Calculating VaR for a Normal Loss Distribution with a Rare Shock
Summary
The document explains a Value at Risk calculation for a mixture of a standard normal variable and an independent, rare discrete shock. In the example, the shock shifts a small-probability component far into the loss tail. Because that component accounts for most of the worst one percent of outcomes, the requested tail quantile can be approximated by finding the corresponding quantile in the unshifted normal component after accounting for the shock probability.
For less widely separated components, the answer gives the mixture cumulative distribution as a weighted sum of normal cumulative probabilities and says to solve for the target quantile numerically. It also corrects an initial approximation for the weight of the normal component, yielding a value that rounds to the reported result. The shortcut relies on the shock component being sufficiently separated from the main distribution; it is not a general closed-form VaR solution. The general method is to find the quantile of the full mixture distribution.
Key ideas
- A normal variable plus an independent discrete shock forms a mixture distribution.
- A rare, distant shock can dominate a tail quantile even when its probability is small.
- The separated-component approximation adjusts the target probability for the shock mass.
- For overlapping components, solve the weighted mixture cumulative distribution for the desired quantile.
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# Value at Risk for normal r.v. with shock (regimes)
# Value at Risk for normal r.v. with shock (regimes)
I am struggling to understand how was this simple Value-at-Risk calculated. It's Example 1 in Daníelsson, Jón, et al. "Fat tails, VaR and subadditivity." Journal of econometrics 172.2 (2013): 283-291 (a draft of which can be found in this link).
The authors consider a random variable $X_i$ defined as the sum of a standard normal random variable and a discrete random variable (which is independent from the normal r.v.).
I have attempted to obtain the distribution of $X_i$ by considering that it can be seen a mixture of gaussian distributions, with each component of the mixture representing a "regime", but I don't arrive at the same value as the authors (3.1).
I am conscious that this is a very basic exercise, but I must be missing something....
Thank you in advance for any suggestions.
## Answer by Attack68 (score 0, accepted)
https://quant.stackexchange.com/a/40847
You can approximate the pdf of the mixture as: $$f_X(x) = 0.009 g(x+10) + 0.991 g(x)$$ for $g(x)$ the pdf of $\mathcal{N}(0,1)$
You will observe that since the standard deviation is 1 almost all of the density attributed to the component $g(x+10)$ is about the point x=-10 and certainly for about 7 standard deviations away about the point x=-3.1 it is effectively all encompassed.
Therefore finding the 1% VaR is equivalent to finding the (1%-0.9%) VaR of g(x), which is in fact -3.09. I suppose the author rounded up.
`=NORMINV(0.01-0.009,0,1) gives -3.0902 in excel`
edit after comments
If the numbers were not no obviously skewed in this case then you would essentially have the equation:
$$ 0.009 * \Phi(\alpha + 10) + 0.991 * \Phi(\alpha) = 1\% $$ for $\Phi$ the distribution function associated with g(x). Your task is to solve for $\alpha$ for which I don't believe there to be an analytical solution and you would have to use an iterative procedure. The estimation above was essentially 1 iteration since the problem lended itself nicely to the scenario. (And actually note that from this explicit formula you can see I made a mistake by not dividing by 0.991: -3.0902 / 0.991 = -3.118, which is still rounded to -3.1)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.