The cash accumulation equation is an equation which calculates how much money will be in a bank account, at any point in time. The account pays interest, and is being fed a steady trickle of money.
Contents
Compound interest
We will approach the development of this equation by first considering the simpler case, that of just placing a lump sum in an account and then making no additions to the sum. With the usual notation, namely
the equation is
and so the sum of money grows exponentially. Differentiating this we derive
and applying the definition of y from eqn (1) to eqn (2), yields
Note that eqn. (1) is a particular solution to the ordinary differential equation in eqn. (3), with y equal to P at t=0.
Cash infeed
Having achieved this we are ready to start feeding money into the account, at a rate of
and accordingly we need to solve the equation
From a table of integrals, the solution is
where
and making this substitution we find that
Using this expression for
gives us the solution :
This is the neatest form of the cash accumulation equation, as we are calling it, but it not the most useful form. Using the exponential instead of the logarithmic function, the equation can be written out like this :
First special case
From this new perspective, eqn (1) is just a special case of eqn (4) - namely with
Second special case
For completeness we will consider the case
One way of evaluating this is to write out the Maclaurin expansion
At a glance we can subtract
With this result the cash accumulation equation now reads
Thus the cash sum just increases linearly, as expected, if no interest is being paid.
Third special case
The only other special case to mention is
Evidently
An alternative interpretation of this special case is that