r t = P t + 1 B P t + 1 − P t B P t P t B P t = P t + 1 B P t + 1 P t B P t − 1 = ( P t + 1 B P t + 1 P t P t B ) − 1 = ( P t + 1 B P t B P t P t + 1 ) − 1 {\displaystyle r_{t}={\frac {{\frac {\!P_{t+1}^{B}}{\!P_{t+1}}}-{\frac {\!P_{t}^{B}}{\!P_{t}}}}{\frac {\!P_{t}^{B}}{\!P_{t}}}}={\frac {\frac {\!P_{t+1}^{B}}{\!P_{t+1}}}{\frac {\!P_{t}^{B}}{\!P_{t}}}}-1=\left({\frac {\!P_{t+1}^{B}}{\!P_{t+1}}}~{\frac {\!P_{t}}{\!P_{t}^{B}}}\right)-1=\left({\frac {\!P_{t+1}^{B}}{\!P_{t}^{B}}}~{\frac {\!P_{t}}{\!P_{t+1}}}\right)-1}
We know that :
i t = P t + 1 B − P t B P t B = P t + 1 B P t B − 1 ⇒ i t + 1 = P t + 1 B P t B {\displaystyle i_{t}={\frac {\!P_{t+1}^{B}-\!P_{t}^{B}}{\!P_{t}^{B}}}={\frac {\!P_{t+1}^{B}}{\!P_{t}^{B}}}-1\quad \Rightarrow \ \quad i_{t}+1={\frac {\!P_{t+1}^{B}}{\!P_{t}^{B}}}}
and
Π t = P t + 1 − P t P t = P t + 1 P t − 1 ⇒ Π t + 1 = P t + 1 P t ⇒ P t P t + 1 = 1 Π t + 1 {\displaystyle \Pi _{t}={\frac {\!P_{t+1}-\!P_{t}}{\!P_{t}}}={\frac {\!P_{t+1}}{\!P_{t}}}-1\quad \Rightarrow \ \quad \Pi _{t}+1={\frac {\!P_{t+1}}{\!P_{t}}}\quad \Rightarrow \ \quad {\frac {\!P_{t}}{\!P_{t+1}}}={\frac {1}{\Pi _{t}+1}}}
So, we have :
r t = ( i t + 1 ) ( 1 Π t + 1 ) − 1 = ( i t + 1 Π t + 1 ) − 1 = ( i t + 1 ) − Π t − 1 Π t + 1 = i t − Π t Π t + 1 {\displaystyle r_{t}=(i_{t}+1)\left({\frac {1}{\Pi _{t}+1}}\right)-1=\left({\frac {i_{t}+1}{\Pi _{t}+1}}\right)-1={\frac {(i_{t}+1)-\Pi _{t}-1}{\Pi _{t}+1}}={\frac {i_{t}-\Pi _{t}}{\Pi _{t}+1}}}
Finally,
r t ≈ i t − Π t i f i t − Π t Π t + 1 ≈ i t − Π t i f Π t + 1 ≈ 1 ⇒ Π t ≈ 0 {\displaystyle r_{t}\approx i_{t}-\Pi _{t}\quad i\!f\quad {\frac {i_{t}-\Pi _{t}}{\Pi _{t}+1}}\approx i_{t}-\Pi _{t}\quad i\!f\quad {\Pi _{t}+1}\approx 1\quad \Rightarrow \quad {\Pi _{t}}\approx 0}
