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.

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>

For various interest-accumulation protocols, the accumulation function is as follows (with ''i'' denoting the interest rate and ''d'' denoting the discount rate): *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 of a positive rate of return, as in the case of interest, the accumulation function is an increasing function.

==Variable rate of return== The logarithmic or continuously compounded return, sometimes called force of interest, is a function of time defined as follows:

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

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

Conversely:

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

reducing to

:<math>a(t)=e^{t \delta}</math> for constant <math>\delta</math>.

The effective annual percentage rate at any time is: :<math> r(t) = e^{\delta_t} - 1</math>

==See also== *Time value of money

==References== {{reflist}}

{{DEFAULTSORT:Accumulation Function}} Category:Mathematical finance