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The challenge of percent change

People are seen through a percentage sign as they stand in front of a shopping window in downtown Cologne
Reuters/Wolfgang Rattay
Taking a percentage.
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Percent change is arithmetic. Determine how much a measure has changed (or perhaps how much it’s expected to change) and divide it by the original level and that’s percent change. Simple enough.

Where it gets complicated is how we talk about this arithmetic. In economics, we don’t talk about one percent change, we talk about many—and the difference is in which data points we choose to compare. For instance, these are a couple of ways you could interpret the US’s second quarter GDP.

In each case, we’re comparing the data point for second quarter GDP to some prior data. The question is which prior data point to pick. Last quarter? A year ago? Something else?

The formula for percent change

Let’s take a step back and start with the basics, the percent-change formula.

(first – last) ÷ first × 100

As you can see, it’s pretty simple. It only has three numbers in it: the starting value, the ending value, and 100! Let’s take a look at what how that appears on our GDP chart by calculating how much the US economy has shrunk since the beginning of the pandemic.

As you can see—and now calculate—the US economy has shrunk 10.24% during the pandemic.

If you have a hard time remembering the formula. It can be helpful to think of it in two steps.

  1. Calculating how much has something changed.
  1. Calculating how big that change is relative to its original value.

You do the subtraction to calculate the raw change. The division makes sure that change is in context to the size of the numbers.

The meaning of month-to-month and quarter-to-quarter percent change

While the formula for percentage change is straightforward the phrases we use to describe those changes can be confusing. When percent change is described as month-to-month or quarter-to-quarter it means that the “first” value in the formula is from the month or quarter immediately preceding the “last” value. If you were reading about US GDP you might have seen sentences like:

  • US GDP dropped 9.09% from last quarter.
  • US GDP fell 9.09% quarter-to-quarter.
  • The US economy contracted 9.09% from the previous period.

They all mean exactly the same thing. This is what that means on our chart:

What year-over-year percent change means

When an indicator has seasonal fluctuations, it doesn’t make much sense to compare one period to the next. Our lives have rhythms that cycle daily, monthly, and annually with commutes, holidays, and weather—and those ebbs and flows show up in data. For example, retailers typically have their highest sales in the fourth quarter because of holiday season shopping, airlines have more passengers on Fridays and Sundays as people make weekend trips, US candy sales spike in October as people buy for halloween.

Comparing a department store’s sales from October to November, or an airline’s passenger totals from Thursday to Friday, or candy sales from September to October would always show a big increase. With data like these, we make sure metrics like percent change are meaningful by comparing values from similar periods: One Friday to the Friday a year before. Compare one quarter or month to the same quarter or month in the past. When we compare a period to the same period the year prior it’s called year-over-year.

Year-over-year metrics are less useful in headline figures like GDP because they have been seasonally adjusted by statisticians to remove the seasonal effects in the data. Nonetheless year-over-year percent change provides a more stable backwards looking measure of annual changes.

As you can see in the chart, these two sentences are equivalent:

  • US GDP fell 9.14% over the last four quarters.
  • In the second quarter, the US economy shrank 9.14% year-over-year.

You might see year-over-year abbreviated as “yoy.” For seasonal businesses, comparing quarterly figures like this is key to understanding their performance trends.

What annualized percent change means

An annualized percent change is a projection of future change over a year or the reduction of a longer-term change to the average year-length change. One example is extrapolating a one-month change to a full-year change. It is also used to take a multi-year change and turn it into the equivalent annual figure. This calculation is not as simple as the others. Calculating an annualized rate requires doing more complex math.

For instance, if a business had monthly sales of $100,000 in January followed by 2% growth in February to bring sales to $102,000, our annualization presumes that in March, sales will grow another 2% from $102,000. So instead of $2,000 in new March sales there would be $2,040. This is the formula for annualized percent change:

[ (last ÷ first)periods per year ÷ periods between first and last - 1 ] × 100

The formula is more complicated because projecting the annual figure is not as simple as extending the line of the trend. The math reflects that we are calculating what happens if the percent change would happen again and again, compounding. That’s why annualized percent change is sometimes called “CAGR” for “Compounded Annualized Growth Rate.” For the sales example above, the formula would work out like this:

[ (102,000 ÷ 100,000)12 ÷ 1 – 1 ] × 100

As you can calculate, a 2% change month to month is a 26.82% change at an annualized rate.

Let’s take a look at how the annualized GDP calculation shows up on our chart. If you look closely can can see that the dashed line is not straight. While the proportional drop in each quarter remains constant, in absolute terms the drop in GDP gets smaller in each successive quarter.

We already calculated how US GDP shrank 9.09% quarter-to-quarter. Using the formula we can calculate that continuing at that pace of decline over the next three quarters would leave GDP 31.7% lower than it was in the first quarter.

[ (17.28 trillion ÷ 19.01 trillion)4 ÷ 1 – 1 ] × 100