CoachHistorical simulation applies each of the last observed risk-factor changes to today’s positions, revalues, and reads VaR from the empirical distribution of the P&Ls: with , the -th largest loss.
Voorbeelden
Example 21.9 (One book, several numbers)
Over the last 500 days, historical simulation gives a one-day 99% VaR of USD 2.10 million; the EWMA delta-normal VaR is 1.58 million, the equal-weighted one 1.60 million, Monte Carlo with full revaluation on the EWMA covariance 1.66 million, and filtered historical simulation 2.09 million (Figure 21.1). Historical simulation with delta–gamma revaluation gives 2.10 million, delta alone 1.91: the short straddle’s gamma adds losses in both tails. The normal methods are lower because the historical moves have fatter tails than the normal.
Example 21.17 (Four years of backtesting)
Recomputing each model every day from 2 September 2022 to 23 September 2026, on the same book revalued at each day’s levels, against the next day’s P&L: historical simulation has 10 exceptions in 1 000 days (Kupiec p-value 1.00) and none in the last 250, green; the EWMA delta-normal model has 28 (Kupiec likelihood ratio 22.0, p-value ) and 8 in the last 250, yellow with a 0.75 increase; filtered historical simulation has 8 and 3, green (Figure 21.3). None fails the independence test at 5%.