title: Accumulation function
text: # Accumulation function
  
  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.
  
  Thus a(0)=1 and the value at time t is given by:
  
      $A(t)=A(0)\cdot a(t)$.
  
  where the initial investment is $A(0).$
  
  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: $a(t)=1+t\cdot i$
  * compound interest: $a(t)=(1+i)^{t}$
  * simple discount: $a(t)=1+{\frac {td}{1-d}}$
  * compound discount: $a(t)=(1-d)^{-t}$
  
  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:
  
      $\delta _{t}={\frac {a'(t)}{a(t)}}\,$
  
  which is the rate of change with time of the natural logarithm of the accumulation function.
  
  Conversely:
  
      $a(t)=e^{\int _{0}^{t}\delta _{u}\,du}$
  
  reducing to
  
      $a(t)=e^{t\delta }$
  
  for constant $\delta$.
  
  The effective annual percentage rate at any time is:
  
      $r(t)=e^{\delta _{t}}-1$
  