Accumulation
September 2018
Say we have an amount function C(t) that tells us how much of something, say capital, we have accumulated at time t. Then C(0) is the principal. There must then be some function a(t) that, based on the initial condition of C(0) = c, tells how the process of accumulation evolves in time all the way to some time point t = p.

Let's assume that we are talking about a financial setting where we accumulate interest because that whole concept works wonderfully with the calculations and is a good basis for intuition.

Then the simplest case is that of simple interest. If we have discrete time, where t belongs to the natural numbers, and one step is one year, the simple interest for t years with interest rate i can be calculated as:

C(t) = a(t) * C(0)
= (1 + it) * C(0)
= C(0) + itC(0)

For practical example, say our principal is 1 000 (of something) and our interest rate is 5% <--> 0.05 per annum. The accumulated capital after 3 years is then:

C(3) = a(3) * C(0)
= (1 + 0.05 * 3) * 1 000
= (1.15) * 1 000
= 1 150

We could have calculated the above also in the following way:

C(3) = a(3) * C(0)
= (1 + 0.05 * 3) * 1 000
= 1 000 + 0.15 * 1 000
= 1 150

and arrived at exactly the same answer. I prefer the latter form because it is strongly aligned with the nature of simple interest. That is, it takes the principal and adds to it the a multiple of the principal in proportion to the interest rate per time unit times the passed time units. From this, it's easy to see why it's called the simple interest. The principal which we are using to calculate the interest does not change within one time period. What if the investor would take the capital out with accumulated interest in the middle of the 3 year period (t = 3/2) and invest it again for 3/2 years?

C(3/2) = a(3/2) * C(0)
= (1 + 0.5 * (3/2)) * 1 000
= 1075

C(3/2) = a(3/2) * C(0)
= (1 + 0.025 * (3/2)) * 1 037.5
= 1 155.625

In this scenario our amount function returned a larger value after the same number of periods, three, had passed. How so? Notice that after the initial 3/2 periods had passed, the amount function returned exactly half of what it returned after 3 periods in the first example. Until that point, the accumulation was exactly equivalent. But because the capital – with interest – was withdrawn after 3/2 periods, and reinvested again with the same terms, then also the interest began earning interest in the latter example. The accumulation of interest generated in the latter 3/2 period by the interest accumulated in the first 3/2 period was responsible for the difference in the amount functions after 3 periods had passed, a curiosity which we can confirm by calculation

C_1(3) – C_2((3/2) + (3/2))
= 1 155.625 – 1 150
= 5.625
= a(3/2) * (1 075 - 1 000) – 75
= (1 + 0.05 * (3/2)) * 75 – 75
= 75 + 0.15 * 75 – 75
= 80.625 – 75
= 5.625

Where we subtracted the amount of interest accumulated in both cases at the latter 3/2 period to highlight the difference that the reinvestment made to the final value of the amount function.

A note of great importance here is that the latter example returned a higher total amount of capital than the first. If we would again reinvest the accumulated interest in the middle of each 3/2 periods, would we get a higher final value? Indeed we would. This leads to the curious question of how long could we keep doing this halving, and if the amount function always returns a higher value after each halving, how large we value we could get the amount function to return?

Yet, the calculations with the simple interest accumulation function would become cumbersome if we increased the halving – or decreased the investment periods – drastically. To circumvent this trouble, we invent a new form for the accumulation function, which is often called the compound interest form:

a(t) = (1 + i)^t

The form is highly intuitive: we exponentiate the translated interest rate for one period to the power of periods the investment accumulated interest. Then our amount function is of the form:

C(t) = (1 + i)^t * C(0)

That is, we scale the principal in exponential proportion to the number of time periods that has passed, where C(0) is our initial value.

Now assume we are interested in some intervals 1/m of some total time period, where m belongs to the natural numbers. By assumption, 1/m is the shortest interval where it is possible to accrue interest, and all other time intervals are some integer multiples of that. Then, from the idea of compound interest above we have

C(1 / m) = (1 + i)^(1 / m) * C

Which we can manipulate into the format of simple interest:

C(1 / m) = (1 + (i(m) / m)) * C

Where i(m) is the nominal interest rate function, accepting input m, for which we can solve for by combining the previous two equations:

1 + (i(m) / m) = (1 + i)^(1 / m)
i(m) = m((1 + i)^(1 / m) - 1)

The limiting value of this function,

lim m--> inf (i(m)) = D

is called the force of interest. At every time point, the increase of the amount function is proportional to the current amount, where the constant is the force of interest. We thus arrive at the differential equation:

C'(t) = D * C(t)

Which we, by remembering that C(0) = C, can solve, the solution being:

C(t) = C * e^(Dt)

We now notice that

1 + i = e^D