Accumulation function: Difference between revisions

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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 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]].

Thus the value at time ''ta''(0)&nbsp;=&nbsp;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>a(0)=1.</math>

* In the case that the accumulation is due to interest it is an [[increasing function]].

The [[Rate_of_return#Logarithmic_or_continuously_compounded_return|logarithmic or continuously compounded return]] is a function of time defined as follows:

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

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Conversely:

:<math>a(nt)=e^{ \exp \left( \int_0^nt \delta_tdelta_u\, dt}\du \right), </math>

reducing to

==Common accumulation functions==

:<math>Aa(t)=\int_0e^{t f(x)\,dxdelta}</math>

The accumulation function for the two common types of interest:

*for constant <math>a(0)=1.\delta</math>.

The effective [[annual percentage rate]] at any time is:

===Simple interest===

:<math>a r(t) =1+t e^{\cdotdelta_t} i.- 1</math>

===CompoundSee interest=also==

*[[timeTime value of money]]

:<math>a(t)=(1+i)^t.</math>

==See alsoReferences==

{{reflist}}

*[[time value of money]]

{{DEFAULTSORT:Accumulation Function}}

[[Category:Mathematical finance]]

[[ro:Funcţie de acumulare]]