Jensen's Inequality

There's a trap that catches people constantly: they compute an average and then plug it into a formula, assuming the answer is the average of what the formula would have given. When the formula is a straight line, that's fine. When it curves, it's wrong — and Jensen's inequality tells you exactly which way it's wrong and by how much.

A function is convex if it bows upward, like a cup or a bowl — think $x^2$, or $e^x$, or $1/x$ for positive $x$. For a convex function $\varphi$, the average of the outputs always sits at or above the output of the average:

$$E\bigl[\varphi(X)\bigr] \ge \varphi\bigl(E[X]\bigr) .$$

If the function bows downward — concave, like $\log x$ or $\sqrt{x}$ — the inequality flips, and the average output sits at or below the output of the average. The two sides are equal only when the function is a straight line, or when $X$ has no randomness at all. Every bit of spread in $X$, combined with every bit of curvature in $\varphi$, opens a gap between the two — the Jensen gap.

The most important special case

Set $\varphi(x) = x^2$, which is convex. Jensen says $E[X^2] \ge (E[X])^2$, and the gap between the two is exactly the variance. So the fact that variance can never be negative is just Jensen's inequality in disguise. This is the template for everything else: curvature turns randomness into a systematic, one-directional bias.

Worked example: volatility drag

This is where Jensen pays rent for a trader. Suppose an investment gains $50\%$ one period and loses $50\%$ the next. The arithmetic average return is $\tfrac{(+50\%) + (-50\%)}{2} = 0\%$, which sounds like you broke even. But wealth compounds by multiplying, and $100 becomes

$$100 \times 1.5 \times 0.5 = 75 .$$

You're down $25\%$. Where did the money go? Growth over time is governed by the log of the return factors, and $\log$ is concave. Jensen (flipped, for concave) says the average of the logs is below the log of the average:

$$E\bigl[\log(1+R)\bigr] \le \log\bigl(1 + E[R]\bigr) .$$

The left side drives compound growth; the right side is what the naive arithmetic mean would promise. The gap between them is volatility drag — the steady tax that swings impose on compounded wealth. Per period here the geometric growth is $\sqrt{0.75} - 1 \approx -13.4\%$, a real loss, even though the arithmetic mean was a cheerful zero. This concavity gap is the mathematical heart of the The Kelly Criterion and of why lower-volatility paths compound faster.

Another quick one: averages of reciprocals

Because $1/x$ is convex for positive $x$, Jensen gives $E[1/X] \ge 1/E[X]$. So the average of reciprocals overshoots the reciprocal of the average — a trap when someone averages ratios, speeds, or per-unit costs and expects the "obvious" answer. If a route is driven at $30$ mph out and $60$ mph back, the average speed is not $45$ but the harmonic mean $40$ mph, and that is Jensen at work.

Where it shows up

Jensen underlies the arithmetic-mean-beats-geometric-mean fact, the existence of a risk premium (a risk-averse investor has concave utility, so they'll pay to remove variance — that's insurance), the non-negativity of KL divergence and entropy bounds, and the log-optimal growth logic behind position sizing. Any time a curved function meets a random input, Jensen is quietly setting the direction of the error.