Jun 05

Normal Distribution: Properties

by David Harper, CFA, FRM, CIPM

FRM | Risk | Quant | |

For the 2007 FRM, learning outcome 2.5 is "Identify the key properties of the normal distribution." The density function for the normal distribution is given by: 

 

And its key properties are...

 

  • the normal only requires two parameters: mean and dispersion (variance or standard deviation)
  • the normal is symmetrical (i.e., skewness = 0)
  • the normal has kurtosis = 3 (or "excess kurtosis = 0" where excess is the difference between three)
  • 90% of its density lies within +/- 1.645 standard deviations of the mean
  • 98% of its density lies within +/- 2.33 standard deviations of the mean

For risk measurement, we typically measure the z-values (or reliability factor) at 90% and 98% confidence intervals. Confidence intervals span both directions from the mean: positive and negative. So, a 90% confidence interval has 10% of its density in the tails, which is 5% to the right and 5% to the mean. Similarly, a 98% interval has 1% in each tail (1% to the left + 1% to the right + 98% density "in the middle" = 100%).

Because we care only about losses, ours is a one-tailed test. That's why the critical z-value is 1.645 for both the 90% confidence interval and the 95% confidence level (or 5% significance) for a one-tailed VaR test:

 

In regard to parametric (e.g., delta-normal VaR), we typically use 5% and 1% significance. If you are sitting for the exam, you should therefore memorize 1.645 and 2.33.

Here is an EditGrid spreadsheet that plots the normal probability density function (pdf). You can open your own read/write copy here. Notice I used both the built-in function and the hand calculated approach.

EditGrid Spreadsheet by bt/admin.

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