Depict and derive the international Fisher effect

Question

Depict & derive the international Fisher effect.

The international Fisher effect can be acquired by combining the Fisher effect and the relative version of PPP in its expectational form. Particularly, the Fisher effect holds that

          E(∏$) = I$ - ρ$,

          E(∏£) = I£ - ρ£.

Supposing that the real interest rate is the similar between the two countries that means ρ$ = ρ£, and substituting the above results into the PPP, i.e., E(e) = E(∏$)- E(∏£), we get the international Fisher effect: E(e) = I_$ - I_£.