If at time the rate of income generated by some enterprise is given by the value of the function , then the total income generated between time and time is .
What value should one assign to an expected future payment?
Recall that if one has a principal of which earns continuously compounded interest at a rate of , then after years, the investment would be worth .
The present value of a payment of $ made years in the future is the amount for which with a principal of $ dollars invested for years with continuously compounded interest of rate one would earn $.
From the formula for continuously compounded interest, we conclude that so that .
Suppose that some enterprise produces income at a steady rate of $ per year. Of course, this income stream over the next years will produce $, but how much is it worth in present dollars?
We may approximate the continous income stream as one that is paid in discrete increments.
Suppose that between now and years from now payments are made at uniform intervals. Then, the length of time between each payment is years.
The payment of is made at time . As such, if we assume an interest rate of , it has a present value of .
So, the sum of the present values is .
The expression
is the right-hand approximation to
Assuming an interest rate of %, compute the present value of a constant income stream of $ per year for years.
If the income stream varies as a function of time, so that at time , is the rate at which the payments are made, and the interest rate also varies (possibly) as a function of time, given by the function , then the present value of the income stream over the next years is
Suppose the interest rate is constantly % and the income stream is given by the function . What is the present value of this income stream over the next years?
We compute that
Integrating by parts, with and , so that and .
So,
Thus, .
So, the total present value is .