"Present value and accumulated value of money flow" problem

Question

Find the present value and accumulated value after 10 years for an income stream with the rate of money flow $f(t) = 200 + 150t$ dollars per year and the rate of interest 12% compounded continuously. Then, generalize the result for an income stream of $f(t) = A + Bt$ and a rate of interest $r$%.

I'm not sure on how to proceed with this problem, I don't know what it's asking for. I tried to find the definite integral for the total money flow between $0$ and $10$, but I don't know what to do with that. I know the formula for continuous compounded interest is $A = A_{0}e^{rt}$ where $A_{0}$ is the initial value, but I'm not sure how to apply it.

Answer 1

For each small element of time $dt'$ at time $t'$ you receive income $(200+150t')dt'$. At a later time $t$, it has become $(200+150t')\exp(0.12(t-t'))dt'$ So at time $t$ you have $$\int_0^t(200+150t')\exp(0.12(t-t'))dt'$$

Answer 2

If the investment horizon is 10 years

Then Accumulated Value = $\Sigma_0^{10} (200+150t)e^{(.12*(10-t))}$ as T is in years and payment stream is in years and are discrete but the money flow is continuously compounded.

Then Present Value = $\Sigma_0^{10} (200+150t)e^{-(.12*t)}$

Extending it to an horizon of T years and at the rate r,

Accumulated Value = $\Sigma_0^{T} (A+Bt)e^{(r*(T-t)/100)}$

Present Value = $\Sigma_0^{T} (A+Bt)e^{-(r*t)/100}$

If on the other hand payment stream is conitnuous, then Ross's answer is just as good.

Thanks

Satish