How do you annualize a multi year number?

When we make investments, we invest our money in different assets and earn returns for different periods of time. For example, an investment in a short-term Treasury bill will be for 3 months. We may invest in a stock and exit after a week for a few days. For the purpose of making the returns on these different investments comparable, we need to annualize the returns. So, all daily, weekly, monthly, or quarterly returns will be converted to annualized returns. The process for annualizing the returns is as follows:

The basic idea is to compound the returns to an annual period. So, if we have monthly returns, we know that there are 12 months in the year, similarly there are 52 weeks, 4 quarters, and 365 days. We compound our returns by the number of periods in the whole year.

Let’s take a few examples to understand this.

Example 1: Quarterly Returns

Let’s say we have 5% quarterly returns. Since there are four quarters in a year, the annual returns will be:

Annual returns = (1+0.05)^4 – 1 = 21.55%

Example 2: Monthly Returns

Let’s say we have 2% monthly returns. Since there are 12 months in a year, the annual returns will be:

Annual returns = (1+0.02)^12 – 1 = 26.8%

Example 3: Weekly Returns

Let’s say we have 0.5% weekly returns. Since there are 52 weeks in a year, the annual returns will be:

Annual returns = (1+0.005)^52 – 1 = 29.6%

Example 4: Daily Returns

Let’s say we have 0.1% daily returns. Since there are 365 days in a year, the annual returns will be:

Annual returns = (1+0.001)^365 – 1 = 44.02%

Example 5: 100 Days Returns

We can actually have returns for any number of days and convert them to annualized returns. Let’s say we have 6% returns over 100 days. The annual returns will be:

Annual returns = (1+0.06)^(365/100) – 1 = 23.69%

Annualized returns however have one limitation – they assume that we will be able to reinvest the money at the same rate. However this may not always be possible. If we earned 5% in a quarter there is no guarantee that we will be able to replicate these returns over the next three quarters in the year.

DataBasics

Annualizing Data

The Economic Problem

Annualizing Data Facilitates Comparison of Growth Rates of Various Time Periods

Suppose Texas employment grew 0.92 percent in the first five months of a particular year. Then in June and July, employment advanced 0.15 percent and 0.22 percent, respectively. Would employment growth in June and July be above or below the pace set in the first five months of the year?

While this simple problem could probably be tackled in a few different ways, the most common one is a process called data annualization. In this method, growth rates are adjusted to reflect the amount a variable would have changed over a year’s time, had it continued to grow at the given rate. The result is a percent change that is easily comparable to other annualized data.

In this case, the 0.92 percent translates into an annualized 2.22 percent. The 0.15 becomes 1.81 percent (annualized), and the 0.22 figure becomes 2.67 percent (annualized). Thus, employment growth in June was below the rate established in the first five months, while the July figure was above it, in annualized terms. This kind of data adjustment is very common in economic analysis. It allows for quick comparison of percent changes, no matter the time period.

The Technical Solution

The formula for annualizing monthly data is straightforward:

where Xm and Xm – 1 are the values of the economic variable in months m and m –1, respectively (for example, m = February, then m – 1 = January), and gm is the annualized percent change.

For year-to-date calculations on monthly data, the formula is:

where XDec is the value of the economic variable in the December of a given year, m is the number of the month in question, Xm is the value of the economic variable in the mth month of the given year, and hm is the annualized year-to-m percent change.

Real-World Example

Table 1 uses these two formulas to calculate the values cited in the Economic Problem section above.

On the July row, 0.22 is found by calculating the percent change between 9,553,800 (June) and 9,574,800 (July). The annualized figure of 2.67 is found by applying Equation 1: Divide 9,574,800 by 9,553,800, raise this quotient by 12, subtract 1, and multiply the whole thing by 100 (Calculation 1). This rate represents the amount employment would have increased for the year had it expanded at that monthly rate all 12 months. The calculation for the other months is the same.

Calculation 1

In the last row, the 0.92 figure is found by calculating the simple percent change between 9,452,500 (December) and 9,539,500 (May). The annualized figure of 2.22 percent is found by applying Equation 2: Divide 9,539,500 by 9,452,500, raise this quotient by 2.4 (12/5), subtract 1, and multiply the whole thing by 100 (Calculation 2). This rate represents the amount employment would have increased for the year had it continued to expand at the pace set between January and May.

Calculation 2

Summary

The annualizing methodology offers a simple way to compare the growth rates of economic variables presented across different periods. Analysts can regularly assess the monthly or quarterly performance of key economic indicators relative to their changes in recent years.

Note

Annualized rates of growth in monthly or quarterly data are generally only calculated for data that are not seasonal, or that have had the seasonality removed.

Glossary at a Glance

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What is the formula to annualize a number?

How do you calculate annualized return over 3 years in Excel?

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