# Accumulation function

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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](/source/future_value)) and the initial investment ([present value](/source/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&dq=%22accumulation+function%22&pg=PA62 |language=en}}</ref><ref name="Chan2021">{{cite book |last1=Chan |first1=Wai-sum |last2=Tse |first2=Yiu-kuen |title=Financial Mathematics For Actuaries |date=14 September 2021 |publisher=World Scientific |isbn=978-981-12-4329-5 |page=2 |edition=Third |url=https://books.google.com/books?id=VoZGEAAAQBAJ&dq=%22accumulation+function%22&pg=PA2 |language=en}}</ref> It is used in [interest theory](/source/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](/source/interest_rate) and ''d'' denoting the [discount rate](/source/annual_effective_discount_rate)):
*[simple interest](/source/simple_interest): <math>a(t)=1+t \cdot i</math>
*[compound interest](/source/compound_interest): <math>a(t)=(1+i)^t</math>
*[simple discount](/source/simple_discount): <math>a(t) = 1+\frac{td}{1-d}</math>
*[compound discount](/source/compound_discount): <math>a(t) = (1-d)^{-t}</math>

In the case of a positive [rate of return](/source/rate_of_return), as in the case of interest, the accumulation function is an [increasing function](/source/increasing_function).

==Variable rate of return==
The [logarithmic or continuously compounded return](/source/Rate_of_return), sometimes called [force of interest](/source/Compound_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](/source/annual_percentage_rate) at any time is:
:<math>  r(t) = e^{\delta_t} - 1</math>

==See also==
*[Time value of money](/source/Time_value_of_money)

==References==
{{reflist}}

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

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Adapted from the Wikipedia article [Accumulation function](https://en.wikipedia.org/wiki/Accumulation_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Accumulation_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.