Two losing bets can make a winning one

Take a fair coin with a strange payout: heads doubles your stake, tails halves it. Should you play?

On paper the flip is worth 25% ($\tfrac{2 + 0.5}{2} = 1.25$). But that average describes a crowd of players, not you. Play repeatedly and each doubling is sooner or later cancelled by a halving; the geometric growth rate is $\sqrt{2 \times 0.5} = 1$ — exactly zero. The coin zig-zags wildly and goes, in the long run, nowhere.

Now take two of these coins and flip them together, independently. Split your money equally between them, and after every flip pool what’s left and split it equally again. Four outcomes, equally likely:

• Both heads: you double. ($2.00$)
• Both tails: you halve. ($0.50$)
• One of each: you end up at $1.25$. ($0.5 \times 2 + 0.5 \times 0.5$)

The mixed outcome — one win, one loss — happens half the time, and it leaves you up 25%. The geometric mean across all four outcomes is $(2 \times 1.25 \times 1.25 \times 0.5)^{1/4} \approx 1.12$. That’s 12% growth per flip, from two coins that individually grow at 0%.¹

This feels like a magic trick. Where does the growth come from? Rebalancing forces you to sell whichever coin just won and buy whichever just lost — mechanically buying low and selling high, every single flip. Slide the rebalancing to “never” and the gold line merely averages the two coins, drifting wherever the luckier one wanders. The growth isn’t hidden in either coin; it lives in the act of rebalancing between them. No individual bet needs to be a winner. The portfolio wins because it harvests the volatility.

The demon in the wild

Double-or-halve coins are a fairground abstraction; markets are where this has to earn its keep. Two ingredients are needed: assets that swing hard, and assets that don’t swing together. The classic pair is stocks and gold. Since gold became easy to hold in 2004, the S&P 500 and gold have returned almost exactly the same — 10.9% and 10.7% a year — while moving almost independently day to day (a correlation of 0.07).²

Hold them 50/50 and never touch the mix, and you get the average: 10.8% a year. Reset the mix to 50/50 once a year and you get 11.5%. Monthly, 11.6%; daily, 11.7%. Most of a percentage point a year, earned by trimming whichever asset just ran ahead rather than by picking a winner. The rebalanced portfolio outgrows both of its ingredients — and with far less pain, since stocks lost 55% of their value through the 2008 crisis while the monthly-rebalanced mix never fell more than a third.

One catch before you rebalance your way to riches: most assets are far more correlated than people assume. Two stock funds — even a world tracker against the S&P 500 — move together with a correlation above 0.9. When both halves of your portfolio win and lose on the same days, the mixed outcome that feeds the demon almost never turns up, and rebalancing just shuffles money between twins. Stocks and gold are the exception, not the rule. The demon only eats where volatility is high and correlation is low — and every trade it makes pays a spread, and often a tax bill.

Diversification, revisited

The demon is the extreme case of something more familiar. How much to bet on a single favourable coin is a solved problem; the Kelly criterion gives you the number. But split the same wager across several independent favourable coins and the variance shrinks — some land heads, some tails, and the round as a whole becomes more predictable.

With enough coins, a single round starts to resemble the ensemble average. You’re turning a sequential bet into something closer to a parallel one. The optimal total fraction increases as you add coins — diversification lets you bet more aggressively because the variance per round falls. The ride gets smoother, and the Kelly-optimal wager grows with it.

The maths behind this is elegant: standard deviation shrinks as $1/\sqrt{n}$, where $n$ is the number of independent bets.³ Five coins doesn’t halve your risk, but it cuts it by more than half ($1/\sqrt{5} \approx 0.45$). This is why diversification works, and why Kelly accounts for it.

The practical upshot: a bet you’d refuse on its own can become worth taking as part of a balanced portfolio. Your gut says “that’s a losing bet, stay away.” The geometry says “that’s a losing bet, but pair it with another losing bet and rebalance.” Your gut is good at protecting you from ruin; it’s less good at seeing when two mediocre bets combine into a good one.