# Compounding Equivalence

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.

• 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. – noob2 Sep 23 '20 at 2:13

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