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Compounding

Returns earned on returns. It makes the path of an equity curve, not just its endpoints, determine the final balance.

Compounding against a flat returnTwo account balances over fifteen years at the same yearly rate: one curve bends upwards as gains are left in, the other rises in a straight line.ACCOUNT VALUE$10k$20k$30k$40k051015YEARSCOMPOUNDED 10% a yearSIMPLE: 10% of the original sumboth start at $10,000 and run 15 years$41,772DIFFERENCE$16,772$25,000
Compounding against a flat return. Two accounts start at $10,000 and earn 10% a year for fifteen years. Leaving the gains in means each year earns on a larger balance, so the curve bends away from the straight line and ends $16,772 higher.

Under compounding a percentage loss needs a larger percentage gain to recover: down 20% needs +25%, down 50% needs +100%, down 80% needs +400%. This asymmetry is the mathematical case for hard drawdown limits.

It also changes how position sizing should work. Fixed-fractional sizing, risking a constant percentage of current equity, compounds naturally and cannot mathematically reach zero from losses alone; fixed-dollar sizing does not adapt and can.

Worked example: 2% growth per month compounds to 26.8% a year, not 24%. Over ten years that difference is the gap between 8.9x and 6.8x on the original stake.

Related: arithmetic-vs-geometric-return, equity-curve, position-sizing, drawdown

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