September 2012 Q1

[Nikita90]

I'm struggling to understand part 2 of this question - value of VaR and TailVaR.

From my understanding for Company AA,

VaR(X) = -t where t=max{x: P(X<x)<=p)
In this case, p=0.05

X (return as %) Prob
6% 0.9811
-100% 0.0189

So here, P(X<6) is not less than 0.05
therefore t = -100%
so VaR(X) = 100%, which I think means that we are 95% certain that the expected loss will be £100. but this doesn't match the answer, but I can't get my head around where I am going wrong. I guess my interpretation isn't quite right.

In the revision book, it uses the risk neutral probabilities:

X (% return) probability
6% 0.9811
-100% 0.0189.
The solution then goes on to state that VaR(X) = -6%. But again, this doesn't make sense since the probability associated with a 6% return is 0.9811, hence greater than 0.05, meaning the condition P(X<x)<= 0.05 is not met.

Also struggling to understand why the VaR(X) for a 50:50 investment in Company AA and BB is 46%

Help please!

 

[Mark Mitchell]

You’re correct that the 95% VaR is -t where t = max{x: P(X < x) <= 0.05}. But the way that you’re working out t is not correct.

First of all, start off with the worst case, ie lowest return. Here, this is -100% (which relates to losing all of the money invested). It is impossible to get a worse return than this. So P(X < -100%) = 0. (Note the strict inequality.)

The next return is 6%. Since we have a discrete distribution, the only return strictly below 6% is -100%, so P(X < 6%) = P(X = -100%) = 0.0189.

And that’s it. There are no more possibilities. Since P(X < 6%) = 0.0189 < 0.05, then 6% is the maximum value of x that satisfies the definition above, meaning the VaR is -6%. This means we are 95% certain of getting back 6% more than we originally invested (since a negative value at risk implies a gain).

For part (c) of the question (the 50:50 split), the returns and associated probabilities are:
7% with prob 0.9448
-46% with prob 0.0182
-47% with prob 0.0363
-100% with prob 0.0007

So this gives:
P(X < -100%) = 0
P(X < -47%) = P(X = -100%) = 0.0007
P(X < -46%) = P(X = -100%) + P(X < -47%) = 0.0007 + 0.0363 = 0.037
P(X < 7%) = P(X = -100%) + P(X < -47%) + P(X < -46%) = 0.0007 + 0.0363 + 0.0182 = 0.0552 > 0.05

So the maximum value of x that satisfies P(X < x) <= 0.05 is -46%. Meaning the VaR is 46%.

 

[anees aslam]

Hi,

I am struggling to understand how the TailVaR and conditional TailVaR are calculated?

 

[Steve Hales]

Hi,

I am struggling to understand how the TailVaR and conditional TailVaR are calculated?

The TailVaR is the expected shortfall below L, ie E[max(L-X,0)], where L is chosen to be a particular percentile point on the distribution. The conditional TailVaR requires this quantity to be divided by the probability of the shortfall occurring.
In this question we have that L is the 95% point of the return distribution.
Let me know where you're struggling.