategies to a significantly smaller number of truly valuable strategy choices. 

In this post, I'll consider another concept of game theory, pure and mixed strategies, and how they might be useful for analyzing retirement income strategies.

According to GameTheory.net, a pure strategy defines a specific move or action that a player will follow in every possible attainable situation in a game. A mixed strategy is created by playing members of a set of available pure strategies at some proportion of each.

Assume a tennis player has two pure strategies available: serve to her opponent's forehand or to her opponent's backhand. She might also attempt to keep her opponent guessing with a mixed strategy by randomly serving to her opponent's backhand or to her opponent's forehand.

Game theory will use the server's success rate serving and the opponent's success rate returning serve from both sides to calculate the optimum proportion of serves to each. Based on probabilities of success for both pure strategies and responses, game theory might tell her, for instance, that the optimum strategy is to randomly serve to a particular opponent's forehand 30% of the time. This is a mixed strategy.

Let's consider some pure retirement income strategies including sustainable withdrawal rates (the dynamic kind, since game theory tells us that SWR-Fixed and SWR-Variable are dominated), a Social Security benefits strategy, an annuity strategy, a time-segmentation strategy and a TIPS bond ladder strategy. Other strategies have been proposed, but let's go with this shorter set of pure strategies for now.

Why isn't the floor-and-upside strategy on the list? Glad you asked. Because floor-and-upside is a mixed strategy consisting of some mixture of pure floor strategies and pure upside strategies.

The floor strategy could consist of life annuities, TIPS bonds held to maturity, Social Security benefits or some combination of these.

The upside strategy contains risky assets like stocks and bonds. SWR portfolios typically recommend something like 50% stocks and 50% bonds. Jason Scott's and John Watson's floor-leverage rule (download a PDF) recommends 15% of assets be invested in a triple-leveraged ETF of derivatives. Zvi Bodie and Nassim Taleb have recommended an upside portfolio of 10% of assets invested in long term index options (LEAPS).

Note that a mixed strategy can allocate zero percent to some available pure strategies, so for instance, an SWR strategy can be considered a floor-and-upside strategy allocated 100% to the upside portfolio and 0% to the floor strategy. More importantly, because nearly all Americans have some Social Security income or public pension income, it will be very rare that a retiree plays a pure upside strategy.

An exception to this observation is retirees who postpone claiming Social Security benefits and spend from a stock and bond portfolio until those benefits start, but by age 70 at the latest, they will likely have a floor-and-upside strategy, though they may not think of it that way.

While it will be rare for a retiree to implement a pure upside strategy with no floor, it is easy enough to implement a pure floor strategy with no upside portfolio. A retirement income plan based solely on pension or Social Security income would qualify as a 0% upside/100% floor portfolio, as would any strategy comprised solely of Social Security benefits, TIPS bond ladders and life annuities.

In other words, nearly all of us will have a floor. Those of us with adequate retirement savings can choose to add more floor, add an upside strategy, or implement some combination of the two. This is the first decision in choosing a retirement income strategy. It answers the question, "how much of your retirement savings are you willing to risk in the stock market in hopes of being able to spend more?"

For those who answer that they wish to take no risk with their standard of living, the next step will be to determine how to most effectively build a floor of income. For the rest, the next step will be to determine how much of their desired standard of living should be locked in with a floor portfolio, with the remainder put at risk in the market.

Viewed from this perspective, sustainable withdrawal rates is a floor-and-upside mixed strategy with a floor consisting of Social Security or pension benefits. A TIPS Bond Ladder strategy is a floor-and-upside mixed strategy of Social Security or pension benefits and a TIPS Bond Ladder with zero percent upside portfolio strategy. Floor-leverage rule is a floor-and-upside mixed strategy with a floor consisting of 85% of our portfolio plus Social Security or pension benefits and an upside portfolio strategy consisting of investing 15% of assets in a triple-leveraged derivatives portfolio.

Most strategies can be viewed as a form of a mixed floor-and-upside strategy and understanding this may simplify your decision of which strategy to implement.

Pure upside strategies will be rare, because most Americans will have Social Security benefits or public pension income at some point. That leaves a floor strategy or a mixed floor-and-upside strategy as the options available to most retirees.

This, of course, is the root of the "safety first" versus "probabilities" divide, but I don't see the divide so much as a disagreement on whether or not to put standard of living at risk as one of how much of our standard of living we should bet in the market. Because most of us are going to have a floor and probably a mixed strategy, the big question is, "how much floor?"

I think this is a far more reasonable approach than having retirees read about a dozen or so strategies to pick the one with which they feel most comfortable.

If you're interested in game theory, William Spaniel has an outstanding series of tutorials on YouTube entitled Game Theory 101.  If the academics of the subject interest you, Yale filmed Professor Ben Polak teaching Econ 159 Game Theory. He is an amazing professor and, although it doesn't use modern on-line teaching technology, it is probably the best on-line class I have ever taken.

Made me wish I'd gone to Yale. Go figure.

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Dominated Strategies and Dynamic Spending

Sharp-eyed readers will notice that I have tweaked my blog format to include some of my favorite posts from other retirement blogs. Retirement blogs may not be the best place to find sharp-eyed readers and I have three pairs of reading glasses here by my keyboard, just in case. Nevertheless, you will find these posts in the sidebar. This week, I included one from the Canadian Couch Potato blog on Spending Dividends Only and another from Wade Pfau's new website. I hope you enjoy them both.

In my last post, Dominated Strategies, I showed that for retiree's who want to keep their risk below a maximum level throughout retirement, game theory tells us that the variable sustainable withdrawal rate strategy (SWR-V) never provides worse payoffs than fixed-dollar withdrawals (SWR-F) and SWR-V provides better payoffs if the portfolio grows.

