Historical simulation in practice · RiskAtoms Academy

By the end you will be able to

1Run historical simulation on a trading book with full revaluation, and read off VaR as a quantile.
2Explain how the window (length, and whether it holds a stressed spell) drives the result, including ghost effects.
3Apply age weighting (BRW) and say what it fixes and what it breaks.
4Compare historical, delta-normal and delta-based historical VaR, and say where each goes wrong.

Why it matters

Same desk, same positions, same day. Delta-normal VaR: £1.94m. Historical simulation over 750 days: £2.27m. Over the last 250 days: £2.86m. Over 750 days without the stressed spell: £1.70m. Each is a correct calculation; they answer different questions about which past is a guide to tomorrow. Choosing the window is a model decision, and the risk manager has to own it.

Core explanation

Replaying 750 days on today's book

Illustrative dataRiskAtoms Bank plc's Markets desk

One made-up trading book on one made-up valuation day, in pounds: eight positions in four books, eight risk factors and 750 days of simulated factor moves. The levels (UK index 8,000, GBP/USD 1.30, Brent $80 and so on) are round illustrative numbers, not market data. The same book runs through every lesson in Module 3.2, so the figures agree from lesson to lesson.

750 days of factor moves (days 0 to 749, yesterday is 749) with a stressed spell on days 480 to 539. One day, pounds; VaR is the k-th worst loss with k = n(1 − α) rounded up.

L2.5.07: Historical VaR replayed a fund's past returns. On a trading desk the replay works one level down. Each scenario is a past day's moves in the eight risk factors, not a past P&L, because the positions have changed since. Apply day $t$ 's moves to today's positions, reprice each one in full (the gilt with its cash flows, the puts with Black-Scholes), and add up. Do that for all 750 days and you have 750 possible P&Ls for tomorrow:

The 99% VaR from 750 scenarios is the 8th worst loss: £2,274,006, from day 513. 7 of the 8 worst days fall in the stressed spell:

Rank	Day	Loss	Equities	Rates	FX & comm.	Credit
1	488	£4.98m	−£2,589k	−£1,293k	−£376k	−£720k
2	494	£3.44m	−£2,155k	+£158k	−£1,184k	−£261k
3	515	£2.93m	+£425k	−£2,002k	−£47k	−£1,305k
4	383 (calm)	£2.92m	−£1,623k	−£383k	−£359k	−£553k
5	520	£2.90m	−£3,137k	−£625k	+£656k	+£211k
6	524	£2.86m	−£2,329k	+£258k	−£1,053k	+£263k
7	486	£2.34m	−£1,751k	+£749k	−£1,147k	−£188k
8	513	£2.27m	−£2,003k	−£235k	−£269k	+£233k

Historical VaR is above delta-normal (£1.94m) because the history has fat tails and a stressed spell, where the volatilities were 2.5 times their calm level. The standard deviation of the 750 P&Ls is £967,082, against delta-normal's £835,845. The worst day is not the jump day 486 but day 488, two days later: on day 486 the rally in rates and the puts cushioned the equity loss.

Why revalue in full?

Run the same 750 days through the sensitivities instead (P&L = δ′x, as in L3.2.08: P&L attribution and explain) and the 99% VaR is £2.82m, and the worst day £5.92m instead of £4.98m. The difference is the puts: on big down days they gain far more than their delta says. For a desk with options, a historical VaR built on deltas overstates losses on falls and understates them on rallies.

Which quantile? (M-C09)

With n = 750 and α = 99%, n(1 − α) = 7.5 sits between two order statistics. This course takes the 8th worst (£2,274,006). Interpolating halfway between the 7th (£2,336,924) and 8th gives £2,305,465. Both are defensible; the risk report must say which (M-C09: Quantiles).

The window

The window decides which past counts. The same book over six windows:

Window	Days	Stressed days	k	99% VaR	Set by day
Last 250 days (500–749)	250	40	3	£2.86m	524
Last 500 days (250–749)	500	60	5	£2.90m	520
All 750 days	750	60	8	£2.27m	513
750 days without the stressed spell (690)	690	0	7	£1.70m	453
250 days ending with the stress (290–539)	250	60	3	£2.93m	515
First 250 days (0–249)	250	0	3	£1.68m	207

Three lessons stand out. Leaving out the stressed spell cuts the VaR to £1.70m, close to delta-normal: in calm times the book is close to normal. A 250-day window is very sensitive to what it holds: £1.68m over the first year, £2.86m over the last. And a short window rests on few tail points: with 250 days the 99% VaR is the 3rd worst loss, so three days decide it.

