The square-root-of-time rule is one line of arithmetic. Whether it holds depends on two things the rule assumes away: independent days and a book you can sell.
Multiply the one-day figure by the square root of the horizon, not by the horizon. A one-day 99 per cent value at risk of 7,242,456 on an 840M credit fund becomes 22,902,656 over ten days (times 3.162), or 2.73 per cent of net assets. The rule holds only if days are independent: with a daily autocorrelation of 0.2, the ten-day figure is 27,459,319.
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 that runs through Financial Risk Management: 840,000,000 of net assets in four sleeves, with the reported correlation matrix (off-diagonal entries between 0.45 and 0.80). Volatilities are annual and the year has 252 trading days. All figures are illustrative.
| Sleeve | Weight | Market value | Volatility |
|---|---|---|---|
| Investment-grade credit | 0.30 | 252,000,000 | 4.5% |
| High yield | 0.35 | 294,000,000 | 8.0% |
| Leveraged loans | 0.25 | 210,000,000 | 6.0% |
| Structured credit | 0.10 | 84,000,000 | 10.0% |
| Fund | 1.00 | 840,000,000 | 5.883% |
The portfolio volatility comes from the quadratic form, weights times volatilities times correlations, and on the reported matrix it is 5.883 per cent a year. Divide by the square root of 252 (15.8745) to get a daily volatility of 0.3706 per cent, then multiply by net assets and by the 99 per cent normal quantile, 2.326.
One-day VaR = net assets × (annual volatility / √252) × z99%
= 840,000,000 × 0.3706% × 2.326 = 7,242,456, or 0.862 per cent of net assets
Excel: =NAV*Vol/SQRT(252)*NORM.S.INV(0.99)
Variances of independent returns add. Ten independent days carry ten times the variance of one day, so the standard deviation grows by the square root of 10, and so does a normal VaR, because the quantile is a fixed multiple of the standard deviation.
h-day VaR = one-day VaR × √h
Ten days: 7,242,456 × 3.1623 = 22,902,656
Excel: =VaR1*SQRT(10)
| Horizon, days | Multiplier | Value at risk | Share of net assets |
|---|---|---|---|
| 1 | 1.0000 | 7,242,456 | 0.86% |
| 5 | 2.2361 | 16,194,623 | 1.93% |
| 10 | 3.1623 | 22,902,656 | 2.73% |
| 20 | 4.4721 | 32,389,246 | 3.86% |
Scaling linearly, ten times the one-day figure, would report 72,424,556, or 8.62 per cent of the fund. That is 3.162 times too much on the model's own assumptions, and it is a surprisingly common error in spreadsheets built from a one-day system output. The zero mean is also an assumption: over ten days the expected return of a credit fund is small enough against 22.90M to ignore, but over a quarter it is not.
The square root of time is a statement about independence. Credit sleeves, and anything marked with a lag or from matrix pricing, tend to show positive autocorrelation in daily returns: a bad day is followed by another bad day as the marks catch up. Under a first-order autoregressive process with daily autocorrelation ρ, the variance of the ten-day sum is not 10 times the daily variance but:
Variance ratio = h + 2 × Σk=1..h−1 (h − k) × ρk
Ten-day VaR = one-day VaR × √(variance ratio)
At ρ = 0.2: √(10 + 2 × (9 × 0.2 + 8 × 0.04 + ...)) = 3.791
| Daily autocorrelation | Ten-day multiplier | Ten-day VaR | Share of net assets | Above square-root rule |
|---|---|---|---|---|
| 0.0 | 3.162 | 22,902,656 | 2.73% | 0.0% |
| 0.1 | 3.461 | 25,062,778 | 2.98% | 9.4% |
| 0.2 | 3.791 | 27,459,319 | 3.27% | 19.9% |
| 0.3 | 4.165 | 30,164,585 | 3.59% | 31.7% |
A modest 0.2 adds almost a fifth to the ten-day figure. Over long horizons the correction tends to √((1 + ρ) / (1 − ρ)), which is 1.225 at 0.2. The one-day VaR does not show it at all, because a one-day measure is blind to what tomorrow inherits from today. Before scaling, measure the first few autocorrelations of the daily return series; if they are positive and stable, use the variance ratio instead of the square root.
A ten-day horizon is a convention. The economically relevant horizon is how long the book takes to exit, and that differs by sleeve. Give each sleeve its own days to sell (illustratively 2 for investment grade, 8 for high yield, 15 for leveraged loans and 25 for structured credit), scale each sleeve's volatility by the square root of its own horizon, and run the same quadratic form. The fund comes out at 22,544,204, the equivalent of a 9.7-day horizon, almost exactly the value-weighted average of 9.7 days.
On these inputs the uniform ten-day figure and the liquidity-horizon figure are within 1.6 per cent of each other. The horizon is rarely the missing piece. What neither figure contains is the cost of selling and the fatter tail of a stressed market, which is why the book sets a stress loss beside the VaR rather than stretching the VaR to cover it.
Three errors recur. The first is linear scaling, which overstates by 3.162 times at ten days. The second is scaling a VaR whose positions will not survive the horizon: square-root-of-time assumes the portfolio is held unchanged for ten days, while a fund meeting redemptions is selling throughout. The third, and the most expensive, is scaling an autocorrelated return series as if it were independent, which understates exactly the funds whose marks are smoothest. The one-day VaR of a smoothed book looks low, and the square root of ten carries that flattering number forward.
For what the same correlation matrix does when it is re-estimated on crisis months, see how much diversification survives a crisis. The site's VaR and expected shortfall model carries the horizon block on any portfolio you type in.
The quadratic form, the correlation switch and the horizon rows are live in the free workbook for this case, so a different sleeve mix or horizon reruns the table.
Because variances of independent returns add. Ten independent days have ten times the variance of one, so the standard deviation, and with it a normal VaR, grows by the square root of 10, which is 3.162. A one-day VaR of 7,242,456 becomes 22,902,656. Scaling by 10 instead would give 72,424,556, overstating the risk 3.162 times.
When daily returns are positively autocorrelated, which is common in credit and private assets marked with a lag. With a daily autocorrelation of 0.2 the ten-day multiplier is 3.791 rather than 3.162, and the ten-day VaR is 27,459,319 rather than 22,902,656: 19.9 per cent higher. At 0.3 the gap is 31.7 per cent.
No. Ten days is a horizon applied to the whole book. A liquidity-adjusted figure gives each sleeve its own days to sell. With illustrative horizons of 2 to 25 days the fund here comes out at 22,544,204, the equivalent of 9.7 days, close to the ten-day figure. Neither includes the cost of selling, which is a separate number.
The one-day construction and the horizon scaling are worked in the value at risk 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.
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