# The Ragged Rock Part 2

Risk in retirement investing*And the ragged rock in the restless waters,*

*Waves wash over it, fogs conceal it;*

*On a halcyon day it is merely a monument,*

*In navigable weather it is always a seamark*

*To lay a course by: but in the sombre season*

*Or the sudden fury is what it always was.*

TS Eliot “The Dry Salvages”

The Ibbotson study that I mentioned in Part 1 as cited by Burton Malkiel, clearly linked asset class returns and risk as measured by the standard deviation of the returns (rewards in Bogle’s parlance). The standard deviation is the most commonly used measure of risk and is often referred to as volatility. It is the basic measure of risk in Modern Portfolio Theory. The standard deviation formula is presented below in two parts. The first formula is the variance. The variance is the sum of the squared deviations from the average return divided by the number of returns. So it is the average of the squared deviations from the average. The standard deviation is the square root of the variance. It has the advantage of being the same units as your returns. So if your returns are percentages, then the standard deviation is in percentage units. Often you see the Greek symbol sigma used instead of ‘s’. So if someone says something is a 2 sigma event that means its probability is determined to be within two standard deviations of the mean, positive or negative.

So just how does this linkage between risk and return work? Modern Portfolio Theory, first articulated by Harry Markowitz in his doctoral dissertation in 1952, in its simple form says that you can set the reward and then have to accept the associated risk or you can set the risk and have to accept the reward for that level of risk. So in essence if you want higher returns, you get higher risk and if you want lower risk you get lower returns.

The beauty of the model for financial theory is that you only need two numbers to make the model work, the average return and the volatility (standard deviation). And with past market data you can compute both. By assuming that the data follows the bell curve you can then compute the probabilities of various losses or gains and in essence ‘predict’ your risk-reward relationship. It is then a matter of personal preference where you choose to position yourself.

Some key assumptions underlie the theory. These assumptions are often unrecognized or ignored by individual investors. That is where the problems begin. All these assumptions have pretty much been proven wrong. So here are the assumptions:

- Each return is independent. That is, a return from yesterday or a year ago does not influence the return today.
- The process generating returns, whatever it is, does not change.
- Returns follow the proportions of the bell curve. That is most returns, measured as positive or negative changes, are small and very large changes have a very low probability.

We can start with the third assumption that risk is adequately represented by the standard deviation.The bell curve is a fundamental component of the model as it is applied to real world returns and used to make asset allocation decisions. This third assumption says that 35.74 percent of all returns are within one standard deviation of the mean; 95.72 percent of all returns are within two standard deviations of the mean; and 99.74 percent of all deviations are within three standard deviations of the mean. The bell curve was discovered about 200 years ago and is credited to Carl Friedrich Gauss. Gauss and others used it to gauge the errors in observations of planetary orbits. In effect, based on this scientists could safely ignore measurements that were far from the average as “outliers” that were such a remote probability. With real world returns these outliers actually occur and are tough to ignore.** **

I spent a lot of time using this model to craft a retirement portfolio that I thought was prudent and at the same time could provide the growth of assets that would maintain sufficient funds for my retirement. I got a big surprise. The 2008 S&P market drop of 37% was the largest since the S&P began in 1957. The previous largest single year drop was 26.3% in 1974. Now, most people are not 100% invested in stocks. But take a typical 60% stock portfolio. A 37% drop in stocks translates to a 30% drop in the total value of your portfolio—still pretty shocking if you are retired or close to retirement and have to depend on that portfolio for income.

What does the bell curve say about a 37% drop? If I take the Standard and Poor’s 500 Stock Index annual returns (from Yahoo Finance) from 1958 to 2008 I can compute the average return and the standard deviation for the 51 year period. This is really easy in Excel. The average return is 11.25% and the SD is 17.60%. (Let me caution you on a couple things. There are two kinds of average return, the arithmetic mean which is what I have presented and the geometric mean, which is the return computed from the actual sequence of returns. This is what an investor would see and is usually less than the arithmetic mean. So the actual return to the investor during the time was not 11.25%. It was more like 9.75%. The second caution is that this return data does not take inflation into account and is termed the nominal return as opposed to the real return.) The normal curve is plotted in the picture. I have retained the decimal values rather than percentages.