Rolled forward day by day, a 250-day VaR moves in steps, with nothing changing in the book. When day 494 entered the window, the VaR rose by £582k overnight. The stressed days began leaving the 250-day window on day 730, and the last leaves on day 790; the 210 calm days since the stress give a VaR of just £1.51m. These are ghost effects.

The line is minus the 250-day 99% VaR known each morning, from the 250 days before. It ranges from £1.58m to £2.93m.

Age weighting

Equal weights treat a day from three years ago like yesterday. Age weighting gives the day $i$ days ago a weight proportional to $\lambda^{i-1}$ . With λ = 0.99, yesterday gets 1.00% (equal weights would give 0.13%), the worst day 488 gets 0.073%, and the weight halves every 69 days (the half-life). VaR is then the loss where the weights of the worst days first add up to 1%:

Weighting	Half-life	Weight on the stressed spell	99% VaR	Set by day
Equal (1/750)	none	8.0%	£2.27m	513
λ = 0.99	69 days	5.5%	£1.82m	679
λ = 0.98	34 days	1.0%	£1.74m	691
λ = 0.995	138 days	9.3%	£2.27m	513

The stressed spell ended 210 days ago, so age weighting lowers the VaR. Had it been last month, the same weights would have raised the VaR faster than equal weights, which is the point: the VaR reacts to recent markets. The cost is a smaller effective sample and a VaR that forgets crises quickly.

Other fixes, and where they live

The formal version

The estimator. With scenarios $x_1, \dots, x_n$ (past factor moves) and today's book value function $V$ :

\begin{aligned} L_t &= -\big[V(x_{\text{today}} + x_t) - V(x_{\text{today}})\big], \qquad t = 1, \dots, n \\ L_{(1)} &\ge L_{(2)} \ge \dots \ge L_{(n)} \quad \text{(sorted worst first)} \\ \widehat{\text{VaR}}_\alpha &= L_{(k)}, \qquad k = \lceil n(1-\alpha) \rceil \end{aligned}

$\widehat{\text{VaR}}_\alpha$ is the empirical $\alpha$ quantile of the loss (M-C09: Quantiles): the smallest loss with at least a share $1-\alpha$ of scenarios at or beyond it.

Age weights. Number the days by age, $i = 1$ for yesterday. The weights must add up to one; a geometric series (M-A07: Geometric series) fixes the constant:

\begin{aligned} w_i &= c\,\lambda^{i-1}, \qquad \sum_{i=1}^{n} c\,\lambda^{i-1} = c\,\frac{1-\lambda^n}{1-\lambda} = 1 \\ \Rightarrow\quad w_i &= \frac{(1-\lambda)\,\lambda^{i-1}}{1-\lambda^n}, \qquad \text{half-life } h:\ \lambda^h = \tfrac12 \Rightarrow h = \frac{\ln 0.5}{\ln \lambda} \end{aligned}

The weighted quantile. Sort losses worst first, carrying each one's weight. VaR is the first loss at which the running total of weights reaches $1-\alpha$ :

\widehat{\text{VaR}}_\alpha = L_{(k^*)}, \qquad k^* = \min\Big\{k : \sum_{j=1}^{k} w_{(j)} \ge 1-\alpha\Big\}, \qquad n_{\text{eff}} = \frac{1}{\sum_i w_i^2}

With equal weights $w_i = 1/n$ , $k^* = \lceil n(1-\alpha)\rceil$ and $n_{\text{eff}} = n$ : plain historical simulation. With λ = 0.99 the worst 14 losses are needed to collect 1% of the weight here, because the old stressed days carry little.

Try it

Start with 250 days ending yesterday. Slide the end of the window back through the stressed spell, then forward past it. Switch to 750 days, tick the box to drop the stress, and try λ = 0.97.

Try itRun historical simulation on the desk

Window length

Confidence

Window ends after day749

Age weighting λ (1 = equal weights)off

Leave out the stressed spell (days 480–539)

99.0% one-day VaR£2.861m

Scenarios in the window250

Of which stressed40

k-th worst loss3

Set by day524

Delta-normal VaR, same α£1.944m

Historical minus delta-normal+£916k

Each scenario is one past day's factor moves applied to today's eight positions, revalued in full. With age weighting the bars show weights rather than counts, and the VaR is the loss at which the weights of the worst days first add up to 1 − α. The effective sample size is 1 / Σw²: the number of equally weighted days that would carry as much information.

Worked example

The 8th worst day, revalued by hand

From one scenario to the VaR

Step 1 of 5

1
Take the scenario
Day 513, in the stressed spell, moved the factors by: UK equity index −4.65%; US equity index −4.31%; GBP 2-year swap rate −17.96 bp; GBP 10-year swap rate −9.80 bp; GBP/USD (dollars per pound) −0.30%; Brent crude −1.71%; UK index implied volatility 4.43 vol pts; Investment-grade credit spread 8.11 bp.

