Monday, October 27, 2014

Spherical Cows

Physicists are known for sometimes oversimplifying assumptions in order to simplify the math required to solve a problem. Physicists refer to these assumptions as "Spherical Cows".

The term comes from a story about a farmer who talks to a physicist about his farm's underproduction of milk and asks if the physicist might be able to offer some advice. The physicist goes away to perform some calculations, but soon returns with an answer.

"I have a solution for your problem," he explains to the farmer, "but it only works for spherical cows in a vacuum."

We encounter a lot of Spherical Cows in retirement finance, huge oversimplifications that make the math easier.

For one, we generally assume that market returns are "normally distributed" even though we have tons of evidence that they are not. If they were normally distributed, we wouldn't see nearly as many market crashes as we do. Often we assume they are log-normally distributed, meaning the logarithms of the returns are normally distributed, but they aren't really that, either.

According to Professors Fama and French, "Distributions of daily and monthly stock returns are rather symmetric about their means, but the tails are fatter (i.e., there are more outliers) than would be expected with normal distributions."

They go on to say that longer periods, like years, conform more to a normal distribution. The 23% drop in the Dow of October 19th, 1987 was something that probably never could have happened in a single day under a normal distribution of returns, but the 37% year-long drop in 2008 was a 2.5 sigma event that might happen once every 80 years.

Their advice to investors is to expect more extreme good and bad returns than a normal distribution would seem to indicate.

So, assuming annual returns are normally distributed works fairly well, but not so with daily or monthly returns.

One of my favorite Spherical Cows is the one used to calculate sustainable withdrawal rates. SWR models assume that a mythical investor will continue to spend the same amount of money each year from savings, even after it becomes obvious that he or she is about to deplete their retirement savings. The models take a percentage, say 4%, of initial portfolio value and subtract that fixed dollar amount ($4,000 from a $100,000 portfolio in this case) from the portfolio balance every year, counting the number of years before the portfolio is depleted.

This assumption makes it far easier to build a spreadsheet than would modeling how a real investor might behave with their spending as their savings grow or dwindle.

I don't think most retirees would behave that way. Would you keep spending the same amount if you saw your savings vaporizing before your eyes? I would expect them to spend a little more when their portfolio grows and a little less when it shrinks. Spending 4% of remaining savings each year instead of a flat $4,000 a year might accomplish that, for example.

In an extreme case, say retirement savings shrink by 50% in the first decade after retiring (or, conversely, grow 50%), I suspect a lot of retirees would not only reduce their spending, but abandon the SWR strategy and look for a new adviser. Of course, by then, the retiree has locked in a lower standard of living for the remainder of her life. The life annuity she took a pass on ten years earlier would start to look pretty sweet in retrospect. Despite what you may have read, a shrunken $50,000 portfolio is not assured of doubling in size because the retiree used to have $100,000.

Every retiree will behave differently, of course, and that would be really hard to implement in a spreadsheet or any other software, so we go with the constant dollar spending models because the oversimplified model makes the math a whole lot easier.

Many financial writers argue that no one really "does it that way", meaning everyone adjusts spending based on their remaining portfolio balance instead of spending a flat amount, but I have two responses to that. If no one does it that way, then everyone in the financial press should stop saying that you can do it that way.

And second, the SWR models predict outcomes for you only if you do "do it that way". (Operations Research guys say that a model is predictive only to the extent that its policies are followed.) The SWR results aren't predictive if you do something else, like adjust spending to portfolio value changes – which apparently is what everyone is actually doing.

(In simpler terms, you can't predict the average height of American men by measuring the height of players in the NBA. That's called the unrepresentative sample fallacy. Likewise, you can't predict portfolio failure rates for people who care about their savings balance from the failure rates of mythical retirees who ignore pending ruin.)

SWR predictions work, but only for spherical cows in a vacuum, or retirees who are oblivious to their current savings balance.

Perhaps the biggest assumption we make to simplify the math is that future stock market returns will look like historical returns.