Game theory principles tell us that SWR-V weakly dominates SWR-F and that we should never play a dominated strategy, so I cross SWR-Fixed off my list of strategies to consider. (As I mentioned in my previous post, even William Bengen stated that SWR's should be revisited throughout retirement and not set in stone.)

SWR-F underperforms SWR-V, which prescribes spending a fixed percentage of an ever-changing portfolio value rather than a fixed dollar amount, because SWR-V uses new information as it develops over time, the current value of a retiree's savings. SWR-F only calculates a spending amount that was safe on the first day of retirement (an a priori expectation).

As conditions change, like our portfolio value, SWR-V takes advantage, increasing spending when it is safe to do so. By decreasing spending when it becomes riskier, SWR-V reduces sequence of returns risk. SWR-F ignores this new information.

There are other important changes besides portfolio balance to the key determinants of the probability of ruin as retirement progresses, including market return expectations, remaining life expectancy, spending needs, and risk tolerance.

As David Blanchett and Sudipto Banerjee have written (both links download PDFs), retirement spending typically declines over time, about 3% a year on average. A retiree's risk tolerance and capacity can also change over time as dependents need less support, for example, or a spouse is lost. And, of course, life expectancy constantly declines at a rate of a little less than a year per year of life. Neither the SWR-F nor the SWR-V strategies account for any of these important changes, leaving open the possibility that there is a strategy that dominates SWR-V.

If considering more data and more timely information improves retirement income spending strategies, then a strategy that considers more new information than SWR-V takes into account could be expected to dominate it. I will refer to this new strategy as "Dynamic Spending."

(David Blanchett and Larry Frank have written about this strategy previously in A Dynamic and Adaptive Approach to Distribution Planning and Monitoring, as has Ken Steiner. Larry Frank provides a nice explanation in a blog post entitled, "How income may compare between Dynamic and Safe approaches.")

Let's consider a version of the Safety First game from Dominated Strategies as a strategic game in which the retiree wishes to maximize available spending while ensuring that risk of ruin never exceeds a desired level. The SWR-Fixed strategy assumes some acceptable probability of ruin, typically 5% to 10%, at the beginning of retirement, but lets the risk drift throughout retirement in order to ensure a predictable, fixed amount of annual spending. Retirees who are happy to see steady spending even when their portfolio declines may not understand that it comes at the cost of increased probability of ruin.

SWR-Variable fixes the variable risk problem of SWR-Fixed but generates unpredictable annual spending. (A retiree spending from a volatile portfolio can have constant risk or constant income, but not both.) In fact, SWR-V "over-fixes" the risk problem because it doesn't consider a declining life expectancy. Over time, risk will decline with SWR-V and SWR-F as the retiree's remaining life expectancy declines. A retiree who thinks a 5% risk of outliving savings is acceptable, for example, might see risk decline to 3% as she ages, which means she will be spending less than she could safely spend.

A Dynamic Spending strategy will recalculate a sustainable withdrawal rate annually by considering updated portfolio balance, an updated life expectancy, changes in risk tolerance over time, changes in expected future returns and changes in spending.

Whether the retiree's portfolio balance trends downward or upward, Dynamic Spending will provide a better payoff than either SWR strategy because it considers remaining life expectancy. As remaining life expectancy declines throughout retirement, risk of ruin is reduced and the sustainable withdrawal rate increases. (The sustainable withdrawal amount will decrease if portfolio losses exceed the benefit of the life expectancy decrease.)

Spending gains due to decreasing life expectancy increase exponentially. Even if portfolio value remained flat throughout retirement, decreasing life expectancy would more than double spending by the end of a long retirement (see chart). SWR-V and SWR-F ignore this increase.

When portfolio values trend upward, Dynamic Spending will have a larger payoff than SWR-V because it will be augmented by a declining life expectancy contribution. When portfolios trend downward, Dynamic Spending will limit increasing risk of ruin by reducing the spending percentage and by adding the declining life expectancy contribution.

As I mentioned in Dominated Strategies, SWR-Variable "over-fixes" risk reduction. Spending a percentage of remaining portfolio balance and ignoring the life expectancy contribution with a declining portfolio eventually lowers risk too much, unnecessarily lowering spending. By considering both, Dynamic Spending adjusts spending to the retiree's current risk tolerance and maximizes spending at that level.

Now, let me try to simplify this rather lengthy post. All three of these strategies use the same basic mechanism. They calculate a sustainable spending amount using Milevsky's formula, simulation or historical data and all three are based on the same information regarding the retiree's financial situation.  The difference is when we recalculate using new data.

SWR-Fixed makes a single calculation at the beginning of retirement and ignores any new information thereafter, no matter how critical that information might be. (Intuitively, this should feel like a bad idea.) The information to calculate the SWR-Fixed sustainable spending amount should include initial portfolio value, expected market returns, life expectancy, and asset allocation based on risk tolerance.

SWR-Variable uses the same information except it recalculates the sustainable spending amount every year, taking into consideration changes to the portfolio value from the previous year, but nothing more. And it assumes that the withdrawal percentage calculated at the beginning of retirement remains the best one. Doing so reduces sequence of returns risk, but it doesn't maximize sustainable spending.

Dynamic Spending recalculates sustainable spending every year, too, but it doesn't stop with updating portfolio values, as does SWR-V. It also updates a decreasing life expectancy, changes in risk tolerance and capacity, and expectations about future market returns. Dynamic Spending maximizes the sustainable spending amount given the retiree's current risk tolerance.

