The risk that matters is not losing money but doing worse than the fund you could have committed to, and it falls far more slowly with the deal count.
Enough that the chance of trailing the fund, not of losing money, is small. On an illustrative spread of no-fee deal outcomes averaging 2.00x, against a fund that returns 1.64x net on the same deals, one co-investment lags the fund 35.0 per cent of the time, ten deals still lag 18.0 per cent of the time and it takes 21 to get below 10 per cent. The risk of losing money is gone much sooner: below 1 per cent by eight deals.
Worked in full in The Co-Investment Practitioner by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A co-investment programme is sold on the fee saving: the same deals as the fund, without the management fee and carried interest, so a better net multiple. That is true on average. Any one investor, though, owns a handful of deals rather than the average, and a programme of three or four can easily finish behind the fund that charged the fees. The question an investment committee should ask is not "will we lose money?" but "how likely are we to have done worse than if we had simply committed to the fund?", and the two have very different answers.
| Multiple | Probability |
|---|---|
| 0.0x | 10% |
| 0.5x | 10% |
| 1.0x | 15% |
| 2.0x | 30% |
| 3.0x | 25% |
| 4.5x | 10% |
| Mean, standard deviation | 2.00x, 1.28x |
The fund invests in the same kind of deals at the same 2.00x gross and returns 1.64x net, so the fee and carry gap is 0.36x. A single deal finishes below 1.64x whenever it lands on 0.0x, 0.5x or 1.0x: a 35.0 per cent chance. Deals are taken as independent, which is the right assumption for this question, and the reason is set out below.
The programme's multiple is the average of its n deals. Its standard deviation falls with the square root of n: 1.28x for one deal, 0.41x for ten. But the probability of lagging the fund is not a smooth function of that figure, because deal outcomes are lumpy. The exact way to get it is to enumerate every combination of outcomes, which a spreadsheet does by convolution.
P(lag) = P(average of n deals < 1.64x) = P(sum of n multiples < 1.64 × n)
Express the outcomes in half-turn units (0, 1, 2, 4, 6, 9). The distribution of the sum of n deals is the distribution for n − 1 deals convolved with one more deal: Pn(s) = Σk Pn−1(s − k) × pk. Then add up the probabilities of every sum below 1.64 × n turns.
In Excel, put the one-deal probabilities in a column indexed by units, and build each further column with =SUMPRODUCT of the previous column against the reversed one-deal column. A normal approximation is quicker and wrong in exactly the range that matters, below ten deals.
| Deals | Std dev of the average | P(lag the fund) | P(below 1.0x) |
|---|---|---|---|
| 1 | 1.28x | 35.0% | 20.0% |
| 2 | 0.91x | 38.2% | 10.0% |
| 3 | 0.74x | 28.5% | 6.7% |
| 4 | 0.64x | 31.5% | 4.4% |
| 5 | 0.57x | 28.2% | 2.9% |
| 8 | 0.45x | 22.9% | 0.9% |
| 10 | 0.41x | 18.0% | 0.4% |
| 15 | 0.33x | 14.6% | 0.1% |
| 20 | 0.29x | 10.3% | 0.0% |
| 30 | 0.23x | 6.2% | 0.0% |
| 40 | 0.20x | 3.8% | 0.0% |
Two curves, two speeds. The chance of losing money at least halves with every doubling of the programme and is negligible by ten deals. The chance of trailing the fund falls slowly: 18.0 per cent at ten deals, 10.3 at twenty, 6.2 at thirty. It also does not fall in a straight line. Two deals lag more often than one, and four more often than three, because with lumpy outcomes the bar moves: three deals need 4.92 turns between them to beat the fund, four need 6.56, and the extra deal does not always bring the extra 1.64.
Why independence is the right assumption here. Deals done in the same vintage share entry prices and exit markets, so their outcomes are correlated. But so are the fund's. A bad vintage drags the fund's net multiple down along with the co-investments, and the comparison between the two is unaffected. What a larger count diversifies away is the idiosyncratic part of each deal, and for a relative question that is the whole of the risk. Read the outcome table in that spirit: as the spread of a deal's result around what the fund's own deals made in the same vintage, not as the spread of buyout outcomes across all market conditions.
| Deals | P(lag), base | P(lag), wider | P(below 1.0x), wider |
|---|---|---|---|
| 1 | 35.0% | 45.0% | 30.0% |
| 5 | 28.2% | 35.5% | 8.1% |
| 10 | 18.0% | 26.0% | 2.4% |
| 20 | 10.3% | 18.0% | 0.2% |
| 30 | 6.2% | 13.1% | 0.0% |
With a standard deviation of 1.75x instead of 1.28x, twenty deals lag as often as ten did before. Growth and venture-style co-investments sit closer to this row than to the base case.
Sizing the programme on the probability of loss. By that test five deals look safe, with a 2.9 per cent chance of finishing below cost. By the test the programme is actually judged on, beating the fund it could have committed to, five deals fail more than a quarter of the time. At an illustrative three deals a year, getting to ten takes 3.3 years and getting to twenty takes 6.7, so the count is as much a pacing decision as an allocation one. The fee saving is the expected reward for taking that risk; how much worse a deal can be before it erases that saving is worked in how much worse co-investment deals can be and still beat the fund, and the equivalent count for whole funds in how many private equity funds you need to diversify.
Count co-investments against the right line. On these illustrative outcomes, ten deals cut the chance of losing money to 0.4 per cent but still trail the fund 18.0 per cent of the time; 21 deals are needed to bring that below 10 per cent, and more if the deals are riskier. The free workbook and cases for the book take the same question on the book's own deal, including a case that prices exactly this four-deal problem.
Yes, often, when it holds few deals. On illustrative outcomes averaging 2.00x with no fees, against a fund returning 1.64x net, a single co-investment lags the fund 35.0 per cent of the time and a five-deal programme 28.2 per cent of the time, even though its expected multiple is 0.36x higher.
Because deal outcomes are lumpy and the bar rises with every deal. Three deals need 4.92 turns between them to beat a 1.64x fund and four need 6.56. On illustrative outcomes, three deals lag 28.5 per cent of the time and four 31.5 per cent, before the count starts to work again.
At an illustrative three deals a year, ten deals take 3.3 years and twenty take 6.7. Ten deals bring the chance of trailing the fund to 18.0 per cent on the illustrative outcomes; twenty bring it to 10.3 per cent, so the deal count is a pacing decision as much as an allocation one.
This article is one calculation from The Co-Investment Practitioner. 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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