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How can I derive this equation:

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

For the love of maths
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user19506
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  • will you please ,define all the parameters?What do they mean by $r$ and $t$? – Rakibul Islam Prince Nov 10 '18 at 07:45
  • r is interest rate and t is time. – user19506 Nov 10 '18 at 07:46
  • I forgot, sorry.... – user19506 Nov 10 '18 at 07:47
  • No idea, could you maybe elaborate on what all of these terms mean (I have a vague understanding, but I'm not an economist). You have an expression in the form: $$s=p \frac{1-e^{-rt}}{r}$$ which is likely obtained by integration: $$s=p \int_0^t e^{-r \tau} d \tau$$ Why is it so? The answer is likely lying in the economics textbook, not a calculus one – Yuriy S Nov 10 '18 at 11:50

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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 $$ PV=p\times \int_0^T e^{-rt}\mathrm d t=p\times\frac{1-e^{-rT}}{r} $$

alexjo
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