Dynamic Spending always provides better payoffs when risk is considered appropriately than does SWR-Fixed or SWR-Variable. SWR-Fixed and SWR-Variable are strategies that are dominated and should never be played. That's a stronger message than "some of these strategies are sometimes better than others."

One of my hobbies is shooting sporting clays, so the following analogy works for me. Hopefully, it will help you visualize the comparison of strategies, too. In sporting clays, the objective is to break a clay target with a shotgun.

Trap and skeet throw targets in predictably similar paths all the time, but sporting clays can come from anywhere and go anywhere, relatively speaking. In retirement finance, breaking the clay is symbolic of reaching the end of retirement with at least a little money to spare. That's our target.

SWR-Fixed is analogous to aiming where targets have ended up most often in the past, yelling "pull" and shooting in that direction.

SWR-Variable adds some data to the calculation: the changing value of your savings over time.

SWR-V is like deciding that you will track every target through its path and shoot a foot in front of it (lead it).  A foot will work for some shots that quarter away from you, but it won't be enough for a target that crosses directly in front of you or is farther away. Nonetheless, you are a bit more likely to hit the shot than by aiming where a lot of targets have gone in the past because you are now considering more information, that being where the target currently is and not just where targets have historically been.

Dynamic Spending is like tracking the target, knowing where it has been and where it is, and consequently where it is likely to soon be, estimating its vertical and horizontal speed and meeting the target with the correct lead. If the target is falling, you shoot below it. If it's a crossing target, you shoot farther ahead. You adjust your aim constantly. You hit a lot more targets that way.

Although all three of these strategies are proposed as viable alternatives, game theory tells us that Dynamic Spending dominates the other two and should always be our choice from among these three.

The explanation may be complex, but the advice is straightforward. If you're going to fund retirement by spending from a volatile portfolio of stocks and bonds, recalculate a sustainable withdrawal amount every year based on your revised expectations of future market returns, life expectancy, risk tolerance and capacity and estimated future spending needs.

Even if you ultimately decide to spend more, you'll at least know how much risk you're taking.

Next, I'll consider the application of game theory's Pure and Mixed Strategies.

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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.

The Sustainable Withdrawal Range

I had an interesting discussion this past week at Adviser Perspectives with two financial advisers who are frustrated by the fact that there are a wide range of recommendations for sustainable withdrawal rates. I sympathize with their frustration, but their suggestion of getting the industry to agree on one specific model of the future that would provide a single, agreed sustainable withdrawal rate isn’t a reasonable solution.

We could get every meteorologist in America to agree that the high temperature in Chapel Hill next Friday will be 42 degrees, but that wouldn’t make it any more likely that the prediction would be correct. In fact, it would be less likely. Different models with different predictions give us a range of possible outcomes to consider. When you can’t accurately predict something, like future market returns or future temperatures, providing upper and lower bounds for the most likely range is the next best information to have.

Let’s look at the current predictions for future sustainable withdrawal rates. The original SWR studies by William Bengen predict a 95%-safe SWR of about 4.4% for a 30-year retirement with a 50% equity portfolio. Wade Pfau et al recently produced a study suggesting that, based on today’s low-return environment, 3.5% might be a better guess. Even Bengen commented that Pfau might be onto something. (If you follow Wade Pfau's blog, by the way, he has a new website at RetirementResearcher.com, where you will need to re-subscribe to his email posts.)

Bengens’s approach uses historical market returns, assuming that the future will look like the past. Pfau et al use Monte Carlo simulation based on lower expected returns in the future than we have seen historically. Moshe Milevsky’s formula for probability of ruin using stochastic calculus calculates a 95% safe withdrawal rate of about 3.25%. In his paper, Milevsky notes that his formula often produces withdrawal rates significantly lower than many advisers recommend. Depending on the spending level, Milevsky’s calculation can differ from simulation results dramatically.

There are other studies that predict safe rates both higher and lower than these. The discussion at Adviser Perspectives was about why we can’t just all decide on one approach using the same assumptions and settle on one sustainable withdrawal rate. In other words, which model is right? The reason we can't is that these are all completely justifiable opinions about the future and we can’t know which model will work best. Any of them might turn out to be right.

The safest bet would be that the future 30-year SWR will not be 3.25%, 3.5% or 4.4% precisely. I would bet, however, that the correct answer will turn out to be not much lower than 3.25% and not much higher than 4.5% because that is the range several models suggest. I would plan for the possibility that it will be significantly lower.

If that sounds like a hedge instead of a commitment, that’s exactly what it is. Financial advisers hoping to hear “3.4%" or even “3.3% to 3.5%” would be disappointed.

These are not insignificant differences. If the actual SWR turns out to be 3.25%, a retiree will need to have saved 31 times his retirement income shortfall after Social Security benefits and pensions. If it is 4.5%, he will “only" need to have saved about 22 times that shortfall. If the shortfall is $10,000 a year, those savings requirements would be $307,692 and $222,222.

The wide discrepancy of recommended sustainable withdrawal rates is not a problem with the models that predict them, it is a result of our inability to predict the future of market returns. It is impossible to prove that any of the models are incorrect. . . well, not for 30 years, anyway.

Human beings have a poor record of predicting the future for even a few years, let alone for thirty. A little more than five years ago, there were widespread predictions that by not taking a path of austerity out of the Great Recession we would soon see rampant inflation. The inflation rate last year was 0.8% and deflation seems possible today. The EU took the austerity path and is trying to avoid an existential deflationary spiral. Both predicted their way would be best.

