Deriving approx. Uncovered Interest Parity

We derive approximated version of Uncoverd Interest Parity (UIP) from original UIP with/without using logarism and Taylor expansion.

In other words, we convert

$$1+i=(1+i^{\ast })E^e/E$$

to

$$i\approx i^{\ast }+\frac{E^e-E}{E}$$

simple algebra

$$1+i=(1+i^{\ast })\frac{E^e}{E}$$

$$i=(1+i^{\ast })\frac{E^e}{E}-1$$

$$i=\frac{E^e}{E}+\frac{E^e}{E}{i}^{\ast }-1$$

$$i=\frac{E^e-E}{E}+\frac{E^e}{E}{i}^{\ast }$$

$$i=\frac{E^e-E}{E}+\frac{E^e}{E}{i}^{\ast }+(-\frac{E}{E}{i}^{\ast }+\frac{E}{E}{i}^{\ast })$$

$$i=\frac{E^e-E}{E}+\frac{E^e-E}{E}{i}^{\ast }+\frac{E}{E}{i}^{\ast }$$

$$\therefore i\approx i^{\ast }+\frac{E^e-E}{E}$$

The term ${\displaystyle \frac{E^e-E}{E}{i}^{\ast }}$ is very small and we can ignore it up to first order.

Using Taylor approximation

Note that $\log (1+x)\approx x$ for small $x$. (See below)

$$1+i=(1+i^{\ast })\frac{E^e}{E}$$

$$\log (1+i)=\log (1+i^{\ast })+\log \left(\frac{E^e}{E}\right)$$

$$i\approx i^{\ast }+\log (\frac{E^e-E}{E}+\frac{E}{E})$$

$$i\approx i^{\ast }+\log (1+\frac{E^e-E}{E})$$

$$i\approx i^{\ast }+\frac{E^e-E}{E}$$

Taylor expansion around $x=x^{\ast }$

Taylor expansion of $f(x)=\log (1+x)$ around $x^{\ast }=0$

See Also International Finance: Theory and Policy.