I was an excellent high school mathematician. I represented Australia at the International Mathematical Olympiad in 1983, and the path ahead was obvious: university professor, blackboards, not much money. I was fine with that.

Two things changed that trajectory. The first was Douglas Hofstadter’s Godel, Escher, Bach, which I read in my early years at university. If you know the book, you know what it does to you. Godel’s incompleteness theorems proved that any consistent mathematical system contains truths it cannot prove within its own rules. Mathematics was a magnificent game, but a game defined by its axioms, and it couldn’t solve everything. I had believed mathematics was the key to understanding the world. Learning that it wasn’t was genuinely devastating for a young idealist.

The second was a recruiter. In the mid-1980s, finance houses started showing up on university campuses looking for mathematicians to price and trade derivatives. They called us “rocket scientists.” Black-Scholes wasn’t yet taught in finance courses, and the banks needed people who could handle the maths. I took the finance path instead of the academic one. It was certainly a sliding doors moment.

I have never regretted it, but I have also never stopped thinking like a mathematician.

The models are only an approximation

The mathematics of derivatives pricing is easy; if you’re a mathematician. Futures pricing, option pricing and the Greeks, bond and swap pricing: I learned the core principles in a few hours. Once you grasp the underlying logic, it’s straightforward. Black-Scholes is elegant and compact and you can see why it won a Nobel Prize.

The problem is that it’s wrong. I’m not saying it’s useless, but it’s wrong in the way a map is wrong. It’s a good-enough approximation that leaves out the things most likely to hurt you. The model assumes normally distributed returns, but markets have fat tails. It assumes continuous trading, but markets gap. It assumes constant volatility, but volatility moves around in ways the model can’t capture. And around expiry, when the theory should be at its most precise, markets can behave chaotically.

You learn this in the first few weeks of trading real money, not in a textbook. And you learn something else: there are scenarios where adjusting the closed-form model simply doesn’t work. The distribution of potential outcomes is too complex, too path-dependent, too sensitive to assumptions you can’t verify.

When that happens, you build a Monte Carlo simulation. You run thousands of possible paths, model the payoff under each one, and look at the resulting distribution. It’s less elegant than a closed-form solution, but it’s honest. Monte Carlo doesn’t give you a single answer. It gives you a distribution of answers, and it forces you to think about the shape of the uncertainty rather than pretending you’ve got a precise number.

The same progression played out when I started thinking about angel investing.

The ten-sided die

When I first learned about angel investing, the statistic everyone quotes is that roughly one in ten investments produces a meaningful return. I immediately thought of it as a die.

Picture a ten-sided die. Nine sides show zero. One side shows 30. Each roll costs you one unit. Do you want to play?

Everyone says yes. The expected value is 3x your cost. Obviously you want to play. But then I ask: do you want to play it once?

That’s a different question. Playing once gives you a 90 percent chance of losing everything. That’s not investing, that’s gambling. And I think this is why a lot of people make one or two angel investments, lose money, and decide the whole thing is broken.

The right answer is you want to play as many times as you can. And working out how many times you need is high-school probability, not rocket science. Run the numbers, and you find you need at least 22 rolls to have a 90 per cent or better chance of coming out ahead overall (29 for 95 per cent). This lines up almost exactly with the academic research on angel portfolio size, which consistently says you need 25 to 30 investments to have a properly diversified portfolio. It’s a handy initial guide to angel investing.

I’ve used this die model in every conversation I’ve had with prospective angel investors. It cuts through the war stories and gets to the maths. The die doesn’t care how good you think your deal selection is. It says: play enough times, or don’t play.

Why 30 on the tenth side? Because it’s a reasonable approximation that lines up with historical angel investing IRRs of 20–30 percent. It’s a toy model, but it’s a useful one.

Where the simple model runs out

The ten-sided die, like Black-Scholes, is a good first-order approximation. It captures the essential shape of the game. And like Black-Scholes, it’s wrong in ways that matter once you look more closely.

Real angel returns aren’t nine zeros and a 30x. In practice, there are investments that return 1x, 2x, 4x – not zeros, not hitting a six, but outcomes that affect the portfolio. And the right tail extends well past 30x. Some investments return 100x. A very small number return 500x. These are rare, but their impact on the portfolio mathematics is enormous, just as fat tails in equity markets dominate the risk in ways Black-Scholes can’t handle.

Then there’s the question of cadence. Making 25+ investments is easy to say. But you can’t make them all at once. Deal flow arrives over time. How many per year? Over how many years? The die model assumes every roll is instantaneous and independent. Real angel investing is neither.

You could hand the money to an early-stage VC and let them build the portfolio for you. The maths would be similar. But where’s the fun in that? There’s a big difference between saying “I invested in Canva” and “I invested in a fund which invested in Canva.” If you’re the kind of person drawn to angel investing in the first place, you want to make your own decisions.

The high-school maths runs out. As I wrote in The Walking Investor, pattern recognition compounds. The die model was where mine started, but real portfolios demanded something better.

Monte Carlo, again

So a few years ago, before AI could do this sort of thing conversationally, I asked my son Patrick Moore, then a university student, and his friend Aidan Knipe to build me a proper angel investing simulation. A Monte Carlo model, the same class of tool I’d used to price derivatives decades earlier, applied to angel portfolio construction.

The model lets you set the key parameters: portfolio size, investment cadence, cheque size, the distribution of returns including the long tail, and how many years you deploy capital over. It runs thousands of simulated portfolios and shows you the distribution of outcomes. Not one number, but the full range of what might happen and the probability of each.

Three things came out of it.

First, the die’s lesson holds up but gets sharper: more investments, suitably diversified, produce reliably better outcomes. The improvement is steep as you move from 10 to 25 investments, and continues at a diminishing rate toward 40 or 50.

Second, the model shows you need to think about deploying capital over about seven years. Angel investments are illiquid and take years to mature. If you put all your money in during year one, you’re concentrated in a single vintage and you’ve got nothing left when new opportunities come along. Spreading deployment over multiple years produces materially better outcomes.

Third – and this is the one that catches people – angel investors should be writing smaller cheques than their instinct tells them. If you’ve got a fixed allocation and the maths says you need 30+ investments over seven years, the arithmetic on cheque size does itself. Most people’s first instinct is fewer, bigger bets. The simulation says the opposite. If you want to understand why, go try the model.

The implementation

I’ve gone all in on these learnings. The mathematics is clear, and I’ve spent enough years watching models meet reality to know when a model is telling you something true.

But building a portfolio of 30+ investments over seven years requires something most individual angel investors don’t have: consistent access to qualified deal flow. You can’t diversify if you only see three deals a year.

This is what MooCoo’s syndicate is built for. Co-investing alongside Brisbane Angels, the syndicate offers access to roughly 15 new investments per year, plus follow-on opportunities in companies that are performing. This is the kind of cadence that the maths requires.

If you’d like to try the simulation model yourself, it’s available to MooCoo members along with some explanatory videos. Change the parameters, watch the distributions shift. I find it genuinely illuminating, though I accept my bar for illuminating may not be the same as everyone else’s!

Godel taught me that mathematics can’t solve everything. He was right. But it can tell you how to play a positive-sum game without turning it into a gamble. For angel investing, that’s more than enough.

Richard Moore is co-founder of MooCoo Ventures, an angel syndicate that co-invests alongside Brisbane Angels, one of Australia’s most active angel groups. He has made over eighty personal angel investments since 2013.

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