Diminishing or non-diminishing returns?

Bitcoin’s price history is best looked at by using a logarithmic scale for the price, giving us as so-called semi-log plot, in which the x-axis represents time and is linear, and the y-axis displays the price of bitcoin, and is scaled logarithmically.

Using a logarithmic scale gives us the advantage of being able to observe bitcoin’s full price history in a single plot. It also has the property that equi-distant movements on the y-axis indicate price changes that are identical in percentage terms. E.g. the price movement from $1 to $10 per bitcoin takes up the same distance on the y-scale as the price movement from $100 to $1000. This property is extremely useful but is not always perfectly understood.

To better understand the properties of a semi-log plot, let’s look at two models:

one with non-diminishing returns (equal expected growth rates over time)

one with diminishing returns (growth rates become smaller over time)

I used the equation described in the previous article for the model with diminishing returns, but a different model with slowing growth rates could have been used as well for the purposes of this article.

The model with non-diminishing returns displays like a straight line in the semi-log plot, whereas the model with diminishing returns displays like a curve that initially grows quickly, and then more slowly.

Which model should we prefer? The difference between the two is important, as the two predict wildly different prices in the future.

In the previous article, the choice for a model with diminishing returns was mostly motivated by the fact that the bitcoin price curve in the semi-log plot appears to be slowing. Also, the regression error for the model with diminishing returns is “good”: It is about 5.3 times lower than for the model with non-diminishing returns. The model with diminishing returns is therefore empirically better at modeling the data. This already tells us that bitcoin’s long-term growth has diminishing returns, but in this article, we will make some observations that give additional weight to the conclusion that bitcoin’s upward price movements face greater and greater resistance.

Expected returns, long term

Diminishing returns means that bitcoin’s growth is slowing. Non-diminishing returns means that bitcoin’s growth is not slowing down, i.e. the expected growth rate stays the same over time. To better understand the difference between the two, let’s take the perspective of fictional investors.

A model with non-diminishing returns

Let’s assume that bitcoin’s price follows a non-diminishing model. How much money can an investor expect to make? The answer depends on the amount of time the investor held his bitcoin before selling them (the “hodling period”). The longer the hodling period, the higher the expected return. What is interesting is that for the model with non-diminishing returns, the profit the investor can expect to make does not depend on when he invested.

This is demonstrated in the plot below. Each colored line represents one hodling period. The x-value of each point on the line represents the time at which the investor sold his bitcoin. The y-value represents the percent profit he made from his investment.

The longer the hodling period, the later the starting point of the line representing that hodling period. This is because bitcoin’s price history is limited and we assume that it was not possible to invest in bitcoin before the 17th of July 2010. According to this, it is not possible to have held bitcoin for eight years before mid-2018, which is why the yellow line above starts in mid-2018.

We can display the same data in a semi-log plot:

We see that according to this model, an investor who bought bitcoin and sold them 3 years later would have made a profit of approximately 2500% no matter when this investor bought his bitcoin. The same holds true for any other hodling period, but the returns are higher for longer hodling periods.

A model with diminishing returns

Using a model with diminishing returns, the situation is different: The expected returns depends on when one invested. The lines in the below plot drop sharply, which means that for the same hodling period, buying earlier gives higher expected returns.

A semi-log plot again makes the data easier to read:

Investor A who bought bitcoin in mid-2011 and sold them three years later in mid-2014 would have made about 10000% profit.

Investor B who bought bitcoin in January 2015 and sold them three years later in January 2018 would have made “only” about 1000% profit, or 10 times less than investor A.

The situation is similar, but even more pronounced for longer hodling periods. A 10x decline in returns occurs faster. Investors A and B invested three and a half years apart, with a 10x difference in returns. For an 8-year hodling period, a 10x decline in returns occurs in about a year.

(Note: For the model with diminishing returns, I used the same model as in my previous article , but the exact choice of the model is not very important here, as the specific numbers are not as interesting as the principle itself.)

Actual returns

The profiles of the expected returns are very different for the model with diminishing compared to the model with non-diminishing returns. The difference is extremely important to anyone who invests in bitcoin. Which of the two models better reflects reality?

To answer this question, we will perform the same exercise as before, but using bitcoin’s actual price history:

What is immediately noticeable are sharp drops in the return curves, which is in agreement with the model with diminishing returns. What is also immediately noticeable is that the curves are much more noisy than those based on model (i.e. simulated) data. The noisiness is due to the wild price swings for which bitcoin is so famous.

In the semi-log chart, we notice very low returns for the 3-year hodlers around the 2017 mark. This is because the price around 2017 was about $1000, approximately the same as during the previous all-time-high, around 2014. Someone who bought at that all-time-high and sold three years later could have made a small loss.