The Expectation Hypothesis and Which Probability Measure Applies

Summary

The document asks how to interpret the expectation hypothesis for interest rates. It distinguishes today’s one- and two-year zero-coupon rates from a one-year forward rate that begins in a year, then asks whether compounding links the two-year accumulation factor to the one-year spot and forward factors. It further asks whether the expected future short rate can be inferred directly from that forward rate, and under which probability measure such an expectation should be taken.

No answer or supporting evidence is included, so the document does not resolve the equation or specify a measure. The conceptual issue is that a forward rate is implied by current market prices, while an expectation of a future spot rate depends on the hypothesis and the measure being used; risk premia can make these differ. The question is a useful prompt for distinguishing the expectations hypothesis from risk-neutral pricing, but readers need an external explanation to obtain a complete derivation or conclusion.

Key ideas

• The question relates spot rates, a forward rate, and compounded accumulation factors.
• It asks whether a forward rate predicts a future spot rate under the expectation hypothesis.
• The choice of probability measure matters when defining an expectation of future rates.
• The document provides no answer, derivation, or evidence resolving these questions.

Tags

# Expectation hypothesis, expectation under which measure?


# Expectation hypothesis, expectation under which measure?












As I understand the expectation hypothesis says that the implied forward rate, can be used to predict future spot rates?

If $r_{0,2}$ is the rate for a zero coupon bond maturing in two years, and the same with $r_{0,1}$. And $r_{1,2}$ is the rate fora zero coupon bond sold at time $1$ and maturing at time $2$. Do we then have

$$1+2r_{0,2}=(1+r_{0,1})(1+r_{1,2})?$$

And by taking expectation we have $$E[r_{1,2}]=\frac{1+2r_{0,2}}{1+r_{0,1}}-1?$$

From what I understand there are 3 possible ways to take the expectation, it is under the real world measure, the risk neutral measure or the forward measure. Which is used?

What exaxtly does the expectation hypothesis mean?

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