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 "E[f(X)] ≥ f(E[X]) for convex functions and vice versa for concave functions," a theorem at the heart of nonlinearities where the order of averaging changes the result. Intuitively, the more convex the curve is bent before averaging (i.e., the more f is applied first), the larger the number becomes, and the smaller if it is concave. It follows from the geometry that the dividing line between two points lies on a convex function, and is generalized in probability theory in the form of an expectation value. The difference between the two sides, E[f(X)] - f(E[X]), is called the "Jensen gap" and is a measure of the degree to which nonlinearity affects the outcome.

This inequality systematically teaches "the phase where the simple average misrepresents reality. For example, since logarithms are concave functions, E[log(1+R)] ≤ log E[1+R]. Hence, compound growth (time averaging) diminishes with volatility for the same average return. The famous relationship "arithmetic mean > geometric mean" is actually a shadow picture of Jensen's inequality. The "volatility drain" in the investment world is a consequence of this concavity, and refers to the phenomenon where the effective growth rate of compound interest is reduced from the average return.

The same picture can be seen in theoretical models. In a geometric Brownian motion with normal price log, E[S] = exp(μ + σ²/2) is larger than exp(E[X]) due to the convexity of S = exp(X), while the time-averaged growth rate, which is of interest to investors, is reduced by the volatility to μ - σ²/2 due to the concavity of log µ - σ²/2 due to the concavity of log. Again, "which side to average on" is decisive and gives the reason for the divergence between the expected value (one time average) and the time average (compound interest).

The range of applications is wide: AM-GM (arithmetic mean ≥ geometric mean) is only a special case of applying Jensen to the concavity of log. At the textbook level of probability theory, one can derive Var(X)=E[X²]-E[X]² ≥ 0 from the convexity of g(x)=x². Such a technique of deciphering the mean relationship from the "shape of the function" also leads to a practical calculation procedure that, when combined with LOTUS (Law of the Statistician's Unconscious), allows direct evaluation of E[g(X)] as long as the distribution is known.

The implications for strategy design are clear. In convex regions (where f bends upward), the more you average through nonlinearities first, the more you gain, and vice versa in concave regions. Taleb's emphasis on "buying convexity and selling concavity" and "bimodality (barbell) over moderation" are precisely the translation of Jensen's order effect into profit/loss shape. The Kelly criterion for maximizing long-term growth also stands in this framework in that it assumes concavity of log and gives "a betting strategy that maximizes the geometric mean (time mean).

In short, there is more than one average. Which function "bends" the world and then averages it--this order determines the meaning of risk and the substance of reward. Jensen's inequality is a practical lesson in designing "shape" rather than forecasting, a nonlinear compass that turns variation into friend or foe.