No-Arbitrage Identity for Forward Rates

Summary

The document presents an identity for forward rates at a given observation time: a forward rate spanning multiple periods is stated to equal the average of the sequence of one-period forward rates over the same horizon. It frames the identity as a no-arbitrage condition and asks how to derive it formally and what arbitrage would arise if it failed.

The document contains only the question and formula; it provides no derivation, trading strategy, example, or answer describing the arbitrage. The relation is therefore useful as a topic for studying consistency among forward rates, but the source alone does not establish its assumptions or conventions. In particular, it does not clarify whether rates are simple, continuously compounded, or otherwise defined, which matters when averaging rates across periods. A reader would need additional context to assess the exact identity and construct a valid arbitrage argument.

Key ideas

• The document states that a multi-period forward rate equals the average of consecutive one-period forward rates.
• It presents the relation as a no-arbitrage condition.
• It asks for a formal derivation and the arbitrage implied by a violation.
• No derivation or arbitrage example is supplied, and the compounding convention is unspecified.

Tags

# Identity for forward rates


# Identity for forward rates












In the context of interest models I came across the following identity for forward rates at time $m$ which, according to my book, has to always be fulfilled due to non-arbitrage:

$$f_m(t, t+s) = \dfrac{1}{s}\big[f_m(t, t+1) + \ldots + f_m(t + s - 1, t + s)\big]$$

The book did not mention why this is the case or even hinted at a derivation. Therefore, I have two questions:

- How do you derive this condition formally?

- What kind of arbitrage exists when the condition is not fulfilled?

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