Background: I'm an engineering student with no experience with a real investment account or anything of that sort.
If we were to extend the concept of compound interest strictly mathematically, the formula would be:
$$\text{Amount} = \text{Principal} \times \text{rate} ^ t$$ where $t$ is time in number of compounding periods passed, including fractional part
For example, a principal of 1, with an interest rate of 1.2 (20% increment), compounded annually, after 2 years and 3 months would be:
$$\text{Amount} = 1 \times (1.2)^{(2+3/12)}$$
But as per my understanding, banks and other such institutes write the change on records only at the end of the compounding period (a year). So it also makes sense for someone to not realise this mathematical extension, and instead use simple interest for the fractional part of the year, following the formula:
$$\text{Amount} = 1 \times (\text{rate} ^ {\text{(completed years)}} + \text{rate} \times \text{remaining fraction of the year})$$
Is there a formal term to differentiate between these two? Or any other phrase that we can use to explicitly describe this, apart from writing out the formula directly?
This is a comment but too long to fit in the allocated space:
There is no widely recognised term, but most would refer to simple interest, as the following:
\begin{align} V_{end} = V_{start} *(1 + i \times t) \end{align}
where $V$ stands for value, $i$ is the quoted annual interest rate (e.g. $0.03$ for 3%) and $t$ the time.
For compound interest it is common to refer to the annual rate (e.g. 3%) and then the compounding period (unless it is already clear), e.g. "3% per annum compounded quarterly" and by convention that means
\begin{align} V_{end} = V_{start} \times \left(1 + i \times p\right)^{t/p} \end{align}
Where $p$ is the time period over which compounding is to be applied (e.g.0.25 for quarterly), $t$ is the time, and should be a whole number of periods and $i$ is the annual interest rate. If the time includes an incomplete period at the end, the treatment is ambiguous, but typically simple interest is applied in for the incomplete period.
Note the addition of $1$ to the interest rate terms. This is different from the formula you posted in your question.