Historical Simulation VaR

You know what VaR says: a horizon, a confidence level, and a dollar figure — “99% confident we won’t lose more than $X tomorrow.” That’s the sentence. This lesson is about the most honest way to actually compute it, and it requires almost no math at all. No bell curve, no algebra, no Greek letters. Just a stack of the portfolio’s real past days, sorted worst to best, and a finger pointed at the right one.

It’s called historical simulation, and its whole philosophy fits on a fortune-cookie slip: the best guess for how bad tomorrow can be is to look at how bad the recent past actually was. Where the parametric method (next lesson) assumes the returns follow a tidy distribution and does algebra on it, historical simulation refuses to assume anything. It lets the data speak — fat tails, skew, ugly clusters and all.

Quick instinct check before we open the toolbox.

Replay the past

Analogy. Imagine you want to know how cold tomorrow morning might get. One approach: assume temperatures follow a neat bell curve, estimate its mean and spread, and calculate. The other approach: pull out your weather diary for the last 250 days, line up every morning’s low temperature from coldest to warmest, and say “the coldest 5% of mornings were below −3°C, so I’ll plan for −3°C.” Historical simulation is the weather-diary method, applied to money.

Definition. Historical-simulation VaR takes the portfolio’s realised daily profit-and-loss over the last $N$ days, treats that empirical histogram as if it were the distribution of tomorrow’s return, and reads the VaR straight off the appropriate percentile of that histogram. There is no fitted curve. The collection of dots is the model.

There’s one subtlety that separates pros from amateurs, so plant it now: you don’t take the P&L your portfolio actually earned on each past day. You take today’s portfolio and ask what it would have gained or lost if each past day’s market moves happened again. You apply the historical returns — the percentage moves in every risk factor — to your current positions. This is the difference between “what did I make last March?” and “what would my book make if last March’s moves repeated tomorrow?” The second is what you want, because the second is the risk you’re carrying now.

So the empirical histogram you sort is a histogram of hypothetical P&L for today’s book under each past day’s market move. Same shape of idea, far sharper in practice.

The recipe, step by step

Three steps. That’s genuinely the whole algorithm.

1. Gather $N$ historical scenarios. For each of the last $N$ trading days (a common choice is $N = 250$, roughly one trading year, or $N = 500$ for two), record the percentage move of every risk factor, apply those moves to today’s positions, and total up the hypothetical P&L. You now have $N$ numbers.
2. Sort them ascending. Worst loss on the left, best gain on the right.
3. Read off the percentile. VaR at confidence $c$ is the loss sitting at the $\lfloor(1-c)\cdot N\rfloor$-th worst observation. With $c = 0.99$ and $N = 250$, that’s $\lfloor 0.01 \times 250\rfloor = 2$ — you walk in to the 2nd-worst-ish day and read the loss there.

The island below is that recipe made tactile. Twenty realised daily P&L outcomes, sorted worst (red, on the left) to best (blue, on the right). Slide the confidence level and watch the dashed line march into the loss tail: higher confidence cuts deeper, further into the disasters.

Picking the percentile, by hand

The only thing that ever trips people up is which sorted observation to grab. Walk three cases:

• $N = 20$ observations, 95% confidence. $(1-0.95)\times 20 = 1$. So VaR is the 1st-worst observation — the single most brutal day in the sample. Cut off the worst 5% of 20 days, and 5% of 20 is exactly 1 day.
• $N = 100$, 99% confidence. $(1-0.99)\times 100 = 1$. The 1st-worst day again. The worst 1% of 100 days is the single worst.
• $N = 100$, 95% confidence. $(1-0.95)\times 100 = 5$. The 5th-worst day: four days were worse, and you’re reading the loss at the boundary between “the worst 5%” and the rest.

When $(1-c)\cdot N$ isn’t a whole number, you’re between two observations — say the 2.5th-worst. Practitioners then interpolate between the neighbouring sorted values (linear interpolation is the common convention; some desks just round to the nearest, or conservatively take the worse of the two). The exact rule is a house choice, but interpolation between adjacent order statistics is the standard. Don’t lose sleep over it — with $N = 250$ it nudges the number by pennies.

Worked example: a $10M book

Suppose you run a $10,000,000 portfolio. You’ve computed the hypothetical daily P&L of today’s book under each of the last 10 trading days’ market moves (tiny $N$, just to keep the arithmetic in view), and sorted them ascending — worst return on top:

Converting each return to dollars is just multiply-by-$10M: a $-1.7\%$ day means $-0.017 \times 10{,}000{,}000 = -170{,}000$, i.e. −$170,000. Now pick the percentiles.

95% VaR. $(1-0.95)\times 10 = 0.5$. That’s between the 0th and 1st observation — effectively the worst day. Reading rank 1: a loss of $3.8\%$, i.e. $380,000. (Some conventions interpolate toward the worst extreme; with the smallest possible tail, the 1st-worst is the natural read.) Plain English: “We are 95% confident we won’t lose more than $380,000 tomorrow.”

