Var of a loss

[sapozan]

Hello,

I am struggling to find an approach to the following:

Probability density function of the daily portfolio loss L:

View attachment 2521
How to calculate VaR and Tail VaR with a confidence level of 0.95?

Can someone help please?

Thank you

 

[Detective]

First check if it's a valid PDF, in this case it is since p(x) >= 0 for all x, and it integrates to 1 across the domain. Area of triangle is (20)*0.10*0.5 = 1.



By inspection this distribution is symmetric around x = 0, so you can focus on the right half of the triangle and find the value of x < 10 such that area equals 0.45.

The math is using just power rule and some algebra.

VaR math: https://www.wolframalpha.com/input/?i=integral+from+0+to+x+of+(1/10-x/100)+dx+=+0.45

Two solutions, pick one in domain, so x = 6.83772.

ES = 1/(0.05) * E(x | x>6.83772)

ES Math: https://www.wolframalpha.com/input/?i=20*integral+from+6.83772+to+10+of+(x/10-x^2/100)+dx+

ES = 7.89183.

 

[sapozan]

Thank you @Detective, just a couple of questions:

Why do I pick an area of 0,45 for VaR? I understand I cannot standardize this and take the value from the tables?

Is Tail VAR exactly the same as Expected Shortfall?

Why is the integral in ES formula multiplied by 20?

Thank you so much for your help.

 

[Detective]

Thank you @Detective, just a couple of questions:

Why do I pick an area of 0,45 for VaR? I understand I cannot standardize this and take the value from the tables?

Is Tail VAR exactly the same as Expected Shortfall?

Why is the integral in ES formula multiplied by 20?

Thank you so much for your help.

>> Why do I pick an area of 0,45 for VaR?

You asked to find at 0.95 confidence (where I assume confidence is from [0,1]) in the question right? There is 1/2 area (0.50 to 1) on the first quadrant, so we want to find the 95th percentile, so 0.50 - 0.05 = 0.45. Note the rest of the 0.5 is actually already in the II quadrant (i.e. -10 <= x < 0).

>> I understand I cannot standardize this and take the value from the tables?

I am not aware of tables for that particular distribution. It is a straightforward integral. You could potentially use simple geometry in this case given the function we are working with. However, using calculus is a general solution.

>> Is Tail VAR exactly the same as Expected Shortfall?

Is this a homework problem, FRM prep, or something else? As far as I know in industry people loosely use tail VaR, CVaR, and ES. So I take them to pretty much mean the same thing, but depending on the textbook or who you ask in academia maybe it has different meaning.

>> Why is the integral in ES formula multiplied by 20?

1/(1-0.95) = 1/0.05 = 20

It's by definition of the measure. I view it as a scaling factor since you want the average of "x" in the 5% tail, so you have multiply by that adjustment factor. ES at same confidence as VaR always has to at least as large as VaR by definition.

 

[sapozan]

Great, thank you @Detective, I just currently study FRM and it is part of my preparation so not sure Tail VaR is exactly the same as ES in context of Market Risk ?

 

[Detective]

Great, thank you @Detective, I just currently study FRM and it is part of my preparation so not sure Tail VaR is exactly the same as ES in context of Market Risk ?

For FRM it is safe assumption to make tail VaR means same thing as conditional VaR, i.e. ES. FYI calculus (taking an integral) and this question is outside scope of FRM exam. You won't see this type of problem on the exam.

 

[sapozan]

Sorry for misunderstanding, I am not exacly preparing for the exam just yet, but doing my frm masters. this question came up in past exams so I am trying to make sure I understand it all correctly