Investments, tenth edition

  Relative Frequency of Large, Negative 3-Sigma Returns 

 Here we concentrate on the relative frequency of large, negative returns compared with those 

frequencies in a normal distribution with the same mean and standard deviation. Extreme 

returns are often called  jumps,  as the stock price makes a big sudden movement. We compare 

the fraction of observations with returns 3 or more standard deviations below the mean to 

the relative frequency of negative 3-sigma returns in the corresponding normal distribution. 

  

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 A formula for the ES in the case of normally distributed returns is given in Jonathan Treussard, “The Non-

monotonicity of Value-at-Risk and the Validity of Risk Measures over Different Horizons,”  IFCAI Journal of 

Financial Risk Management,  March 2007. The formula is   

ES 5

1

.05

 exp(m)N

32s 2 F(.95)4 2 1  

 where   m  is the mean of the continuously compounded returns,  s  is the SD,  N (• ) is the cumulative standard nor-

mal, and  F  is its inverse. In the sample above,  m  and  s  were estimated as 5.47% and 19.54%. Assuming normal-

ity, we would have ES  5   2 30.57%, suggesting that this distribution has a larger left tail than the normal. It should 

be noted, however, that estimates of VaR and ES from historical samples, while unbiased, are subject to large 

estimation errors because they are estimated from a small number of extreme returns. 

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6/18/13   8:03 PM

6/18/13   8:03 PM

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