The argument that they will look similar is an inductive argument that is not strong. Inductive arguments can't prove something is true, they can only argue that something is probably true. They are also defeasible, meaning that future information can prove the conclusion wrong. As Nassem Taleb would say, it was accepted as fact that all swans were white until someone found a black one. Future market returns will mirror past market returns until they don't.

It is interesting that some authors choose various periods of U.S. historical market data upon which to base their studies instead of using it all. They say things like, "We used historical data for the post-World War II era, because market data prior to that period is not representative of the current era." If one past period of history was not representative of this one, how do we know that the author's chosen data is representative of the future era, which is, after all, the one we need to know about?

It does make the math easier, though, when we toss in that little assumption.

On the other hand, there are many strong arguments that the future won't look like the past. Wade Pfau showed that 4% sustainable withdrawals only worked in 4 of 17 developed market nations (Canada, Sweden, Denmark and the U.S., in that order), lending credence to the argument that high equity returns in the 20th Century may be an anomaly of American history not to be repeated. Wade also recently argued effectively that future safe spending rates will be closer to 3% than 4% because the current risk-free rate in the U.S. is so low. That means that both stock and bond returns will be lower in the future than they have been.

Is it safe to assume that the worst thirty-year period of stock returns in our limited history is the worst that will ever happen? Well, no, because a black swan could reset the bottom. The bottom was reset in October 1929, for example.

The 30-year period beginning in 1966 was rough on retirees, but 2007 through 2009 were bad years and their returns are currently showing up only at the end of 30-year periods, like 1979 through 2008. With sequence of returns risk, however, we know that the real damage from these years will show up in studies that begin, not end, around 2008. That will be in 2037 and, again, that's the period recent retirees should wonder about.

But it certainly makes the math easier when we assume that we have already seen the worst.

I'm not saying that the work based on these Spherical Cows is without value, because sometimes having a questionable forecast is better than having none. Sometimes, it's the best we can do, given the shortage of reliable fortune tellers. As a friend of mine is fond of saying, "Bad breath is better than no breath at all."

But I also think it's important to understand the strength of the arguments and the assumptions upon which our plans are based. Assuming you're safe because your portfolio would have survived the worst bear market in the past 50 years is a big assumption.


  1. The part of Bill Bernstein's "Four Pillars of Investing" book that has stuck with me the most was his deep dive into financial history. Ever since then, I have paid a lot of attention to the time series that the various recommendations for investing strategies are based on. In particular, I have been concerned with the focus on the 1946 to present "Pax Americana" period, mainly because it is easy and pretty to the eye.

    However, the push towards de-regulation and wealth/income inequality over the last couple of decades has made me wonder if the 1870s to 1930s is becoming just as appropriate time period to consider. The stock market volatility over the past 20 years and the decline in interest rates looks much more similar to that period than the post WW II era. I don't think anybody but Shiller has put the entirety of that period into their finance models, probably because the result would scare a lot of people due to the high degree of volatility.

    It is nice that North America is surrounded by a moat which is a luxury that Europe and Asia don't have. Also, I think we will be heading into an era of more favorable demographics over the next decade which should help with the overall economic and financial trend.

    So I am not paralyzed with fear about the future as I head towards retirement, but it is clear that optimism should be tempered with a fair amount of caution without too much dependence on having spherical cows.

  2. A very nice post Dirk! Returns (or anything else) are unpredictable. The goal of dynamic updating (the other end of the spectrum to the SWR approach you refer to above) is to start each year with what is now known. Items such as this year's portfolio balance, new rolling adjustment to the risk and returns data because now another year may be added to the historical record, updating the time period remaining for the calculation this year using life tables that may have been updated too, etc. Modeling has been too stuck for too long on choosing data and time remaining periods ... and missing the elephant in the room ... the calculation or simulation is NOT a prediction! It simply is an indication on what may be prudent this year. AND, that too may be subject to adjustments during the coming year, when markets effect the portfolio you actually have (not the hypothetical portfolio). The longer it has been since the last time one looked at their prudent income for the coming calendar year, the more likely that past projection will be wrong. Spot on Dirk! [with pun intended for spotted Holstein spherical cows]!

