Monday, January 26, 2015

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.


Thursday, January 22, 2015

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.

Thursday, January 8, 2015

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.


Tuesday, December 30, 2014

Happy New Year 2015

A few thoughts to wrap up 2014 and then on to what I hope is a happy and prosperous New Year for us all.

My last post on Game Theory and Social Security Benefits, in which I showed that there is no dominant strategy across the board for claiming benefits, ironically grew into a discussion of which strategies people feel certain are dominant. It was a fun discussion, nonetheless, and your participation is greatly appreciated. I'm happy to keep the discussion of that post open as long as you have questions or opinions. I will tie up the topic (Social Security, not game theory) for now with a couple of thoughts.

First, since most Americans have under-saved for retirement, most will need to claim benefits right away. If you have the luxury of choice, consider yourself very fortunate and then be advised that the rules are quite complex. Unless you’re willing to spend a lot of time studying the subject, buy some software like Maximize My Social Security or find a professional adviser you can trust. I’d do both. This is one of the most important financial decisions you will make and it is, for all practical purposes, a permanent decision. You need to get it right.

Second, be cautious of analyses you read that are based on life expectancy. About half of the population of any age will live longer than their life expectancy and our goal in retirement is to be able to pay for even a very long retirement, not just until our life expectancy.

It is correct to say that if you don’t live beyond your life expectancy there is little to be gained by delaying your benefits in terms of total lifetime payments. You will receive about the same total payments if you claim at 62 and live to your life expectancy that you would receive if you claimed at 66 and lived to the life expectancy of a 66-year old. As retirees, however, we need to protect against the risk that we will live well beyond our life expectancy and that’s when delaying benefits pays off.

Planning on living to your life expectancy is like forgoing homeowner’s insurance because your house probably won’t burn down.

But there are implications of early claiming beyond total lifetime payments. If you claim retirement benefits before your full retirement age (66 for most of us Boomers), you cannot take advantage of a higher retirement benefit that might become available later, for instance. Your benefit is locked in. If you are the higher earner of a married couple and claim early, your spouse’s survivor benefit is also locked in.

If you claim at 62 and live to 70 but your widow lives to 95, your legacy might also be at risk. Imagine her picking up a smallish benefit check at 90, shaking her head and saying, “My poor departed Harry was such a sweet man, but he royally screwed my benefits.”

If any of this is news to you, get some help before claiming.

Next topic, for those of you who showed interest in Moshe Milevsky’s probability of ruin formula, remember to use real (after inflation) returns when running the model. Long term historical real stock returns, for example, should be in the 6%-ish range. Also, the results are not directly comparable to other studies, such as those by William Bengen, that use the SWR model. Those studies assume different fixed life expectancies, like 15, 20 or 30 years. Milevsky uses a life expectancy probability. While Bengen assume a life expectancy of exactly 30 years, for example, Milevsky assumes that the length of retirement is a random variable with a mean of 30 years. They’re not the same thing.

I thank everyone for reading this past year. I especially thank the reader who sent a surprise Christmas gift – it made my holiday season. I hope to see all of you in 2015 when we can continue to try to figure this thing out together.

Our goal is to make sure we can feel secure and be happy in retirement. Make sure you don't forget the “happy” part.

Happy New Year!

Friday, December 19, 2014

Game Theory and Social Security Benefits


In A Tiny Bit of Game Theory, I explained a few basics of this study of decision theory. The Social Security claiming decision provides a good example of how to analyze a financial decision with game theory.

Our Social Security game will be a stochastic game against nature in which nature decides your life expectancy, which is, when you think about it, pretty realistic. Unrealistically, we are going to assume that you will live to age 64, to age 70, or to age 95 to simplify the game.

Your choices as the player are to claim benefits at age 62, full retirement age of 66, or at the maximum age of 70. We will assume that you are a single retiree with a typical lifetime record of FICA payments. Having a spouse makes this a very different game, of course, and a lot more complex. So would adding all the claiming age options.

For payoffs, I’ll use the total estimated lifetime benefits for each claiming option according to the Social Security website at SSA.gov for a single person born in 1955 and currently earning $75,000 annually. In this first game example, we will further assume that the retiree has adequate retirement savings to support her lifestyle between retirement at age 62 and the age at which she will claim benefits.

This simplified game in matrix form with lifetime Social Security benefits payoffs in 2014 dollars will look like this:


The retiree will need to also make a decision about her overall objectives. Many game theory analyses select strategies that will avoid the worst-case loss. Prisoner’s Dilemma, for example, encourages each perpetrator to confess first and avoid the longest prison sentence. Mutually-Assured Destruction was also an attempt to minimize the worst-case scenario, a nuclear war. These are referred to as “maximin” strategies because they seek to maximize the minimum outcomes. In other words, they seek the strategy that has the best payoff from among worst-case scenarios.

Some retirees want to minimize the chances of “leaving benefits money on the table.” They decide to claim as early as possible in case they don’t live long enough to “break even”. This strategy seems wrong to me on so many levels, but to each his own. Game theory allows us to analyze the problem with a wide range of potential objectives.

