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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:

  1. How do you derive this condition formally?
  2. What kind of arbitrage exists when the condition is not fulfilled?
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  • $\begingroup$ Can you elaborate a bit on the notation. If $m$ is the time, what are $t$ and $s$? Also, what book is this? Simple or compound rate? $\endgroup$ – Raskolnikov Nov 27 '17 at 17:46
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    $\begingroup$ $m$ is the current point in time. $t$ is the beginning of the period in consideration and $s$ its length, i.e. $t + s$ is the second point in time. I read it in a German textbook, I'll look up the title. It's a compound interest rate as far as I know. $\endgroup$ – Taufi Nov 27 '17 at 18:19
  • $\begingroup$ Usually this kind of thing is written multiplicatively. But it could be written additively like this (1) as an approximation, (2) if the f's are logarithms of interest rates, i.e. $f=\ln(1+r)$ $\endgroup$ – noob2 Nov 27 '17 at 22:24

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