Studies show that “experts” are no better at predicting the future than us non-experts. Investment manager, Ken Fisher, used to project the coming year’s market return by looking at the projections of the same handful of “market experts” every year. He noticed that actual returns usually fell in the gap that no expert had predicted and that there was always such a gap. In other words, he simply chose the return that no one else had chosen. This worked eerily well for several years until others caught on. (Once everyone is playing the same game, no one can win.)

The wide range of projections is a result of our inability to predict market returns and the length of retirement, and thereby SWR’s, not an ability to agree on a model.

Financial risk is defined as the uncertainty of outcomes. Future sustainable withdrawal rates cannot be identified with a great deal of precision, so different models and different assumptions, all reasonable, produce widely disparate estimates of sustainable rates. This is just proof of what we already knew – sustainable withdrawal rates is a risky strategy.

Some advisers at Adviser Perspectives asked how they should communicate this complicated information when a client asks, “How much can I spend each year for the next 30 years and be 95% certain that I won't outlive my savings?" Here is what I would say to a client (or reader):

That amount is impossible to identify with any accuracy because we can’t predict future market returns or know how long you and your spouse will live. The current estimates from a wide range of models and assumptions range from about 3.25% to about 4.5% of your initial portfolio value for the first year, assuming your life expectancy is about 30 years. That percentage, by the way, increases as you age. It could approach 10% of your remaining portfolio balance near the end of your retirement. I would recommend a guess near the low end of the range because that will be safest, but that will also significantly reduce the amount you can spend. I would also recommend that you have a backup plan in case the sustainable rate turns out to be even lower than we expect, because it certainly could. I realize this is a broad estimate, but that’s because SWR is unpredictable, which is the financial definition of “risky.” If that’s more uncertainty than you are comfortable with, there are safer, more predictable spending strategies we can discuss.

Yes, its complicated and probably not what a client wants to hear. But, it is honest and that’s what clients need to hear.

Long Ladders

Long TIPS bond ladders demonstrate the challenge of matching liabilities in the more distant future, say, funding the last half of a 30-year retirement.

This is one of the toughest pieces of retirement funding to figure out for several reasons. First, we don't know if we will still be alive when it begins, which is a major reason people don't like annuities. Retirees who don't live beyond their life expectancy won't get much benefit from an annuity.

Or, we might live that 15 years and then some, possibly outliving a bond ladder and wishing we had purchased the annuity. Inflation has a much greater impact on more distant years of spending, and even when inflation protection can be purchased it is quite expensive.

In short, the further into the future we plan, the more uncertainty we must deal with. Life annuities remove the uncertainty of living a very long time (longevity risk). Long TIPS bond ladders provide an alternative, but come with a different set of risks, including some degree of longevity risk.

I'm a big advocate of floor-and-upside strategies that secure an acceptable level of income before investing in a risky portfolio. A lot of really bright people, like Zvi Bodie (Risk Less and Prosper), Nassim Taleb (The Black Swan), William Bernstein (too many to list) and Wade Pfau (How Do I Build a TIPS Bond Ladder for Retirement Income?) like TIPS bonds in the safe "floor" portfolio.

TIPS bonds held to maturity are considered risk-free assets – they have no default risk, no interest rate risk, no inflation risk and no correlation to market returns – but no asset is absolutely risk-free. With a TIPS bond ladder, there is the aforementioned risk that you might live longer than the ladder you buy and there is also a risk that you won't be able to hold all of your TIPS bonds to maturity, despite your intentions, in which case you will have interest rate risk.

The risk that you will not be able to hold all the bonds to maturity is obviously greater for a 30-year ladder than for a 5-year ladder and that is one reason I separate this discussion of long ladders from the previous post addressing short ladders. Short ladders are about as risk-free as investments assets can be, but risk grows with the length of the ladder.

Here is the scenario that makes me waiver just a bit. Let's say I buy a 30-year TIPS bond ladder today. Since yields are at record lows currently, I will likely lock in low interest rates for the next thirty years and when interest rates begin to rise in a few years, which seems more likely than not, I will regret not having waited.

I shouldn't regret the purchase because I will ultimately get what I want, a near-certain match of those future liabilities. It's just that the price for this income will decline in this scenario and I'll feel like the guy whose neighbor gets a better deal on a car identical to his. It shouldn't make me feel any worse about my own car, but it does.

If I buy a TIPS bond ladder, my goal isn't to invest to optimize my return or to get the best deal (risk-free assets never achieve that over time), it's to provide certainty of future income.

A second concern I have is that I would need to buy several long bonds.

I hate long bonds.

They're almost as risky as stocks and their return doesn't adequately compensate for that risk. As I mentioned in my last post, Funds and Ladders: What Matters?, in 2013, a bad year for bonds, iShares intermediate ETF TIP lost 8.65%, while long duration (27) bond ETF PIMCO ZROZ lost 22% of its value.

Long bonds fall much faster in value when yields increase than short or intermediate bonds do. To build a long ladder, I'll need to purchase 15 to 20 years of expenses in long bonds. And speaking of the stock-like risk of long bonds, after losing 21% of its value in 2013, ZROZ gained 49% in 2014. PIMCO LTPZ, not limited to zero coupon bonds, fell 20% in 2013 and gained 20% in 2014, still a wild ride. Don't try to match long-duration liabilities with long-duration bond funds. Long bond funds have no place in a safe floor portfolio.

Many advisers make what I call the "mark-to-market" argument that funds and ladders are identical. This argument says that a ladder has the same volatility as the fund but that the ladder-holder simply ignores daily price volatility. That is correct, but the ladder offers the possibility of ignoring volatility by holding bonds to maturity while the fund does not, and if the investor is able to hold the bonds to maturity, that volatility is irrelevant.

While I am generally not swayed by this argument, its advocates do have a point. Even if I plan to hold all those bonds to maturity, there is a risk that I won't be able to, and this risk should be considered.

