Curvature is destiny - Jensen's inequality and the meaning of variation (November 2025)

Curvature is destiny - Jensen's inequality and the meaning of variation (November 2025)

Jensen's inequality is a theorem that states a non-linear property whose result depends on the order in which the averages are taken. It reveals that E[f(X)] ≥ f(E[X]) for convex functions and the converse holds for concave functions, indicating that variability affects the result. This inequality is especially important in finance and economics because it teaches the danger that simple averages misrepresent reality. For example, due to the concavity of the logarithmic function, the compound growth rate (time average) decreases with volatility. The so-called "volatility drag" or "arithmetic mean > geometric mean" relationship is a consequence of this inequality. Moreover, since fluctuations in convex functions push up expectations and those in concave functions push down expectations, which "curvature" one views the world with determines the meaning of risk and the reality of reward. Taleb developed this as the "buy convexity, sell concavity" philosophy, which he used as th
e theoretical basis for his barbell strategy. Jensen's inequality is not just a mathematical fact, but as "the wisdom of designing shapes," it is at the heart of the anti-fragility that makes fluctuation an ally.