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Please take a look at the following snippet of a paper. According to the underlined part, eq(3)(4) can be obtained by taking logs of eq(1)(2), so by taking log of eq(1) (and ignore expectation) I get

$$\frac{1+r_{t}}{1+r_{t}^{i}}=\frac{S_{t+1}^{i}}{S_{t}^{i}}\Rightarrow$$ $$ln(\frac{1+r_{t}}{1+r_{t}^{i}})=ln(\frac{S_{t+1}^{i}}{S_{t}^{i}})=ln(S_{t+1}^{i})-ln(S_{t}^{i}) $$

However please help me understand how to convert $$ln(\frac{1+r_{t}}{1+r_{t}^{i}})$$ to $$r_{t}-r_{t}^{i}$$

Thanks a lot!

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user39086
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    It looks like they are using the approximation $\ln(1+x)\approx x$. This is valid when $x$ is small, as interest rates usually are. – Mike Earnest Jun 18 '18 at 13:19
  • Thanks @MikeEarnest Please make your comment an answer if you like, so that I can mark it as the best answer. – user39086 Jun 19 '18 at 09:59

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