The reason a compound rate deceives over long periods

Seven per cent a year sounds like a modest figure and a small annual change. Over thirty years it multiplies the quantity by more than seven. Asked to estimate the endpoint without arithmetic, most people give a number closer to three.

What everyone sees

A percentage is treated as a description of size — seven per cent is small, so its consequences should be small. The rate describes a process, and the process feeds its own base. Intuition extrapolates by repeated addition, which tracks the curve closely for a few periods and then diverges without limit.

What is actually happening

Wagenaar and Sagaristano tested extrapolation of exponential series and found systematic underestimation that grew with the number of periods, with performance improving only slightly when participants were told the growth was exponential. Stango and Zinman documented the same bias in household decisions, finding that people who linearised growth systematically misjudged the long-run consequences of repeated rates, and that the tendency was stable enough to predict behaviour across unrelated choices. Nothing here is about arithmetic ability. The error persists in people who can compute the answer when asked to, because the intuitive estimate arrives first and feels sufficient, and nobody reaches for a calculator to check a figure that already seems reasonable.

Why it stays hidden

The gap hides because short-horizon experience confirms the linear intuition. Over three or four periods the two curves are nearly identical, so the habit of extending in a straight line is repeatedly rewarded, and the divergence only becomes visible at horizons long enough that nobody is checking their earlier estimate.