VaR at level p% corresponds to which percentile of the underlying loss distribution?

• A. The p-th percentile of the random variable X
• B. The mean of the random variable X
• C. The p-th percentile of the random variable X minus its mean
• D. The upper p-th percentile of X

Answer: (A)

VaR at level p% is the p-th percentile of the loss distribution. By definition, VaR_p is the value v such that the probability of losses not exceeding v is p, i.e., P(L ≤ v) = p. This makes VaR_p the p-th percentile (the p-quantile) of the loss variable L. For example, with p = 0.95, 95% of losses are at or below VaR_0.95, and only 5% exceed it. The mean is a measure of center, not a boundary of outcomes, and centering by subtracting the mean or using the upper tail percentile does not describe VaR on the original loss scale.