Which of the following statements are correct?

The daily VaR of an investor’s commodity position is $10m. The annual VaR, assuming daily returns are independent, is ~$158m (using the square root of time rule).

Which of the following statements are correct?

I. If daily returns are not independent and show mean-reversion, the actual annual VaR will be higher than $158m.

II. If daily returns are not independent and show mean-reversion, the actual annual VaR will be lower than $158m.

III. If daily returns are not independent and exhibit trending (autocorrelation), the actual annual VaR will be higher than $158m.

IV. If daily returns are not independent and exhibit trending (autocorrelation), the actual annual VaR will be lower than $158m.
A . I and IV
B . I and III
C . II and III
D . II and IV

View Answer

Answer: C

Explanation:

In the case of mean reversion, the actual VaR would be lower than that estimated using the square root of time rule. This is because gains over a period would be followed by losses so that the price can revert to the mean. In such cases, the autocorrelation between subsequent periods is effectively negative. This means the combined VaR over the periods would be lower.

In the case of positive autocorrelation, the actual VaR would be higher than that estimated using the square root of time rule for exactly the opposite reason than that described for the mean-reverting case.

(Recall that Variance (A + B) = Variance(A) + Variance(B) + 2*Correlation*StdDev(A)*StdDev(B). In cases where correlation is zero, the variance can simply be added together (which is the case for iid observations). In cases where the correlation is negative, the combined variance (and therefore standard deviation and also VaR) will be lower; and where correlation is positive, the combined variance (and therefore standard deviation and also VaR) will be higher.)

Therefore statement II is correct, and so is statement III. Choice ‘c’ is the correct answer.