    1. Well said, Larry. I plan to write on this topic in the near future to explain that retirement is best viewed as a dynamic process. In my experience, however, most spherical cows are charolais and not spotted, at all.

    2. hahaha ... and that too depends on assumptions and past experiences (paradigm)!

  3. Your post today reminded me of this quote:
    "A physicist, a chemist, and an economist are stranded on an island, with nothing to eat. A can of soup washes ashore. The physicist says, "Let's smash the can open with a rock." The chemist says, "Let's build a fire and heat the can first." The economist says, "Assume a can opener."

  4. TIPS and annuities have their problems also. TIPS assumes that the govt will calculate inflation honestly and accurately. A big if considering the type of sociopaths that seek public office. Annuities transfer the risk to insurance companies that can go bankrupt. If you envision poor long term returns, high inflation, etc then the insurance companies that guarantee the annuities will suffer just like any portfolio and may not be paid. I think that it is best to be prudent and enjoy life. If the zombie apocalypse comes your TIPS ladder won't do you much good

    1. Working 40 years to fund 70 is a tall order. There are no ideal solutions.

      Thanks for writing. And steer clear of zombies.

    2. I posted Bill Bernstein's blog post on this a while back where he concluded that you only have about an 80% chance of success in a 30 year retirement no matter what the Monte Carlo analyses come up with.

      There were a lot of Europeans who thought they were doing well in 1913 who soon discovered that their lives would be quite different than they imagined within a few years. A Monte Carlo analysis in 1913 based on the previous 50 years of data would not have remotely predicted their next 30 to 50 years.

    3. That point has come up on this blog several times and is a great one to keep in mind. Americans were in a similar situation at the beginning of October 1929.

      I think you should also keep in mind that, if Bernstein is accurate, 80% of the time your standard of living will be bounded by your finances and not by societal changes. It would be unwise to plan finances that are only 80% safe because there is a 20% chance of a larger societal risk.

      Bernstein is saying that no matter how safe your retirement plan might be financially, it won't stop a meteor.

    4. I think one of the biggest challenges that comes out of the 80% to 95% range is how to represent the difference in the risk for possibilities that do not exist in the data sets due to their short history.

      For example, one item that I have been pondering is the market disruption risk analysis of international equities when the databases of data used in the models are typically based on post-1946 data, which is a somewhat artificially stable period in the Western World. Right now, my equity portfolio has a reasonably high percentage of international equities in it. However, as I move towards retirement and the "pig" period instead of the "chicken" period, the apparently increasing instability in Europe and possibly parts of Asia, has led me to wonder about the potential for the investments to simply get cutoff behind a wall, figuratively or literally. This could cause a significant loss of capital, permanently or temporarily. The WW II period in Europe with the subsequent nationalization of industries in many countries behind the Iron Curtain is an example of this. Qualitative analysis indicates to me that a reduction in the percentage of international equities to the minimum amount that would provide significant diversification benefits of portfolio volatility due to their relatively low correlation in the models would be prudent for the long-term. For equity funds, I have stayed away from any single country funds (except the US) and look for very broadly diversified international funds that invest in multiple countries and continents.

      I generally stay away from international bonds as that is supposed to be the "safe" money. The Vanguard Target Date and LifeStrategy funds have a small percentage of the bond allocation in international bonds and that allocation percentage of about 20% is about as high as I would want to go due to the increase of default risk aas well as currency risk when international relationships go south. I have been seeing a lot of marketing of international income funds, which is probably counter to what most retirees need.

      It is nice living on a continent that has a moat. From a qualitative standpoint, that does reduce North America's risk profile for somebody who lives here. I don't think that is adequately reflected in MPT models. I don't think getting 95% modeled success instead of 90% by taking a lot of additional international risk would be an appropriate application of the models.

