Growth rates computed with different time units s are not directly comparable. For example, monthly growth rates are much smaller than annual growth rates. Growth rates can be *annualized *to make them comparable. This article discusses how:

- Growth rates with different time units cannot be compared
- Annualizing growth rates by compounding
- Annualizing "complements", such as churn and contraction
- Annualizing by restricting the analysis to the relevant sub-group
- Annualizing Net Recurring Revenue (NRR)
- Annualizing using year-on-year comparisons
- The relationship between quarterly and annual growth rates

## Growth rates with different time units cannot be compared

Growth rates cannot be directly computed across different time units. The problem is illustrated below, which shows a software company's quarterly and annual growth rates. The annual rates are more than four times the quarterly rates.

The cause of the problem can understood by looking at how growth rates are calculated. Where *Value 1 *is a result taken at one time and *Value 2 *is a result at a later time, the growth rate over the two times is:

For example, if revenue is $2M at the end of 2022 and $3M at the end of 2023, the growth rate is (3 – 2) / 2 = ½ = 50%. If the revenue was at $2.75M at the end of Q3, the growth rate for Q4 = (3 - 2.75) / 2.75 = 9%. The 9% looks much smaller than the 50%, but the cause of the difference is that the quarterly growth rate necessarily has a much smaller numerator and usually a bigger denominator.

## Annualizing growth rates by compounding

A growth rate between two points in time can be converted into an annual growth rate (i.e., be *annualized*) using the same approach used to calculate compound interest rates. Where *Proportion *refers to the proportion of the year that has been used to compute the growth rate:

For example, if revenue on 31 March 2022 is $100, and revenue on 30 June 2022 is $110 (i.e., one quarter later), then:

- The growth rate is (110 – 100)/100 = 10%.
- A quarter of a year represent approximately s ¼ of a year (i.e., Proportion = 0.25)
- 1/.25 = 4
- Annualized growth rate by compounding = (.1 + 1)^(1/.25) -1 = 1.1^4 - 1 = 46.4%

We can improve the accuracy of such calculations by computing the proportion more accurately. For example, the second quarter in 2023 represents 0.249 of the year 2023, so a more accurate calculation is 46.6%. While with quarterly data, this additional precision is unlikely to be useful, it should be considered when:

- Annualizing incomplete periods (e.g., halfway through a quarter).
- Annualizing monthly data as, otherwise, February consistently appears poor.
- Annualizing weekly sales data, performing the calculations based on the number of shopping days each week (e.g., excluding public holidays).

The chart below shows the annualized quarterly data, which is now directly comparable with the annual data and reveals more insight than was evident before. For example, we can now see that the company’s growth rate slowed to around 30% in Q4 2022, where it's been hovering close to ever since, except in Q4 of 2022.

## Annualizing "complements", such as churn and contraction

When a proportion is subtracted from 1, it's known as a "complement" in mathematics. For example, churn complements retention (i.e., 100% - % Retention = % Churn). When annualizing a complement, we need to modify the formula. For example:

Consider a company with $100 in recurring revenue at the end of the previous year and churn of $10 in Q1, leading to a churn rate of 10% for the quarter (i.e., 10 / 100). The annualized rate is 1 - (1 - .1)^4 = 34.39%.

## Annualizing by restricting the analysis to the relevant sub-group

With some rates, such as churn and contraction, there is another way of annualizing the data. Consider again the customer with $100 in recurring revenue and $10 in churn in Q1. Where the customers are all on annual subscriptions, only a subset of the customers can renew in Q1. If we restrict the analysis to the customers that can renew, we automatically annualize the data.

In the previous section, we calculated churn as 10 / 100 = 10%. If we know that only $29.08 of the recurring revenue was available to renew in Q1, we can replace the denominator of 100 with 29.08. As, by definition, the companies that did churn must have renewed, the numerator of 10 remains the same, resulting in the annualized churn being 10 / 29.08 = 34.39%. While the example has been contrived to get the same result, when applied appropriately, both methods of annualizing give the same result.

Annualizing by restricting the analysis to the relevant sub-group is preferable to the compounding approach, as it automatically considers some forms of seasonality in the data (e.g., if in one quarter there is a much higher proportion of renewals than in other quarters). This method is typically appropriate for churn, contraction, and price effects. It is typically inappropriate for expansion and new sales, as there is no way of knowing who will and will not buy ahead of time.

## Annualizing Net Recurring Revenue (NRR)

When annualizing NRR (e.g., calculating quarterly annualized NRR), it is advisable to annualize each component and then add them up, using different methods for annualizing some components. % Churn, % Contraction, and % Price relating to renewal should be annualized by restricting to the relevant sub-group, while the other components are better compounded.

## Annualizing using year-on-year comparisons

An alternative approach to annualizing growth rates is to perform year-on-year comparisons. For example, the growth rate for Q1 2022 is computed as (Q1 2022 – Q1 2021)/Q1 2021.

Year-on-year comparisons are already annualized and are thus already on the same scale as annual growth rates. A second advantage is that they automatically remove seasonality from data (e.g., as summer one year is compared against summer in the previous year).

However, year-on-year comparisons are, by definition, comparing with data that is more than a year old (e.g., if performing year-on-year comparisons of quarters, the earliest data is potentially 15 months older than the most recent data, meaning that the analysis is telling a story that’s quite old. There is no free lunch with analysis - by using such old data, we make newer results less visible. For example, the massive drop in growth in Q4 2022 is easy to see in the annualized data below but largely invisible in the year-on-year growth rates.

## The relationship between quarterly and annual growth rates

The sections above describe how to annualize growth rates. A related question is how to compute an annual growth rate from a quarterly growth rate.

Before describing how to do it, it is worth reviewing two common wrong assumptions:

- Assumption 1: Annual growth is the sum of quarterly growth. For example, the 2022 data shown above shows that 7% + 8% + 7% + 3% = 23%, well below the 2022 annual growth rate of less than 27%.
- Assumption 2: Annual growth is the average of annualized quarterly growth. The detail of this calculation is not shown, but the average annualized growth is $27.5%, which is close to, but not the same as, the actual annual growth rate of 27.3%.

The second assumption is the better, as it addresses compounding. However, both approaches start from the fundamentally incorrect assumption that annual growth is determined by the growth in the four quarters within the year. To see why this is wrong, consider the formulas for growth of the four quarters in 2022:

Q1 Growth = (Q1 2022 - Q4 2021) / Q4 2021

Q2 Growth = (Q2 2022 - Q1 2022) / Q1 2022

Q3 Growth = (Q3 2022 - Q3 2022) / Q2 2022

Q4 Growth = (Q4 2022 - Q4 2022) / Q3 2022

These growth rates use data from only five quarters, Q4 2021 to Q4 2022. By contrast, the annual growth rate uses data from all eight quarters in the two years, as shown below, meaning that we generally cannot compute annual growth rates using only the growth rates of the four quarters.

(Q1 2022 + Q2 2022 + Q3 2022 + Q4 2022 - Q1 2021 - Q2 2021 - Q3 2021 - Q4 2021) /

(Q1 2021 + Q2 2021 + Q3 2021 + Q4 2021)

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