Cash accumulation equation
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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 current sum (dollars)|
|= principal (dollars)|
|= force of interest (per year)|
|= time (years)|
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.
Having achieved this we are ready to start feeding money into the account, at a rate of dollars/year. This is effected by making a small change to eqn (3) as follows
and accordingly we need to solve the equation
From a table of integrals, the solution is
where is the constant of integration. The initial sum deposited was so we know one point on the curve :
and making this substitution we find that
Using this expression for , and recalling that
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 , and specifically the expression
One way of evaluating this is to write out the Maclaurin expansion
At a glance we can subtract from this series and divide by , to find out that
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 . Upon making this substitution, eqn (4) becomes simply
Evidently is negative, and money is being withdrawn rather than deposited. Specifically, the interest is being withdrawn as fast as it is being earned.
An alternative interpretation of this special case is that is negative - the account is overdrawn - and money is being fed in at a rate which just meets the interest charges. A force of interest value is always positive.