Why Reciprocal Exchange Rates Can Have Different Expectations

Summary

The question compares expected EUR/USD and USD/EUR exchange rates under a two-outcome scenario. It assigns equal probability to EUR/USD ending at 1.25 or 0.8, then observes that each outcome's reciprocal is the corresponding USD/EUR quote. The apparent puzzle is that averaging the two quotes in either direction seems to imply both currencies appreciate.

The key issue is that expectation does not commute with taking a reciprocal: the expected value of an inverse exchange rate is generally not the inverse of the expected exchange rate. In the stated scenario, each quote has an arithmetic mean of 1.025 in its own units, but the two quoted rates describe reciprocal quantities and cannot both be read as evidence that both currencies strengthen. The example highlights the importance of quote convention and nonlinear transformations. It does not specify a pricing measure, interest rates, or a broader forecast framework, so these arithmetic expectations alone do not establish a tradable view.

Key ideas

• An exchange rate and its reciprocal use opposite quote conventions.
• The expectation of a reciprocal is generally not the reciprocal of the expectation.
• Averaging each quote in its own units can produce values above the starting quote in both directions.
• Quote direction must be checked before interpreting currency appreciation or depreciation.
• The two-outcome arithmetic example is not by itself a market forecast or valuation framework.

Tags

# what is the exchange rate Expectation? I am confused about it


# what is the exchange rate Expectation? I am confused about it












Suppose now the rate of EUR/USD is 1. and in half-year, maybe become 1.25EUR/USD, or 0.8 EUR/USD, the probability of each case is 0.5. what's the Expectation of EUR/USD in half-year? And, what's the Expectation of USD/EUR in half-year? answer 1:

```
E(EUR/USD) = 0.5*1.25 + 0.5*0.8 = 1.025 EUR/USD。
```

answer 2: since 1.25EUR/USD is equivalent to 0.8 USD/EUR , and 0.8EUR/USD is equivalent to 1.25 USD/EUR, so

```
E(USD/EUR) = 0.5*1.25 + 0.5*0.8 = 1.025 USD/EUR。
```

Both USD and EUR ate expected to be appreciated. It's confusing me.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.