The Historical Simulation Method

Value at Risk is defined in terms of a whole probability distribution of tomorrow's loss — but nobody hands a risk manager that distribution on a silver platter. Where does it actually come from? The simplest, and still enormously popular, answer is: steal it from history. Take the market's own recent behaviour — the last few hundred trading days — and ask "what would happen to today's book if each of those days repeated tomorrow?" No bell curve, no covariance matrix, no assumption about the shape of anything: just the market's own track record, replayed against the positions you actually hold right now.

The method, step by step

Collect the daily percentage returns of every risk factor in the portfolio (each stock, FX rate, interest rate…) over the last M trading days — typically 250 to 500 days, roughly one to two years.
For each historical day t, apply that day's percentage moves to today's position sizes, producing a hypothetical profit-and-loss \text{P\&L}_t. If the portfolio holds weights w_i in risk factors with historical returns r_{i,t},

\text{P\&L}_t = \sum_i w_i\, r_{i,t}, \qquad t = 1, \dots, M.

Sort the M hypothetical P&Ls from worst to best.
Read the c\% VaR off the appropriate point in that sorted list: the loss such that only (1-c)\% of the scenarios were worse. With M scenarios, that is (approximately) the k-th worst outcome, where k = \lceil (1-c)\,M \rceil.

Nothing here required deciding whether returns are normal, log-normal, or anything else — the empirical sample is the distribution. If the historical window happens to include a crash, that crash's severity is already baked into your VaR estimate, fat tail and all.

Worked example: 20 days, reweighted onto a $10 million book

Suppose the last 20 trading days produced the following daily percentage returns for a single equity index (already sorted worst to best, for convenience):

Rank (worst → best)	Return	Rank	Return
1	−3.4%	11	+0.1%
2	−2.2%	12	+0.2%
3	−1.5%	13	+0.3%
4	−1.1%	14	+0.4%
5	−0.9%	15	+0.5%
6	−0.8%	16	+0.7%
7	−0.6%	17	+0.9%
8	−0.4%	18	+1.2%
9	−0.3%	19	+1.5%
10	−0.2%	20	+2.0%

Reweight each return onto today's \$10{,}000{,}000 position by multiplying: the worst day, -3.4\%, becomes a hypothetical loss of 0.034 \times \$10{,}000{,}000 = \$340{,}000. With M=20, the 95% VaR needs k = \lceil 0.05 \times 20 \rceil = 1 — the single worst scenario. So the 1-day 95% VaR is $340,000. The 90% VaR needs k = \lceil 0.10 \times 20 \rceil = 2, the second-worst day (-2.2\%): $220,000.

Try asking for the 99% VaR from this same sample: k = \lceil 0.01 \times 20 \rceil = 1 — it lands on the very same single worst day as the 95% figure. With only 20 scenarios, the 95% and 99% estimates collapse onto the same observation, which should make you deeply suspicious of the 99% number. That's not a coincidence — it's a sample-size problem, and it's the subject of the first "Watch out!" below.

Reading VaR off a bigger sample

With a realistic sample of a few hundred days the sorted hypothetical returns trace out a smooth empirical cumulative distribution — the fraction of historical days with a return at or below each level on the horizontal axis. VaR is simply where that curve crosses the height 1-c, read straight off the picture.

Historical simulation's great appeal is also its great limitation: it assumes the future will resemble the recent past. If your 500-day window happens to fall entirely within a calm bull market, its empirical tail will look thin — not because risk is genuinely low, but because your sample got lucky. And a 500-day window is a compromise: shorter windows react quickly to new market conditions but give a noisy, poorly-populated tail (as the worked example just showed for 99% VaR on only 20 days); longer windows populate the tail better but drag stale, possibly irrelevant history along for the ride — a distant crash can keep inflating your VaR for years after the market has moved on. This exact tension — "the recent past may not contain the worst case" — is precisely why risk managers don't stop at VaR and layer on deliberately extreme scenarios drawn from further back in history.

Reweight returns, not dollars. The classic beginner's mistake is to replay the actual historical dollar P&L from those 20 or 500 days, rather than applying each day's percentage move to today's position sizes. Your portfolio was a different size and shape back then — reusing old dollar figures silently answers the wrong question.
A thin sample can't support a confident tail estimate. Asking for a 99% VaR from 20 days of history (as above) is asking one single worst observation to represent the entire worst 1%; the estimate is little more than noise. A useful rule of thumb: you need at least 100/(1-c) scenarios before the c\% tail contains even a single observed data point.