Retirees who are building a 4- or 5-year ladder to fund a gap or pay for college, for instance, are far less likely to be forced to sell bonds they intended to hold to maturity than are retirees who hold a 30-year ladder simply because there is less time for something to go wrong.

A retiree might be forced to sell bonds sooner than planned due to a financial crisis, such as a medical emergency, but there is also a significant risk that the bonds will be sold, not by the retiree, but by her estate or her heirs. I suppose this could be called "reverse longevity risk."

Longevity risk is the risk of outliving our savings. Buying a 30-year TIPS ladders and living 35 years would be an example of longevity risk. But, there is also a risk that a retiree might buy a 30-year ladder and live only 15 years. The bonds could then be sold at a loss by her estate if yields have risen, or by her young heirs who don't have a lot of need for a portfolio of long TIPS bonds at the age of 25.

Of course, should interest rates fall over time, the bonds might be sold before maturity at a profit, but I can live with that risk.

Put these three factors together and you see my concern: I buy a 30-year TIPS bond ladder today and lock in historically low interest rates. Rates rise for the next ten years, lowering the market value of my bonds, especially the long ones, and I die soon after that. The remaining bonds are inherited by my children, whose financial needs aren't well met by holding long TIPS bonds to maturity, so they sell them at a loss. Since the basis of these bonds is stepped up, they won't even get a tax break.

(I would suffer the same fate in this scenario if I funded those liabilities with a long TIPS bond fund instead of a ladder. So, this is also a concern with funding distant future liabilities with a bond fund.)

I have to weigh this risk of unplanned sales against the certainty, offered by a TIPS ladder held to maturity, of meeting future liabilities. Many factors would exacerbate or mitigate this risk. A married couple is much more likely to have at least one spouse who will survive long enough to use most of the ladder. The longer at least one spouse survives, the less likely the ladder will contain bonds with a large loss, because a bond's price will approach its face value as time passes.

Retirees with no bequest motive may care less about these risks than those who wish to leave an inheritance. (With no bequest motive, however, they might find a life annuity a better fit.) Retirees who have lots of other retirement income (over-savers) are less likely to need to sell bonds from their ladder in an emergency. Retirees who fund a lot of annual income with a ladder will have greater risk exposure than those that need only fund a small annual shortfall. The risk of needing to sell bonds before maturity varies significantly based on the household's individual situation.

Is there a way to fix this problem? Not a good one. We could use an annuity, but it will have even less liquidity than a ladder. A retiree who insists on following the daily market value of a TIPS fund should also want to follow the resale value of an annuity, and it will be even worse. The heirs of the TIPS bond ladder may see a loss, but the heirs of the retiree with an annuity will receive nothing at all.

Ultimately, I believe that Bernstein, Bodie, et al have it right. The safest way to provide certain future income is to purchase TIPS bonds and hold them to maturity. Yes, you may lock in low rates for a long time if you're unlucky, the strategy has opportunity cost and you might need to sell some bonds at a loss before they mature, but if your goal is strictly to provide income with certainty, this strategy is the best bet.

It is only when you add additional requirements, like a goal of maximizing yield or one of maximizing a bequest, or a concern about the market value of your assets should you have to sell them in a fire sale tomorrow, that the strategy shows some weaknesses. None of these are great objectives for a floor portfolio, by the way.

Still, on an individual basis, those additional requirements might be important to you and should be given consideration, especially for long ladders. They shouldn't be an issue for short ladders.

For most do-it-yourself retirees, I would summarize the last few posts on bond ladders and funds as follows. Retirees who aren't using bonds to match future liabilities will probably realize little advantage from buying a ladder instead of a fund. I believe TIPS ladders are the way to go for funding a few years, but using a short-duration fund for this purpose, instead, probably isn't a deal-breaker.

Long ladders and long bond funds are a different story. For the safest approach to providing certain income, ladders are the solution, especially if there is little risk that you will need to sell the bonds before maturity or if the residual value of the ladder isn't a concern. As I mentioned, long bond funds can be extremely volatile and do not belong in a safe floor portfolio.

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P.S. Several readers asked after my recent discussion of bond funds versus ladders why I focussed on TIPS bond ladders. Wouldn't duration- and convexity-matching arguments also hold for funds and ladders of say, corporate bonds or munis?

They would, but Treasury bonds have no default risk so there is no need to diversify among issues. Other bonds, including corporates, do have default risk and we need to diversify among many issues for an acceptable level of safety. It would be difficult for most retirees to buy enough individual corporate bonds to adequately diversify, so mutual funds win over ladders in non-Treasury bond asset classes right off the bat based on their diversification advantage.

Funds and Ladders: What Matters?

In my previous post, First Derivatives and Second Moments, I showed, with more math than should be tolerated on this blog, that a bond fund and a bond ladder have different expected returns and risks unless they contain virtually the same bonds in the same proportions. A TIPS bond fund and a ladder of individual TIPS bonds held to maturity are rarely identical even if they have the same duration.

Two things I did not show are when one is better than the other and when the difference is enough to matter.

There are many factors we could use to compare ladders to funds beyond interest rate risk and return. There is the convenience issue, though I hope I have explained that tax reporting is no longer a big issue and that bond desks can do most of the legwork for you. There is the familiarity issue for those who have never purchased individual bonds. TIPS bonds aren't available for every year and some funds can be bought in smaller price increments. There are many ways we could compare bonds and ladders, but not all of them are critical.

One of my objectives for this blog is to simplify retirement finance so it makes sense to most do-it-yourself retirement planners. In that spirit, I will try to unravel this issue by considering three common retirement scenarios for bond funds or bond ladders: funding known liabilities, funding a few years of living expenses, and funding a long retirement of living expenses.

