Compound interest formula is not understandable

Question

I am struggling to understand the formula for compound interest. More specifically what the $n$ stands for.

The formula is as follows according to the wikipedia: $$ A = P(1 + \frac{r}{n})^{nt} $$

Where
$A$ = final amount
$P$ = initial principal balance (money invested)
$r$ = interest rate
$n$ = number of times interest applied per time period
$t$ = number of time periods elapsed

But we can twist the parameters so that the interest is not $r$.

Let's take an example: annual interest rate of 20%, compounded quarterly. This means that the parameters are $t = 1$ year, $r = 20$%, $n = 4$

If I invest 1 USD for a year ($P = 1$) it should be 1.20 USD at the end of the year by the definition of annual interest rate, but based on the formula, I calculate something different: The annual interest rate is $21.55$%, because by investing 1 USD, I will have earned 1.2155 USD by the end of the year.

$$ A = 1(1 + \frac{0.2}{4})^{4 \cdot 1} = 1.21550625 $$ Which is approximately 21.55%, not 20% annually.

The continuous compounding interest is derived from this formula, so I would like to understand this before understanding that.

Answer 1

An annual interest rate of $20\%$ compounded quarterly means what the formula reflects, namely that your investment grows once every quarter at a quarterly rate of $20\%/4=5\%.$ This is not the same thing as an annual interest rate of $20\%$ (no compounding), which means your investment grows once every year at the yearly rate of $20\%$. As you have noted, the former gives an effective annual rate of approximately $21.55\%$ while the latter simply gives an effective annual rate of $20\%.$

Compounding more frequently thus leads to a higher annual return, i.e.

$$P(1+r/n)^{nt}\quad (1)$$

increases in $n$ for $r,t>0$. Intuitively, the exponential effect of more compounding periods outweighs the linear effect of the lower per-period interest rate. As you suggestively note, taking the limit of $(1)$ as $n\to \infty$ gives the continuous compounding formula

$$Pe^{rt}.$$

Answer 2

If $n=1$ then the interest is applied once per year. So you'd get $20$ cents in your example, as you expect. But if $n=4$, you get $1/4$ of the interest each quarter of the year. So after $3$ months, you get $5$ cents. But now you have \$1.05 in the account, so at the end of the next quarter, you get 5 cents of interest on the dollar, but also 5% interest on the extra nickel. After you do this 4 times, you have "interest on the interest" adding up to an extra 1.55 cents.

Answer 3

The reason you end up with $21.55\%$ interest and not just $20\%$ is precisely because of the $n=4$ that you do not understand.

After the first quarter ($1/4$)of the year, the interest you earned so far ($20\%/4 = 5\%$) is added to your balance. Then you earn interest on that interest for the rest of the year.