Continuous Compounding
Rates are sometimes converted into the continuous compound interest rate equivalent because the continuous equivalent is more convenient (for example, more easily differentiated). Each of the formulæ above may be restated in their continuous equivalents. For example, the present value at time 0 of a future payment at time t can be restated in the following way, where e is the base of the natural logarithm and r is the continuously compounded rate:
This can be generalized to discount rates that vary over time: instead of a constant discount rate r, one uses a function of time r(t). In that case the discount factor, and thus the present value, of a cash flow at time T is given by the integral of the continuously compounded rate r(t):
Indeed, a key reason for using continuous compounding is to simplify the analysis of varying discount rates and to allow one to use the tools of calculus. Further, for interest accrued and capitalized overnight (hence compounded daily), continuous compounding is a close approximation for the actual daily compounding. More sophisticated analysis includes the use of differential equations, as detailed below.
Read more about this topic: Time Value Of Money
Famous quotes containing the word continuous:
“The gap between ideals and actualities, between dreams and achievements, the gap that can spur strong men to increased exertions, but can break the spirit of others—this gap is the most conspicuous, continuous land mark in American history. It is conspicuous and continuous not because Americans achieve little, but because they dream grandly. The gap is a standing reproach to Americans; but it marks them off as a special and singularly admirable community among the world’s peoples.”
—George F. Will (b. 1941)