- Beatrice Badalotti

# Now let’s talk about Standard Deviation from a “non-standard” point of view.

The Standard Deviation is an indicator that looks at dispersed data around a position index; it is one of the few statistical indicators that are able to measure fluctuation around the mean.

In finance, especially in Italy, this indicator has become increasingly used to assess the risk of a financial instrument, illustrating that the higher the standard deviation, the higher the risk the investor runs.

This association is very approximative and misleading; the standard deviation is not an indicator of risk but one of uncertainty since when it’s very high, estimates on a given financial instrument are not too reliable, and when it is low, they can be considered more accurate.

First introduced by Pearson, the standard deviation is nothing but the square root of the variance, see Wikipedia for the mathematical formula.

The main issue with the standard deviation from a financial use point-of-view, is not so much the fact that it is associated with a normal distribution of returns (randomness), but that it is a symmetrical indicator, a squared number in the math formula, and unable to distinguish positive from negative returns.

This is a strong defect in the evaluation of any investment risk, considering that a historical series yielding 0.5%, -2%, 0.5%, -2% has the same volatility as one with -0.5% , 2%, -0.5%, 2% even if their trajectories and final results are completely different.

Knowing this we can interpret that when used alone, the standard deviation is not a useful indicator for quantifying the qualities of a financial instrument.

It explains why other statistical indicators were born based on evaluating the mean and the standard deviation, that is, to better assess financial instruments. It’s the same case with the Sharpe index, which we’ll get to later.

Another major weakness, or strength of the standard deviation, depending on how it is used, is that it is not stable and constant throughout time. Of course it’s more stable than the returns are, but it varies according to the timing parameters, the frequency (daily, weekly, monthly, etc.) and the historical period of observation. For example, the standard deviation of a balanced fund in 2008 was greater than that of a 2006 equity fund. Though financial intermediaries seem to not have fully grasped the concept, solely using the standard deviation to determine the degree of risk of an investment instrument or portfolio, while meeting MiFID standards, can create big problems.

There are other more obscure and less promoted indicators, which we will discuss in this blog, such as the Ulcer Index, that can be used to obtain a greater amount of information than the standard deviation provides.

Standard Deviation, Volatility and VIX.

In the previous post we discussed some of the flaws and limitations of the Standard Deviation.

To understand just how dissimilar the topic of financial statistics in Italy is from the Anglo-Saxon interpretation, look no further than Wikipedia and its definition of Volatility in both languages.

In the Italian version, Volatility is defined as the percentage of price variation over time, and up to this point the definitions coincide. Following that, volatility is described as the variance, which is the square of the standard deviation, while the English definition appropriately states that it is the square root of the variance.

Years ago, Prof. Francesco Corielli of Bocconi University, told me that Volatility is the price we pay for keeping the stock markets open every day; never had a definition given me such a clear idea of what price volatility truly meant.

If every day outside your house there was a display screen scrolling offers by passers-by to buy your house, the price fluctuations would probably be similar to those of the stock market. That’s because each person would have their own opinions and diverse ideas, and people can be influenced by mood, emotion, and current events. Just like that, the financial markets must price the value of an asset on a daily basis, which is much more difficult to estimate than the price of a house.

Using this example, we can further guess that should potential buyers be uncertain of the value of your home and the buying opportunity, perhaps fearing the real estate market will lose value, we would see some very high price fluctuations on the display screen. With the price volatility being higher, the risk of that price dropping would increase as well. The price wouldn’t necessarily fall, but the fluctuations (therefore the price range) would be greater. The risk at this point really lies in selling the house in a panic at a low price for fear that it will collapse even more…

However, I’d like to draw attention to the fact that almost all financial operators think that there is a perfect negative correlation between increasing volatility and a falling market.

If it is always true that volatility, by pure structure of calculation rises when the market loses, it is just as true that a growth phase in the market can correspond with a growth in volatility.

Anyone feeling skeptical about this statement can go see what happened in 1999 and 2007.

In 1993, the CBOE (Chicago Board Option Exchange) constructed the VIX (acronym of Volatility Index), thanks to Professor Whaley. The VIX is an index of the implied volatility derived from the prices of options at various maturities on the American stock market.

Implied volatility is completely different from historical volatility in the way it is calculated, though it’s interesting to note the similarity in results obtained over time from the estimates of the two indicators.

As of 2004, it’s been possible to buy and/or sell futures on the VIX index, betting on future fluctuations in the markets; strategies on this will be discussed in this blog.