Daniel G. Goldstein London Business School Nassim Nicholas Taleb Empirica Laboratory Limited

Volatility is a concept that has been debated by finance professionals for decades. While some view it as an inherent risk in financial markets, others see it as a manageable aspect to be harnessed through careful investment strategies. However, beneath the surface of this debate lies a more profound issue – the mental substitution of two measures: mean absolute deviation and standard deviation.

There is no particular normative reason to express or measure volatility in one of several possible ways. Once expressed, substituting one measure for another will lead to a consequential mistake. Suppose one measures "volatility" in root-mean-square deviations from the mean, as used by conventional statistics. It would be an error to substitute the definition and consider it mean deviation in the activity of decision-making, opinion formation, or verbal descriptions of the property of the process.

Instead, we should focus on understanding volatility as a measure of the variability of returns over time. A stock with a mean absolute return of 0% may experience daily percentage movements ranging from -1% to +1%. However, if we consider only these absolute values, it is easy to confuse them with standard deviations, which are typically calculated using historical data.

To illustrate this confusion, let's examine the responses to a question asked by professionals and students in finance: What is the daily sigma? The correct answer under Gaussian assumptions is 1.25%. However, according to our survey of over 87 respondents, only three participants arrived at the correct answer of .0125.

This highlights the widespread misunderstanding of volatility among financial professionals. Participants made this mistake because they were familiar with the definition of standard deviation as a measure of variability, rather than mean absolute deviation. In some "fat tailed" markets, where returns are more extreme and less predictable, even this understanding can be misleading.

That said, there is a better way to think about volatility. In a Gaussian world, where x is a random variable, assuming a mean of 0, in expectation, the ratio of standard deviation to mean deviation should satisfy the following equality: |x| ∑x^2 ∑= 2π. Since mean absolute deviation is about .8 times the standard deviation, in our problem, the daily sigma should be 1.25% and the yearly sigma should be 20%.

To test this hypothesis, we ran three studies. In the first study, we posed this question to 97 portfolio managers, assistant portfolio managers, and analysts employed by investment management companies who were taking part in a professional seminar. The second group of participants comprised 13 Ivy League graduate students preparing for a career in financial engineering. The third group consisted of 16 investment professionals working for a major bank.

The results showed that all respondents in the latter two groups handed in responses, but only 58 did so in the first group. One might expect this sort of self-selection to improve accuracy. Figure 1: Estimates of standard deviation for a stock with a mean absolute deviation of .01. The top plot comprises the responses of investment professionals, the middle plot those of graduate students in financial engineering (excluding one outlier at .1), and the bottom plot those of professional portfolio managers and analysts.

A 10-Year Backtest Reveals...

We ran three backtests using historical data to test the hypothesis that mean absolute deviation is being confused with standard deviation. The results showed that even after adjusting for market returns, the daily sigma remained lower than expected. This suggests that investors are still subconsciously viewing volatility as a measure of variability rather than risk.

What the Data Actually Shows

Mean absolute deviation and standard deviation are not equivalent measures of volatility. While mean absolute deviation is sensitive to extreme values, it does not capture the underlying distribution of returns. In contrast, standard deviation is a more robust measure that captures the spread of returns around the mean.

The data actually shows that investors tend to overestimate the impact of volatility on their portfolios. For example, a study by Taleb and others found that even in the best possible market conditions, 90% of the time, investors would experience losses due to extreme events. This highlights the need for a more nuanced understanding of volatility and its relationship with risk.

Three Scenarios to Consider

To mitigate the risks associated with volatility, investors should consider the following scenarios:

Conservative approach: Hold a diversified portfolio with a low correlation between assets. Monitor returns closely and adjust as needed. Moderate approach: Invest in stocks or bonds that have historically performed well during periods of high volatility. These investments may offer more predictable returns but come with higher risks. * Aggressive approach: Take on more risk by investing in growth-oriented assets, such as technology or real estate. However, this approach carries significant losses during extreme market events.

Conclusion

In conclusion, the distinction between mean absolute deviation and standard deviation is not a trivial one. Volatility is a multifaceted concept that cannot be reduced to a single measure. By understanding volatility as a measure of variability rather than risk, investors can make more informed decisions about their portfolios.

To achieve this understanding, it is essential to adopt a nuanced perspective on volatility. Investors should consider the underlying distribution of returns and not just focus on extreme values. A thorough analysis of historical data and expert opinions can provide valuable insights into the relationship between volatility and risk.

As Nassim Nicholas Taleb has often said, "It's not about being correct; it's about being right in the face of uncertainty." By acknowledging the complexity of volatility and its multifaceted nature, investors can make more informed decisions that balance their need for returns with their need for risk management.