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Question

# Find the future value of an income stream of \$2000 per year, deposited into an account paying 2% interest per year, compounded continuously, over a 15-year period.

Solution

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The deposit of $P(t) \Delta t$ has a time period of $(M-t)$ years to earn interest and

Future Value of Money deposited inn interval $t \text { to } t+\Delta t \approx[P(t) \Delta t] e^{r(M-t)}$

Summing all over subintervals

$\text { Total future value } \approx \sum P(t) \Delta t e^{r(u-t)} \text { dollars. }$

As the length of the subdivisions tend towards zero

$\text { Future value }=\int_{0}^{M} P(t) e^{(u-1)} d t \quad \text { dollars }$

In other words

$\text { Future Value }=e^{n t} . \text { Present Value }$

Now calculating future value

\begin{align*} \int_{0}^{10} 1000 e^{0.08(10-0)} d t \text { dollars }&=1000 e^{0.8} \int_{0}^{10} e^{-0.08 t} d t \text { dollars }\\ &=1000 e^{0.8}\left[\frac{-e^{-0.88 t}}{0.08}\right]_{0}^{10} \text { dollars }\\ &=-\frac{1000 e^{103}}{0.08}\left[e^{-0.8}-1\right] \text { dollars }\\ &=12500(2.225541)(1-0.449329)\\ &=\ 15,319.26 \end{align*}

The future value of income stream is $=\ 15,319.26$

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