Posted March 15, 2012 By Presh Talwalkar. Read about me , or email me .

Exchange rates always seemed simple to me, until I learned a bit more finance.

There are some quirky things that can happen, as today’s problem will illustrate.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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Exchange rate problem

Bob frequently travels betwen the U.S. and Canada, so he keeps an eye out for exchange rate changes.

Currently the exchange rate is 1 ($/C$), so Bob can trade 1 dollar U.S. for 1 dollar Canadian.

The market expects that in one year there is a 50 percent chance the U.S. dollar will be stronger by 10 percent, and a 50 percent chance the U.S. dollar will be weaker by 10 percent.

Question: what is the expected exchange rate on year from now? Solve the problem for both a U.S. and Canadian investor.

Siegel’s paradox

If Bob lived in America, he would see the problem as follows. There is a 50 percent chance the exchange rate will be 1.1 ($/C$), and there is a 50 percent chance the exchange rate will be 0.9 ($/C$).

Thus, the expected exchange rate one year from now would be:

E($/C$) = (0.5)(1.1) + (0.5)(0.9) = 1

As an American investor, Bob expects the future exchange rate to be 1 ($/C$).

But what if Bob lived in Canada? How would he see the problem?

In that case, he would have the following numbers in terms of Canadian dollars per U.S. dollar. There is a 50 percent chance the exchange rate will be 1/1.1 (C$/$), and there is a 50 percent chance the exchange rate will be 1/0.9 (C$/$).

Thus, the expected exchange rate one year from now, in $/C$, would be this reciprocal:

1 / E(C$/$) = 1 / [(0.5)(1/1.1) + (0.5)(1/0.9)] = 0.99

If Bob lived in America, he expects the future rate to be 1 ($/C$). But living in Canada, using the same assumptions, the future rate would be 0.99 ($/C$).

The exchange rates are very close, but they are not the same! Why is there any difference?

It turns out this same phenomenon happens for different expectations of how the currency might change. This is known as Siegel’s paradox.

Why Siegel’s paradox happens

There is a mathematical explanation for why the two expectations are not equal.

Specifically, if the exchange rate is the random variable X, then the American investor calculates E(X). The Canadian investor is calculating 1/E(1/X).

These two values will not be equal due to something called Jensen’s inequality, which implies that

E(X) > 1/E(1/X) or alternately E(1/X) > 1/E(X)

The math explanation is the easier part of resolving the paradox. The bigger question is whether investors can profit on this information.

The long and short is there is no real economic impact of Siegel’s paradox. It turns out to be a monetary illusion. What happens in real markets is that prices and interest rates in both countries typically adjust to prevent such profitable anomalies.

For more on Siegel’s paradox, consult this paper, or check out the first chapter of the book Puzzles of Finance.