Editing Time value of money (section)

==Continuous compounding==
Rates are sometimes converted into the [[continuous compounding|continuous compound interest]] rate equivalent because the continuous equivalent is more convenient (for example, more easily differentiated). Each of the formulas 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 (mathematical constant)|e]]''' is the base of the [[natural logarithm]] and '''r''' is the continuously compounded rate:
:<math> \text{PV}  = \text{FV}\cdot e^{-rt} </math>
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''):
:<math> \text{PV}  = \text{FV}\cdot \exp\left(-\int_0^T r(t)\,dt\right)</math>
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|differential equations]], as detailed below.

===Examples===
Using continuous compounding yields the following formulas for various instruments:
;Annuity:
:<math> \ PV \ = \ {A(1-e^{-rt}) \over e^r -1}</math>
;Perpetuity:
:<math> \ PV  \ = \  {A \over e^r - 1} </math>
;Growing annuity:
:<math> \ PV  \ = \  {Ae^{-g}(1-e^{-(r-g)t}) \over e^{(r-g)} - 1} </math>
;Growing perpetuity:
:<math> \ PV  \ = \  {Ae^{-g} \over e^{(r-g)} - 1} </math>
;Annuity with continuous payments:
:<math> \ PV  \ = \  { 1 - e^{(-rt)} \over r } </math>
These formulas assume that payment A is made in the first payment period and annuity ends at time ''t''.<ref>{{cite web|url=http://baselineeducation.blogspot.co.uk/2012/10/annuities-and-perpetuities-with.html|title=Annuities and perpetuities with continuous compounding|date=11 October 2012 }}</ref>