I explained the formula for doubling your money through investment, as it was surprisingly little known

Hello everyone. My name is Fujiwara.

This is a story from a little while ago, but when I mentioned that since the interest rate is 5%, it would take about 14 years to roughly double, I got a reaction like, "Huh?" It occurred to me that perhaps this convenient "rough doubling formula" isn't very well known, so I decided to write this note.

The Magic Number 70

The "rough doubling formula" is the idea that if you multiply the interest rate (using % as the unit) by the duration, you get roughly 70, which represents the number of years required to double your money.

In the previous example, the interest rate is 5%, so since $${5\times14=70}$$, it takes 14 years to double. If the interest rate were lower, say 2%, then $${2\times35=70}$$, meaning it would take 35 years to double.

The previous example considers when you know the interest rate and want to know when it will roughly double, but you can also use it in reverse to determine the minimum interest rate required to double your money by a set deadline.

If you want to double your funds in 20 years and are wondering what interest rate you need, since $${3.5\times20=70}$$, you would need an interest rate of at least 3.5%.

It's a convenient formula, but since it seems surprisingly unknown, I will explain it, including why it works.

The relationship between interest rate and duration to double your money

If you invest funds $${M}$$ at an interest rate $${r}$$ for $${n}$$ years with compound interest, the result will be double, or $${2M}$$, so it becomes
$${M( 1 + r )^{n} = 2M}$$
.

We solve this for $${n}$$. Taking the logarithm of both sides,
$${n \cdot \ln{( 1 + r )} = \ln{2}}$$
$${n = \cfrac{\ln{2}}{\ln{( 1 + r )}}}$$

This is the relational expression between $${r}$$ and $${n}$$ to double your money. ($${\ln}$$ is the natural logarithm)

Trying to calculate the product of r and n in Excel

$${n = \cfrac{\ln{2}}{\ln{( 1 + r )}}}$$
Naturally, in the formula above, if you change $${\ln{2}}$$ to $${\ln{3}}$$ it becomes the relational expression for tripling, and if you change it to $${\ln{4}}$$ it becomes the expression for quadrupling.

The following table shows the results of listing interest rates $${r}$$ in Excel, calculating the corresponding $${n}$$, and finding the product of $${r}$$ and $${n}$$. (Please note that the interest rate $${r}$$ is converted to a percentage unit before calculating the product with $${n}$$.)

Looking at the table on the left for doubling, you can see that for realistic interest rates (roughly between 0.1% and 5.0%), $${r \times n}$$ is roughly 70.

Similarly, when tripling, $${r \times n}$$ is roughly 110, and when quadrupling, $${r \times n}$$ is roughly 140, so it might be useful to know these as well, but I think the doubling one is the most realistic.

Also, if the interest rate is too high, $${r \times n}$$ will deviate significantly from 70, so please do not assume it applies to any interest rate and use it correctly within realistic ranges.

See you later!