Historical simulation VaR worked on one fund: the ranking, the three percentile conventions, the window length and why the same volatility gives a larger number than the normal curve.
Historical VaR applies today's portfolio to each of the last N days' market moves, ranks the resulting profits and losses, and reads off the loss at the confidence percentile. With 500 days and 99 per cent confidence that is the fifth or sixth worst day. On an 840M credit fund whose parametric one-day VaR is 7,242,456, an illustrative 500-day history with exactly the same volatility gives a historical VaR of 8,244,223, 13.8 per cent higher, because the history has fatter tails than the normal curve.
Worked in full in Financial Risk Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The fund is Halverton, the open-ended credit fund of Financial Risk Management, with 840,000,000 of net assets and a one-day 99 per cent parametric VaR of 7,242,456. The return history is synthetic and illustrative: 500 daily returns drawn from a fat-tailed distribution (Student t with 4 degrees of freedom) and rescaled to a mean of zero and the same daily standard deviation the parametric figure uses. Any difference between the two VaRs therefore comes from the shape of the tail, not from the volatility.
| Input | Value |
|---|---|
| Net assets | 840,000,000 |
| Annual volatility (parametric model) | 5.883% |
| Daily standard deviation | 0.3706% |
| One-day standard deviation in money | 3,113,230 |
| Observations | 500 days |
| Confidence | 99% |
| Parametric VaR, z = 2.3263 | 7,242,456 |
For a portfolio of linear positions, each scenario P&L is today's market value times that day's return, sleeve by sleeve, summed. Positions with options or call features are fully revalued under each day's moves instead. The point is that today's holdings are used, not the fund's historical P&L, which mixes in positions it no longer holds.
P&Lt = Σi MVi × ri,t, for t = 1 to 500
Excel, market values in a row B1:E1 and daily returns in B2:E501: =SUMPRODUCT($B$1:$E$1,B2:E2), filled down.
| Rank | Day | Loss | Return | Daily sd |
|---|---|---|---|---|
| 1 | 125 | 14,186,327 | −1.689% | 4.56 |
| 2 | 127 | 12,430,635 | −1.480% | 3.99 |
| 3 | 88 | 11,716,874 | −1.395% | 3.76 |
| 4 | 101 | 9,706,355 | −1.156% | 3.12 |
| 5 | 393 | 9,564,234 | −1.139% | 3.07 |
| 6 | 271 | 8,244,223 | −0.981% | 2.65 |
| 7 | 380 | 7,401,469 | −0.881% | 2.38 |
| 8 | 423 | 7,326,003 | −0.872% | 2.35 |
| 9 | 426 | 7,297,036 | −0.869% | 2.34 |
| 10 | 120 | 7,071,848 | −0.842% | 2.27 |
One per cent of 500 is five. The sixth-worst loss, 8,244,223, is exceeded on exactly five days, 1 per cent of the sample, which makes it the 99th percentile of the loss distribution. That is the historical VaR: 0.981 per cent of net assets, against 0.862 per cent on the parametric model.
Historical VaR99% = −PERCENTILE(P&L, 1%) = 8,244,223
Excel: =-PERCENTILE.INC(F2:F501,0.01) returns 8,257,423, interpolating at rank 5.99; =-SMALL(F2:F501,6) returns the sixth-worst exactly.
The same 500 numbers support three answers, depending on a line nobody puts in the risk report.
| Convention | VaR | vs parametric |
|---|---|---|
| Sixth-worst loss (exceeded on 1% of days) | 8,244,223 | 1.138 |
| PERCENTILE.INC, rank 5.99 | 8,257,423 | 1.140 |
| PERCENTILE.EXC, rank 5.01 | 9,551,034 | 1.319 |
| Fifth-worst loss (conservative) | 9,564,234 | 1.321 |
The gap between the fifth and sixth observations is 16.0 per cent. In a fat tail neighbouring observations are far apart, which is exactly where the percentile is read. Either convention is defensible; switching between them from one report to the next is not.
| Measure | Historical | Parametric | Ratio |
|---|---|---|---|
| 99%, 500 days, sixth-worst | 8,244,223 | 7,242,456 | 1.138 |
| 99%, first 250 days, third-worst | 11,716,874 | 7,242,456 | 1.618 |
| 99%, last 250 days, third-worst | 7,401,469 | 7,242,456 | 1.022 |
| 95%, 500 days, 26th-worst | 5,079,829 | 5,120,808 | 0.992 |
Two lessons. First, at 95 per cent the two methods agree to within 1 per cent; the normal curve gets the middle of the distribution right and the tail wrong, so the confidence level decides how much the method matters. Second, the window. The three worst days in the whole sample fall in the first 250 days, so a one-year window computed then reads 11,716,874 and one computed a year later reads 7,401,469: a ratio of 1.58 for the same fund with the same holdings, only because the bad days aged out. On 250 days the 99 per cent quantile rests on the second and third observations, and the VaR jumps on the day a large loss leaves the window.
A one-day VaR, historical or parametric, assumes the position can be closed or hedged within the day. A fund that takes eleven days on average to sell its book is exposed to eleven days of moves, which no ranking of one-day returns captures.
The parametric construction of the 7,242,456, with expected shortfall and the backtest counter beside it, is in the free workbook for this case; for your own book use the VaR and expected shortfall model.
Both are used. 1 per cent of 500 is 5, and the sixth-worst loss is the one exceeded on exactly five days, which matches the definition of the 99th percentile. The fifth-worst is the conservative reading. On this fund they are 8,244,223 and 9,564,234, 16.0 per cent apart, so the convention has to be written down.
Because real return distributions have fatter tails than the normal curve. With identical daily volatility of 0.3706 per cent, the normal model gives 7,242,456 at 99 per cent; the 500-day history gives 8,244,223. At 95 per cent the two almost agree, 5,079,829 against 5,120,808: the normal model fails in the tail, not in the middle.
Regulatory VaR uses at least one year, about 250 trading days; many desks use 500. A short window reacts faster but its 99 per cent quantile rests on two or three observations. Here the first 250 days give 11,716,874 and the last 250 give 7,401,469, a ratio of 1.58 for the same fund.
The parametric VaR this history is set against is built step by step in the workbook for chapters 4, 6 and 7 of Financial Risk Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
Get the book on Amazon →Free companion files
Also on Amazon UK · Amazon Germany · Amazon France · Amazon Canada
Reading guide: corporate finance, valuation and markets → · All 1340 articles →
If this book helped, or didn’t, a few lines on Amazon are worth more than they look: they are what the next reader goes on. Write a review. The workbook stays free either way.