Would you play this game?
A coin is flipped. Heads, your money grows 50%. Tails, it shrinks 40%. A fair coin, over and over, with your whole stack each round.
On average it looks like a winner: half the time +50%, half the time −40%, so the "expected" round adds 5%. Free money? Press the button. Flip a few times. Then flip a hundred. Watch what actually happens to your pile.
The expected value of one round is 0.5·1.5 + 0.5·0.6 = 1.05 — plus 5% per round. But you never get the average of the two outcomes; you get the product of the ones that land. Over many rounds your per-round multiplier converges to the geometric mean, not the arithmetic one.
This is Ole Peters' multiplicative coin toss. Wealth follows Wt+1 = Wt(1+rt), a multiplicative process. The decision-relevant quantity is not the expectation but the time-average growth rate g = E[ln(1+r)].
The ensemble grows at +5%/round while every single path decays at −5.13%/round. Time average ≠ ensemble average: the process is non-ergodic. The expectation is carried by a vanishing fraction of ever-luckier paths — true of the crowd, false for you. (Peters, Nature Physics 15, 2019.)
Now the twist. Put a thousand people in the same casino and let each one flip a hundred times. The average of all their piles really does climb — a handful get fabulously lucky. But the person in the middle? Ruined.
Ensemble vs. trajectory. Average across a thousand simultaneous players and you recover the +5%: a few runaway winners drag the mean up. Follow any one player through time and you get −5.13%. The two averages answer different questions.
Where the expectation hides. Below, a thousand simulated players. The green line is the theoretical expectation 1.05t; the red line is the time-average path 0.9487t. Note how few players live anywhere near the green line — and how much of the total wealth a single top player ends up holding.
“Risk isn't what you think is going to happen. It's what hurts if it happens.”
David Dredge · Convex Strategies
The same average, a worse life
Take two portfolios with the identical average return. The only difference is how wildly they swing. The calm one ends up richer — every time. The gap between them has a name.
Lose 50% and you need a 100% gain just to break even, not 50%. Big swings dig holes that are expensive to climb out of. So volatility quietly skims a tax off your compounding — and the wilder the ride, the higher the tax. Drag the slider and watch the calm-return line sink below the average.
Hold the arithmetic mean fixed at +5% and only widen the swings. The compound (geometric) return falls below it by roughly half the variance:
Same average, more variance, less money in your pocket. This is why “−50% then +100%” averages +25% per year yet leaves you exactly where you started.
For multiplicative returns the compound growth rate is g = E[ln(1+r)]. A second-order expansion gives the closed form of the drag:
Exact for log-normal / GBM, an approximation in discrete time — excellent for small moves, looser for large ones. For a symmetric ±s bet the exact drag is √((1+μ)²−s²) computed below, not the σ²⁄2 shorthand. The chart plots the exact geometric return; the readout shows σ²⁄2 alongside so you can see them part company as σ grows.
Why insurance that loses money makes you richer
Here is the punchline the first two games were built for. You hold a market that returns +25% in good years and −50% in a crash. You can buy a slice of crash insurance — it bleeds a little almost every year and only pays off when things fall apart. On its own it is a guaranteed money-loser. Add a sliver of it anyway, and your long-run wealth goes up.
It feels like throwing money away nine years out of ten. But the one year it pays, it stops the −50% wipeout that the first two games showed is so ruinously expensive to recover from. Cap the disaster and your compounding survives to do its work. Move the slider: a little insurance lifts your final wealth above holding none at all.
Good year (90%): market ×1.25, insurance worthless. Crash (10%): market ×0.50, insurance pays 8×. Standalone, the insurance loses 20% a year on average. Yet allocating a fraction x to it and rebalancing each year, the crash year softens from −50% to a survivable −20% — buying back far more compounding than the premiums cost.
Maximize the time-average growth rate over the allocation:
Setting g′(x)=0 gives an interior optimum at x* = 4%, where the portfolio's two outcomes collapse to a clean ±20% (Rup=1.20, Rcrash=0.80). At that point compound growth rises from 14.06% to 15.23% even as the arithmetic mean falls from 17.5% to 16.0%. Over 100 periods (90 up, 10 crash) the hedged investor ends with 2.79× the wealth of the unhedged one. Arithmetic logic rejects the trade; geometric logic demands it.
The interior optimum exists only in a window. Open the advanced controls and push the payoff multiple or crash probability around — too cheap a payoff and insurance helps everywhere; too expensive and the optimum collapses to zero. The math is unconditional; whether a real hedge delivers this convexity is the honest empirical question.
“The one who wins the race is the one with the best brakes — so you can confidently drive as fast as you want.”
David Dredge · Convex Strategies
Protect the downside. The upside takes care of itself.
The crowd's average is not your path. Big losses cost more than equal gains return. And a small, convex hedge that loses money most of the time can still be the thing that lets your capital compound. Three games, one conclusion — and, as the man says, it's just math.
“Understanding is a poor substitute for convexity.”
Nassim Nicholas Taleb · Antifragile
The mathematics here is exact and reproducible. The non-ergodic coin toss follows Ole Peters, “The ergodicity problem in economics,” Nature Physics 15 (2019). The volatility-tax framing follows Mark Spitznagel, Safe Haven (2021). The convexity argument is the one David Dredge of Convex Strategies makes month after month — that resilience comes first, and compounding follows.