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TheIn actuarial mathematics, the '''accumulation function''' ''a''(''t'') is a function defined in terms of time ''t'' expressing the ratio of the value at time ''t'' ([[future value]]) and the initial investment ([[present value]]).<ref (seename="Vaaler2009">{{cite alsobook [[time|last1=Vaaler value|first1=Leslie ofJane money]])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]].
Thus the value at time ''ta''(0) = 1 whereand the initialvalue investmentat istime ''kt'' is given by:
:<math>A(t) = kA(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]]):
Accumulation functions can be expressed for complex functions (not merely linear) using integration, in the following set up
*[[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 thatof thea accumulationpositive is[[rate dueof toreturn]], as in the case of interest , the accumulation itfunction is an [[increasing function]]. ▼:<math>A(t)=\int_0^t f(x)\,dx</math> ▼where "t" is the finishing point. Visually, the total amount of accumulation is the area between the function and the x-axis between the bounds given.
==Variable rate of return==
The accumulation function has the following two properties:
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'(0t)=1.}{a(t)}\,</math>
▲* In the case that the accumulation is due to interest it is an [[increasing function]].
which is the rate of change with time of the natural logarithm of the accumulation function.
==Common accumulation functions==
Conversely:
The accumulation function for the two common types of interest:
:<math>a(t)= \exp \left( \int_0^t \delta_u\, du \right), </math>
===Simple interest===
▲:<math>a(t)=1+t \cdot i.</math>
reducing to
===Compound interest===
▲:<math>a(t)=(1+i)^t.</math>
▲:<math> Aa(t)= \int_0e^ {t f(x)\ ,dxdelta}</math>
[[Category:Mathematical finance]] ▼for constant <math>\delta</math>.
The effective [[annual percentage rate]] at any time is:
[[ro:Funcţie de acumulare]]
:<math> r(t) = e^{\delta_t} - 1</math>
==See also==
*[[Time value of money]]
==References==
{{reflist}}
{{DEFAULTSORT:Accumulation Function}}
▲[[Category:Mathematical finance]]
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