Miller - End of Chapter question - Hypothesis testing and Confidence Intervals


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Question 4:
Suppose you invest in a product whose returns follow a uniform distribution between −40% and
+60%. What is the expected return? What is the 95% VaR? The expected shortfall?
The expected return is +10%. The 95% VaR is 35% (i.e., 5% of the returns are expected to be
worse than –35%). The expected shortfall is 37.5% (again the negative is implied).

Can you please explain how the 95% VAR is calculated?

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @Fab It is a uniform distribution which not only has a "flat" density but in this case conveniently spans 100% = + 60 - (-40%). If we think about this density as a rectangle with height of 1/[60% -(-40%)] = 1.0, then the 95% VaR is just 5% from the worst outcome of -40%, which literally requires only adding 5%.

Put another way -40% + 5%*(60% - -40%) =-40% + 5% = -35%. The density between -35% and -40% is 5.0% of the total 100% density between 60% and -40%. It is where the "worst 5% loss tail" begins, so to speak. BTW, the expected shortfall (ES) is just the (conditional) average of that 5% tail, which spans from -35% to -45%. But since this is a uniform distribution, it's just the average of the endpoints, average{-40, -35} = -37.5%; this is the answer to the question, if we know that our outcome is already in the worst 5.0% ("conditional on a loss worse than -35.0%") what is the average of such an already-bad loss? I hope that helps!