A quartile boundary is an estimate from a sample: the formula for its precision, the cohort size it needs, and how often a second-quartile fund ranks top quartile.
More than most peer groups have. The standard error of an upper-quartile boundary is 1.3626 × σ ÷ √n, where σ is the spread of fund returns in the vintage. With an illustrative 10-point spread of net IRRs, a 20-fund cohort places the top-quartile line within ±6.0 points at 95 per cent confidence, and pinning it to ±3 points takes 80 funds.
Worked in full in Private Markets Performance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Quartile rankings are quoted as if the boundary were a known number. It is an estimate from a sample, and its precision depends on two things only: how many funds are in the cohort and how widely their returns are spread. Both can be put into one formula, and the answer explains why so many managers can call themselves top quartile.
| Input | Value |
|---|---|
| Mean net IRR of the vintage | 12.0% |
| Standard deviation of net IRRs across funds, σ | 10.0 points |
| Quantile of interest, p (upper quartile) | 0.75 |
| True upper-quartile boundary, 12.0 + 0.6745 × 10.0 | 18.74% |
| Confidence level | 95% (z = 1.96) |
Step 1: the standard error of a sample quantile. For a sample of n funds, the estimated p-quantile has a standard error of √(p(1 − p) ÷ n) divided by the density of the distribution at that quantile. For a normal distribution the density at the upper quartile is φ(0.6745) ÷ σ = 0.3178 ÷ σ.
SE(upper quartile) = √(0.75 × 0.25) ÷ 0.3178 × σ ÷ √n = 1.3626 × σ ÷ √n
For the median the factor is 1.2533, so the median is barely more precise than the quartile.
In Excel: =SQRT(p*(1-p))/NORM.S.DIST(NORM.S.INV(p),FALSE)*sigma/SQRT(n).
Step 2: apply it to cohort sizes. With σ at 10.0 points, a 20-fund cohort gives 1.3626 × 10.0 ÷ √20 = 3.05 points of standard error, and a 95 per cent band of 1.96 × 3.05 = ±6.0 points around the published boundary.
| Funds in cohort | SE, upper quartile | 95% band, ± | SE, median |
|---|---|---|---|
| 10 | 4.31 | 8.4 | 3.96 |
| 20 | 3.05 | 6.0 | 2.80 |
| 30 | 2.49 | 4.9 | 2.29 |
| 40 | 2.15 | 4.2 | 1.98 |
| 60 | 1.76 | 3.4 | 1.62 |
| 100 | 1.36 | 2.7 | 1.25 |
| 200 | 0.96 | 1.9 | 0.89 |
Step 3: invert it for the size you need. n = (1.96 × 1.3626 × σ ÷ tolerance)2. For ±4 points that is 44.6, so 45 funds; for ±3 points, 79.3, so 80; for ±2 points, 179. Precision improves only with the square root of the cohort: halving the band takes four times the funds.
A simulation confirms the formula and shows what it means for a manager. Drawing 20,000 cohorts of 20 funds from the illustrative vintage and applying Excel's QUARTILE.INC, the estimated boundary has a standard deviation of 2.90 points (the formula's 3.05 is the large-sample value) and a 95 per cent range of 12.7 to 24.1 per cent. The estimate also sits about 0.4 points below the true 18.74 on average at that size.
| True percentile of the fund | Its IRR | Cohort of 20 | Cohort of 60 |
|---|---|---|---|
| 65th (second quartile) | 15.9% | 20.3% | 5.3% |
| 70th (second quartile) | 17.2% | 36.0% | 22.0% |
| 80th (top quartile) | 20.4% | 76.6% | 85.1% |
| 85th (top quartile) | 22.4% | 91.5% | 98.4% |
In a 20-fund cohort, a fund that is truly at the 70th percentile is labelled top quartile more than a third of the time, and one at the 65th a fifth of the time. A genuinely top-quartile fund at the 80th percentile misses the label in almost one cohort in four. Combine that with the several databases, measures (IRR, TVPI, DPI) and quartile conventions a manager can choose from, as shown in how private equity quartile rankings are calculated, and a second-quartile fund will usually find a cell where it ranks top quartile.
| σ of net IRRs | SE at 20 funds | 95% band at 20, ± | Funds for ±3 points |
|---|---|---|---|
| 6 points | 1.83 | 3.6 | 29 |
| 10 points | 3.05 | 6.0 | 80 |
| 15 points | 4.57 | 9.0 | 179 |
| 20 points | 6.09 | 11.9 | 318 |
The required cohort grows with the square of the dispersion. Strategies with wide outcomes, venture above all, need several hundred funds per vintage for a boundary precise to 3 points, which no single vintage cell provides. Measure σ on the cohort you are given: the sample standard deviation of its IRRs is the input.
Reading a fund as top quartile because its IRR is above the published line, without asking how many funds drew the line. A fund 2 points above the boundary of a 20-fund cohort is well inside a ±6.0 point band: the ranking says nothing either way. The second mistake is treating the median as safer. Its standard error is only 8 per cent smaller (1.2533 against 1.3626).
The test to run on any quartile claim: count the cohort, compute the standard deviation of its returns, and calculate 1.96 × 1.3626 × σ ÷ √n. If the fund's distance from the boundary is smaller than that, the claim is a draw.
A quartile boundary from 20 funds with a 10-point spread is uncertain by ±6.0 points; reaching ±3 takes 80 funds and ±2 takes 179. Below that, a quartile label measures the cohort as much as the manager. The standard error of a boundary at any cohort size is computed in the free workbooks for Private Markets Performance.
For a quartile boundary, not very. With an illustrative 10-point spread of net IRRs, the standard error of the upper quartile at 20 funds is 3.05 points and the 95 per cent band is plus or minus 6.0. A fund within 6 points of the published line cannot be placed on either side of it with confidence.
Often, in small cohorts. Simulating 20,000 cohorts of 20 funds with a 10-point spread, a fund at the true 70th percentile ranks above the estimated boundary 36.0 per cent of the time, and one at the 65th percentile 20.3 per cent. In cohorts of 60 those rates fall to 22.0 and 5.3 per cent.
Only slightly. For normally distributed returns the median's standard error is 1.2533 x sigma / sqrt(n), against 1.3626 for the upper quartile, about 8 per cent smaller. At 20 funds and a 10-point spread that is 2.80 points against 3.05.
This article is one calculation from Private Markets Performance. 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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