Calculating Compound Interest

How Fast can One's Fortune Grow?

Assume an ideal investor with those variables:

$P$ → original principal sum
$r$ → nominal annual interest rate
$c$ → number of compounding
$A_c$ → amount after compounding c times
$n$ → compounding frequency for each year (annually compounding means n=1 and monthly compounding means n=12)
$\frac{r}{n}$ → nominal interest rate for each compounding
$t$ → overall length in year
$t\times n$ → total number of compounding

Case 1: without monthly money inflow

Initial condition:

\[A_0 = P\]

For each compounding:

\[A_{n+1} = A_{n} \times (1+r/n)\]

Hence, after $t$ years or compounding $c=t\times n$ times:

\[A_{t\times n} = P(1+r/n)^{t\times n}\]

Assuming an investor is investing low-to-medium risk assets, and luckily he has a stable yearly interest rate of $r=2\% =0.02$.

For doubling his initial investment, the final amount would be $A_{t\times n}=2P$.

\[A_{t\times n}=2P = P(1+r/1)^{t\times 1}\]

\[2=(1+r)^t\]

\[t=\frac{\ln(2)}{\ln(1+r)}\]

Thus, it takes the investor $t\approx 0.69/\ln(1+0.02) \approx 35.00$ years to double his fortune.

Note that the $\ln$, instead of for example $\log_{10}$, was used because we can get a simple empirical relationship.

Because $\ln(1+r)$ approximates $r$ when the value of r is very small.

The equation can simplify into

$$ t\approx \frac{0.69}{r} \approx \frac{70}{r\%}, $$ which is also called the Rule of 70.

One's fortune doubles per $\frac{70}{\text{interest percentage point}}$ years.

The initial asset is still $P$ but in each month, the investor adds $w$ into his investment.

For each month, the final amount is

\[ A_{c+1} = (A_{c}+w)\times(1+r) \]

Sequentially, the amount after each month's compounding is

This can be summerized into

\[ A_{t\times n} = P(1+r/n)^{t\times n} + \sum_{d=0}^{t\times n-1}w(1+r/n)^d, \]

where $n=12$.

So after one year or 12 compounding, one's fortune would be

\[ A_{12} = P(1+r/12)^{12} + w\sum_{d=0}^{11}(1+r/12)^d. \]

https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator

(TODO: Approximation of sequences?)