Siegel's paradox

Siegel's Paradox is a way of investing in foreign investments to make money. The technique works (mathematically), but it is called a paradox because it sounds as if it should not. Fischer Black credits the idea to Jeremy Siegel.^[1]

It is closely related to the two envelopes problem.

Example

Assume two countries, where one eats only apples and one eats only oranges.

These countries have their own currencies and the initial exchange rate is 1:1. But in the future it will change to either 1:2 or 2:1, with equal probability.

The apple country could buy all the apples now at 1:1 and the orange country could buy all the oranges at 1:1 and they'd have no risk for the future.

However, if the apple country buys an orange now instead of an apple, what will happen to its value? If the exchange rate goes 1:2, it can get 0.5 of an apple. If the exchange rate goes to 2:1, it can get 2 apples. So the expected value of the orange is (0.5 + 2)/2 = 1.25 apples!

Why?

Reportedly, this is due to the (expected value of 1/x) being not equal to 1/(expected value of x).

Consequences

Fischer Black concluded that when investing overseas, investors should not hedge all their currency risk.

References