    5. All in all, a very defensible argument. I will add a couple of notes. First, Bill Bernstein provides an excellent argument for skipping international bonds altogether here.

      Second, while MPT (Modern Portfolio Theory) models would reflect the impact of international bonds on overall portfolio risk, it is SWR models that predict portfolio survival rates of 95% over time.

    6. Thanks for the link. The most important thing I found there is Bill Bernstein's "If You Can" for millenials which is already winging its way to my kids and a nephew. No 1 and 2 kids have generally been following his saving and allocation advice (at my insistence as the baby boom generation generally did not get this type of advice). No 3 & 4 kids have just been finishing up schooling in the past 12 months and are just starting to get into the real world game. Any message reinforcement is useful.

      BTW - one child is in Canada which seems to be a couple of decades behind the US for low-cost consumer-friendly retirement investment options although Vanguard has now shown up there over the past couple of years which should change the landscape. It does require a rethinking of investment strategy since the Canadian stock market is much smaller and wilder than the US one so accounting for currency risks is a major process. The currency tends to track commodity prices which is also the driver for a significant percentage of the Cdn stock market and home prices, so they can have triple whammies periodically.

  5. Set the stage for non-engineers: Engineers are typically known as practical, physicists as theoretical (per Dirk's blog article), and geologists as skeptical of limited evidence.

    A civil engineer and a geologist were driving through the countryside enjoying the scenery, when the civil engineer looks out he window and says, "Hey, look, a herd of dairy cattle, and they're all spotted." The geologist replies dryly, "Yes, on this side."

  6. Interesting article, but I'm not sure why you think that "the math" for modeling non-SWR portfolios is difficult. Perhaps it's hard to do in Excel, but even a marginally competent programmer like me wouldn't find it difficult to do using a language like Python, R, or Matlab.

    1. I don't think that's difficult at all. I think what's difficult is building a model that predicts how individual investors will behave in various market scenarios. Each will make his or her own spending decision at varying points. The SWR models oversimplify this by assuming all investors will simply decide to just keep spending the same amount when they are obviously going broke.

      You would need to develop a mathematical model that predicts when a retiree will decide to increase or decrease spending and by how much. I would probably decide at a different point than you, for example. That's difficult (I'd argue practically impossible to do well) no matter what programming language you choose.

      By the way, I've been developing software since 1972 and consider myself more than marginally competent. I have a computer science degree, 12 hours of post-grad work and a decade of code-writing experience before I moved into management.

      Thanks for writing!

  7. Here's random thought on your point about the difficulty of modeling withdrawal rates that factor in real world investor behavioral changes based on changing portfolio values. I found myself thinking about how traffic engineers develop models that predict future traffic congestion on a given roadway. What they don't do, as I understand it, is simply add predicted future car trips to a given roadway until at some simple to calculate future date the roadway's capacity is exceeded and gridlock permanently ensues. In other words, the roadway fails. Instead, they build into their models some interesting complexity based on anticipated human behavioral changes in the face of changing traffic conditions. For example, as a given roadway gets more and more crowded during peak traffic hours, the traffic model will assume that some number of drivers will choose alternate routes, that some will start their commute earlier or later, and that some will switch to carpools or public transit. Somehow they have a way, or claim to have a way, to build these behavioral changes into their modeling. I have no idea how they do this or if they do it well, but it struck me that it might be interesting to lock Wade Phau and a bevy of traffic engineers in a room for a week and see what they come up with.

    1. And here's another random thought. Why does everyone want to lock Wade in a room?

      I haven't given a lot of thought to the impact of human behavioral expectations on retirement planning, except to the extent of Bernstein's early pieces about how the amount you save isn't as important as saving more than everyone else saves. I think one of the challenges would be that a highway is something shared by many individuals, while a retirement plan is not. It's a point that I think deserves further exploration and might be an area where game theory could help.

      I emailed Wade and asked his thoughts, but I got back one of those canned "away" messages saying, "I can't respond right now. I'm locked in a room. But I'll get back to you as soon as I can."

      I'll keep you posted!