You hold 250 daily P&L observations. The three worst are -8.2m, -6.9m and -6.1m. What is the historical 99% one day value at risk under linear interpolation, and how precise is that estimate?

You hold 250 daily P&L observations. The three worst are -8.2m, -6.9m and -6.1m. What is the historical 99% one day value at risk under linear interpolation, and how precise is that estimate?

Approach: Locate the quantile by rank in the ordered losses and interpolate. Then count how many observations actually determine the figure.

6.5. The historical value at risk is $6.5m. A 99% one day value at risk from 250 observations sits at rank 0.01*250 = 2.5 in the ordered losses, so linear interpolation between the second worst 6.9 and the third worst 6.1 gives 6.5. Three numbers determine the whole estimate. The standard error of an empirical quantile is sqrt(p*(1 - p)/n)/f(q), and with a plausible fitted tail density the sampling error here exceeds $1m, so 6.5 and 8.0 are not distinguishable at this sample length. A single day rolling out of the trailing window can move the figure by 0.4m with no position change at all, which is why historical simulation is reported beside a parametric figure and beside stress scenarios that do not depend on the last 250 days having contained a crisis.

Follow-up: How would exponentially weighting the 250 observations change both the estimate and its standard error?

Key concepts: historical simulation, quantile estimate, value at risk, sampling error.