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I am trying to understand under what circumstances or transformations would $[1+(E_2-E_1)*\frac{d}{360}]$ equal to $(\frac{1+E_2}{1+E_1})^{\frac{d}{360}}$.

For context, $E_2, E_1$ are interest rates.

Any help is very much appreciated.

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  • $\begingroup$ They are only approximately equal if $E_1,E_2,\frac{d}{360}$ are small enough. The first expression is a linearization of the other. $\endgroup$ – noob2 Sep 23 at 2:13
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They're not equivalent, but you can use log identities to derive something similar after applying a log.

Eg,

$ln\left(\left(\frac{1+E_2}{1+E_1}\right)^{\frac{d}{360}}\right)$

$\frac{d}{360}*\left(1+E_2-\left(1+E_1\right)\right) $

$\left(E_2-E_1\right)*\frac{d}{360}$

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