Siegel’s Paradox, Currency Expectations, and Forward Pricing

# Possibility of obtaining a positive mathematical expectation in a quoted currency


# Possibility of obtaining a positive mathematical expectation in a quoted currency












There is a currency pair C/USD = 1. C - currency in which I want to invest in order to make a profit in USD.

Suppose its price changes discretely: 50% - increases by 20%, 50% - decreases by 20%. This situation is similar to a game with a 50% probability of winning and k = 2. The expectation of such a game is 0.

But let's consider the reverse quote. If C = 1.2USD, then USD = 1/1.2C, and if C = 0.8USD, then USD = 1.25C. This situation is similar to a game with a 50% probability of winning and k = 2.5. The expectation here is positive, applying the Kelly criterion to this game the bank will grow with a geometric mean of 1/0.96.

This is due to the fact that the geometric mean of the base currency is less than one - 0.96. If we assume that its geometric mean is equal to 1, then a positive mathematical expectation can be obtained both in the base currency and in the quote currency. I wrote some simple Python code to test both hypotheses:

```
import random

rounds = 1000000
n = 1
avg_price = 0

for i in range(rounds):
  price = 1
  for j in range(n):
    if random.random() > 0.5: price *= 1.2
    else: price *= 0.8
  avg_price += price / rounds

print(avg_price)
```

If you try with values 1.2 and 0.8, then the average price will be very close to 1. If 0.8 is changed to 1/1.2, then the average price will converge to 1.0168. The second hypothesis seems strange, because it contradicts the unpredictability of prices.

Thus, I see a paradox here in that obtaining a positive mathematical expectation is simply inevitable in at least one of the currencies. Can I read about this somewhere?

## Answer by nbbo2 (score 1, accepted)

https://quant.stackexchange.com/a/78540

When Siegel wrote his famous article in 1972, a widely accepted theory of forward rates was the Unbiased Expectations Hypothesis, that the forward price (or futures price) of a currency was equal to the expectation of its value N months ahead.

Siegel showed that this could not be true in both directions, looking first at USD as the foreign currency with EUR as the domestic and then vice versa. This is because, as a consequence of Jensen's Inequality, the inverse of the expectation of X is not equal to the expectation of the inverse of X. So there seemed to be a conceptual problem with the theory.

However, empirical research in the 1980's and 1990's showed that the UEH theory did not hold up, and it was abandoned. Attention focused on another theory, the Covered Interest Parity theory that the forward price is determined by interest rates in both countries, with the currency at higher rates trading at a discount in the forward market. This theory can be applied in both directions. After 2005 the theory had to be modified to reflect another smaller factor called the Cross Currency Basis, in addition to interest rate effects. And here we are.

Now that expectations do not play any role in the theory of forward rates, the significance of Siegel's Paradox is much less. Yes, the expectations are not equal due to Jensen's inequality, but this does not really affect forward rate determination. Or AFAIK have any practical implications. (It is still intellectually interesting, I will admit).