What is a constant investment chart?

 

Charts are indispensable, especially in finance: they can reveal patterns and all sorts of information hidden in data. But the utility of such graphical elements is as good as the proper construction of the chart. A poorly designed chart will misinform while losing all its purpose. Here we examine a useful way to represent graphically how a (long-term) investment performs over time.

To illustrate the use and methodology, we can take as example one well known millionaire making strategy (in hindsight of course): HODL 1 bitcoin since 2013, i.e. buy 1 bitcoin in 2013 and keep it until Q1 2021.

If we display a chart of how much money we made in US dollars over time of the above strategy (i.e., data points (t,p_t) representing the prices p_t for each day t in a linear scale), we obtain the following:

 

Can you see what happened in 2013? What about 2014? Was the volatility very small then and is now insanely large? Well, no, it was not. We are looking at the wrong chart for this purpose; in 2013 there was a small amount of money in the chart compared to the more recent past, therefore the moves in money terms appear bigger recently.

To have a better visual representation of the strategy’s performance we can plot a constant investment chart.

The first step is to calculate the returns of the strategy for day t (instead of day, you can use any frequency you would like as well, such as weeks or months):

 The data points (t,c_t) for the constant investment chart will be recursively calculated:

Where you can choose the initial value (your constant investment) to be c₀ = 1.

Now we can have a look at the constant investment chart:

 

As you can see, this methodology is similar to a semi-log chart or some other type of coordinate transformation.

But why is it called a constant investment chart? Because instead of compounding the returns, you are investing the same amount c₀ = 1 every day t, and adding the latter returns to your performance.

Also, notice that there are two y-axis labels on the chart. The right labels correspond to the chart’s actual values c_t, while the left labels correspond to the prices p_t. The c_t are not very informative about the actual money or compound returns, we are only interested in the visuals, hence there is no point in using its labels for the chart (in this piece, I left them in the right label for informative purposes). So, one can pick which label makes sense to use and match the data points in the chart; as stated, I matched the prices p_t with the data points in the chart and used them in the left y-axis labels. Since with this transformation, the shape of the plot changes, the match is not exact. It is more an indication of key levels of prices p_t in time for this chart (i.e., the transforamtion is not an isomorphism).

With this method, we can see now what happened in the distant past and compare apples to apples within the timeline. It can be applied to review the performance of any instrument or portfolio in a very clear and straightforward way.