VaR/TailVar confusion - discrete case - Speciman Exam Paper Q5

[Sandor Kelemen]

Hello everyone,

Computing Question 5 ii) a) (parts b) & c) have the same issue) from https://www.actuaries.org.uk/studying/curriculum-2019/actuarial-mathematics I arrive at stochastic return r_A as follows:

r_A = -1 (i.e. loosing all the investment) if company A defaults (probability q=0.01887)
r_A = 0.06 if company A survives (prob 1-q)​

Hence

P(r_A<x) = 0 for x<=-1
P(r_A<x) = q for 1<x<=0.06
P(r_A<x) = 1 for 0.06<x​

As q < 0.05 (95% confidence is required) is is evident that max { x : P(X<x) <= 0.05} = 0.06. So VaR_95% = - 0.06. I.e. in monetary terms noting 100 original investment the maximum loss on a 95% degree of confidence are -6 pounds. However the official solution (available via https://www.actuaries.org.uk/studying/curriculum-2019/actuarial-mathematics) says that VaR_95% = 0.

Then the TailVar is also calculated in the official solution in my opininion as a conditional TailVar (conditioned that there is a shorfall). According to my calculations the TailVar should be as follows:

A) using my VaR_95%
TailVar = (0.06 - (-1))q=0.02 so the expected loss in excess of the VaR_95% is 100x0.02 = 2. In absolute thinking the loss is -6+2 = -4 (still a profit).

B)
Using theii zero VaR_95%
TailVar = (0 - (-1))q=q=0.01887 so the expected loss in excess of the VaR_95% is 1.887. In absolute thinking the loss is -6+2 = -4 (still a profit).​

Could anybody reinforce me in thinking that something went wrong in their official calculation?

Side comment: in the CMP VaR's are calculated always for returns (then as an answer the monetary approach is prefered). I thought I will work accordingly also during the exam. But I start to feel here that in some cases it is better to directly work in monetary terms.

For your convenience, I also attached the mentioned paper and the corresponding solution file.

Thx. in advance for your help,

S.

 

[Anna Bishop]

Hi Sandor, your approach is fine. If you look at CT8 Sep 2012 Q1 (which is where this question was examined), the Examiners Report says, after the solution:

The question was framed sufficiently openly that candidates could quote values at risk

relative to the maximum return, the expected return or the initial investment. Full marks

were available for any approach if it was followed through correctly although the below sets

out answers relative to the maximum return.

Your approach considers the return relative to the initial investment so it is fine.

Equally with TVaR, your approach involves a non-conditional TailVar whereas the examiners have gone for a conditional TVaR in their solution. Either is fine, as long as you state what you are doing.

Always state the formula that you are using for VaR and TVaR in your answer, then it's absolutely clear what you are trying to do.

Monetary or % terms is fine. Hope this helps assuage the panic.

Anna