To simplify the explanation, when I use the term "ladder" I will refer to a series of TIPS bonds maturing annually. Ladders can be constructed with many types of bonds, but I will only refer to U.S. Treasury Inflation Protected Securities ladders. I will use the term "fund" or "bond fund" to refer to a fund or ETF that predominantly consists of these TIPS bonds. Furthermore, when I compare a fund to a ladder, I am referring to a fund with an average duration similar to the average duration of the ladder.

Why do I limit the discussion to Treasury bonds when we could ladder most any kind of bond? Two reasons. First, I believe that only Treasury bonds, and especially TIPS, are safe enough for the risk-free portion of a retiree's portfolio. Second, Treasuries presumably have no credit risk, so diversification of individual bonds is unnecessary. The diversification advantage of a fund of corporate bonds, for example, would generally outweigh any advantages of a corporate bond ladder, in my opinion. If you don't buy Treasury bonds, you're probably better off with a fund for its diversification.

Let's look first at the scenario of funding known future liabilities. We might wish to fund four years of a child's college education, for example. We have a good estimate of the cost and the years those expenses will be incurred. Or, we might plan to fund five years of living expenses between retiring at age 65 and claiming Social Security benefits. Finally, we might want to plan funding for 30 years or more of retirement, in which case we plan for 30 or more known future liabilities, our living expenses.

If bonds aren't meant to fund a known future liability, they still have great diversification value in a portfolio. Consider a retiree whose living expenses are completely covered by a pension and Social Security benefits, but who has also saved a large investment portfolio. She will likely need to diversify that portfolio into bonds to manage risk, but holding individual bonds to maturity with no liability to match would provide little additional benefit. That need would be better met by a diversified bond fund, and one not limited to Treasury bonds.

But, for known future liabilities, a ladder of bonds held to maturity has an economic benefit created by the option to hold bonds to maturity that match those liabilities. We can know with relative certainty how much they will be worth in real dollars at maturity. Funds don't provide that option.

If you're investing in bonds for diversification and not liability-matching, a ladder has no economic advantage and a fund should be fine.

The second scenario, such as funding the gap between retiring and claiming Social Security benefits or funding college, is a liability-matching problem, but for a limited time. I believe there are signification differences between brief liability-matching scenarios and liability-matching scenarios that could last thirty years or more. Let's consider the former.

Short-duration (about 2.5 for a five-year period, in this scenario) TIPS funds are relatively safe and will not lose much in just a few years, nor do they provide much upside potential. In 2013, a bad year for bonds, iShares intermediate ETF TIP lost 8.65%, and long duration (27) PIMCO ZROZ lost 22% of its value. Short-duration Vanguard TIPS fund VTIP lost just 1.55%.

In 2014, an up year for ZROZ that returned over 49%, VTIP lost 1.2%. At the short end, there isn't a lot of risk, nor is there much upside.

While a ladder would seem an obvious choice in this scenario, the fact that the term of the ladder is short means that you won't do a whole lot worse in a fund and there is some potential to do better. If you can tolerate a small shortfall in the worst case, using a bond fund instead of a ladder should work fine for short periods. On the other hand, the ladder is safer and buying a five-year ladder does not entail a lot of inconvenience. The fund has sequence of returns risk. If the possibility of a shortfall is a concern, go with a ladder.

The last scenario I will consider is that of funding 30 years or more of retirement with a TIPS bond ladder. At first glance, it might resemble the 5-year scenario, but funding a much longer period has important differences. I'll cover that in my next post.

To summarize, if you are not trying to match a known future liability, holding bonds to maturity doesn't have a clear economic advantage over a fund. The returns and risks will be different, but we can't predict which will do better. Go with a diverse bond fund.

For matching a few years of a known liability, I much prefer a ladder of TIPS bonds to a TIPS bond fund. But, in this scenario, making the wrong choice is unlikely to make or break your retirement plan. Don't lose sleep over it.

First Derivatives and Second Moments

A common argument that bond funds and bond ladders are identical involves their duration. Duration, though precisely defined mathematically, is roughly the number of years it would take a bond or fund to recover the capital loss from a 1% increase in interest rates (yield).

An increase in interest rates would lower the price of the bond but the bond's future interest payments could then be reinvested at the higher yield and would eventually make up for the capital loss. A bond with a duration of 5 would recover a 1% capital loss in about 5 years.

Another way to look at duration is that is the percentage capital loss one would expect from a 1% increase in interest rate. So, that same bond with a duration of 5 would lose about 5% of its value if interest rates rose 1%. (The opposite would happen if rates fell 1%.)

The argument goes like this. If the duration of the TIPS bond ladder and the average duration of the bonds in the fund are the same, then the risk of the ladder and the fund are identical. Therefore, it doesn't matter which you buy.

There are a couple of problems with this argument. First, because the ladder locks in yields and the fund doesn't, their returns won't be the same unless interest rates happen to remain unchanged over time. And second, two investments with the same duration don't necessarily have the same risk.

I recently had a brief chat with Professor Moshe Milevsky at York University in Toronto. I told him that I had read opinions that a TIPS bond fund with the same duration as a TIPS ladder provides equal risk for retirees. In other words, it was suggested that owning a 5-year ladder of individual TIPS bonds with an average duration of about 2.5 years has the same risk as owning a TIPS bond fund with a duration of about 2.5 years.

Dr. Milevsky responded, "Duration is just one moment. [I] would like to match [the] second moment (convexity) and perhaps higher, before I agree.”

