What's the $1$ in the compound interest formula for? I'm aware of how rudimentary the question per se sounds, but please explain it in childspeak. I just started learning how to trade, and it'd be of great help if I can understand the concept behind this. Please try to use words instead of numbers or symbols.
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3$\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$– Community BotCommented Mar 19, 2022 at 12:18
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1$\begingroup$ In all cases, you retain the principle. That is to say, if you start with $P$ and wait one period you wind up with $P+rP=(1+r)P$. Where $r$ is the interest rate over the compounding period. Hence the $1$. $\endgroup$– luluCommented Mar 19, 2022 at 12:20
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$\begingroup$ Mathematically, $A=P(1+\frac{r}{n})^{nt}=P(1+nt\cdot\frac{r}{n}+\cdots)=P+Pnt\cdot\frac{r}{n}+\cdots$ so the $1$ indicates the principal amount treated as a unit. $\endgroup$– user10575Commented Mar 19, 2022 at 12:25
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$\begingroup$ I understand that the next year's compounding will be done on the basis of the principle amount +the interest of the first year, so is the 1 here a placeholder for the original principle amount? If then wouldn't it be better to just add the principle there instead of the 1. My understanding of this seems to be behind by leaps and bounds, please clarify. Thank you. $\endgroup$– Leoleo123Commented Mar 19, 2022 at 12:27
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$\begingroup$ We are indeed adding the principle instead of $1$ in the alternate formula . $A=P+Pnt\cdot\frac{r}{n}+\cdots$. It's only that, then the formula becomes cumbersome, so we 'take $P$ common' and arrive at the formula in vogue. $\endgroup$– user10575Commented Mar 19, 2022 at 12:33
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2 Answers
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I know this isn't what you asked for, but using the math is the easiest way for you to understand what is happening. Before showing you why the number $1$ is present in the formula, you've got to understand what compound interest means:
Compound interest (or compounding interest) is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods.
Say you've put in $\$100$ in the bank and the compound interest is $\%5$ or $\dfrac{5}{100}$ or $0.05$. When the interest is first applied, you get:
$$100 + 100(0.05) = 105$$
Note that $A + A(B)$ = $A(1) + A(B)$ and by the distributive property, that is equal to $A(1+B)$.
$100 + 100(0.05) = 105$ is hence equal to (here's where the $1$ makes its appearance): $$100(1+0.05) = 105$$
(The answer ends here, but you can read on if you want to see the formula taking shape)
If the interest is applied again, we get:
$$105(1+0.05) = 110.25$$
The $105$ is equal to our original $100(1+0.05)$, and therefore when replacing we get:
$$100(1+0.05)(1+0.05) = 110.25$$
Or more generally, for $n$ times the interest is applied...:
$$100(1+0.05)^n$$
With our initial principal balance being $P$ and rate $r$:
$$P(1+r\%)^{n} = \text{final amount}$$
(assuming the interest is only applied once per time period $n$)
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$\begingroup$ I understood this, but say we changed the topic to the present value calculation of money. Then, the formula would be(Amount/(1+discount rate)^no.of years. Can you explain what happened here?@πππ $\endgroup$ Commented Mar 19, 2022 at 13:44
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Say the return rate is 10% per year, that's $i = \frac1{10}$. Say you have an initial investment of \$100, that's $P=100$. How much is your portfolio worth after one year?
Is it $P\cdot i$? No, that would be \$10. $P\cdot i$ is how much additional value your investment gained in the first year. But you still have your original investment of $P$. All together, at the end of the first year, your portfolio is worth $P + P\cdot i$.
Factoring out the $P$ we get $$P\cdot(1+i).$$ (Maybe this is the part that is giving you trouble? Let me know in the comments.) After one year your portfolio has increased to whatever it was before, plus $i$ fraction more.
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$\begingroup$ So, my P=110 after a year.I'm confused as to why it's P+P.i , which would be principle +principle with first yr interest absorbed.Isn't that redundant, since we already have the interest absorbed amount, and why does it then just change into 1 of all things. I'm aware that these sound dumb to you, but I genuinely find these confusing. Please simplify it further Thank you. $\endgroup$ Commented Mar 19, 2022 at 13:01
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$\begingroup$ $P \cdot i$ is not "principle with first year interest" as you say. It's the additional value your investment gained. For example, in this case, $P \cdot i=100\cdot \frac1{10}$ which is \$10, not \$110. You still have to add on the principle. $\endgroup$ Commented Mar 19, 2022 at 13:05
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$\begingroup$ Yes, that's the part I'm having trouble with.So, P.i is the interest and not the interest absorbed amount. I still fail to understand as to why it's 1 cause, I mean, if you let one be added to the i fraction from the first year, wouldn't it skew your calculation for the second year by the same 1? $\endgroup$ Commented Mar 19, 2022 at 13:06
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$\begingroup$ Did you know that, for any three numbers, $$a\cdot(b+c) =(a\cdot b)+(a\cdot c).$$ This is called the distributive law. $$P\cdot(1+i) = (P\cdot1)+(P\cdot i)$$ is an example of that. $\endgroup$– MJDCommented Mar 19, 2022 at 13:08
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$\begingroup$ If you expand P(1 + i) you get P.1 + P.i, which is P+P.i, of course. Conversely, if you want to factorise out P from P + P.i, you can write it first as P.1 + P.i and then factorise out P to get P(1 + i) $\endgroup$ Commented Mar 19, 2022 at 13:12