Three scenarios, one expected multiple, and three different expected IRRs: which one belongs in the investment committee memo.
A probability-weighted MOIC is each scenario's multiple times its probability, summed. On an illustrative $20M co-investment with a 25 per cent chance of 0.5x, a 50 per cent chance of 2.2x and a 25 per cent chance of 3.5x, the expected MOIC is 2.10x. The IRR needs more care: averaging the three scenario IRRs gives 15.0 per cent, while the IRR of the probability-weighted cash flows is 17.4 per cent.
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 comes with the sponsor's model, and the sponsor's model has one case. A co-investor that underwrites independently builds at least three: a downside where the plan fails, a base where it broadly works, and an upside. The decision rests on the combination, because the co-investor will make many such investments and its programme return converges on the expected value, not on any single case. The base case tells the committee what is most likely; the expected multiple tells it what the programme should earn from deals like this one.
| Scenario | Probability | MOIC | Exit year | Proceeds, $M | Scenario IRR |
|---|---|---|---|---|---|
| Downside | 25% | 0.50x | 6 | 10.0 | −10.9% |
| Base | 50% | 2.20x | 5 | 44.0 | 17.1% |
| Upside | 25% | 3.50x | 4 | 70.0 | 36.8% |
The exit years differ on purpose. Good outcomes tend to be sold early, and problem companies are held while the sponsor tries to fix them, so the downside is both smaller and later.
1. Expected MOIC: 25% × 0.50 + 50% × 2.20 + 25% × 3.50 = 0.125 + 1.100 + 0.875 = 2.10x, or $42.0M of expected proceeds.
2. Expected cash flows: −20.0 at year 0; 25% × 70.0 = 17.5 at year 4; 50% × 44.0 = 22.0 at year 5; 25% × 10.0 = 2.5 at year 6.
3. IRR of the expected cash flows: =IRR({-20,0,0,0,17.5,22,2.5}) = 17.4%.
4. For comparison, the average of the scenario IRRs: 25% × (−10.9%) + 50% × 17.1% + 25% × 36.8% = 15.0%.
In Excel: =SUMPRODUCT(Probabilities,Multiples) for the expected MOIC.
The expected multiple is unambiguous: 2.10x. Multiples are linear in the cash flows, so weighting them is exactly the same as weighting the proceeds. IRRs are not. Three figures can claim to be the expected IRR of this deal:
The differences are not rounding: 2.4 points separate the first and the last. The conversion from multiple to IRR at different holding periods is laid out in how to convert MOIC to IRR.
| Downside probability | Base probability | Expected MOIC | IRR of expected cash flows | Average of IRRs |
|---|---|---|---|---|
| 10% | 65% | 2.36x | 20.3% | 19.2% |
| 25% | 50% | 2.10x | 17.4% | 15.0% |
| 40% | 35% | 1.84x | 14.2% | 10.8% |
Every 15 points of probability moved from base to downside costs 0.255x of expected multiple. Against a 2.00x programme target, the deal clears as long as the downside probability is no more than 30.9 per cent. That break-even is the number worth arguing about in committee, because it turns a debate about the business plan into a question with an answer: does anyone believe the chance of the plan failing is above or below 31 per cent? The gap between the two IRR methods also widens as the downside becomes more likely, from 1.1 to 3.4 points, because averaging gives full weight to a large negative rate earned over a long period.
Severity matters as much as probability. If the downside is a write-off rather than 0.5x, the expected MOIC falls from 2.10x to 1.98x, below the 2.00x target with no change to any probability. How many deals' fee savings a single zero erases is in how many co-investments one write-off erases.
Presenting the base case as the expected return. The base is 2.2x; the expected multiple is 2.10x. The 0.10 gap looks small, but it is the whole difference between a deal that clears a 2.15x bar and one that does not, and it grows with every point of downside probability. The second mistake is assigning probabilities after the outcomes, so that they come out to whatever makes the deal work. Set the probabilities from the programme's own record of how often plans fail, then build the scenarios, and keep the scenario years honest: a downside that exits as fast as the base case flatters the expected IRR.
Expected MOIC = Σ probability × multiple: 2.10x here. Expected IRR = IRR of the probability-weighted cash flows: 17.4 per cent, not the 15.0 per cent average of the scenario IRRs. The decision memo template and the return bridge with its modelling traps are in the free companion files for this book.
No. IRR is not linear in the cash flows, so the average of scenario IRRs is not the IRR of the average outcome. In this case the downside, base and upside IRRs of minus 10.9, 17.1 and 36.8 per cent average to 15.0 per cent, while the expected cash flows return 17.4. Weight the multiples or the cash flows, then compute one IRR.
Solve for the probability at which the expected multiple equals the target. With a 2.2x base, a 0.5x downside and a 25 per cent chance of 3.5x, the expected MOIC stays at or above 2.00x as long as the downside probability is no more than 30.9 per cent. At 40 per cent it falls to 1.84x.
Because the downside is deeper below the base than the upside is above it, once each is weighted by its probability. Here the 2.2x base sits 0.10 above the 2.10x expected multiple. A memo that leads with the base case presents the most likely outcome, not the average one, and the gap grows with downside probability.
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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