Is fisher equation a definition, identity? Or is it rather a very good estimate So let us agree that the fisher equation is $1 + i = (1 + r)(1 + \pi).$ Is it a definiton or good estimate. Intuition tells me this make so much sense and almost qualify as an equation that must hold. However, sources online tell me that it is a good approximation. If we accept that, can anyone suggest examples when the equation fails and why? Thank you!
The fisher equation has its basis in the fact that the real return on an asset is the nominal return divided by the inflation rate. If you hold a bond today, it gives you back $1+r_{t+1}$ tomorrow. This is basically $\frac{1+\iota_{t}}{1+\pi_{t+1}}$ such that the promised nominal rate is deflated by the inflation rate. Rearranging terms, you get your definition. This is exact and always holds conceptually. The approximation is that $i\approx r+\pi$  which is obtained via a Taylor series expansion.