While most financial and accounting operations use the end of period compound interest model it is convenient to consider Cash Flows as
continuous streams.
The continuous compounding model is: F = Pqern
where:
F = Future Value
P = Present Value
q = the rate of flow
e = Base of natural logarithms = 2.71828......
r = the discount (interest) rate as a fraction.
n = length of the flow in periods.
The Present Value model of a uniform flow series becomes:
P = q/r(1 - e-rn)
as compared to the end of period Present Value discrete model:
P = q(((1 + r) n - 1)/( r(1 + r) n ))
e.g. A continus cash flow of 1000 per period for 5 periods discounted
at 10% is as follows:
P = 1000 / 0.1 x (1 - e -0.1 x 5)
P = 10000 x (1 -2.718283 -0.5 )
P = 10000 x (1 - .6065305)
P = 3934.69
Continuous compound at the same nominal rate accumulates more quickly. The difference is attributable to the having the cash flows ffrom the rate happen continuously threough a time period rather than being postponed to the end of the period.
It can be argued that a continuous flow model more accurately reflects actual
performance of the cash flow stream for many situations. The difficulties of
using the end of period model can be overcome by shortening the length of the
periods. Discounting over considerable lengths of time using periods as short
as a week or a day can be overcome using the power of even small computers.
This is especially true if arrray processing languages such as APL are used to
carry out the discounting. The major problem remains the estimation of the
cash flows and their pattern.
A number of other expressions have been derrived for various continuous
flow streams. These can be found by a dilligent search of the literature.