The table below shows how much Social Security benefits a retiree might “leave on the table” by waiting to claim but dying before the break-even age, which in this example ranges from ages 75 to 78 depending on the claiming ages.



As you can see from the payoffs, if you won’t live very long, you will maximize your total lifetime benefits by claiming as early as possible (Table 1) and if your objective is to wring every available dollar out of the U.S. Treasury (Table 2), claiming early would be the way to go. Of course, if you’re wrong about your checkout date, you might have done significantly better by claiming at a later age.

If you live to be very old, then you will receive the greatest lifetime benefit by claiming at age 70, when benefits top out. If you plan to live a long time but don’t, you will have missed years of benefits by not claiming early.

For retirees with the “maximin” objective of protecting against the worst-case scenario, claiming at 70 is the best choice, because minimizing your benefits by claiming them at age 62 and then living well into your 90's will be very painful for a very long time. The formal name for Social Security retirement benefits is Old Age and Survivors Insurance (OASI) and claiming as late as possible is the best use of benefits if you view them as insurance. Delaying the claim date for your benefits is the cheapest way to purchase longevity insurance.

I mentioned earlier that for this example game we would assume that the retiree has adequate resources to retire at age 62 and pay for her standard of living until she claims benefits. Another way to implement this strategy is to work longer, if you have the option.

Retirees who don’t have the option to work longer and don’t have substantial retirement savings can’t play this game. They will need to claim early because they will need the income immediately. So, you have more options with Social Security if you also have a lot of money.

I’m sure you’re shocked.

There is one other game theory concept we can introduce with this example, that of dominant strategies.

If you were offered two bets and the first bet always paid at least as much as the second bet and sometimes more, you would always choose the first bet, right? Game theory refers to the first bet as a dominant strategy and the second as a dominated strategy. Game theory tells us never to play a dominated strategy. (And, it tells us that there usually isn’t a dominant one.)

In the Social Security benefits game I have described, there is no dominant strategy that always provides the best results under all circumstances. Sometimes claiming at age 62 pays more lifetime benefits and sometimes claiming at age 70 does, depending on how long you live.

However, claiming at age 62 is a dominant strategy if the objective is merely to leave the minimum amount of benefits on the table and claiming at age 70 is a dominant strategy if the objective is to minimize longevity risk.

Note that I’m not trying to use game theory to explain the best Social Security benefits-claiming strategy. That will depend on your individual resources and goals. I’m suggesting that it provides a good framework for laying out all the options and outcomes and for clearly identifying our objectives so we don’t focus only on the most likely outcomes.

Hopefully, that supports a better decision.


Monday, December 15, 2014

A Tiny Bit of Game Theory

I’m fascinated by game theory and I’ve lately been thinking about retirement finances through that lens.

You may be familiar with three products of game theory, whether you realize it or not. The first is the strategy of Mutually-Assured Destruction, with the appropriate acronym MAD, that was developed from game theory in the 1960's as a response to the threat of nuclear war. The second is "Nash equilibrium", suggested in the book and movie, "A Beautiful Mind". (John Nash won a Nobel Prize for his work on game theory.) The other is called "Prisoner's Dilemma", a game that pits two "perps" against one another to obtain a confession that seems to part of every TV crime drama ever created.

Game theory is the study of strategic decision-making or, according to expert Roger Myerson, "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.”

I keep his book, Game Theory: Analysis of Conflict, on my desk. On days when I want to humble myself, I try to understand the math. But, there is a lot to learn from game theory even if you wouldn’t touch linear algebra with a ten-foot pole.

Game theory can model strategic decisions in a number of ways, but the simplest is by using a matrix of Player A’s strategies versus those of Player B’s. The cells of the matrix contain the payoffs for each player when each chooses a particular strategy. A pair of numbers describes the payoffs. The first of the pair (boldface) is Player A’s payoff and the second is Player B’s.

Here’s an example. In this “game”, if Player A chooses Strategy 1 and Player B chooses Strategy 2, then Player A will receive a payoff of 3 “points” and Player B will receive 0 points. Each player will look at the potential payoffs for each strategy available to her, guess what Player B might do, and choose a strategy accordingly. The outcome of the game will be determined by the contents of the cell at the intersection of the two strategies.


(In case you're interested, the game above is “Prisoner’s Dilemma”, where each player’s Strategy 2 is to confess and rat out his partner in crime. Strategy 1 is to keep silent. The payoffs are the number of years in prison, so I suppose they should be negative numbers.)

In financial planning, we rarely are interested in a game between two individuals, but a game, instead, of an individual against a system of markets with random outcomes. Game theory refers to these as “stochastic games against nature”, a phrase you may never need to hear again. On the other hand, when someone asks what you're doing about retirement, you could impress them by answering, "I'm playing a stochastic game against nature."

In these games, there will be only one payoff in each cell, since “nature” doesn’t need payoffs.

Here’s an example. Let’s say that nature has two possible strategies in a game: it can rain or not rain where you are. You, in turn have two strategies. You can carry an umbrella, or leave it at home.

