Certain things can be proven statistically, even if the cause is unknown.

Consider what happens when your roll a pair of dice. There are 36 possible combinations, e.g., (1 + 1), (3 + 4), (6 + 5). Only one combination—(1 + 1)—produces an outcome of 2. Therefore, you can expect to roll a 2 only one out of 36 times, on average. On the other hand, six combinations produce an outcome of 7— (1 + 6), (2 + 5), (3 + 4), (4 +3), (5 + 2), (6 + 1). Accordingly, you can expect to roll a 7 once in every six rolls, on average. The table below shows the number of times each outcome should occur in 100,000 rolls.

If the results of your dice-rolling experiment diverge from these figures by more than a tiny, tiny bit you can be certain that you were not using fair dice. (We rule out the possibility of psychokinesis.) Note as well that the results do not tell you *why* the dice were not fair. There may have been a defect in the manufacturing process. Alternatively, someone may have “fixed” the dice by loading, shaving, or some other method. Whatever the cause, the fact that the dice are not fair is incontrovertible, given the laws of probability and the very large number of trials.

Let us now consider price moves on bonds. Unless something unusual is going on, a histogram of monthly price changes should produce a bell-shaped curve as in the illustration (non-bond-related) below (see note 1). This pattern is intuitive to seasoned market participants: In most months, prices go up or down by about an average amount. Once in a while the price change is much greater or much smaller than average. Even more infrequently the monthly price change is *very *much greater or smaller than average. (By the way, this same reasoning applies to total returns. We confirmed that our key result holds for total returns as well as price changes.)

The notion that price moves should follow a normal distribution is not a matter of academic theorizing with no connection to the real world. According to the math that formally describes the bell-shaped curve, price changes in 68.2% of all months in the sample of observations should fall in a range of plus/minus one standard deviation from the mean (simple average) return. Price changes between one and two standard deviations greater than the mean or between one and two standard deviations smaller than the mean should account for 27.2% of the months. That leaves 4.6% of all months in which to expect price moves more than two standard deviations above or below the mean. In the top panel of the table below, we apply those percentages to the 372 months in our 1987–2017 observation period to predict how many months will be found in each of those three ranges, if the monthly price changes are in fact normally distributed.

In the case of investment-grade corporates, the actual distribution among the three ranges almost perfectly matches the predicted distribution—253/101/18 versus 254/101/17. That result should lay to rest any notion that the normal distribution is just a theoretical model of how the things would work in an idealized world. Investment-grade corporate behavior demonstrates that following a bell-shaped curve with a specific mathematical description is how the real world works.

Government bonds (Treasuries and agencies) veer a bit more from the predicted distribution. All told, 125 months lie outside the range of plus/minus one standard deviation versus the predicted total of 118. Note that research has found that sort of pattern in equity returns, popularly referred to “fat tails.” That abnormality is commonly attributed to momentum trading. The notion is that instead of settling down to normal swings after major positive or negative news comes into the market, stocks continue to gyrate wildly without any additional, major news hitting the market, thanks to aggressive traders who jump on the recent trend and overpower value-oriented traders.

High-yield bonds display the opposite sort of abnormality. Instead of having too many extreme moves, the asset class has too few. Prices in just 83 months moved up or down by one standard deviation or more, some 30% less than the predicted count of 118. The “missing” months were all in the plus/minus 1–2 standard deviations range—just 64 actual versus 101 predicted. Extreme observations of plus/minus 2 standard deviations or more were about right—19 actual versus 17 predicted.

These are not inconsequential divergences from the standard bell curve. With the statistical technique known as the Jarque-Bera(JB) test we can confirm that the data shown for the ICE BAML High Yield Index do not represent a normal distribution. The calculation produces a very high JB value indicative of non-normality and a p-value far below 0.01. Similarly non-normal are the distributions shown, in the second panel of the table, for the BB, B, and CCC-C sub-indexes. Of the three, the BB sub-index is the most overconcentrated in the plus/minus 1 standard deviation range—202 actual months versus 172 predicted.

