Accumulation function: Difference between revisions

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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]]) 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]].

The '''accumulation function''' ''a''(''t'') is a function defined in terms of time ''t'' expressing the [[time value of money]]. It is used in [[interest theory]].

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

:<math>A(t) = kA(0) \cdot a(t). </math>.

The accumulation function assumes the initial investment to be 1. To obtain value of money where the initial investment is ''k'', simply have

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]]):

:<math>A(t) = k \cdot a(t)</math>.

*[[simple interest]]: <math>a(t)=1+t \cdot i.</math>

*[[compound interest]]: <math>a(0t)=(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]].

Accumulation functions can be expressed for complex functions (not merely linear) using integration, in the following set up

==Variable rate of return==

:<math>A(t)=\int_0^t f(x)\,dx</math>

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:

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.

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

The accumulation function has the following two properties:

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

* <math>a(0)=1.</math>

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

Conversely:

==Common accumulation functions==

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

The accumulation function for the two common types of interest:

reducing to

===Simple interest===

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

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

===Compound interest===

:for constant <math>a(t)=(1+i)^t.\delta</math>.

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

[[Category:Mathematical finance]]

:<math> r(t) = e^{\delta_t} - 1</math>

==See also==

[[ro:Funcţie de acumulare]]

*[[Time value of money]]

==References==

{{reflist}}

{{DEFAULTSORT:Accumulation Function}}

[[Category:Mathematical finance]]