That response didn’t help me a lot, because he seemed to be saying that he would agree that there is no significant risk difference between the two so long as I could match several risk factors. But, I could only match several risk factors by holding nearly identical bonds in both the ladder and the fund, and of course those would have equal risk.

Notice what Dr. Milevsky didn't say – that it makes no difference because ladders and funds are identical.

"To match all those moments, don't you effectively need to hold the same bonds in the ladder as in the fund?” I then asked.

"Good point,” he replied. "Match all moments and you get the same portfolio (of strips.) To get "reasonably" close, give me two moments.”

Now, that was something I could work with.

I have often written that the duration of a bond is the percentage loss of a bond's value that would result from a 1% increase in interest rates, but that is only precisely correct for small interest rate movements. The real amount of loss (or gain) a bond will experience also depends on how much rates change. Duration is just an estimate of bond price sensitivity when the interest rate change is very small.

In order to compare the risk of a bond fund to a bond ladder, or to a different bond fund for that matter, simply knowing their durations isn’t enough information. We also need to understand at least their convexities. Higher moments would allow us to make an even better comparison of risks, but duration and convexity get us “reasonably close.”

I try to avoid the weeds on this blog, but please bear with me and I promise to bring us back out of them and onto smoothly-mowed lawn as quickly as possible.

The following chart from Investopedia.com shows how much a change in interest rates (yield) along the x-axis changes the price of the bond along the y-axis. Duration and convexity can be calculated for both bonds and bonds fund.

The red line shows bond duration and illustrates the fact that bond prices move in the opposite direction of yields. Duration is one estimate of interest rate risk. A bond with a duration of 5 will decline in value about 5% for every 1% decline in interest rates. But duration is just a first-order estimate of the impact of an interest rate change on a bond’s price.

(Duration is the first derivative of the price/yield curve, the blue curve, for anyone who remembers first semester calculus. And because it is the first derivative, it is the calculation of duration at a single point along that curve. At any other point on the curve this is an estimate of the curve's slope. Convexity is the second derivative.)

The precise change in the bond’s price as yields change is shown by the blue curve. The yellow area shows the estimation error of the bond’s duration. The larger the yield change, the greater the error.

The next chart is similar, but adds a second bond. Bond A has greater convexity (a sharper curve) than Bond B.

Notice the red arrow. Near the current yield and price at point (*Y,*P), the duration and convexity of both bonds are identical. But, as the yield moves farther to the right or left along the x-axis, Bond B’s price changes differently than its duration predicts.

Duration predicts that the price change will be linear, but it will not be. In fact, though this simplified chart shows the curves as symmetrical, there is more error when bond yields decline than when they rise.

In other words, duration is a good estimate of expected price change if yields increase or decrease just a little, but the difference (estimation error) becomes substantial if yields change a lot in either direction.

Also notice that Bond A’s price (**P) changes less than Bond B’s price (**P) for the same change of yield. Bond A has greater convexity than Bond B.

So, if your bond fund looks like Bond B and a ladder looks like Bond A, they both have the same duration but your fund has more interest rate risk. A rate increase from *Y to **Y will cause Bond A (the ladder) to fall from price *P to price **P, but Bond B (your fund) will fall farther, to price **P.

Of course, you might be comparing a ladder with greater convexity than your fund. Your fund could look like Bond A and the ladder like Bond B, in which case the opposite would be true, but the point here is only that they are different.

There are a lot of ways to build bond portfolios (funds or ladders) with the same duration. It is far more challenging to build two bond portfolios with both the same duration and the same convexity and, therefore, the same risk. (More challenging unless, as I suggested to Dr. Milevsky, we put the same bonds in both the ladder and the fund, but that isn't an interesting scenario.)

What does this have to do with the ladder-versus-fund debate? It throws a monkey wrench into the argument that a TIPS bond fund has the same risk as a TIPS bond ladder if the duration of the two is the same. The risk is “reasonably close”, according to Dr. Milevsky, only if the convexity of the fund also matches that of the ladder.

Some advisers suggest mixing funds of different durations to achieve the duration you need. Mix a fund with a duration of 4 years with an equal amount of a fund with a duration of two years, for example, to create a fund of funds with a 3-year duration. This might work for duration, depending on your needs and whether workable funds exist, but that math doesn’t work with convexity. (An excellent Powerpoint lecture explaining why can be found here.) It will be impractical to combine funds to generate both the duration and convexity of the ladder you seek to replace. And at some point, buying the individual bonds is just a lot less work.

This all assumes, of course, that you can learn the convexity of a fund and that the manager will hold it steady while you own it. Bond durations are fairly easy to find online, even if they aren’t guaranteed over time, at places like Morningstar.com, but funds don’t typically even report their convexity.

Here is where I honor my promise to return from the weeds. A TIPS bond fund and a TIPS ladder won’t have the same risk unless they both hold about the same bonds in the same proportion. Good luck finding a bond fund that needs the same bonds as you. A fund and a ladder can have the same price, yield and duration but if one has lower convexity, their risk is different.

The intent of my post today is to dispel the notion that a TIPS bond fund and a TIPS ladder are no different so long as they have the same duration. Duration is a first-order estimate of interest rate risk. A ladder and a fund can have the same duration but different amounts of risk. Even if the risk is similar enough, they won't have the same expected return.

A bond fund is not a bond ladder unless they effectively hold the same bonds. A bond fund is not even another bond fund.

Intermediate TIPS bond fund IPE has a duration of 6.84, a 5-year return of 4.08% and a 5-year standard deviation of 5.54%. Intermediate TIPS bond ETF TIP has a duration of 7.62, a return of 3.97% and standard deviation 5.08. Though they are both “intermediate term TIPS bond funds”, TIP has a longer duration, similar return and 8% less volatility. We don't know the convexity of either.