**Why are high-yield price moves not normally distributed?**

If a dice-rolling experiment fails to produce the predicted distribution of outcomes, as discussed above, we know that the dice are not fair. That information does not explain why the dice are not fair, that is, whether the manufacturing process was defective or whether somebody altered them. Similarly, the fact that price changes on the high-yield index are not normally distributed does not tell us *why *they are not normally distributed. We can, however, generate hypotheses and, to the extent feasible, test them.

One hypothesis we thought of is that the Federal Reserve has created an artificially stable financial environment through its quantitative easing (QE) policy. The third and fourth panels of the table display the results of our test of this hypothesis. They show far fewer months outside the plus/minus one standard deviation range in both the pre-QE era (42 actual versus 84 predicted) and the QE era (25 actual versus 35 predicted). The JB test confirms that in both sub-periods the distributions are non-normal. In short, we can reject the hypothesis that quantitative easing artificially stabilized the high-yield market, resulting in fewer extreme monthly price changes than ought to have been observed. If anything, high-yield price changes have been closer to normal during the QE period. (Note that the pre-QE versus QE testing is based on means and standard deviations within those sub-periods.)

We have come up with only one other hypothesis to explain the shortfall of extreme price changes in the high-yield market. Other market participants may find it unpalatable and propose other possible explanations. We encourage discussion and debate on this topic.

Our remaining hypothesis is that reported prices understate the high-yield market’s true volatility. This does not necessarily imply that traders are deliberately understating the magnitude of price declines during major market declines, although we cannot readily disprove that possibility.

Understatement of price declines could also result from good-faith efforts to mark to market the many issues that do not trade in any given month.

Note that if price declines are understated in downturns, price rises will be understated in subsequent upturns. This can explain why in the underlying data we find a shortfall of large up moves as well as a shortfall of large down moves. Also, as noted above, the shortfall of extreme high-yield price changes was entirely in the plus/minus 1 to 2 standard deviations range. We might reasonably infer that the flaw in high-yield pricing, whatever it turns out to be, understates price changes in “somewhat extreme” market declines, but cannot hide the most massive sell-offs.

Our finding of a non-normal distribution of high-yield price changes, with fewer extreme changes than are expected to occur in a properly functioning market, has important implications for asset allocation. It implies that the index-derived standard deviations and, by extension, Sharpe ratios that institutional investors are using to evaluate the high-yield asset class paint too rosy a picture of its risk-reward ratio. Further exploration of this important question seems warranted.

**Indexes used in this report:**

ICE BofA Merrill Lynch BB US High Yield Index

ICE BofA Merrill Lynch B US High Yield Index

ICE BofA Merrill Lynch CCC-C US High Yield Index

ICE BofA Merrill Lynch US Corporate Index

ICE BofA Merrill Lynch US High Yield Index

ICE BofA Merrill Lynch US Treasury & Agency Index

Thanks to John Finnerty and Yuewu Xu for their kind assistance in this analysis. Any errors or omissions are the author’s.

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*Marty Fridson, Chief Investment Officer of Lehmann Livian Fridson Advisors LLC, is a contributing analyst to S&P Global Market Intelligence. His weekly leveraged finance commentary appears exclusively on LCD, an offering of S&P Global Market Intelligence. Marty can be reached at marty@fridson.com.*

*Research assistance provided by Kai Zhao and Yaxian Li.*

*ICE BofAML Index System data is used by permission. Copyright © 2018 ICE Data Services. The use of the above in no way implies that ICE Data Services or any of its affiliates endorses the views or interpretation or the use of such information or acts as any endorsement of Lehmann, Livian, Fridson Advisors, LLC’s use of such information. The information is provided “as is” and none of ICE Data Services or any of its affiliates warrants the accuracy or completeness of the information**.*