Historical VaR · RiskAtoms Academy

By the end you will be able to

1Compute historical VaR by replaying past days on today's portfolio.
2Read VaR off the sorted losses, and state which order statistic you used.
3Compare historical and normal VaR, and explain the difference with fat tails.
4Explain how the window drives the result, including cliff effects.

Why it matters

The normal model in L2.5.06: Value at risk: the idea gave the fund a 99% VaR of $148,044. But the fund's own history holds days of $412,600 and $319,600. What if, instead of trusting a bell curve, we simply asked the data?

Core explanation

Let the past vote

Historical simulation needs no distribution at all. Take today's positions. For each of the last $n$ trading days, apply that day's market moves and work out what the portfolio would have made or lost. You now have $n$ possible tomorrows. Sort the losses from worst to best and count in from the worst end.

Illustrative dataThe Module 2.5 data set

500 trading days of daily returns for an equity index fund, a bond fund and a bank share, generated for this module by a seeded model with a calm stretch, a turbulent spell and a few extreme days. It is not real market history. The same series runs through every lesson in the module, so the numbers line up. The fund we follow is worth $10 million: $6 million in the equity index fund and $4 million in the bond fund.

The fund's positions have not changed over the 500 days, so the replayed P&L on each day is simply $6,000,000 × equity return + $4,000,000 × bond return. For a real portfolio that trades, you would apply each past day's returns to today's holdings.

With 500 days and 99% confidence, the worst 1% is 500 × 0.01 = 5 days. The VaR is the 5th worst loss, an order statistic (M-C09: Quantiles). Here are the worst six days:

Rank	Day	Equity	Bonds	Fund P&L
1	388	−6.27%	−0.91%	−$412,600
2	394	−4.94%	−0.58%	−$319,600
3	97	−4.38%	+0.36%	−$248,400
4	385	−3.37%	−0.75%	−$232,064
5 ← 99% VaR	386	−3.16%	+0.12%	−$184,550
6	187	−2.77%	+0.07%	−$163,676

So the historical 99% VaR is $184,550, 25% above the normal figure. At 95% (the 25th worst) it is $85,204, 19% below the normal $104,675. That is the fat-tail signature from L2.5.02: Return distributions: ordinary bad days are milder than a normal curve predicts, and the extremes are worse.

Strengths

No distribution to assume. Fat tails, skew and the way assets moved together in a crisis all come in automatically. Easy to explain: these are real days.

Weaknesses

Only knows its window. Few observations in the tail: at 99% on 500 days, the answer rests on 5 days. Slow to react when volatility changes, and jumps when big days enter or leave.

The window decides. Use only the last 250 days and the 99% VaR becomes the 3rd worst of those days: $232,064, because the turbulent spell sits inside them. The longer window dilutes it: $184,550. Neither is wrong; they answer different questions. And the cliff effect follows: in a rolling 250-day window, day 97's loss dropped out on day 347, and the VaR fell from $118,908 to $111,538 that morning, although nothing happened in the market. In the other direction, as the turbulent days arrived, the rolling VaR rose from $111,538 on day 384 to $232,064 by day 394.

Improvements. Volatility-weighted historical simulation (Hull and White, 1998) rescales each past day by how volatile markets are now compared with then, so calm-period data are scaled up in a storm and vice versa. Filtered historical simulation and Monte Carlo methods go further (M5.2: Market risk modelling).

The formal version

Let $L_{(1)} \ge L_{(2)} \ge \dots \ge L_{(n)}$ be the replayed losses sorted from worst to best (a loss is minus the P&L). With confidence $\alpha$ :

k = \big\lceil n(1-\alpha) \big\rceil, \qquad \widehat{\text{VaR}}_{\alpha} = L_{(k)}

where $\lceil \cdot \rceil$ rounds up. This is the empirical quantile of the distribution of losses (M-D02: Histograms and empirical distributions). With 500 days that gives the 5th worst at 99%, the 13th at 97.5% and the 25th at 95%.

Conventions differ: state yours

There is no single agreed order statistic. This course uses the k-th worst loss, with

k = \lceil n(1-\alpha)\rceil

. Some banks use the next one in, the (k+1)-th, so that exactly

k

losses lie strictly beyond the VaR. Spreadsheets and the RiskAtoms calculator (PERCENTILE.INC, histvar) interpolate between two order statistics at rank

(n-1)(1-\alpha) + 1

from the bottom. On the fund at 99% the three give $184,550, $163,676 and $163,885. With few tail observations the choice matters, so a VaR figure should say which it used.

Try it

The explorer opens on all 500 days at 99%. Switch to the last 250 days and watch the VaR jump. Change the convention and see which loss is picked. Slide the confidence and compare the solid historical line with the dashed normal one in the rolling chart: the steps in the solid line are cliff effects.