Now make it concrete with a cleaner cut. Say instead $N = 20$ at 95%: $(1-0.95)\times 20 = 1$, so VaR is the 1st-worst of 20 days. With $N = 100$ at 95%, it’d be the 5th-worst — and if that 5th-worst day was, say, $-1.7\%$, your 95% VaR would be $170,000.

99% VaR with $N = 100$. $(1-0.99)\times 100 = 1$ → the 1st-worst of 100 days. If that single worst day was $-3.8\%$, your 99% VaR is $0.038 \times 10{,}000{,}000 =$ $380,000. Notice the pattern that bites later: at 99% with only 100 days, your entire VaR estimate rests on one observation — the worst day in the window. Lose or gain that one day and the number lurches.

Why it’s loved

For all its simplicity, historical simulation earns its keep, and risk desks adore it for three concrete reasons.

No distribution assumption. The parametric method has to assume a shape — almost always the normal bell curve — and real financial returns famously aren’t normal. Historical simulation sidesteps the whole question. It never says “returns are normal” or anything else; it just uses the dots you have. If the truth is lumpy, skewed, and mean, the dots are lumpy, skewed, and mean too.

It captures the actual fat tails, skew, and correlations in the window — for free. This is the quiet superpower. Markets crash harder and more often than a bell curve predicts (fat tails), losses and gains aren’t symmetric (skew), and assets move together in a panic (correlation spikes). A parametric model has to be told about each of these, painstakingly. Historical simulation gets all of them automatically, because if those features were present in the last $N$ days, they’re sitting right there in the sorted P&L. The correlation between your stocks and your bonds on a crash day isn’t estimated — it’s baked into that day’s joint move, which you replayed in one shot.

It’s simple to explain and audit. You can sit a regulator, a board member, or a junior analyst down and explain it in two sentences: “these are the last 250 days; we sorted them; here’s the bad one we’re worried about.” No covariance matrix, no Cholesky decomposition, no Greek letters. When VaR has to survive an audit — and under Basel it does — being explainable is worth a great deal.

Where it bites

Now the hard truths. Historical simulation’s great virtue — “the future looks like the chosen past” — is also its single deepest flaw, because sometimes the future emphatically does not.

Four specific failure modes, each worth naming:

The window-length trade-off. How many days, $N$, should you use? It’s a genuine dilemma with no free answer:

Short windows are twitchy and statistically thin in the tail (the place you care about most); long windows are stable but slow, mixing in regimes that no longer exist. There is no universally correct $N$ — it’s a judgement call about how fast your market’s risk regime turns over.

Equal-weighting old and new days. Plain historical simulation treats a move from 250 days ago as exactly as relevant as yesterday’s. But markets have memory — recent volatility predicts near-term volatility (volatility clusters). Giving a sleepy day from ten months back the same vote as a turbulent one from this morning understates risk when trouble is brewing. The standard fix is the exponentially-weighted (or “BRW” — Boudoukh–Richardson–Whitelaw) variant: weight recent days more heavily and let old ones decay, so the percentile leans on what’s relevant now. Same recipe, weighted sort.

It can never produce a loss bigger than the worst day in the window. This one is structural and worth saying plainly: historical VaR is bounded above by the single worst observation in your sample. If the worst day in your 250-day window was $-4\%$, your historical VaR cannot exceed $4\%$ — not at 99%, not at 99.9%, not ever. The method has literally never seen anything worse, so it cannot imagine anything worse. For a portfolio whose true catastrophe is a 10% gap that simply hasn’t happened yet in your window, historical simulation will quietly tell you everything’s fine. (This is exactly why Expected Shortfall and Monte Carlo exist — later lessons.)

The ghost / echo effect. Picture one monstrous $-9\%$ day sitting in your 250-day window. For 250 days it props up your VaR — a permanent reminder of pain. Then, on day 251, it rolls out of the window. Nothing changed in the market, but your VaR suddenly drops sharply, purely because one bad memory aged out of the sample. This artificial jump is the ghost (or echo) effect: VaR moving for calendar reasons rather than market reasons. Exponential weighting softens it (the day was already fading), but plain equal-weighted historical simulation has a hard, discontinuous cliff the day a big move exits.

When to use it

Reach for historical simulation when your portfolio’s risk factors have a decent, relevant price history and you want a transparent, assumption-light number you can defend in an audit — it’s the default first VaR engine at most banks for exactly that reason. It shines when the recent past genuinely resembles the near future (stable regimes) and when capturing messy real-world correlations matters more than modelling the far tail precisely.

Be wary — and reach instead for parametric or Monte Carlo methods — when you need to stress beyond anything in your window (true tail catastrophes), when you hold new instruments with no price history, or when you’re in a regime shift where the recent past is actively misleading. The honest move on a real desk is to run more than one method and worry hard when they disagree.

Historical simulation is the honest, assumption-light first answer — but it’s hostage to its window, blind beyond the worst day it’s seen, and statistically thin in the tail. The next lesson, Parametric (Variance–Covariance) VaR, swaps that empirical record for a confident assumption — returns are normal — and buys back speed, smoothness, and the ability to extrapolate past anything you’ve witnessed. Whether that assumption is a gift or a lie is exactly the question we’ll interrogate.