What's the $1$ in the compound interest formula for?

Question

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.

Answer 1

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.

Investopedia

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)

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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 $n$)

Answer 2

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.