Try itHistorical VaR explorer

Confidence level99.0%

Window

Convention

The k-th worst loss, k = n(1 − α) rounded up (this course).

Daily P&L, days 1 to 500 ($20k bins). Red: the losses at or beyond the VaR.

Rolling 250-day VaR at 99.0%, days 250 to 500. Solid: historical. Dashed: normal.

Shaded: the turbulent spell, days 381 to 440. Steps in the solid line happen when a big loss enters the window or, 250 days later, leaves it.

99.0% one-day historical VaR

$184,550

Days in window, n500

k = n(1 − α), up5

Normal VaR$148,044

Historical ÷ normal1.25

Worst losses in the window

1. day 388$412,600
2. day 394$319,600
3. day 97$248,400
4. day 385$232,064
5. day 386$184,550
6. day 187$163,676
7. day 70$124,156
8. day 239$118,908
9. day 51$118,272
10. day 233$111,538

Worked example

Historical VaR by hand

Twenty days are enough to see the method, and the calculator can check them. Take the fund's P&L on days 381 to 400, the start of the turbulent spell, and find the one-day 90% VaR. Try each step before revealing it.

Twenty days, 90% confidence

Step 1 of 5

1
The data
+$146,988, +$141,910, +$14,812, −$54,764, −$232,064, −$184,550, +$19,150, −$412,600, +$225,400, +$15,708, +$10,838, +$200,676, +$85,200, −$319,600, +$46,428, +$21,282, +$35,228, +$155,994, +$62,620, −$22,628

Twenty days is far too few for a real VaR: at 90% the answer rests on two days. It is used here only so the arithmetic fits on the page. Note also that 5 of 500 is a small sample for a 99% figure.

Regulatory lens

The Basel market risk rules have always let banks choose their VaR method, provided it used at least one year (about 250 trading days) of data. After 2008, Basel 2.5 (July 2009) added a “stressed VaR”: the same 99% 10-day VaR, but computed on a 12-month period of significant stress for the bank's portfolio. It was a direct answer to the window problem: a calm window had made VaR look small just before the crisis. The FRTB (L2.5.08: Expected shortfall) keeps the idea by calibrating expected shortfall to a stress period. The detail is in M4.3: Market risk capital (FRTB) and CVA, part of Level 4: Regulation, Capital & Accounting.

Sparks

SparkThe industry's favourite

Christophe Pérignon and Daniel Smith studied the VaR disclosures of large commercial banks from 1996 to 2005. Of the banks that said which method they used, 73% used historical simulation. Its appeal is that everyone can see where the number comes from: real days, replayed.

SparkGiving history a pulse

In 1998 John Hull and Alan White proposed scaling each historical day by the ratio of today's volatility to the volatility on that day. A 2% fall from a calm year then counts for more in a stormy year. The fix is simple, and it removes much of the slow reaction of plain historical simulation.

Key takeaways

Historical VaR replays past days on today's positions, sorts the losses and reads off the k-th worst.
This course uses k = n(1 − α) rounded up; others use the (k+1)-th or interpolate. Always say which.
It captures fat tails and real co-movements, but only those inside the window.
The window choice can change VaR a lot, and cliff effects make it jump when big days enter or leave.
On the fund, historical VaR is above normal VaR at 99% and below it at 95%: the fat-tail pattern.

Reaction

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Glossary

Historical simulation: Estimating the P&L distribution by applying each past day's market moves to today's positions, then reading VaR off the sorted results. Also called historical VaR.
Window: The run of past days used, such as the last 250 or 500 trading days.
Order statistic: The value in a given position once the data are sorted: the 5th worst loss is an order statistic.
Cliff effect: A jump in historical VaR caused only by a big day entering or leaving the window, not by anything happening in the market that day. Also called the ghost effect.
Volatility-weighted historical simulation: A version of historical simulation that rescales each past day's moves by the ratio of today's volatility to the volatility at the time.

Deep Dives & Origins

Deep Dives

J. C. Hull, Risk Management and Financial Institutions, Wiley, the chapter on historical simulation. Intermediate.
K. Dowd, Measuring Market Risk, Wiley, 2nd edition, 2005, the chapter on non-parametric approaches. Advanced.

Origins

C. Pérignon and D. R. Smith, “The level and quality of Value-at-Risk disclosure by commercial banks”, Journal of Banking & Finance 34(2), 2010, 362–377.
J. Hull and A. White, “Incorporating volatility updating into the historical simulation method for value-at-risk”, Journal of Risk 1(1), 1998, 5–19.
Basel Committee on Banking Supervision, Revisions to the Basel II market risk framework, July 2009.

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