I am trying to derive the log-linear version of the uncovered interest rate parity under complete asset markets.

I know that the UIP condition is given by


I have seen in papers that the log-linear version is simply given by


but I cannot see how this is derived. According to me, the log-linear version should be


Could someone please write down all steps to obtain the log-linear version of the UIP?

Thank you very much


1 Answer 1


It’s derived as follows. First start with original equation.


Take natural logs of both sides:

$$\ln(1+i_t)=\ln(1+i^*_t)+\ln(S_{t+1}) -\ln(S_t)$$

Now you just use the following:


And use the well known fact that for small values of $i$ the following approximation holds:

$$\ln(1+i)\approx i$$

And rearrange to get


PS: Also actually the equation you derived is not log-linearized as it’s not linear in its log parameters


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