Regulatory lens · looking ahead

The 1996 Market Risk Amendment required at least one year of history (about 250 trading days) for VaR models, updated at least quarterly. After 2008, Basel 2.5 added a stressed VaR from a 12-month stress period, because windows of calm years had produced low VaRs going into the crisis. FRTB replaces both with an expected shortfall calibrated to stress (M4.3: Market risk capital (FRTB) and CVA). Filtered historical simulation and volatility weighting are in M6.2: Market risk modelling.

Spark Atoms

Spark AtomThe best of both worlds

Boudoukh, Richardson and Whitelaw called their 1998 paper on age weighting “The Best of Both Worlds”. It borrowed the exponential decay that J.P. Morgan's RiskMetrics used for volatilities (a daily λ of 0.94) and applied it to the scenarios themselves, so historical simulation kept its fat tails but reacted faster.

Spark AtomGhosts on the calendar

With equal weights, a crash day raises the VaR for exactly as long as the window and then vanishes overnight. Risk managers watch the calendar of big days due to leave the window, so the drop does not get mistaken for a fall in risk.

Key takeaways

On a desk, historical scenarios are past factor moves applied to today's positions, revalued in full.
The desk's 99% historical VaR over 750 days is £2,274,006 (8th worst), against £1,944,466 delta-normal.
Full revaluation matters for options: a delta-based replay gets the puts wrong on big days.
The window is a model choice: length, and whether it holds a stress, can move VaR by a factor of almost two.
Equal weights cause ghost effects as big days enter and leave the window.
Age weighting (BRW) makes VaR react faster but forgets crises; volatility weighting (M6.2) rescales instead.

Reaction

Check what stuck

Question 1 · objective 1

You run historical simulation over 500 days. Which worst loss is the 99% VaR, with this course's convention?

Question 2 · objective 2

And the 97.5% VaR from 250 days?

Question 3 · objective 2

The desk's 250-day VaR is £2.86m today. Nothing in the market or the book changes, but in 40 trading days the VaR will be far lower. Why?

AThe stressed days leave the 250-day windowBThe positions will have agedCVolatility will be rescaledDThe confidence level changes

Question 4 · objective 3

With age weighting over 250 days and λ = 0.98, what weight does the most recent day get, in %?

%

Question 5 · objective 3

A stressed spell ended a year ago. Compared with equal weights, what does age weighting with λ = 0.99 do to the 99% VaR?

ALowers it, because the stressed days now carry less weightBRaises it, because old days count moreCNothing: weights do not change a quantileDMakes it equal to the delta-normal VaR

Glossary

Scenario (historical): One past day's moves in every risk factor, applied to today's positions.
Full revaluation: Repricing every position with its own pricing model at the shocked factor levels, rather than using sensitivities.
Window: The run of past days used as scenarios, for example the last 250 or 750 trading days.
Ghost effect: A jump in historical VaR when an extreme day enters the window, or a drop when it leaves, with nothing changing in the book or the market that day.
Age weighting (BRW): Age weighting (Boudoukh, Richardson and Whitelaw, 1998): recent days get more weight, falling geometrically with age by a factor λ a day.
Half-life: The age at which a day's weight has halved: ln 0.5 / ln λ days.
Effective sample size: 1 / Σwᵢ²: the number of equally weighted days that would carry the same information as a weighted sample.
Volatility weighting: Volatility weighting (Hull and White, 1998): rescale each past move by today's volatility over the volatility on that day.
Stressed VaR: VaR calibrated to a 12-month period of significant stress for the bank's portfolio, added to the capital requirement in Basel 2.5.

Deep Dives & Origins

Deep Dives

K. Dowd, Measuring Market Risk, 2nd ed., Wiley, 2005, the chapter on non-parametric approaches. Intermediate.
C. Alexander, Market Risk Analysis, Volume IV: Value-at-Risk Models, Wiley, 2008, on historical simulation. Advanced.

Origins

J. Boudoukh, M. Richardson and R. Whitelaw, “The Best of Both Worlds: A Hybrid Approach to Calculating Value at Risk”, Risk 11(5), 1998.
J. Hull and A. White, “Incorporating Volatility Updating into the Historical Simulation Method for Value-at-Risk”, Journal of Risk 1(1), 1998.
Basel Committee on Banking Supervision, Amendment to the Capital Accord to incorporate market risks, BIS, January 1996.
Basel Committee on Banking Supervision, Revisions to the Basel II market risk framework, BIS, July 2009.
J.P. Morgan/Reuters, RiskMetrics: Technical Document, 4th ed., 1996.

Watch this lesson

Video version

Prefer to watch? The same lesson as a 12-minute video will sit here, made from this page's script.