5% VaR of Normal P/L

[alexwallace]

Dear @David Harper CFA FRM

I came across the following question in the Schweser notes to Market Risk book:



It is stated that correct answer is D but I think it is A, could you please help me?

Thanks !

 

[David Harper CFA FRM]

Hi @alexwallace It's a lazy question that might penalize candidates who know too much. Ironically, those without preparation can probably just visualize to see what they want: the distribution has mean of +20 such that the left-side 5% quantile (0.050) is located at 20 - 1.65*10 = + 3.5, so the answer is supposed to be that 95.0% of the distribution lies to the right of +3.50. Hence, it's clear why D is meant to be the answer.

But the confusion might be that they ask for the 5.0% VaR which is unusual, but we should understand that 0.050 is sometimes used as a synonym for 0.950 in the VaR context because VaR only cares about the adverse side of the distribution. The alternative way to express this, and I suspect what rightly confuses you, is that we could also say "a 5.0% probability of earnings less than $3.50" or really "a 5.0% probability of profits less than $3.50". But if I were to edit the question, I would look to be more precise (e.g., earnings versus losses should be "profit" versus losses).

(but a literal person would be tempted to claim, given the 5%, that the answer should be "95% probability that profits do not exceed 36.5").

Further, given the question is lazy, I doubt the writer realizes that profit/loss is a format (P/L) which alternatives with loss/profit (P/L). This doesn't really matter as the +20 is a positive mean, such that implicitly the format is P/L.

But I would just visualize a distribution with positive mean of +20.0 and sigma = 10. On the left side, the 0.050 quantile is 20 - 10*1.65 = +3.50 and on the right side is 20 + 10*1.65 = +36.50. This is straightforward. We would not take that distribution and say "5% probability of losses at least $3.50 million" because, let's see, Z = (-3.5 - 20)/10 = -2.35 and N(-2.35) = 0.94%, so we could say "0.94% probability of losses that exceed $3.50 million". I hope that's helpful, but again, the question is language lazy so I wouldn't get too stuck on it. Thanks,

 

[alexwallace]

Hi @alexwallace It's a lazy question that might penalize candidates who know too much. Ironically, those without preparation can probably just visualize to see what they want: the distribution has mean of +20 such that the left-side 5% quantile (0.050) is located at 20 - 1.65*10 = + 3.5, so the answer is supposed to be that 95.0% of the distribution lies to the right of +3.50. Hence, it's clear why D is meant to be the answer.

But the confusion might be that they ask for the 5.0% VaR which is unusual, but we should understand that 0.050 is sometimes used as a synonym for 0.950 in the VaR context because VaR only cares about the adverse side of the distribution. The alternative way to express this, and I suspect what rightly confuses you, is that we could also say "a 5.0% probability of earnings less than $3.50" or really "a 5.0% probability of profits less than $3.50". But if I were to edit the question, I would look to be more precise (e.g., earnings versus losses should be "profit" versus losses).

(but a literal person would be tempted to claim, given the 5%, that the answer should be "95% probability that profits do not exceed 36.5").

Further, given the question is lazy, I doubt the writer realizes that profit/loss is a format (P/L) which alternatives with loss/profit (P/L). This doesn't really matter as the +20 is a positive mean, such that implicitly the format is P/L.

But I would just visualize a distribution with positive mean of +20.0 and sigma = 10. On the left side, the 0.050 quantile is 20 - 10*1.65 = +3.50 and on the right side is 20 + 10*1.65 = +36.50. This is straightforward. We would not take that distribution and say "5% probability of losses at least $3.50 million" because, let's see, Z = (-3.5 - 20)/10 = -2.35 and N(-2.35) = 0.94%, so we could say "0.94% probability of losses that exceed $3.50 million". I hope that's helpful, but again, the question is language lazy so I wouldn't get too stuck on it. Thanks,

Many thanks David, as always your comments are very useful. This was exactly what was bothering me