Because ladders lock in current interest rates and bond funds continue to track rate changes after shares are purchased, a TIPS ladder will have a different expected return than a TIPS bond fund. A fund could have the same risk as a ladder if duration and convexity match, but achieving this with mutual funds is not practical. A fund's convexity is not generally available information and, even if it were, the fund manager makes no commitment to maintain it over time.

This shows that ladders and funds are not identical unless they hold very similar bonds in the same proportion, but it doesn't show which is better, under what conditions it is better, and when it is superior enough to matter.

It largely depends on how you will use them. More on that next time. (See Funds and Ladders: What Matters?) And, if you don't enjoy the math, it should be more interesting.

Funding the Gap

I interrupt my current wanderings through Game Theory to re-address a question I have discussed in the past regarding whether a TIPS bond ladder held to maturity can safely be replaced by a TIPS bond fund.

Bond ladders can be set up in a couple of ways, fixed length and rolling. A 5-year fixed length bond ladder, for example, will be depleted in five years as each of the rungs matures. We replace the longest rung of a rolling ladder each year as the shortest bond matures, so a 5-year rolling ladder always contains five rungs.

Today I'll talk about short fixed-length ladders. A retiree might use a 5-year, fixed-length ladder, for example, to bridge the gap between retirement at age 65 and claiming Social Security benefits at age 70.

The key to this discussion is that a ladder of TIPS bonds held to maturity isn't a do-it-yourself bond fund. When held to maturity, TIPS bonds are risk-free assets that have more in common with cash than with bond funds.

A ladder of TIPS bonds held to maturity has no volatility, interest rate risk or correlation to the stock market, nor does it have inflation risk. These bonds are a contract with the U.S. Treasury to pay specified amounts of interest in each year (the "coupon") until maturity and then return the face value of the bond plus additional principle to compensate for inflation. In other words, you will receive the face value in inflation-adjusted dollars. You have no opportunity for capital gain or risk of capital loss with a ladder of TIPS bonds held to maturity; you would have both with a TIPS bond fund.

The absolute safest way to insure that you will have the cash you need for each of those five years is to purchase a ladder of TIPS bonds and hold them to maturity. That's the only way to automatically adjust your bond holdings' durations to match the dates when you will need the money.

A fund manager will try to keep its duration fairly constant, at about 2.5 years for a short term TIPS bond fund, for example. That won't precisely match the ideal durations of 1, 2, 3, 4 and 5 years in our example, as individual bonds could. An investor could mix portions of funds with different durations to better match the duration of the spending, but that seems like a lot more trouble than buying individual bonds without a lot of improvement. Still, bond funds are at best an approximation of expense durations.

Why would you consider alternatives to a TIPS ladder? Many people are unfamiliar with purchasing individual bonds and prefer the simplicity of investing in a bond fund. I don't find ladders difficult to purchase. I ask the bond desk at Vanguard, Fidelity or Schwab to find the bonds for me. Schwab charges $1 per bond and they all do the search for free, but I understand that some might find this daunting.

You might also figure that you could receive higher returns from the bond fund if interest rates increase. That is a possibility, but how much profit can you make by investing in a short term TIPS bond fund like Vanguard Short-Term Inflation-Protected Securities Index Fund Investor Shares (VTIPX)?

And, if this is money that you want to be truly safe, would you risk it to earn a little more interest?

VTIPX has a duration at present of 2.4, which is about the same as a 5-year ladder of zero coupon TIPS bonds (2.5). You can expect a 1% increase in interest rates of similar duration bonds to result in a capital loss of about 2.4% with VTIPX. That loss would be recovered by the additional interest in 2.4 years, assuming you hold the investment at least that long. A 1% decline in rates would result in a capital gain of about 2.4%. In other words, there isn't a lot of profit to be earned or capital to be lost whether you invest in short term TIPS bonds or a fund made up of them.

If that's the case, then why not leave the funds in a money market account or certificates of deposit? VTIPX has a current yield of 0.72%, Vanguard money market account yields are barely observable with the naked eye. You can buy a 1-year CD that pays around 1% and a 5-year CD can earn 1.8%.

(Don't spend it all in one place.)

The answer, of course, is that TIPS bonds and funds provide inflation protection. But, how much inflation risk do we expect for the next five years? The current rate in 2015 is only about 1.8% a year and many predict that we will experience low inflation for quite some time. If inflation averages 1.8% a year, the real return on the 1-year CD is negative 0.8% and the real return on the 5-year CD is zero, so there are worse things than a 0.72% return.

Inflation would seem to be a bigger concern than nominal returns at present, since most nominal returns for short term, low-volatility investments are currently near zero. Even a low rate of 1.8% annual inflation means that the dollar you want to spend in 5 years will be worth only about 91 cents in today's dollars.

So, a TIPS bond fund makes sense to me right now. As I said, the absolute safest alternative is a ladder of TIPS bonds held to maturity, but the TIPS bond fund doesn't add much risk. You can't really lose much money (or make much) at this duration in a Treasury bond or fund. This is a situation in which we should probably be more concerned about not losing money than making more, anyway.

A fixed length (non-rolling) TIPS bond ladder is not the same asset as a TIPS bond fund. The former is a risk-free asset and the latter has volatility of returns.

Rolling ladders and longer non-rolling ladders have other risks that concern me more, but if you want to go the more convenient fund route, I don't see a compelling reason to buy individual bonds in this scenario with today's interest rates unless very small losses would make a difference in your situation.

I'll talk more about ladders and funds next time in First Moments and Second Derivatives.