## Economies of Scales: How to Chart Long-Term Time Series

Charts are widely used in finance. As well as representing information efficiently, they can reveal relationships or anomalies in data. But their effectiveness or otherwise is a function of the choices made during chart construction.

*A poorly calibrated chart can mislead rather than illuminate. This underscores the importance of using appropriate scales and judicious layouts.*

It is not immediately obvious how to do this well, however. Here we examine some of the most important considerations for creating good charts of long-term financial market time series.

## The Linear Scale

Time series are most commonly plotted on a linear scale, as shown in this chart of the Dow Jones Industrial Average since 1900.

## Dow Jones Industrial Average – Linear Scale

There are several problems with this approach. For a series with increasing values, a linear scale will overemphasise the most recent changes. In the case of the example above, the linear scale measures the daily change in the Dow’s absolute value, which has become larger with time, rather than daily changes in relative, or percentage, terms.

Plotting nominal prices on a linear scale, therefore, obscures most of the information in earlier periods. Important events such as the 1920s bull market and Great Depression are barely observable.

## Real vs. Nominal Prices

Part of the reason for this distortion is inflation. The following chart shows what happens when adjusting for consumer and producer price changes.

## Dow Jones Industrial Average – Linear Scale, Inflation-Adjusted

Market events before 1980 are now discernible, and less information is lost visually in the early period than when nominal prices are used.

But some limitations remain. First, the linear scale still over-emphasises recent price movements relative to earlier ones. And adjusting for inflation introduces new complications, both in terms of standardisation with other charts and the potential for bias, since there are many deflators from which to choose.

## The Logarithmic Scale

Winton’s preferred method is to plot nominal prices on a logarithmic scale.

## Dow Jones Industrial Average – Logarithmic Scale

A logarithmic, or log, scale gives equal visual weight to equal relative changes: a 10% move looks the same whether it is from a high or a low base.

The result is that the magnitudes of earlier and later booms and busts can be compared on a level playing field. In the case of the Dow Jones since 1900, information in the first half of the chart has now been fully brought to light. Further, with log scales, series experiencing exponential growth appear as straight lines, making charts easier to interpret.

Finally, the use of nominal prices absolves researchers of the need to select a deflator and deal with the theoretical issues such a choice would entail. Researchers are often still interested in inflation as one economic phenomenon among many, and they can incorporate that into charts separately. But for a quick summary of a market, there is value in plotting raw price data.

It is worth noting that, for all their advantages, log scale charts have at least one failing: they cannot show negative values, even though prices can fall below zero. Indeed, this regularly occurs in European power markets. Separately, natural gas has traded as low as -$.29 MMBTU on a North American market amid transportation capacity constraints.

Other complications with log scales can occur with axis labelling. Most charting software automatically labels logarithmic axes with exponentially increasing values: for example, 10^0, 10^1, 10^2, and so on. In time series with meaningful units – such as exchange rates, bond prices and commodity prices - these are not the increments that are most relevant to investors. For instance, on an oil chart, we may want to label in multiples of $25 per barrel. To do this, we need to turn to more flexible charting technologies.

## Crude Oil – Logarithmic Chart with Custom Labels

Y-Axis Considerations

Careful judgment is also necessary when choosing where to crop y-axes. Charts are often criticised when the y-axis begins at a value higher than 0. This can give the impression that variation in the data is more significant than it is.

A subtler issue is where to end the y-axis. Decades ago, financial market charts were drawn by hand, with new data points filled in daily. Ranges were by necessity left open, since automatic rescaling was not possible.

Today, many charting applications and libraries scale axes by putting the maximum of the series close to the top of the chart by default. This gives the impression that the future trading range is constrained and limits readers’ imagination about the potential magnitude of future booms.

The price history of Amazon.com offers an instructive example. During the autumn of 2009, shares approached their dotcom bubble-era highs amid excitement about the Kindle. A linear scale chart presented with the 15 October closing price near the top of the y-axis might have implied limited room to run. By contrast, allowing for more room on the y-axis would have immediately suggested greater potential for the shares to rise.

## Two Views of Amazon.com Shares, October 2009

Fast forward eight years and Amazon’s time series plotted on a linear scale looks like this:

## Amazon.com - Today

It is easy with hindsight to demonstrate how the framing of a chart could mislead a reader. Nonetheless, it serves as a reminder that a chart’s design can itself introduce accidental biases.

It also highlights an additional benefit of the log scale: charts can be displayed with more space above the maximum y-axis value with minimal visual disruption:

## Amazon.com – Today, Log Scale

Scaling and axis truncation are two basic considerations that are easy to overlook and important to get right, alongside other decisions, such as whether to use a line or another visual representation with additional information, such as a candlestick chart. Convenient charting applications and libraries such as Excel, matplotlib and ggplot2 have made it easy to produce attractive visualisations quickly. With that convenience comes the temptation to go along with defaults that may introduce subtle biases. Reflecting on the basics is an important first step to charting time series well.