Editing Accumulation function

In actuarial mathematics, the '''accumulation function''' ''a''(''t'') is a function of time ''t'' expressing the ratio of the value at time ''t'' ([[future value]]) and the initial investment ([[present value]]).<ref name="Vaaler2009">{{cite book |last1=Vaaler |first1=Leslie Jane Federer |last2=Daniel |first2=James |title=Mathematical Interest Theory |date=19 February 2009 |publisher=MAA |isbn=978-0-88385-754-0 |page=11-61 |url=https://books.google.com/books?id=1lLsmGVj2HIC&pg=PA62&dq=%22accumulation+function%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjW1MvvmZOLAxXYweYEHZVSHMIQ6AF6BAgGEAM#v=onepage&q=%22accumulation%20function%22&f=false |language=en}}</ref><ref name="Chan2021">{{cite book |last1=Chan |first1=Wai-sum |last2=Tse |first2=Yiu-kuen |title=Financial Mathematics For Actuaries (Third Edition) |date=14 September 2021 |publisher=World Scientific |isbn=978-981-12-4329-5 |page=2 |url=https://books.google.com/books?id=VoZGEAAAQBAJ&pg=PA2&dq=%22accumulation+function%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjW1MvvmZOLAxXYweYEHZVSHMIQ6AF6BAgMEAM#v=onepage&q=%22accumulation%20function%22&f=false |language=en}}</ref> It is used in [[interest theory]]. 

Thus ''a''(0)&nbsp;=&nbsp;1 and the value at time ''t'' is given by:

:<math>A(t) = A(0) \cdot a(t). </math>
where the initial investment is <math>A(0).</math>

For various interest-accumulation protocols, the accumulation function is as follows (with ''i'' denoting the [[interest rate]] and ''d'' denoting the [[annual effective discount rate|discount rate]]):
*[[simple interest]]: <math>a(t)=1+t \cdot i</math>
*[[compound interest]]: <math>a(t)=(1+i)^t</math>
*[[simple discount]]: <math>a(t) = 1+\frac{td}{1-d}</math>
*[[compound discount]]: <math>a(t) = (1-d)^{-t}</math>

In the case of a positive [[rate of return]], as in the case of interest, the accumulation function is an [[increasing function]].

==Variable rate of return==
The [[Rate_of_return#Logarithmic_or_continuously_compounded_return|logarithmic or continuously compounded return]], sometimes called [[Compound interest#Force of interest|force of interest]], is a function of time defined as follows: 

:<math>\delta_{t}=\frac{a'(t)}{a(t)}\,</math>

which is the rate of change with time of the natural logarithm of the accumulation function.

Conversely:

:<math>a(t)= \exp \left( \int_0^t \delta_u\, du \right), </math>

reducing to

:<math>a(t)=e^{t \delta}</math>
for constant <math>\delta</math>.

The effective [[annual percentage rate]] at any time is:
:<math>  r(t) = e^{\delta_t} - 1</math>

==See also==
*[[Time value of money]]

==References==
{{reflist}}

{{DEFAULTSORT:Accumulation Function}}
[[Category:Mathematical finance]]