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
.