Why Value at Risk Cannot Be Calculated from an Incomplete Distribution

Summary

The question considers whether a 98% value at risk can be found when a discrete set of asset outcomes is accompanied by probabilities that sum to only 70%. The accepted answer says the distribution is invalid as provided, so the requested VaR is undefined from those inputs. The missing probability mass matters because it could correspond to outcomes in the loss tail and therefore affect the quantile used for VaR.

The example highlights a basic data requirement for probability-based risk measures: the stated probabilities must describe a complete distribution, or the omitted outcomes and their probabilities must be supplied. Without that information, a calculated tail threshold would depend on assumptions not contained in the question. The reply is brief and does not discuss how to model missing outcomes, normalize probabilities, or handle conditional distributions, so it establishes the limitation rather than offering a recovery method.

Key ideas

• A discrete probability distribution must account for all probability mass before its quantiles can be determined.
• An omitted portion of the distribution may contain severe losses and change the VaR threshold.
• The stated inputs do not define the requested VaR when their probabilities sum to less than one.

Tags

# Calculating VaR of an Incomplete Distribution


# Calculating VaR of an Incomplete Distribution












I am currently completing a multiple choice question that has stumped me.

> An asset has its price and its corresponding probability described as: 100, 0, -50, -70 and -90 with probabilities 50%, 12%, 6.5%, 1% and 0.5% respectively. Calculate the absolute value at risk with a loss probability of 2%.

To begin, we have not been given the complete probability distribution of the price of the asset; the probabilities add up to 70%. Is it still possible to calculate the 98% VaR?

It doesn't make too much sense to me as we don't know what's happening at the loss tail of the distribution. For all we know a loss of -100,000,000 can occur with probability 30%.

There is a choice of "None of the answers is correct", which is what I am leaning for.

I am hoping for a better explanation.

Kind regards,

## Answer by Aldanor (score 1, accepted)

https://quant.stackexchange.com/a/26009

The answer is undefined as the probability distribution provided is invalid, the probabilities don't sum up to one, so there's not much to expand on here.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.