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VaR and Standard Deviation

 

How is VaR related to Standard Deviation?


VaR and Standard Deviation are both related measures of a Distribution of Returns.  Standard Deviation is designed to measure the overall width of a distribution and therefore considers both positive and negative returns.   It is usually represented by the Greek letter sigma (s) such that 1 sigma is one Standard Deviation, 2 sigma is twice the Standard Deviation, and so forth.

 

VaR, on the other hand, seeks to measure just the size of the losing (left) tail.  It is characterized by a percentage that represents the area under the curve not considered as VaR.  Thus a 95% VaR means that 95% of the area under the curve is to the right, with 5% of the area to the left.  These percentages are of course directly related to probability that you will lose an amount equal to or greater than the computed VaR.

 

The graph below shows a typical Distribution of Returns with its 1-sigma width and 95% VaR levels indicated:

For many types of financial instruments, the Distribution of Returns is found to be perfectly "Normal".  A Normal Distribution follows a precise mathematical formula (known as a Gaussian Curve) which has many interesting features.  For our purpose, the most interesting feature is the fact that the tail of a Normal Distribution is directly proportional to its width.  This means that VaR levels can be found by simply multiplying the measured Standard Deviation by an appropriate factor as follows:

 

Number of Sigma

Value-at-Risk

1.00

84.1%

1.28

90.0%

1.64

95.0%

2.00

97.7%

2.33

99.0%

3.00

99.9%

 

When RiskManager computes VaR from a Distribution of Returns, it employs one of three methodologies: Monte Carlo Simulation, Historical Simulation, or Parametric Modeling.  If you select Parametric Modeling when computing VaR, RM automatically assumes the Distribution of Returns will be Normal.  VaR at different confidence levels are therefore calculated directly from the Standard Deviation of the Distribution using the table above.

However, not all financial instruments follow a Normal Distribution.  Many have fatter tails than would be expected from a Normal Curve.  Others, such as options, have a skewed Distribution as illustrated below:

For Distributions like this, there is no direct relationship between VaR and Standard Deviation.  Qualitatively, one could say that the higher the sigma, the higher the confidence level, but there is no simple table as above.  In one case we might find that 95% VaR falls out at 1.8 sigma while in another case it might be 1.3 sigma.  VaR must therefore be computed for each confidence level desired by direct measurement of the area under the curve.  This procedure is used whenever you choose to compute VaR using the Monte Carlo or Historical Simulation models.

Finally, please note that the units of VaR are generally the same as the units of Standard Deviation, and both are determined by the units that the Distribution of Returns is calculated.  For example, if the Distribution of Returns is in JPY won and lost, then VaR and Standard Deviation will be in JPY.  If the Distribution of Returns is computed as a percentage of current Present Value, then so will the VaR and Standard Deviation.  For convenience, RM includes Percent VaR as one of its predefined statistics.  For more details, please refer to How is Percent VaR calculated?