VaR Methods

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This Demonstration shows a comparison between calculating value at risk (VaR) with the delta-normal method and the lognormal distribution method.

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The VaR is defined as a threshold loss value, such that the probability that the loss on the portfolio over the given time horizon will exceed the value of .

Various VaR methods are used in finance by portfolio managers in order to assess the exposure of their portfolio, or the exposure of individual assets in big losses.

The lognormal VaR method is based on the geometric Brownian process. In addition, the delta-normal method for VaR calculation assumes that the coefficient of variation of the asset returns is described with the normal distribution.

As you can see, there are significant differences between the two methods. Choosing one model instead of another can lead to large changes in the variables and thus, the decision made may be different.

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Contributed by: Dimos Karaflos (January 2015)
Open content licensed under CC BY-NC-SA


Snapshots


Details

As you can see, the lognormal VaR method is more sensitive to volatility changes! This is due to the fact that the mean of the distribution depends on volatility. Specifically, this is due to the factor .

The difference is that the same input values in these models assess different measures of loss over the same time horizon. This depends on the fact that the mathematical calculation is different in each case.

The lognormal VaR method is calculated with the code:

stockPrice-Quantile[LogNormalDistribution[Log[stockPrice]+(r-volatility^2/2) time,volatility Sqrt[time]],confidenceLevel]],

which is based on the geometric Brownian process.

The delta-normal method for VaR calculation is calculated with the code:

-Quantile[NormalDistribution[0,1],level] price volatility Sqrt[time],

where it is assumed that the coefficient of variation of the asset returns is described with the normal distribution N[0,1].

The delta-normal method is related to the hedging factor and sensitivity measure delta, which is multiplied in the formula of the delta-normal approach. In this Demonstration we refer to stock VaR, thus delta is equal to 1.

Choosing one model instead of another can lead to large changes in the variables.

The negative values of VaR denote that there is no potential of loss!

References

[1] J. C. Hull, Options, Futures, and Other Derivatives, 7th ed., Upper Saddle River, NJ: Prentice Hall, 2009.

[2] P. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd ed., New York: McGraw-Hill, 2007.



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