The Merchant's Paradox — Volatility's Tax & Diversification's Free Lunch

— A Note from 1738 —

In 1738 in St. Petersburg, the Swiss mathematician Daniel Bernoulli wrote a paper on a merchant's puzzle: a cargo from Amsterdam might arrive at a fortune or sink at the bottom of the Baltic. The expected money-value, Bernoulli noticed, was beside the point. Wealth's value is not linear. Doubling from 1,000 to 2,000 rubles means more than doubling from 100,000 to 200,000. Volatility is therefore not just risk — it is a tax on compounding. Two and a half centuries later, the same arithmetic still bankrupts gamblers and rewards the diversified.

The Merchants

Boris

— The Brave —

Each year, Boris stakes all his wealth on a single voyage. +60% if the cargo arrives. −40% if the ship sinks. He thinks the expected return is what matters.

Expected: +10%/yr · Realized: −2%/yr

Stefan

— The Steady —

Stefan keeps his wealth in Hanseatic T-bills earning a steady +2% a year. No voyages. No drama. He earns less than Boris's expected return — but more than Boris's realized one.

Expected: +2%/yr · Realized: +2%/yr

Hanna

— The Prudent —

Each year, Hanna splits her wealth across two independent voyages — same odds as Boris's, just two ships instead of one. Same expected return per ruble at risk. Different volatility.

Expected: +10%/yr · Realized: +3.4%/yr

II.

The Stage

— Before You Watch · Place Your Bet —

Three merchants. Each starts with $10,000. Each has the same +10% expected return per ruble at risk (Stefan: +2%). After thirty years — who has the most?

Boris bets full on one voyage. Hanna splits across two. Stefan stays in T-bills. The mathematics of compounding favors one of them more than feels obvious.

A. Boris — full exposure to the +10% bet wins long-run B. Stefan — boring +2% beats both gamblers C. Boris > Hanna > Stefan — more risk, more reward, in order D. Boris & Hanna tie — same expected return, same outcome E. Hanna > Stefan > Boris — diversification beats both

Wealth Over Time · 30 Voyages | St. Petersburg, 1738 — 1768

Boris — full bet, +60% / −40%

Stefan — T-bills, +2%/yr

Hanna — split across two voyages

Year

1738

VOYAGE 0 / 30

Merchant	Balance	Realized CAGR	This year
Boris	$10K	—	—
Stefan	$10K	—	+2% T-bill
Hanna	$10K	—	—

Scrub

— The Verdict —

…

III.

The Reckoning

1768 Same expected return. Same thirty years. The merchant who took on the most risk ends with the least. The merchant who diversified the same risk across two ships ends with the most. The arithmetic mean lied. The geometric mean told the truth.

Merchant	Strategy	Arithmetic E[r]	Geometric (CAGR)	Balance · 1768
Boris	Full bet, one voyage per year	+10.0%	—	—
Stefan	Hanseatic T-bills, +2% guaranteed	+2.0%	—	—
Hanna	Split across two independent voyages	+10.0%	—	—