Suppose someone asked you the following question:
A stock (or a fund) has an average return of 0%. It moves on average 1% a day in absolute value; the average up move is 1% and the average down move is 1%. It does not mean that all up moves are 1%–some are .6%, others 1.45%, etc. Assume that we live in the Gaussian world in which the returns (or daily percentage moves) can be safely modeled using a Normal Distribution. Assume that a year has 256 business days. The following questions concern the standard deviation of returns (i.e., of the percentage moves), the “sigma” that is used for volatility in financial applications. What is the daily sigma?
If you answered 1%, you are in good company. Daniel Goldstein of the London Business School and Nassim Taleb posed this question to a group of portfolio managers, analysts, graduate students in financial engineering, and investment professionals. Over half of the respondents chose 1%.
Unfortunately, they are all wrong. The correct answer is the mean absolute deviation is approximately 0.8 the magnitude of the standard deviation (reference post three of this thread to see a proof). Therefore, 1.25% is the correct answer. The following is a plot of the responses:
Goldstein and Taleb:
The error is more consequential than it seems. The dominant response of 1% shown in Figure 1 suggests that even financially- and mathematically-savvy decision makers treat mean absolute deviation and standard deviation as the same thing. Though a Gaussian random variable that has a daily percentage move in absolute terms of 1% has a standard deviation of about 1.25%, it can reach up to 1.9% in empirical distributions (emerging market currencies and bonds).
Debriefings with respondents revealed that they rarely had an immediate understanding of the error when it was pointed out to them. However when asked to present the equation for “standard deviation” they expressed it flawlessly as the root mean square of deviations from the mean. Whatever reason there was for their error, it did not result from ignorance of the concept. Indeed, most participants would have failed a basic statistics course had they not been aware of the mathematical definition. And yet, when given data that is clearly not a standard deviation, they treat it as one. Kahneman and Frederick (2002) discuss a similar problem of statisticians making basic statistical mistakes outside the classroom, “the mathematical psychologists who participated in the survey not only should have known better–they did know better…most of them would have computed the correct answers on the back of an envelope”.
While I have serious concerns on how the question is phrased (it’s entirely possible that the respondents have a strong grasp on statistics), it does suggest one thing: humans chronically underestimate volatility and risk. This is one of the re-occuring themes in that appear in Nassim Taleb’s research.
Read the full paper here.