Historical simulation VaR

Historical simulation takes today's portfolio and runs it through the last 500 market days, one at a time. Each past day gives a scenario for moves in rates, equities and currencies, the positions are revalued, and the 99% VaR is read off the fifth loss in the sorted list, under the simplest convention; others take the sixth or interpolate between the two. No distribution is assumed: the fat tails and correlations of the period come through untouched. The parametric method, the variance-covariance approach popularized by J.P. Morgan's RiskMetrics in 1994, assumes normal returns instead and derives VaR from a standard deviation; Monte Carlo draws thousands of scenarios from a calibrated model, more flexible, more costly, and only as good as that model. Historical simulation has a known flaw. The whole tail rests on a handful of points, and it reacts late: a shock enters the window all at once and drops out all at once 500 days later. Hull and White, in the Fall 1998 Journal of Risk, rescale each past return by the ratio of current volatility to volatility on the original day, estimated with GARCH or EWMA. Barone-Adesi, Giannopoulos and Vosper (Journal of Futures Markets, 1999, vol. 19, no. 5) go further with filtered historical simulation: they draw the standardized residuals of a GARCH to build multi-day paths and fully reprice options, with no correlation matrix. Basel allowed it from the 1996 amendment, at 99% over ten days with at least one year of history. In a Solvency II internal model, the one-year horizon at 99.5% is where it struggles: thirty years of data give thirty independent annual changes, for a one-in-200 quantile. PRIIPs use it in their own way. Category 2 reads the moments of five years of returns before a Cornish-Fisher adjustment; only category 3 truly resamples those daily returns, over at least 10,000 simulations.