Calculating VaR for a Discrete Poisson Distribution

Summary

The document asks how to calculate a 95% Value at Risk for a Poisson random variable, with its mean and variance both given as 10. The accepted response clarifies that a Poisson distribution is discrete, so its cumulative distribution is obtained by summing probability masses rather than integrating a density. The relevant quantile can therefore be found from the discrete cumulative probabilities.

For a loss convention where VaR is reported as a positive loss amount, the response describes 95% VaR as the opposite sign of the distribution’s 5% quantile. It suggests consulting a statistical table to identify that quantile, but does not give a numerical VaR for the stated parameter or discuss alternative loss definitions. The explanation is brief and assumes the sign convention is understood; the tail probability and loss interpretation should be checked against the variable’s definition.

Key ideas

• For a Poisson variable, calculate cumulative probabilities by summing discrete probability masses rather than integrating.
• A 95% VaR corresponds to the opposite sign of the 5% quantile under the stated loss convention.
• A statistical table can be used to look up the required quantile.
• The response does not calculate a numerical VaR for the example parameter.

Tags

# Determining Value at Risk of a Poisson distribution


# Determining Value at Risk of a Poisson distribution












If my discrete random variable had a poisson distribution with both moments say equal to 10, how can I find the Value at Risk for a 95 percent confidence interval?

I have seen that I need to integrate the PDF from the lower limit up until $L$ so I am trying to integrate that from $0$ to $L$ then equating it to 0.5 but its an absolute mess. Any help would be appreciated.

## Answer by Martin Vesely (score 0, accepted)

https://quant.stackexchange.com/a/53915

Firstly, a Poisson distribution is discrete one, so you can get CDF by sumation instead of integration. See here how CDF looks like.

Secondly, 95 % VaR is a opposite value to 5 % quantile of the distribution (i.e. if quantile is -10, VaR is +10, meaning loss 10), so have a look at some statistical table. For example here.

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