What is a symmetrical distribution?

Symmetry implies that the magnitude and frequency of upside gains is a mirror image of downside losses. Of course, that isn't a reality in passive market index returns. For example, looking at the decade 2000 - 2010 the S&P 500 stock index ended with no upside gain, but it had two downside loss periods of -50% or more along the way. 0% upside gain vs. -50% downside loss does not meet the definition of symmetry. Some passive asset allocators portend to offer symmetry by their asset allocation and balancing, yet their actual performance doesn’t provide evidence they have been able to balance risk and reward, either. But the real issue is: why would you want to?

A symmetrical distribution is also called a symmetric or normal distribution or referred to as symmetry. A data set of values or variables that occur at a regular frequency with an average (mean), mode, and median occurring at the same point. A symmetrical distribution has balance and symmetry. A symmetrical distribution has the shape of a bell curve when plotted on a graph. It is not skewed to the left or right like an asymmetric distribution. The majority of the data is stacked in the middle at or near the mean, average, and mode. If a line is drawn in the middle, the right and left side mirror each other. Symmetry has very little observations that fit into the tails to the right or left (known as "fat tails"). 

Market returns are not a symmetric distribution. Market returns do not fit into a normal bell curve. Market returns are skewed and have many "fat tails" and the data in the tails are much larger than predicted by Gaussian mathematics. 

An interesting observation about symmetry is that market returns can have decades that end without a positive advance. Over such a period, we could say risk and reward was symmetrical: the profits and losses were balanced. With symmetry, the gains in bull markets are balanced from the losses from bear markets. A 100% gain from a multi-year bull market may get wiped out by a -50% loss of a bear market. A -50% loss of a bear market then needs a 100% gain just to get back to breakeven. 

Symmetry: academic theory based risk measurement.

Academic theory based risk measurement defines risk as volatility, regardless of direction. Academic theory based risk measurement uses backward-looking annualized standard deviation of historical returns. Standard deviation assumes that market returns conform to a normal bell-shaped distribution. Two characteristics of a normal distribution, a normal bell-curve, are skinny "tails" and perfect symmetry. That is, the left and the right are a mirror of each other so they balance and there are very few, and small, outliers.

Because standard deviation assumes that market returns conform to a normal bell-shaped distribution the use of academic theory risk measurement is a serious risk in itself because returns are not symmetrical. Since market returns do not display such perfect symmetry necessary for academic theory based risk measurement, their losses routinely exceed what is expected. In many cases, their losses far exceed expectations to the point of panic selling. 

The normal distribution model provides for simple guides.

1.      68.3% of the time returns should fall within one standard deviations

2.      95% of the time returns should fall within two standard deviations.

3.      .03% of the time returns should fall within three standard deviations.

The last one, which is expected to capture the outsized risk, is a real problem for investors using this academic theory based risk measurement. Symmetry implies a very low, almost non-existent, occurrence of returns that are three standard deviations away from average. However, actual market losses exceed three standard deviations far more often and with greater magnitude than predicted by standard deviation.

The Normal Distribution Image (click to view)

The use of academic theory based risk measurement is a risk itself and largely the cause of larger market declines.

To learn more, read about asymmetrical distribution. 

Asymmetry™ is a registered trademark of Shell Capital Management, LLC