If you leave your umbrella at home, there are two possible outcomes. It may rain, in which case you will get wet, or it may not rain, and you will have a good outcome. You stay dry and don’t have to lug an umbrella around for no reason.

If you choose the umbrella strategy instead of leaving it at home, you also have two possible outcomes. If it rains, you stay dry. If it doesn’t rain, you will have to carry an umbrella around all day, looking stupid and encumbered for no good reason.

A matrix to describe this game might look like this:


The correct strategy choice in this game, of course, depends on the weather forecast’s probability of rain and its accuracy. It is called a stochastic game because the outcome depends on chance. It is called a “game against nature”, not because we’re talking about rain, but because we are playing against a complex system and not against an individual. The stock market, for example, would also be included in this definition of nature.

If these were the payoffs (I made them up), at what probability of rain would you switch strategies?

I think we can gain some insight into some retirement financial decisions if we look at them from a game theory perspective. In particular, I think game theory can make us focus on all possible results of our financial decisions and not just the most likely outcomes. In my next few blogs, I’ll provide some examples from retirement finance and we’ll find out if you agree.



           


Friday, December 5, 2014

Think Like a Bayesian Pig

OK, one more barnyard animal theme and I promise to move on.

I spoke at the RIIA Fall Conference of retirement planners a few weeks back on the topic, "Think Like a Pig". I suggested that they view retirement from the perspective of a retiree who would actually feel pain if their retirement plan failed as opposed to the perspective of a somewhat-interested third-party. I suggest you do the same with your own retirement planning because being retired isn't quite the same as thinking about retiring one day.

It's for real.

Now, I would like to recommend a further adjustment to your view of retirement planning.

Academics often treat retirement as if it is one integrated whole that begins around age 65 and could last thirty years (spherical cow alert!). This often makes sense in an academic environment when we are trying to understand the financial process involved.

Systematic withdrawals of constant dollar amounts are a good example. We can use the strategy to study the probability of failure over 30-year periods and learn about sequence of returns risk, but implementing that strategy doesn't make sense in real life. Calculating that you can spend 4% of a million dollar nest egg, or $40,000 a year for the next 30 years with little chance of outliving your savings and then actually doing that requires that you ignore any new information along the way.

When was ignoring new data ever a good idea?

At the beginning of World War I, horse-mounted cavalry ignored new data and charged machine guns.

That $40,000 spending estimate is based on what statisticians call "prior probabilities," meaning it's the best guess from the starting gate. After retirement begins, things happen that change your probability of success. The updated probability is called the "conditional probability."

Here's an example I used in a post some time ago. Let's say that you leave Los Angeles on a flight to Honolulu and you learn from the airlines that they have attempted this flight 1,000 times and only 10 of those flights didn't reach Honolulu because mechanical problems, weather or something else forced them to return. Your prior probability of reaching Honolulu would be 99%. That looks pretty darned good.

During the flight, your crew will constantly update their forecasts based on new information, running into headwinds, perhaps, or needing to fly around storms (a good model for your retirement plan). They will continuously create a conditional probability of reaching Honolulu and if that probability drops below a certain threshold, they will return to Los Angeles.

At least, you hope they will.

Should you find yourself halfway to Honolulu and discover that a wing has fallen off your plane, the conditional probability of reaching your planned destination has just declined considerably. (That's why I hate when flight attendants announce, "We'll be on the ground shortly." I need more details than that.) Once the wing is gone, you should take little comfort from the fact that your prior probability of reaching Honolulu was actually quite high.

Retirement works the same way. You might start retirement with a million bucks and a safe spending amount of $40,000, but if your portfolio declines 50% in a bear market you need to start spending less. That original $40,000 safe spending amount flew out the window with your bear market losses. To continue spending the same $40,000 after a large decline in your savings balance is simply ignoring new information, to wit, that you have less money.

In the 1700's, Thomas Bayes thought about how new information should be used to adjust our previous expectations. Bayes Theorem essentially says that we should begin with a prior probability, like a sustainable withdrawal rate or the percent of successful flights to Honolulu in the past, and modify that original expectation in light of any relevant new data that comes along.

Relevant new data for an airplane would be like, remaining fuel, unexpected headwinds and structural integrity of the wings.

This Bayesian approach is the way we retirees should view a retirement plan. Rather than view it as one integrated whole, we should think of it as planning for a 30-year retirement based on some set of prior assumptions. After a year, we should take stock of our new life expectancy, new portfolio balance, and any changes in expected spending along with several other variables and use that new information to plan a 29-year retirement.

Rinse and repeat.

That isn't what we do when we plan on a constant-dollar spending SWR strategy. Instead, it is what Larry Frank refers to when he describes Dynamic Updating and what Ken Steiner is getting at when he explains how to re-budget your spending every year with actuarial techniques.

And, it's what Moshe Milevski's equation for the probability of ruin (also an actuarial approach, by the way) tells us: it is a function of current retirement savings balance, expected spending, expected market returns and volatility (asset allocation) and remaining life expectancy. It doesn't matter what those were back on the day you retired.

What matters is what they are today.