seek the convex side

When the relationship between what you put in and what you get out bends instead of running straight, volatility stops being neutral and picks a side.

\[ \begin{aligned} \mathbb{E}[f(x)] &\ge f(\mathbb{E}[x]) \qquad (f \text{ convex}) \\ f(x) = x^2, \quad & x = 0 \text{ or } 10 \text{ with equal odds} \\ f(\mathbb{E}[x]) &= f(5) = 25 \\ \mathbb{E}[f(x)] &= \tfrac{1}{2}(0)^2 + \tfrac{1}{2}(10)^2 = 50 \end{aligned} \]

x is the uncertain input, f the payoff it produces, and E[x] the expected value of x, meaning its probability-weighted average. when f curves upward (convex), the average of the payoffs is at least the payoff of the average. here both have the same expected input of 5, but spreading x to 0 or 10 lifts the expected payoff from 25 to 50. cap the downside, keep the upside open, and the swings work for you.

spread of outcomes s 3.0

payoff of the average = 25, average of the payoffs = 34, the gap is 9 and it only grows with the swings.

the outcome averages 5 either way; s is how far it swings to each side. the black dot is the payoff of the average, the blue dot above it is the average of the two payoffs. on an upward-bending curve the blue always sits higher, and that gap is yours for free.

If your downside is small and capped while your upside stays open, the more the world shakes the better you do. If the curve runs the other way, the same shaking hurts you. So set things up where the worst that can happen is small and fixed and the best is open ended -- and then you actually want the swings that scare everyone else.