Q: Assuming a normal distribution, calculate the Vaule at Risk for each of the following situations: The 95% VaR for an investor with a 10m portfolio where the average annual return is 6% and there is a 5% chance that the value of the porfolio witll fall by more than 10% over a year. (Though not explicitly stated, it is assumed in the solution that they want the VaR over a year). Their answer: =(0.06-(-0.1))*10m = 1.6m My issue: doesn't saying "there is a 5% chance that the value of the porfolio will fall by more than 10% over a year" include the six percent gain? Why should I factor this in at all? (I just did 10%*10m=1m) Thanks,
IIUIC, there is 1 5% chance that the portfolio will lose 10% of its value, which means that you lose both the expected income of 6% as well as the portfolio shrinkage of 10%, for a 16% total loss.
But you never had the 6% to lose it, so it wouldn't factor into VaR except to offset losses (i.e. VaR at 50% might be -6%*1M=600,000, meaning at the 50th c.i. you expect a 600,000 loss
It was your EXPECTED value. At no change to the portfolio, you expect to make $6M. I understood the question as you did originally (which is part of the confusion around VaR, especially for those of us who approach it from the loss side and not the banking side) but there is a 5% change that the financial situation will by $1.6M less than expected, which is what they are calling VaR. See Sweeting pg 394 formula 15.13, and go with that definition for this exam
Thanks, that certainly helps my understanding of what they are calling VaR. I think its defined differently in the CAS exams
Hi. In the real world, there are any number of ways of calculating and expressing VaR. Whenever a VaR is quoted it is useful to know how it is being measured (relative to the current position? an expected position? over what time period? at what level of confidence etc). In the exam, we would suggest you stick to the bookwork definition, but the examiners recognise there are different definitions. As an example, Q5 in April 2012 asked for a VaR to be calculated. The examiners expressed their solution as the loss in excess of the mean liabilities. The bookwork solution gives the loss including the mean liabilities. Credit was given for either approach.
We recently had this same issue in a CT8 exam question about VaR. The examiners accepted that the question had been ambiguous and awarded full credit for any possible interpretation. So, with the question in the original post, any of 1m, 1.6m or even 9m would have been accepted (although personally I was less happy with 9m). The key thing is that you explain what you're doing, so the examiners can follow your workings and give you appropriate credit.
Assuming we agree on the assumptions that lead to the 1.6m answer, I think this question is a heck of a lot easier to answer working from first principles. The (annual) 95% VaR = 10m * annual volatility * Z(95%) Let X represent annual return. We're told that: P(X<-.1)=.05 Standardizing gives us: P(Z<(-.1-.06)/annual volatility) = .05 This implies that: (-.1-.06)/annual volatility = Z(5%) --> annual volatility = (-.1-.06) / Z(5%) Thus, The (annual) 95% VaR = 10m * annual volatility * Z(95%) = 10m * (-.1-.06) / Z(5%) * Z(95%) = 10m * -(-.1-.06) = 10m * (.1+.06) = 1.6m I don't believe that the solution provided in ActEd is that intuitive.
