Accumulation function

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).^[1][2] It is used in interest theory.

Thus a(0) = 1 and the value at time t is given by:

A ( t ) = A ( 0 ) ⋅ a ( t ) . {\displaystyle A(t)=A(0)\cdot a(t).}

where the initial investment is A ( 0 ) . {\displaystyle 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 ⋅ i {\displaystyle a(t)=1+t\cdot i}
• compound interest: a ( t ) = ( 1 + i ) t {\displaystyle a(t)=(1+i)^{t}}
• simple discount: a ( t ) = 1 + t d 1 − d {\displaystyle a(t)=1+{\frac {td}{1-d}}}
• compound discount: a ( t ) = ( 1 − d ) − t {\displaystyle 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.

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

δ t = a ′ ( t ) a ( t ) {\displaystyle \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 ) = exp ⁡ ( ∫ 0 t δ u d u ) , {\displaystyle a(t)=\exp \left(\int _{0}^{t}\delta _{u}\,du\right),}

reducing to

a ( t ) = e t δ {\displaystyle a(t)=e^{t\delta }}

for constant δ {\displaystyle \delta } .

The effective annual percentage rate at any time is:

r ( t ) = e δ t − 1 {\displaystyle r(t)=e^{\delta _{t}}-1}

• Time value of money