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\begin{align*}BW_0=\frac{(1+i)^T-1}{i(1+i)^T}r&\Longrightarrow\frac{iBW_0}{r}=\frac{(1+i)^T-1}{(1+i)^T}=1-\frac1{(1+i)^T}\Longrightarrow\frac1{(1+i)^T}=1-\frac{iBW_0}r=\frac{r-iBW_0}{r}\\&\Longrightarrow(1+i)^T=\frac r{r-iBW_0}\Longrightarrow T\ln(1+i)=\ln\left((1+i)^T\right)=\ln\left(\frac r{r-iBW_0}\right)\\&\Longrightarrow T=\frac{\ln\left(\frac r{r-iBW_0}\right)}{\ln(1+i)}\end{align*}
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