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]].

{{Unreferenced|date=December 2009}}

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

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]]). It is used in [[interest theory]].

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

Thus ''a''(0)=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>

'''A'''(a,b) = A(b)÷A(a) where 0 < a < b

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]]):

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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==

Derivation of Compounded Interest rate function:

Assume an investment of 1 unit at time T₀ .

At time T₁ the invest ment increases 1 × i , thus the value at T₁ =1 + i.

At time T₂ the invest ment increases with (1 + i) i , thus the value at T₂ = (1 +i ) + (1+i)i = (1+i) (1+i) = (1+i)²

We can continue with this pattern up until time T_k thus the value at time T_k = (1 + i )^k

We can then define a function that finds the value of an investment 1 at time t as the following a(t) = (1 + i)^t where i is the fixed compounded interest rate.

===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:

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Conversely:

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

reducing to

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==See also==

*[[Time value of money]]

==References==

{{reflist}}

{{DEFAULTSORT:Accumulation Function}}