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 $A(0).$

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 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.

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 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:

$r(t)=e^{\delta _{t}}-1$

See also

• Time value of money