Standard deviation does provide a good summary of the spread and movement of past asset prices and provides a means for comparing one asset class to another. Whether or not it is truly a comprehensive measure of the risks enumerated earlier remains highly debatable.

I went looking for help on this and read Nassim Nicholas Taleb’s *The Black Swan. *It is subtitled ‘The Impact of the Highly Improbable’. He conveniently has a chapter, *The Bell Curve, That Great Intellectual Fraud*. He says “Shockingly, the bell curve is used as a risk-measurement tool by those regulators and central bankers who wear dark suits and talk in a boring way about currencies.” (p230). He says the main point of the Gaussian…is that most observations hover around the mediocre, the average; the odds of a deviation decline faster and faster (exponentially) as you move away from the average.” Later he says “Measures of uncertainty that are based on the bell curve simply disregard the possibility, and the impact of sharp jumps or discontinuities …Although unpredictable large deviations are rare, they cannot be dismissed as outliers because, cumulatively, their impact is so dramatic. ” (p 236)

One mathematician that Taleb has high regard for is Benoit Mandelbrot and he references a recent book, *The (Mis)Behavior of Markets* by Mandelbrot and Rick Hudson (The ideas are Mandelbrot’s. Hudson helps make the concepts understandable.) Madelbrot is most famous for the discovery of fractals and fractal geometry. So I ordered the book. You should get it too. It is interesting to read and has very little math. As it turns out, Mandelbrot has been a long time student of pricing and market theory. With his studies of cotton pricing in the early 1960’s Mandelbrot shook up the economics establishment for a short time. His findings were never refuted, but merely ignored and so the orthodox thinking still dominates. One of his big issues is with the measurement of risk.

As noted, one of the key ideas of modern finance is that through the use of the laws of chance risk can be measured and managed. Mandelbrot doesn’t really argue with using the laws of chance to model financial markets. He just disputes the theorists’ use of the bell curve as that model. He uses daily price changes to make his point. Using the Gaussian formula for the bell curve Mandelbrot calculates the probability of three successive daily declines of 3.5%, 4.4% and 6.8% in the Dow Jones Industrial Average in August 1998 which is fairly recent history. Mandelbrot calculates odds of the 6.8% decline at one in 20 million and he adds that if you trade daily it still wouldn’t happen once in 100,000 years. “The odds of getting three such declines in one month are about 1 in 50 billion. He then calculates the odds of the October 19, 1997 crash at less than one in. You have to agree with Mandelbrot when he says: “The seemingly improbable happens all the time in financial markets.” (p4)

One last point from Taleb: “In the last fifty years, the ten most extreme days in the financial markets represents half the returns.” (p 275) So on the positive side of the curve as well as the negative, highly improbable (according to the model) events have a huge impact.

So there is a huge disconnect between the theory and the empirical reality. But you know that the difference between theory and reality is that in theory there is no difference. John Maynard Keynes said that “all models are wrong, but some models are useful.” So the real question are MPT and the bell curve useful models to construct an effective portfolio for your retirement?

Use them but don’t rely on them. At the end of 2008, it was problematic whether or not my portfolio would sufficiently recover. Fortunately it has. But I believe the current investment and risk models represent “financial orthodoxy” and lull us into a false sense of security. The current formulation of risk in financial theory is proving inadequate and increasingly is being called into question by real events and by critical thinkers.

I would rather rely on seasoned investors and practical traders who have long experience. This includes John Bogle and Burton Malkiel. Another example is David Swensen, the head of Yale University’s endowment. He is a legend in his own time. Over twenty-five years of steady exceptional performance have made him hugely influential on Wall Street and especially within the endowment and public funds community. In an interview with Chrystia Freeland in the Financial Times (10/10/2009) he offers some investing advice: ‘”you should invest only in things that you understand. That should be the starting point and the finishing point….The overwhelming number of investors, individual and institutional, should be completely in low-cost index funds because that’s easy to understand.”’