Siegel's paradox

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Siegel's paradox is the idea that when we aren’t sure what future prices will be, rational people might temporarily trade away their favorite goods (like apples for oranges or dollars for euros) with the plan to trade back later once prices are clearer. The surprising part is that this can look like a real gain on average, even though no one has made a mistake.

A classic apple-oranges version: today the exchange rate is 1 apple = 1 orange. Next year it could swing to 2 oranges per apple or 1 apple per 2 oranges, each with 50/50 probability. If an apple lover trades an apple for an orange now, next year the orange they got is worth an expected 1.25 apples, and the apple they gave up is worth an expected 1.25 oranges. So both sides seem to gain on average. This apparent gain comes from a mathematical idea called Jensen’s inequality, which means that “averages of outcomes” can behave oddly when the future is uncertain.

A wine example with Americans and Germans shows a similar point. In November the two wines trade 1:1, but in December the trendiness of the wines could flip so that the fashionable wine becomes 2:1 or the other way around, with equal chances. Loyalists who only drink their own nationality’s wine can end up with an expected advantage (about 1.25 of their own wine) by temporarily trading away from their preferred drink, hoping to buy more if its price drops later. Trendy buyers who don’t care about nationality, however, often don’t gain anything real from a similar strategy—the apparent upside is usually balanced by possible downsides.

What this means for real-world investing is nuanced. Sometimes you should take on a bit of currency risk rather than hedge everything away, especially when price uncertainty isn’t perfectly explained by inflation or purchasing-power differences. Other times, the risk isn’t worth it and you won’t gain by waiting on future rates.

Some researchers, like Mallahi-Karai and Safari, propose a way to avoid guaranteed arbitrage from future price bets: settle on a single fair rate calculated from the weighted geometric mean of the possible future rates (possibly with a simple reciprocity adjustment). In the apple-orange case, this approach would give a common, risk-free exchange rate that prevents one side from gaining just by hoping prices swing a certain way.