Two LP bases with the same expected re-up rate, computed outcome by outcome, and why the concentrated one needs a bigger pipeline of new investors.
LP concentration does not change the re-up a GP should expect, but it widens the range around it. In an illustrative $500m fund where each LP re-ups with a 70 per cent probability, a base with one $150m LP and a base of twenty $25m LPs both expect $350.0m back. The concentrated base, however, has a 12.7 per cent chance of re-ups below $250m, against 1.7 per cent for the diversified one, and its 10th-percentile outcome is $225m rather than $275m.
Worked in full in Private Equity Investor Relations by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Successor fund planning usually starts from one number: the share of the current fund expected to come back. The arithmetic of turning it into new money is worked in what re-up rate a successor fund needs. But LPs decide one by one, and the money arrives in lumps the size of their tickets. When one LP holds 30 per cent of the fund, its single decision moves the outcome by 50m, as much as six of the 5m LPs deciding the same way at once. Investor relations is the team that knows this distribution, and the one that should show it to the partners before a target is set.
| Base | LPs | Largest LP | Herfindahl index | Effective LPs |
|---|---|---|---|---|
| Concentrated: 150, two of 50, ten of 25 | 13 | 30% | 0.1350 | 7.4 |
| Diversified: twenty of 25 | 20 | 5% | 0.0500 | 20.0 |
Both funds total $500m. Each LP re-ups at its Fund I size with probability 70 per cent, independently of the others. The successor targets $600m. The Herfindahl index is the sum of squared shares, 0.302 + 2 × 0.102 + 10 × 0.052 = 0.1350 for the concentrated base; its inverse is the number of equal LPs that would give the same concentration.
Step 1, the expectation. Expected re-up = Σ p × ticket = 0.70 × 500 = $350.0m in both bases. Concentration does not touch it.
Step 2, the spread. With independent decisions, the variance is Σ p(1 − p) × ticket2:
Concentrated: √(0.21 × (1502 + 2 × 502 + 10 × 252)) = $84.2m, 16.8% of the fund
Diversified: √(0.21 × 20 × 252) = $51.2m, 10.2% of the fund
Excel, tickets in B2:B21 and p in E1: =SQRT(E1*(1-E1)*SUMSQ(B2:B21))
Step 3, the whole distribution. The standard deviation is a summary; the tails are what the plan needs. Build the distribution exactly by adding one LP at a time: each existing outcome splits into "returns" with probability 0.70 and "does not" with 0.30. Thirteen LPs give 8,192 combinations, small enough to enumerate in a spreadsheet or a few lines of code.
| Measure | Concentrated | Diversified |
|---|---|---|
| Expected re-up, $m | 350.0 | 350.0 |
| Standard deviation, $m | 84.2 | 51.2 |
| Probability re-ups reach $400m | 40.6% | 23.8% |
| Probability re-ups fall below $300m | 25.3% | 11.3% |
| Probability re-ups fall below $250m | 12.7% | 1.7% |
| 10th-percentile re-up, $m | 225 | 275 |
The concentrated base is more likely to deliver a strong re-up and much more likely to deliver a weak one. Almost all of the downside is the anchor. If it declines, which happens 30 per cent of the time, the most the other twelve LPs can bring back is $350m, their expected re-up is $245.0m, and the chance of ending below $250m is 42.0 per cent.
| Anchor commitment | Share of fund | Effective LPs | Std dev, $m | P(below $250m) | P($400m or more) |
|---|---|---|---|---|---|
| 25 | 5% | 20.0 | 51.2 | 1.7% | 23.8% |
| 50 | 10% | 18.2 | 53.7 | 2.2% | 25.1% |
| 100 | 20% | 12.5 | 64.8 | 5.4% | 31.6% |
| 150 | 30% | 8.0 | 81.0 | 12.5% | 40.9% |
| 200 | 40% | 5.3 | 99.9 | 22.4% | 50.7% |
| 250 | 50% | 3.6 | 120.2 | 29.2% | 59.5% |
The 150 row differs slightly from the base case because the other LPs here are all 5m tickets rather than two of 0m and ten of 5m. Up to a tenth of the fund, an anchor barely changes the downside; at a fifth, the chance of a weak re-up already roughly triples, and above that it rises steeply: at 40 per cent, re-ups fall below half of Fund I more than one time in five.
The same arithmetic works in reverse when the GP decides how much to accept from a new anchor in the successor fund. A large commitment fills the target quickly and raises the effective size of the next re-up decision. Capping any single LP at a fifth of the fund, or securing an early indication from the largest LPs before the first close, keeps the downside of the following raise within reach.
A $600m target with an expected $350.0m of re-ups leaves $250m of new money to raise, in both bases. Built on that number, either GP's pipeline is too thin whenever re-ups come in below the mean, and the concentrated GP's is badly short in the worst one outcome in ten, when re-ups reach only 25m. Planning for the 10th percentile instead means $375m of new money for the concentrated base and $325m for the diversified one, a $50m difference that translates directly into more first meetings, worked as a weighted pipeline in how to calculate a probability-weighted fundraising pipeline.
Independence is the generous assumption. LP decisions are correlated: a weak DPI, a key person departure or a denominator effect hits many LPs at once. Correlation widens both distributions, and it cannot be diversified away by adding names. The anchor's decision is also the one most worth knowing early, before a target is announced.
Concentration leaves the expected re-up at $350.0m and moves the tails: a standard deviation of $84.2m against $51.2m, and a 12.7 per cent chance of re-ups below half the fund against 1.7. Measure the effective number of LPs, plan new money on a low percentile rather than the mean, and find out the anchor's intention first. The fund-level measures an LP weighs before that decision are worked end to end in the free workbook for this book. All figures are illustrative.
With the Herfindahl index: the sum of each LP's squared share of commitments. Its inverse is the effective number of LPs. A $500m fund with one $150m LP, two of $50m and ten of $25m has an index of 0.1350, the same as 7.4 equal LPs, although it has thirteen. Twenty equal LPs give 0.0500, or 20.0.
It widens the range of outcomes in both directions. With a $150m anchor, the chance that re-ups reach $400m rises to 40.6 per cent from 23.8 for an evenly spread base, but the chance they fall below $250m rises to 12.7 per cent from 1.7. If the anchor declines, the expected re-up from the rest is only $245.0m.
Plan for a bad but plausible outcome rather than the average. For a $600m successor target, the expected re-up of $350m in both bases leaves $250m to find. At the 10th percentile, re-ups are $225m for the concentrated base and $275m for the diversified one, so the pipeline should be built for $375m and $325m of new money respectively.
This article is one calculation from Private Equity Investor Relations. 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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