Accumulation function: Difference between revisions

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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&pg=PA62&dq=%22accumulation+function%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjW1MvvmZOLAxXYweYEHZVSHMIQ6AF6BAgGEAM#v=onepage&q=%22accumulation%20function%22&f=false |language=en}}</ref><ref name="Chan2021">{{cite book |last1=Chan |first1=Wai-sum |last2=Tse |first2=Yiu-kuen |title=Financial Mathematics For Actuaries (Third Edition) |date=14 September 2021 |publisher=World Scientific |isbn=978-981-12-4329-5 |page=2 |url=https://books.google.com/books?id=VoZGEAAAQBAJ&pg=PA2&dq=%22accumulation+function%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjW1MvvmZOLAxXYweYEHZVSHMIQ6AF6BAgMEAM#v=onepage&q=%22accumulation%20function%22&f=false |language=en}}</ref> It is used in [[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>

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Conversely:

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

reducing to