4.2.3 Historical Simulation VaR (Empirical Quantile & Filtered Historical Simulation)

Basic Prerequisites: Statistics & Inference (order statistics).“

1. Intuition & Practical Objective

Historical simulation (HS) is the most honest of the three VaR methods: don’t assume a distribution at all - replay last week/month/year through today’s portfolio and see how it would have done. The practical objective: build $n$ hypothetical P&L scenarios (“what if today’s positions had been held on each of the last $n$ days?”) and take the empirical quantile. No normality assumption, fat tails for free, options handled by brute-force full revaluation if you want.

The method in three steps:

Collect factor-return history: the last $n$ days of daily returns on each risk factor (e.g. $n = 500$ ).
Apply to today’s positions: for each historical day, shock today’s factor levels by that day’s returns and revalue the portfolio - producing $n$ hypothetical daily P&L numbers.
Read the quantile: the 99% VaR is (roughly) the 5th-worst of those 500 scenario P&Ls (Hull Ch 22 §22.2 uses 501 days → 500 returns).

Its great strengths: it is non-parametric (it reproduces whatever fat tails, skew, and jumps happened in the window) and trivially generalizable to options (each scenario just revalues). Its fundamental weakness is the window: it can only “remember” the crises that are in the last $n$ days, and a single old crash casts a ghost into every scenario until it scrolls off (the ghost effect, verified in 05).

Filtered Historical Simulation (FHS, Hull & White 1998; McNeil & Frey 2000) fixes the “yesterday = 3 years ago volatility” naivety by scaling each historical return by the ratio of today’s (conditional GARCH) vol to the vol on that historical day:

r_{i, t}^{*} = r_{i, t} \cdot \frac{σ _{i, today}}{σ _{i, t}},

so recent history reflects current market volatility while keeping the empirical tail shape. It is the bridge from this folder to EVT (McNeil & Frey fuse a GARCH filter with a generalized-Pareto tail).

2. Mathematical Ground Truth & Derivations

The empirical CDF and its quantile. Let ${L_{1}, \dots, L_{n}}$ be the $n$ hypothetical portfolio losses (or, equivalently, rank the P&L). The empirical distribution function assigns mass $1/ n$ to each observation; the HS VaR at confidence $α$ is the empirical $α$ -quantile. Rank the losses ascending: $L_{(1)} \leq \dots \leq L_{(n)}$ ( $L_{(n)}$ = worst). Then

VaR_{α}^{(H S)} = L_{(⌈ n (1 - α)⌉)}

e.g. $n = 500, α = 0.99 \Rightarrow$ rank $⌈ 500 \cdot 0.01 ⌉ = 5$ , the 5th-worst loss. (Hull’s 501-day/5th-worst convention and RiskMetrics variants differ by interpolation convention; that is immaterial vs the method’s real limits.)

Why it carries the tail correctly. Because HS uses realized daily moves, a 1997 crash and a fat-tailed equity skew are in the sample, so the quantile reflects them - no Gaussian assumption. This is exactly what delta-normal VaR sacrifices.

Assumptions you are really making:

(H1) The past window is representative - the next day’s risk is drawn from the same distribution as the last $n$ days (stationarity).
(H2) Returns are i.i.d. across the window - each historical day is equally predictive, no time-variation in vol (FHS relaxes this in the simplest way).
(H3) Full revaluation is affordable - every historical scenario revalues every position (HS gets expensive with many options; a delta/moment shortcut loses the richness it’s valued for).

The empirical-quantile sampling error (Glasserman Ch 9 §9.1). The HS VaR estimate has asymptotic variance

n (x_{p} - x_{p}) \Rightarrow N (0, \frac{p ( 1 - p )}{f ( x _{p} ) ^{2}}),

with $p = 1 - α$ . The density $f (x_{p})^{2}$ in the denominator magnifies the noise exactly in the far tail - the rarer the exceedance you ask about, the noisier the estimate, because few observations live there. This is the formal reason 99.9% HS VaR is unstable.

3. Computational Implementation - HS vs FHS on the shared portfolio, stdlib only

We generate 500 days of factor-history, replay it through the $40 k /60 k$ portfolio (this is the 3,289.54 hub figure), then apply a volatility filter to show FHS moves the VaR with regime.

The exact 3,289.54 reproduces the hub historical-simulation column (5th-worst of 500, i.e. $k = ⌈ 500 \cdot 0.01 ⌉ = 5$ ). The EWMA recursion shows the mechanism behind FHS: when today’s conditional vol (say ~0.024) exceeds the calm historical average (0.012), filtered HS up-weights the biggest recent moves so VaR bends toward the current regime - while plain equal-weight HS keeps saying “the last 500 days looked like this,” which is precisely the ghost/window problem.

4. Failure Modes & First-Principles Breakdowns

The window is everything - and it’s arbitrary. $n = 250$ days misses a crisis from 400 days ago; $n = 1500$ days dilutes recent vol with ancient calm. Either way HS is only as good as the crises in the window. This is the ghost effect (verified: crash-in-window VaR 3,465, after it exits 3,196 - 05).
The i.i.d. assumption (H2). Equal-weight HS treats a serene day the same as a crash day; if vol is currently high, plain HS understates risk (yesterday’s calm dominates). FHS fixes the level but not the tail dependence.
Fat tails are only sampled, not modeled. HS shows you the crashes that happened, but the next crash may be worse than any in the window - HS has no way to extrapolate a “worse than ever” tail. That is EVT’s job (EVT).
Empirical-quantile noise in the tail. The $\propto p / f (x_{p})^{2}$ variance (Glasserman) means HS VaR at 99.9%, built from a few extreme days, is one of the noisiest numbers in all of risk.
Cost of full revaluation. Revaluing every option/structured position under 500–1000 historical scenarios is slow; shortcuts (delta/moment) erode exactly the nonlinearity HS exists to capture.

5. References

Hull, Options, Futures, and Other Derivatives
Glasserman, Monte Carlo Methods in Financial Engineering
Hull & White: Incorporating Volatility Updating into the Historical Simulation Method for Value-at-Risk (1998)
McNeil & Frey: Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series (2000)

6. Connected Graph Bridges

Base: Statistics & Inference (order statistics, empirical quantiles) · Econometrics & Time Series (EWMA/GARCH vol filtering).
Back: 02 · Parametric VaR.
Forward: 04 · Monte Carlo VaR · Index Hub.
Tails: Extreme Value Theory & Fat Tails.

Kwant Atlas

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