How to understand "force of interest" in simple interest?

Question

I am struggling to understand "force of interest" in simple interest. I understand that "force of interest" can be interpreted as compound interest with infinitely small time interval. That is, the new interest generated any infinitely short moment is added to the value.

However, as simple interest means that the interest has a linear continuous relationship with time, I am not sure how to understand "force of interest" in simple interest. For example, how can I interpret the equation $\delta _t = \frac{i}{1+it}$ realistically in simple interest?

Answer 1

You seem to have taken the force of interest to be $\dfrac{a'(t)}{a(t)}$ where $a(t)$ is the accumulated amount at time $t$

So your expression $\delta _t = \dfrac{i}{1+it}$ seems to be correct for simple interest with an interest rate of $i$, since $a(t)=a(0) (1+it)$ and $a'(t) = a(0)i$

In this case, $\delta_t$ is the momentary compounding rate, and is a decreasing function of $t$:

• when $t=0$ and the starting amount is $a(0)$, it is $\delta_0=i$,
• while for example when $t=\frac1i$ and the accumulated amount is $2a(0)$, it is $\delta_{1/i}=\frac12 i$.

Answer 2

late to the party but; combining the two answers above clarifies beautifully.

1. simple interest implies that as time increases, the degree interest has on accumulated value decreases, because AV increases with t(linear continuous relationship) but it increases continuously by the same amount. a larger quantity of time applies a proportionate level of change as a lesser amount. so the force((i feel like this should be “momentum“) of interest has to decrease as time increases such that the amount(monetary, which i feel should be where “force” comes into play) of change is constant.

2. although interest is simple(product of length of period), the period in question is not discrete, even if it is an instant(momentary). and if the interest were applied at discrete intervals, the case would then become that the force of interest could only exist during interest respective moments of application.