How to derive the equation $\text{Present value} = \text{profit} \times \dfrac {1-e^{-\text{interest rate}\times \text{time}}}{\text{interest rate}}$

Question

How can I derive this equation:

$\text{Present value} = \text{profit} \times \dfrac {1-e^{-\text{interest rate}\times \text{time}}}{\text{interest rate}}$

Answer 1

We can compute the present value of a continuous income stream that will generate $I(t)$ dollars in year $t$, with interest compounded continuously at an annual rate $r$. More specifically, if $I$ open an account that gives interest compounded continuously at the rate $r$, with the objective of earning $I(t)$ dollars in year $t$ for every year until the year $T$, the money that $I$ need to put in today is $$ \text{present value of a continuous income stream} =\int_0^T I(t)e^{-rt}\mathrm d t $$ If $I(t)$ is constant and equal to the profit $p$ we have $$ PV=p\times \int_0^T e^{-rt}\mathrm d t=p\times\frac{1-e^{-rT}}{r} $$