Risk is usually defined in finance as volatility — how much a price moves around. That definition is useful because it is measurable, and misleading because it is not what most people mean. To an investor, risk is the chance of ending up with materially less than they need, at the moment they need it.
The clearest illustration is the arithmetic of recovery. A loss and the gain that undoes it are not symmetric, and the asymmetry gets worse the deeper the loss goes.
| Loss | Gain needed to get back to even |
|---|---|
| −10% | +11.1% |
| −20% | +25.0% |
| −33% | +49.3% |
| −50% | +100% |
| −80% | +400% |
| −90% | +900% |
gain needed = 1 / (1 − loss) − 1
Losing half means the remaining half must double. This is why avoiding large drawdowns matters more than capturing every gain: the maths of recovery is steeply against you.
This is the arithmetic reason for position sizing, and it holds regardless of how good an idea looks. If a single position can lose 80% of its value and it is 60% of your portfolio, you have accepted a scenario that needs a 400% gain to undo. No amount of conviction changes that arithmetic.
Common belief
"Higher risk means higher returns."
What is actually true
Higher risk means a wider range of outcomes, including much worse ones. Compensation for bearing risk is expected, on average, over long periods, in efficiently priced markets — it is not delivered to any individual holder as a rule. Plenty of very risky assets simply lose money.
Portfolio A returns 8% every year. Portfolio B returns +40%, −20%, +40%, −20%, and so on. Their simple averages look similar. Compounded over four years, A turns $100 into $136. B turns $100 into $125.44 — and along the way B halves in value relative to its peak twice, which is when most people abandon a plan.