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Value at Risk: Methodsintermediate

Historical Simulation VaR

Historical simulation estimates VaR with no distributional assumption. Take the portfolio today, apply each of the last NN historical return scenarios, build the resulting profit-and-loss distribution, sort it, and read VaR off the relevant percentile (the 5th percentile for a 95% VaR). Its strength is that fat tails, skew, and real co-movements are inherited straight from the data. Its weaknesses are a heavy reliance on the chosen window and the assumption that the past sample represents the future, so it is blind to shocks absent from the window.

Try it yourself

Value at Risk — three ways

How much could this book lose on a bad day? VaR draws a line in the loss tail at confidence 95%. The parametric line assumes a normal curve, VaR = (z·σ·√h − μ·h)·V with z = 1.645 (one-tailed). The historical and Monte-Carlo methods read VaR off a simulated loss distribution instead. Under normality the three roughly agree; fat tails pull the historical figure deeper.

1-day 95% parametric VaRUS$3.29M
-6.0%-4.0%-2.0%0.0%2.0%4.0%6.0%Portfolio return over 1 day (%)VaR cutoff5% loss tail
Parametric (normal) · largestUS$3.29M
Historical (fat tails)US$3.05M
Monte-Carlo (5,000 draws)US$3.27M
z (95%, one-tailed) 1.645Horizon σ√h 2.00%Largest VaR Parametric
Portfolio value VUS$100M
Daily volatility σ2.0%
Daily expected return μ+0.00%
Horizon h1 day
Confidence level c
The three estimates land near US$3.29M, but the historical figure of US$3.05M is about the same, since this sample happens not to bite past the Gaussian tail here. The largest of the three is Parametric.
Discuss: which method would you report to a regulator, and which to your own risk committee?

Why it matters

Instead of assuming a bell curve, you let history vote. Replay every day of the past year on today's portfolio, line up the outcomes from worst to best, and the loss at your confidence percentile is the VaR. The data carry whatever fat tails and asymmetry the market actually showed, no formula required. The flip side is that the method only knows what is in its window: if the last year was calm, it will quietly assume calm continues and underestimate a crisis it never saw.

Formulas

VaR as an empirical percentile
VaRc=− Percentile(1−c){ r1V,…,rNV }\mathrm{VaR}_c = -\,\text{Percentile}_{(1-c)}\{\,r_1 V, \dots, r_N V\,\}
Sort the NN simulated profit-and-loss outcomes and take the (1−c)(1-c) quantile. For 95% with 100 scenarios, VaR is roughly the 5th-worst loss.

Worked examples

Scenario

You hold US$100M and have 100 daily return scenarios from the past 100 trading days. The five worst simulated daily losses are −6.0%,−5.5%,−5.2%,−4.8%,−4.5%-6.0\%, -5.5\%, -5.2\%, -4.8\%, -4.5\% of value. Find the one-day 95% VaR.

Solution

A 95% VaR is the 5th-percentile loss. With 100 ordered outcomes, that is the 5th-worst, here a return of −4.5%-4.5\%. So VaR is 0.045 times US$100M, which is US$4.5M. Notice the worst day, a −6.0%-6.0\% return or US$6.0M loss, is a 99%-level event beyond the 95% threshold. No normal curve was assumed; the number came straight from the empirical ordering.

Common mistakes

  • ✗Historical simulation assumes normal returns. It assumes nothing about the shape; the distribution is whatever the historical sample delivers, fat tails and all.
  • ✗A longer window is always better. A longer window adds data but blends old regimes with current conditions; a window too long can mask a recent volatility shift.
  • ✗It can predict losses larger than anything in the sample. Plain historical simulation cannot exceed the worst scenario in its window, so it is blind to unprecedented shocks.
  • ✗Each historical day should be reweighted by recency automatically. Basic historical simulation weights all days equally; age-weighting is a deliberate extension, not the default.

Revision bullets

  • •Replay past return scenarios on today's portfolio
  • •VaR is the (1−c)(1-c) percentile of the simulated P&L
  • •No distributional assumption; inherits fat tails and skew
  • •Heavily dependent on the chosen historical window
  • •Blind to shocks not present in the sample window

Quick check

With 100 daily return scenarios, the one-day 95% historical-simulation VaR is read off the

A key limitation of plain historical simulation is that it

Connected topics

VaR DefinitionMonte Carlo VaRFat TailsMethod Tradeoffs

Sources

  1. Jorion (2007), Ch. 10
    Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007. ISBN 978-0-07-146495-6.
    Develops historical-simulation VaR and its reliance on the sampling window.
  2. Hull (2018), Ch. 13
    Hull, J. C. Risk Management and Financial Institutions. 5th ed. Wiley, 2018. ISBN 978-1-119-44811-2.
    Describes building the empirical loss distribution and reading VaR from its percentile.
How to cite this page
Dr. Phil's Quant Lab. (2026). Historical Simulation VaR. Derivatives Atlas. https://phucnguyenvan.com/concept/frm-historical-simulation
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