[jason1990]

You are measuring one stochastic process in terms of another, which tends to complicate things. It may be clearer if you measure both USD and EUR in terms of some fixed quantity, say the 2000 USD. Let n denote the year 2000 + n, and let U_n denote the value of the USD at time n, measured in 2000 USDs. In particular, U_0 = 1. Suppose that

U_n = (1 + X_n)U_{n-1},

where {X_n} is an iid sequence with mean r. For simplicity, assume X_n = r + 0.1 or X_n = r - 0.1 with equal probability. Note that we can write

U_n = U_0*e^{S_n},

where

S_n = \sum_{j=1}^n \log(1 + X_j).

The long term behavior of U_n is determined by the long term behavior of S_n. Since S_n is a random walk, its long term behavior is determined by whether E[\log(1 + X_j)] is positive, negative, or 0.

Now let V_n denote the value of the EUR at time n in 2000 USDs. Suppose that

V_n = (1 + Y_n)V_{n-1},

where the sequence {Y_n} is independent of and has the same distribution as {X_n}. We can write

V_n = V_0*e^{T_n},

where

T_n = \sum_{j=1}^n \log(1 + Y_j).

You are interested in the exchange rates W_n = U_n/V_n and M_n = V_n/U_n. Note that

W_n = W_0*e^{S_n - T_n}.

Since S_n - T_n is a random walk with mean 0, W_n will not converge to zero. Similarly, neither will M_n. Interestingly, we can write

W_n = [(1 + X_n)/(1 + Y_n)]W_{n-1}
= (1 + Z_n)W_{n-1},

where

Z_n = (X_n - Y_n)/(1 + Y_n).

The mean growth rate of W_n is E[Z_n], and you can check for yourself that E[Z_n] is positive. Similarly, the mean growth rate of M_n is also positive. In other words,

E[(1 + X_n)/(1 + Y_n)] and
E[(1 + Y_n)/(1 + X_n)]

are both greater than 1. (This is surprising only if you forget that the expected value of a reciprocal is not the reciprocal of the expected value.)

[ QUOTE ]
Okay so basically we buy 100 USD worth of EUR today, and put the EUR in the bank earning interest.

We buy back USD once the exchange rate is such that we make ten times as much USD than we would have if we had just put it into a US bank account to begin with
...
This seems counterintuitive - it's like we are beating the perfectly efficient market. Thoughts?

[/ QUOTE ]
For simplicity, assume that interest rates are 0. At the time you make your trade, the USD is worth X 2000 USDs and the EUR is worth 100X 2000 USDs. If you have Y EUR, then you have something worth 100XY 2000 USDs. If you convert, then you will have 100Y USDs, which is again something that is worth 100XY 2000 USDs. Whether you convert back to USD or keep it in EUR, you have the same thing. Trading back and forth in this setting corresponds to sometimes following the random walk S_n, then sometimes following T_n, then back to S_n for a while, and so on. The result is just another random walk.