I'm preparing an exam, Quantitative Methods for Financial Markets. My book is not really clear for what concerns the calculation of normal distribution quantiles that have to be used in VaR's formula.
1) I'm asked to find the standard normal distribution quantile corresponding to a Confidence Level of 97,5%. Looking at the normal distribution table it's really easy to find it's equal to -1,96 but I doubt I'll be given the opportunity to use it during the exam so my problem is… how can I calculate it?
2) I'm asked to find 2% and 98% daily VaR of an asset whose value is 80.000 knowing that the normal distribution's variance is 20%. The first problem is that I don't know the mean of the distribution. The second problem is that, while the quantile value for a Confidence Level of 98% is known to be -2,06, I can't really figure out how to find the quantile for 2%… is it just the inverse value? 2,06? If so, I would obtain a negative and a positive VaR with the same value; on a financial point of view, what would that mean?
Many thanks!
Best Answer
1) It could be that it's one of those values you should just remember. 1.96 is so commonly used that it might be considered "common knowledge" in a statistics class. Same for 1.6449, which corresponds to a tail probability of 5 %.
2) I don't know much about the finance part of this, but I think you are just supposed to calculate the 2 % tail probability for a normally distributed random variable with mean 80,000 and variance 20 % of that (i.e. 0.2*80,000=16,000). That is, with $X\sim N(80,000; 16,000)$,
$$ Pr(X>c)=0.02 \Leftrightarrow Pr\left(\frac{X-80,000}{4,000}>\frac{c-80,0000}{4,000}\right)=0.02 $$
which amounts to setting the right hand side of the inequality equal to 2.06 and solve for $c$. Remember that the Normal distribution is symmetric, so $$ Pr(Z>2.06)=0.02\Leftrightarrow Pr(Z<-2.06)=0.02=1-Pr(Z<2.06) $$ where $Z\sim N(0,1)$.