Jensen's Inequality
Jensen's Inequality is a fundamental theorem of probability and convex analysis stating that for a convex function φ and a random variable X, the expectation of the function is greater than or equal to the function of the expectation: E[φ(X)] ≥ φ(E[X]), with strict inequality when X is non-degenerate (has positive variance) and φ is strictly convex. In finance, Jensen's Inequality underlies convexity adjustments in fixed income, explains why the arithmetic mean return exceeds the geometric mean return, and provides theoretical grounding for the value of optionality.
Key takeaways
- For a convex function φ(x), Jensen's Inequality states E[φ(X)] ≥ φ(E[X]); for a concave function, the inequality reverses: E[φ(X)] ≤ φ(E[X]).
- Bond price as a function of yield is convex: E[P(y)] > P(E[y]), meaning that the expected bond price in a world of uncertain yields exceeds the price computed at the expected yield—the 'convexity advantage' of bonds.
- The geometric mean return always falls below the arithmetic mean return due to Jensen's Inequality (ln is a concave function of returns): geometric mean ≈ arithmetic mean - σ²/2, where σ² is return variance.
- Jensen's Inequality justifies the theoretical value of financial options: an option payoff is a convex function of the underlying price, so E[max(S-K,0)] > max(E[S]-K, 0)—options have positive expected value even when the forward price equals the strike.
- Convexity adjustments in interest rate derivatives (futures vs. forwards, CMS rates vs. swap rates) are direct applications of Jensen's Inequality to the non-linear price-yield or payment-rate relationships.
Explanation
Jensen's Inequality, formulated by Danish mathematician Johan Jensen in 1906, is one of the most widely applied theorems in probability theory and mathematical finance. Its central insight—that the expected value of a non-linear function of a random variable differs from the function of the expected value—appears throughout quantitative finance wherever convexity or concavity creates differences between expected outcomes and point estimates.
The mathematical statement is precise: if φ is a convex function (φ''(x) ≥ 0 everywhere) and X is a random variable with finite expectation, then E[φ(X)] ≥ φ(E[X]). The inequality is strict if φ is strictly convex and X has positive variance. For concave functions (φ''(x) ≤ 0), the inequality reverses: E[φ(X)] ≤ φ(E[X]). Geometrically, convexity means the function lies below any chord connecting two points on the curve; this means the expected value of the function, which averages across multiple realizations, lies above the function of the average of those realizations.
The bond price-yield relationship provides the most important application in fixed income. Bond price P is a strictly convex function of yield y: P = Σ CF_i / (1+y)^i has positive second derivative (∂²P/∂y² > 0). By Jensen's Inequality, E[P(y)] > P(E[y]): if yields are random, the expected price of a bond is higher than the price computed at the expected yield. This convexity advantage benefits holders of long-duration bonds: in a world of yield uncertainty, the asymmetric price-yield relationship means that yield decreases cause larger price increases than equivalent yield increases cause price decreases, creating a positive expected price return from convexity even if yields are expected to remain constant on average.
In derivatives pricing, Jensen's Inequality explains why the value of an option is strictly positive even when it is at-the-money (forward price equals strike). The option payoff max(S-K, 0) is a convex function of S. By Jensen's Inequality, E[max(S-K, 0)] > max(E[S]-K, 0). If E[S] = K (at-the-money forward), then max(E[S]-K, 0) = 0, but E[max(S-K, 0)] > 0 because of S's randomness. The option has positive expected payoff—and hence positive value—precisely because its payoff function is convex in the underlying price. This is the mathematical foundation of the time value of options.
Convexity adjustments in interest rate derivatives arise directly from Jensen's Inequality. The most common example involves Eurodollar futures versus FRA (Forward Rate Agreement) pricing. A Eurodollar futures contract settles linearly in rates (daily mark-to-market on a futures P&L of $25 per basis point), while a FRA settles non-linearly (the payment is discounted at the actual rate, creating a convex relationship). Because futures settlement is linear and FRA payoff is convex in the future rate, Jensen's Inequality implies futures rates differ from forward rates by a convexity adjustment. For long-dated contracts (10+ year expiry), this adjustment can be substantial—tens of basis points—and must be applied when bootstrapping forward rate curves from Eurodollar or SOFR futures prices.
Jensen's Inequality also explains the arithmetic-geometric mean return gap, crucial for long-term investment analysis. The geometric mean return G satisfies: G = (Π(1+r_t))^(1/T) - 1, while the arithmetic mean A = (Σr_t)/T. Since the log function is concave, Jensen's Inequality gives E[ln(1+r)] ≤ ln(1+E[r]), implying G ≈ A - σ²/2 for continuously compounded returns. For a portfolio with 15% arithmetic annual return and 20% annual volatility, the geometric mean is approximately 15% - (0.20²/2) = 13%. Over long horizons, the geometric mean—not the arithmetic mean—determines terminal wealth, making this Jensen's Inequality application directly relevant for long-term portfolio construction and asset allocation decisions.
Formula
Jensen's Inequality: E[φ(X)] ≥ φ(E[X]) for convex φ; Geometric-Arithmetic Mean: G ≈ A - σ²/2 (log-normal returns)
Example
Consider a zero-coupon bond with 10-year maturity and a yield that is either 3% or 7% with equal probability (expected yield = 5%). The bond prices are: P(3%) = 100/(1.03)^10 = 74.41 and P(7%) = 100/(1.07)^10 = 50.83. Expected bond price = (74.41 + 50.83)/2 = 62.62. However, P(E[y]) = P(5%) = 100/(1.05)^10 = 61.39. By Jensen's Inequality, E[P(y)] = 62.62 > P(E[y]) = 61.39—a convexity advantage of $1.23 per $100 face value. A bond trader who valued the bond at $61.39 (using the expected yield) would underprice it by $1.23, ignoring the convexity benefit from yield uncertainty. This convexity advantage is larger for longer-duration bonds and higher yield volatility—precisely the conditions where bond convexity is most commercially significant.
Related terms
Asset Allocation At The Money Basis Bond Convexity Convexity Adjustment Copula Duration Eurodollar Face Value Forward Rate Agreement Futures Contract