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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?

Andhavarapu Balu
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  • No. Just like in the rest of all sciences using mathematics, no formal term will tell you what exactly is going on, unless it comes along with its precise definition. – Kurt G. Mar 07 '24 at 08:49
  • @KurtG. -- We all do know how James Clerk Maxwell predictged the electromagnetic radiation, don't we? – m-stgt Mar 07 '24 at 11:30
  • @m-stgt Do we? And if so: what has that to do with OP? – Kurt G. Mar 07 '24 at 13:04
  • @KurtG Can you recommend any phrase that clarifies this instead? I initially used the word "continual" but realised that the word is used for continual compounding (the formula with Euler's constant as the base of exponentiation) – Andhavarapu Balu Mar 07 '24 at 14:20
  • I believe I was quite clear that "phrases" do not clarify anything. This is particularly true of phrases that were recommended to you by someone else. – Kurt G. Mar 07 '24 at 17:00

1 Answers1

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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.

WA Don
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  • This is the same formula that I have provided, just in a different way (and more concretely presented by you). As I have highlighted with my example of "20% increment", my rate term presumes that the +1 is already done – Andhavarapu Balu Mar 07 '24 at 14:16
  • Thank you for the comment. "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." Can you please provide examples of what kind of fields you have seen it been applied this way? Was it in a classroom, a banking application, an investment analyser, or anything else? – Andhavarapu Balu Mar 07 '24 at 14:18
  • @AndhavarapuBalu You are close to asking what banks or some apps actually do under these circumstances. That's off topic for this site. All we can do is write the formulas for each case. The one that concerns you does not have a common name. – Ethan Bolker Mar 07 '24 at 14:35
  • @Balu: sorry I didn't understand your term increment the "interest rate" in your case is 0.2 or 20% - 1 + interest rate doesn't have a universal name. The treatment of incomplete periods depends on terms for the account. In most loans, interest is paid in cash at the end of each period so never compounds and the loan document will specify how interest may be calculated if terminated mid-period. UK's NS&I ISA does add interest once a year and so does compound, but if you withdraw mid year a pro-rated interest is given. But lender / borrower can agree anything they want. – WA Don Mar 07 '24 at 15:18
  • @WADon Got it, thank you – Andhavarapu Balu Mar 07 '24 at 16:33