You have probably heard the expression that “the first million is the hardest”. This has nothing to do with learning how to make money and somehow mastering it after the first million. In fact, it is equally hard to save the first 1 million as it is to save the first 100000 USD; or more counterintuitively, the first 1 million or 100000 GBP, respectively. On the other hand, we never say that the first $100 or the first $1000 is the hardest. Why is this?

It has to do with Benford’s law. The simplest explanation I can think of is that the increase from 1 to 2 million represents an increase of 100%. Conversely, the increase from 2 million to 3 million is 50%, 3 to 4 is 33%, …. and 9 to 10 is 11.1%. Hence, if wealth grows exponentially by compound interest, the first million is indeed the hardest, because imagine starting from 100000, then 1000000 means increasing it by a factor 10 whereas the second million is only an increase by a factor 2.

Conversely, if money is accumulated linearly by working and stuffing money in the mattress rather than exponentially growing it by investing, the second million is as hard as the first million, much like the second $1000 is as hard as the first $1000.

Let’s get technical (see how much you remember from high school). If you start at 100000 and compound at 5% and plot a graph make a dot for each compounding period (each year) until you reach 1000000, and then count up how many periods start with the number 1, how many start with the number 2, … you will see that there are more numbers starting with 1 than 2, more of 2 than 3.

*Interestingly this will be true no matter what currency you use(!). Hence, if you find yourself being unmotivated because your networth is one hundred and something … convert it into a currency where it is sixty something thousand or seven hundred something thousand and you’re flying again 😉*

Exactly how many more instances of numbers beginning by 1 than 2 can be found from plotting the curve on a logarithmic scale. This will make an exponential curve linear. Since the dots are uniformly (equidistantly) distributed on the time axis and the line is straight they will be equidistantly distributed on the line. Now count how many dots fall on the logarithmic y-axis (the money axis). You will find that the fraction of dots starting with 1 is log(2)-log(1) = 30.1% which is also the fraction of space that range takes up on the paper. The number of dots starting with 2 is log(3)-log(2) = 17.6%. Thus for almost half the time your net worth will start with either a 1 or a 2 whether that is 100000USD or 200000USD or 1000000GDP or 2000000GDP.

Once you leave the doldrums of ones and twos, it will go up quickly. Three-something will take up only 12% of the time and nine-something will only take up 4.5% of the time.

Originally posted 2010-03-18 09:59:17.